LMFDB - Embedded newform 400.10.c.c.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.c (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:	$206.014334466$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.10.c.c.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+174.000i q^{3} -4658.00i q^{7} -10593.0 q^{9} +O(q^{10})$	$q+174.000i q^{3} -4658.00i q^{7} -10593.0 q^{9} -28992.0 q^{11} +164446. i q^{13} -594822. i q^{17} -295780. q^{19} +810492. q^{21} +2.54453e6i q^{23} +1.58166e6i q^{27} +3.72297e6 q^{29} -2.33577e6 q^{31} -5.04461e6i q^{33} +1.08404e7i q^{37} -2.86136e7 q^{39} +2.15939e7 q^{41} +1.08323e7i q^{43} -5.17214e6i q^{47} +1.86566e7 q^{49} +1.03499e8 q^{51} -9.81797e7i q^{53} -5.14657e7i q^{57} +1.61629e7 q^{59} -4.39282e7 q^{61} +4.93422e7i q^{63} +8.15574e7i q^{67} -4.42749e8 q^{69} -1.61308e8 q^{71} +2.47148e8i q^{73} +1.35045e8i q^{77} -5.83346e8 q^{79} -4.83711e8 q^{81} -1.45718e7i q^{83} +6.47797e8i q^{87} -4.70134e8 q^{89} +7.65989e8 q^{91} -4.06424e8i q^{93} -1.17838e8i q^{97} +3.07112e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 21186 q^{9}+O(q^{10}) $ 2 * q - 21186 * q^9	$ 2 q - 21186 q^{9} - 57984 q^{11} - 591560 q^{19} + 1620984 q^{21} + 7445940 q^{29} - 4671544 q^{31} - 57227208 q^{39} + 43187724 q^{41} + 37313286 q^{49} + 206998056 q^{51} + 32325720 q^{59} - 87856316 q^{61} - 885497832 q^{69} - 322615464 q^{71} - 1166691440 q^{79} - 967421718 q^{81} - 940267380 q^{89} + 1531978936 q^{91} + 614224512 q^{99}+O(q^{100}) $ 2 * q - 21186 * q^9 - 57984 * q^11 - 591560 * q^19 + 1620984 * q^21 + 7445940 * q^29 - 4671544 * q^31 - 57227208 * q^39 + 43187724 * q^41 + 37313286 * q^49 + 206998056 * q^51 + 32325720 * q^59 - 87856316 * q^61 - 885497832 * q^69 - 322615464 * q^71 - 1166691440 * q^79 - 967421718 * q^81 - 940267380 * q^89 + 1531978936 * q^91 + 614224512 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	174.000i	1.24023i	0.784509	+	0.620117i	$0.212915\pi$
$3$	174.000i	1.24023i	−0.784509	+	0.620117i	$0.787085\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4658.00i	− 0.733261i	−0.930367	−	0.366630i	$-0.880511\pi$
$7$	− 4658.00i	− 0.733261i	0.930367	−	0.366630i	$-0.119489\pi$
$8$	0	0
$9$	−10593.0	−0.538180
$10$	0	0
$11$	−28992.0	−0.597051	−0.298525	−	0.954402i	$-0.596495\pi$
$11$	−28992.0	−0.597051	−0.298525	+	0.954402i	$0.596495\pi$
$12$	0	0
$13$	164446.i	1.59690i	0.602060	+	0.798451i	$0.294347\pi$
$13$	164446.i	1.59690i	−0.602060	+	0.798451i	$0.705653\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 594822.i	− 1.72730i	−0.504095	−	0.863648i	$-0.668174\pi$
$17$	− 594822.i	− 1.72730i	0.504095	−	0.863648i	$-0.331826\pi$
$18$	0	0
$19$	−295780.	−0.520688	−0.260344	−	0.965516i	$-0.583836\pi$
$19$	−295780.	−0.520688	−0.260344	+	0.965516i	$0.583836\pi$
$20$	0	0
$21$	810492.	0.909415
$22$	0	0
$23$	2.54453e6i	1.89598i	0.318305	+	0.947988i	$0.396886\pi$
$23$	2.54453e6i	1.89598i	−0.318305	+	0.947988i	$0.603114\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.58166e6i	0.572765i
$28$	0	0
$29$	3.72297e6	0.977459	0.488729	−	0.872435i	$-0.337461\pi$
$29$	3.72297e6	0.977459	0.488729	+	0.872435i	$0.337461\pi$
$30$	0	0
$31$	−2.33577e6	−0.454258	−0.227129	−	0.973865i	$-0.572934\pi$
$31$	−2.33577e6	−0.454258	−0.227129	+	0.973865i	$0.572934\pi$
$32$	0	0
$33$	− 5.04461e6i	− 0.740482i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.08404e7i	0.950907i	0.879741	+	0.475454i	$0.157716\pi$
$37$	1.08404e7i	0.950907i	−0.879741	+	0.475454i	$0.842284\pi$
$38$	0	0
$39$	−2.86136e7	−1.98053
$40$	0	0
$41$	2.15939e7	1.19345	0.596723	−	0.802447i	$-0.296469\pi$
$41$	2.15939e7	1.19345	0.596723	+	0.802447i	$0.296469\pi$
$42$	0	0
$43$	1.08323e7i	0.483184i	0.970378	+	0.241592i	$0.0776695\pi$
$43$	1.08323e7i	0.483184i	−0.970378	+	0.241592i	$0.922330\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.17214e6i	− 0.154607i	−0.997008	−	0.0773036i	$-0.975369\pi$
$47$	− 5.17214e6i	− 0.154607i	0.997008	−	0.0773036i	$-0.0246311\pi$
$48$	0	0
$49$	1.86566e7	0.462329
$50$	0	0
$51$	1.03499e8	2.14225
$52$	0	0
$53$	− 9.81797e7i	− 1.70915i	−0.519328	−	0.854575i	$-0.673818\pi$
$53$	− 9.81797e7i	− 1.70915i	0.519328	−	0.854575i	$-0.326182\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.14657e7i	− 0.645775i
$58$	0	0
$59$	1.61629e7	0.173654	0.0868269	−	0.996223i	$-0.472327\pi$
$59$	1.61629e7	0.173654	0.0868269	+	0.996223i	$0.472327\pi$
$60$	0	0
$61$	−4.39282e7	−0.406218	−0.203109	−	0.979156i	$-0.565105\pi$
$61$	−4.39282e7	−0.406218	−0.203109	+	0.979156i	$0.565105\pi$
$62$	0	0
$63$	4.93422e7i	0.394626i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.15574e7i	0.494455i	0.968957	+	0.247228i	$0.0795195\pi$
$67$	8.15574e7i	0.494455i	−0.968957	+	0.247228i	$0.920480\pi$
$68$	0	0
$69$	−4.42749e8	−2.35145
$70$	0	0
$71$	−1.61308e8	−0.753343	−0.376671	−	0.926347i	$-0.622931\pi$
$71$	−1.61308e8	−0.753343	−0.376671	+	0.926347i	$0.622931\pi$
$72$	0	0
$73$	2.47148e8i	1.01860i	0.860589	+	0.509301i	$0.170096\pi$
