LMFDB - Embedded newform 38.9.b.a.37.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,9,Mod(37,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("38.37");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	38.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:	$15.4803871823$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 $ x^12 + 46118x^10 + 738386961x^8 + 5214446299656x^6 + 17370647163698184x^4 + 24830681474333400768*x^2 + 9218779084612644462864
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{21} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.12
Root			$145.414i$ of defining polynomial
Character	$\chi$	$=$	38.37
Dual form			38.9.b.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3137i q^{2} +132.686i q^{3} -128.000 q^{4} -12.7536 q^{5} -1501.17 q^{6} -1217.09 q^{7} -1448.15i q^{8} -11044.5 q^{9} +O(q^{10})$	\(q+11.3137i q^{2} +132.686i q^{3} -128.000 q^{4} -12.7536 q^{5} -1501.17 q^{6} -1217.09 q^{7} -1448.15i q^{8} -11044.5 q^{9} -144.290i q^{10} -8287.40 q^{11} -16983.8i q^{12} +7541.52i q^{13} -13769.8i q^{14} -1692.22i q^{15} +16384.0 q^{16} +38006.8 q^{17} -124955. i q^{18} +(130262. - 3934.45i) q^{19} +1632.46 q^{20} -161491. i q^{21} -93761.2i q^{22} -128007. q^{23} +192150. q^{24} -390462. q^{25} -85322.6 q^{26} -594903. i q^{27} +155787. q^{28} -166361. i q^{29} +19145.3 q^{30} -1.19799e6i q^{31} +185364. i q^{32} -1.09962e6i q^{33} +429998. i q^{34} +15522.3 q^{35} +1.41370e6 q^{36} +1.58309e6i q^{37} +(44513.2 + 1.47374e6i) q^{38} -1.00065e6 q^{39} +18469.2i q^{40} +2.10564e6i q^{41} +1.82706e6 q^{42} -1.97470e6 q^{43} +1.06079e6 q^{44} +140858. q^{45} -1.44824e6i q^{46} -6.30119e6 q^{47} +2.17393e6i q^{48} -4.28349e6 q^{49} -4.41758e6i q^{50} +5.04296e6i q^{51} -965315. i q^{52} +1.41058e7i q^{53} +6.73056e6 q^{54} +105694. q^{55} +1.76253e6i q^{56} +(522045. + 1.72839e7i) q^{57} +1.88216e6 q^{58} -1.66817e7i q^{59} +216604. i q^{60} -517902. q^{61} +1.35537e7 q^{62} +1.34422e7 q^{63} -2.09715e6 q^{64} -96181.5i q^{65} +1.24408e7 q^{66} +3.30572e6i q^{67} -4.86487e6 q^{68} -1.69848e7i q^{69} +175614. i q^{70} -1.89129e7i q^{71} +1.59942e7i q^{72} -4.16435e7 q^{73} -1.79107e7 q^{74} -5.18088e7i q^{75} +(-1.66735e7 + 503609. i) q^{76} +1.00865e7 q^{77} -1.13211e7i q^{78} -3.09516e7i q^{79} -208955. q^{80} +6.47201e6 q^{81} -2.38226e7 q^{82} -4.97694e7 q^{83} +2.06708e7i q^{84} -484723. q^{85} -2.23411e7i q^{86} +2.20738e7 q^{87} +1.20014e7i q^{88} +3.06123e7i q^{89} +1.59362e6i q^{90} -9.17871e6i q^{91} +1.63849e7 q^{92} +1.58956e8 q^{93} -7.12899e7i q^{94} +(-1.66130e6 + 50178.3i) q^{95} -2.45952e7 q^{96} +1.74140e7i q^{97} -4.84622e7i q^{98} +9.15305e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) $ 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9	$ 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) $ 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9 - 12546 * q^11 + 196608 * q^16 + 270810 * q^17 + 41512 * q^19 - 71424 * q^20 - 823956 * q^23 - 229376 * q^24 + 865538 * q^25 - 431616 * q^26 + 694016 * q^28 + 71168 * q^30 - 1194378 * q^35 + 1995776 * q^36 + 998784 * q^38 + 5786100 * q^39 - 8383744 * q^42 + 7586646 * q^43 + 1605888 * q^44 + 2226046 * q^45 - 20260530 * q^47 - 19498842 * q^49 + 16933888 * q^54 - 14858554 * q^55 + 14430564 * q^57 - 5506560 * q^58 - 41363266 * q^61 + 32266752 * q^62 + 84235798 * q^63 - 25165824 * q^64 + 14371328 * q^66 - 34663680 * q^68 + 87906498 * q^73 - 2149632 * q^74 - 5313536 * q^76 - 78817962 * q^77 + 9142272 * q^80 - 100904812 * q^81 - 49609728 * q^82 - 55944960 * q^83 + 25440254 * q^85 + 119189604 * q^87 + 105466368 * q^92 + 105500856 * q^93 + 81396774 * q^95 + 29360128 * q^96 - 85554938 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$.

$n$	$21$
$\chi(n)$	$-1$

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 $

$2$			11.3137i			0.707107i
$2$			11.3137i			0.707107i
$3$			132.686i			1.63810i	0.573724	+	0.819049i	$0.305498\pi$
$3$			132.686i			1.63810i	−0.573724	+	0.819049i	$0.694502\pi$
$4$	−128.000			−0.500000
$5$	−12.7536			−0.0204057			−0.0102029	−	0.999948i	$-0.503248\pi$
$5$	−12.7536			−0.0204057			−0.0102029	+	0.999948i	$0.503248\pi$
$6$	−1501.17			−1.15831
$7$	−1217.09			−0.506909			−0.253455	−	0.967347i	$-0.581567\pi$
$7$	−1217.09			−0.506909			−0.253455	+	0.967347i	$0.581567\pi$
$8$		−	1448.15i		−	0.353553i
$9$	−11044.5			−1.68336
$10$		−	144.290i		−	0.0144290i
$11$	−8287.40			−0.566040			−0.283020	−	0.959114i	$-0.591336\pi$
$11$	−8287.40			−0.566040			−0.283020	+	0.959114i	$0.591336\pi$
$12$		−	16983.8i		−	0.819049i
$13$			7541.52i			0.264050i	0.991246	+	0.132025i	$0.0421479\pi$
$13$			7541.52i			0.264050i	−0.991246	+	0.132025i	$0.957852\pi$
$14$		−	13769.8i		−	0.358439i
$15$		−	1692.22i		−	0.0334266i
$16$	16384.0			0.250000
$17$	38006.8			0.455056			0.227528	−	0.973771i	$-0.426936\pi$
$17$	38006.8			0.455056			0.227528	+	0.973771i	$0.426936\pi$
$18$		−	124955.i		−	1.19032i
$19$	130262.	−	3934.45i	0.999544	−	0.0301904i
$19$	130262.	−	3934.45i	0.999544	−	0.0301904i
$20$	1632.46			0.0102029
$21$		−	161491.i		−	0.830367i
$22$		−	93761.2i		−	0.400251i
$23$	−128007.			−0.457429			−0.228714	−	0.973494i	$-0.573452\pi$
$23$	−128007.			−0.457429			−0.228714	+	0.973494i	$0.573452\pi$
$24$	192150.			0.579155
$25$	−390462.			−0.999584
$26$	−85322.6			−0.186711
$27$		−	594903.i		−	1.11942i
$28$	155787.			0.253455
$29$		−	166361.i		−	0.235212i	−0.993060	−	0.117606i	$-0.962478\pi$
$29$		−	166361.i		−	0.235212i	0.993060	−	0.117606i	$-0.0375221\pi$
$30$	19145.3			0.0236362
$31$		−	1.19799e6i		−	1.29720i	−0.761130	−	0.648599i	$-0.775355\pi$
$31$		−	1.19799e6i		−	1.29720i	0.761130	−	0.648599i	$-0.224645\pi$
$32$			185364.i			0.176777i
$33$		−	1.09962e6i		−	0.927229i
$34$			429998.i			0.321773i
$35$	15522.3			0.0103439
$36$	1.41370e6			0.841682
$37$			1.58309e6i			0.844694i	0.906434	+	0.422347i	$0.138794\pi$
$37$			1.58309e6i			0.844694i	−0.906434	+	0.422347i	$0.861206\pi$
$38$	44513.2	+	1.47374e6i	0.0213478	+	0.706784i
$39$	−1.00065e6			−0.432539
$40$			18469.2i			0.00721452i
$41$			2.10564e6i			0.745159i	0.928000	+	0.372580i	$0.121527\pi$
$41$			2.10564e6i			0.745159i	−0.928000	+	0.372580i	$0.878473\pi$
$42$	1.82706e6			0.587158
$43$	−1.97470e6			−0.577599			−0.288800	−	0.957390i	$-0.593256\pi$
$43$	−1.97470e6			−0.577599			−0.288800	+	0.957390i	$0.593256\pi$
$44$	1.06079e6			0.283020
$45$	140858.			0.0343503
$46$		−	1.44824e6i		−	0.323451i
$47$	−6.30119e6			−1.29131			−0.645656	−	0.763628i	$-0.723416\pi$
$47$	−6.30119e6			−1.29131			−0.645656	+	0.763628i	$0.723416\pi$
$48$			2.17393e6i			0.409524i
$49$	−4.28349e6			−0.743043
$50$		−	4.41758e6i		−	0.706812i
$51$			5.04296e6i			0.745427i
$52$		−	965315.i		−	0.132025i
$53$			1.41058e7i			1.78770i	0.448369	+	0.893848i	$0.352005\pi$
$53$			1.41058e7i			1.78770i	−0.448369	+	0.893848i	$0.647995\pi$
$54$	6.73056e6			0.791546
$55$	105694.			0.0115505
$56$			1.76253e6i			0.179220i
$57$	522045.	+	1.72839e7i	0.0494548	+	1.63735i
$58$	1.88216e6			0.166320
$59$		−	1.66817e7i		−	1.37668i	−0.725390	−	0.688338i	$-0.758341\pi$
$59$		−	1.66817e7i		−	1.37668i	0.725390	−	0.688338i	$-0.241659\pi$
