LMFDB - Embedded newform 75.15.d.b.74.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,15,Mod(74,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 15, names="a")

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

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	75.d (of order $2$, degree $1$, not 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:	$93.2467261139$
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} + 31304x^{4} + 828100 $ x^8 + 31304*x^4 + 828100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{14}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			74.8
Root			$-9.40358 - 9.40358i$ of defining polynomial
Character	$\chi$	$=$	75.74
Dual form			75.15.d.b.74.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+225.686 q^{2} +(-1042.26 + 1922.67i) q^{3} +34550.1 q^{4} +(-235223. + 433920. i) q^{6} -478389. i q^{7} +4.09983e6 q^{8} +(-2.61037e6 - 4.00784e6i) q^{9} +O(q^{10})$	\(q+225.686 q^{2} +(-1042.26 + 1922.67i) q^{3} +34550.1 q^{4} +(-235223. + 433920. i) q^{6} -478389. i q^{7} +4.09983e6 q^{8} +(-2.61037e6 - 4.00784e6i) q^{9} +1.60910e7i q^{11} +(-3.60101e7 + 6.64285e7i) q^{12} +3.37094e7i q^{13} -1.07966e8i q^{14} +3.59205e8 q^{16} -6.70754e8 q^{17} +(-5.89124e8 - 9.04512e8i) q^{18} -3.92753e8 q^{19} +(9.19786e8 + 4.98604e8i) q^{21} +3.63152e9i q^{22} +3.24506e7 q^{23} +(-4.27308e9 + 7.88264e9i) q^{24} +7.60773e9i q^{26} +(1.04264e10 - 8.41689e8i) q^{27} -1.65284e10i q^{28} +2.47510e10i q^{29} -3.17682e10 q^{31} +1.38959e10 q^{32} +(-3.09378e10 - 1.67710e10i) q^{33} -1.51380e11 q^{34} +(-9.01885e10 - 1.38471e11i) q^{36} +4.38690e10i q^{37} -8.86387e10 q^{38} +(-6.48121e10 - 3.51338e10i) q^{39} +1.11156e11i q^{41} +(2.07583e11 + 1.12528e11i) q^{42} +1.00533e11i q^{43} +5.55947e11i q^{44} +7.32365e9 q^{46} -2.14354e11 q^{47} +(-3.74384e11 + 6.90634e11i) q^{48} +4.49367e11 q^{49} +(6.99098e11 - 1.28964e12i) q^{51} +1.16466e12i q^{52} -5.81823e11 q^{53} +(2.35310e12 - 1.89957e11i) q^{54} -1.96132e12i q^{56} +(4.09349e11 - 7.55135e11i) q^{57} +5.58594e12i q^{58} -2.31936e12i q^{59} -4.22999e12 q^{61} -7.16964e12 q^{62} +(-1.91731e12 + 1.24877e12i) q^{63} -2.74911e12 q^{64} +(-6.98222e12 - 3.78498e12i) q^{66} +2.32014e12i q^{67} -2.31746e13 q^{68} +(-3.38219e10 + 6.23919e10i) q^{69} -1.42262e13i q^{71} +(-1.07021e13 - 1.64315e13i) q^{72} -1.06676e13i q^{73} +9.90062e12i q^{74} -1.35696e13 q^{76} +7.69778e12 q^{77} +(-1.46272e13 - 7.92921e12i) q^{78} -3.21572e13 q^{79} +(-9.24873e12 + 2.09239e13i) q^{81} +2.50864e13i q^{82} +1.21884e13 q^{83} +(3.17787e13 + 1.72268e13i) q^{84} +2.26889e13i q^{86} +(-4.75880e13 - 2.57969e13i) q^{87} +6.59706e13i q^{88} -3.00312e13i q^{89} +1.61262e13 q^{91} +1.12117e12 q^{92} +(3.31107e13 - 6.10799e13i) q^{93} -4.83768e13 q^{94} +(-1.44831e13 + 2.67173e13i) q^{96} +5.28428e13i q^{97} +1.01416e14 q^{98} +(6.44903e13 - 4.20035e13i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 78592 q^{4} - 628992 q^{6} + 3249720 q^{9}+O(q^{10}) $ 8 * q + 78592 * q^4 - 628992 * q^6 + 3249720 * q^9	$ 8 q + 78592 q^{4} - 628992 q^{6} + 3249720 q^{9} + 2227994624 q^{16} - 2629894480 q^{19} + 3895487568 q^{21} - 26850410496 q^{24} - 69941420144 q^{31} - 557694498816 q^{34} - 564781849344 q^{36} - 37777373712 q^{39} - 546920285184 q^{46} + 4493154241128 q^{49} + 5312833763328 q^{51} + 10055727182592 q^{54} - 8343283252784 q^{61} - 19748163682304 q^{64} - 28743720445440 q^{66} - 30454662970368 q^{69} - 38498990571008 q^{76} - 82430885157520 q^{79} - 34777637555832 q^{81} + 123890760580608 q^{84} + 46891488783776 q^{91} - 101904474986496 q^{94} + 163635660324864 q^{96} + 172210751032320 q^{99}+O(q^{100}) $ 8 * q + 78592 * q^4 - 628992 * q^6 + 3249720 * q^9 + 2227994624 * q^16 - 2629894480 * q^19 + 3895487568 * q^21 - 26850410496 * q^24 - 69941420144 * q^31 - 557694498816 * q^34 - 564781849344 * q^36 - 37777373712 * q^39 - 546920285184 * q^46 + 4493154241128 * q^49 + 5312833763328 * q^51 + 10055727182592 * q^54 - 8343283252784 * q^61 - 19748163682304 * q^64 - 28743720445440 * q^66 - 30454662970368 * q^69 - 38498990571008 * q^76 - 82430885157520 * q^79 - 34777637555832 * q^81 + 123890760580608 * q^84 + 46891488783776 * q^91 - 101904474986496 * q^94 + 163635660324864 * q^96 + 172210751032320 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$-1$

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 $

$2$	225.686			1.76317			0.881585	−	0.472025i	$-0.156477\pi$
$2$	225.686			1.76317			0.881585	+	0.472025i	$0.156477\pi$
$3$	−1042.26	+	1922.67i	−0.476569	+	0.879137i
$3$	−1042.26	+	1922.67i	−0.476569	+	0.879137i
$4$	34550.1			2.10877
$5$		0			0
$5$		0			0
$6$	−235223.	+	433920.i	−0.840273	+	1.55007i
$7$		−	478389.i		−	0.580892i	−0.956891	−	0.290446i	$-0.906196\pi$
$7$		−	478389.i		−	0.580892i	0.956891	−	0.290446i	$-0.0938036\pi$
$8$	4.09983e6			1.95495
$9$	−2.61037e6	−	4.00784e6i	−0.545763	−	0.837939i
$10$		0			0
$11$			1.60910e7i			0.825725i	0.910793	+	0.412862i	$0.135471\pi$
$11$			1.60910e7i			0.825725i	−0.910793	+	0.412862i	$0.864529\pi$
$12$	−3.60101e7	+	6.64285e7i	−1.00498	+	1.85390i
$13$			3.37094e7i			0.537214i	0.963250	+	0.268607i	$0.0865633\pi$
$13$			3.37094e7i			0.537214i	−0.963250	+	0.268607i	$0.913437\pi$
$14$		−	1.07966e8i		−	1.02421i
$15$		0			0
$16$	3.59205e8			1.33814
$17$	−6.70754e8			−1.63464			−0.817318	−	0.576187i	$-0.804540\pi$
$17$	−6.70754e8			−1.63464			−0.817318	+	0.576187i	$0.804540\pi$
$18$	−5.89124e8	−	9.04512e8i	−0.962274	−	1.47743i
$19$	−3.92753e8			−0.439384			−0.219692	−	0.975569i	$-0.570505\pi$
$19$	−3.92753e8			−0.439384			−0.219692	+	0.975569i	$0.570505\pi$
$20$		0			0
$21$	9.19786e8	+	4.98604e8i	0.510683	+	0.276835i
$22$			3.63152e9i			1.45589i
$23$	3.24506e7			0.00953077			0.00476539	−	0.999989i	$-0.498483\pi$
$23$	3.24506e7			0.00953077			0.00476539	+	0.999989i	$0.498483\pi$
$24$	−4.27308e9	+	7.88264e9i	−0.931670	+	1.71867i
$25$		0			0
$26$			7.60773e9i			0.947200i
$27$	1.04264e10	−	8.41689e8i	0.996757	−	0.0804647i
$28$		−	1.65284e10i		−	1.22497i
$29$			2.47510e10i			1.43485i	0.696637	+	0.717424i	$0.254679\pi$
$29$			2.47510e10i			1.43485i	−0.696637	+	0.717424i	$0.745321\pi$
$30$		0			0
$31$	−3.17682e10			−1.15468			−0.577340	−	0.816504i	$-0.695909\pi$
$31$	−3.17682e10			−1.15468			−0.577340	+	0.816504i	$0.695909\pi$
$32$	1.38959e10			0.404424
$33$	−3.09378e10	−	1.67710e10i	−0.725925	−	0.393515i
$34$	−1.51380e11			−2.88214
$35$		0			0
$36$	−9.01885e10	−	1.38471e11i	−1.15089	−	1.76702i
$37$			4.38690e10i			0.462111i	0.972941	+	0.231055i	$0.0742179\pi$
$37$			4.38690e10i			0.462111i	−0.972941	+	0.231055i	$0.925782\pi$
$38$	−8.86387e10			−0.774709
$39$	−6.48121e10	−	3.51338e10i	−0.472285	−	0.256020i
$40$		0			0
$41$			1.11156e11i			0.570751i	0.958416	+	0.285376i	$0.0921184\pi$
$41$			1.11156e11i			0.570751i	−0.958416	+	0.285376i	$0.907882\pi$
$42$	2.07583e11	+	1.12528e11i	0.900422	+	0.488107i
$43$			1.00533e11i			0.369853i	0.982752	+	0.184927i	$0.0592048\pi$
$43$			1.00533e11i			0.369853i	−0.982752	+	0.184927i	$0.940795\pi$
$44$			5.55947e11i			1.74126i
$45$		0			0
$46$	7.32365e9			0.0168044
$47$	−2.14354e11			−0.423104			−0.211552	−	0.977367i	$-0.567852\pi$
$47$	−2.14354e11			−0.423104			−0.211552	+	0.977367i	$0.567852\pi$
$48$	−3.74384e11	+	6.90634e11i	−0.637718	+	1.17641i
$49$	4.49367e11			0.662565
$50$		0			0
$51$	6.99098e11	−	1.28964e12i	0.779017	−	1.43707i
$52$			1.16466e12i			1.13286i
$53$	−5.81823e11			−0.495290			−0.247645	−	0.968851i	$-0.579657\pi$
