LMFDB - Embedded newform 208.10.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,10,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.a (trivial)

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:	yes
Analytic conductor:	$107.127453922$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 1602x^{2} + 1544x + 342272 $ x^4 - x^3 - 1602x^2 + 1544x + 342272
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-15.3567$ of defining polynomial
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-42.6243 q^{3} -1236.25 q^{5} -892.010 q^{7} -17866.2 q^{9} +O(q^{10})$	$q-42.6243 q^{3} -1236.25 q^{5} -892.010 q^{7} -17866.2 q^{9} +27149.8 q^{11} -28561.0 q^{13} +52694.2 q^{15} -34643.4 q^{17} -428885. q^{19} +38021.3 q^{21} -2.03704e6 q^{23} -424814. q^{25} +1.60051e6 q^{27} -5.26400e6 q^{29} +4.15910e6 q^{31} -1.15724e6 q^{33} +1.10275e6 q^{35} -7.58854e6 q^{37} +1.21739e6 q^{39} -4.92536e6 q^{41} -1.71882e7 q^{43} +2.20870e7 q^{45} +2.95568e7 q^{47} -3.95579e7 q^{49} +1.47665e6 q^{51} -2.72331e7 q^{53} -3.35640e7 q^{55} +1.82809e7 q^{57} +1.13602e8 q^{59} -3.76868e7 q^{61} +1.59368e7 q^{63} +3.53085e7 q^{65} -1.90094e8 q^{67} +8.68273e7 q^{69} -6.87130e7 q^{71} +3.61495e8 q^{73} +1.81074e7 q^{75} -2.42179e7 q^{77} +1.42229e8 q^{79} +2.83439e8 q^{81} +5.80240e7 q^{83} +4.28279e7 q^{85} +2.24374e8 q^{87} -8.59928e8 q^{89} +2.54767e7 q^{91} -1.77279e8 q^{93} +5.30208e8 q^{95} +1.46970e9 q^{97} -4.85064e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 163 q^{3} + 471 q^{5} + 11241 q^{7} - 29953 q^{9}+O(q^{10}) $ 4 * q + 163 * q^3 + 471 * q^5 + 11241 * q^7 - 29953 * q^9	$ 4 q + 163 q^{3} + 471 q^{5} + 11241 q^{7} - 29953 q^{9} + 40140 q^{11} - 114244 q^{13} - 83307 q^{15} + 78717 q^{17} - 209664 q^{19} + 1138431 q^{21} + 4257444 q^{23} - 2900157 q^{25} + 2077801 q^{27} - 1647936 q^{29} + 11366002 q^{31} - 14413222 q^{33} + 13789797 q^{35} + 4636891 q^{37} - 4655443 q^{39} + 13859538 q^{41} + 33368081 q^{43} - 17423928 q^{45} + 3943005 q^{47} + 23294923 q^{49} + 19664471 q^{51} - 171019326 q^{53} + 121160538 q^{55} - 47829030 q^{57} + 63389388 q^{59} + 77050190 q^{61} + 155695476 q^{63} - 13452231 q^{65} + 41174072 q^{67} + 546642556 q^{69} - 252460989 q^{71} + 594415068 q^{73} - 533318748 q^{75} + 561950454 q^{77} - 115998984 q^{79} + 437803700 q^{81} + 79577862 q^{83} + 549463469 q^{85} + 1087526510 q^{87} - 1152240276 q^{89} - 