$60$			216604.i			0.0167133i
$61$	−517902.			−0.0374049			−0.0187025	−	0.999825i	$-0.505954\pi$
$61$	−517902.			−0.0374049			−0.0187025	+	0.999825i	$0.505954\pi$
$62$	1.35537e7			0.917257
$63$	1.34422e7			0.853313
$64$	−2.09715e6			−0.125000
$65$		−	96181.5i		−	0.00538813i
$66$	1.24408e7			0.655650
$67$			3.30572e6i			0.164046i	0.996630	+	0.0820231i	$0.0261381\pi$
$67$			3.30572e6i			0.164046i	−0.996630	+	0.0820231i	$0.973862\pi$
$68$	−4.86487e6			−0.227528
$69$		−	1.69848e7i		−	0.749313i
$70$			175614.i			0.00731421i
$71$		−	1.89129e7i		−	0.744259i	−0.928181	−	0.372129i	$-0.878628\pi$
$71$		−	1.89129e7i		−	0.744259i	0.928181	−	0.372129i	$-0.121372\pi$
$72$			1.59942e7i			0.595159i
$73$	−4.16435e7			−1.46641			−0.733205	−	0.680007i	$-0.761977\pi$
$73$	−4.16435e7			−1.46641			−0.733205	+	0.680007i	$0.761977\pi$
$74$	−1.79107e7			−0.597289
$75$		−	5.18088e7i		−	1.63742i
$76$	−1.66735e7	+	503609.i	−0.499772	+	0.0150952i
$77$	1.00865e7			0.286931
$78$		−	1.13211e7i		−	0.305851i
$79$		−	3.09516e7i		−	0.794649i	−0.917678	−	0.397325i	$-0.869939\pi$
$79$		−	3.09516e7i		−	0.794649i	0.917678	−	0.397325i	$-0.130061\pi$
$80$	−208955.			−0.00510144
$81$	6.47201e6			0.150348
$82$	−2.38226e7			−0.526907
$83$	−4.97694e7			−1.04870			−0.524349	−	0.851504i	$-0.675691\pi$
$83$	−4.97694e7			−1.04870			−0.524349	+	0.851504i	$0.675691\pi$
$84$			2.06708e7i			0.415183i
$85$	−484723.			−0.00928576
$86$		−	2.23411e7i		−	0.408424i
$87$	2.20738e7			0.385301
$88$			1.20014e7i			0.200125i
$89$			3.06123e7i			0.487906i	0.969787	+	0.243953i	$0.0784443\pi$
$89$			3.06123e7i			0.487906i	−0.969787	+	0.243953i	$0.921556\pi$
$90$			1.59362e6i			0.0242893i
$91$		−	9.17871e6i		−	0.133849i
$92$	1.63849e7			0.228714
$93$	1.58956e8			2.12494
$94$		−	7.12899e7i		−	0.913096i
$95$	−1.66130e6	+	50178.3i	−0.0203964	+	0.000616058i
$96$	−2.45952e7			−0.289577
$97$			1.74140e7i			0.196704i	0.995152	+	0.0983518i	$0.0313570\pi$
$97$			1.74140e7i			0.196704i	−0.995152	+	0.0983518i	$0.968643\pi$
$98$		−	4.84622e7i		−	0.525411i
$99$	9.15305e7			0.952851
$100$	4.99792e7			0.499792
$101$	1.44641e8			1.38997			0.694986	−	0.719023i	$-0.255410\pi$
$101$	1.44641e8			1.38997			0.694986	+	0.719023i	$0.255410\pi$
$102$	−5.70546e7			−0.527096
$103$			1.21419e8i			1.07879i	0.842053	+	0.539394i	$0.181347\pi$
$103$			1.21419e8i			1.07879i	−0.842053	+	0.539394i	$0.818653\pi$
$104$	1.09213e7			0.0933557
$105$			2.05958e6i			0.0169443i
$106$	−1.59589e8			−1.26409
$107$			1.59301e8i			1.21530i	0.794205	+	0.607650i	$0.207888\pi$
$107$			1.59301e8i			1.21530i	−0.794205	+	0.607650i	$0.792112\pi$
$108$			7.61476e7i			0.559708i
$109$			2.04141e8i			1.44619i	0.690750	+	0.723094i	$0.257281\pi$
$109$			2.04141e8i			1.44619i	−0.690750	+	0.723094i	$0.742719\pi$
$110$			1.19579e6i			0.00816742i
$111$	−2.10054e8			−1.38369
$112$	−1.99408e7			−0.126727
$113$		−	1.38877e8i		−	0.851761i	−0.904779	−	0.425881i	$-0.859964\pi$
$113$		−	1.38877e8i		−	0.851761i	0.904779	−	0.425881i	$-0.140036\pi$
$114$	−1.95545e8	+	5.90627e6i	−1.15778	+	0.0349699i
$115$	1.63255e6			0.00933418
$116$			2.12942e7i			0.117606i
$117$		−	8.32927e7i		−	0.444492i
$118$	1.88732e8			0.973456
$119$	−4.62576e7			−0.230672
$120$	−2.45060e6			−0.0118181
$121$	−1.45678e8			−0.679598
$122$		−	5.85940e6i		−	0.0264493i
$123$	−2.79389e8			−1.22064
$124$			1.53343e8i			0.648599i
$125$	9.96167e6			0.0408030
$126$			1.52081e8i			0.603383i
$127$			2.03065e8i			0.780586i	0.920691	+	0.390293i	$0.127626\pi$
$127$			2.03065e8i			0.780586i	−0.920691	+	0.390293i	$0.872374\pi$
$128$		−	2.37266e7i		−	0.0883883i
$129$		−	2.62014e8i		−	0.946164i
$130$	1.08817e6			0.00380998
$131$	4.95298e8			1.68183			0.840914	−	0.541169i	$-0.182018\pi$
$131$	4.95298e8			1.68183			0.840914	+	0.541169i	$0.182018\pi$
$132$			1.40751e8i			0.463615i
$133$	−1.58540e8	+	4.78857e6i	−0.506678	+	0.0153038i
$134$	−3.73999e7			−0.115998
$135$			7.58715e6i			0.0228425i
$136$		−	5.50397e7i		−	0.160887i
$137$	6.69842e8			1.90147			0.950736	−	0.310002i	$-0.100330\pi$
$137$	6.69842e8			1.90147			0.950736	+	0.310002i	$0.100330\pi$
$138$	1.92161e8			0.529844
$139$	1.00947e8			0.270416			0.135208	−	0.990817i	$-0.456830\pi$
$139$	1.00947e8			0.270416			0.135208	+	0.990817i	$0.456830\pi$
$140$	−1.98685e6			−0.00517193
$141$		−	8.36080e8i		−	2.11530i
$142$	2.13975e8			0.526270
$143$		−	6.24996e7i		−	0.149463i
$144$	−1.80954e8			−0.420841
$145$			2.12170e6i			0.00479969i
$146$		−	4.71142e8i		−	1.03691i
$147$		−	5.68359e8i		−	1.21718i
$148$		−	2.02636e8i		−	0.422347i
$149$	−6.35720e8			−1.28979			−0.644897	−	0.764269i	$-0.723100\pi$
$149$	−6.35720e8			−1.28979			−0.644897	+	0.764269i	$0.723100\pi$
$150$	5.86150e8			1.15783
$151$			8.90416e8i			1.71271i	0.516384	+	0.856357i	$0.327278\pi$
$151$			8.90416e8i			1.71271i	−0.516384	+	0.856357i	$0.672722\pi$
$152$	−5.69769e6	−	1.88639e8i	−0.0106739	−	0.353392i
$153$	−4.19768e8			−0.766025
$154$			1.14116e8i			0.202891i
$155$			1.52787e7i			0.0264703i
$156$	1.28084e8			0.216270
$157$	6.88287e8			1.13285			0.566423	−	0.824115i	$-0.308327\pi$
$157$	6.88287e8			1.13285			0.566423	+	0.824115i	$0.308327\pi$
$158$	3.50178e8			0.561902
$159$	−1.87164e9			−2.92842
$160$		−	2.36405e6i		−	0.00360726i
$161$	1.55796e8			0.231875
$162$			7.32224e7i			0.106312i
$163$	−8.41015e8			−1.19139			−0.595694	−	0.803211i	$-0.703123\pi$
$163$	−8.41015e8			−1.19139			−0.595694	+	0.803211i	$0.703123\pi$
$164$		−	2.69522e8i		−	0.372580i
$165$			1.40241e7i			0.0189208i
$166$		−	5.63077e8i		−	0.741541i
$167$			1.63087e8i			0.209678i	0.994489	+	0.104839i	$0.0334327\pi$
$167$			1.63087e8i			0.209678i	−0.994489	+	0.104839i	$0.966567\pi$
$168$	−2.33863e8			−0.293579
$169$	7.58856e8			0.930278
$170$		−	5.48401e6i		−	0.00656603i
$171$	−1.43868e9	+	4.34542e7i	−1.68260	+	0.0508214i
$172$	2.52761e8			0.288800
$173$			2.78758e8i			0.311203i	0.987820	+	0.155601i	$0.0497315\pi$
$173$			2.78758e8i			0.311203i	−0.987820	+	0.155601i	$0.950268\pi$
$174$			2.49737e8i			0.272449i
$175$	4.75228e8			0.506698
$176$	−1.35781e8			−0.141510
$177$	2.21342e9			2.25513
$178$	−3.46339e8			−0.345002
$179$		−	1.02487e9i		−	0.998287i	−0.866519	−	0.499144i	$-0.833648\pi$
$179$		−	1.02487e9i		−	0.998287i	0.866519	−	0.499144i	$-0.166352\pi$
$180$	−1.80298e7			−0.0171751
$181$			6.16982e8i			0.574854i	0.957802	+	0.287427i	$0.0927999\pi$
$181$			6.16982e8i			0.574854i	−0.957802	+	0.287427i	$0.907200\pi$
$182$	1.03845e8			0.0946457
$183$		−	6.87184e7i		−	0.0612729i
$184$			1.85374e8i			0.161726i
$185$		−	2.01901e7i		−	0.0172366i
$186$			1.79838e9i			1.50256i
$187$	−3.14977e8			−0.257580
$188$	8.06553e8			0.645656
$189$			7.24050e8i			0.567442i
$190$	−567703.	−	1.87955e7i	−0.000435619	−	0.0144225i
$191$	−2.10144e8			−0.157901			−0.0789503	−	0.996879i	$-0.525157\pi$
$191$	−2.10144e8			−0.157901			−0.0789503	+	0.996879i	$0.525157\pi$
$192$		−	2.78262e8i		−	0.204762i
$193$			1.45217e9i			1.04662i	0.852142	+	0.523310i	$0.175303\pi$
$193$			1.45217e9i			1.04662i	−0.852142	+	0.523310i	$0.824697\pi$
$194$	−1.97017e8			−0.139090
$195$	1.27619e7			0.00882628
$196$	5.48287e8			0.371521
