LMFDB - Embedded newform 15.10.b.a.4.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,10,Mod(4,15)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("15.4");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	15.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.72553754246$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 939x^{6} + 217699x^{4} + 14559561x^{2} + 31136400 $ x^8 + 939x^6 + 217699x^4 + 14559561*x^2 + 31136400
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{12}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4.7
Root			$-1.48693i$ of defining polynomial
Character	$\chi$	$=$	15.4
Dual form			15.10.b.a.4.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+29.7516i q^{2} -81.0000i q^{3} -373.156 q^{4} +(343.445 - 1354.68i) q^{5} +2409.88 q^{6} -10707.3i q^{7} +4130.82i q^{8} -6561.00 q^{9} +O(q^{10})$	\(q+29.7516i q^{2} -81.0000i q^{3} -373.156 q^{4} +(343.445 - 1354.68i) q^{5} +2409.88 q^{6} -10707.3i q^{7} +4130.82i q^{8} -6561.00 q^{9} +(40304.0 + 10218.0i) q^{10} +54425.4 q^{11} +30225.7i q^{12} +53507.4i q^{13} +318559. q^{14} +(-109729. - 27819.0i) q^{15} -313954. q^{16} -644691. i q^{17} -195200. i q^{18} +210320. q^{19} +(-128159. + 505509. i) q^{20} -867290. q^{21} +1.61924e6i q^{22} +95214.7i q^{23} +334596. q^{24} +(-1.71722e6 - 930519. i) q^{25} -1.59193e6 q^{26} +531441. i q^{27} +3.99549e6i q^{28} +2.25330e6 q^{29} +(827660. - 3.26462e6i) q^{30} -548893. q^{31} -7.22566e6i q^{32} -4.40846e6i q^{33} +1.91806e7 q^{34} +(-1.45050e7 - 3.67736e6i) q^{35} +2.44828e6 q^{36} +1.19273e7i q^{37} +6.25734e6i q^{38} +4.33410e6 q^{39} +(5.59595e6 + 1.41871e6i) q^{40} +1.48678e7 q^{41} -2.58033e7i q^{42} +2.80141e7i q^{43} -2.03092e7 q^{44} +(-2.25334e6 + 8.88809e6i) q^{45} -2.83279e6 q^{46} -5.85976e6i q^{47} +2.54303e7i q^{48} -7.42924e7 q^{49} +(2.76844e7 - 5.10899e7i) q^{50} -5.22200e7 q^{51} -1.99666e7i q^{52} +5.05528e7i q^{53} -1.58112e7 q^{54} +(1.86921e7 - 7.37293e7i) q^{55} +4.42298e7 q^{56} -1.70359e7i q^{57} +6.70392e7i q^{58} +5.84637e6 q^{59} +(4.09463e7 + 1.03809e7i) q^{60} -1.07474e7 q^{61} -1.63304e7i q^{62} +7.02505e7i q^{63} +5.42302e7 q^{64} +(7.24856e7 + 1.83768e7i) q^{65} +1.31159e8 q^{66} +7.46365e7i q^{67} +2.40571e8i q^{68} +7.71239e6 q^{69} +(1.09407e8 - 4.31547e8i) q^{70} +4.06259e7 q^{71} -2.71023e7i q^{72} -3.31290e8i q^{73} -3.54857e8 q^{74} +(-7.53720e7 + 1.39095e8i) q^{75} -7.84821e7 q^{76} -5.82749e8i q^{77} +1.28946e8i q^{78} +8.40451e7 q^{79} +(-1.07826e8 + 4.25309e8i) q^{80} +4.30467e7 q^{81} +4.42342e8i q^{82} +6.88232e8i q^{83} +3.23635e8 q^{84} +(-8.73353e8 - 2.21416e8i) q^{85} -8.33465e8 q^{86} -1.82517e8i q^{87} +2.24821e8i q^{88} +1.04545e9 q^{89} +(-2.64435e8 - 6.70405e7i) q^{90} +5.72919e8 q^{91} -3.55300e7i q^{92} +4.44603e7i q^{93} +1.74337e8 q^{94} +(7.22332e7 - 2.84917e8i) q^{95} -5.85279e8 q^{96} +1.28245e9i q^{97} -2.21032e9i q^{98} -3.57085e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 1194 q^{4} - 690 q^{5} + 486 q^{6} - 52488 q^{9}+O(q^{10}) $ 8 * q - 1194 * q^4 - 690 * q^5 + 486 * q^6 - 52488 * q^9	\( 8 q - 1194 q^{4} - 690 q^{5} + 486 q^{6} - 52488 q^{9} + 67090 q^{10} - 71988 q^{11} + 416364 q^{14} + 80190 q^{15} - 1505630 q^{16} + 851584 q^{19} + 2078100 q^{20} - 1593108 q^{21} + 1242702 q^{24} + 1695500 q^{25} - 877524 q^{26} - 73572 q^{29} + 3086100 q^{30} + 474088 q^{31} - 8124388 q^{34} - 36357180 q^{35} + 7833834 q^{36} + 12959676 q^{39} - 15313390 q^{40} + 93320088 q^{41} - 74555892 q^{44} + 4527090 q^{45} - 9664072 q^{46} + 51329600 q^{49} + 67798200 q^{50} - 108196236 q^{51} - 3188646 q^{54} + 64428480 q^{55} - 67781220 q^{56} + 236526036 q^{59} + 63172710 q^{60} - 357427760 q^{61} - 12137026 q^{64} + 19848300 q^{65} + 23317308 q^{66} + 167059584 q^{69} + 200900520 q^{70} - 156890664 q^{71} - 1523381796 q^{74} - 528573600 q^{75} + 1098697344 q^{76} + 863922280 q^{79} + 630213180 q^{80} + 344373768 q^{81} + 529023636 q^{84} - 2223350420 q^{85} + 997642392 q^{86} + 357382224 q^{89} - 440177490 q^{90} + 214754328 q^{91} - 721679824 q^{94} + 1698584640 q^{95} - 475022718 q^{96} + 472313268 q^{99}+O(q^{100}) \) 8 * q - 1194 * q^4 - 690 * q^5 + 486 * q^6 - 52488 * q^9 + 67090 * q^10 - 71988 * q^11 + 416364 * q^14 + 80190 * q^15 - 1505630 * q^16 + 851584 * q^19 + 2078100 * q^20 - 1593108 * q^21 + 1242702 * q^24 + 1695500 * q^25 - 877524 * q^26 - 73572 * q^29 + 3086100 * q^30 + 474088 * q^31 - 8124388 * q^34 - 36357180 * q^35 + 7833834 * q^36 + 12959676 * q^39 - 15313390 * q^40 + 93320088 * q^41 - 74555892 * q^44 + 4527090 * q^45 - 9664072 * q^46 + 51329600 * q^49 + 67798200 * q^50 - 108196236 * q^51 - 3188646 * q^54 + 64428480 * q^55 - 67781220 * q^56 + 236526036 * q^59 + 63172710 * q^60 - 357427760 * q^61 - 12137026 * q^64 + 19848300 * q^65 + 23317308 * q^66 + 167059584 * q^69 + 200900520 * q^70 - 156890664 * q^71 - 1523381796 * q^74 - 528573600 * q^75 + 1098697344 * q^76 + 863922280 * q^79 + 630213180 * q^80 + 344373768 * q^81 + 529023636 * q^84 - 2223350420 * q^85 + 997642392 * q^86 + 357382224 * q^89 - 440177490 * q^90 + 214754328 * q^91 - 721679824 * q^94 + 1698584640 * q^95 - 475022718 * q^96 + 472313268 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$.

$n$	$7$	$11$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			29.7516i			1.31485i	0.753522	+	0.657423i	$0.228354\pi$
$2$			29.7516i			1.31485i	−0.753522	+	0.657423i	$0.771646\pi$
$3$		−	81.0000i		−	0.577350i
$3$		−	81.0000i		−	0.577350i
$4$	−373.156			−0.728821
$5$	343.445	−	1354.68i	0.245749	−	0.969333i
$5$	343.445	−	1354.68i	0.245749	−	0.969333i
$6$	2409.88			0.759127
$7$		−	10707.3i		−	1.68554i	−0.538276	−	0.842768i	$-0.680924\pi$
$7$		−	10707.3i		−	1.68554i	0.538276	−	0.842768i	$-0.319076\pi$
$8$			4130.82i			0.356559i
$9$	−6561.00			−0.333333
$10$	40304.0	+	10218.0i	1.27452	+	0.323122i
$11$	54425.4			1.12082			0.560409	−	0.828216i	$-0.310644\pi$
$11$	54425.4			1.12082			0.560409	+	0.828216i	$0.310644\pi$
$12$			30225.7i			0.420785i
$13$			53507.4i			0.519599i	0.965663	+	0.259800i	$0.0836565\pi$
$13$			53507.4i			0.519599i	−0.965663	+	0.259800i	$0.916343\pi$
$14$	318559.			2.21622
$15$	−109729.	−	27819.0i	−0.559645	−	0.141883i
$16$	−313954.			−1.19764
$17$		−	644691.i		−	1.87211i	−0.351852	−	0.936056i	$-0.614448\pi$
$17$		−	644691.i		−	1.87211i	0.351852	−	0.936056i	$-0.385552\pi$
$18$		−	195200.i		−	0.438282i
$19$	210320.			0.370244			0.185122	−	0.982716i	$-0.440732\pi$
$19$	210320.			0.370244			0.185122	+	0.982716i	$0.440732\pi$
$20$	−128159.	+	505509.i	−0.179107	+	0.706471i
$21$	−867290.			−0.973145
$22$			1.61924e6i			1.47370i
$23$			95214.7i			0.0709461i	0.999371	+	0.0354731i	$0.0112938\pi$
$23$			95214.7i			0.0709461i	−0.999371	+	0.0354731i	$0.988706\pi$
$24$	334596.			0.205859
$25$	−1.71722e6	−	930519.i	−0.879215	−	0.476426i
$26$	−1.59193e6			−0.683193
$27$			531441.i			0.192450i
$28$			3.99549e6i			1.22845i
$29$	2.25330e6			0.591600			0.295800	−	0.955250i	$-0.404414\pi$
$29$	2.25330e6			0.591600			0.295800	+	0.955250i	$0.404414\pi$
$30$	827660.	−	3.26462e6i	0.186555	−	0.735847i
$31$	−548893.			−0.106748			−0.0533741	−	0.998575i	$-0.516998\pi$
$31$	−548893.			−0.106748			−0.0533741	+	0.998575i	$0.516998\pi$
$32$		−	7.22566e6i		−	1.21816i
$33$		−	4.40846e6i		−	0.647104i
$34$	1.91806e7			2.46154
$35$	−1.45050e7	−	3.67736e6i	−1.63385	−	0.414219i
$36$	2.44828e6			0.242940
$37$			1.19273e7i			1.04625i	0.852256	+	0.523125i	$0.175234\pi$
$37$			1.19273e7i			1.04625i	−0.852256	+	0.523125i	$0.824766\pi$
$38$			6.25734e6i			0.486815i
$39$	4.33410e6			0.299991
$40$	5.59595e6	+	1.41871e6i	0.345624	+	0.0876240i
$41$	1.48678e7			0.821714			0.410857	−	0.911700i	$-0.365230\pi$
$41$	1.48678e7			0.821714			0.410857	+	0.911700i	$0.365230\pi$
$42$		−	2.58033e7i		−	1.27954i
$43$			2.80141e7i			1.24960i	0.780787	+	0.624798i	$0.214819\pi$
$43$			2.80141e7i			1.24960i	−0.780787	+	0.624798i	$0.785181\pi$
$44$	−2.03092e7			−0.816875