$73$	2.47148e8i	1.01860i	−0.860589	+	0.509301i	$0.829904\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.35045e8i	0.437794i
$78$	0	0
$79$	−5.83346e8	−1.68502	−0.842508	−	0.538684i	$-0.818922\pi$
$79$	−5.83346e8	−1.68502	−0.842508	+	0.538684i	$0.818922\pi$
$80$	0	0
$81$	−4.83711e8	−1.24854
$82$	0	0
$83$	− 1.45718e7i	− 0.0337024i	−0.999858	−	0.0168512i	$-0.994636\pi$
$83$	− 1.45718e7i	− 0.0337024i	0.999858	−	0.0168512i	$-0.00536416\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.47797e8i	1.21228i
$88$	0	0
$89$	−4.70134e8	−0.794267	−0.397133	−	0.917761i	$-0.629995\pi$
$89$	−4.70134e8	−0.794267	−0.397133	+	0.917761i	$0.629995\pi$
$90$	0	0
$91$	7.65989e8	1.17095
$92$	0	0
$93$	− 4.06424e8i	− 0.563386i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.17838e8i	− 0.135149i	−0.997714	−	0.0675747i	$-0.978474\pi$
$97$	− 1.17838e8i	− 0.135149i	0.997714	−	0.0675747i	$-0.0215261\pi$
$98$	0	0
$99$	3.07112e8	0.321321
$100$	0	0
$101$	−8.60927e7	−0.0823228	−0.0411614	−	0.999153i	$-0.513106\pi$
$101$	−8.60927e7	−0.0823228	−0.0411614	+	0.999153i	$0.513106\pi$
$102$	0	0
$103$	1.92872e9i	1.68850i	0.535947	+	0.844252i	$0.319955\pi$
$103$	1.92872e9i	1.68850i	−0.535947	+	0.844252i	$0.680045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.39685e9i	− 1.03020i	−0.857130	−	0.515100i	$-0.827755\pi$
$107$	− 1.39685e9i	− 1.03020i	0.857130	−	0.515100i	$-0.172245\pi$
$108$	0	0
$109$	6.04327e8	0.410065	0.205033	−	0.978755i	$-0.434270\pi$
$109$	6.04327e8	0.410065	0.205033	+	0.978755i	$0.434270\pi$
$110$	0	0
$111$	−1.88623e9	−1.17935
$112$	0	0
$113$	1.68580e9i	0.972643i	0.873780	+	0.486322i	$0.161662\pi$
$113$	1.68580e9i	0.972643i	−0.873780	+	0.486322i	$0.838338\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.74198e9i	− 0.859421i
$118$	0	0
$119$	−2.77068e9	−1.26656
$120$	0	0
$121$	−1.51741e9	−0.643531
$122$	0	0
$123$	3.75733e9i	1.48015i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 3.70716e9i	− 1.26452i	−0.774758	−	0.632258i	$-0.782128\pi$
$127$	− 3.70716e9i	− 1.26452i	0.774758	−	0.632258i	$-0.217872\pi$
$128$	0	0
$129$	−1.88482e9	−0.599261
$130$	0	0
$131$	−2.54080e9	−0.753789	−0.376895	−	0.926256i	$-0.623008\pi$
$131$	−2.54080e9	−0.753789	−0.376895	+	0.926256i	$0.623008\pi$
$132$	0	0
$133$	1.37774e9i	0.381800i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.15390e9i	− 0.279849i	−0.990162	−	0.139925i	$-0.955314\pi$
$137$	− 1.15390e9i	− 0.279849i	0.990162	−	0.139925i	$-0.0446860\pi$
$138$	0	0
$139$	−5.62721e9	−1.27858	−0.639288	−	0.768968i	$-0.720771\pi$
$139$	−5.62721e9	−1.27858	−0.639288	+	0.768968i	$0.720771\pi$
$140$	0	0
$141$	8.99952e8	0.191749
$142$	0	0
$143$	− 4.76762e9i	− 0.953431i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.24626e9i	0.573396i
$148$	0	0
$149$	2.13688e9	0.355174	0.177587	−	0.984105i	$-0.443171\pi$
$149$	2.13688e9	0.355174	0.177587	+	0.984105i	$0.443171\pi$
$150$	0	0
$151$	9.67515e6	0.00151447	0.000757236	−	1.00000i	$-0.499759\pi$
$151$	9.67515e6	0.00151447	0.000757236		1.00000i	$0.499759\pi$
$152$	0	0
$153$	6.30095e9i	0.929597i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.88488e8i	− 0.0904373i	−0.998977	−	0.0452187i	$-0.985602\pi$
$157$	− 6.88488e8i	− 0.0904373i	0.998977	−	0.0452187i	$-0.0143985\pi$
$158$	0	0
$159$	1.70833e10	2.11975
$160$	0	0
$161$	1.18524e10	1.39024
$162$	0	0
$163$	− 1.43082e10i	− 1.58759i	−0.608182	−	0.793797i	$-0.708101\pi$
$163$	− 1.43082e10i	− 1.58759i	0.608182	−	0.793797i	$-0.291899\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 9.98735e9i	− 0.993633i	−0.867856	−	0.496817i	$-0.834502\pi$
$167$	− 9.98735e9i	− 0.993633i	0.867856	−	0.496817i	$-0.165498\pi$
$168$	0	0
$169$	−1.64380e10	−1.55010
$170$	0	0
$171$	3.13320e9	0.280224
$172$	0	0
$173$	− 3.51396e9i	− 0.298256i	−0.988818	−	0.149128i	$-0.952353\pi$
$173$	− 3.51396e9i	− 0.298256i	0.988818	−	0.149128i	$-0.0476466\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.81234e9i	0.215371i
$178$	0	0
$179$	1.19502e9	0.0870038	0.0435019	−	0.999053i	$-0.486149\pi$
$179$	1.19502e9	0.0870038	0.0435019	+	0.999053i	$0.486149\pi$
$180$	0	0
$181$	−9.12053e9	−0.631635	−0.315818	−	0.948820i	$-0.602279\pi$
$181$	−9.12053e9	−0.631635	−0.315818	+	0.948820i	$0.602279\pi$
$182$	0	0
$183$	− 7.64350e9i	− 0.503805i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.72451e10i	1.03128i
$188$	0	0
$189$	7.36737e9	0.419986
$190$	0	0
$191$	9.37431e9	0.509670	0.254835	−	0.966985i	$-0.417979\pi$
$191$	9.37431e9	0.509670	0.254835	+	0.966985i	$0.417979\pi$
$192$	0	0
$193$	− 2.40000e10i	− 1.24510i	−0.782580	−	0.622550i	$-0.786097\pi$
$193$	− 2.40000e10i	− 1.24510i	0.782580	−	0.622550i	$-0.213903\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.56124e8i	− 0.0263071i	−0.999913	−	0.0131536i	$-0.995813\pi$
$197$	− 5.56124e8i	− 0.0263071i	0.999913	−	0.0131536i	$-0.00418703\pi$
$198$	0	0
$199$	−2.51255e10	−1.13573	−0.567866	−	0.823121i	$-0.692231\pi$
$199$	−2.51255e10	−1.13573	−0.567866	+	0.823121i	$0.692231\pi$
$200$	0	0
$201$	−1.41910e10	−0.613240
$202$	0	0