$197$	−3.35462e8			−0.222730			−0.111365	−	0.993780i	$-0.535522\pi$
$197$	−3.35462e8			−0.222730			−0.111365	+	0.993780i	$0.535522\pi$
$198$			1.03555e9i			0.673768i
$199$	−1.16499e8			−0.0742867			−0.0371434	−	0.999310i	$-0.511826\pi$
$199$	−1.16499e8			−0.0742867			−0.0371434	+	0.999310i	$0.511826\pi$
$200$			5.65450e8i			0.353406i
$201$	−4.38622e8			−0.268724
$202$			1.63643e9i			0.982859i
$203$			2.02477e8i			0.119231i
$204$		−	6.45499e8i		−	0.372713i
$205$		−	2.68545e7i		−	0.0152055i
$206$	−1.37369e9			−0.762819
$207$	1.41378e9			0.770019
$208$			1.23560e8i			0.0660124i
$209$	−1.07953e9	+	3.26063e7i	−0.565782	+	0.0170890i
$210$	−2.33015e7			−0.0119814
$211$		−	2.63636e8i		−	0.133007i	−0.997786	−	0.0665035i	$-0.978816\pi$
$211$		−	2.63636e8i		−	0.133007i	0.997786	−	0.0665035i	$-0.0211844\pi$
$212$		−	1.80554e9i		−	0.893848i
$213$	2.50947e9			1.21917
$214$	−1.80229e9			−0.859347
$215$	2.51845e7			0.0117863
$216$	−8.61512e8			−0.395773
$217$			1.45806e9i			0.657562i
$218$	−2.30959e9			−1.02261
$219$		−	5.52550e9i		−	2.40212i
$220$	−1.35288e7			−0.00577524
$221$			2.86629e8i			0.120158i
$222$		−	2.37649e9i		−	0.978418i
$223$			3.71646e9i			1.50283i	0.659828	+	0.751416i	$0.270629\pi$
$223$			3.71646e9i			1.50283i	−0.659828	+	0.751416i	$0.729371\pi$
$224$		−	2.25604e8i		−	0.0896098i
$225$	4.31248e9			1.68266
$226$	1.57122e9			0.602286
$227$		−	2.97089e9i		−	1.11888i	−0.828872	−	0.559439i	$-0.811017\pi$
$227$		−	2.97089e9i		−	1.11888i	0.828872	−	0.559439i	$-0.188983\pi$
$228$	−6.68218e7	−	2.21234e9i	−0.0247274	−	0.818675i
$229$	−2.40867e9			−0.875860			−0.437930	−	0.899009i	$-0.644288\pi$
$229$	−2.40867e9			−0.875860			−0.437930	+	0.899009i	$0.644288\pi$
$230$			1.84702e7i			0.00660026i
$231$			1.33834e9i			0.470021i
$232$	−2.40917e8			−0.0831602
$233$	6.31227e8			0.214172			0.107086	−	0.994250i	$-0.465848\pi$
$233$	6.31227e8			0.214172			0.107086	+	0.994250i	$0.465848\pi$
$234$	9.42349e8			0.314303
$235$	8.03628e7			0.0263502
$236$			2.13525e9i			0.688338i
$237$	4.10685e9			1.30171
$238$		−	5.23345e8i		−	0.163110i
$239$	−2.12116e9			−0.650102			−0.325051	−	0.945696i	$-0.605381\pi$
$239$	−2.12116e9			−0.650102			−0.325051	+	0.945696i	$0.605381\pi$
$240$		−	2.77254e7i		−	0.00835665i
$241$			5.36737e9i			1.59108i	0.605899	+	0.795542i	$0.292814\pi$
$241$			5.36737e9i			1.59108i	−0.605899	+	0.795542i	$0.707186\pi$
$242$		−	1.64816e9i		−	0.480549i
$243$		−	3.04442e9i		−	0.873130i
$244$	6.62915e7			0.0187025
$245$	5.46299e7			0.0151623
$246$		−	3.16093e9i		−	0.863125i
$247$	2.96717e7	+	9.82371e8i	0.00797177	+	0.263929i
$248$	−1.73487e9			−0.458629
$249$		−	6.60370e9i		−	1.71787i
$250$			1.12703e8i			0.0288521i
$251$	−5.20809e9			−1.31215			−0.656075	−	0.754695i	$-0.727785\pi$
$251$	−5.20809e9			−1.31215			−0.656075	+	0.754695i	$0.727785\pi$
$252$	−1.72060e9			−0.426656
$253$	1.06085e9			0.258923
$254$	−2.29742e9			−0.551957
$255$		−	6.43159e7i		−	0.0152110i
$256$	2.68435e8			0.0625000
$257$		−	6.29253e9i		−	1.44242i	−0.692715	−	0.721211i	$-0.743586\pi$
$257$		−	6.29253e9i		−	1.44242i	0.692715	−	0.721211i	$-0.256414\pi$
$258$	2.96435e9			0.669039
$259$		−	1.92677e9i		−	0.428183i
$260$			1.23112e7i			0.00269406i
$261$			1.83739e9i			0.395948i
$262$			5.60366e9i			1.18923i
$263$	3.69162e9			0.771603			0.385802	−	0.922582i	$-0.373925\pi$
$263$	3.69162e9			0.771603			0.385802	+	0.922582i	$0.373925\pi$
$264$	−1.59242e9			−0.327825
$265$		−	1.79899e8i		−	0.0364793i
$266$	−5.41765e7	−	1.79368e9i	−0.0108214	−	0.358276i
$267$	−4.06182e9			−0.799238
$268$		−	4.23132e8i		−	0.0820231i
$269$			6.61298e8i			0.126296i	0.998004	+	0.0631478i	$0.0201140\pi$
$269$			6.61298e8i			0.126296i	−0.998004	+	0.0631478i	$0.979886\pi$
$270$	−8.58388e7			−0.0161521
$271$	−8.93376e9			−1.65637			−0.828184	−	0.560456i	$-0.810626\pi$
$271$	−8.93376e9			−1.65637			−0.828184	+	0.560456i	$0.810626\pi$
$272$	6.22703e8			0.113764
$273$	1.21789e9			0.219258
$274$			7.57839e9i			1.34454i
$275$	3.23592e9			0.565805
$276$			2.17405e9i			0.374657i
$277$	7.50670e9			1.27506			0.637529	−	0.770427i	$-0.279957\pi$
$277$	7.50670e9			1.27506			0.637529	+	0.770427i	$0.279957\pi$
$278$			1.14208e9i			0.191213i
$279$			1.32312e10i			2.18365i
$280$		−	2.24786e7i		−	0.00365711i
$281$			1.58420e9i			0.254088i	0.991897	+	0.127044i	$0.0405489\pi$
$281$			1.58420e9i			0.254088i	−0.991897	+	0.127044i	$0.959451\pi$
$282$	9.45916e9			1.49574
$283$	1.89421e9			0.295313			0.147656	−	0.989039i	$-0.452827\pi$
$283$	1.89421e9			0.295313			0.147656	+	0.989039i	$0.452827\pi$
$284$			2.42085e9i			0.372129i
$285$	−6.65795e6	−	2.20431e8i	−0.00100916	−	0.0334114i
$286$	7.07102e8			0.105686
$287$		−	2.56275e9i		−	0.377728i
$288$		−	2.04726e9i		−	0.297579i
$289$	−5.53124e9			−0.792924
$290$	−2.40043e7			−0.00339389
$291$	−2.31060e9			−0.322220
$292$	5.33037e9			0.733205
$293$		−	1.07432e10i		−	1.45768i	−0.684684	−	0.728840i	$-0.740060\pi$
$293$		−	1.07432e10i		−	1.45768i	0.684684	−	0.728840i	$-0.259940\pi$
$294$	6.43025e9			0.860674
$295$			2.12751e8i			0.0280921i
$296$	2.29256e9			0.298644
$297$			4.93020e9i			0.633634i
$298$		−	7.19235e9i		−	0.912023i
$299$		−	9.65371e8i		−	0.120784i
$300$			6.63153e9i			0.818708i
$301$	2.40338e9			0.292790
$302$	−1.00739e10			−1.21107
$303$			1.91918e10i			2.27691i
$304$	2.13421e9	−	6.44620e7i	0.249886	−	0.00754760i
$305$	6.60512e6			0.000763275
$306$		−	4.74913e9i		−	0.541662i
$307$			7.21305e9i			0.812018i	0.913869	+	0.406009i	$0.133080\pi$
$307$			7.21305e9i			0.812018i	−0.913869	+	0.406009i	$0.866920\pi$
$308$	−1.29107e9			−0.143466
$309$	−1.61105e10			−1.76716
$310$	−1.72858e8			−0.0187173
$311$	1.32885e10			1.42048			0.710241	−	0.703959i	$-0.248586\pi$
$311$	1.32885e10			1.42048			0.710241	+	0.703959i	$0.248586\pi$
$312$			1.44910e9i			0.152926i
$313$	−8.92884e9			−0.930288			−0.465144	−	0.885235i	$-0.653997\pi$
$313$	−8.92884e9			−0.930288			−0.465144	+	0.885235i	$0.653997\pi$
$314$			7.78707e9i			0.801043i
$315$	−1.71436e8			−0.0174125
$316$			3.96181e9i			0.397325i
$317$			7.39046e9i			0.731871i	0.930640	+	0.365935i	$0.119251\pi$
$317$			7.39046e9i			0.731871i	−0.930640	+	0.365935i	$0.880749\pi$
$318$		−	2.11752e10i		−	2.07071i
$319$			1.37870e9i			0.133140i
$320$	2.67462e7			0.00255072
$321$	−2.11370e10			−1.99078
$322$			1.76264e9i			0.163960i
$323$	4.95082e9	−	1.49536e8i	0.454849	−	0.0137383i
$324$	−8.28417e8			−0.0751742
$325$		−	2.94468e9i		−	0.263940i
$326$		−	9.51500e9i		−	0.842439i
$327$	−2.70867e10			−2.36900
$328$	3.04930e9			0.263454
$329$	7.66912e9			0.654578
$330$	−1.58665e8			−0.0133790
$331$		−	1.83234e10i		−	1.52649i	−0.646110	−	0.763244i	$-0.723605\pi$
$331$		−	1.83234e10i		−	1.52649i	0.646110	−	0.763244i	$-0.276395\pi$
$332$	6.37049e9			0.524349
$333$		−	1.74845e10i		−	1.42193i
$334$	−1.84512e9			−0.148265
$335$		−	4.21597e7i		−	0.00334749i
$336$		−	2.64586e9i		−	0.207592i
$337$		−	1.08774e10i		−	0.843344i	−0.906749	−	0.421672i	$-0.861443\pi$
$337$		−	1.08774e10i		−	0.843344i	0.906749	−	0.421672i	$-0.138557\pi$
$338$			8.58548e9i			0.657806i
$339$	1.84271e10			1.39527
$340$	6.20445e7			0.00464288
$341$			9.92821e9i			0.734266i
$342$	−4.91628e8	−	1.62768e10i	−0.0359362	−	1.18977i