$45$	−2.25334e6	+	8.88809e6i	−0.0819164	+	0.323111i
$46$	−2.83279e6			−0.0932832
$47$		−	5.85976e6i		−	0.175162i	−0.996157	−	0.0875810i	$-0.972086\pi$
$47$		−	5.85976e6i		−	0.175162i	0.996157	−	0.0875810i	$-0.0279136\pi$
$48$			2.54303e7i			0.691458i
$49$	−7.42924e7			−1.84103
$50$	2.76844e7	−	5.10899e7i	0.626427	−	1.15603i
$51$	−5.22200e7			−1.08086
$52$		−	1.99666e7i		−	0.378695i
$53$			5.05528e7i			0.880042i	0.897987	+	0.440021i	$0.145029\pi$
$53$			5.05528e7i			0.880042i	−0.897987	+	0.440021i	$0.854971\pi$
$54$	−1.58112e7			−0.253042
$55$	1.86921e7	−	7.37293e7i	0.275440	−	1.08645i
$56$	4.42298e7			0.600993
$57$		−	1.70359e7i		−	0.213761i
$58$			6.70392e7i			0.777863i
$59$	5.84637e6			0.0628134			0.0314067	−	0.999507i	$-0.490001\pi$
$59$	5.84637e6			0.0628134			0.0314067	+	0.999507i	$0.490001\pi$
$60$	4.09463e7	+	1.03809e7i	0.407881	+	0.103408i
$61$	−1.07474e7			−0.0993849			−0.0496925	−	0.998765i	$-0.515824\pi$
$61$	−1.07474e7			−0.0993849			−0.0496925	+	0.998765i	$0.515824\pi$
$62$		−	1.63304e7i		−	0.140357i
$63$			7.02505e7i			0.561846i
$64$	5.42302e7			0.404046
$65$	7.24856e7	+	1.83768e7i	0.503665	+	0.127691i
$66$	1.31159e8			0.850843
$67$			7.46365e7i			0.452496i	0.974070	+	0.226248i	$0.0726460\pi$
$67$			7.46365e7i			0.452496i	−0.974070	+	0.226248i	$0.927354\pi$
$68$			2.40571e8i			1.36443i
$69$	7.71239e6			0.0409608
$70$	1.09407e8	−	4.31547e8i	0.544635	−	2.14826i
$71$	4.06259e7			0.189732			0.0948660	−	0.995490i	$-0.469758\pi$
$71$	4.06259e7			0.189732			0.0948660	+	0.995490i	$0.469758\pi$
$72$		−	2.71023e7i		−	0.118853i
$73$		−	3.31290e8i		−	1.36538i	−0.730706	−	0.682692i	$-0.760809\pi$
$73$		−	3.31290e8i		−	1.36538i	0.730706	−	0.682692i	$-0.239191\pi$
$74$	−3.54857e8			−1.37566
$75$	−7.53720e7	+	1.39095e8i	−0.275065	+	0.507615i
$76$	−7.84821e7			−0.269842
$77$		−	5.82749e8i		−	1.88918i
$78$			1.28946e8i			0.394442i
$79$	8.40451e7			0.242767			0.121384	−	0.992606i	$-0.461267\pi$
$79$	8.40451e7			0.242767			0.121384	+	0.992606i	$0.461267\pi$
$80$	−1.07826e8	+	4.25309e8i	−0.294319	+	1.16091i
$81$	4.30467e7			0.111111
$82$			4.42342e8i			1.08043i
$83$			6.88232e8i			1.59178i	0.605440	+	0.795891i	$0.292997\pi$
$83$			6.88232e8i			1.59178i	−0.605440	+	0.795891i	$0.707003\pi$
$84$	3.23635e8			0.709249
$85$	−8.73353e8	−	2.21416e8i	−1.81470	−	0.460070i
$86$	−8.33465e8			−1.64303
$87$		−	1.82517e8i		−	0.341560i
$88$			2.24821e8i			0.399637i
$89$	1.04545e9			1.76624			0.883119	−	0.469149i	$-0.155439\pi$
$89$	1.04545e9			1.76624			0.883119	+	0.469149i	$0.155439\pi$
$90$	−2.64435e8	−	6.70405e7i	−0.424842	−	0.107707i
$91$	5.72919e8			0.875804
$92$		−	3.55300e7i		−	0.0517070i
$93$			4.44603e7i			0.0616310i
$94$	1.74337e8			0.230311
$95$	7.22332e7	−	2.84917e8i	0.0909873	−	0.358890i
$96$	−5.85279e8			−0.703302
$97$			1.28245e9i			1.47085i	0.677608	+	0.735423i	$0.263017\pi$
$97$			1.28245e9i			1.47085i	−0.677608	+	0.735423i	$0.736983\pi$
$98$		−	2.21032e9i		−	2.42068i
$99$	−3.57085e8			−0.373606
$100$	6.40790e8	+	3.47229e8i	0.640790	+	0.347229i
$101$	1.58042e9			1.51121			0.755606	−	0.655027i	$-0.227343\pi$
$101$	1.58042e9			1.51121			0.755606	+	0.655027i	$0.227343\pi$
$102$		−	1.55363e9i		−	1.42117i
$103$		−	1.32982e9i		−	1.16420i	−0.813118	−	0.582098i	$-0.802232\pi$
$103$		−	1.32982e9i		−	1.16420i	0.813118	−	0.582098i	$-0.197768\pi$
$104$	−2.21029e8			−0.185268
$105$	−2.97866e8	+	1.17490e9i	−0.239150	+	0.943302i
$106$	−1.50402e9			−1.15712
$107$			4.42707e8i			0.326505i	0.986584	+	0.163252i	$0.0521985\pi$
$107$			4.42707e8i			0.326505i	−0.986584	+	0.163252i	$0.947802\pi$
$108$		−	1.98311e8i		−	0.140262i
$109$	1.29332e9			0.877580			0.438790	−	0.898590i	$-0.355407\pi$
$109$	1.29332e9			0.877580			0.438790	+	0.898590i	$0.355407\pi$
$110$	2.19356e9	+	5.56121e8i	1.42851	+	0.362161i
$111$	9.66115e8			0.604053
$112$			3.36160e9i			2.01867i
$113$			1.61237e8i			0.0930276i	0.998918	+	0.0465138i	$0.0148112\pi$
$113$			1.61237e8i			0.0930276i	−0.998918	+	0.0465138i	$0.985189\pi$
$114$	5.06845e8			0.281063
$115$	1.28986e8	+	3.27010e7i	0.0687704	+	0.0174349i
$116$	−8.40833e8			−0.431170
$117$		−	3.51062e8i		−	0.173200i
$118$			1.73939e8i			0.0825899i
$119$	−6.90289e9			−3.15551
$120$	1.14915e8	−	4.53272e8i	0.0505897	−	0.199546i
$121$	6.04181e8			0.256232
$122$		−	3.19753e8i		−	0.130676i
$123$		−	1.20430e9i		−	0.474417i
$124$	2.04823e8			0.0778003
$125$	−1.85033e9	+	2.00670e9i	−0.677882	+	0.735171i
$126$	−2.09006e9			−0.738741
$127$		−	3.57556e9i		−	1.21963i	−0.792544	−	0.609814i	$-0.791244\pi$
$127$		−	3.57556e9i		−	1.21963i	0.792544	−	0.609814i	$-0.208756\pi$
$128$		−	2.08611e9i		−	0.686897i
$129$	2.26915e9			0.721454
$130$	−5.46740e8	+	2.15656e9i	−0.167894	+	0.662242i
$131$	1.55233e9			0.460535			0.230267	−	0.973127i	$-0.426040\pi$
$131$	1.55233e9			0.460535			0.230267	+	0.973127i	$0.426040\pi$
$132$			1.64505e9i			0.471623i
$133$		−	2.25195e9i		−	0.624061i
$134$	−2.22055e9			−0.594963
$135$	7.19935e8	+	1.82521e8i	0.186548	+	0.0472944i
$136$	2.66310e9			0.667517
$137$		−	4.50293e8i		−	0.109208i	−0.998508	−	0.0546038i	$-0.982610\pi$
$137$		−	4.50293e8i		−	0.109208i	0.998508	−	0.0546038i	$-0.0173896\pi$
$138$			2.29456e8i			0.0538571i
$139$	−6.12996e9			−1.39281			−0.696404	−	0.717650i	$-0.745218\pi$
$139$	−6.12996e9			−1.39281			−0.696404	+	0.717650i	$0.745218\pi$
$140$	5.41263e9	+	1.37223e9i	1.19078	+	0.301892i
$141$	−4.74641e8			−0.101130
$142$			1.20869e9i			0.249469i
$143$			2.91216e9i			0.582376i
$144$	2.05985e9			0.399214
$145$	7.73884e8	−	3.05251e9i	0.145385	−	0.573457i
$146$	9.85639e9			1.79527
$147$			6.01768e9i			1.06292i
$148$		−	4.45076e9i		−	0.762530i
$149$	1.55432e9			0.258346			0.129173	−	0.991622i	$-0.458768\pi$
$149$	1.55432e9			0.258346			0.129173	+	0.991622i	$0.458768\pi$
$150$	−4.13828e9	−	2.24244e9i	−0.667436	−	0.361668i
$151$	−5.43848e9			−0.851298			−0.425649	−	0.904888i	$-0.639954\pi$
$151$	−5.43848e9			−0.851298			−0.425649	+	0.904888i	$0.639954\pi$
$152$			8.68792e8i			0.132014i
$153$			4.22982e9i			0.624037i
$154$	1.73377e10			2.48398
$155$	−1.88515e8	+	7.43577e8i	−0.0262333	+	0.103475i
$156$	−1.61730e9			−0.218640
$157$		−	1.26222e10i		−	1.65801i	−0.559243	−	0.829003i	$-0.688908\pi$
$157$		−	1.26222e10i		−	1.65801i	0.559243	−	0.829003i	$-0.311092\pi$
$158$			2.50047e9i			0.319202i
$159$	4.09477e9			0.508093
$160$	−9.78849e9	−	2.48162e9i	−1.18080	−	0.299361i
$161$	1.01949e9			0.119582
$162$			1.28071e9i			0.146094i
$163$			7.13851e9i			0.792070i	0.918235	+	0.396035i	$0.129614\pi$
$163$			7.13851e9i			0.792070i	−0.918235	+	0.396035i	$0.870386\pi$
$164$	−5.54803e9			−0.598883
$165$	−5.97207e9	−	1.51406e9i	−0.627260	−	0.159025i
$166$	−2.04760e10			−2.09295
$167$			6.87603e9i			0.684091i	0.939683	+	0.342045i	$0.111120\pi$
$167$			6.87603e9i			0.684091i	−0.939683	+	0.342045i	$0.888880\pi$
$168$		−	3.58262e9i		−	0.346983i
$169$	7.74146e9			0.730017
$170$	6.58747e9	−	2.59836e10i	0.604921	−	2.38605i
$171$	−1.37991e9			−0.123415
$172$		−	1.04537e10i		−	0.910731i
$173$			5.24945e9i			0.445560i	0.974869	+	0.222780i	$0.0715131\pi$
$173$			5.24945e9i			0.445560i	−0.974869	+	0.222780i	$0.928487\pi$
$174$	5.43018e9			0.449099
$175$	−9.96333e9	+	1.83867e10i	−0.803033	+	1.48195i
$176$	−1.70871e10			−1.34234
$177$		−	4.73556e8i		−	0.0362653i
$178$			3.11039e10i			2.32233i
$179$	−1.26741e10			−0.922740			−0.461370	−	0.887208i	$-0.652642\pi$
$179$	−1.26741e10			−0.922740			−0.461370	+	0.887208i	$0.652642\pi$
$180$	8.40849e8	−	3.31665e9i	0.0597024	−	0.235490i
$181$	2.31189e10			1.60108			0.800541	−	0.599278i	$-0.204545\pi$
$181$	2.31189e10			1.60108			0.800541	+	0.599278i	$0.204545\pi$
$182$			1.70452e10i			1.15155i
$183$			8.70542e8i			0.0573799i
$184$	−3.93314e8			−0.0252964
$185$	1.61578e10	+	4.09638e9i	1.01417	+	0.257115i
$186$	−1.32277e9			−0.0810354
$187$		−	3.50876e10i		−	2.09829i
$188$			2.18661e9i			0.127662i
$189$	5.69029e9			0.324382
$190$	8.47672e9	+	2.14905e9i	0.471886	+	0.119634i