$203$	− 1.73416e10i	− 0.716732i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.69542e10i	− 1.02038i
$208$	0	0
$209$	8.57525e9	0.310877
$210$	0	0
$211$	1.63915e10	0.569309	0.284654	−	0.958630i	$-0.408121\pi$
$211$	1.63915e10	0.569309	0.284654	+	0.958630i	$0.408121\pi$
$212$	0	0
$213$	− 2.80675e10i	− 0.934321i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.08800e10i	0.333090i
$218$	0	0
$219$	−4.30037e10	−1.26330
$220$	0	0
$221$	9.78161e10	2.75832
$222$	0	0
$223$	4.65257e10i	1.25986i	0.776654	+	0.629928i	$0.216916\pi$
$223$	4.65257e10i	1.25986i	−0.776654	+	0.629928i	$0.783084\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.91415e9i	0.0478475i	0.999714	+	0.0239237i	$0.00761589\pi$
$227$	1.91415e9i	0.0478475i	−0.999714	+	0.0239237i	$0.992384\pi$
$228$	0	0
$229$	1.45825e10	0.350406	0.175203	−	0.984532i	$-0.443942\pi$
$229$	1.45825e10	0.350406	0.175203	+	0.984532i	$0.443942\pi$
$230$	0	0
$231$	−2.34978e10	−0.542966
$232$	0	0
$233$	− 4.12790e10i	− 0.917545i	−0.888554	−	0.458773i	$-0.848289\pi$
$233$	− 4.12790e10i	− 0.917545i	0.888554	−	0.458773i	$-0.151711\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.01502e11i	− 2.08981i
$238$	0	0
$239$	3.65502e10	0.724602	0.362301	−	0.932061i	$-0.381991\pi$
$239$	3.65502e10	0.724602	0.362301	+	0.932061i	$0.381991\pi$
$240$	0	0
$241$	−8.66070e10	−1.65377	−0.826887	−	0.562368i	$-0.809890\pi$
$241$	−8.66070e10	−1.65377	−0.826887	+	0.562368i	$0.809890\pi$
$242$	0	0
$243$	− 5.30339e10i	− 0.975720i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.86398e10i	− 0.831488i
$248$	0	0
$249$	2.53549e9	0.0417989
$250$	0	0
$251$	7.20769e10	1.14621	0.573105	−	0.819482i	$-0.305739\pi$
$251$	7.20769e10	1.14621	0.573105	+	0.819482i	$0.305739\pi$
$252$	0	0
$253$	− 7.37711e10i	− 1.13199i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.12729e11i	− 1.61190i	−0.591986	−	0.805948i	$-0.701656\pi$
$257$	− 1.12729e11i	− 1.61190i	0.591986	−	0.805948i	$-0.298344\pi$
$258$	0	0
$259$	5.04947e10	0.697263
$260$	0	0
$261$	−3.94374e10	−0.526049
$262$	0	0
$263$	5.55225e10i	0.715596i	0.933799	+	0.357798i	$0.116472\pi$
$263$	5.55225e10i	0.715596i	−0.933799	+	0.357798i	$0.883528\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 8.18033e10i	− 0.985076i
$268$	0	0
$269$	−2.79726e10	−0.325722	−0.162861	−	0.986649i	$-0.552072\pi$
$269$	−2.79726e10	−0.325722	−0.162861	+	0.986649i	$0.552072\pi$
$270$	0	0
$271$	−3.32884e10	−0.374914	−0.187457	−	0.982273i	$-0.560025\pi$
$271$	−3.32884e10	−0.374914	−0.187457	+	0.982273i	$0.560025\pi$
$272$	0	0
$273$	1.33282e11i	1.45225i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.46240e10i	0.557474i	0.960367	+	0.278737i	$0.0899158\pi$
$277$	5.46240e10i	0.557474i	−0.960367	+	0.278737i	$0.910084\pi$
$278$	0	0
$279$	2.47428e10	0.244473
$280$	0	0
$281$	−8.37818e10	−0.801625	−0.400813	−	0.916160i	$-0.631272\pi$
$281$	−8.37818e10	−0.801625	−0.400813	+	0.916160i	$0.631272\pi$
$282$	0	0
$283$	− 8.36086e10i	− 0.774840i	−0.921903	−	0.387420i	$-0.873366\pi$
$283$	− 8.36086e10i	− 0.774840i	0.921903	−	0.387420i	$-0.126634\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.00584e11i	− 0.875107i
$288$	0	0
$289$	−2.35225e11	−1.98355
$290$	0	0
$291$	2.05039e10	0.167617
$292$	0	0
$293$	2.28547e11i	1.81164i	0.423666	+	0.905819i	$0.360743\pi$
$293$	2.28547e11i	1.81164i	−0.423666	+	0.905819i	$0.639257\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.58555e10i	− 0.341969i
$298$	0	0
$299$	−4.18438e11	−3.02769
$300$	0	0
$301$	5.04568e10	0.354300
$302$	0	0
$303$	− 1.49801e10i	− 0.102100i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.95064e11i	− 1.25330i	−0.779302	−	0.626648i	$-0.784426\pi$
$307$	− 1.95064e11i	− 1.25330i	0.779302	−	0.626648i	$-0.215574\pi$
$308$	0	0
$309$	−3.35598e11	−2.09414
$310$	0	0
$311$	−2.15637e11	−1.30708	−0.653540	−	0.756892i	$-0.726717\pi$
$311$	−2.15637e11	−1.30708	−0.653540	+	0.756892i	$0.726717\pi$
$312$	0	0
$313$	− 1.91755e11i	− 1.12927i	−0.825341	−	0.564635i	$-0.809017\pi$
$313$	− 1.91755e11i	− 1.12927i	0.825341	−	0.564635i	$-0.190983\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.38886e11i	− 1.88489i	−0.334358	−	0.942446i	$-0.608519\pi$
$317$	− 3.38886e11i	− 1.88489i	0.334358	−	0.942446i	$-0.391481\pi$
$318$	0	0
$319$	−1.07936e11	−0.583592
$320$	0	0
$321$	2.43051e11	1.27769
$322$	0	0
$323$	1.75936e11i	0.899383i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.05153e11i	0.508577i
$328$	0	0
$329$	−2.40918e10	−0.113367
$330$	0	0
$331$	1.78427e11	0.817022	0.408511	−	0.912753i	$-0.366048\pi$
$331$	1.78427e11	0.817022	0.408511	+	0.912753i	$0.366048\pi$
$332$	0	0
$333$	− 1.14833e11i	− 0.511760i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.64693e11i	1.11791i	0.829196	+	0.558957i	$0.188798\pi$
$337$	2.64693e11i	1.11791i	−0.829196	+	0.558957i	$0.811202\pi$
$338$	0	0
$339$	−2.93330e11	−1.20631
$340$	0	0
$341$	6.77187e10	0.271215
$342$	0	0
$343$	− 2.74870e11i	− 1.07227i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.11912e10i	0.300626i	0.988638	+	0.150313i	$0.0480281\pi$