$53$	−5.81823e11			−0.495290			−0.247645	+	0.968851i	$0.579657\pi$
$54$	2.35310e12	−	1.89957e11i	1.75745	−	0.141873i
$55$		0			0
$56$		−	1.96132e12i		−	1.13562i
$57$	4.09349e11	−	7.55135e11i	0.209397	−	0.386278i
$58$			5.58594e12i			2.52988i
$59$		−	2.31936e12i		−	0.931973i	−0.884792	−	0.465986i	$-0.845700\pi$
$59$		−	2.31936e12i		−	0.931973i	0.884792	−	0.465986i	$-0.154300\pi$
$60$		0			0
$61$	−4.22999e12			−1.34595			−0.672977	−	0.739663i	$-0.734985\pi$
$61$	−4.22999e12			−1.34595			−0.672977	+	0.739663i	$0.734985\pi$
$62$	−7.16964e12			−2.03590
$63$	−1.91731e12	+	1.24877e12i	−0.486752	+	0.317029i
$64$	−2.74911e12			−0.625076
$65$		0			0
$66$	−6.98222e12	−	3.78498e12i	−1.27993	−	0.693834i
$67$			2.32014e12i			0.382817i	0.981510	+	0.191408i	$0.0613055\pi$
$67$			2.32014e12i			0.382817i	−0.981510	+	0.191408i	$0.938695\pi$
$68$	−2.31746e13			−3.44707
$69$	−3.38219e10	+	6.23919e10i	−0.00454207	+	0.00837886i
$70$		0			0
$71$		−	1.42262e13i		−	1.56415i	−0.623182	−	0.782077i	$-0.714160\pi$
$71$		−	1.42262e13i		−	1.56415i	0.623182	−	0.782077i	$-0.285840\pi$
$72$	−1.07021e13	−	1.64315e13i	−1.06694	−	1.63813i
$73$		−	1.06676e13i		−	0.965622i	−0.875725	−	0.482811i	$-0.839616\pi$
$73$		−	1.06676e13i		−	0.965622i	0.875725	−	0.482811i	$-0.160384\pi$
$74$			9.90062e12i			0.814780i
$75$		0			0
$76$	−1.35696e13			−0.926560
$77$	7.69778e12			0.479656
$78$	−1.46272e13	−	7.92921e12i	−0.832718	−	0.451406i
$79$	−3.21572e13			−1.67451			−0.837256	−	0.546812i	$-0.815841\pi$
$79$	−3.21572e13			−1.67451			−0.837256	+	0.546812i	$0.815841\pi$
$80$		0			0
$81$	−9.24873e12	+	2.09239e13i	−0.404284	+	0.914633i
$82$			2.50864e13i			1.00633i
$83$	1.21884e13			0.449159			0.224580	−	0.974456i	$-0.427899\pi$
$83$	1.21884e13			0.449159			0.224580	+	0.974456i	$0.427899\pi$
$84$	3.17787e13	+	1.72268e13i	1.07691	+	0.583782i
$85$		0			0
$86$			2.26889e13i			0.652114i
$87$	−4.75880e13	−	2.57969e13i	−1.26143	−	0.683805i
$88$			6.59706e13i			1.61425i
$89$		−	3.00312e13i		−	0.678957i	−0.940614	−	0.339479i	$-0.889749\pi$
$89$		−	3.00312e13i		−	0.678957i	0.940614	−	0.339479i	$-0.110251\pi$
$90$		0			0
$91$	1.61262e13			0.312063
$92$	1.12117e12			0.0200982
$93$	3.31107e13	−	6.10799e13i	0.550285	−	1.01512i
$94$	−4.83768e13			−0.746005
$95$		0			0
$96$	−1.44831e13	+	2.67173e13i	−0.192736	+	0.355544i
$97$			5.28428e13i			0.654009i	0.945023	+	0.327004i	$0.106039\pi$
$97$			5.28428e13i			0.654009i	−0.945023	+	0.327004i	$0.893961\pi$
$98$	1.01416e14			1.16822
$99$	6.44903e13	−	4.20035e13i	0.691907	−	0.450650i
$100$		0			0
$101$			9.00471e13i			0.839886i	0.907551	+	0.419943i	$0.137950\pi$
$101$			9.00471e13i			0.839886i	−0.907551	+	0.419943i	$0.862050\pi$
$102$	1.57777e14	−	2.91054e14i	1.37354	−	2.53380i
$103$			9.84098e13i			0.800162i	0.916480	+	0.400081i	$0.131018\pi$
$103$			9.84098e13i			0.800162i	−0.916480	+	0.400081i	$0.868982\pi$
$104$			1.38203e14i			1.05023i
$105$		0			0
$106$	−1.31309e14			−0.873281
$107$	1.29015e14			0.803440			0.401720	−	0.915763i	$-0.368413\pi$
$107$	1.29015e14			0.803440			0.401720	+	0.915763i	$0.368413\pi$
$108$	3.60234e14	−	2.90804e13i	2.10193	−	0.169682i
$109$	−1.17591e14			−0.643265			−0.321633	−	0.946865i	$-0.604232\pi$
$109$	−1.17591e14			−0.643265			−0.321633	+	0.946865i	$0.604232\pi$
$110$		0			0
$111$	−8.43458e13	−	4.57228e13i	−0.406259	−	0.220228i
$112$		−	1.71840e14i		−	0.777317i
$113$	2.84987e14			1.21137			0.605684	−	0.795705i	$-0.292899\pi$
$113$	2.84987e14			1.21137			0.605684	+	0.795705i	$0.292899\pi$
$114$	9.23843e13	−	1.70423e14i	0.369202	−	0.681075i
$115$		0			0
$116$			8.55148e14i			3.02577i
$117$	1.35102e14	−	8.79939e13i	0.450153	−	0.293192i
$118$		−	5.23446e14i		−	1.64323i
$119$			3.20882e14i			0.949546i
$120$		0			0
$121$	1.20828e14			0.318179
$122$	−9.54649e14			−2.37315
$123$	−2.13717e14	−	1.15853e14i	−0.501769	−	0.272003i
$124$	−1.09760e15			−2.43495
$125$		0			0
$126$	−4.32709e14	+	2.81830e14i	−0.858227	+	0.558977i
$127$			1.21233e14i			0.227506i	0.993509	+	0.113753i	$0.0362873\pi$
$127$			1.21233e14i			0.227506i	−0.993509	+	0.113753i	$0.963713\pi$
$128$	−8.48106e14			−1.50654
$129$	−1.93292e14	−	1.04781e14i	−0.325152	−	0.176261i
$130$		0			0
$131$			4.29478e14i			0.648697i	0.945938	+	0.324348i	$0.105145\pi$
$131$			4.29478e14i			0.648697i	−0.945938	+	0.324348i	$0.894855\pi$
$132$	−1.06890e15	−	5.79440e14i	−1.53081	−	0.829833i
$133$			1.87889e14i			0.255234i
$134$			5.23623e14i			0.674971i
$135$		0			0
$136$	−2.74998e15			−3.19564
$137$	7.44475e14			0.821876			0.410938	−	0.911663i	$-0.365201\pi$
$137$	7.44475e14			0.821876			0.410938	+	0.911663i	$0.365201\pi$
$138$	−7.63312e12	+	1.40810e13i	−0.00800845	+	0.0147734i
$139$	1.29615e15			1.29286			0.646431	−	0.762972i	$-0.276261\pi$
$139$	1.29615e15			1.29286			0.646431	+	0.762972i	$0.276261\pi$
$140$		0			0
$141$	2.23412e14	−	4.12133e14i	0.201639	−	0.371967i
$142$		−	3.21065e15i		−	2.75787i
$143$	−5.42419e14			−0.443591
$144$	−9.37659e14	−	1.43964e15i	−0.730310	−	1.12128i
$145$		0			0
$146$		−	2.40753e15i		−	1.70256i
$147$	−4.68356e14	+	8.63985e14i	−0.315758	+	0.582485i
$148$			1.51568e15i			0.974486i
$149$			9.69877e14i			0.594857i	0.954744	+	0.297429i	$0.0961290\pi$
$149$			9.69877e14i			0.594857i	−0.954744	+	0.297429i	$0.903871\pi$
$150$		0			0
$151$	−1.01305e15			−0.565966			−0.282983	−	0.959125i	$-0.591324\pi$
$151$	−1.01305e15			−0.565966			−0.282983	+	0.959125i	$0.591324\pi$
$152$	−1.61022e15			−0.858974
$153$	1.75092e15	+	2.68827e15i	0.892125	+	1.36973i
$154$	1.73728e15			0.845716
$155$		0			0
$156$	−2.23926e15	−	1.21388e15i	−0.995940	−	0.539887i
$157$		−	1.64534e15i		−	0.699776i	−0.936792	−	0.349888i	$-0.886220\pi$
$157$		−	1.64534e15i		−	0.699776i	0.936792	−	0.349888i	$-0.113780\pi$
$158$	−7.25742e15			−2.95245
$159$	6.06409e14	−	1.11865e15i	0.236040	−	0.435428i
$160$		0			0
$161$		−	1.55240e13i		−	0.00553635i
$162$	−2.08731e15	+	4.72222e15i	−0.712823	+	1.61265i
$163$			2.55006e15i			0.834137i	0.908875	+	0.417068i	$0.136942\pi$
$163$			2.55006e15i			0.834137i	−0.908875	+	0.417068i	$0.863058\pi$
$164$			3.84046e15i			1.20358i
$165$		0			0
$166$	2.75075e15			0.791945
$167$	8.83395e14			0.243859			0.121930	−	0.992539i	$-0.461092\pi$
$167$	8.83395e14			0.243859			0.121930	+	0.992539i	$0.461092\pi$
$168$	3.77097e15	+	2.04419e15i	0.998361	+	0.541199i
$169$	2.80105e15			0.711401
$170$		0			0
$171$	1.02523e15	+	1.57409e15i	0.239800	+	0.368177i
$172$			3.47343e15i			0.779936i
$173$	8.97288e14			0.193468			0.0967340	−	0.995310i	$-0.469160\pi$
$173$	8.97288e14			0.193468			0.0967340	+	0.995310i	$0.469160\pi$
$174$	−1.07399e16	−	5.82199e15i	−2.22411	−	1.20566i
$175$		0			0
$176$			5.77999e15i			1.10494i
$177$	4.45936e15	+	2.41736e15i	0.819332	+	0.444150i
$178$		−	6.77761e15i		−	1.19712i
$179$			1.09489e16i			1.85951i	0.368178	+	0.929755i	$0.379982\pi$
$179$			1.09489e16i			1.85951i	−0.368178	+	0.929755i	$0.620018\pi$
$180$		0			0
$181$	−6.47973e15			−1.01814			−0.509070	−	0.860725i	$-0.670010\pi$
$181$	−6.47973e15			−1.01814			−0.509070	+	0.860725i	$0.670010\pi$
$182$	3.63945e15			0.550220
$183$	4.40874e15	−	8.13289e15i	0.641441	−	1.18328i
$184$	1.33042e14			0.0186322
$185$		0			0
$186$	7.47261e15	−	1.37849e16i	0.970246	−	1.78983i
$187$		−	1.07931e16i		−	1.34976i
$188$	−7.40597e15			−0.892230
$189$	−4.02655e14	−	4.98789e15i	−0.0467413	−	0.579008i
$190$		0			0
$191$			1.10814e16i			1.19498i	0.801877	+	0.597490i	$0.203835\pi$
$191$			1.10814e16i			1.19498i	−0.801877	+	0.597490i	$0.796165\pi$