$191$	−4.12970e9			−0.224527			−0.112263	−	0.993678i	$-0.535810\pi$
$191$	−4.12970e9			−0.224527			−0.112263	+	0.993678i	$0.535810\pi$
$192$		−	4.39264e9i		−	0.233276i
$193$			1.18524e10i			0.614890i	0.951566	+	0.307445i	$0.0994739\pi$
$193$			1.18524e10i			0.614890i	−0.951566	+	0.307445i	$0.900526\pi$
$194$	−3.81549e10			−1.93394
$195$	1.48852e9	−	5.87133e9i	0.0737225	−	0.290791i
$196$	2.77227e10			1.34178
$197$		−	2.02139e10i		−	0.956208i	−0.878303	−	0.478104i	$-0.841324\pi$
$197$		−	2.02139e10i		−	0.956208i	0.878303	−	0.478104i	$-0.158676\pi$
$198$		−	1.06239e10i		−	0.491234i
$199$	4.29336e9			0.194070			0.0970351	−	0.995281i	$-0.469064\pi$
$199$	4.29336e9			0.194070			0.0970351	+	0.995281i	$0.469064\pi$
$200$	3.84380e9	−	7.09350e9i	0.169874	−	0.313492i
$201$	6.04556e9			0.261249
$202$			4.70199e10i			1.98701i
$203$		−	2.41267e10i		−	0.997163i
$204$	1.94862e10			0.787756
$205$	5.10629e9	−	2.01412e10i	0.201936	−	0.796515i
$206$	3.95643e10			1.53074
$207$		−	6.24703e8i		−	0.0236487i
$208$		−	1.67989e10i		−	0.622293i
$209$	1.14467e10			0.414976
$210$	−3.49553e10	−	8.86199e9i	−1.24030	−	0.314445i
$211$	−3.98374e10			−1.38363			−0.691816	−	0.722074i	$-0.743189\pi$
$211$	−3.98374e10			−1.38363			−0.691816	+	0.722074i	$0.743189\pi$
$212$		−	1.88641e10i		−	0.641394i
$213$		−	3.29070e9i		−	0.109542i
$214$	−1.31712e10			−0.429303
$215$	3.79503e10	+	9.62132e9i	1.21127	+	0.307087i
$216$	−2.19528e9			−0.0686197
$217$			5.87716e9i			0.179928i
$218$			3.84783e10i			1.15388i
$219$	−2.68345e10			−0.788305
$220$	−6.97509e9	+	2.75126e10i	−0.200746	+	0.791825i
$221$	3.44957e10			0.972748
$222$			2.87434e10i			0.794237i
$223$			5.47781e10i			1.48332i	0.670775	+	0.741661i	$0.265962\pi$
$223$			5.47781e10i			1.48332i	−0.670775	+	0.741661i	$0.734038\pi$
$224$	−7.73672e10			−2.05325
$225$	1.12667e10	+	6.10514e9i	0.293072	+	0.158809i
$226$	−4.79706e9			−0.122317
$227$			6.69846e9i			0.167440i	0.996489	+	0.0837198i	$0.0266801\pi$
$227$			6.69846e9i			0.167440i	−0.996489	+	0.0837198i	$0.973320\pi$
$228$			6.35705e9i			0.155793i
$229$	1.12592e10			0.270551			0.135275	−	0.990808i	$-0.456808\pi$
$229$	1.12592e10			0.270551			0.135275	+	0.990808i	$0.456808\pi$
$230$	−9.72906e8	+	3.83753e9i	−0.0229243	+	0.0904226i
$231$	−4.72026e10			−1.09072
$232$			9.30797e9i			0.210940i
$233$		−	4.81544e10i		−	1.07037i	−0.844735	−	0.535185i	$-0.820242\pi$
$233$		−	4.81544e10i		−	1.07037i	0.844735	−	0.535185i	$-0.179758\pi$
$234$	1.04446e10			0.227731
$235$	−7.93813e9	−	2.01251e9i	−0.169790	−	0.0430459i
$236$	−2.18161e9			−0.0457797
$237$		−	6.80765e9i		−	0.140162i
$238$		−	2.05372e11i		−	4.14901i
$239$	1.44449e10			0.286369			0.143184	−	0.989696i	$-0.454266\pi$
$239$	1.44449e10			0.286369			0.143184	+	0.989696i	$0.454266\pi$
$240$	3.44500e10	+	8.73391e9i	0.670254	+	0.169925i
$241$	−5.48372e10			−1.04713			−0.523563	−	0.851987i	$-0.675397\pi$
$241$	−5.48372e10			−1.04713			−0.523563	+	0.851987i	$0.675397\pi$
$242$			1.79753e10i			0.336905i
$243$		−	3.48678e9i		−	0.0641500i
$244$	4.01047e9			0.0724338
$245$	−2.55153e10	+	1.00643e11i	−0.452433	+	1.78458i
$246$	3.58297e10			0.623785
$247$			1.12536e10i			0.192379i
$248$		−	2.26738e9i		−	0.0380620i
$249$	5.57468e10			0.919016
$250$	−5.97026e10	−	5.50502e10i	−0.966637	−	0.891310i
$251$	−8.03717e10			−1.27812			−0.639060	−	0.769157i	$-0.720676\pi$
$251$	−8.03717e10			−1.27812			−0.639060	+	0.769157i	$0.720676\pi$
$252$		−	2.62144e10i		−	0.409485i
$253$			5.18210e9i			0.0795176i
$254$	1.06379e11			1.60362
$255$	−1.79347e10	+	7.07416e10i	−0.265621	+	1.04772i
$256$	8.98308e10			1.30721
$257$		−	5.62547e10i		−	0.804377i	−0.915557	−	0.402189i	$-0.868250\pi$
$257$		−	5.62547e10i		−	0.804377i	0.915557	−	0.402189i	$-0.131750\pi$
$258$			6.75107e10i			0.948601i
$259$	1.27709e11			1.76349
$260$	−2.70485e10	−	6.85743e9i	−0.367082	−	0.0930640i
$261$	−1.47839e10			−0.197200
$262$			4.61842e10i			0.605532i
$263$			4.68527e10i			0.603856i	0.953331	+	0.301928i	$0.0976303\pi$
$263$			4.68527e10i			0.603856i	−0.953331	+	0.301928i	$0.902370\pi$
$264$	1.82105e10			0.230731
$265$	6.84831e10	+	1.73621e10i	0.853055	+	0.216270i
$266$	6.69991e10			0.820544
$267$		−	8.46816e10i		−	1.01974i
$268$		−	2.78511e10i		−	0.329789i
$269$	−3.49439e10			−0.406898			−0.203449	−	0.979086i	$-0.565215\pi$
$269$	−3.49439e10			−0.406898			−0.203449	+	0.979086i	$0.565215\pi$
$270$	−5.43028e9	+	2.14192e10i	−0.0621849	+	0.245282i
$271$	−7.70873e10			−0.868203			−0.434101	−	0.900864i	$-0.642934\pi$
$271$	−7.70873e10			−0.868203			−0.434101	+	0.900864i	$0.642934\pi$
$272$			2.02404e11i			2.24212i
$273$		−	4.64064e10i		−	0.505645i
$274$	1.33969e10			0.143591
$275$	−9.34602e10	−	5.06439e10i	−0.985439	−	0.533986i
$276$	−2.87793e9			−0.0298531
$277$			1.86937e10i			0.190781i	0.995440	+	0.0953907i	$0.0304100\pi$
$277$			1.86937e10i			0.190781i	−0.995440	+	0.0953907i	$0.969590\pi$
$278$		−	1.82376e11i		−	1.83133i
$279$	3.60129e9			0.0355827
$280$	1.51905e10	−	5.99175e10i	0.147693	−	0.582562i
$281$	6.30122e9			0.0602901			0.0301451	−	0.999546i	$-0.490403\pi$
$281$	6.30122e9			0.0602901			0.0301451	+	0.999546i	$0.490403\pi$
$282$		−	1.41213e10i		−	0.132970i
$283$			2.42010e10i			0.224282i	0.993692	+	0.112141i	$0.0357708\pi$
$283$			2.42010e10i			0.224282i	−0.993692	+	0.112141i	$0.964229\pi$
$284$	−1.51598e10			−0.138281
$285$	−2.30783e10	−	5.85089e9i	−0.207205	−	0.0525315i
$286$	−8.66414e10			−0.765735
$287$		−	1.59194e11i		−	1.38503i
$288$			4.74076e10i			0.406052i
$289$	−2.97039e11			−2.50480
$290$	9.08170e10	+	2.30243e10i	0.754008	+	0.191159i
$291$	1.03878e11			0.849194
$292$			1.23623e11i			0.995121i
$293$			1.20415e11i			0.954504i	0.878767	+	0.477252i	$0.158367\pi$
$293$			1.20415e11i			0.954504i	−0.878767	+	0.477252i	$0.841633\pi$
$294$	−1.79036e11			−1.39758
$295$	2.00791e9	−	7.91999e9i	0.0154363	−	0.0608871i
$296$	−4.92696e10			−0.373050
$297$			2.89239e10i			0.215701i
$298$			4.62434e10i			0.339685i
$299$	−5.09469e9			−0.0368635
$300$	2.81256e10	−	5.19040e10i	0.200473	−	0.369960i
$301$	2.99956e11			2.10624
$302$		−	1.61804e11i		−	1.11933i
$303$		−	1.28014e11i		−	0.872498i
$304$	−6.60308e10			−0.443420
$305$	−3.69115e9	+	1.45594e10i	−0.0244238	+	0.0963371i
$306$	−1.25844e11			−0.820513
$307$			1.50323e11i			0.965836i	0.875666	+	0.482918i	$0.160423\pi$
$307$			1.50323e11i			0.965836i	−0.875666	+	0.482918i	$0.839577\pi$
$308$			2.17456e11i			1.37687i
$309$	−1.07716e11			−0.672149
$310$	−2.21226e10	−	5.60861e9i	−0.136053	−	0.0344927i
$311$	−1.39646e10			−0.0846460			−0.0423230	−	0.999104i	$-0.513476\pi$
$311$	−1.39646e10			−0.0846460			−0.0423230	+	0.999104i	$0.513476\pi$
$312$			1.79034e10i			0.106964i
$313$			1.79204e11i			1.05536i	0.849444	+	0.527678i	$0.176937\pi$
$313$			1.79204e11i			1.05536i	−0.849444	+	0.527678i	$0.823063\pi$
$314$	3.75530e11			2.18002
$315$	9.51673e10	+	2.41272e10i	0.544616	+	0.138073i
$316$	−3.13620e10			−0.176934
$317$		−	1.42332e11i		−	0.791655i	−0.918325	−	0.395828i	$-0.870458\pi$
$317$		−	1.42332e11i		−	0.791655i	0.918325	−	0.395828i	$-0.129542\pi$
$318$			1.21826e11i			0.668064i
$319$	1.22637e11			0.663075
$320$	1.86251e10	−	7.34648e10i	0.0992940	−	0.391656i
$321$	3.58593e10			0.188508
$322$			3.03315e10i			0.157232i
$323$		−	1.35591e11i		−	0.693139i
$324$	−1.60632e10			−0.0809801
$325$	4.97896e10	−	9.18837e10i	0.247550	−	0.456839i
$326$	−2.12382e11			−1.04145
$327$		−	1.04759e11i		−	0.506671i
$328$			6.14163e10i			0.292989i
$329$	−6.27422e10			−0.295242
$330$	4.50458e10	−	1.77679e11i	0.209094	−	0.824750i
$331$	1.82835e11			0.837209			0.418605	−	0.908169i	$-0.362519\pi$
$331$	1.82835e11			0.837209			0.418605	+	0.908169i	$0.362519\pi$
$332$		−	2.56818e11i		−	1.16012i
$333$		−	7.82553e10i		−	0.348750i
$334$	−2.04573e11			−0.899474
$335$	1.01109e11	+	2.56335e10i	0.438620	+	0.111201i
$336$	2.72290e11			1.16548
$337$		−	3.69501e11i		−	1.56056i	−0.625428	−	0.780282i	$-0.715076\pi$
$337$		−	3.69501e11i		−	1.56056i	0.625428	−	0.780282i	$-0.284924\pi$