$347$	8.11912e10i	0.300626i	−0.988638	+	0.150313i	$0.951972\pi$
$348$	0	0
$349$	2.26688e11	0.817926	0.408963	−	0.912551i	$-0.365890\pi$
$349$	2.26688e11	0.817926	0.408963	+	0.912551i	$0.365890\pi$
$350$	0	0
$351$	−2.60098e11	−0.914649
$352$	0	0
$353$	− 3.28733e10i	− 0.112683i	−0.998412	−	0.0563413i	$-0.982056\pi$
$353$	− 3.28733e10i	− 0.112683i	0.998412	−	0.0563413i	$-0.0179435\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.82098e11i	− 1.57083i
$358$	0	0
$359$	−3.12096e11	−0.991661	−0.495831	−	0.868419i	$-0.665136\pi$
$359$	−3.12096e11	−0.991661	−0.495831	+	0.868419i	$0.665136\pi$
$360$	0	0
$361$	−2.35202e11	−0.728884
$362$	0	0
$363$	− 2.64030e11i	− 0.798129i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6.28209e11i	− 1.80762i	−0.427934	−	0.903810i	$-0.640759\pi$
$367$	− 6.28209e11i	− 1.80762i	0.427934	−	0.903810i	$-0.359241\pi$
$368$	0	0
$369$	−2.28744e11	−0.642289
$370$	0	0
$371$	−4.57321e11	−1.25325
$372$	0	0
$373$	9.84770e10i	0.263418i	0.991288	+	0.131709i	$0.0420464\pi$
$373$	9.84770e10i	0.263418i	−0.991288	+	0.131709i	$0.957954\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.12228e11i	1.56091i
$378$	0	0
$379$	2.89024e11	0.719544	0.359772	−	0.933040i	$-0.382855\pi$
$379$	2.89024e11	0.719544	0.359772	+	0.933040i	$0.382855\pi$
$380$	0	0
$381$	6.45046e11	1.56830
$382$	0	0
$383$	2.01350e11i	0.478141i	0.971002	+	0.239071i	$0.0768427\pi$
$383$	2.01350e11i	0.478141i	−0.971002	+	0.239071i	$0.923157\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.14746e11i	− 0.260040i
$388$	0	0
$389$	−3.93769e11	−0.871904	−0.435952	−	0.899970i	$-0.643588\pi$
$389$	−3.93769e11	−0.871904	−0.435952	+	0.899970i	$0.643588\pi$
$390$	0	0
$391$	1.51354e12	3.27491
$392$	0	0
$393$	− 4.42099e11i	− 0.934875i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.15625e11i	− 1.04178i	−0.853623	−	0.520891i	$-0.825600\pi$
$397$	− 5.15625e11i	− 1.04178i	0.853623	−	0.520891i	$-0.174400\pi$
$398$	0	0
$399$	−2.39727e11	−0.473521
$400$	0	0
$401$	−8.43309e11	−1.62869	−0.814343	−	0.580384i	$-0.802902\pi$
$401$	−8.43309e11	−1.62869	−0.814343	+	0.580384i	$0.802902\pi$
$402$	0	0
$403$	− 3.84108e11i	− 0.725406i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.14285e11i	− 0.567740i
$408$	0	0
$409$	−5.64549e11	−0.997577	−0.498789	−	0.866724i	$-0.666222\pi$
$409$	−5.64549e11	−0.997577	−0.498789	+	0.866724i	$0.666222\pi$
$410$	0	0
$411$	2.00778e11	0.347078
$412$	0	0
$413$	− 7.52866e10i	− 0.127333i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 9.79134e11i	− 1.58573i
$418$	0	0
$419$	6.92475e11	1.09759	0.548796	−	0.835956i	$-0.315086\pi$
$419$	6.92475e11	1.09759	0.548796	+	0.835956i	$0.315086\pi$
$420$	0	0
$421$	4.08125e11	0.633176	0.316588	−	0.948563i	$-0.397463\pi$
$421$	4.08125e11	0.633176	0.316588	+	0.948563i	$0.397463\pi$
$422$	0	0
$423$	5.47885e10i	0.0832065i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.04617e11i	0.297863i
$428$	0	0
$429$	8.29566e11	1.18248
$430$	0	0
$431$	−4.41055e11	−0.615665	−0.307833	−	0.951441i	$-0.599604\pi$
$431$	−4.41055e11	−0.615665	−0.307833	+	0.951441i	$0.599604\pi$
$432$	0	0
$433$	7.15390e10i	0.0978019i	0.998804	+	0.0489009i	$0.0155719\pi$
$433$	7.15390e10i	0.0978019i	−0.998804	+	0.0489009i	$0.984428\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.52622e11i	− 0.987212i
$438$	0	0
$439$	4.74967e11	0.610342	0.305171	−	0.952298i	$-0.401286\pi$
$439$	4.74967e11	0.610342	0.305171	+	0.952298i	$0.401286\pi$
$440$	0	0
$441$	−1.97630e11	−0.248816
$442$	0	0
$443$	− 1.97072e11i	− 0.243113i	−0.992585	−	0.121556i	$-0.961211\pi$
$443$	− 1.97072e11i	− 0.243113i	0.992585	−	0.121556i	$-0.0387885\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.71817e11i	0.440499i
$448$	0	0
$449$	6.20156e11	0.720100	0.360050	−	0.932933i	$-0.382760\pi$
$449$	6.20156e11	0.720100	0.360050	+	0.932933i	$0.382760\pi$
$450$	0	0
$451$	−6.26049e11	−0.712548
$452$	0	0
$453$	1.68348e9i	0.00187830i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.88006e11i	0.845097i	0.906340	+	0.422549i	$0.138864\pi$
$457$	7.88006e11i	0.845097i	−0.906340	+	0.422549i	$0.861136\pi$
$458$	0	0
$459$	9.40806e11	0.989334
$460$	0	0
$461$	1.53496e10	0.0158286	0.00791429	−	0.999969i	$-0.497481\pi$
$461$	1.53496e10	0.0158286	0.00791429	+	0.999969i	$0.497481\pi$
$462$	0	0
$463$	1.94494e11i	0.196694i	0.995152	+	0.0983471i	$0.0313555\pi$
$463$	1.94494e11i	0.196694i	−0.995152	+	0.0983471i	$0.968644\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.06503e11i	− 0.103618i	−0.998657	−	0.0518089i	$-0.983501\pi$
$467$	− 1.06503e11i	− 0.103618i	0.998657	−	0.0518089i	$-0.0164987\pi$
$468$	0	0
$469$	3.79894e11	0.362564
$470$	0	0
$471$	1.19797e11	0.112163
$472$	0	0
$473$	− 3.14050e11i	− 0.288485i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.04002e12i	0.919831i
$478$	0	0
$479$	−8.31146e11	−0.721386	−0.360693	−	0.932685i	$-0.617460\pi$
$479$	−8.31146e11	−0.721386	−0.360693	+	0.932685i	$0.617460\pi$
$480$	0	0
$481$	−1.78266e12	−1.51851
$482$	0	0