$343$	1.22297e10			0.883565
$344$			2.85967e9i			0.204212i
$345$			2.16617e8i			0.0152903i
$346$	−3.15379e9			−0.220054
$347$	−2.72593e9			−0.188017			−0.0940084	−	0.995571i	$-0.529968\pi$
$347$	−2.72593e9			−0.188017			−0.0940084	+	0.995571i	$0.529968\pi$
$348$	−2.82545e9			−0.192650
$349$	−8.99555e8			−0.0606354			−0.0303177	−	0.999540i	$-0.509652\pi$
$349$	−8.99555e8			−0.0606354			−0.0303177	+	0.999540i	$0.509652\pi$
$350$			5.37659e9i			0.358290i
$351$	4.48648e9			0.295581
$352$		−	1.53618e9i		−	0.100063i
$353$	2.46411e10			1.58694			0.793472	−	0.608607i	$-0.208271\pi$
$353$	2.46411e10			1.58694			0.793472	+	0.608607i	$0.208271\pi$
$354$			2.50420e10i			1.59462i
$355$			2.41207e8i			0.0151872i
$356$		−	3.91838e9i		−	0.243953i
$357$		−	6.13774e9i		−	0.377864i
$358$	1.15951e10			0.705896
$359$	2.82706e10			1.70199			0.850994	−	0.525175i	$-0.176000\pi$
$359$	2.82706e10			1.70199			0.850994	+	0.525175i	$0.176000\pi$
$360$		−	2.03984e8i		−	0.0121447i
$361$	1.69526e10	−	1.02501e9i	0.998177	−	0.0603533i
$362$	−6.98035e9			−0.406483
$363$		−	1.93294e10i		−	1.11325i
$364$			1.17487e9i			0.0669246i
$365$	5.31104e8			0.0299232
$366$	7.77459e8			0.0433265
$367$	−1.92447e10			−1.06083			−0.530417	−	0.847737i	$-0.677965\pi$
$367$	−1.92447e10			−1.06083			−0.530417	+	0.847737i	$0.677965\pi$
$368$	−2.09727e9			−0.114357
$369$		−	2.32559e10i		−	1.25437i
$370$	2.28425e8			0.0121881
$371$		−	1.71680e10i		−	0.906200i
$372$	−2.03464e10			−1.06247
$373$			2.45634e10i			1.26897i	0.772934	+	0.634487i	$0.218788\pi$
$373$			2.45634e10i			1.26897i	−0.772934	+	0.634487i	$0.781212\pi$
$374$		−	3.56356e9i		−	0.182137i
$375$			1.32177e9i			0.0668393i
$376$			9.12510e9i			0.456548i
$377$	1.25462e9			0.0621078
$378$	−8.19170e9			−0.401242
$379$		−	1.48216e10i		−	0.718354i	−0.933269	−	0.359177i	$-0.883057\pi$
$379$		−	1.48216e10i		−	0.718354i	0.933269	−	0.359177i	$-0.116943\pi$
$380$	2.12647e8	−	6.42282e6i	0.0101982	−	0.000308029i
$381$	−2.69439e10			−1.27868
$382$		−	2.37751e9i		−	0.111653i
$383$			6.09696e9i			0.283347i	0.989913	+	0.141673i	$0.0452483\pi$
$383$			6.09696e9i			0.283347i	−0.989913	+	0.141673i	$0.954752\pi$
$384$	3.14818e9			0.144789
$385$	−1.28639e8			−0.00585504
$386$	−1.64295e10			−0.740072
$387$	2.18096e10			0.972309
$388$		−	2.22900e9i		−	0.0983518i
$389$	−1.36807e10			−0.597459			−0.298730	−	0.954338i	$-0.596563\pi$
$389$	−1.36807e10			−0.597459			−0.298730	+	0.954338i	$0.596563\pi$
$390$			1.44385e8i			0.00624112i
$391$	−4.86515e9			−0.208156
$392$			6.20316e9i			0.262705i
$393$			6.57191e10i			2.75500i
$394$		−	3.79532e9i		−	0.157494i
$395$			3.94745e8i			0.0162154i
$396$	−1.17159e10			−0.476426
$397$	−3.87817e10			−1.56122			−0.780611	−	0.625018i	$-0.785092\pi$
$397$	−3.87817e10			−1.56122			−0.780611	+	0.625018i	$0.785092\pi$
$398$		−	1.31804e9i		−	0.0525286i
$399$	−6.35376e8	−	2.10360e10i	−0.0250691	−	0.829988i
$400$	−6.39734e9			−0.249896
$401$			3.02540e10i			1.17005i	0.811015	+	0.585026i	$0.198916\pi$
$401$			3.02540e10i			1.17005i	−0.811015	+	0.585026i	$0.801084\pi$
$402$		−	4.96244e9i		−	0.190016i
$403$	9.03466e9			0.342525
$404$	−1.85141e10			−0.694986
$405$	−8.25413e7			−0.00306797
$406$	−2.29076e9			−0.0843093
$407$		−	1.31197e10i		−	0.478131i
$408$	7.30299e9			0.263548
$409$		−	3.38079e10i		−	1.20816i	−0.796923	−	0.604081i	$-0.793540\pi$
$409$		−	3.38079e10i		−	1.20816i	0.796923	−	0.604081i	$-0.206460\pi$
$410$	3.03824e8			0.0107519
$411$			8.88785e10i			3.11480i
$412$		−	1.55416e10i		−	0.539394i
$413$			2.03031e10i			0.697850i
$414$			1.59951e10i			0.544486i
$415$	6.34739e8			0.0213995
$416$	−1.39793e9			−0.0466778
$417$			1.33942e10i			0.442968i
$418$	−3.68898e8	−	1.22135e10i	−0.0120837	−	0.400068i
$419$	4.33217e10			1.40556			0.702779	−	0.711408i	$-0.251942\pi$
$419$	4.33217e10			1.40556			0.702779	+	0.711408i	$0.251942\pi$
$420$		−	2.63627e8i		−	0.00847213i
$421$			1.45910e10i			0.464468i	0.972660	+	0.232234i	$0.0746035\pi$
$421$			1.45910e10i			0.464468i	−0.972660	+	0.232234i	$0.925397\pi$
$422$	2.98270e9			0.0940502
$423$	6.95938e10			2.17375
$424$	2.04274e10			0.632046
$425$	−1.48402e10			−0.454867
$426$			2.83914e10i			0.862082i
$427$	6.30334e8			0.0189609
$428$		−	2.03905e10i		−	0.607650i
$429$	8.29281e9			0.244835
$430$			2.84930e8i			0.00833420i
$431$			3.84218e10i			1.11344i	0.830699	+	0.556722i	$0.187941\pi$
$431$			3.84218e10i			1.11344i	−0.830699	+	0.556722i	$0.812059\pi$
$432$		−	9.74690e9i		−	0.279854i
$433$			4.20484e9i			0.119618i	0.998210	+	0.0598092i	$0.0190492\pi$
$433$			4.20484e9i			0.119618i	−0.998210	+	0.0598092i	$0.980951\pi$
$434$	−1.64961e10			−0.464966
$435$	−2.81520e8			−0.00786235
$436$		−	2.61301e10i		−	0.723094i
$437$	−1.66744e10	+	5.03638e8i	−0.457220	+	0.0138100i
$438$	6.25139e10			1.69856
$439$		−	5.89148e10i		−	1.58623i	−0.609071	−	0.793116i	$-0.708457\pi$
$439$		−	5.89148e10i		−	1.58623i	0.609071	−	0.793116i	$-0.291543\pi$
$440$		−	1.53061e8i		−	0.00408371i
$441$	4.73093e10			1.25081
$442$	−3.24284e9			−0.0849642
$443$	−4.60523e7			−0.00119574			−0.000597869	−	1.00000i	$-0.500190\pi$
$443$	−4.60523e7			−0.00119574			−0.000597869		1.00000i	$0.500190\pi$
$444$	2.68869e10			0.691846
$445$		−	3.90417e8i		−	0.00995608i
$446$	−4.20470e10			−1.06266
$447$		−	8.43510e10i		−	2.11281i
$448$	2.55242e9			0.0633637
$449$		−	1.30171e10i		−	0.320278i	−0.987094	−	0.160139i	$-0.948806\pi$
$449$		−	1.30171e10i		−	0.320278i	0.987094	−	0.160139i	$-0.0511943\pi$
$450$			4.87901e10i			1.18982i
$451$		−	1.74503e10i		−	0.421790i
$452$			1.77763e10i			0.425881i
$453$	−1.18146e11			−2.80559
$454$	3.36117e10			0.791166
$455$			1.17061e8i			0.00273129i
$456$	2.50297e10	−	7.56003e8i	0.578891	−	0.0174849i
$457$	−3.40030e10			−0.779564			−0.389782	−	0.920907i	$-0.627450\pi$
$457$	−3.40030e10			−0.779564			−0.389782	+	0.920907i	$0.627450\pi$
$458$		−	2.72510e10i		−	0.619327i
$459$		−	2.26104e10i		−	0.509397i
$460$	−2.08967e8			−0.00466709
$461$	4.03527e10			0.893447			0.446723	−	0.894672i	$-0.352591\pi$
$461$	4.03527e10			0.893447			0.446723	+	0.894672i	$0.352591\pi$
$462$	−1.51415e10			−0.332355
$463$	−2.75138e10			−0.598723			−0.299362	−	0.954140i	$-0.596774\pi$
$463$	−2.75138e10			−0.598723			−0.299362	+	0.954140i	$0.596774\pi$
$464$		−	2.72566e9i		−	0.0588031i
$465$	−2.02726e9			−0.0433609
$466$			7.14152e9i			0.151442i
$467$	−6.55796e10			−1.37880			−0.689400	−	0.724381i	$-0.742126\pi$
$467$	−6.55796e10			−1.37880			−0.689400	+	0.724381i	$0.742126\pi$
$468$			1.06615e10i			0.222246i
$469$		−	4.02335e9i		−	0.0831566i
$470$			9.09202e8i			0.0186324i
$471$			9.13259e10i			1.85571i
$472$	−2.41576e10			−0.486728
$473$	1.63651e10			0.326944
$474$			4.64637e10i			0.920450i
$475$	−5.08622e10	+	1.53625e9i	−0.999128	+	0.0301778i
$476$	5.92098e9			0.115336
$477$		−	1.55792e11i		−	3.00934i
$478$		−	2.39982e10i		−	0.459692i
$479$	2.61049e10			0.495884			0.247942	−	0.968775i	$-0.420246\pi$
$479$	2.61049e10			0.495884			0.247942	+	0.968775i	$0.420246\pi$
$480$	3.13677e8			0.00590904
$481$	−1.19389e10			−0.223041
$482$	−6.07248e10			−1.12507
$483$			2.06720e10i			0.379834i
$484$	1.86468e10			0.339799