$338$			2.30321e11i			0.959860i
$339$	1.30602e10			0.0537095
$340$	3.25897e11	+	8.26228e10i	1.32259	+	0.335309i
$341$	−2.98737e10			−0.119645
$342$		−	4.10544e10i		−	0.162272i
$343$			3.63392e11i			1.41759i
$344$	−1.15721e11			−0.445554
$345$	2.64878e9	−	1.04479e10i	0.0100661	−	0.0397046i
$346$	−1.56179e11			−0.585843
$347$			4.74482e11i			1.75686i	0.477870	+	0.878431i	$0.341409\pi$
$347$			4.74482e11i			1.75686i	−0.477870	+	0.878431i	$0.658591\pi$
$348$			6.81075e10i			0.248936i
$349$	1.62373e11			0.585867			0.292934	−	0.956133i	$-0.405368\pi$
$349$	1.62373e11			0.585867			0.292934	+	0.956133i	$0.405368\pi$
$350$	−5.47034e11	−	2.96425e11i	−1.94853	−	1.05587i
$351$	−2.84360e10			−0.0999969
$352$		−	3.93260e11i		−	1.36533i
$353$			7.78808e10i			0.266959i	0.991052	+	0.133479i	$0.0426150\pi$
$353$			7.78808e10i			0.266959i	−0.991052	+	0.133479i	$0.957385\pi$
$354$	1.40890e10			0.0476833
$355$	1.39528e10	−	5.50353e10i	0.0466265	−	0.183914i
$356$	−3.90117e11			−1.28727
$357$			5.59134e11i			1.82184i
$358$		−	3.77075e11i		−	1.21326i
$359$	3.04177e11			0.966500			0.483250	−	0.875482i	$-0.339456\pi$
$359$	3.04177e11			0.966500			0.483250	+	0.875482i	$0.339456\pi$
$360$	−3.67150e10	−	9.30814e9i	−0.115208	−	0.0292080i
$361$	−2.78453e11			−0.862919
$362$			6.87824e11i			2.10518i
$363$		−	4.89386e10i		−	0.147935i
$364$	−2.13788e11			−0.638304
$365$	−4.48793e11	−	1.13780e11i	−1.32351	−	0.335542i
$366$	−2.59000e10			−0.0754458
$367$		−	4.13105e11i		−	1.18868i	−0.804216	−	0.594338i	$-0.797414\pi$
$367$		−	4.13105e11i		−	1.18868i	0.804216	−	0.594338i	$-0.202586\pi$
$368$		−	2.98931e10i		−	0.0849680i
$369$	−9.75480e10			−0.273905
$370$	−1.21874e11	+	4.80720e11i	−0.338067	+	1.33347i
$371$	5.41283e11			1.48334
$372$		−	1.65907e10i		−	0.0449180i
$373$			2.15288e11i			0.575877i	0.957649	+	0.287938i	$0.0929698\pi$
$373$			2.15288e11i			0.575877i	−0.957649	+	0.287938i	$0.907030\pi$
$374$	1.04391e12			2.75894
$375$	1.62543e11	+	1.49877e11i	0.424451	+	0.391375i
$376$	2.42056e10			0.0624555
$377$			1.20568e11i			0.307395i
$378$			1.69295e11i			0.426512i
$379$	−2.82401e11			−0.703055			−0.351528	−	0.936177i	$-0.614338\pi$
$379$	−2.82401e11			−0.703055			−0.351528	+	0.936177i	$0.614338\pi$
$380$	−2.69543e10	+	1.06319e11i	−0.0663134	+	0.261567i
$381$	−2.89621e11			−0.704153
$382$		−	1.22865e11i		−	0.295218i
$383$		−	7.06442e11i		−	1.67757i	−0.544459	−	0.838787i	$-0.683265\pi$
$383$		−	7.06442e11i		−	1.67757i	0.544459	−	0.838787i	$-0.316735\pi$
$384$	−1.68975e11			−0.396580
$385$	−7.89441e11	−	2.00142e11i	−1.83124	−	0.464264i
$386$	−3.52627e11			−0.808485
$387$		−	1.83801e11i		−	0.416532i
$388$		−	4.78554e11i		−	1.07198i
$389$	3.18498e11			0.705234			0.352617	−	0.935768i	$-0.385292\pi$
$389$	3.18498e11			0.705234			0.352617	+	0.935768i	$0.385292\pi$
$390$	1.74681e11	+	4.42859e10i	0.382346	+	0.0969337i
$391$	6.13841e10			0.132819
$392$		−	3.06888e11i		−	0.656436i
$393$		−	1.25738e11i		−	0.265890i
$394$	6.01396e11			1.25727
$395$	2.88648e10	−	1.13855e11i	0.0596599	−	0.235323i
$396$	1.33249e11			0.272292
$397$		−	3.67747e11i		−	0.743006i	−0.928432	−	0.371503i	$-0.878843\pi$
$397$		−	3.67747e11i		−	0.743006i	0.928432	−	0.371503i	$-0.121157\pi$
$398$			1.27734e11i			0.255173i
$399$	−1.82408e11			−0.360302
$400$	5.39128e11	+	2.92141e11i	1.05298	+	0.570587i
$401$	6.06906e11			1.17212			0.586059	−	0.810268i	$-0.300679\pi$
$401$	6.06906e11			1.17212			0.586059	+	0.810268i	$0.300679\pi$
$402$			1.79865e11i			0.343502i
$403$		−	2.93698e10i		−	0.0554662i
$404$	−5.89742e11			−1.10140
$405$	1.47842e10	−	5.83147e10i	0.0273055	−	0.107704i
$406$	7.17808e11			1.31112
$407$			6.49151e11i			1.17266i
$408$		−	2.15711e11i		−	0.385391i
$409$	−4.36411e11			−0.771153			−0.385576	−	0.922676i	$-0.625997\pi$
$409$	−4.36411e11			−0.771153			−0.385576	+	0.922676i	$0.625997\pi$
$410$	5.99234e11	+	1.51920e11i	1.04730	+	0.265514i
$411$	−3.64737e10			−0.0630510
$412$			4.96232e11i			0.848491i
$413$		−	6.25987e10i		−	0.105874i
$414$	1.85859e10			0.0310944
$415$	9.32338e11	+	2.36370e11i	1.54297	+	0.391179i
$416$	3.86626e11			0.632953
$417$			4.96527e11i			0.804138i
$418$			3.40558e11i			0.545630i
$419$	−6.90916e11			−1.09512			−0.547561	−	0.836766i	$-0.684443\pi$
$419$	−6.90916e11			−1.09512			−0.547561	+	0.836766i	$0.684443\pi$
$420$	1.11151e11	−	4.38423e11i	0.174297	−	0.687498i
$421$	−7.41180e11			−1.14988			−0.574942	−	0.818194i	$-0.694976\pi$
$421$	−7.41180e11			−1.14988			−0.574942	+	0.818194i	$0.694976\pi$
$422$		−	1.18523e12i		−	1.81926i
$423$			3.84459e10i			0.0583873i
$424$	−2.08824e11			−0.313787
$425$	−5.99898e11	+	1.10707e12i	−0.891922	+	1.64599i
$426$	9.79035e10			0.144031
$427$			1.15076e11i			0.167517i
$428$		−	1.65199e11i		−	0.237963i
$429$	2.35885e11			0.336235
$430$	−2.86249e11	+	1.12908e12i	−0.403772	+	1.59264i
$431$	−4.80814e11			−0.671166			−0.335583	−	0.942011i	$-0.608933\pi$
$431$	−4.80814e11			−0.671166			−0.335583	+	0.942011i	$0.608933\pi$
$432$		−	1.66848e11i		−	0.230486i
$433$		−	7.82797e11i		−	1.07017i	−0.844797	−	0.535086i	$-0.820279\pi$
$433$		−	7.82797e11i		−	1.07017i	0.844797	−	0.535086i	$-0.179721\pi$
$434$	−1.74855e11			−0.236577
$435$	−2.47253e11	−	6.26846e10i	−0.331086	−	0.0839382i
$436$	−4.82610e11			−0.639599
$437$			2.00255e10i			0.0262674i
$438$		−	7.98368e11i		−	1.03650i
$439$	−2.64583e11			−0.339995			−0.169997	−	0.985445i	$-0.554376\pi$
$439$	−2.64583e11			−0.339995			−0.169997	+	0.985445i	$0.554376\pi$
$440$	3.04562e11	+	7.72138e10i	0.387382	+	0.0982105i
$441$	4.87432e11			0.613678
$442$			1.02630e12i			1.27901i
$443$		−	9.95347e11i		−	1.22788i	−0.789351	−	0.613942i	$-0.789583\pi$
$443$		−	9.95347e11i		−	1.22788i	0.789351	−	0.613942i	$-0.210417\pi$
$444$	−3.60512e11			−0.440247
$445$	3.59055e11	−	1.41626e12i	0.434051	−	1.71207i
$446$	−1.62974e12			−1.95034
$447$		−	1.25900e11i		−	0.149156i
$448$		−	5.80658e11i		−	0.681035i
$449$	−2.20812e11			−0.256398			−0.128199	−	0.991748i	$-0.540920\pi$
$449$	−2.20812e11			−0.256398			−0.128199	+	0.991748i	$0.540920\pi$
$450$	−1.81637e11	+	3.35201e11i	−0.208809	+	0.385344i
$451$	8.09189e11			0.920992
$452$		−	6.01667e10i		−	0.0678005i
$453$			4.40517e11i			0.491497i
$454$	−1.99290e11			−0.220157
$455$	1.96766e11	−	7.76124e11i	0.215228	−	0.848946i
$456$	7.03721e10			0.0762182
$457$			1.65804e12i			1.77816i	0.457750	+	0.889081i	$0.348655\pi$
$457$			1.65804e12i			1.77816i	−0.457750	+	0.889081i	$0.651345\pi$
$458$			3.34979e11i			0.355732i
$459$	3.42615e11			0.360288
$460$	−4.81319e10	−	1.22026e10i	−0.0501214	−	0.0127070i
$461$	−6.56687e11			−0.677180			−0.338590	−	0.940934i	$-0.609950\pi$
$461$	−6.56687e11			−0.677180			−0.338590	+	0.940934i	$0.609950\pi$
$462$		−	1.40435e12i		−	1.43413i
$463$			1.34486e12i			1.36007i	0.733178	+	0.680036i	$0.238036\pi$
$463$			1.34486e12i			1.36007i	−0.733178	+	0.680036i	$0.761964\pi$
$464$	−7.07433e11			−0.708524
$465$	6.02297e10	+	1.52697e10i	0.0597410	+	0.0151458i
$466$	1.43267e12			1.40737
$467$			6.86309e11i			0.667719i	0.942623	+	0.333860i	$0.108351\pi$
$467$			6.86309e11i			0.667719i	−0.942623	+	0.333860i	$0.891649\pi$
$468$			1.31001e11i			0.126232i
$469$	7.99154e11			0.762699
$470$	5.98752e10	−	2.36172e11i	0.0565987	−	0.223248i
$471$	−1.02240e12			−0.957251
$472$			2.41503e10i			0.0223966i
$473$			1.52468e12i			1.40057i
$474$	2.02538e11			0.184291
$475$	−3.61164e11	−	1.95706e11i	−0.325524	−	0.176394i
$476$	2.57586e12			2.29980
$477$		−	3.31677e11i		−	0.293347i
$478$			4.29760e11i			0.376531i
$479$	1.45334e12			1.26141			0.630705	−	0.776023i	$-0.282766\pi$
$479$	1.45334e12			1.26141			0.630705	+	0.776023i	$0.282766\pi$
$480$	−2.01011e11	+	7.92868e11i	−0.172836	+	0.681734i
$481$	−6.38201e11			−0.543631
$482$		−	1.63149e12i		−	1.37681i
$483$		−	8.25787e10i		−	0.0690409i
$484$	−2.25454e11			−0.186747
$485$	1.73731e12	+	4.40451e11i	1.42574	+	0.361459i
$486$	1.03737e11			0.0843474
$487$			3.49907e10i			0.0281885i	0.999901	+	0.0140943i	$0.00448649\pi$