$192$	2.86528e15	−	5.28564e15i	0.297892	−	0.549527i
$193$		−	9.76570e15i		−	0.979044i	−0.871991	−	0.489522i	$-0.837171\pi$
$193$		−	9.76570e15i		−	0.979044i	0.871991	−	0.489522i	$-0.162829\pi$
$194$			1.19259e16i			1.15313i
$195$		0			0
$196$	1.55257e16			1.39720
$197$	1.03403e16			0.897984			0.448992	−	0.893536i	$-0.351783\pi$
$197$	1.03403e16			0.897984			0.448992	+	0.893536i	$0.351783\pi$
$198$	1.45545e16	−	9.47961e15i	1.21995	−	0.794573i
$199$	−1.32910e16			−1.07544			−0.537721	−	0.843123i	$-0.680715\pi$
$199$	−1.32910e16			−1.07544			−0.537721	+	0.843123i	$0.680715\pi$
$200$		0			0
$201$	−4.46087e15	−	2.41818e15i	−0.336548	−	0.182439i
$202$			2.03224e16i			1.48086i
$203$	1.18406e16			0.833491
$204$	2.41539e16	−	4.45572e16i	1.64277	−	3.03045i
$205$		0			0
$206$			2.22097e16i			1.41082i
$207$	−8.47081e13	−	1.30057e14i	−0.00520155	−	0.00798621i
$208$			1.21086e16i			0.718870i
$209$		−	6.31980e15i		−	0.362810i
$210$		0			0
$211$	1.38905e16			0.746003			0.373001	−	0.927831i	$-0.378329\pi$
$211$	1.38905e16			0.746003			0.373001	+	0.927831i	$0.378329\pi$
$212$	−2.01020e16			−1.04445
$213$	2.73523e16	+	1.48273e16i	1.37511	+	0.745428i
$214$	2.91168e16			1.41660
$215$		0			0
$216$	4.27466e16	−	3.45078e15i	1.94861	−	0.157305i
$217$			1.51976e16i			0.670743i
$218$	−2.65387e16			−1.13419
$219$	2.05103e16	+	1.11184e16i	0.848914	+	0.460186i
$220$		0			0
$221$		−	2.26107e16i		−	0.878149i
$222$	−1.90357e16	−	1.03190e16i	−0.716303	−	0.388299i
$223$		−	1.51743e14i		−	0.00553317i	−0.999996	−	0.00276659i	$-0.999119\pi$
$223$		−	1.51743e14i		−	0.00553317i	0.999996	−	0.00276659i	$-0.000880633\pi$
$224$		−	6.64765e15i		−	0.234926i
$225$		0			0
$226$	6.43176e16			2.13585
$227$	5.26622e16			1.69558			0.847790	−	0.530332i	$-0.177933\pi$
$227$	5.26622e16			1.69558			0.847790	+	0.530332i	$0.177933\pi$
$228$	1.41431e16	−	2.60900e16i	0.441570	−	0.814573i
$229$	1.16961e16			0.354154			0.177077	−	0.984197i	$-0.443336\pi$
$229$	1.16961e16			0.354154			0.177077	+	0.984197i	$0.443336\pi$
$230$		0			0
$231$	−8.02306e15	+	1.48003e16i	−0.228589	+	0.421684i
$232$			1.01475e17i			2.80506i
$233$	8.01172e14			0.0214899			0.0107450	−	0.999942i	$-0.496580\pi$
$233$	8.01172e14			0.0214899			0.0107450	+	0.999942i	$0.496580\pi$
$234$	3.04905e16	−	1.98590e16i	0.793696	−	0.516947i
$235$		0			0
$236$		−	8.01340e16i		−	1.96532i
$237$	3.35160e16	−	6.18277e16i	0.798021	−	1.47212i
$238$			7.24184e16i			1.67421i
$239$			6.36561e16i			1.42907i	0.699598	+	0.714536i	$0.253362\pi$
$239$			6.36561e16i			1.42907i	−0.699598	+	0.714536i	$0.746638\pi$
$240$		0			0
$241$	−6.17574e16			−1.30789			−0.653943	−	0.756544i	$-0.726886\pi$
$241$	−6.17574e16			−1.30789			−0.653943	+	0.756544i	$0.726886\pi$
$242$	2.72693e16			0.561004
$243$	−3.05902e16	−	3.95903e16i	−0.611418	−	0.791308i
$244$	−1.46147e17			−2.83831
$245$		0			0
$246$	−4.82329e16	−	2.61465e16i	−0.884704	−	0.479587i
$247$		−	1.32394e16i		−	0.236043i
$248$	−1.30244e17			−2.25734
$249$	−1.27035e16	+	2.34343e16i	−0.214056	+	0.394873i
$250$		0			0
$251$		−	3.71900e16i		−	0.592530i	−0.955106	−	0.296265i	$-0.904259\pi$
$251$		−	3.71900e16i		−	0.592530i	0.955106	−	0.296265i	$-0.0957411\pi$
$252$	−6.62431e16	+	4.31452e16i	−1.02645	+	0.668542i
$253$			5.22164e14i			0.00786979i
$254$			2.73605e16i			0.401132i
$255$		0			0
$256$	−1.46364e17			−2.03121
$257$	−1.85887e15			−0.0251025			−0.0125512	−	0.999921i	$-0.503995\pi$
$257$	−1.85887e15			−0.0251025			−0.0125512	+	0.999921i	$0.503995\pi$
$258$	−4.36233e16	−	2.36476e16i	−0.573298	−	0.310778i
$259$	2.09865e16			0.268436
$260$		0			0
$261$	9.91978e16	−	6.46091e16i	1.20232	−	0.783088i
$262$			9.69271e16i			1.14376i
$263$	−8.73906e16			−1.00409			−0.502047	−	0.864840i	$-0.667420\pi$
$263$	−8.73906e16			−1.00409			−0.502047	+	0.864840i	$0.667420\pi$
$264$	−1.26840e17	−	6.87583e16i	−1.41915	−	0.769303i
$265$		0			0
$266$			4.24038e16i			0.450022i
$267$	5.77401e16	+	3.13002e16i	0.596896	+	0.323570i
$268$			8.01611e16i			0.807273i
$269$			1.26379e17i			1.23996i	0.784616	+	0.619982i	$0.212860\pi$
$269$			1.26379e17i			1.23996i	−0.784616	+	0.619982i	$0.787140\pi$
$270$		0			0
$271$	5.86413e16			0.546284			0.273142	−	0.961974i	$-0.411937\pi$
$271$	5.86413e16			0.546284			0.273142	+	0.961974i	$0.411937\pi$
$272$	−2.40939e17			−2.18738
$273$	−1.68076e16	+	3.10054e16i	−0.148720	+	0.274346i
$274$	1.68017e17			1.44911
$275$		0			0
$276$	−1.16855e15	+	2.15565e15i	−0.00957819	+	0.0176691i
$277$		−	1.76150e17i		−	1.40774i	−0.710327	−	0.703872i	$-0.751453\pi$
$277$		−	1.76150e17i		−	1.40774i	0.710327	−	0.703872i	$-0.248547\pi$
$278$	2.92523e17			2.27954
$279$	8.29269e16	+	1.27322e17i	0.630182	+	0.967551i
$280$		0			0
$281$			1.87231e17i			1.35342i	0.736249	+	0.676711i	$0.236595\pi$
$281$			1.87231e17i			1.35342i	−0.736249	+	0.676711i	$0.763405\pi$
$282$	5.04210e16	−	9.30127e16i	0.355523	−	0.655841i
$283$		−	1.05273e16i		−	0.0724121i	−0.999344	−	0.0362060i	$-0.988473\pi$
$283$		−	1.05273e16i		−	0.0724121i	0.999344	−	0.0362060i	$-0.0115273\pi$
$284$		−	4.91516e17i		−	3.29844i
$285$		0			0
$286$	−1.22416e17			−0.782126
$287$	5.31760e16			0.331545
$288$	−3.62734e16	−	5.56925e16i	−0.220720	−	0.338883i
$289$	2.81534e17			1.67203
$290$		0			0
$291$	−1.01599e17	−	5.50758e16i	−0.574963	−	0.311680i
$292$		−	3.68567e17i		−	2.03627i
$293$	2.07675e17			1.12024			0.560121	−	0.828411i	$-0.310755\pi$
$293$	2.07675e17			1.12024			0.560121	+	0.828411i	$0.310755\pi$
$294$	−1.05701e17	+	1.94989e17i	−0.556736	+	1.02702i
$295$		0			0
$296$			1.79856e17i			0.903405i
$297$	1.35436e16	+	1.67772e17i	0.0664417	+	0.823047i
$298$			2.18887e17i			1.04883i
$299$			1.09389e15i			0.00512006i
$300$		0			0
$301$	4.80939e16			0.214845
$302$	−2.28630e17			−0.997895
$303$	−1.73131e17	−	9.38523e16i	−0.738375	−	0.400264i
$304$	−1.41079e17			−0.587959
$305$		0			0
$306$	3.95157e17	+	6.06706e17i	1.57297	+	2.41506i
$307$		−	3.26458e17i		−	1.27016i	−0.772446	−	0.635081i	$-0.780967\pi$
$307$		−	3.26458e17i		−	1.27016i	0.772446	−	0.635081i	$-0.219033\pi$
$308$	2.65959e17			1.01149
$309$	−1.89210e17	−	1.02568e17i	−0.703452	−	0.381333i
$310$		0			0
$311$		−	3.94473e17i		−	1.40183i	−0.713245	−	0.700915i	$-0.752775\pi$
$311$		−	3.94473e17i		−	1.40183i	0.713245	−	0.700915i	$-0.247225\pi$
$312$	−2.65719e17	−	1.44043e17i	−0.923294	−	0.500506i
$313$		−	4.19357e17i		−	1.42486i	−0.701741	−	0.712432i	$-0.747594\pi$
$313$		−	4.19357e17i		−	1.42486i	0.701741	−	0.712432i	$-0.252406\pi$
$314$		−	3.71331e17i		−	1.23382i
$315$		0			0
$316$	−1.11103e18			−3.53116
$317$	3.68072e17			1.14424			0.572120	−	0.820170i	$-0.306121\pi$
$317$	3.68072e17			1.14424			0.572120	+	0.820170i	$0.306121\pi$
$318$	1.36858e17	−	2.52465e17i	0.416179	−	0.767734i
$319$	−3.98268e17			−1.18479
$320$		0			0
$321$	−1.34467e17	+	2.48053e17i	−0.382895	+	0.706334i
$322$		−	3.50355e15i		−	0.00976152i
$323$	2.63441e17			0.718232
$324$	−3.19545e17	+	7.22922e17i	−0.852543	+	1.92875i
$325$		0			0
$326$			5.75513e17i			1.47073i
$327$	1.22561e17	−	2.26090e17i	0.306561	−	0.565518i
$328$			4.55722e17i			1.11579i
$329$			1.02545e17i			0.245778i
$330$		0			0
$331$	1.46374e17			0.336255			0.168127	−	0.985765i	$-0.446228\pi$
$331$	1.46374e17			0.336255			0.168127	+	0.985765i	$0.446228\pi$
$332$	4.21111e17			0.947174
$333$	1.75820e17	−	1.14514e17i	0.387221	−	0.252203i
$334$	1.99370e17			0.429966
$335$		0			0
$336$	3.30392e17	+	1.79101e17i	0.683368	+	0.370445i
$337$			5.56586e17i			1.12752i	0.825940	+	0.563758i	$0.190645\pi$