$485$		−	2.22091e8i		−	0.00401388i
$486$	3.44436e10			0.617396
$487$		−	4.17323e10i		−	0.741918i	−0.928649	−	0.370959i	$-0.879029\pi$
$487$		−	4.17323e10i		−	0.741918i	0.928649	−	0.370959i	$-0.120971\pi$
$488$			7.50003e8i			0.0132246i
$489$		−	1.11591e11i		−	1.95161i
$490$			6.18067e8i			0.0107214i
$491$	−3.70917e10			−0.638191			−0.319096	−	0.947722i	$-0.603379\pi$
$491$	−3.70917e10			−0.638191			−0.319096	+	0.947722i	$0.603379\pi$
$492$	3.57618e10			0.610322
$493$		−	6.32286e9i		−	0.107035i
$494$	−1.11143e10	+	3.35697e8i	−0.186626	+	0.00563689i
$495$	−1.16734e9			−0.0194436
$496$		−	1.96279e10i		−	0.324299i
$497$			2.30186e10i			0.377272i
$498$	7.47123e10			1.21472
$499$	5.59412e10			0.902255			0.451128	−	0.892459i	$-0.351022\pi$
$499$	5.59412e10			0.902255			0.451128	+	0.892459i	$0.351022\pi$
$500$	−1.27509e9			−0.0204015
$501$	−2.16393e10			−0.343473
$502$		−	5.89228e10i		−	0.927831i
$503$	2.42435e9			0.0378725			0.0189362	−	0.999821i	$-0.493972\pi$
$503$	2.42435e9			0.0378725			0.0189362	+	0.999821i	$0.493972\pi$
$504$		−	1.94664e10i		−	0.301692i
$505$	−1.84469e9			−0.0283634
$506$			1.20021e10i			0.183086i
$507$			1.00690e11i			1.52389i
$508$		−	2.59923e10i		−	0.390293i
$509$			9.39660e10i			1.39991i	0.714188	+	0.699954i	$0.246796\pi$
$509$			9.39660e10i			1.39991i	−0.714188	+	0.699954i	$0.753204\pi$
$510$	7.27651e8			0.0107558
$511$	5.06838e10			0.743337
$512$			3.03700e9i			0.0441942i
$513$	−2.34061e9	−	7.74931e10i	−0.0337956	−	1.11891i
$514$	7.11918e10			1.01995
$515$		−	1.54852e9i		−	0.0220135i
$516$			3.35378e10i			0.473082i
$517$	5.22205e10			0.730935
$518$	2.17989e10			0.302771
$519$	−3.69873e10			−0.509780
$520$	−1.39286e8			−0.00190499
$521$			6.87008e10i			0.932419i	0.884674	+	0.466209i	$0.154381\pi$
$521$			6.87008e10i			0.932419i	−0.884674	+	0.466209i	$0.845619\pi$
$522$	−2.07876e10			−0.279978
$523$			1.78438e10i			0.238496i	0.992865	+	0.119248i	$0.0380483\pi$
$523$			1.78438e10i			0.238496i	−0.992865	+	0.119248i	$0.961952\pi$
$524$	−6.33981e10			−0.840914
$525$			6.30560e10i			0.830021i
$526$			4.17659e10i			0.545606i
$527$		−	4.55317e10i		−	0.590298i
$528$		−	1.80162e10i		−	0.231807i
$529$	−6.19251e10			−0.790759
$530$	2.03533e9			0.0257947
$531$			1.84241e11i			2.31744i
$532$	2.02931e10	−	6.12937e8i	0.253339	−	0.00765190i
$533$	−1.58797e10			−0.196759
$534$		−	4.59543e10i		−	0.565146i
$535$		−	2.03166e9i		−	0.0247991i
$536$	4.78719e9			0.0579991
$537$	1.35985e11			1.63529
$538$	−7.48174e9			−0.0893045
$539$	3.54990e10			0.420592
$540$		−	9.71155e8i		−	0.0114213i
$541$	−4.08082e10			−0.476385			−0.238192	−	0.971218i	$-0.576555\pi$
$541$	−4.08082e10			−0.476385			−0.238192	+	0.971218i	$0.576555\pi$
$542$		−	1.01074e11i		−	1.17123i
$543$	−8.18647e10			−0.941668
$544$			7.04508e9i			0.0804434i
$545$		−	2.60353e9i		−	0.0295105i
$546$			1.37788e10i			0.155039i
$547$			4.56308e10i			0.509693i	0.966982	+	0.254846i	$0.0820249\pi$
$547$			4.56308e10i			0.509693i	−0.966982	+	0.254846i	$0.917975\pi$
$548$	−8.57397e10			−0.950736
$549$	5.72000e9			0.0629661
$550$			3.66102e10i			0.400084i
$551$	−6.54540e8	−	2.16705e10i	−0.00710116	−	0.235105i
$552$	−2.45966e10			−0.264922
$553$			3.76709e10i			0.402815i
$554$			8.49286e10i			0.901602i
$555$	2.67894e9			0.0282352
$556$	−1.29212e10			−0.135208
$557$	5.73755e10			0.596081			0.298041	−	0.954553i	$-0.403667\pi$
$557$	5.73755e10			0.596081			0.298041	+	0.954553i	$0.403667\pi$
$558$	−1.49694e11			−1.54408
$559$		−	1.48922e10i		−	0.152515i
$560$	2.54317e8			0.00258597
$561$		−	4.17930e10i		−	0.421942i
$562$	−1.79231e10			−0.179667
$563$			1.05916e11i			1.05421i	0.849799	+	0.527107i	$0.176723\pi$
$563$			1.05916e11i			1.05421i	−0.849799	+	0.527107i	$0.823277\pi$
$564$			1.07018e11i			1.05765i
$565$			1.77119e9i			0.0173808i
$566$			2.14305e10i			0.208818i
$567$	−7.87701e9			−0.0762130
$568$	−2.73888e10			−0.263135
$569$		−	1.03973e10i		−	0.0991911i	−0.998769	−	0.0495956i	$-0.984207\pi$
$569$		−	1.03973e10i		−	0.0991911i	0.998769	−	0.0495956i	$-0.0157932\pi$
$570$	2.49390e9	−	7.53261e7i	0.0236254	−	0.000713586i
$571$	5.16002e10			0.485408			0.242704	−	0.970100i	$-0.421966\pi$
$571$	5.16002e10			0.485408			0.242704	+	0.970100i	$0.421966\pi$
$572$			7.99995e9i			0.0747314i
$573$		−	2.78831e10i		−	0.258656i
$574$	2.89943e10			0.267094
$575$	4.99821e10			0.457238
$576$	2.31621e10			0.210420
$577$	1.16635e11			1.05226			0.526131	−	0.850404i	$-0.323642\pi$
$577$	1.16635e11			1.05226			0.526131	+	0.850404i	$0.323642\pi$
$578$		−	6.25789e10i		−	0.560682i
$579$	−1.92683e11			−1.71447
$580$		−	2.71578e8i		−	0.00239984i
$581$	6.05738e10			0.531595
$582$		−	2.61414e10i		−	0.227844i
$583$		−	1.16900e11i		−	1.01191i
$584$			6.03062e10i			0.518454i
$585$			1.06228e9i			0.00907018i
$586$	1.21545e11			1.03074
$587$	5.52320e10			0.465199			0.232599	−	0.972573i	$-0.425277\pi$
$587$	5.52320e10			0.465199			0.232599	+	0.972573i	$0.425277\pi$
$588$			7.27500e10i			0.608588i
$589$	−4.71342e9	−	1.56052e11i	−0.0391629	−	1.29661i
$590$	−2.40700e9			−0.0198641
$591$		−	4.45110e10i		−	0.364853i
$592$			2.59374e10i			0.211174i
$593$	8.65658e10			0.700047			0.350024	−	0.936741i	$-0.386174\pi$
$593$	8.65658e10			0.700047			0.350024	+	0.936741i	$0.386174\pi$
$594$	−5.57788e10			−0.448047
$595$	5.89951e8			0.00470704
$596$	8.13721e10			0.644897
$597$		−	1.54578e10i		−	0.121689i
$598$	1.09219e10			0.0854072
$599$		−	1.74590e11i		−	1.35616i	−0.734987	−	0.678081i	$-0.762812\pi$
$599$		−	1.74590e11i		−	1.35616i	0.734987	−	0.678081i	$-0.237188\pi$
$600$	−7.50272e10			−0.578914
$601$			2.16063e11i			1.65608i	0.560666	+	0.828042i	$0.310545\pi$
$601$			2.16063e11i			1.65608i	−0.560666	+	0.828042i	$0.689455\pi$
$602$			2.71912e10i			0.207034i
$603$		−	3.65101e10i		−	0.276149i
$604$		−	1.13973e11i		−	0.856357i
$605$	1.85792e9			0.0138677
$606$	−2.17131e11			−1.61002
$607$		−	1.21678e11i		−	0.896305i	−0.893957	−	0.448152i	$-0.852082\pi$
$607$		−	1.21678e11i		−	0.896305i	0.893957	−	0.448152i	$-0.147918\pi$
$608$	7.29304e8	+	2.41458e10i	0.00533696	+	0.176696i
$609$	−2.68658e10			−0.195313
$610$			7.47283e7i			0.000539717i
$611$		−	4.75206e10i		−	0.340971i
$612$	5.37302e10			0.383013
$613$	−6.28473e10			−0.445087			−0.222543	−	0.974923i	$-0.571436\pi$
$613$	−6.28473e10			−0.445087			−0.222543	+	0.974923i	$0.571436\pi$
$614$	−8.16063e10			−0.574183
$615$	3.56321e9			0.0249081
$616$		−	1.46068e10i		−	0.101445i
$617$	1.32336e11			0.913137			0.456569	−	0.889688i	$-0.349078\pi$
$617$	1.32336e11			0.913137			0.456569	+	0.889688i	$0.349078\pi$
$618$		−	1.82270e11i		−	1.24957i
$619$	6.13179e10			0.417662			0.208831	−	0.977952i	$-0.433034\pi$
$619$	6.13179e10			0.417662			0.208831	+	0.977952i	$0.433034\pi$
$620$		−	1.95567e9i		−	0.0132351i
$621$			7.61520e10i			0.512053i
$622$			1.50343e11i			1.00443i
$623$		−	3.72579e10i		−	0.247324i
$624$	−1.63947e10			−0.108135
$625$	1.52397e11			0.998751
$626$		−	1.01018e11i		−	0.657813i
$627$	−4.32640e9	−	1.43238e11i	−0.0279934	−	0.926806i
$628$	−8.81007e10			−0.566423
$629$			6.01682e10i			0.384384i
$630$		−	1.93958e9i		−	0.0123125i
$631$	−3.09769e11			−1.95398			−0.976992	−	0.213277i	$-0.931586\pi$