$487$			3.49907e10i			0.0281885i	−0.999901	+	0.0140943i	$0.995514\pi$
$488$		−	4.43957e10i		−	0.0354366i
$489$	5.78219e11			0.457302
$490$	−2.99428e12	−	7.59121e11i	−2.34644	−	0.594879i
$491$	7.80089e11			0.605728			0.302864	−	0.953034i	$-0.402057\pi$
$491$	7.80089e11			0.605728			0.302864	+	0.953034i	$0.402057\pi$
$492$			4.49391e11i			0.345765i
$493$		−	1.45268e12i		−	1.10754i
$494$	−3.34814e11			−0.252948
$495$	−1.22639e11	+	4.83738e11i	−0.0918133	+	0.362149i
$496$	1.72327e11			0.127846
$497$		−	4.34993e11i		−	0.319800i
$498$			1.65856e12i			1.20836i
$499$	1.83310e12			1.32353			0.661765	−	0.749711i	$-0.269807\pi$
$499$	1.83310e12			1.32353			0.661765	+	0.749711i	$0.269807\pi$
$500$	6.90462e11	−	7.48815e11i	0.494055	−	0.535808i
$501$	5.56959e11			0.394960
$502$		−	2.39119e12i		−	1.68053i
$503$		−	1.49423e12i		−	1.04079i	−0.853927	−	0.520393i	$-0.825786\pi$
$503$		−	1.49423e12i		−	1.04079i	0.853927	−	0.520393i	$-0.174214\pi$
$504$	−2.90192e11			−0.200331
$505$	5.42786e11	−	2.14097e12i	0.371379	−	1.46487i
$506$	−1.54176e11			−0.104553
$507$		−	6.27058e11i		−	0.421475i
$508$			1.33424e12i			0.888891i
$509$	−5.89858e11			−0.389509			−0.194754	−	0.980852i	$-0.562391\pi$
$509$	−5.89858e11			−0.389509			−0.194754	+	0.980852i	$0.562391\pi$
$510$	−2.10468e12	−	5.33585e11i	−1.37759	−	0.349251i
$511$	−3.54721e12			−2.30141
$512$			1.60452e12i			1.03188i
$513$			1.11772e11i			0.0712536i
$514$	1.67367e12			1.05763
$515$	−1.80149e12	−	4.56721e11i	−1.12850	−	0.286100i
$516$	−8.46746e11			−0.525811
$517$		−	3.18920e11i		−	0.196325i
$518$			3.79956e12i			2.31872i
$519$	4.25205e11			0.257244
$520$	−7.59113e10	+	2.99425e11i	−0.0455293	+	0.179586i
$521$	−3.61567e11			−0.214991			−0.107495	−	0.994206i	$-0.534283\pi$
$521$	−3.61567e11			−0.214991			−0.107495	+	0.994206i	$0.534283\pi$
$522$		−	4.39844e11i		−	0.259288i
$523$		−	9.32025e10i		−	0.0544716i	−0.999629	−	0.0272358i	$-0.991330\pi$
$523$		−	9.32025e10i		−	0.0544716i	0.999629	−	0.0272358i	$-0.00867050\pi$
$524$	−5.79261e11			−0.335647
$525$	1.48932e12	+	8.07030e11i	0.855603	+	0.463631i
$526$	−1.39394e12			−0.793978
$527$			3.53867e11i			0.199844i
$528$			1.38406e12i			0.774998i
$529$	1.79209e12			0.994967
$530$	−5.16550e11	+	2.03748e12i	−0.284361	+	1.12164i
$531$	−3.83580e10			−0.0209378
$532$			8.40330e11i			0.454829i
$533$			7.95540e11i			0.426962i
$534$	2.51941e12			1.34080
$535$	5.99728e11	+	1.52045e11i	0.316492	+	0.0802382i
$536$	−3.08310e11			−0.161341
$537$			1.02660e12i			0.532744i
$538$		−	1.03964e12i		−	0.535009i
$539$	−4.04339e12			−2.06346
$540$	−2.68648e11	−	6.81088e10i	−0.135960	−	0.0344692i
$541$	1.31586e12			0.660422			0.330211	−	0.943907i	$-0.392880\pi$
$541$	1.31586e12			0.660422			0.330211	+	0.943907i	$0.392880\pi$
$542$		−	2.29347e12i		−	1.14155i
$543$		−	1.87263e12i		−	0.924385i
$544$	−4.65832e12			−2.28052
$545$	4.44184e11	−	1.75204e12i	0.215664	−	0.850667i
$546$	1.38066e12			0.664846
$547$			3.42187e12i			1.63426i	0.576455	+	0.817129i	$0.304436\pi$
$547$			3.42187e12i			1.63426i	−0.576455	+	0.817129i	$0.695564\pi$
$548$			1.68030e11i			0.0795928i
$549$	7.05139e10			0.0331283
$550$	1.50674e12	−	2.78059e12i	0.702110	−	1.29570i
$551$	4.73913e11			0.219037
$552$			3.18585e10i			0.0146049i
$553$		−	8.99894e11i		−	0.409193i
$554$	−5.56167e11			−0.250848
$555$	3.31807e11	−	1.30878e12i	0.148446	−	0.585529i
$556$	2.28744e12			1.01511
$557$		−	1.29602e12i		−	0.570511i	−0.958452	−	0.285255i	$-0.907922\pi$
$557$		−	1.29602e12i		−	0.570511i	0.958452	−	0.285255i	$-0.0920784\pi$
$558$			1.07144e11i			0.0467858i
$559$	−1.49896e12			−0.649289
$560$	4.55391e12	+	1.15452e12i	1.95676	+	0.496086i
$561$	−2.84210e12			−1.21145
$562$			1.87471e11i			0.0792722i
$563$			3.87326e12i			1.62476i	0.583128	+	0.812380i	$0.301829\pi$
$563$			3.87326e12i			1.62476i	−0.583128	+	0.812380i	$0.698171\pi$
$564$	1.77115e11			0.0737055
$565$	2.18425e11	+	5.53761e10i	0.0901748	+	0.0228615i
$566$	−7.20018e11			−0.294896
$567$		−	4.60914e11i		−	0.187282i
$568$			1.67818e11i			0.0676506i
$569$	−4.17226e12			−1.66865			−0.834326	−	0.551271i	$-0.814143\pi$
$569$	−4.17226e12			−1.66865			−0.834326	+	0.551271i	$0.814143\pi$
$570$	1.74073e11	−	6.86615e11i	0.0690709	−	0.272443i
$571$	3.13278e12			1.23330			0.616648	−	0.787239i	$-0.288490\pi$
$571$	3.13278e12			1.23330			0.616648	+	0.787239i	$0.288490\pi$
$572$		−	1.08669e12i		−	0.424448i
$573$			3.34505e11i			0.129631i
$574$	4.73628e12			1.82110
$575$	8.85991e10	−	1.63504e11i	0.0338006	−	0.0623769i
$576$	−3.55804e11			−0.134682
$577$		−	2.59568e12i		−	0.974900i	−0.873151	−	0.487450i	$-0.837927\pi$
$577$		−	2.59568e12i		−	0.974900i	0.873151	−	0.487450i	$-0.162073\pi$
$578$		−	8.83738e12i		−	3.29343i
$579$	9.60042e11			0.355007
$580$	−2.88780e11	+	1.13906e12i	−0.105960	+	0.417948i
$581$	7.36910e12			2.68301
$582$			3.09055e12i			1.11656i
$583$			2.75136e12i			0.986367i
$584$	1.36850e12			0.486840
$585$	−4.75578e11	−	1.20570e11i	−0.167888	−	0.0425637i
$586$	−3.58255e12			−1.25503
$587$		−	4.58789e11i		−	0.159493i	−0.996815	−	0.0797465i	$-0.974589\pi$
$587$		−	4.58789e11i		−	0.159493i	0.996815	−	0.0797465i	$-0.0254111\pi$
$588$		−	2.24554e12i		−	0.774680i
$589$	−1.15443e11			−0.0395229
$590$	2.35632e11	+	5.97384e10i	0.0800572	+	0.0202964i
$591$	−1.63733e12			−0.552067
$592$		−	3.74464e12i		−	1.25303i
$593$			1.45638e12i			0.483646i	0.970320	+	0.241823i	$0.0777453\pi$
$593$			1.45638e12i			0.483646i	−0.970320	+	0.241823i	$0.922255\pi$
$594$	−8.60532e11			−0.283614
$595$	−2.37076e12	+	9.35124e12i	−0.775465	+	3.05874i
$596$	−5.80003e11			−0.188288
$597$		−	3.47762e11i		−	0.112046i
$598$		−	1.51575e11i		−	0.0484699i
$599$	5.06991e11			0.160909			0.0804544	−	0.996758i	$-0.474363\pi$
$599$	5.06991e11			0.160909			0.0804544	+	0.996758i	$0.474363\pi$
$600$	−5.74574e11	−	3.11348e11i	−0.180994	−	0.0980766i
$601$	−3.08264e12			−0.963802			−0.481901	−	0.876226i	$-0.660053\pi$
$601$	−3.08264e12			−0.963802			−0.481901	+	0.876226i	$0.660053\pi$
$602$			8.92415e12i			2.76938i
$603$		−	4.89690e11i		−	0.150832i
$604$	2.02941e12			0.620444
$605$	2.07503e11	−	8.18474e11i	0.0629687	−	0.248374i
$606$	3.80861e12			1.14720
$607$			2.83474e12i			0.847549i	0.905768	+	0.423774i	$0.139295\pi$
$607$			2.83474e12i			0.847549i	−0.905768	+	0.423774i	$0.860705\pi$
$608$		−	1.51970e12i		−	0.451015i
$609$	−1.95426e12			−0.575712
$610$	−4.33165e11	−	1.09818e11i	−0.126669	−	0.0321135i
$611$	3.13541e11			0.0910140
$612$		−	1.57838e12i		−	0.454811i
$613$			9.08777e11i			0.259947i	0.991517	+	0.129974i	$0.0414893\pi$
$613$			9.08777e11i			0.259947i	−0.991517	+	0.129974i	$0.958511\pi$
$614$	−4.47235e12			−1.26993
$615$	−1.63144e12	−	4.13609e11i	−0.459868	−	0.116588i
$616$	2.40723e12			0.673603
$617$		−	5.40631e11i		−	0.150182i	−0.997177	−	0.0750910i	$-0.976075\pi$
$617$		−	5.40631e11i		−	0.150182i	0.997177	−	0.0750910i	$-0.0239247\pi$
$618$		−	3.20471e12i		−	0.883773i
$619$	−5.69902e12			−1.56024			−0.780121	−	0.625629i	$-0.784843\pi$
$619$	−5.69902e12			−1.56024			−0.780121	+	0.625629i	$0.784843\pi$
$620$	7.03454e10	−	2.77471e11i	0.0191194	−	0.0754144i
$621$	−5.06010e10			−0.0136536
$622$		−	4.15468e11i		−	0.111296i
$623$		−	1.11940e13i		−	2.97706i
$624$	−1.36071e12			−0.359281
$625$	2.08297e12	+	3.19580e12i	0.546037	+	0.837761i
$626$	−5.33161e12			−1.38763
$627$		−	9.27186e11i		−	0.239587i
$628$			4.71005e12i			1.20839i
$629$	7.68945e12			1.95870
$630$	−7.17822e11	+	2.83138e12i	−0.181545	+	0.716086i
$631$	−2.97266e12			−0.746470			−0.373235	−	0.927737i	$-0.621752\pi$
$631$	−2.97266e12			−0.746470			−0.373235	+	0.927737i	$0.621752\pi$
$632$			3.47175e11i			0.0865608i
$633$			3.22683e12i			0.798840i
$634$	4.23460e12			1.04091
$635$	−4.84376e12	−	1.22801e12i	−1.18223	−	0.299723i
$636$	−1.52799e12			−0.370309
$637$		−	3.97519e12i		−	0.956600i
$638$			3.64864e12i			0.871842i
$639$	−2.66547e11			−0.0632440
$640$	−2.82601e12	−	7.16462e11i	−0.665832	−	0.168804i