$483$	2.06232e12i	1.72423i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.38566e12i	1.11629i	0.829743	+	0.558146i	$0.188487\pi$
$487$	1.38566e12i	1.11629i	−0.829743	+	0.558146i	$0.811513\pi$
$488$	0	0
$489$	2.48962e12	1.96899
$490$	0	0
$491$	−3.92804e11	−0.305007	−0.152503	−	0.988303i	$-0.548734\pi$
$491$	−3.92804e11	−0.305007	−0.152503	+	0.988303i	$0.548734\pi$
$492$	0	0
$493$	− 2.21450e12i	− 1.68836i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.51371e11i	0.552397i
$498$	0	0
$499$	−7.52029e11	−0.542978	−0.271489	−	0.962442i	$-0.587516\pi$
$499$	−7.52029e11	−0.542978	−0.271489	+	0.962442i	$0.587516\pi$
$500$	0	0
$501$	1.73780e12	1.23234
$502$	0	0
$503$	− 1.83429e12i	− 1.27765i	−0.769351	−	0.638826i	$-0.779420\pi$
$503$	− 1.83429e12i	− 1.27765i	0.769351	−	0.638826i	$-0.220580\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.86021e12i	− 1.92248i
$508$	0	0
$509$	6.19864e11	0.409323	0.204662	−	0.978833i	$-0.434391\pi$
$509$	6.19864e11	0.409323	0.204662	+	0.978833i	$0.434391\pi$
$510$	0	0
$511$	1.15122e12	0.746900
$512$	0	0
$513$	− 4.67823e11i	− 0.298232i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.49951e11i	0.0923083i
$518$	0	0
$519$	6.11428e11	0.369907
$520$	0	0
$521$	5.25683e11	0.312575	0.156287	−	0.987712i	$-0.450047\pi$
$521$	5.25683e11	0.312575	0.156287	+	0.987712i	$0.450047\pi$
$522$	0	0
$523$	1.68426e11i	0.0984353i	0.998788	+	0.0492177i	$0.0156728\pi$
$523$	1.68426e11i	0.0984353i	−0.998788	+	0.0492177i	$0.984327\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.38937e12i	0.784639i
$528$	0	0
$529$	−4.67350e12	−2.59473
$530$	0	0
$531$	−1.71213e11	−0.0934570
$532$	0	0
$533$	3.55102e12i	1.90582i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.07934e11i	0.107905i
$538$	0	0
$539$	−5.40893e11	−0.276034
$540$	0	0
$541$	2.20612e12	1.10724	0.553620	−	0.832769i	$-0.313246\pi$
$541$	2.20612e12	1.10724	0.553620	+	0.832769i	$0.313246\pi$
$542$	0	0
$543$	− 1.58697e12i	− 0.783375i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.51639e12i	1.67940i	0.543049	+	0.839701i	$0.317270\pi$
$547$	3.51639e12i	1.67940i	−0.543049	+	0.839701i	$0.682730\pi$
$548$	0	0
$549$	4.65331e11	0.218618
$550$	0	0
$551$	−1.10118e12	−0.508951
$552$	0	0
$553$	2.71722e12i	1.23556i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.29996e11i	0.277325i	0.990340	+	0.138663i	$0.0442803\pi$
$557$	6.29996e11i	0.277325i	−0.990340	+	0.138663i	$0.955720\pi$
$558$	0	0
$559$	−1.78133e12	−0.771597
$560$	0	0
$561$	−3.00064e12	−1.27903
$562$	0	0
$563$	4.02619e11i	0.168891i	0.996428	+	0.0844455i	$0.0269119\pi$
$563$	4.02619e11i	0.168891i	−0.996428	+	0.0844455i	$0.973088\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.25313e12i	0.915507i
$568$	0	0
$569$	−2.55482e12	−1.02177	−0.510887	−	0.859648i	$-0.670683\pi$
$569$	−2.55482e12	−1.02177	−0.510887	+	0.859648i	$0.670683\pi$
$570$	0	0
$571$	−1.16243e12	−0.457620	−0.228810	−	0.973471i	$-0.573484\pi$
$571$	−1.16243e12	−0.457620	−0.228810	+	0.973471i	$0.573484\pi$
$572$	0	0
$573$	1.63113e12i	0.632110i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.66332e12i	1.75148i	0.482787	+	0.875738i	$0.339625\pi$
$577$	4.66332e12i	1.75148i	−0.482787	+	0.875738i	$0.660375\pi$
$578$	0	0
$579$	4.17601e12	1.54421
$580$	0	0
$581$	−6.78754e10	−0.0247127
$582$	0	0
$583$	2.84643e12i	1.02045i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.88422e12i	− 1.35030i	−0.737678	−	0.675152i	$-0.764078\pi$
$587$	− 3.88422e12i	− 1.35030i	0.737678	−	0.675152i	$-0.235922\pi$
$588$	0	0
$589$	6.90875e11	0.236527
$590$	0	0
$591$	9.67655e10	0.0326270
$592$	0	0
$593$	2.01341e12i	0.668631i	0.942461	+	0.334315i	$0.108505\pi$
$593$	2.01341e12i	0.668631i	−0.942461	+	0.334315i	$0.891495\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4.37184e12i	− 1.40857i
$598$	0	0
$599$	2.10578e12	0.668333	0.334166	−	0.942514i	$-0.391545\pi$
$599$	2.10578e12	0.668333	0.334166	+	0.942514i	$0.391545\pi$
$600$	0	0
$601$	4.74058e12	1.48216	0.741082	−	0.671415i	$-0.234313\pi$
$601$	4.74058e12	1.48216	0.741082	+	0.671415i	$0.234313\pi$
$602$	0	0
$603$	− 8.63938e11i	− 0.266106i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.01476e12i	1.20036i	0.799866	+	0.600179i	$0.204904\pi$
$607$	4.01476e12i	1.20036i	−0.799866	+	0.600179i	$0.795096\pi$
$608$	0	0
$609$	3.01744e12	0.888915
$610$	0	0
$611$	8.50537e11	0.246893
$612$	0	0
$613$	− 1.30399e12i	− 0.372993i	−0.982456	−	0.186496i	$-0.940287\pi$
$613$	− 1.30399e12i	− 0.372993i	0.982456	−	0.186496i	$-0.0597133\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.10164e12i	1.41719i	0.705618	+	0.708593i	$0.250670\pi$
$617$	5.10164e12i	1.41719i	−0.705618	+	0.708593i	$0.749330\pi$
$618$	0	0
$619$	−4.50630e11	−0.123371	−0.0616854	−	0.998096i	$-0.519648\pi$
$619$	−4.50630e11	−0.123371	−0.0616854	+	0.998096i	$0.519648\pi$
$620$	0	0
$621$	−4.02459e12	−1.08595
$622$	0	0
$623$	2.18988e12i	0.582404i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.49209e12i	0.385560i
$628$	0	0
$629$	6.44812e12	1.64250
$630$	0	0