$631$	−3.09769e11			−1.95398			−0.976992	+	0.213277i	$0.931586\pi$
$632$	−4.48228e10			−0.280951
$633$	3.49808e10			0.217878
$634$	−8.36135e10			−0.517511
$635$		−	2.58981e9i		−	0.0159284i
$636$	2.39570e11			1.46421
$637$		−	3.23041e10i		−	0.196200i
$638$	−1.55982e10			−0.0941440
$639$			2.08884e11i			1.25286i
$640$			3.02599e8i			0.00180363i
$641$		−	1.72998e11i		−	1.02473i	−0.858768	−	0.512364i	$-0.828770\pi$
$641$		−	1.72998e11i		−	1.02473i	0.858768	−	0.512364i	$-0.171230\pi$
$642$		−	2.39138e11i		−	1.40769i
$643$	−4.72012e10			−0.276127			−0.138064	−	0.990423i	$-0.544088\pi$
$643$	−4.72012e10			−0.276127			−0.138064	+	0.990423i	$0.544088\pi$
$644$	−1.99419e10			−0.115938
$645$			3.34162e9i			0.0193072i
$646$	1.69180e9	+	5.60122e10i	0.00971448	+	0.321627i
$647$	−5.19437e10			−0.296426			−0.148213	−	0.988955i	$-0.547352\pi$
$647$	−5.19437e10			−0.296426			−0.148213	+	0.988955i	$0.547352\pi$
$648$		−	9.37247e9i		−	0.0531562i
$649$			1.38248e11i			0.779254i
$650$	3.33153e10			0.186634
$651$	−1.93464e11			−1.07715
$652$	1.07650e11			0.595694
$653$	−5.46418e10			−0.300519			−0.150260	−	0.988647i	$-0.548011\pi$
$653$	−5.46418e10			−0.300519			−0.150260	+	0.988647i	$0.548011\pi$
$654$		−	3.06451e11i		−	1.67513i
$655$	−6.31683e9			−0.0343189
$656$			3.44988e10i			0.186290i
$657$	4.59933e11			2.46850
$658$			8.67661e10i			0.462857i
$659$		−	2.81797e11i		−	1.49415i	−0.664738	−	0.747077i	$-0.731457\pi$
$659$		−	2.81797e11i		−	1.49415i	0.664738	−	0.747077i	$-0.268543\pi$
$660$		−	1.79509e9i		−	0.00946040i
$661$			3.42983e11i			1.79667i	0.439315	+	0.898333i	$0.355221\pi$
$661$			3.42983e11i			1.79667i	−0.439315	+	0.898333i	$0.644779\pi$
$662$	2.07305e11			1.07939
$663$	−3.80316e10			−0.196830
$664$			7.20738e10i			0.370771i
$665$	2.02195e9	−	6.10715e7i	0.0103391	−	0.000312285i
$666$	1.97815e11			1.00545
$667$			2.12955e10i			0.107593i
$668$		−	2.08751e10i		−	0.104839i
$669$	−4.93122e11			−2.46179
$670$	4.76983e8			0.00236703
$671$	4.29206e9			0.0211727
$672$	2.99345e10			0.146790
$673$			1.13379e11i			0.552680i	0.961060	+	0.276340i	$0.0891215\pi$
$673$			1.13379e11i			0.552680i	−0.961060	+	0.276340i	$0.910878\pi$
$674$	1.23063e11			0.596334
$675$			2.32287e11i			1.11895i
$676$	−9.71336e10			−0.465139
$677$		−	2.42947e10i		−	0.115653i	−0.998327	−	0.0578265i	$-0.981583\pi$
$677$		−	2.42947e10i		−	0.115653i	0.998327	−	0.0578265i	$-0.0184170\pi$
$678$			2.08479e11i			0.986604i
$679$		−	2.11944e10i		−	0.0997109i
$680$			7.01953e8i			0.00328301i
$681$	3.94195e11			1.83283
$682$	−1.12325e11			−0.519205
$683$			7.20712e9i			0.0331191i	0.999863	+	0.0165596i	$0.00527131\pi$
$683$			7.20712e9i			0.0331191i	−0.999863	+	0.0165596i	$0.994729\pi$
$684$	1.84151e11	−	5.56213e9i	0.841298	−	0.0254107i
$685$	−8.54288e9			−0.0388009
$686$			1.38363e11i			0.624775i
$687$		−	3.19596e11i		−	1.43474i
$688$	−3.23534e10			−0.144400
$689$	−1.06379e11			−0.472041
$690$	−2.45074e9			−0.0108119
$691$	2.63378e11			1.15523			0.577613	−	0.816310i	$-0.303984\pi$
$691$	2.63378e11			1.15523			0.577613	+	0.816310i	$0.303984\pi$
$692$		−	3.56811e10i		−	0.155601i
$693$	−1.11401e11			−0.483009
$694$		−	3.08404e10i		−	0.132948i
$695$	−1.28743e9			−0.00551805
$696$		−	3.19663e10i		−	0.136224i
$697$			8.00286e10i			0.339090i
$698$		−	1.01773e10i		−	0.0428757i
$699$			8.37549e10i			0.350834i
$700$	−6.08291e10			−0.253349
$701$	−1.32699e11			−0.549537			−0.274768	−	0.961510i	$-0.588601\pi$
$701$	−1.32699e11			−0.549537			−0.274768	+	0.961510i	$0.588601\pi$
$702$			5.07587e10i			0.209008i
$703$	6.22859e9	+	2.06216e11i	0.0255017	+	0.844309i
$704$	1.73799e10			0.0707550
$705$			1.06630e10i			0.0431642i
$706$			2.78782e11i			1.12214i
$707$	−1.76041e11			−0.704590
$708$	−2.83318e11			−1.12756
$709$	−4.21128e11			−1.66659			−0.833296	−	0.552826i	$-0.813549\pi$
$709$	−4.21128e11			−1.66659			−0.833296	+	0.552826i	$0.813549\pi$
$710$	−2.72894e9			−0.0107389
$711$			3.41847e11i			1.33768i
$712$	4.43314e10			0.172501
$713$			1.53351e11i			0.593376i
$714$	6.94405e10			0.267190
$715$			7.97094e8i			0.00304990i
$716$			1.31183e11i			0.499144i
$717$		−	2.81448e11i		−	1.06493i
$718$			3.19845e11i			1.20349i
$719$	2.14519e11			0.802694			0.401347	−	0.915926i	$-0.368542\pi$
$719$	2.14519e11			0.802694			0.401347	+	0.915926i	$0.368542\pi$
$720$	2.30781e9			0.00858757
$721$		−	1.47777e11i		−	0.546848i
$722$	1.15967e10	+	1.91797e11i	0.0426762	+	0.705818i
$723$	−7.12174e11			−2.60635
$724$		−	7.89736e10i		−	0.287427i
$725$			6.49578e10i			0.235115i
$726$	2.18687e11			0.787185
$727$	−1.90053e11			−0.680358			−0.340179	−	0.940361i	$-0.610488\pi$
$727$	−1.90053e11			−0.680358			−0.340179	+	0.940361i	$0.610488\pi$
$728$	−1.32922e10			−0.0473229
$729$	4.46414e11			1.58062
$730$			6.00875e9i			0.0211589i
$731$	−7.50518e10			−0.262840
$732$			8.79595e9i			0.0306364i
$733$	−1.59763e11			−0.553425			−0.276713	−	0.960953i	$-0.589245\pi$
$733$	−1.59763e11			−0.553425			−0.276713	+	0.960953i	$0.589245\pi$
$734$		−	2.17729e11i		−	0.750123i
$735$			7.24862e9i			0.0248374i
$736$		−	2.37279e10i		−	0.0808628i
$737$		−	2.73958e10i		−	0.0928568i
$738$	2.63110e11			0.886976
$739$	−4.25579e11			−1.42693			−0.713464	−	0.700692i	$-0.752875\pi$
$739$	−4.25579e11			−1.42693			−0.713464	+	0.700692i	$0.752875\pi$
$740$			2.58433e9i			0.00861831i
$741$	−1.30347e11	+	3.93702e9i	−0.432342	+	0.0130585i
$742$	1.94234e11			0.640780
$743$		−	3.34285e11i		−	1.09689i	−0.836188	−	0.548443i	$-0.815221\pi$
$743$		−	3.34285e11i		−	1.09689i	0.836188	−	0.548443i	$-0.184779\pi$
$744$		−	2.30193e11i		−	0.751278i
$745$	8.10771e9			0.0263192
$746$	−2.77903e11			−0.897300
$747$	5.49681e11			1.76534
$748$	4.03171e10			0.128790
$749$		−	1.93884e11i		−	0.616047i
$750$	−1.49541e10			−0.0472625
$751$			3.48163e11i			1.09452i	0.836964	+	0.547259i	$0.184329\pi$
$751$			3.48163e11i			1.09452i	−0.836964	+	0.547259i	$0.815671\pi$
$752$	−1.03239e11			−0.322828
$753$		−	6.91040e11i		−	2.14943i
$754$			1.41944e10i			0.0439168i
$755$		−	1.13560e10i		−	0.0349492i
$756$		−	9.26785e10i		−	0.283721i
$757$	4.43308e11			1.34996			0.674982	−	0.737834i	$-0.264151\pi$
$757$	4.43308e11			1.34996			0.674982	+	0.737834i	$0.264151\pi$
$758$	1.67687e11			0.507953
$759$			1.40760e11i			0.424141i
$760$	7.26659e7	+	2.40582e9i	0.000217809	+	0.00721123i
$761$	−6.31113e10			−0.188178			−0.0940889	−	0.995564i	$-0.529994\pi$
$761$	−6.31113e10			−0.188178			−0.0940889	+	0.995564i	$0.529994\pi$
$762$		−	3.04835e11i		−	0.904160i
$763$		−	2.48458e11i		−	0.733086i
$764$	2.68984e10			0.0789503
$765$	5.35354e9			0.0156313
$766$	−6.89793e10			−0.200356
$767$	1.25805e11			0.363511
$768$			3.56176e10i			0.102381i
$769$	3.96997e11			1.13522			0.567612	−	0.823296i	$-0.307867\pi$
$769$	3.96997e11			1.13522			0.567612	+	0.823296i	$0.307867\pi$
$770$		−	1.45539e9i		−	0.00414014i
$771$	8.34929e11			2.36283
$772$		−	1.85878e11i		−	0.523310i
$773$			1.01575e11i			0.284490i	0.989831	+	0.142245i	$0.0454322\pi$
$773$			1.01575e11i			0.284490i	−0.989831	+	0.142245i	$0.954568\pi$
$774$			2.46748e11i			0.687526i
$775$			4.67770e11i			1.29666i
$776$	2.52182e10			0.0695452
$777$	2.55655e11			0.701406