321054201 q^{91} + 1618266556 q^{93} + 1273705170 q^{95} + 1049098084 q^{97} - 2132181050 q^{99}+O(q^{100}) $ 4 * q + 163 * q^3 + 471 * q^5 + 11241 * q^7 - 29953 * q^9 + 40140 * q^11 - 114244 * q^13 - 83307 * q^15 + 78717 * q^17 - 209664 * q^19 + 1138431 * q^21 + 4257444 * q^23 - 2900157 * q^25 + 2077801 * q^27 - 1647936 * q^29 + 11366002 * q^31 - 14413222 * q^33 + 13789797 * q^35 + 4636891 * q^37 - 4655443 * q^39 + 13859538 * q^41 + 33368081 * q^43 - 17423928 * q^45 + 3943005 * q^47 + 23294923 * q^49 + 19664471 * q^51 - 171019326 * q^53 + 121160538 * q^55 - 47829030 * q^57 + 63389388 * q^59 + 77050190 * q^61 + 155695476 * q^63 - 13452231 * q^65 + 41174072 * q^67 + 546642556 * q^69 - 252460989 * q^71 + 594415068 * q^73 - 533318748 * q^75 + 561950454 * q^77 - 115998984 * q^79 + 437803700 * q^81 + 79577862 * q^83 + 549463469 * q^85 + 1087526510 * q^87 - 1152240276 * q^89 - 321054201 * q^91 + 1618266556 * q^93 + 1273705170 * q^95 + 1049098084 * q^97 - 2132181050 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−42.6243	−0.303817	−0.151908	−	0.988395i	$-0.548542\pi$
$3$	−42.6243	−0.303817	−0.151908	+	0.988395i	$0.548542\pi$
$4$	0	0
$5$	−1236.25	−0.884588	−0.442294	−	0.896870i	$-0.645835\pi$
$5$	−1236.25	−0.884588	−0.442294	+	0.896870i	$0.645835\pi$
$6$	0	0
$7$	−892.010	−0.140420	−0.0702099	−	0.997532i	$-0.522367\pi$
$7$	−892.010	−0.140420	−0.0702099	+	0.997532i	$0.522367\pi$
$8$	0	0
$9$	−17866.2	−0.907695
$10$	0	0
$11$	27149.8	0.559114	0.279557	−	0.960129i	$-0.409812\pi$
$11$	27149.8	0.559114	0.279557	+	0.960129i	$0.409812\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	52694.2	0.268753
$16$	0	0
$17$	−34643.4	−0.100601	−0.0503003	−	0.998734i	$-0.516018\pi$
$17$	−34643.4	−0.100601	−0.0503003	+	0.998734i	$0.516018\pi$
$18$	0	0
$19$	−428885.	−0.755004	−0.377502	−	0.926009i	$-0.623217\pi$
$19$	−428885.	−0.755004	−0.377502	+	0.926009i	$0.623217\pi$
$20$	0	0
$21$	38021.3	0.0426619
$22$	0	0
$23$	−2.03704e6	−1.51783	−0.758916	−	0.651188i	$-0.774271\pi$
$23$	−2.03704e6	−1.51783	−0.758916	+	0.651188i	$0.774271\pi$
$24$	0	0
$25$	−424814.	−0.217505
$26$	0	0
$27$	1.60051e6	0.579590
$28$	0	0
$29$	−5.26400e6	−1.38205	−0.691026	−	0.722830i	$-0.742841\pi$
$29$	−5.26400e6	−1.38205	−0.691026	+	0.722830i	$0.742841\pi$
$30$	0	0
$31$	4.15910e6	0.808856	0.404428	−	0.914570i	$-0.367471\pi$
$31$	4.15910e6	0.808856	0.404428	+	0.914570i	$0.367471\pi$
$32$	0	0