$631$	−7.97105e10	−0.0200163	−0.0100081	−	0.999950i	$-0.503186\pi$
$631$	−7.97105e10	−0.0200163	−0.0100081	+	0.999950i	$0.503186\pi$
$632$	0	0
$633$	2.85212e12i	0.706076i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.06801e12i	0.738294i
$638$	0	0
$639$	1.70873e12	0.405434
$640$	0	0
$641$	−3.12648e12	−0.731468	−0.365734	−	0.930719i	$-0.619182\pi$
$641$	−3.12648e12	−0.731468	−0.365734	+	0.930719i	$0.619182\pi$
$642$	0	0
$643$	− 5.94982e12i	− 1.37263i	−0.727303	−	0.686316i	$-0.759227\pi$
$643$	− 5.94982e12i	− 1.37263i	0.727303	−	0.686316i	$-0.240773\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.56187e12i	− 0.799115i	−0.916708	−	0.399557i	$-0.869164\pi$
$647$	− 3.56187e12i	− 0.799115i	0.916708	−	0.399557i	$-0.130836\pi$
$648$	0	0
$649$	−4.68594e11	−0.103680
$650$	0	0
$651$	−1.89312e12	−0.413109
$652$	0	0
$653$	1.37883e11i	0.0296757i	0.999890	+	0.0148379i	$0.00472321\pi$
$653$	1.37883e11i	0.0296757i	−0.999890	+	0.0148379i	$0.995277\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.61804e12i	− 0.548191i
$658$	0	0
$659$	−4.16654e12	−0.860581	−0.430290	−	0.902691i	$-0.641589\pi$
$659$	−4.16654e12	−0.860581	−0.430290	+	0.902691i	$0.641589\pi$
$660$	0	0
$661$	−3.29083e12	−0.670500	−0.335250	−	0.942129i	$-0.608821\pi$
$661$	−3.29083e12	−0.670500	−0.335250	+	0.942129i	$0.608821\pi$
$662$	0	0
$663$	1.70200e13i	3.42097i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.47322e12i	1.85324i
$668$	0	0
$669$	−8.09547e12	−1.56252
$670$	0	0
$671$	1.27357e12	0.242532
$672$	0	0
$673$	− 5.74732e12i	− 1.07993i	−0.841686	−	0.539967i	$-0.818437\pi$
$673$	− 5.74732e12i	− 1.07993i	0.841686	−	0.539967i	$-0.181563\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.31245e12i	1.70379i	0.523716	+	0.851893i	$0.324545\pi$
$677$	9.31245e12i	1.70379i	−0.523716	+	0.851893i	$0.675455\pi$
$678$	0	0
$679$	−5.48892e11	−0.0990998
$680$	0	0
$681$	−3.33062e11	−0.0593421
$682$	0	0
$683$	9.19967e12i	1.61763i	0.588063	+	0.808815i	$0.299891\pi$
$683$	9.19967e12i	1.61763i	−0.588063	+	0.808815i	$0.700109\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.53735e12i	0.434585i
$688$	0	0
$689$	1.61453e13	2.72934
$690$	0	0
$691$	−1.03125e13	−1.72074	−0.860369	−	0.509672i	$-0.829767\pi$
$691$	−1.03125e13	−1.72074	−0.860369	+	0.509672i	$0.829767\pi$
$692$	0	0
$693$	− 1.43053e12i	− 0.235612i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.28445e13i	− 2.06144i
$698$	0	0
$699$	7.18255e12	1.13797
$700$	0	0
$701$	−9.52566e12	−1.48992	−0.744961	−	0.667108i	$-0.767532\pi$
$701$	−9.52566e12	−1.48992	−0.744961	+	0.667108i	$0.767532\pi$
$702$	0	0
$703$	− 3.20638e12i	− 0.495126i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.01020e11i	0.0603641i
$708$	0	0
$709$	8.38537e12	1.24628	0.623138	−	0.782112i	$-0.285858\pi$
$709$	8.38537e12	1.24628	0.623138	+	0.782112i	$0.285858\pi$
$710$	0	0
$711$	6.17938e12	0.906842
$712$	0	0
$713$	− 5.94345e12i	− 0.861263i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.35974e12i	0.898676i
$718$	0	0
$719$	6.94013e12	0.968473	0.484236	−	0.874937i	$-0.339097\pi$
$719$	6.94013e12	0.968473	0.484236	+	0.874937i	$0.339097\pi$
$720$	0	0
$721$	8.98398e12	1.23811
$722$	0	0
$723$	− 1.50696e13i	− 2.05107i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 7.74984e12i	− 1.02893i	−0.857510	−	0.514467i	$-0.827990\pi$
$727$	− 7.74984e12i	− 1.02893i	0.857510	−	0.514467i	$-0.172010\pi$
$728$	0	0
$729$	−2.92986e11	−0.0384214
$730$	0	0
$731$	6.44329e12	0.834602
$732$	0	0
$733$	− 4.12555e12i	− 0.527854i	−0.964543	−	0.263927i	$-0.914982\pi$
$733$	− 4.12555e12i	− 0.527854i	0.964543	−	0.263927i	$-0.0850178\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.36451e12i	− 0.295215i
$738$	0	0
$739$	3.33931e12	0.411867	0.205933	−	0.978566i	$-0.433977\pi$
$739$	3.33931e12	0.411867	0.205933	+	0.978566i	$0.433977\pi$
$740$	0	0
$741$	8.46333e12	1.03124
$742$	0	0
$743$	− 1.26471e13i	− 1.52245i	−0.648489	−	0.761224i	$-0.724599\pi$
$743$	− 1.26471e13i	− 1.52245i	0.648489	−	0.761224i	$-0.275401\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.54359e11i	0.0181380i
$748$	0	0
$749$	−6.50651e12	−0.755405
$750$	0	0
$751$	−1.56552e12	−0.179588	−0.0897941	−	0.995960i	$-0.528621\pi$
$751$	−1.56552e12	−0.179588	−0.0897941	+	0.995960i	$0.528621\pi$
$752$	0	0
$753$	1.25414e13i	1.42157i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.54781e11i	0.105675i	0.998603	+	0.0528375i	$0.0168265\pi$
$757$	9.54781e11i	0.105675i	−0.998603	+	0.0528375i	$0.983173\pi$
$758$	0	0
$759$	1.28362e13	1.40394
$760$	0	0
$761$	5.27330e12	0.569969	0.284985	−	0.958532i	$-0.408012\pi$
$761$	5.27330e12	0.569969	0.284985	+	0.958532i	$0.408012\pi$
$762$	0	0
$763$	− 2.81496e12i	− 0.300685i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.65792e12i	0.277308i
$768$	0	0
$769$	6.96923e12	0.718648	0.359324	−	0.933213i	$-0.383007\pi$
$769$	6.96923e12	0.718648	0.359324	+	0.933213i	$0.383007\pi$
$770$	0	0
$771$	1.96149e13	1.99913
$772$	0	0
$773$	3.79383e12i	0.382182i	0.981572	+	0.191091i	$0.0612025\pi$