$337$			5.56586e17i			1.12752i	−0.825940	+	0.563758i	$0.809355\pi$
$338$	6.32158e17			1.25432
$339$	−2.97030e17	+	5.47937e17i	−0.577301	+	1.06496i
$340$		0			0
$341$		−	5.11184e17i		−	0.953447i
$342$	2.31380e17	+	3.55250e17i	0.422808	+	0.649159i
$343$		−	5.39427e17i		−	0.965770i
$344$			4.12169e17i			0.723046i
$345$		0			0
$346$	2.02505e17			0.341117
$347$	−5.48397e16			−0.0905292			−0.0452646	−	0.998975i	$-0.514413\pi$
$347$	−5.48397e16			−0.0905292			−0.0452646	+	0.998975i	$0.514413\pi$
$348$	−1.64417e18	−	8.91284e17i	−2.66006	−	1.44199i
$349$	−5.77040e17			−0.915014			−0.457507	−	0.889206i	$-0.651258\pi$
$349$	−5.77040e17			−0.915014			−0.457507	+	0.889206i	$0.651258\pi$
$350$		0			0
$351$	2.83728e16	+	3.51469e17i	0.0432267	+	0.535472i
$352$			2.23599e17i			0.333943i
$353$	−8.97000e17			−1.31332			−0.656658	−	0.754188i	$-0.728031\pi$
$353$	−8.97000e17			−1.31332			−0.656658	+	0.754188i	$0.728031\pi$
$354$	1.00641e18	+	5.45565e17i	1.44462	+	0.783112i
$355$		0			0
$356$		−	1.03758e18i		−	1.43177i
$357$	−6.16950e17	−	3.34441e17i	−0.834781	−	0.452525i
$358$			2.47101e18i			3.27863i
$359$		−	1.18291e18i		−	1.53918i	−0.638540	−	0.769589i	$-0.720461\pi$
$359$		−	1.18291e18i		−	1.53918i	0.638540	−	0.769589i	$-0.279539\pi$
$360$		0			0
$361$	−6.44752e17			−0.806942
$362$	−1.46238e18			−1.79515
$363$	−1.25934e17	+	2.32313e17i	−0.151634	+	0.279723i
$364$	5.57162e17			0.658069
$365$		0			0
$366$	9.94989e17	−	1.83548e18i	1.13097	−	2.08632i
$367$			1.25177e18i			1.39592i	0.716135	+	0.697962i	$0.245909\pi$
$367$			1.25177e18i			1.39592i	−0.716135	+	0.697962i	$0.754091\pi$
$368$	1.16564e16			0.0127535
$369$	4.45496e17	−	2.90159e17i	0.478255	−	0.311495i
$370$		0			0
$371$			2.78338e17i			0.287710i
$372$	1.14398e18	−	2.11032e18i	1.16042	−	2.14066i
$373$			1.64900e18i			1.64157i	0.571240	+	0.820783i	$0.306462\pi$
$373$			1.64900e18i			1.64157i	−0.571240	+	0.820783i	$0.693538\pi$
$374$		−	2.43586e18i		−	2.37986i
$375$		0			0
$376$	−8.78817e17			−0.827149
$377$	−8.34339e17			−0.770820
$378$	−9.08735e16	−	1.12570e18i	−0.0824128	−	1.02089i
$379$	2.49403e17			0.222038			0.111019	−	0.993818i	$-0.464589\pi$
$379$	2.49403e17			0.222038			0.111019	+	0.993818i	$0.464589\pi$
$380$		0			0
$381$	−2.33091e17	−	1.26356e17i	−0.200009	−	0.108422i
$382$			2.50091e18i			2.10695i
$383$	7.17438e17			0.593462			0.296731	−	0.954961i	$-0.404104\pi$
$383$	7.17438e17			0.593462			0.296731	+	0.954961i	$0.404104\pi$
$384$	8.83944e17	−	1.63063e18i	0.717970	−	1.32445i
$385$		0			0
$386$		−	2.20398e18i		−	1.72622i
$387$	4.02920e17	−	2.62428e17i	0.309915	−	0.201852i
$388$			1.82572e18i			1.37915i
$389$			2.36642e18i			1.75568i	0.478958	+	0.877838i	$0.341015\pi$
$389$			2.36642e18i			1.75568i	−0.478958	+	0.877838i	$0.658985\pi$
$390$		0			0
$391$	−2.17664e16			−0.0155793
$392$	1.84233e18			1.29528
$393$	−8.25745e17	−	4.47626e17i	−0.570293	−	0.309149i
$394$	2.33365e18			1.58330
$395$		0			0
$396$	2.22815e18	−	1.45123e18i	1.45907	−	0.950318i
$397$		−	9.86307e17i		−	0.634569i	−0.948330	−	0.317284i	$-0.897229\pi$
$397$		−	9.86307e17i		−	0.634569i	0.948330	−	0.317284i	$-0.102771\pi$
$398$	−2.99960e18			−1.89619
$399$	−3.61248e17	−	1.95828e17i	−0.224386	−	0.121637i
$400$		0			0
$401$			2.15290e18i			1.29126i	0.763651	+	0.645629i	$0.223405\pi$
$401$			2.15290e18i			1.29126i	−0.763651	+	0.645629i	$0.776595\pi$
$402$	−1.00676e18	−	5.45750e17i	−0.593392	−	0.321671i
$403$		−	1.07089e18i		−	0.620310i
$404$			3.11114e18i			1.77113i
$405$		0			0
$406$	2.67225e18			1.46959
$407$	−7.05898e17			−0.381576
$408$	2.86619e18	−	5.28731e18i	1.52294	−	2.80940i
$409$	−3.32383e18			−1.73610			−0.868052	−	0.496474i	$-0.834628\pi$
$409$	−3.32383e18			−1.73610			−0.868052	+	0.496474i	$0.834628\pi$
$410$		0			0
$411$	−7.75934e17	+	1.43138e18i	−0.391681	+	0.722541i
$412$			3.40007e18i			1.68736i
$413$	−1.10955e18			−0.541375
$414$	−1.91174e16	−	2.93520e16i	−0.00917122	−	0.0140811i
$415$		0			0
$416$			4.68422e17i			0.217262i
$417$	−1.35092e18	+	2.49208e18i	−0.616139	+	1.13660i
$418$		−	1.42629e18i		−	0.639696i
$419$		−	2.14089e18i		−	0.944272i	−0.881526	−	0.472136i	$-0.843483\pi$
$419$		−	2.14089e18i		−	0.944272i	0.881526	−	0.472136i	$-0.156517\pi$
$420$		0			0
$421$	8.73440e17			0.372613			0.186307	−	0.982492i	$-0.440348\pi$
$421$	8.73440e17			0.372613			0.186307	+	0.982492i	$0.440348\pi$
$422$	3.13489e18			1.31533
$423$	5.59544e17	+	8.59098e17i	0.230915	+	0.354536i
$424$	−2.38538e18			−0.968269
$425$		0			0
$426$	6.17302e18	+	3.34632e18i	2.42455	+	1.31432i
$427$			2.02358e18i			0.781854i
$428$	4.45748e18			1.69427
$429$	5.65340e17	−	1.04289e18i	0.211402	−	0.389977i
$430$		0			0
$431$			5.74588e17i			0.207977i	0.994578	+	0.103989i	$0.0331605\pi$
$431$			5.74588e17i			0.207977i	−0.994578	+	0.103989i	$0.966839\pi$
$432$	3.74523e18	−	3.02339e17i	1.33381	−	0.107673i
$433$		−	1.73636e18i		−	0.608450i	−0.952600	−	0.304225i	$-0.901602\pi$
$433$		−	1.73636e18i		−	0.608450i	0.952600	−	0.304225i	$-0.0983975\pi$
$434$			3.42988e18i			1.18264i
$435$		0			0
$436$	−4.06280e18			−1.35650
$437$	−1.27451e16			−0.00418767
$438$	4.62889e18	+	2.50926e18i	1.49678	+	0.811386i
$439$	−2.13182e17			−0.0678419			−0.0339209	−	0.999425i	$-0.510799\pi$
$439$	−2.13182e17			−0.0678419			−0.0339209	+	0.999425i	$0.510799\pi$
$440$		0			0
$441$	−1.17301e18	−	1.80099e18i	−0.361604	−	0.555189i
$442$		−	5.10292e18i		−	1.54833i
$443$	−1.81656e18			−0.542530			−0.271265	−	0.962505i	$-0.587442\pi$
$443$	−1.81656e18			−0.542530			−0.271265	+	0.962505i	$0.587442\pi$
$444$	−2.91416e18	−	1.57973e18i	−0.856707	−	0.464410i
$445$		0			0
$446$		−	3.42462e16i		−	0.00975593i
$447$	−1.86476e18	−	1.01086e18i	−0.522961	−	0.283491i
$448$			1.31515e18i			0.363101i
$449$		−	4.45737e18i		−	1.21158i	−0.795623	−	0.605792i	$-0.792856\pi$
$449$		−	4.45737e18i		−	1.21158i	0.795623	−	0.605792i	$-0.207144\pi$
$450$		0			0
$451$	−1.78862e18			−0.471283
$452$	9.84634e18			2.55450
$453$	1.05585e18	−	1.94775e18i	0.269722	−	0.497562i
$454$	1.18851e19			2.98960
$455$		0			0
$456$	1.67826e18	−	3.09593e18i	0.409361	−	0.755156i
$457$		−	1.94241e18i		−	0.466581i	−0.972407	−	0.233290i	$-0.925051\pi$
$457$		−	1.94241e18i		−	0.466581i	0.972407	−	0.233290i	$-0.0749493\pi$
$458$	2.63964e18			0.624434
$459$	−6.99358e18	+	5.64567e17i	−1.62934	+	0.131530i
$460$		0			0
$461$			1.12266e18i			0.253713i	0.991921	+	0.126856i	$0.0404888\pi$
$461$			1.12266e18i			0.253713i	−0.991921	+	0.126856i	$0.959511\pi$
$462$	−1.81069e18	+	3.34022e18i	−0.403042	+	0.743500i
$463$			3.87736e18i			0.850098i	0.905170	+	0.425049i	$0.139743\pi$
$463$			3.87736e18i			0.850098i	−0.905170	+	0.425049i	$0.860257\pi$
$464$			8.89068e18i			1.92003i
$465$		0			0
$466$	1.80813e17			0.0378904
$467$	−2.71349e18			−0.560156			−0.280078	−	0.959977i	$-0.590360\pi$
$467$	−2.71349e18			−0.560156			−0.280078	+	0.959977i	$0.590360\pi$
$468$	4.66778e18	−	3.04020e18i	0.949269	−	0.618274i
$469$	1.10993e18			0.222375
$470$		0			0
$471$	3.16346e18	+	1.71487e18i	0.615199	+	0.333492i
$472$		−	9.50897e18i		−	1.82196i
$473$	−1.61768e18			−0.305397
$474$	7.56409e18	−	1.39536e19i	1.40705	−	2.59561i
$475$		0			0
$476$			1.10865e19i			2.00238i
$477$	1.51877e18	+	2.33185e18i	0.270311	+	0.415023i
$478$			1.43663e19i			2.51970i
$479$			9.98368e18i			1.72561i	0.505538	+	0.862804i	$0.331294\pi$
$479$			9.98368e18i			1.72561i	−0.505538	+	0.862804i	$0.668706\pi$
$480$		0			0
$481$	−1.47880e18			−0.248252
$482$	−1.39378e19			−2.30603