$33$	−1.15724e6	−0.169868
$34$	0	0
$35$	1.10275e6	0.124214
$36$	0	0
$37$	−7.58854e6	−0.665657	−0.332829	−	0.942987i	$-0.608003\pi$
$37$	−7.58854e6	−0.665657	−0.332829	+	0.942987i	$0.608003\pi$
$38$	0	0
$39$	1.21739e6	0.0842636
$40$	0	0
$41$	−4.92536e6	−0.272214	−0.136107	−	0.990694i	$-0.543459\pi$
$41$	−4.92536e6	−0.272214	−0.136107	+	0.990694i	$0.543459\pi$
$42$	0	0
$43$	−1.71882e7	−0.766696	−0.383348	−	0.923604i	$-0.625229\pi$
$43$	−1.71882e7	−0.766696	−0.383348	+	0.923604i	$0.625229\pi$
$44$	0	0
$45$	2.20870e7	0.802936
$46$	0	0
$47$	2.95568e7	0.883522	0.441761	−	0.897133i	$-0.354354\pi$
$47$	2.95568e7	0.883522	0.441761	+	0.897133i	$0.354354\pi$
$48$	0	0
$49$	−3.95579e7	−0.980282
$50$	0	0
$51$	1.47665e6	0.0305642
$52$	0	0
$53$	−2.72331e7	−0.474084	−0.237042	−	0.971499i	$-0.576178\pi$
$53$	−2.72331e7	−0.474084	−0.237042	+	0.971499i	$0.576178\pi$
$54$	0	0
$55$	−3.35640e7	−0.494585
$56$	0	0
$57$	1.82809e7	0.229383
$58$	0	0
$59$	1.13602e8	1.22054	0.610272	−	0.792192i	$-0.291060\pi$
$59$	1.13602e8	1.22054	0.610272	+	0.792192i	$0.291060\pi$
$60$	0	0
$61$	−3.76868e7	−0.348502	−0.174251	−	0.984701i	$-0.555750\pi$
$61$	−3.76868e7	−0.348502	−0.174251	+	0.984701i	$0.555750\pi$
$62$	0	0
$63$	1.59368e7	0.127458
$64$	0	0
$65$	3.53085e7	0.245340
$66$	0	0
$67$	−1.90094e8	−1.15247	−0.576237	−	0.817283i	$-0.695479\pi$
$67$	−1.90094e8	−1.15247	−0.576237	+	0.817283i	$0.695479\pi$
$68$	0	0
$69$	8.68273e7	0.461143
$70$	0	0
$71$	−6.87130e7	−0.320905	−0.160453	−	0.987044i	$-0.551295\pi$
$71$	−6.87130e7	−0.320905	−0.160453	+	0.987044i	$0.551295\pi$
$72$	0	0
$73$	3.61495e8	1.48987	0.744937	−	0.667135i	$-0.232480\pi$
$73$	3.61495e8	1.48987	0.744937	+	0.667135i	$0.232480\pi$
$74$	0	0
$75$	1.81074e7	0.0660816
$76$	0	0
$77$	−2.42179e7	−0.0785107
$78$	0	0
$79$	1.42229e8	0.410834	0.205417	−	0.978675i	$-0.434145\pi$
$79$	1.42229e8	0.410834	0.205417	+	0.978675i	$0.434145\pi$
$80$	0	0
$81$	2.83439e8	0.731606
$82$	0	0
$83$	5.80240e7	0.134201	0.0671006	−	0.997746i	$-0.478625\pi$
$83$	5.80240e7	0.134201	0.0671006	+	0.997746i	$0.478625\pi$
$84$	0	0
$85$	4.28279e7	0.0889901
$86$	0	0
$87$	2.24374e8	0.419891
$88$	0	0
$89$	−8.59928e8	−1.45280	−0.726402	−	0.687270i	$-0.758809\pi$
$89$	−8.59928e8	−1.45280	−0.726402	+	0.687270i	$0.758809\pi$
$90$	0	0
$91$	2.54767e7	0.0389454
$92$	0	0
$93$	−1.77279e8	−0.245744
$94$	0	0
$95$	5.30208e8	0.667867
$96$	0	0
$97$	1.46970e9	1.68560	0.842802	−	0.538223i	$-0.180904\pi$