$773$	3.79383e12i	0.382182i	−0.981572	+	0.191091i	$0.938797\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.78607e12i	0.864769i
$778$	0	0
$779$	−6.38703e12	−0.621413
$780$	0	0
$781$	4.67663e12	0.449784
$782$	0	0
$783$	5.88847e12i	0.559854i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.05939e13i	1.91361i	0.290736	+	0.956803i	$0.406100\pi$
$787$	2.05939e13i	1.91361i	−0.290736	+	0.956803i	$0.593900\pi$
$788$	0	0
$789$	−9.66092e12	−0.887507
$790$	0	0
$791$	7.85247e12	0.713201
$792$	0	0
$793$	− 7.22381e12i	− 0.648690i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.45507e12i	0.127738i	0.997958	+	0.0638690i	$0.0203440\pi$
$797$	1.45507e12i	0.127738i	−0.997958	+	0.0638690i	$0.979656\pi$
$798$	0	0
$799$	−3.07650e12	−0.267052
$800$	0	0
$801$	4.98013e12	0.427459
$802$	0	0
$803$	− 7.16531e12i	− 0.608156i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 4.86722e12i	− 0.403971i
$808$	0	0
$809$	−1.31749e13	−1.08138	−0.540690	−	0.841222i	$-0.681837\pi$
$809$	−1.31749e13	−1.08138	−0.540690	+	0.841222i	$0.681837\pi$
$810$	0	0
$811$	−9.97833e12	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$811$	−9.97833e12	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$812$	0	0
$813$	− 5.79219e12i	− 0.464981i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.20398e12i	− 0.251588i
$818$	0	0
$819$	−8.11413e12	−0.630179
$820$	0	0
$821$	−1.80913e13	−1.38972	−0.694859	−	0.719146i	$-0.744533\pi$
$821$	−1.80913e13	−1.38972	−0.694859	+	0.719146i	$0.744533\pi$
$822$	0	0
$823$	1.39801e12i	0.106221i	0.998589	+	0.0531107i	$0.0169136\pi$
$823$	1.39801e12i	0.106221i	−0.998589	+	0.0531107i	$0.983086\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.26201e13i	− 1.68159i	−0.541354	−	0.840794i	$-0.682088\pi$
$827$	− 2.26201e13i	− 1.68159i	0.541354	−	0.840794i	$-0.317912\pi$
$828$	0	0
$829$	−1.72646e13	−1.26959	−0.634793	−	0.772682i	$-0.718915\pi$
$829$	−1.72646e13	−1.26959	−0.634793	+	0.772682i	$0.718915\pi$
$830$	0	0
$831$	−9.50458e12	−0.691398
$832$	0	0
$833$	− 1.10974e13i	− 0.798579i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.69440e12i	− 0.260183i
$838$	0	0
$839$	−1.76607e12	−0.123049	−0.0615245	−	0.998106i	$-0.519596\pi$
$839$	−1.76607e12	−0.123049	−0.0615245	+	0.998106i	$0.519596\pi$
$840$	0	0
$841$	−6.46640e11	−0.0445739
$842$	0	0
$843$	− 1.45780e13i	− 0.994203i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.06810e12i	0.471876i
$848$	0	0
$849$	1.45479e13	0.960983
$850$	0	0
$851$	−2.75838e13	−1.80290
$852$	0	0
$853$	2.37844e13i	1.53823i	0.639109	+	0.769116i	$0.279303\pi$
$853$	2.37844e13i	1.53823i	−0.639109	+	0.769116i	$0.720697\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.61941e12i	0.482511i	0.970462	+	0.241256i	$0.0775593\pi$
$857$	7.61941e12i	0.482511i	−0.970462	+	0.241256i	$0.922441\pi$
$858$	0	0
$859$	−1.27695e13	−0.800212	−0.400106	−	0.916469i	$-0.631027\pi$
$859$	−1.27695e13	−0.800212	−0.400106	+	0.916469i	$0.631027\pi$
$860$	0	0
$861$	1.75017e13	1.08534
$862$	0	0
$863$	2.63038e13i	1.61425i	0.590382	+	0.807124i	$0.298977\pi$
$863$	2.63038e13i	1.61425i	−0.590382	+	0.807124i	$0.701023\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 4.09292e13i	− 2.46007i
$868$	0	0
$869$	1.69124e13	1.00604
$870$	0	0
$871$	−1.34118e13	−0.789596
$872$	0	0
$873$	1.24826e12i	0.0727347i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.76832e11i	0.0272187i	0.999907	+	0.0136093i	$0.00433212\pi$
$877$	4.76832e11i	0.0272187i	−0.999907	+	0.0136093i	$0.995668\pi$
$878$	0	0
$879$	−3.97672e13	−2.24685
$880$	0	0
$881$	1.74839e13	0.977791	0.488896	−	0.872342i	$-0.337400\pi$
$881$	1.74839e13	0.977791	0.488896	+	0.872342i	$0.337400\pi$
$882$	0	0
$883$	− 6.48321e12i	− 0.358894i	−0.983768	−	0.179447i	$-0.942569\pi$
$883$	− 6.48321e12i	− 0.358894i	0.983768	−	0.179447i	$-0.0574309\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.92450e12i	0.538335i	0.963093	+	0.269167i	$0.0867485\pi$
$887$	9.92450e12i	0.538335i	−0.963093	+	0.269167i	$0.913252\pi$
$888$	0	0
$889$	−1.72679e13	−0.927220
$890$	0	0
$891$	1.40237e13	0.745443
$892$	0	0
$893$	1.52981e12i	0.0805021i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 7.28083e13i	− 3.75504i
$898$	0	0
$899$	−8.69601e12	−0.444019
$900$	0	0
$901$	−5.83994e13	−2.95221
$902$	0	0
$903$	8.77949e12i	0.439414i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.40768e12i	− 0.0690671i	−0.999404	−	0.0345335i	$-0.989005\pi$
$907$	− 1.40768e12i	− 0.0690671i	0.999404	−	0.0345335i	$-0.0109946\pi$
$908$	0	0
$909$	9.11980e11	0.0443045
$910$	0	0
$911$	−6.91343e12	−0.332553	−0.166277	−	0.986079i	$-0.553174\pi$
$911$	−6.91343e12	−0.332553	−0.166277	+	0.986079i	$0.553174\pi$
$912$	0	0
$913$	4.22465e11i	0.0201221i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.18351e13i	0.552724i
$918$	0	0
$919$	−8.50189e12	−0.393184	−0.196592	−	0.980485i	$-0.562987\pi$
$919$	−8.50189e12	−0.393184	−0.196592	+	0.980485i	$0.562987\pi$
$920$	0	0
$921$	3.39411e13	1.55438
$922$	0	0
$923$	− 2.65264e13i	− 1.20301i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.04309e13i	− 0.908719i