$641$	−8.52908e11			−0.199545			−0.0997725	−	0.995010i	$-0.531812\pi$
$641$	−8.52908e11			−0.199545			−0.0997725	+	0.995010i	$0.531812\pi$
$642$			1.06687e12i			0.247858i
$643$		−	7.69389e12i		−	1.77499i	−0.460815	−	0.887496i	$-0.652443\pi$
$643$		−	7.69389e12i		−	1.77499i	0.460815	−	0.887496i	$-0.347557\pi$
$644$	−3.80430e11			−0.0871541
$645$	7.79327e11	−	3.07398e12i	0.177297	−	0.699330i
$646$	4.03405e12			0.911371
$647$			1.89987e11i			0.0426239i	0.999773	+	0.0213120i	$0.00678433\pi$
$647$			1.89987e11i			0.0426239i	−0.999773	+	0.0213120i	$0.993216\pi$
$648$			1.77818e11i			0.0396176i
$649$	3.18191e11			0.0704023
$650$	2.73369e12	+	1.48132e12i	0.600674	+	0.325491i
$651$	4.76050e11			0.103881
$652$		−	2.66378e12i		−	0.577277i
$653$			4.76733e12i			1.02604i	0.858376	+	0.513022i	$0.171474\pi$
$653$			4.76733e12i			1.02604i	−0.858376	+	0.513022i	$0.828526\pi$
$654$	3.11674e12			0.666194
$655$	5.33139e11	−	2.10291e12i	0.113176	−	0.446412i
$656$	−4.66783e12			−0.984119
$657$			2.17359e12i			0.455128i
$658$		−	1.86668e12i		−	0.388198i
$659$	2.23969e12			0.462597			0.231299	−	0.972883i	$-0.425703\pi$
$659$	2.23969e12			0.462597			0.231299	+	0.972883i	$0.425703\pi$
$660$	2.22852e12	+	5.64982e11i	0.457160	+	0.115901i
$661$	2.99782e12			0.610800			0.305400	−	0.952224i	$-0.401210\pi$
$661$	2.99782e12			0.610800			0.305400	+	0.952224i	$0.401210\pi$
$662$			5.43964e12i			1.10080i
$663$		−	2.79415e12i		−	0.561616i
$664$	−2.84296e12			−0.567563
$665$	−3.05069e12	−	7.73421e11i	−0.604923	−	0.153362i
$666$	2.32822e12			0.458553
$667$			2.14547e11i			0.0419717i
$668$		−	2.56584e12i		−	0.498580i
$669$	4.43703e12			0.856396
$670$	−7.62638e11	+	3.00815e12i	−0.146212	+	0.576717i
$671$	−5.84934e11			−0.111392
$672$			6.26674e12i			1.18544i
$673$			1.56664e12i			0.294375i	0.989109	+	0.147188i	$0.0470221\pi$
$673$			1.56664e12i			0.294375i	−0.989109	+	0.147188i	$0.952978\pi$
$674$	1.09932e13			2.05190
$675$	4.94516e11	−	9.12599e11i	0.0916882	−	0.169205i
$676$	−2.88878e12			−0.532052
$677$			4.59871e12i			0.841370i	0.907207	+	0.420685i	$0.138210\pi$
$677$			4.59871e12i			0.841370i	−0.907207	+	0.420685i	$0.861790\pi$
$678$			3.88562e11i			0.0706198i
$679$	1.37315e13			2.47917
$680$	9.14628e11	−	3.60766e12i	0.164042	−	0.647047i
$681$	5.42575e11			0.0966713
$682$		−	8.88791e11i		−	0.157315i
$683$			3.82576e12i			0.672704i	0.941736	+	0.336352i	$0.109193\pi$
$683$			3.82576e12i			0.672704i	−0.941736	+	0.336352i	$0.890807\pi$
$684$	5.14921e11			0.0899473
$685$	−6.10005e11	−	1.54651e11i	−0.105859	−	0.0268377i
$686$	−1.08115e13			−1.86392
$687$		−	9.11996e11i		−	0.156202i
$688$		−	8.79516e12i		−	1.49657i
$689$	−2.70495e12			−0.457269
$690$	3.10840e11	+	7.88054e10i	0.0522055	+	0.0132353i
$691$	6.34475e12			1.05868			0.529338	−	0.848411i	$-0.322440\pi$
$691$	6.34475e12			1.05868			0.529338	+	0.848411i	$0.322440\pi$
$692$		−	1.95886e12i		−	0.324733i
$693$			3.82341e12i			0.629726i
$694$	−1.41166e13			−2.31000
$695$	−2.10530e12	+	8.30417e12i	−0.342281	+	1.35010i
$696$	7.53945e11			0.121786
$697$		−	9.58517e12i		−	1.53834i
$698$			4.83085e12i			0.770326i
$699$	−3.90050e12			−0.617979
$700$	3.71788e12	−	6.86112e12i	0.585268	−	1.08008i
$701$	8.80702e12			1.37752			0.688760	−	0.724989i	$-0.258155\pi$
$701$	8.80702e12			1.37752			0.688760	+	0.724989i	$0.258155\pi$
$702$		−	8.46016e11i		−	0.131481i
$703$			2.50855e12i			0.387369i
$704$	2.95150e12			0.452862
$705$	−1.63013e11	+	6.42989e11i	−0.0248526	+	0.0980285i
$706$	−2.31708e12			−0.351010
$707$		−	1.69220e13i		−	2.54720i
$708$			1.76710e11i			0.0264309i
$709$	−6.64691e12			−0.987897			−0.493948	−	0.869491i	$-0.664447\pi$
$709$	−6.64691e12			−0.987897			−0.493948	+	0.869491i	$0.664447\pi$
$710$	1.63739e12	+	4.15117e11i	0.241818	+	0.0613067i
$711$	−5.51420e11			−0.0809224
$712$			4.31857e12i			0.629767i
$713$		−	5.22627e10i		−	0.00757336i
$714$	−1.66351e13			−2.39543
$715$	3.94506e12	+	1.00017e12i	0.564516	+	0.143118i
$716$	4.72943e12			0.672513
$717$		−	1.17004e12i		−	0.165335i
$718$			9.04976e12i			1.27080i
$719$	−1.31268e13			−1.83180			−0.915899	−	0.401409i	$-0.868520\pi$
$719$	−1.31268e13			−1.83180			−0.915899	+	0.401409i	$0.868520\pi$
$720$	7.07447e11	−	2.79045e12i	0.0981064	−	0.386971i
$721$	−1.42388e13			−1.96230
$722$		−	8.28443e12i		−	1.13461i
$723$			4.44181e12i			0.604558i
$724$	−8.62697e12			−1.16690
$725$	−3.86940e12	−	2.09674e12i	−0.520143	−	0.281853i
$726$	1.45600e12			0.194512
$727$			9.94672e12i			1.32061i	0.750997	+	0.660306i	$0.229573\pi$
$727$			9.94672e12i			1.32061i	−0.750997	+	0.660306i	$0.770427\pi$
$728$			2.36662e12i			0.312275i
$729$	−2.82430e11			−0.0370370
$730$	3.38513e12	−	1.33523e13i	0.441186	−	1.74022i
$731$	1.80605e13			2.33938
$732$		−	3.24848e11i		−	0.0418197i
$733$			9.56225e11i			0.122347i	0.998127	+	0.0611734i	$0.0194843\pi$
$733$			9.56225e11i			0.122347i	−0.998127	+	0.0611734i	$0.980516\pi$
$734$	1.22905e13			1.56293
$735$	8.15206e12	+	2.06674e12i	1.03033	+	0.261212i
$736$	6.87989e11			0.0864234
$737$			4.06213e12i			0.507165i
$738$		−	2.90221e12i		−	0.360143i
$739$	−1.87544e11			−0.0231315			−0.0115657	−	0.999933i	$-0.503682\pi$
$739$	−1.87544e11			−0.0231315			−0.0115657	+	0.999933i	$0.503682\pi$
$740$	−6.02938e12	−	1.52859e12i	−0.739146	−	0.187391i
$741$	9.11546e11			0.111070
$742$			1.61040e13i			1.95037i
$743$			1.61144e13i			1.93983i	0.243446	+	0.969914i	$0.421722\pi$
$743$			1.61144e13i			1.93983i	−0.243446	+	0.969914i	$0.578278\pi$
$744$	−1.83657e11			−0.0219751
$745$	5.33822e11	−	2.10561e12i	0.0634883	−	0.250423i
$746$	−6.40515e12			−0.757189
$747$		−	4.51549e12i		−	0.530594i
$748$			1.30932e13i			1.52928i
$749$	4.74019e12			0.550336
$750$	−4.45907e12	+	4.83591e12i	−0.514598	+	0.558088i
$751$	6.53503e12			0.749666			0.374833	−	0.927092i	$-0.377700\pi$
$751$	6.53503e12			0.749666			0.374833	+	0.927092i	$0.377700\pi$
$752$			1.83970e12i			0.209781i
$753$			6.51011e12i			0.737923i
$754$	−3.58709e12			−0.404177
$755$	−1.86782e12	+	7.36743e12i	−0.209206	+	0.825192i
$756$	−2.12337e12			−0.236416
$757$		−	8.36186e12i		−	0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$757$		−	8.36186e12i		−	0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$758$		−	8.40187e12i		−	0.924410i
$759$	4.19750e11			0.0459095
$760$	1.17694e12	+	2.98382e11i	0.127965	+	0.0324423i
$761$	−1.39641e13			−1.50932			−0.754660	−	0.656116i	$-0.772198\pi$
$761$	−1.39641e13			−1.50932			−0.754660	+	0.656116i	$0.772198\pi$
$762$		−	8.61667e12i		−	0.925853i
$763$		−	1.38479e13i		−	1.47919i
$764$	1.54102e12			0.163640
$765$	5.73007e12	+	1.45271e12i	0.604900	+	0.153357i
$766$	2.10178e13			2.20575
$767$			3.12824e11i			0.0326378i
$768$		−	7.27629e12i		−	0.754718i
$769$	−3.54841e12			−0.365903			−0.182951	−	0.983122i	$-0.558565\pi$
$769$	−3.54841e12			−0.365903			−0.182951	+	0.983122i	$0.558565\pi$
$770$	5.95454e12	−	2.34871e13i	0.610436	−	2.40780i
$771$	−4.55663e12			−0.464408
$772$		−	4.42279e12i		−	0.448145i
$773$		−	6.64958e12i		−	0.669864i	−0.942242	−	0.334932i	$-0.891287\pi$
$773$		−	6.64958e12i		−	0.669864i	0.942242	−	0.334932i	$-0.108713\pi$
$774$	5.46836e12			0.547675
$775$	9.42568e11	+	5.10756e11i	0.0938545	+	0.0508575i
$776$	−5.29756e12			−0.524443
$777$		−	1.03445e13i		−	1.01815i
$778$			9.47581e12i			0.927274i
$779$	3.12700e12			0.304235
$780$	−5.55452e11	+	2.19093e12i	−0.0537305	+	0.211935i
$781$	2.21108e12			0.212655
$782$			1.82627e12i			0.174637i
$783$			1.19750e12i			0.113853i
$784$	2.33244e13			2.20490
$785$	−1.70991e13	−	4.33503e12i	−1.60716	−	0.407454i
$786$	3.74092e12			0.349604
$787$			8.56342e12i			0.795721i	0.917446	+	0.397861i	$0.130247\pi$
$787$			8.56342e12i			0.795721i	−0.917446	+	0.397861i	$0.869753\pi$
$788$			7.54295e12i			0.696905i
$789$	3.79507e12			0.348637
$790$	3.38735e12	+	8.58775e11i	0.309413	+	0.0784436i
$791$	1.72641e12			0.156802
$792$		−	1.47505e12i		−	0.133212i
$793$		−	5.75067e11i		−	0.0516403i
$794$	1.09411e13			0.976939