$483$	2.98476e16	+	1.61800e16i	0.00486721	+	0.00263845i
$484$	4.17463e18			0.670967
$485$		0			0
$486$	−6.90378e18	−	8.93498e18i	−1.07803	−	1.39521i
$487$			7.04157e18i			1.08384i	0.840429	+	0.541922i	$0.182303\pi$
$487$			7.04157e18i			1.08384i	−0.840429	+	0.541922i	$0.817697\pi$
$488$	−1.73423e19			−2.63128
$489$	−4.90293e18	−	2.65782e18i	−0.733320	−	0.397524i
$490$		0			0
$491$		−	3.57849e18i		−	0.520151i	−0.965588	−	0.260076i	$-0.916253\pi$
$491$		−	3.57849e18i		−	0.520151i	0.965588	−	0.260076i	$-0.0837475\pi$
$492$	−7.38395e18	−	4.00275e18i	−1.05812	−	0.573591i
$493$		−	1.66018e19i		−	2.34545i
$494$		−	2.98796e18i		−	0.416184i
$495$		0			0
$496$	−1.14113e19			−1.54513
$497$	−6.80564e18			−0.908604
$498$	−2.86699e18	+	5.28880e18i	−0.377416	+	0.696228i
$499$	−1.27672e19			−1.65726			−0.828629	−	0.559797i	$-0.810879\pi$
$499$	−1.27672e19			−1.65726			−0.828629	+	0.559797i	$0.810879\pi$
$500$		0			0
$501$	−9.20724e17	+	1.69848e18i	−0.116216	+	0.214386i
$502$		−	8.39326e18i		−	1.04473i
$503$	−1.14021e19			−1.39962			−0.699809	−	0.714330i	$-0.746731\pi$
$503$	−1.14021e19			−1.39962			−0.699809	+	0.714330i	$0.746731\pi$
$504$	−7.86063e18	+	5.11976e18i	−0.951577	+	0.619777i
$505$		0			0
$506$			1.17845e17i			0.0138758i
$507$	−2.91942e18	+	5.38551e18i	−0.339032	+	0.625419i
$508$			4.18860e18i			0.479759i
$509$		−	5.16333e18i		−	0.583319i	−0.956522	−	0.291659i	$-0.905793\pi$
$509$		−	5.16333e18i		−	0.583319i	0.956522	−	0.291659i	$-0.0942073\pi$
$510$		0			0
$511$	−5.10327e18			−0.560921
$512$	−1.91369e19			−2.07483
$513$	−4.09501e18	+	3.30576e17i	−0.437959	+	0.0353549i
$514$	−4.19520e17			−0.0442599
$515$		0			0
$516$	−6.67826e18	−	3.62020e18i	−0.685670	−	0.371693i
$517$		−	3.44919e18i		−	0.349368i
$518$	4.73635e18			0.473299
$519$	−9.35205e17	+	1.72519e18i	−0.0922009	+	0.170085i
$520$		0			0
$521$		−	1.00727e19i		−	0.966675i	−0.875434	−	0.483338i	$-0.839424\pi$
$521$		−	1.00727e19i		−	0.966675i	0.875434	−	0.483338i	$-0.160576\pi$
$522$	2.23875e19	−	1.45814e19i	2.11989	−	1.38072i
$523$		−	7.35018e18i		−	0.686730i	−0.939202	−	0.343365i	$-0.888433\pi$
$523$		−	7.35018e18i		−	0.686730i	0.939202	−	0.343365i	$-0.111567\pi$
$524$			1.48385e19i			1.36795i
$525$		0			0
$526$	−1.97228e19			−1.77039
$527$	2.13087e19			1.88748
$528$	−1.11130e19	−	6.02423e18i	−0.971392	−	0.526580i
$529$	−1.15918e19			−0.999909
$530$		0			0
$531$	−9.29560e18	+	6.05438e18i	−0.780937	+	0.508637i
$532$			6.49157e18i			0.538231i
$533$	−3.74701e18			−0.306616
$534$	1.30311e19	+	7.06401e18i	1.05243	+	0.570509i
$535$		0			0
$536$			9.51219e18i			0.748388i
$537$	−2.10511e19	−	1.14115e19i	−1.63476	−	0.886185i
$538$			2.85220e19i			2.18627i
$539$			7.23078e18i			0.547096i
$540$		0			0
$541$	−3.58953e18			−0.264640			−0.132320	−	0.991207i	$-0.542243\pi$
$541$	−3.58953e18			−0.264640			−0.132320	+	0.991207i	$0.542243\pi$
$542$	1.32345e19			0.963192
$543$	6.75355e18	−	1.24584e19i	0.485214	−	0.895084i
$544$	−9.32074e18			−0.661086
$545$		0			0
$546$	−3.79325e18	+	6.99748e18i	−0.262218	+	0.483719i
$547$		−	4.53669e18i		−	0.309619i	−0.987944	−	0.154809i	$-0.950524\pi$
$547$		−	4.53669e18i		−	0.309619i	0.987944	−	0.154809i	$-0.0494763\pi$
$548$	2.57217e19			1.73315
$549$	1.10418e19	+	1.69531e19i	0.734573	+	1.12783i
$550$		0			0
$551$		−	9.72100e18i		−	0.630449i
$552$	−1.38664e17	+	2.55796e17i	−0.00887954	+	0.0163803i
$553$			1.53836e19i			0.972709i
$554$		−	3.97545e19i		−	2.48209i
$555$		0			0
$556$	4.47822e19			2.72635
$557$	6.89338e18			0.414425			0.207212	−	0.978296i	$-0.433561\pi$
$557$	6.89338e18			0.414425			0.207212	+	0.978296i	$0.433561\pi$
$558$	1.87154e19	+	2.87348e19i	1.11112	+	1.70596i
$559$	−3.38890e18			−0.198690
$560$		0			0
$561$	2.07517e19	+	1.12492e19i	1.18662	+	0.643254i
$562$			4.22554e19i			2.38631i
$563$	1.45922e18			0.0813883			0.0406942	−	0.999172i	$-0.487043\pi$
$563$	1.45922e18			0.0813883			0.0406942	+	0.999172i	$0.487043\pi$
$564$	7.71892e18	−	1.42393e19i	0.425210	−	0.784393i
$565$		0			0
$566$		−	2.37586e18i		−	0.127675i
$567$	1.00098e19	+	4.42449e18i	0.531303	+	0.234845i
$568$		−	5.83249e19i		−	3.05785i
$569$		−	4.36385e18i		−	0.225987i	−0.993596	−	0.112994i	$-0.963956\pi$
$569$		−	4.36385e18i		−	0.225987i	0.993596	−	0.112994i	$-0.0360440\pi$
$570$		0			0
$571$	1.61706e19			0.817099			0.408549	−	0.912736i	$-0.366035\pi$
$571$	1.61706e19			0.817099			0.408549	+	0.912736i	$0.366035\pi$
$572$	−1.87406e19			−0.935431
$573$	−2.13059e19	−	1.15496e19i	−1.05055	−	0.569490i
$574$	1.20011e19			0.584570
$575$		0			0
$576$	7.17620e18	+	1.10180e19i	0.341143	+	0.523775i
$577$		−	8.26639e17i		−	0.0388227i	−0.999812	−	0.0194113i	$-0.993821\pi$
$577$		−	8.26639e17i		−	0.0388227i	0.999812	−	0.0194113i	$-0.00617921\pi$
$578$	6.35381e19			2.94808
$579$	1.87762e19	+	1.01784e19i	0.860714	+	0.466582i
$580$		0			0
$581$		−	5.83080e18i		−	0.260913i
$582$	−2.29295e19	−	1.24298e19i	−1.01376	−	0.549546i
$583$		−	9.36213e18i		−	0.408973i
$584$		−	4.37354e19i		−	1.88774i
$585$		0			0
$586$	4.68694e19			1.97518
$587$	3.82350e19			1.59219			0.796095	−	0.605172i	$-0.206896\pi$
$587$	3.82350e19			1.59219			0.796095	+	0.605172i	$0.206896\pi$
$588$	−1.61817e19	+	2.98508e19i	−0.665862	+	1.22833i
$589$	1.24771e19			0.507347
$590$		0			0
$591$	−1.07772e19	+	1.98810e19i	−0.427952	+	0.789451i
$592$			1.57580e19i			0.618371i
$593$	2.10110e19			0.814826			0.407413	−	0.913244i	$-0.366431\pi$
$593$	2.10110e19			0.814826			0.407413	+	0.913244i	$0.366431\pi$
$594$	3.05661e18	+	3.78638e19i	0.117148	+	1.45117i
$595$		0			0
$596$			3.35093e19i			1.25442i
$597$	1.38527e19	−	2.55543e19i	0.512523	−	0.945461i
$598$			2.46876e17i			0.00902755i
$599$			4.94075e19i			1.78569i	0.450367	+	0.892844i	$0.351293\pi$
$599$			4.94075e19i			1.78569i	−0.450367	+	0.892844i	$0.648707\pi$
$600$		0			0
$601$	−2.29665e19			−0.810912			−0.405456	−	0.914115i	$-0.632887\pi$
$601$	−2.29665e19			−0.810912			−0.405456	+	0.914115i	$0.632887\pi$
$602$	1.08541e19			0.378808
$603$	9.29875e18	−	6.05643e18i	0.320777	−	0.208927i
$604$	−3.50008e19			−1.19349
$605$		0			0
$606$	−3.90733e19	−	2.11811e19i	−1.30188	−	0.705733i
$607$		−	2.40723e19i		−	0.792862i	−0.918065	−	0.396431i	$-0.870249\pi$
$607$		−	2.40723e19i		−	0.792862i	0.918065	−	0.396431i	$-0.129751\pi$
$608$	−5.45765e18			−0.177697
$609$	−1.23409e19	+	2.27656e19i	−0.397216	+	0.732753i
$610$		0			0
$611$		−	7.22575e18i		−	0.227298i
$612$	6.04944e19	+	9.28802e19i	1.88129	+	2.88844i
$613$		−	4.49657e19i		−	1.38248i	−0.722626	−	0.691239i	$-0.757065\pi$
$613$		−	4.49657e19i		−	1.38248i	0.722626	−	0.691239i	$-0.242935\pi$
$614$		−	7.36770e19i		−	2.23951i
$615$		0			0
$616$	3.15596e19			0.937705
$617$	1.15025e18			0.0337907			0.0168954	−	0.999857i	$-0.494622\pi$
$617$	1.15025e18			0.0337907			0.0168954	+	0.999857i	$0.494622\pi$
$618$	−4.27020e19	−	2.31482e19i	−1.24031	−	0.672355i
$619$	−1.85323e19			−0.532224			−0.266112	−	0.963942i	$-0.585739\pi$
$619$	−1.85323e19			−0.532224			−0.266112	+	0.963942i	$0.585739\pi$
$620$		0			0
$621$	3.38344e17	−	2.73133e16i	0.00949987	−	0.000766891i
$622$		−	8.90271e19i		−	2.47166i
$623$	−1.43666e19			−0.394400
$624$	−2.32808e19	−	1.26203e19i	−0.631985	−	0.342591i
$625$		0			0
$626$		−	9.46430e19i		−	2.51228i
$627$	1.21509e19	+	6.58685e18i	0.318960	+	0.172904i
$628$		−	5.68468e19i		−	1.47567i
$629$		−	2.94254e19i		−	0.755383i
$630$		0			0
$631$	2.33455e18			0.0586136			0.0293068	−	0.999570i	$-0.490670\pi$
$631$	2.33455e18			0.0586136			0.0293068	+	0.999570i	$0.490670\pi$