$928$	0	0
$929$	−8.90639e12	−0.392311	−0.196156	−	0.980573i	$-0.562846\pi$
$929$	−8.90639e12	−0.392311	−0.196156	+	0.980573i	$0.562846\pi$
$930$	0	0
$931$	−5.51826e12	−0.240729
$932$	0	0
$933$	− 3.75209e13i	− 1.62109i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.40064e12i	0.313647i	0.987627	+	0.156824i	$0.0501254\pi$
$937$	7.40064e12i	0.313647i	−0.987627	+	0.156824i	$0.949875\pi$
$938$	0	0
$939$	3.33654e13	1.40056
$940$	0	0
$941$	−8.13867e12	−0.338377	−0.169188	−	0.985584i	$-0.554115\pi$
$941$	−8.13867e12	−0.338377	−0.169188	+	0.985584i	$0.554115\pi$
$942$	0	0
$943$	5.49463e13i	2.26275i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.16975e13i	0.472627i	0.971677	+	0.236313i	$0.0759392\pi$
$947$	1.16975e13i	0.472627i	−0.971677	+	0.236313i	$0.924061\pi$
$948$	0	0
$949$	−4.06425e13	−1.62661
$950$	0	0
$951$	5.89661e13	2.33771
$952$	0	0
$953$	2.50410e13i	0.983406i	0.870763	+	0.491703i	$0.163625\pi$
$953$	2.50410e13i	0.983406i	−0.870763	+	0.491703i	$0.836375\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.87809e13i	− 0.723791i
$958$	0	0
$959$	−5.37485e12	−0.205202
$960$	0	0
$961$	−2.09838e13	−0.793649
$962$	0	0
$963$	1.47968e13i	0.554433i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.68904e13i	− 0.988960i	−0.869189	−	0.494480i	$-0.835359\pi$
$967$	− 2.68904e13i	− 0.988960i	0.869189	−	0.494480i	$-0.164641\pi$
$968$	0	0
$969$	−3.06129e13	−1.11544
$970$	0	0
$971$	−3.28780e13	−1.18691	−0.593456	−	0.804866i	$-0.702237\pi$
$971$	−3.28780e13	−1.18691	−0.593456	+	0.804866i	$0.702237\pi$
$972$	0	0
$973$	2.62115e13i	0.937529i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 8.13505e11i	− 0.0285650i	−0.999898	−	0.0142825i	$-0.995454\pi$
$977$	− 8.13505e11i	− 0.0285650i	0.999898	−	0.0142825i	$-0.00454642\pi$
$978$	0	0
$979$	1.36301e13	0.474217
$980$	0	0
$981$	−6.40164e12	−0.220689
$982$	0	0
$983$	− 4.13240e13i	− 1.41160i	−0.708412	−	0.705800i	$-0.750588\pi$
$983$	− 4.13240e13i	− 1.41160i	0.708412	−	0.705800i	$-0.249412\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.19198e12i	− 0.140602i
$988$	0	0
$989$	−2.75631e13	−0.916105
$990$	0	0
$991$	3.61227e13	1.18973	0.594866	−	0.803825i	$-0.297205\pi$
$991$	3.61227e13	1.18973	0.594866	+	0.803825i	$0.297205\pi$
$992$	0	0
$993$	3.10462e13i	1.01330i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 3.07166e13i	− 0.984565i	−0.870435	−	0.492283i	$-0.836163\pi$
$997$	− 3.07166e13i	− 0.984565i	0.870435	−	0.492283i	$-0.163837\pi$
$998$	0	0
$999$	−1.71459e13	−0.544646

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.10.c.c.49.2		2
4.3	odd	2		50.10.b.d.49.2		2
5.2	odd	4		400.10.a.j.1.1		1
5.3	odd	4		80.10.a.a.1.1		1
5.4	even	2	inner	400.10.c.c.49.1		2
20.3	even	4		10.10.a.c.1.1	✓	1
20.7	even	4		50.10.a.a.1.1		1
20.19	odd	2		50.10.b.d.49.1		2
40.3	even	4		320.10.a.b.1.1		1
40.13	odd	4		320.10.a.i.1.1		1
60.23	odd	4		90.10.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.10.a.c.1.1	✓	1	20.3	even	4
50.10.a.a.1.1		1	20.7	even	4
50.10.b.d.49.1		2	20.19	odd	2
50.10.b.d.49.2		2	4.3	odd	2
80.10.a.a.1.1		1	5.3	odd	4
90.10.a.e.1.1		1	60.23	odd	4
320.10.a.b.1.1		1	40.3	even	4
320.10.a.i.1.1		1	40.13	odd	4
400.10.a.j.1.1		1	5.2	odd	4
400.10.c.c.49.1		2	5.4	even	2	inner
400.10.c.c.49.2		2	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+174.000i q^{3} -4658.00i q^{7} -10593.0 q^{9} +O(q^{10})\)	\(q+174.000i q^{3} -4658.00i q^{7} -10593.0 q^{9} -28992.0 q^{11} +164446. i q^{13} -594822. i q^{17} -295780. q^{19} +810492. q^{21} +2.54453e6i q^{23} +1.58166e6i q^{27} +3.72297e6 q^{29} -2.33577e6 q^{31} -5.04461e6i q^{33} +1.08404e7i q^{37} -2.86136e7 q^{39} +2.15939e7 q^{41} +1.08323e7i q^{43} -5.17214e6i q^{47} +1.86566e7 q^{49} +1.03499e8 q^{51} -9.81797e7i q^{53} -5.14657e7i q^{57} +1.61629e7 q^{59} -4.39282e7 q^{61} +4.93422e7i q^{63} +8.15574e7i q^{67} -4.42749e8 q^{69} -1.61308e8 q^{71} +2.47148e8i q^{73} +1.35045e8i q^{77} -5.83346e8 q^{79} -4.83711e8 q^{81} -1.45718e7i q^{83} +6.47797e8i q^{87} -4.70134e8 q^{89} +7.65989e8 q^{91} -4.06424e8i q^{93} -1.17838e8i q^{97} +3.07112e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 21186 q^{9}+O(q^{10}) \) 2 * q - 21186 * q^9	\( 2 q - 21186 q^{9} - 57984 q^{11} - 591560 q^{19} + 1620984 q^{21} + 7445940 q^{29} - 4671544 q^{31} - 57227208 q^{39} + 43187724 q^{41} + 37313286 q^{49} + 206998056 q^{51} + 32325720 q^{59} - 87856316 q^{61} - 885497832 q^{69} - 322615464 q^{71} - 1166691440 q^{79} - 967421718 q^{81} - 940267380 q^{89} + 1531978936 q^{91} + 614224512 q^{99}+O(q^{100}) \) 2 * q - 21186 * q^9 - 57984 * q^11 - 591560 * q^19 + 1620984 * q^21 + 7445940 * q^29 - 4671544 * q^31 - 57227208 * q^39 + 43187724 * q^41 + 37313286 * q^49 + 206998056 * q^51 + 32325720 * q^59 - 87856316 * q^61 - 885497832 * q^69 - 322615464 * q^71 - 1166691440 * q^79 - 967421718 * q^81 - 940267380 * q^89 + 1531978936 * q^91 + 614224512 * q^99

Introduction

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