$97$	1.46970e9	1.68560	0.842802	+	0.538223i	$0.180904\pi$
$98$	0	0
$99$	−4.85064e8	−0.507505
$100$	0	0
$101$	−4.15100e8	−0.396923	−0.198462	−	0.980109i	$-0.563595\pi$
$101$	−4.15100e8	−0.396923	−0.198462	+	0.980109i	$0.563595\pi$
$102$	0	0
$103$	−1.86377e9	−1.63164	−0.815821	−	0.578305i	$-0.803714\pi$
$103$	−1.86377e9	−1.63164	−0.815821	+	0.578305i	$0.803714\pi$
$104$	0	0
$105$	−4.70038e7	−0.0377382
$106$	0	0
$107$	7.50777e8	0.553712	0.276856	−	0.960911i	$-0.410707\pi$
$107$	7.50777e8	0.553712	0.276856	+	0.960911i	$0.410707\pi$
$108$	0	0
$109$	2.07010e9	1.40467	0.702333	−	0.711849i	$-0.252142\pi$
$109$	2.07010e9	1.40467	0.702333	+	0.711849i	$0.252142\pi$
$110$	0	0
$111$	3.23456e8	0.202238
$112$	0	0
$113$	−2.10155e9	−1.21251	−0.606257	−	0.795268i	$-0.707330\pi$
$113$	−2.10155e9	−1.21251	−0.606257	+	0.795268i	$0.707330\pi$
$114$	0	0
$115$	2.51829e9	1.34266
$116$	0	0
$117$	5.10276e8	0.251749
$118$	0	0
$119$	3.09023e7	0.0141263
$120$	0	0
$121$	−1.62083e9	−0.687392
$122$	0	0
$123$	2.09940e8	0.0827033
$124$	0	0
$125$	2.93972e9	1.07699
$126$	0	0
$127$	4.77696e9	1.62943	0.814713	−	0.579865i	$-0.196895\pi$
$127$	4.77696e9	1.62943	0.814713	+	0.579865i	$0.196895\pi$
$128$	0	0
$129$	7.32636e8	0.232935
$130$	0	0
$131$	4.62884e9	1.37326	0.686628	−	0.727009i	$-0.259090\pi$
$131$	4.62884e9	1.37326	0.686628	+	0.727009i	$0.259090\pi$
$132$	0	0
$133$	3.82569e8	0.106018
$134$	0	0
$135$	−1.97863e9	−0.512698
$136$	0	0
$137$	3.85972e9	0.936080	0.468040	−	0.883707i	$-0.344960\pi$
$137$	3.85972e9	0.936080	0.468040	+	0.883707i	$0.344960\pi$
$138$	0	0
$139$	8.38068e9	1.90420	0.952100	−	0.305786i	$-0.0989191\pi$
$139$	8.38068e9	1.90420	0.952100	+	0.305786i	$0.0989191\pi$
$140$	0	0
$141$	−1.25984e9	−0.268429
$142$	0	0
$143$	−7.75427e8	−0.155070
$144$	0	0
$145$	6.50761e9	1.22255
$146$	0	0
$147$	1.68613e9	0.297826
$148$	0	0
$149$	3.96089e9	0.658347	0.329174	−	0.944269i	$-0.393230\pi$
$149$	3.96089e9	0.658347	0.329174	+	0.944269i	$0.393230\pi$
$150$	0	0
$151$	−1.11419e10	−1.74407	−0.872036	−	0.489442i	$-0.837201\pi$
$151$	−1.11419e10	−1.74407	−0.872036	+	0.489442i	$0.837201\pi$
$152$	0	0
$153$	6.18946e8	0.0913148
$154$	0	0
$155$	−5.14168e9	−0.715504
$156$	0	0
$157$	−7.39104e9	−0.970861	−0.485430	−	0.874275i	$-0.661337\pi$
$157$	−7.39104e9	−0.970861	−0.485430	+	0.874275i	$0.661337\pi$
$158$	0	0
$159$	1.16079e9	0.144035
$160$	0	0
$161$	1.81706e9	0.213134