$632$	−1.31839e20			−3.27359
$633$	−1.44775e19	+	2.67069e19i	−0.355522	+	0.655838i
$634$	8.30685e19			2.01749
$635$		0			0
$636$	2.09515e19	−	3.86496e19i	0.497754	−	0.918217i
$637$			1.51479e19i			0.355939i
$638$	−8.98836e19			−2.08899
$639$	−5.70162e19	+	3.71356e19i	−1.31067	+	0.853658i
$640$		0			0
$641$			2.31344e19i			0.520298i	0.965568	+	0.260149i	$0.0837717\pi$
$641$			2.31344e19i			0.520298i	−0.965568	+	0.260149i	$0.916228\pi$
$642$	−3.03472e19	+	5.59821e19i	−0.675109	+	1.24539i
$643$			3.27850e19i			0.721438i	0.932675	+	0.360719i	$0.117469\pi$
$643$			3.27850e19i			0.721438i	−0.932675	+	0.360719i	$0.882531\pi$
$644$		−	5.36357e17i		−	0.0116749i
$645$		0			0
$646$	5.94548e19			1.26637
$647$	2.52933e19			0.532937			0.266468	−	0.963844i	$-0.414143\pi$
$647$	2.52933e19			0.532937			0.266468	+	0.963844i	$0.414143\pi$
$648$	−3.79183e19	+	8.57844e19i	−0.790357	+	1.78806i
$649$	3.73208e19			0.769553
$650$		0			0
$651$	−2.92200e19	−	1.58398e19i	−0.589675	−	0.319656i
$652$			8.81048e19i			1.75900i
$653$	2.21595e19			0.437692			0.218846	−	0.975759i	$-0.429771\pi$
$653$	2.21595e19			0.437692			0.218846	+	0.975759i	$0.429771\pi$
$654$	2.76602e19	−	5.10253e19i	0.540519	−	0.997106i
$655$		0			0
$656$			3.99279e19i			0.763748i
$657$	−4.27540e19	+	2.78464e19i	−0.809132	+	0.527001i
$658$			2.31429e19i			0.433348i
$659$		−	1.61070e19i		−	0.298413i	−0.988806	−	0.149207i	$-0.952328\pi$
$659$		−	1.61070e19i		−	0.298413i	0.988806	−	0.149207i	$-0.0476719\pi$
$660$		0			0
$661$	−3.88817e19			−0.705236			−0.352618	−	0.935767i	$-0.614709\pi$
$661$	−3.88817e19			−0.705236			−0.352618	+	0.935767i	$0.614709\pi$
$662$	3.30346e19			0.592875
$663$	4.34730e19	+	2.35662e19i	0.772013	+	0.418499i
$664$	4.99704e19			0.878085
$665$		0			0
$666$	3.96801e19	−	2.58443e19i	0.682736	−	0.444677i
$667$			8.03184e17i			0.0136752i
$668$	3.05214e19			0.514244
$669$	2.91752e17	+	1.58155e17i	0.00486442	+	0.00263694i
$670$		0			0
$671$		−	6.80649e19i		−	1.11139i
$672$	1.27813e19	+	6.92856e18i	0.206533	+	0.111959i
$673$		−	3.72205e19i		−	0.595219i	−0.954688	−	0.297609i	$-0.903811\pi$
$673$		−	3.72205e19i		−	0.595219i	0.954688	−	0.297609i	$-0.0961893\pi$
$674$			1.25614e20i			1.98800i
$675$		0			0
$676$	9.67767e19			1.50018
$677$	8.23009e19			1.26265			0.631326	−	0.775517i	$-0.282511\pi$
$677$	8.23009e19			1.26265			0.631326	+	0.775517i	$0.282511\pi$
$678$	−6.70355e19	+	1.23662e20i	−1.01788	+	1.87770i
$679$	2.52794e19			0.379908
$680$		0			0
$681$	−5.48876e19	+	1.01252e20i	−0.808061	+	1.49065i
$682$		−	1.15367e20i		−	1.68109i
$683$	6.26033e19			0.902927			0.451463	−	0.892290i	$-0.350902\pi$
$683$	6.26033e19			0.902927			0.451463	+	0.892290i	$0.350902\pi$
$684$	3.54218e19	+	5.43849e19i	0.505682	+	0.776401i
$685$		0			0
$686$		−	1.21741e20i		−	1.70282i
$687$	−1.21903e19	+	2.24877e19i	−0.168779	+	0.311350i
$688$			3.61120e19i			0.494917i
$689$		−	1.96129e19i		−	0.266077i
$690$		0			0
$691$	2.70281e19			0.359309			0.179655	−	0.983730i	$-0.442502\pi$
$691$	2.70281e19			0.359309			0.179655	+	0.983730i	$0.442502\pi$
$692$	3.10014e19			0.407980
$693$	−2.00940e19	−	3.08514e19i	−0.261779	−	0.401923i
$694$	−1.23765e19			−0.159618
$695$		0			0
$696$	−1.95103e20	−	1.05763e20i	−2.46603	−	1.33681i
$697$		−	7.45586e19i		−	0.932971i
$698$	−1.30230e20			−1.61333
$699$	−8.35028e17	+	1.54039e18i	−0.0102414	+	0.0188926i
$700$		0			0
$701$		−	9.70271e19i		−	1.16645i	−0.812310	−	0.583226i	$-0.801790\pi$
$701$		−	9.70271e19i		−	1.16645i	0.812310	−	0.583226i	$-0.198210\pi$
$702$	6.40334e18	+	7.93215e19i	0.0762161	+	0.944128i
$703$		−	1.72297e19i		−	0.203044i
$704$		−	4.42360e19i		−	0.516140i
$705$		0			0
$706$	−2.02440e20			−2.31560
$707$	4.30776e19			0.487883
$708$	1.54071e20	+	8.35202e19i	1.72778	+	0.936610i
$709$	−1.02481e20			−1.13794			−0.568968	−	0.822360i	$-0.692657\pi$
$709$	−1.02481e20			−1.13794			−0.568968	+	0.822360i	$0.692657\pi$
$710$		0			0
$711$	8.39421e19	+	1.28881e20i	0.913887	+	1.40314i
$712$		−	1.23123e20i		−	1.32733i
$713$	−1.03090e18			−0.0110050
$714$	−1.39237e20	−	7.54786e19i	−1.47186	−	0.797878i
$715$		0			0
$716$			3.78285e20i			3.92128i
$717$	−1.22390e20	−	6.63460e19i	−1.25635	−	0.681052i
$718$		−	2.66965e20i		−	2.71383i
$719$		−	1.67928e19i		−	0.169052i	−0.996421	−	0.0845260i	$-0.973062\pi$
$719$		−	1.67928e19i		−	0.169052i	0.996421	−	0.0845260i	$-0.0269376\pi$
$720$		0			0
$721$	4.70782e19			0.464807
$722$	−1.45511e20			−1.42278
$723$	6.43671e19	−	1.18739e20i	0.623298	−	1.14981i
$724$	−2.23875e20			−2.14702
$725$		0			0
$726$	−2.84216e19	+	5.24299e19i	−0.267357	+	0.493199i
$727$			1.97576e20i			1.84074i	0.391047	+	0.920371i	$0.372113\pi$
$727$			1.97576e20i			1.84074i	−0.391047	+	0.920371i	$0.627887\pi$
$728$	6.61147e19			0.610068
$729$	1.08002e20	−	1.75516e19i	0.987051	−	0.160408i
$730$		0			0
$731$		−	6.74329e19i		−	0.604575i
$732$	1.52322e20	−	2.80992e20i	1.35265	−	2.49526i
$733$			2.03135e20i			1.78672i	0.449341	+	0.893360i	$0.351659\pi$
$733$			2.03135e20i			1.78672i	−0.449341	+	0.893360i	$0.648341\pi$
$734$			2.82506e20i			2.46125i
$735$		0			0
$736$	4.50931e17			0.00385447
$737$	−3.73335e19			−0.316101
$738$	1.00542e20	−	6.54848e19i	0.843245	−	0.549219i
$739$	−3.77054e19			−0.313251			−0.156625	−	0.987658i	$-0.550062\pi$
$739$	−3.77054e19			−0.313251			−0.156625	+	0.987658i	$0.550062\pi$
$740$		0			0
$741$	2.54551e19	+	1.37989e19i	0.207514	+	0.112491i
$742$			6.28169e19i			0.507282i
$743$	−1.51734e20			−1.21384			−0.606918	−	0.794764i	$-0.707595\pi$
$743$	−1.51734e20			−1.21384			−0.606918	+	0.794764i	$0.707595\pi$
$744$	1.35748e20	−	2.50417e20i	1.07578	−	1.98451i
$745$		0			0
$746$			3.72156e20i			2.89436i
$747$	−3.18163e19	−	4.88492e19i	−0.245135	−	0.376368i
$748$		−	3.72904e20i		−	2.84633i
$749$		−	6.17193e19i		−	0.466711i
$750$		0			0
$751$	−1.19860e20			−0.889600			−0.444800	−	0.895630i	$-0.646725\pi$
$751$	−1.19860e20			−0.889600			−0.444800	+	0.895630i	$0.646725\pi$
$752$	−7.69973e19			−0.566175
$753$	7.15042e19	+	3.87615e19i	0.520915	+	0.282381i
$754$	−1.88299e20			−1.35909
$755$		0			0
$756$	−1.39118e19	−	1.72332e20i	−0.0985666	−	1.22100i
$757$		−	6.29864e19i		−	0.442156i	−0.975256	−	0.221078i	$-0.929042\pi$
$757$		−	6.29864e19i		−	0.442156i	0.975256	−	0.221078i	$-0.0709576\pi$
$758$	5.62867e19			0.391491
$759$	−1.00395e18	−	5.44229e17i	−0.00691863	−	0.00375050i
$760$		0			0
$761$			7.74499e19i			0.523997i	0.965068	+	0.261998i	$0.0843815\pi$
$761$			7.74499e19i			0.523997i	−0.965068	+	0.261998i	$0.915619\pi$
$762$	−5.26052e19	−	2.85167e19i	−0.352650	−	0.191167i
$763$			5.62545e19i			0.373667i
$764$			3.82863e20i			2.51994i
$765$		0			0
$766$	1.61916e20			1.04638
$767$	7.81840e19			0.500669
$768$	1.52549e20	−	2.81410e20i	0.968012	−	1.78571i
$769$	1.63707e20			1.02939			0.514697	−	0.857372i	$-0.327904\pi$
$769$	1.63707e20			1.02939			0.514697	+	0.857372i	$0.327904\pi$
$770$		0			0
$771$	1.93742e18	−	3.57399e18i	0.0119631	−	0.0220685i
$772$		−	3.37406e20i		−	2.06458i
$773$	−1.95306e20			−1.18430			−0.592148	−	0.805829i	$-0.701720\pi$
$773$	−1.95306e20			−1.18430			−0.592148	+	0.805829i	$0.701720\pi$
$774$	9.09333e19	−	5.92264e19i	0.546432	−	0.355900i
$775$		0			0
$776$			2.16647e20i			1.27856i
$777$	−2.18733e19	+	4.03501e19i	−0.127928	+	0.235992i
$778$			5.34067e20i			3.09556i
$779$		−	4.36569e19i		−	0.250779i
$780$		0			0
$781$	2.28914e20			1.29156
$782$	−4.91237e18			−0.0274690
$783$	2.08326e19	+	2.58064e20i	0.115455	+	1.43020i
$784$	1.61415e20			0.886608