$778$		−	1.54779e11i		−	0.422468i
$779$	8.28453e9	+	2.74284e11i	0.0224967	+	0.744820i
$780$	−1.63353e9			−0.00441314
$781$			1.56738e11i			0.421280i
$782$		−	5.50429e10i		−	0.147189i
$783$	−9.89689e10			−0.263301
$784$	−7.01808e10			−0.185761
$785$	−8.77812e9			−0.0231166
$786$	−7.43526e11			−1.94808
$787$		−	1.64530e11i		−	0.428891i	−0.976736	−	0.214445i	$-0.931206\pi$
$787$		−	1.64530e11i		−	0.428891i	0.976736	−	0.214445i	$-0.0687944\pi$
$788$	4.29391e10			0.111365
$789$			4.89826e11i			1.26396i
$790$	−4.46602e9			−0.0114660
$791$			1.69026e11i			0.431766i
$792$		−	1.32550e11i		−	0.336884i
$793$		−	3.90577e9i		−	0.00987676i
$794$		−	4.38764e11i		−	1.10395i
$795$	2.38701e10			0.0597566
$796$	1.49119e10			0.0371434
$797$		−	4.34638e11i		−	1.07720i	−0.842563	−	0.538598i	$-0.818954\pi$
$797$		−	4.34638e11i		−	1.07720i	0.842563	−	0.538598i	$-0.181046\pi$
$798$	2.37995e11	−	7.18846e9i	0.586890	−	0.0177265i
$799$	−2.39488e11			−0.587620
$800$		−	7.23776e10i		−	0.176703i
$801$		−	3.38099e11i		−	0.821323i
$802$	−3.42285e11			−0.827352
$803$	3.45116e11			0.830048
$804$	5.61436e10			0.134362
$805$	−1.98696e9			−0.00473158
$806$			1.02216e11i			0.242201i
$807$	−8.77450e10			−0.206885
$808$		−	2.09463e11i		−	0.491430i
$809$	−5.76106e11			−1.34496			−0.672478	−	0.740117i	$-0.734770\pi$
$809$	−5.76106e11			−1.34496			−0.672478	+	0.740117i	$0.734770\pi$
$810$		−	9.33848e8i		−	0.00216938i
$811$		−	5.80887e11i		−	1.34279i	−0.741099	−	0.671396i	$-0.765695\pi$
$811$		−	5.80887e11i		−	1.34279i	0.741099	−	0.671396i	$-0.234305\pi$
$812$		−	2.59170e10i		−	0.0596157i
$813$		−	1.18538e12i		−	2.71329i
$814$	1.48433e11			0.338090
$815$	1.07260e10			0.0243112
$816$			8.26239e10i			0.186357i
$817$	−2.57227e11	+	7.76934e9i	−0.577336	+	0.0174380i
$818$	3.82493e11			0.854300
$819$			1.01375e11i			0.225317i
$820$			3.43737e9i			0.00760276i
$821$	−7.00549e11			−1.54193			−0.770967	−	0.636875i	$-0.780227\pi$
$821$	−7.00549e11			−1.54193			−0.770967	+	0.636875i	$0.780227\pi$
$822$	−1.00555e12			−2.20249
$823$	4.23605e11			0.923339			0.461670	−	0.887052i	$-0.347251\pi$
$823$	4.23605e11			0.923339			0.461670	+	0.887052i	$0.347251\pi$
$824$	1.75833e11			0.381409
$825$			4.29360e11i			0.926843i
$826$	−2.29703e11			−0.493454
$827$		−	5.84073e11i		−	1.24866i	−0.781159	−	0.624332i	$-0.785371\pi$
$827$		−	5.84073e11i		−	1.24866i	0.781159	−	0.624332i	$-0.214629\pi$
$828$	−1.80964e11			−0.385010
$829$		−	1.13733e11i		−	0.240806i	−0.992725	−	0.120403i	$-0.961581\pi$
$829$		−	1.13733e11i		−	0.240806i	0.992725	−	0.120403i	$-0.0384187\pi$
$830$			7.18125e9i			0.0151317i
$831$			9.96033e11i			2.08867i
$832$		−	1.58157e10i		−	0.0330062i
$833$	−1.62802e11			−0.338126
$834$	−1.51538e11			−0.313226
$835$		−	2.07994e9i		−	0.00427864i
$836$	1.38180e11	−	4.17361e9i	0.282891	−	0.00854450i
$837$	−7.12688e11			−1.45210
$838$			4.90129e11i			0.993880i
$839$			3.01770e11i			0.609016i	0.952510	+	0.304508i	$0.0984920\pi$
$839$			3.01770e11i			0.609016i	−0.952510	+	0.304508i	$0.901508\pi$
$840$	2.98260e9			0.00599070
$841$	4.72570e11			0.944675
$842$	−1.65078e11			−0.328428
$843$	−2.10200e11			−0.416220
$844$			3.37454e10i			0.0665035i
$845$	−9.67814e9			−0.0189830
$846$			7.87364e11i			1.53707i
$847$	1.77303e11			0.344495
$848$			2.31109e11i			0.446924i
$849$			2.51335e11i			0.483751i
$850$		−	1.67898e11i		−	0.321640i
$851$		−	2.02648e11i		−	0.386388i
$852$	−3.21212e11			−0.609584
$853$	5.88205e11			1.11105			0.555524	−	0.831501i	$-0.312518\pi$
$853$	5.88205e11			1.11105			0.555524	+	0.831501i	$0.312518\pi$
$854$			7.13141e9i			0.0134074i
$855$	1.83483e10	−	5.54197e8i	0.0343346	−	0.00103705i
$856$	2.30692e11			0.429673
$857$			7.39597e11i			1.37111i	0.728022	+	0.685554i	$0.240440\pi$
$857$			7.39597e11i			1.37111i	−0.728022	+	0.685554i	$0.759560\pi$
$858$			9.38225e10i			0.173124i
$859$	7.08708e11			1.30165			0.650826	−	0.759227i	$-0.274423\pi$
$859$	7.08708e11			1.30165			0.650826	+	0.759227i	$0.274423\pi$
$860$	−3.22361e9			−0.00589317
$861$	3.40041e11			0.618756
$862$	−4.34693e11			−0.787324
$863$			6.75039e10i			0.121699i	0.998147	+	0.0608493i	$0.0193809\pi$
$863$			6.75039e10i			0.121699i	−0.998147	+	0.0608493i	$0.980619\pi$
$864$	1.10274e11			0.197887
$865$		−	3.55517e9i		−	0.00635032i
$866$	−4.75723e10			−0.0845830
$867$		−	7.33918e11i		−	1.29889i
$868$		−	1.86632e11i		−	0.328781i
$869$			2.56509e11i			0.449803i
$870$		−	3.18504e9i		−	0.00555952i
$871$	−2.49301e10			−0.0433164
$872$	2.95628e11			0.511305
$873$		−	1.92330e11i		−	0.331124i
$874$	−5.69801e9	−	1.88650e11i	−0.00976512	−	0.323304i
$875$	−1.21242e10			−0.0206834
$876$			7.07264e11i			1.20106i
$877$			7.27667e11i			1.23008i	0.788495	+	0.615041i	$0.210861\pi$
$877$			7.27667e11i			1.23008i	−0.788495	+	0.615041i	$0.789139\pi$
$878$	6.66545e11			1.12164
$879$	1.42547e12			2.38782
$880$	1.73169e9			0.00288762
$881$	3.14989e11			0.522868			0.261434	−	0.965221i	$-0.415805\pi$
$881$	3.14989e11			0.522868			0.261434	+	0.965221i	$0.415805\pi$
$882$			5.35243e11i			0.884457i
$883$	3.06152e11			0.503609			0.251805	−	0.967778i	$-0.418976\pi$
$883$	3.06152e11			0.503609			0.251805	+	0.967778i	$0.418976\pi$
$884$		−	3.66885e10i		−	0.0600788i
$885$	−2.82291e10			−0.0460176
$886$		−	5.21022e8i		−	0.000845515i
$887$		−	1.30227e11i		−	0.210381i	−0.994452	−	0.105190i	$-0.966455\pi$
$887$		−	1.30227e11i		−	0.210381i	0.994452	−	0.105190i	$-0.0335452\pi$
$888$			3.04191e11i			0.489209i
$889$		−	2.47148e11i		−	0.395686i
$890$	4.41706e9			0.00704001
$891$	−5.36361e10			−0.0851033
$892$		−	4.75708e11i		−	0.751416i
$893$	−8.20804e11	+	2.47917e10i	−1.29072	+	0.0389853i
$894$	9.54323e11			1.49398
$895$			1.30707e10i			0.0203708i
$896$			2.88774e10i			0.0448049i
$897$	1.28091e11			0.197856
$898$	1.47271e11			0.226471
$899$	−1.99299e11			−0.305117
$900$	−5.51997e11			−0.841331
$901$			5.36115e11i			0.813503i
$902$	1.97427e11			0.298251
$903$			3.18895e11i			0.479619i
$904$	−2.01116e11			−0.301143
$905$		−	7.86873e9i		−	0.0117303i
$906$		−	1.33666e12i		−	1.98385i
$907$			8.81128e11i			1.30200i	0.759079	+	0.650998i	$0.225649\pi$
$907$			8.81128e11i			1.30200i	−0.759079	+	0.650998i	$0.774351\pi$
$908$			3.80273e11i			0.559439i
$909$	−1.59750e12			−2.33983
$910$	−1.32440e9			−0.00193132
$911$			3.85324e11i			0.559438i	0.960082	+	0.279719i	$0.0902414\pi$
$911$			3.85324e11i			0.559438i	−0.960082	+	0.279719i	$0.909759\pi$
$912$	8.55319e9	+	2.83179e11i	0.0123637	+	0.409338i
$913$	4.12459e11			0.593605
$914$		−	3.84700e11i		−	0.551235i
$915$			8.76406e8i			0.00125032i
$916$	3.08309e11			0.437930
$917$	−6.02822e11			−0.852534
$918$	2.55807e11			0.360198
$919$	−2.62551e11			−0.368089			−0.184044	−	0.982918i	$-0.558919\pi$
$919$	−2.62551e11			−0.368089			−0.184044	+	0.982918i	$0.558919\pi$
$920$		−	2.36419e9i		−	0.00330013i
$921$	−9.57070e11			−1.33016
$922$			4.56538e11i			0.631762i
$923$	1.42632e11			0.196521
$924$		−	1.71307e11i		−	0.235011i
$925$		−	6.18138e11i		−	0.844342i
$926$		−	3.11283e11i		−	0.423361i
$927$		−	1.34101e12i		−	1.81599i
$928$	3.08374e10			0.0415801
$929$	5.89484e11			0.791424			0.395712	−	0.918375i	$-0.370498\pi$