$795$	1.40633e12	−	5.54713e12i	0.124863	−	0.492511i
$796$	−1.60210e12			−0.141442
$797$		−	1.62073e12i		−	0.142281i	−0.997466	−	0.0711405i	$-0.977336\pi$
$797$		−	1.62073e12i		−	0.142281i	0.997466	−	0.0711405i	$-0.0226639\pi$
$798$		−	5.42693e12i		−	0.473741i
$799$	−3.77774e12			−0.327923
$800$	−6.72362e12	+	1.24080e13i	−0.580361	+	1.07102i
$801$	−6.85921e12			−0.588746
$802$			1.80564e13i			1.54116i
$803$		−	1.80306e13i		−	1.53035i
$804$	−2.25594e12			−0.190404
$805$	3.50139e11	−	1.38109e12i	0.0293872	−	0.115915i
$806$	8.73799e11			0.0729296
$807$			2.83045e12i			0.234923i
$808$			6.52841e12i			0.538835i
$809$	−5.47852e12			−0.449671			−0.224836	−	0.974397i	$-0.572184\pi$
$809$	−5.47852e12			−0.449671			−0.224836	+	0.974397i	$0.572184\pi$
$810$	1.73496e12	+	4.39853e11i	0.141614	+	0.0359025i
$811$	−2.16799e13			−1.75980			−0.879901	−	0.475157i	$-0.842391\pi$
$811$	−2.16799e13			−1.75980			−0.879901	+	0.475157i	$0.842391\pi$
$812$			9.00304e12i			0.726754i
$813$			6.24407e12i			0.501257i
$814$	−1.93133e13			−1.54186
$815$	9.67043e12	+	2.45169e12i	0.767780	+	0.194650i
$816$	1.63947e13			1.29449
$817$			5.89192e12i			0.462656i
$818$		−	1.29839e13i		−	1.01395i
$819$	−3.75892e12			−0.291935
$820$	−1.90544e12	+	7.51584e12i	−0.147175	+	0.580517i
$821$	−2.04442e13			−1.57045			−0.785227	−	0.619208i	$-0.787454\pi$
$821$	−2.04442e13			−1.57045			−0.785227	+	0.619208i	$0.787454\pi$
$822$		−	1.08515e12i		−	0.0829024i
$823$		−	1.10110e13i		−	0.836618i	−0.908305	−	0.418309i	$-0.862623\pi$
$823$		−	1.10110e13i		−	0.836618i	0.908305	−	0.418309i	$-0.137377\pi$
$824$	5.49326e12			0.415104
$825$	−4.10216e12	+	7.57028e12i	−0.308297	+	0.568944i
$826$	1.86241e12			0.139208
$827$		−	1.39884e13i		−	1.03991i	−0.854194	−	0.519954i	$-0.825949\pi$
$827$		−	1.39884e13i		−	1.03991i	0.854194	−	0.519954i	$-0.174051\pi$
$828$			2.33112e11i			0.0172357i
$829$	1.63169e13			1.19989			0.599946	−	0.800041i	$-0.295189\pi$
$829$	1.63169e13			1.19989			0.599946	+	0.800041i	$0.295189\pi$
$830$	−7.03238e12	+	2.77385e13i	−0.514340	+	2.02877i
$831$	1.51419e12			0.110148
$832$			2.90171e12i			0.209942i
$833$			4.78956e13i			3.44662i
$834$	−1.47725e13			−1.05732
$835$	9.31486e12	+	2.36154e12i	0.663112	+	0.168115i
$836$	−4.27142e12			−0.302444
$837$		−	2.91704e11i		−	0.0205437i
$838$		−	2.05558e13i		−	1.43992i
$839$	1.80495e13			1.25758			0.628791	−	0.777575i	$-0.283550\pi$
$839$	1.80495e13			1.25758			0.628791	+	0.777575i	$0.283550\pi$
$840$	−4.85331e12	−	1.23043e12i	−0.336342	−	0.0852708i
$841$	−9.42979e12			−0.650010
$842$		−	2.20513e13i		−	1.51192i
$843$		−	5.10399e11i		−	0.0348085i
$844$	1.48656e13			1.00842
$845$	2.65877e12	−	1.04872e13i	0.179401	−	0.707630i
$846$	−1.14383e12			−0.0767703
$847$		−	6.46914e12i		−	0.431888i
$848$		−	1.58713e13i		−	1.05397i
$849$	1.96028e12			0.129489
$850$	−3.29372e13	−	1.78479e13i	−2.16422	−	1.17274i
$851$	−1.13566e12			−0.0742274
$852$			1.22795e12i			0.0798364i
$853$		−	8.55906e12i		−	0.553548i	−0.960935	−	0.276774i	$-0.910735\pi$
$853$		−	8.55906e12i		−	0.553548i	0.960935	−	0.276774i	$-0.0892654\pi$
$854$	−3.42369e12			−0.220259
$855$	−4.73922e11	+	1.86934e12i	−0.0303291	+	0.119630i
$856$	−1.82874e12			−0.116418
$857$			1.62561e13i			1.02945i	0.857356	+	0.514723i	$0.172105\pi$
$857$			1.62561e13i			1.02945i	−0.857356	+	0.514723i	$0.827895\pi$
$858$			7.01795e12i			0.442097i
$859$	−8.02355e12			−0.502803			−0.251401	−	0.967883i	$-0.580891\pi$
$859$	−8.02355e12			−0.502803			−0.251401	+	0.967883i	$0.580891\pi$
$860$	−1.41614e13	−	3.59026e12i	−0.882802	−	0.223812i
$861$	−1.28947e13			−0.799647
$862$		−	1.43050e13i		−	0.882480i
$863$		−	2.48286e13i		−	1.52372i	−0.647744	−	0.761858i	$-0.724287\pi$
$863$		−	2.48286e13i		−	1.52372i	0.647744	−	0.761858i	$-0.275713\pi$
$864$	3.84001e12			0.234434
$865$	7.11135e12	+	1.80290e12i	0.431896	+	0.109496i
$866$	2.32895e13			1.40711
$867$			2.40602e13i			1.44615i
$868$		−	2.19310e12i		−	0.131135i
$869$	4.57419e12			0.272098
$870$	1.86497e12	−	7.35618e12i	0.110366	−	0.435327i
$871$	−3.99360e12			−0.235117
$872$			5.34246e12i			0.312909i
$873$		−	8.41415e12i		−	0.490282i
$874$	−5.95791e11			−0.0345376
$875$	2.14864e13	+	1.98120e13i	1.23916	+	1.14259i
$876$	1.00135e13			0.574534
$877$		−	8.76193e12i		−	0.500152i	−0.968226	−	0.250076i	$-0.919544\pi$
$877$		−	8.76193e12i		−	0.500152i	0.968226	−	0.250076i	$-0.0804555\pi$
$878$		−	7.87178e12i		−	0.447041i
$879$	9.75364e12			0.551083
$880$	−5.86848e12	+	2.31476e13i	−0.329878	+	1.30117i
$881$	3.11897e13			1.74429			0.872147	−	0.489244i	$-0.162727\pi$
$881$	3.11897e13			1.74429			0.872147	+	0.489244i	$0.162727\pi$
$882$			1.45019e13i			0.806892i
$883$		−	1.75532e13i		−	0.971701i	−0.874042	−	0.485850i	$-0.838510\pi$
$883$		−	1.75532e13i		−	0.971701i	0.874042	−	0.485850i	$-0.161490\pi$
$884$	−1.28723e13			−0.708959
$885$	−6.41519e11	−	1.62640e11i	−0.0351532	−	0.00891217i
$886$	2.96131e13			1.61448
$887$			1.54419e13i			0.837613i	0.908076	+	0.418806i	$0.137551\pi$
$887$			1.54419e13i			0.837613i	−0.908076	+	0.418806i	$0.862449\pi$
$888$			3.99084e12i			0.215380i
$889$	−3.82846e13			−2.05573
$890$	4.21359e13	+	1.06825e13i	2.25111	+	0.570711i
$891$	2.34284e12			0.124535
$892$		−	2.04408e13i		−	1.08108i
$893$		−	1.23242e12i		−	0.0648527i
$894$	3.74571e12			0.196117
$895$	−4.35286e12	+	1.71694e13i	−0.226763	+	0.894443i
$896$	−2.23365e13			−1.15779
$897$			4.12670e11i			0.0212832i
$898$		−	6.56950e12i		−	0.337123i
$899$	−1.23682e12			−0.0631522
$900$	−4.20422e12	−	2.27817e12i	−0.213597	−	0.115743i
$901$	3.25909e13			1.64754
$902$			2.40747e13i			1.21096i
$903$		−	2.42964e13i		−	1.21604i
$904$	−6.66041e11			−0.0331698
$905$	7.94007e12	−	3.13188e13i	0.393465	−	1.55198i
$906$	−1.31061e13			−0.646243
$907$			1.13359e13i			0.556189i	0.960554	+	0.278094i	$0.0897028\pi$
$907$			1.13359e13i			0.556189i	−0.960554	+	0.278094i	$0.910297\pi$
$908$		−	2.49957e12i		−	0.122034i
$909$	−1.03691e13			−0.503737
$910$	2.30909e13	+	5.85410e12i	1.11623	+	0.282992i
$911$	1.02158e13			0.491405			0.245703	−	0.969345i	$-0.420981\pi$
$911$	1.02158e13			0.491405			0.245703	+	0.969345i	$0.420981\pi$
$912$			5.34849e12i			0.256009i
$913$			3.74573e13i			1.78410i
$914$	−4.93292e13			−2.33801
$915$	1.17931e12	+	2.98983e11i	0.0556203	+	0.0141011i
$916$	−4.20145e12			−0.197183
$917$		−	1.66212e13i		−	0.776248i
$918$			1.01933e13i			0.473723i
$919$	2.53449e13			1.17212			0.586058	−	0.810269i	$-0.300679\pi$
$919$	2.53449e13			1.17212			0.586058	+	0.810269i	$0.300679\pi$
$920$	−1.35082e11	+	5.32817e11i	−0.00621658	+	0.0245207i
$921$	1.21762e13			0.557626
$922$		−	1.95375e13i		−	0.890388i
$923$			2.17379e12i			0.0985847i
$924$	1.76140e13			0.794938
$925$	1.10986e13	−	2.04818e13i	0.498461	−	0.919879i
$926$	−4.00117e13			−1.78829
$927$			8.72497e12i			0.388066i
$928$		−	1.62816e13i		−	0.720660i
$929$	−3.30986e13			−1.45794			−0.728970	−	0.684546i	$-0.760000\pi$
$929$	−3.30986e13			−1.45794			−0.728970	+	0.684546i	$0.760000\pi$
$930$	−4.54297e11	+	1.79193e12i	−0.0199144	+	0.0785503i
$931$	−1.56251e13			−0.681633
$932$			1.79691e13i			0.780109i
$933$			1.13113e12i			0.0488704i
$934$	−2.04188e13			−0.877949
$935$	−4.75326e13	−	1.20507e13i	−2.03395	−	0.515654i
$936$	1.45017e12			0.0617559
$937$		−	7.99630e11i		−	0.0338892i	−0.999856	−	0.0169446i	$-0.994606\pi$
$937$		−	7.99630e11i		−	0.0338892i	0.999856	−	0.0169446i	$-0.00539389\pi$
$938$			2.37761e13i			1.00283i
$939$	1.45156e13			0.609310
$940$	2.96216e12	+	7.50979e11i	0.123747	+	0.0313728i
$941$	3.52253e13			1.46454			0.732271	−	0.681013i	$-0.238460\pi$
$941$	3.52253e13			1.46454			0.732271	+	0.681013i	$0.238460\pi$
$942$		−	3.04179e13i		−	1.25864i
$943$			1.41564e12i			0.0582974i
$944$	−1.83549e12			−0.0752279
$945$	1.95430e12	−	7.70855e12i	0.0797165	−	0.314434i
$946$	−4.53617e13			−1.84153
$947$			3.82364e13i			1.54490i	0.635073	+	0.772452i	$0.280970\pi$
$947$			3.82364e13i			1.54490i	−0.635073	+	0.772452i	$0.719030\pi$