$785$		0			0
$786$	−1.86359e20	−	1.01023e20i	−1.00552	−	0.545082i
$787$			1.79888e20i			0.962007i	0.876719	+	0.481004i	$0.159728\pi$
$787$			1.79888e20i			0.962007i	−0.876719	+	0.481004i	$0.840272\pi$
$788$	3.57258e20			1.89364
$789$	9.10834e19	−	1.68023e20i	0.478520	−	0.882736i
$790$		0			0
$791$		−	1.36335e20i		−	0.703674i
$792$	2.64399e20	−	1.72208e20i	1.35265	−	0.881000i
$793$		−	1.42590e20i		−	0.723066i
$794$		−	2.22596e20i		−	1.11885i
$795$		0			0
$796$	−4.59206e20			−2.26786
$797$	−2.77052e20			−1.35629			−0.678146	−	0.734927i	$-0.737216\pi$
$797$	−2.77052e20			−1.35629			−0.678146	+	0.734927i	$0.737216\pi$
$798$	−8.15286e19	−	4.41957e19i	−0.395631	−	0.214466i
$799$	1.43779e20			0.691622
$800$		0			0
$801$	−1.20360e20	+	7.83925e19i	−0.568925	+	0.370550i
$802$			4.85879e20i			2.27671i
$803$	1.71653e20			0.797337
$804$	−1.54124e20	−	8.35485e19i	−0.709703	−	0.384721i
$805$		0			0
$806$		−	2.41684e20i		−	1.09371i
$807$	−2.42985e20	−	1.31719e20i	−1.09010	−	0.590929i
$808$			3.69178e20i			1.64194i
$809$		−	1.89882e20i		−	0.837229i	−0.908164	−	0.418615i	$-0.862516\pi$
$809$		−	1.89882e20i		−	0.837229i	0.908164	−	0.418615i	$-0.137484\pi$
$810$		0			0
$811$	2.25399e20			0.976802			0.488401	−	0.872619i	$-0.337580\pi$
$811$	2.25399e20			0.976802			0.488401	+	0.872619i	$0.337580\pi$
$812$	4.09094e20			1.75764
$813$	−6.11193e19	+	1.12748e20i	−0.260342	+	0.480259i
$814$	−1.59311e20			−0.672784
$815$		0			0
$816$	2.51120e20	−	4.63246e20i	1.04244	−	1.92301i
$817$		−	3.94846e19i		−	0.162508i
$818$	−7.50141e20			−3.06105
$819$	−4.20953e19	−	6.46312e19i	−0.170313	−	0.261490i
$820$		0			0
$821$		−	4.00570e20i		−	1.59322i	−0.604492	−	0.796611i	$-0.706624\pi$
$821$		−	4.00570e20i		−	1.59322i	0.604492	−	0.796611i	$-0.293376\pi$
$822$	−1.75117e20	+	3.23042e20i	−0.690600	+	1.27396i
$823$			1.34324e20i			0.525235i	0.964900	+	0.262617i	$0.0845857\pi$
$823$			1.34324e20i			0.525235i	−0.964900	+	0.262617i	$0.915414\pi$
$824$			4.03464e20i			1.56428i
$825$		0			0
$826$	−2.50411e20			−0.954537
$827$	9.76761e19			0.369190			0.184595	−	0.982815i	$-0.440903\pi$
$827$	9.76761e19			0.369190			0.184595	+	0.982815i	$0.440903\pi$
$828$	−2.92667e18	−	4.49348e18i	−0.0109689	−	0.0168411i
$829$	−1.39626e20			−0.518900			−0.259450	−	0.965757i	$-0.583541\pi$
$829$	−1.39626e20			−0.518900			−0.259450	+	0.965757i	$0.583541\pi$
$830$		0			0
$831$	3.38678e20	+	1.83593e20i	1.23760	+	0.670888i
$832$		−	9.26708e19i		−	0.335799i
$833$	−3.01415e20			−1.08305
$834$	−3.04884e20	+	5.62426e20i	−1.08636	+	2.00403i
$835$		0			0
$836$		−	2.18350e20i		−	0.765083i
$837$	−3.31230e20	+	2.67390e19i	−1.15094	+	0.0929109i
$838$		−	4.83169e20i		−	1.66491i
$839$			3.23963e19i			0.110703i	0.998467	+	0.0553517i	$0.0176280\pi$
$839$			3.23963e19i			0.110703i	−0.998467	+	0.0553517i	$0.982372\pi$
$840$		0			0
$841$	−3.15052e20			−1.05879
$842$	1.97123e20			0.656981
$843$	−3.59984e20	−	1.95143e20i	−1.18984	−	0.644999i
$844$	4.79918e20			1.57315
$845$		0			0
$846$	1.26281e20	+	1.93886e20i	0.407142	+	0.625107i
$847$		−	5.78030e19i		−	0.184827i
$848$	−2.08994e20			−0.662770
$849$	2.02405e19	+	1.09721e19i	0.0636601	+	0.0345094i
$850$		0			0
$851$			1.42358e18i			0.00440427i
$852$	9.45024e20	+	5.12286e20i	2.89978	+	1.57194i
$853$		−	3.08044e19i		−	0.0937495i	−0.998901	−	0.0468748i	$-0.985074\pi$
$853$		−	3.08044e19i		−	0.0937495i	0.998901	−	0.0468748i	$-0.0149262\pi$
$854$			4.56694e20i			1.37854i
$855$		0			0
$856$	5.28940e20			1.57069
$857$	−4.87680e19			−0.143638			−0.0718190	−	0.997418i	$-0.522880\pi$
$857$	−4.87680e19			−0.143638			−0.0718190	+	0.997418i	$0.522880\pi$
$858$	1.27589e20	−	2.35366e20i	0.372737	−	0.687596i
$859$	−1.49706e20			−0.433799			−0.216899	−	0.976194i	$-0.569594\pi$
$859$	−1.49706e20			−0.433799			−0.216899	+	0.976194i	$0.569594\pi$
$860$		0			0
$861$	−5.54230e19	+	1.02240e20i	−0.158004	+	0.291473i
$862$			1.29676e20i			0.366699i
$863$	3.15069e20			0.883751			0.441875	−	0.897076i	$-0.354313\pi$
$863$	3.15069e20			0.883751			0.441875	+	0.897076i	$0.354313\pi$
$864$	1.44885e20	−	1.16960e19i	0.403113	−	0.0325419i
$865$		0			0
$866$		−	3.91872e20i		−	1.07280i
$867$	−2.93430e20	+	5.41297e20i	−0.796840	+	1.46995i
$868$			5.25078e20i			1.41444i
$869$		−	5.17442e20i		−	1.38269i
$870$		0			0
$871$	−7.82105e19			−0.205654
$872$	−4.82105e20			−1.25755
$873$	2.11785e20	−	1.37939e20i	0.548020	−	0.356934i
$874$	−2.87638e18			−0.00738357
$875$		0			0
$876$	7.08633e20	+	3.84141e20i	1.79016	+	0.970426i
$877$			3.62517e20i			0.908512i	0.890871	+	0.454256i	$0.150095\pi$
$877$			3.62517e20i			0.908512i	−0.890871	+	0.454256i	$0.849905\pi$
$878$	−4.81121e19			−0.119617
$879$	−2.16451e20	+	3.99292e20i	−0.533873	+	0.984846i
$880$		0			0
$881$		−	1.15407e20i		−	0.280157i	−0.990140	−	0.140078i	$-0.955265\pi$
$881$		−	1.15407e20i		−	0.280157i	0.990140	−	0.140078i	$-0.0447355\pi$
$882$	−2.64733e20	−	4.06458e20i	−0.637569	−	0.978893i
$883$			3.76570e20i			0.899748i	0.893092	+	0.449874i	$0.148531\pi$
$883$			3.76570e20i			0.899748i	−0.893092	+	0.449874i	$0.851469\pi$
$884$		−	7.81202e20i		−	1.85182i
$885$		0			0
$886$	−4.09972e20			−0.956573
$887$	4.49245e20			1.03996			0.519982	−	0.854177i	$-0.325939\pi$
$887$	4.49245e20			1.03996			0.519982	+	0.854177i	$0.325939\pi$
$888$	−3.45804e20	−	1.87456e20i	−0.794216	−	0.430535i
$889$	5.79964e19			0.132156
$890$		0			0
$891$	−3.36687e20	−	1.48822e20i	−0.755235	−	0.333828i
$892$		−	5.24273e18i		−	0.0116682i
$893$	8.41883e19			0.185905
$894$	−4.20849e20	−	2.28137e20i	−0.922069	−	0.499842i
$895$		0			0
$896$			4.05725e20i			0.875136i
$897$	−2.10319e18	−	1.14011e18i	−0.00450124	−	0.00244006i
$898$		−	1.00596e21i		−	2.13623i
$899$		−	7.86294e20i		−	1.65679i
$900$		0			0
$901$	3.90260e20			0.809619
$902$	−4.03666e20			−0.830953
$903$	−5.01262e19	+	9.24688e19i	−0.102388	+	0.188878i
$904$	1.16840e21			2.36817
$905$		0			0
$906$	2.38291e20	−	4.39581e20i	0.475566	−	0.877286i
$907$		−	9.99929e20i		−	1.98024i	−0.140215	−	0.990121i	$-0.544779\pi$
$907$		−	9.99929e20i		−	1.98024i	0.140215	−	0.990121i	$-0.455221\pi$
$908$	1.81949e21			3.57559
$909$	3.60894e20	−	2.35056e20i	0.703773	−	0.458379i
$910$		0			0
$911$		−	5.11050e20i		−	0.981374i	−0.871336	−	0.490687i	$-0.836746\pi$
$911$		−	5.11050e20i		−	0.981374i	0.871336	−	0.490687i	$-0.163254\pi$
$912$	1.47040e20	−	2.71248e20i	0.280203	−	0.516896i
$913$			1.96124e20i			0.370882i
$914$		−	4.38374e20i		−	0.822661i
$915$		0			0
$916$	4.04101e20			0.746830
$917$	2.05458e20			0.376822
$918$	−1.57835e21	+	1.27415e20i	−2.87280	+	0.231911i
$919$	4.38250e20			0.791613			0.395807	−	0.918334i	$-0.370465\pi$
$919$	4.38250e20			0.791613			0.395807	+	0.918334i	$0.370465\pi$
$920$		0			0
$921$	6.27672e20	+	3.40253e20i	1.11665	+	0.605320i
$922$			2.53369e20i			0.447339i
$923$	4.79555e20			0.840285
$924$	−2.77198e20	+	5.11352e20i	−0.482043	+	0.889234i
$925$		0			0
$926$			8.75065e20i			1.49887i
$927$	3.94411e20	−	2.56886e20i	0.670487	−	0.436699i
$928$			3.43937e20i			0.580287i
$929$			6.29570e20i			1.05423i	0.849795	+	0.527113i	$0.176726\pi$
$929$			6.29570e20i			1.05423i	−0.849795	+	0.527113i	$0.823274\pi$
$930$		0			0
$931$	−1.76490e20			−0.291120
$932$	2.76806e19			0.0453173
$933$	7.58443e20	+	4.11143e20i	1.23240	+	0.668069i
$934$	−6.12395e20			−0.987651
$935$		0			0
$936$	5.53894e20	−	3.60760e20i	0.880027	−	0.573176i
$937$		−	4.29592e20i		−	0.677454i	−0.940885	−	0.338727i	$-0.890004\pi$
$937$		−	4.29592e20i		−	0.677454i	0.940885	−	0.338727i	$-0.109996\pi$