$929$	5.89484e11			0.791424			0.395712	+	0.918375i	$0.370498\pi$
$930$		−	2.29359e10i		−	0.0306608i
$931$	−5.57975e11	+	1.68532e10i	−0.742704	+	0.0224328i
$932$	−8.07971e10			−0.107086
$933$			1.76320e12i			2.32689i
$934$		−	7.41949e11i		−	0.974959i
$935$	4.01709e9			0.00525612
$936$	−1.20621e11			−0.157151
$937$	−1.01168e11			−0.131246			−0.0656228	−	0.997844i	$-0.520903\pi$
$937$	−1.01168e11			−0.131246			−0.0656228	+	0.997844i	$0.520903\pi$
$938$	4.55190e10			0.0588006
$939$		−	1.18473e12i		−	1.52390i
$940$	−1.02864e10			−0.0131751
$941$			5.01460e11i			0.639555i	0.947493	+	0.319778i	$0.103608\pi$
$941$			5.01460e11i			0.639555i	−0.947493	+	0.319778i	$0.896392\pi$
$942$	−1.03323e12			−1.31219
$943$		−	2.69538e11i		−	0.340857i
$944$		−	2.73313e11i		−	0.344169i
$945$		−	9.23424e9i		−	0.0115791i
$946$			1.85150e11i			0.231185i
$947$	−6.95220e11			−0.864415			−0.432207	−	0.901774i	$-0.642265\pi$
$947$	−6.95220e11			−0.864415			−0.432207	+	0.901774i	$0.642265\pi$
$948$	−5.25676e11			−0.650856
$949$		−	3.14055e11i		−	0.387205i
$950$	−1.73807e10	−	5.75441e11i	−0.0213390	−	0.706490i
$951$	−9.80609e11			−1.19888
$952$			6.69882e10i			0.0815550i
$953$			9.41947e11i			1.14197i	0.820960	+	0.570985i	$0.193439\pi$
$953$			9.41947e11i			1.14197i	−0.820960	+	0.570985i	$0.806561\pi$
$954$	1.76259e12			2.12793
$955$	2.68009e9			0.00322208
$956$	2.71508e11			0.325051
$957$	−1.82934e11			−0.218096
$958$			2.95343e11i			0.350643i
$959$	−8.15257e11			−0.963874
$960$			3.54885e9i			0.00417832i
$961$	−5.82287e11			−0.682721
$962$		−	1.35074e11i		−	0.157714i
$963$		−	1.75941e12i		−	2.04579i
$964$		−	6.87023e11i		−	0.795542i
$965$		−	1.85204e10i		−	0.0213571i
$966$	−2.33877e11			−0.268583
$967$	1.44496e12			1.65253			0.826267	−	0.563278i	$-0.190460\pi$
$967$	1.44496e12			1.65253			0.826267	+	0.563278i	$0.190460\pi$
$968$			2.10964e11i			0.240274i
$969$	1.98413e10	+	6.56904e11i	0.0225047	+	0.745087i
$970$	2.51268e9			0.00283824
$971$			2.50166e11i			0.281418i	0.990051	+	0.140709i	$0.0449382\pi$
$971$			2.50166e11i			0.281418i	−0.990051	+	0.140709i	$0.955062\pi$
$972$			3.89685e11i			0.436565i
$973$	−1.22861e11			−0.137077
$974$	4.72147e11			0.524615
$975$	3.90718e11			0.432359
$976$	−8.48531e9			−0.00935123
$977$		−	1.53618e12i		−	1.68602i	−0.537898	−	0.843010i	$-0.680781\pi$
$977$		−	1.53618e12i		−	1.68602i	0.537898	−	0.843010i	$-0.319219\pi$
$978$	1.26251e12			1.38000
$979$		−	2.53696e11i		−	0.276174i
$980$	−6.99263e9			−0.00758117
$981$		−	2.25465e12i		−	2.43446i
$982$		−	4.19645e11i		−	0.451269i
$983$			5.10742e11i			0.547000i	0.961872	+	0.273500i	$0.0881814\pi$
$983$			5.10742e11i			0.547000i	−0.961872	+	0.273500i	$0.911819\pi$
$984$			4.04598e11i			0.431563i
$985$	4.27834e9			0.00454496
$986$	7.15350e10			0.0756851
$987$			1.01758e12i			1.07226i
$988$	−3.79798e9	−	1.25743e11i	−0.00398589	−	0.131965i
$989$	2.52776e11			0.264211
$990$		−	1.32070e10i		−	0.0137487i
$991$		−	1.27935e12i		−	1.32646i	−0.748414	−	0.663232i	$-0.769184\pi$
$991$		−	1.27935e12i		−	1.32646i	0.748414	−	0.663232i	$-0.230816\pi$
$992$	2.22064e11			0.229314
$993$	2.43125e12			2.50054
$994$	−2.60426e11			−0.266771
$995$	1.48578e9			0.00151588
$996$			8.45274e11i			0.858934i
$997$	−1.57993e12			−1.59904			−0.799518	−	0.600643i	$-0.794911\pi$
$997$	−1.57993e12			−1.59904			−0.799518	+	0.600643i	$0.794911\pi$
$998$			6.32902e11i			0.637991i
$999$	9.41787e11			0.945564

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.b.a.37.12	yes	12
3.2	odd	2		342.9.d.a.37.3		12
4.3	odd	2		304.9.e.e.113.1		12
19.18	odd	2	inner	38.9.b.a.37.1	✓	12
57.56	even	2		342.9.d.a.37.9		12
76.75	even	2		304.9.e.e.113.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.1	✓	12	19.18	odd	2	inner
38.9.b.a.37.12	yes	12	1.1	even	1	trivial
304.9.e.e.113.1		12	4.3	odd	2
304.9.e.e.113.12		12	76.75	even	2
342.9.d.a.37.3		12	3.2	odd	2
342.9.d.a.37.9		12	57.56	even	2

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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	38.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.3137i q^{2} +132.686i q^{3} -128.000 q^{4} -12.7536 q^{5} -1501.17 q^{6} -1217.09 q^{7} -1448.15i q^{8} -11044.5 q^{9} +O(q^{10})\)	\(q+11.3137i q^{2} +132.686i q^{3} -128.000 q^{4} -12.7536 q^{5} -1501.17 q^{6} -1217.09 q^{7} -1448.15i q^{8} -11044.5 q^{9} -144.290i q^{10} -8287.40 q^{11} -16983.8i q^{12} +7541.52i q^{13} -13769.8i q^{14} -1692.22i q^{15} +16384.0 q^{16} +38006.8 q^{17} -124955. i q^{18} +(130262. - 3934.45i) q^{19} +1632.46 q^{20} -161491. i q^{21} -93761.2i q^{22} -128007. q^{23} +192150. q^{24} -390462. q^{25} -85322.6 q^{26} -594903. i q^{27} +155787. q^{28} -166361. i q^{29} +19145.3 q^{30} -1.19799e6i q^{31} +185364. i q^{32} -1.09962e6i q^{33} +429998. i q^{34} +15522.3 q^{35} +1.41370e6 q^{36} +1.58309e6i q^{37} +(44513.2 + 1.47374e6i) q^{38} -1.00065e6 q^{39} +18469.2i q^{40} +2.10564e6i q^{41} +1.82706e6 q^{42} -1.97470e6 q^{43} +1.06079e6 q^{44} +140858. q^{45} -1.44824e6i q^{46} -6.30119e6 q^{47} +2.17393e6i q^{48} -4.28349e6 q^{49} -4.41758e6i q^{50} +5.04296e6i q^{51} -965315. i q^{52} +1.41058e7i q^{53} +6.73056e6 q^{54} +105694. q^{55} +1.76253e6i q^{56} +(522045. + 1.72839e7i) q^{57} +1.88216e6 q^{58} -1.66817e7i q^{59} +216604. i q^{60} -517902. q^{61} +1.35537e7 q^{62} +1.34422e7 q^{63} -2.09715e6 q^{64} -96181.5i q^{65} +1.24408e7 q^{66} +3.30572e6i q^{67} -4.86487e6 q^{68} -1.69848e7i q^{69} +175614. i q^{70} -1.89129e7i q^{71} +1.59942e7i q^{72} -4.16435e7 q^{73} -1.79107e7 q^{74} -5.18088e7i q^{75} +(-1.66735e7 + 503609. i) q^{76} +1.00865e7 q^{77} -1.13211e7i q^{78} -3.09516e7i q^{79} -208955. q^{80} +6.47201e6 q^{81} -2.38226e7 q^{82} -4.97694e7 q^{83} +2.06708e7i q^{84} -484723. q^{85} -2.23411e7i q^{86} +2.20738e7 q^{87} +1.20014e7i q^{88} +3.06123e7i q^{89} +1.59362e6i q^{90} -9.17871e6i q^{91} +1.63849e7 q^{92} +1.58956e8 q^{93} -7.12899e7i q^{94} +(-1.66130e6 + 50178.3i) q^{95} -2.45952e7 q^{96} +1.74140e7i q^{97} -4.84622e7i q^{98} +9.15305e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) \) 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9	\( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) \) 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9 - 12546 * q^11 + 196608 * q^16 + 270810 * q^17 + 41512 * q^19 - 71424 * q^20 - 823956 * q^23 - 229376 * q^24 + 865538 * q^25 - 431616 * q^26 + 694016 * q^28 + 71168 * q^30 - 1194378 * q^35 + 1995776 * q^36 + 998784 * q^38 + 5786100 * q^39 - 8383744 * q^42 + 7586646 * q^43 + 1605888 * q^44 + 2226046 * q^45 - 20260530 * q^47 - 19498842 * q^49 + 16933888 * q^54 - 14858554 * q^55 + 14430564 * q^57 - 5506560 * q^58 - 41363266 * q^61 + 32266752 * q^62 + 84235798 * q^63 - 25165824 * q^64 + 14371328 * q^66 - 34663680 * q^68 + 87906498 * q^73 - 2149632 * q^74 - 5313536 * q^76 - 78817962 * q^77 + 9142272 * q^80 - 100904812 * q^81 - 49609728 * q^82 - 55944960 * q^83 + 25440254 * q^85 + 119189604 * q^87 + 105466368 * q^92 + 105500856 * q^93 + 81396774 * q^95 + 29360128 * q^96 - 85554938 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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