$948$			2.54032e12i			0.102153i
$949$	1.77264e13			0.709453
$950$	5.82257e12	−	1.07452e13i	0.231931	−	0.428015i
$951$	−1.15289e13			−0.457062
$952$		−	2.85146e13i		−	1.12512i
$953$		−	2.48978e13i		−	0.977784i	−0.872344	−	0.488892i	$-0.837401\pi$
$953$		−	2.48978e13i		−	0.977784i	0.872344	−	0.488892i	$-0.162599\pi$
$954$	9.86791e12			0.385707
$955$	−1.41832e12	+	5.59444e12i	−0.0551772	+	0.217641i
$956$	−5.39023e12			−0.208712
$957$		−	9.93358e12i		−	0.382827i
$958$			4.32390e13i			1.65856i
$959$	−4.82142e12			−0.184073
$960$	−5.95065e12	−	1.50863e12i	−0.226122	−	0.0573274i
$961$	−2.61383e13			−0.988605
$962$		−	1.89875e13i		−	0.714792i
$963$		−	2.90460e12i		−	0.108835i
$964$	2.04629e13			0.763167
$965$	1.60562e13	+	4.07063e12i	0.596033	+	0.151109i
$966$	2.45685e12			0.0907781
$967$		−	4.13240e13i		−	1.51979i	−0.650047	−	0.759894i	$-0.725251\pi$
$967$		−	4.13240e13i		−	1.51979i	0.650047	−	0.759894i	$-0.274749\pi$
$968$			2.49576e12i			0.0913616i
$969$	−1.09829e13			−0.400184
$970$	−1.31041e13	+	5.16878e13i	−0.475263	+	1.87463i
$971$	−8.49069e12			−0.306518			−0.153259	−	0.988186i	$-0.548977\pi$
$971$	−8.49069e12			−0.306518			−0.153259	+	0.988186i	$0.548977\pi$
$972$			1.30112e12i			0.0467539i
$973$			6.56353e13i			2.34763i
$974$	−1.04103e12			−0.0370636
$975$	−7.44258e12	−	4.03296e12i	−0.263756	−	0.142923i
$976$	3.37420e12			0.119027
$977$		−	2.41147e13i		−	0.846753i	−0.905954	−	0.423377i	$-0.860845\pi$
$977$		−	2.41147e13i		−	0.846753i	0.905954	−	0.423377i	$-0.139155\pi$
$978$			1.72029e13i			0.601281i
$979$	5.68992e13			1.97963
$980$	9.52121e12	−	3.75555e13i	0.329742	−	1.30064i
$981$	−8.48547e12			−0.292527
$982$			2.32089e13i			0.796439i
$983$		−	2.05699e13i		−	0.702655i	−0.936253	−	0.351328i	$-0.885730\pi$
$983$		−	2.05699e13i		−	0.702655i	0.936253	−	0.351328i	$-0.114270\pi$
$984$	4.97472e12			0.169157
$985$	−2.73835e13	−	6.94237e12i	−0.926885	−	0.234987i
$986$	4.32196e13			1.45625
$987$			5.08211e12i			0.170458i
$988$		−	4.19937e12i		−	0.140210i
$989$	−2.66736e12			−0.0886539
$990$	−1.43920e13	−	3.64871e12i	−0.476170	−	0.120720i
$991$	−6.67869e12			−0.219968			−0.109984	−	0.993933i	$-0.535080\pi$
$991$	−6.67869e12			−0.219968			−0.109984	+	0.993933i	$0.535080\pi$
$992$			3.96612e12i			0.130036i
$993$		−	1.48097e13i		−	0.483363i
$994$	1.29417e13			0.420488
$995$	1.47453e12	−	5.81615e12i	0.0476926	−	0.188119i
$996$	−2.08023e13			−0.669798
$997$			4.38644e13i			1.40600i	0.711192	+	0.702998i	$0.248156\pi$
$997$			4.38644e13i			1.40600i	−0.711192	+	0.702998i	$0.751844\pi$
$998$			5.45377e13i			1.74024i
$999$	−6.33868e12			−0.201351

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.10.b.a.4.7	yes	8
3.2	odd	2		45.10.b.c.19.2		8
4.3	odd	2		240.10.f.c.49.7		8
5.2	odd	4		75.10.a.i.1.1		4
5.3	odd	4		75.10.a.l.1.4		4
5.4	even	2	inner	15.10.b.a.4.2	✓	8
15.2	even	4		225.10.a.u.1.4		4
15.8	even	4		225.10.a.q.1.1		4
15.14	odd	2		45.10.b.c.19.7		8
20.19	odd	2		240.10.f.c.49.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.b.a.4.2	✓	8	5.4	even	2	inner
15.10.b.a.4.7	yes	8	1.1	even	1	trivial
45.10.b.c.19.2		8	3.2	odd	2
45.10.b.c.19.7		8	15.14	odd	2
75.10.a.i.1.1		4	5.2	odd	4
75.10.a.l.1.4		4	5.3	odd	4
225.10.a.q.1.1		4	15.8	even	4
225.10.a.u.1.4		4	15.2	even	4
240.10.f.c.49.3		8	20.19	odd	2
240.10.f.c.49.7		8	4.3	odd	2

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	15.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+29.7516i q^{2} -81.0000i q^{3} -373.156 q^{4} +(343.445 - 1354.68i) q^{5} +2409.88 q^{6} -10707.3i q^{7} +4130.82i q^{8} -6561.00 q^{9} +O(q^{10})\)	\(q+29.7516i q^{2} -81.0000i q^{3} -373.156 q^{4} +(343.445 - 1354.68i) q^{5} +2409.88 q^{6} -10707.3i q^{7} +4130.82i q^{8} -6561.00 q^{9} +(40304.0 + 10218.0i) q^{10} +54425.4 q^{11} +30225.7i q^{12} +53507.4i q^{13} +318559. q^{14} +(-109729. - 27819.0i) q^{15} -313954. q^{16} -644691. i q^{17} -195200. i q^{18} +210320. q^{19} +(-128159. + 505509. i) q^{20} -867290. q^{21} +1.61924e6i q^{22} +95214.7i q^{23} +334596. q^{24} +(-1.71722e6 - 930519. i) q^{25} -1.59193e6 q^{26} +531441. i q^{27} +3.99549e6i q^{28} +2.25330e6 q^{29} +(827660. - 3.26462e6i) q^{30} -548893. q^{31} -7.22566e6i q^{32} -4.40846e6i q^{33} +1.91806e7 q^{34} +(-1.45050e7 - 3.67736e6i) q^{35} +2.44828e6 q^{36} +1.19273e7i q^{37} +6.25734e6i q^{38} +4.33410e6 q^{39} +(5.59595e6 + 1.41871e6i) q^{40} +1.48678e7 q^{41} -2.58033e7i q^{42} +2.80141e7i q^{43} -2.03092e7 q^{44} +(-2.25334e6 + 8.88809e6i) q^{45} -2.83279e6 q^{46} -5.85976e6i q^{47} +2.54303e7i q^{48} -7.42924e7 q^{49} +(2.76844e7 - 5.10899e7i) q^{50} -5.22200e7 q^{51} -1.99666e7i q^{52} +5.05528e7i q^{53} -1.58112e7 q^{54} +(1.86921e7 - 7.37293e7i) q^{55} +4.42298e7 q^{56} -1.70359e7i q^{57} +6.70392e7i q^{58} +5.84637e6 q^{59} +(4.09463e7 + 1.03809e7i) q^{60} -1.07474e7 q^{61} -1.63304e7i q^{62} +7.02505e7i q^{63} +5.42302e7 q^{64} +(7.24856e7 + 1.83768e7i) q^{65} +1.31159e8 q^{66} +7.46365e7i q^{67} +2.40571e8i q^{68} +7.71239e6 q^{69} +(1.09407e8 - 4.31547e8i) q^{70} +4.06259e7 q^{71} -2.71023e7i q^{72} -3.31290e8i q^{73} -3.54857e8 q^{74} +(-7.53720e7 + 1.39095e8i) q^{75} -7.84821e7 q^{76} -5.82749e8i q^{77} +1.28946e8i q^{78} +8.40451e7 q^{79} +(-1.07826e8 + 4.25309e8i) q^{80} +4.30467e7 q^{81} +4.42342e8i q^{82} +6.88232e8i q^{83} +3.23635e8 q^{84} +(-8.73353e8 - 2.21416e8i) q^{85} -8.33465e8 q^{86} -1.82517e8i q^{87} +2.24821e8i q^{88} +1.04545e9 q^{89} +(-2.64435e8 - 6.70405e7i) q^{90} +5.72919e8 q^{91} -3.55300e7i q^{92} +4.44603e7i q^{93} +1.74337e8 q^{94} +(7.22332e7 - 2.84917e8i) q^{95} -5.85279e8 q^{96} +1.28245e9i q^{97} -2.21032e9i q^{98} -3.57085e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 1194 q^{4} - 690 q^{5} + 486 q^{6} - 52488 q^{9}+O(q^{10}) \) 8 * q - 1194 * q^4 - 690 * q^5 + 486 * q^6 - 52488 * q^9	\( 8 q - 1194 q^{4} - 690 q^{5} + 486 q^{6} - 52488 q^{9} + 67090 q^{10} - 71988 q^{11} + 416364 q^{14} + 80190 q^{15} - 1505630 q^{16} + 851584 q^{19} + 2078100 q^{20} - 1593108 q^{21} + 1242702 q^{24} + 1695500 q^{25} - 877524 q^{26} - 73572 q^{29} + 3086100 q^{30} + 474088 q^{31} - 8124388 q^{34} - 36357180 q^{35} + 7833834 q^{36} + 12959676 q^{39} - 15313390 q^{40} + 93320088 q^{41} - 74555892 q^{44} + 4527090 q^{45} - 9664072 q^{46} + 51329600 q^{49} + 67798200 q^{50} - 108196236 q^{51} - 3188646 q^{54} + 64428480 q^{55} - 67781220 q^{56} + 236526036 q^{59} + 63172710 q^{60} - 357427760 q^{61} - 12137026 q^{64} + 19848300 q^{65} + 23317308 q^{66} + 167059584 q^{69} + 200900520 q^{70} - 156890664 q^{71} - 1523381796 q^{74} - 528573600 q^{75} + 1098697344 q^{76} + 863922280 q^{79} + 630213180 q^{80} + 344373768 q^{81} + 529023636 q^{84} - 2223350420 q^{85} + 997642392 q^{86} + 357382224 q^{89} - 440177490 q^{90} + 214754328 q^{91} - 721679824 q^{94} + 1698584640 q^{95} - 475022718 q^{96} + 472313268 q^{99}+O(q^{100}) \) 8 * q - 1194 * q^4 - 690 * q^5 + 486 * q^6 - 52488 * q^9 + 67090 * q^10 - 71988 * q^11 + 416364 * q^14 + 80190 * q^15 - 1505630 * q^16 + 851584 * q^19 + 2078100 * q^20 - 1593108 * q^21 + 1242702 * q^24 + 1695500 * q^25 - 877524 * q^26 - 73572 * q^29 + 3086100 * q^30 + 474088 * q^31 - 8124388 * q^34 - 36357180 * q^35 + 7833834 * q^36 + 12959676 * q^39 - 15313390 * q^40 + 93320088 * q^41 - 74555892 * q^44 + 4527090 * q^45 - 9664072 * q^46 + 51329600 * q^49 + 67798200 * q^50 - 108196236 * q^51 - 3188646 * q^54 + 64428480 * q^55 - 67781220 * q^56 + 236526036 * q^59 + 63172710 * q^60 - 357427760 * q^61 - 12137026 * q^64 + 19848300 * q^65 + 23317308 * q^66 + 167059584 * q^69 + 200900520 * q^70 - 156890664 * q^71 - 1523381796 * q^74 - 528573600 * q^75 + 1098697344 * q^76 + 863922280 * q^79 + 630213180 * q^80 + 344373768 * q^81 + 529023636 * q^84 - 2223350420 * q^85 + 997642392 * q^86 + 357382224 * q^89 - 440177490 * q^90 + 214754328 * q^91 - 721679824 * q^94 + 1698584640 * q^95 - 475022718 * q^96 + 472313268 * q^99

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