$938$	2.50496e20			0.392085
$939$	8.06286e20	+	4.37078e20i	1.25265	+	0.679047i
$940$		0			0
$941$		−	2.95757e20i		−	0.452697i	−0.974046	−	0.226349i	$-0.927321\pi$
$941$		−	2.95757e20i		−	0.452697i	0.974046	−	0.226349i	$-0.0726789\pi$
$942$	7.13947e20	+	3.87022e20i	1.08470	+	0.588003i
$943$			3.60709e18i			0.00543970i
$944$		−	8.33125e20i		−	1.24711i
$945$		0			0
$946$	−3.65088e20			−0.538467
$947$	1.58610e20			0.232210			0.116105	−	0.993237i	$-0.462959\pi$
$947$	1.58610e20			0.232210			0.116105	+	0.993237i	$0.462959\pi$
$948$	1.15798e21	−	2.13615e21i	1.68284	−	3.10437i
$949$	3.59598e20			0.518745
$950$		0			0
$951$	−3.83625e20	+	7.07681e20i	−0.545310	+	1.00594i
$952$			1.31556e21i			1.85632i
$953$	8.31966e20			1.16535			0.582673	−	0.812707i	$-0.302007\pi$
$953$	8.31966e20			1.16535			0.582673	+	0.812707i	$0.302007\pi$
$954$	3.42765e20	+	5.26266e20i	0.476605	+	0.731756i
$955$		0			0
$956$			2.19932e21i			3.01359i
$957$	4.15098e20	−	7.65740e20i	0.564634	−	1.04159i
$958$			2.25318e21i			3.04254i
$959$		−	3.56149e20i		−	0.477421i
$960$		0			0
$961$	2.52277e20			0.333284
$962$	−3.33744e20			−0.437711
$963$	−3.36777e20	−	5.17071e20i	−0.438488	−	0.673234i
$964$	−2.13372e21			−2.75803
$965$		0			0
$966$	6.73618e18	+	3.65160e18i	0.00858171	+	0.00465204i
$967$			1.46184e21i			1.84890i	0.381299	+	0.924452i	$0.375477\pi$
$967$			1.46184e21i			1.84890i	−0.381299	+	0.924452i	$0.624523\pi$
$968$	4.95376e20			0.622025
$969$	−2.74573e20	+	5.06510e20i	−0.342287	+	0.631425i
$970$		0			0
$971$			3.08836e20i			0.379484i	0.981834	+	0.189742i	$0.0607652\pi$
$971$			3.08836e20i			0.379484i	−0.981834	+	0.189742i	$0.939235\pi$
$972$	−1.05689e21	−	1.36785e21i	−1.28934	−	1.66869i
$973$		−	6.20065e20i		−	0.751013i
$974$			1.58918e21i			1.91100i
$975$		0			0
$976$	−1.51943e21			−1.80108
$977$	2.08527e20			0.245414			0.122707	−	0.992443i	$-0.460842\pi$
$977$	2.08527e20			0.245414			0.122707	+	0.992443i	$0.460842\pi$
$978$	−1.10652e21	−	5.99832e20i	−1.29297	−	0.700902i
$979$	4.83233e20			0.560632
$980$		0			0
$981$	3.06957e20	+	4.71287e20i	0.351071	+	0.539017i
$982$		−	8.07615e20i		−	0.917115i
$983$	−1.09204e21			−1.23130			−0.615649	−	0.788020i	$-0.711106\pi$
$983$	−1.09204e21			−1.23130			−0.615649	+	0.788020i	$0.711106\pi$
$984$	−8.76204e20	−	4.74980e20i	−0.980934	−	0.531752i
$985$		0			0
$986$		−	3.74679e21i		−	4.13544i
$987$	−1.97160e20	−	1.06878e20i	−0.216072	−	0.117130i
$988$		−	4.57424e20i		−	0.497761i
$989$			3.26236e18i			0.00352499i
$990$		0			0
$991$	5.22556e20			0.556695			0.278347	−	0.960480i	$-0.410213\pi$
$991$	5.22556e20			0.556695			0.278347	+	0.960480i	$0.410213\pi$
$992$	−4.41448e20			−0.466980
$993$	−1.52559e20	+	2.81430e20i	−0.160249	+	0.295614i
$994$	−1.53594e21			−1.60202
$995$		0			0
$996$	−4.38906e20	+	8.09658e20i	−0.451394	+	0.832696i
$997$			8.85987e20i			0.904818i	0.891811	+	0.452409i	$0.149435\pi$
$997$			8.85987e20i			0.904818i	−0.891811	+	0.452409i	$0.850565\pi$
$998$	−2.88137e21			−2.92203
$999$	3.69241e19	+	4.57398e20i	0.0371836	+	0.460612i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.15.d.b.74.8		8
3.2	odd	2	inner	75.15.d.b.74.2		8
5.2	odd	4		3.15.b.a.2.4	yes	4
5.3	odd	4		75.15.c.d.26.1		4
5.4	even	2	inner	75.15.d.b.74.1		8
15.2	even	4		3.15.b.a.2.1	✓	4
15.8	even	4		75.15.c.d.26.4		4
15.14	odd	2	inner	75.15.d.b.74.7		8
20.7	even	4		48.15.e.b.17.1		4
60.47	odd	4		48.15.e.b.17.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.15.b.a.2.1	✓	4	15.2	even	4
3.15.b.a.2.4	yes	4	5.2	odd	4
48.15.e.b.17.1		4	20.7	even	4
48.15.e.b.17.2		4	60.47	odd	4
75.15.c.d.26.1		4	5.3	odd	4
75.15.c.d.26.4		4	15.8	even	4
75.15.d.b.74.1		8	5.4	even	2	inner
75.15.d.b.74.2		8	3.2	odd	2	inner
75.15.d.b.74.7		8	15.14	odd	2	inner
75.15.d.b.74.8		8	1.1	even	1	trivial

Overview	Random
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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+225.686 q^{2} +(-1042.26 + 1922.67i) q^{3} +34550.1 q^{4} +(-235223. + 433920. i) q^{6} -478389. i q^{7} +4.09983e6 q^{8} +(-2.61037e6 - 4.00784e6i) q^{9} +O(q^{10})\)	\(q+225.686 q^{2} +(-1042.26 + 1922.67i) q^{3} +34550.1 q^{4} +(-235223. + 433920. i) q^{6} -478389. i q^{7} +4.09983e6 q^{8} +(-2.61037e6 - 4.00784e6i) q^{9} +1.60910e7i q^{11} +(-3.60101e7 + 6.64285e7i) q^{12} +3.37094e7i q^{13} -1.07966e8i q^{14} +3.59205e8 q^{16} -6.70754e8 q^{17} +(-5.89124e8 - 9.04512e8i) q^{18} -3.92753e8 q^{19} +(9.19786e8 + 4.98604e8i) q^{21} +3.63152e9i q^{22} +3.24506e7 q^{23} +(-4.27308e9 + 7.88264e9i) q^{24} +7.60773e9i q^{26} +(1.04264e10 - 8.41689e8i) q^{27} -1.65284e10i q^{28} +2.47510e10i q^{29} -3.17682e10 q^{31} +1.38959e10 q^{32} +(-3.09378e10 - 1.67710e10i) q^{33} -1.51380e11 q^{34} +(-9.01885e10 - 1.38471e11i) q^{36} +4.38690e10i q^{37} -8.86387e10 q^{38} +(-6.48121e10 - 3.51338e10i) q^{39} +1.11156e11i q^{41} +(2.07583e11 + 1.12528e11i) q^{42} +1.00533e11i q^{43} +5.55947e11i q^{44} +7.32365e9 q^{46} -2.14354e11 q^{47} +(-3.74384e11 + 6.90634e11i) q^{48} +4.49367e11 q^{49} +(6.99098e11 - 1.28964e12i) q^{51} +1.16466e12i q^{52} -5.81823e11 q^{53} +(2.35310e12 - 1.89957e11i) q^{54} -1.96132e12i q^{56} +(4.09349e11 - 7.55135e11i) q^{57} +5.58594e12i q^{58} -2.31936e12i q^{59} -4.22999e12 q^{61} -7.16964e12 q^{62} +(-1.91731e12 + 1.24877e12i) q^{63} -2.74911e12 q^{64} +(-6.98222e12 - 3.78498e12i) q^{66} +2.32014e12i q^{67} -2.31746e13 q^{68} +(-3.38219e10 + 6.23919e10i) q^{69} -1.42262e13i q^{71} +(-1.07021e13 - 1.64315e13i) q^{72} -1.06676e13i q^{73} +9.90062e12i q^{74} -1.35696e13 q^{76} +7.69778e12 q^{77} +(-1.46272e13 - 7.92921e12i) q^{78} -3.21572e13 q^{79} +(-9.24873e12 + 2.09239e13i) q^{81} +2.50864e13i q^{82} +1.21884e13 q^{83} +(3.17787e13 + 1.72268e13i) q^{84} +2.26889e13i q^{86} +(-4.75880e13 - 2.57969e13i) q^{87} +6.59706e13i q^{88} -3.00312e13i q^{89} +1.61262e13 q^{91} +1.12117e12 q^{92} +(3.31107e13 - 6.10799e13i) q^{93} -4.83768e13 q^{94} +(-1.44831e13 + 2.67173e13i) q^{96} +5.28428e13i q^{97} +1.01416e14 q^{98} +(6.44903e13 - 4.20035e13i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 78592 q^{4} - 628992 q^{6} + 3249720 q^{9}+O(q^{10}) \) 8 * q + 78592 * q^4 - 628992 * q^6 + 3249720 * q^9	\( 8 q + 78592 q^{4} - 628992 q^{6} + 3249720 q^{9} + 2227994624 q^{16} - 2629894480 q^{19} + 3895487568 q^{21} - 26850410496 q^{24} - 69941420144 q^{31} - 557694498816 q^{34} - 564781849344 q^{36} - 37777373712 q^{39} - 546920285184 q^{46} + 4493154241128 q^{49} + 5312833763328 q^{51} + 10055727182592 q^{54} - 8343283252784 q^{61} - 19748163682304 q^{64} - 28743720445440 q^{66} - 30454662970368 q^{69} - 38498990571008 q^{76} - 82430885157520 q^{79} - 34777637555832 q^{81} + 123890760580608 q^{84} + 46891488783776 q^{91} - 101904474986496 q^{94} + 163635660324864 q^{96} + 172210751032320 q^{99}+O(q^{100}) \) 8 * q + 78592 * q^4 - 628992 * q^6 + 3249720 * q^9 + 2227994624 * q^16 - 2629894480 * q^19 + 3895487568 * q^21 - 26850410496 * q^24 - 69941420144 * q^31 - 557694498816 * q^34 - 564781849344 * q^36 - 37777373712 * q^39 - 546920285184 * q^46 + 4493154241128 * q^49 + 5312833763328 * q^51 + 10055727182592 * q^54 - 8343283252784 * q^61 - 19748163682304 * q^64 - 28743720445440 * q^66 - 30454662970368 * q^69 - 38498990571008 * q^76 - 82430885157520 * q^79 - 34777637555832 * q^81 + 123890760580608 * q^84 + 46891488783776 * q^91 - 101904474986496 * q^94 + 163635660324864 * q^96 + 172210751032320 * q^99

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