$162$	0	0
$163$	7.32911e8	0.0813218	0.0406609	−	0.999173i	$-0.487054\pi$
$163$	7.32911e8	0.0813218	0.0406609	+	0.999173i	$0.487054\pi$
$164$	0	0
$165$	1.43064e9	0.150263
$166$	0	0
$167$	1.23516e10	1.22885	0.614427	−	0.788974i	$-0.289387\pi$
$167$	1.23516e10	1.22885	0.614427	+	0.788974i	$0.289387\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	7.66252e9	0.685314
$172$	0	0
$173$	1.12727e10	0.956794	0.478397	−	0.878144i	$-0.341218\pi$
$173$	1.12727e10	0.956794	0.478397	+	0.878144i	$0.341218\pi$
$174$	0	0
$175$	3.78938e8	0.0305420
$176$	0	0
$177$	−4.84222e9	−0.370822
$178$	0	0
$179$	3.32764e9	0.242269	0.121134	−	0.992636i	$-0.461347\pi$
$179$	3.32764e9	0.242269	0.121134	+	0.992636i	$0.461347\pi$
$180$	0	0
$181$	1.56098e10	1.08104	0.540521	−	0.841330i	$-0.318227\pi$
$181$	1.56098e10	1.08104	0.540521	+	0.841330i	$0.318227\pi$
$182$	0	0
$183$	1.60637e9	0.105881
$184$	0	0
$185$	9.38133e9	0.588832
$186$	0	0
$187$	−9.40564e8	−0.0562472
$188$	0	0
$189$	−1.42767e9	−0.0813859
$190$	0	0
$191$	−2.64757e10	−1.43945	−0.719725	−	0.694259i	$-0.755732\pi$
$191$	−2.64757e10	−1.43945	−0.719725	+	0.694259i	$0.755732\pi$
$192$	0	0
$193$	−2.67204e9	−0.138623	−0.0693114	−	0.997595i	$-0.522080\pi$
$193$	−2.67204e9	−0.138623	−0.0693114	+	0.997595i	$0.522080\pi$
$194$	0	0
$195$	−1.50500e9	−0.0745385
$196$	0	0
$197$	1.38233e10	0.653903	0.326951	−	0.945041i	$-0.393979\pi$
$197$	1.38233e10	0.653903	0.326951	+	0.945041i	$0.393979\pi$
$198$	0	0
$199$	1.21811e10	0.550616	0.275308	−	0.961356i	$-0.411220\pi$
$199$	1.21811e10	0.550616	0.275308	+	0.961356i	$0.411220\pi$
$200$	0	0
$201$	8.10261e9	0.350141
$202$	0	0
$203$	4.69554e9	0.194068
$204$	0	0
$205$	6.08897e9	0.240797
$206$	0	0
$207$	3.63941e10	1.37773
$208$	0	0
$209$	−1.16441e10	−0.422133
$210$	0	0
$211$	3.16880e9	0.110058	0.0550292	−	0.998485i	$-0.482475\pi$
$211$	3.16880e9	0.110058	0.0550292	+	0.998485i	$0.482475\pi$
$212$	0	0
$213$	2.92885e9	0.0974963
$214$	0	0
$215$	2.12489e10	0.678210
$216$	0	0
$217$	−3.70996e9	−0.113579
$218$	0	0
$219$	−1.54085e10	−0.452648
$220$	0	0
$221$	9.89451e8	0.0279016
$222$	0	0
$223$	1.00184e10	0.271285	0.135642	−	0.990758i	$-0.456690\pi$
$223$	1.00184e10	0.271285	0.135642	+	0.990758i	$0.456690\pi$
$224$	0	0
$225$	7.58980e9	0.197428
$226$	0	0
$227$	5.23965e10	1.30974	0.654871	−	0.755741i	$-0.272723\pi$
$227$	5.23965e10	1.30974	0.654871	+	0.755741i	$0.272723\pi$
$228$	0	0
$229$	−2.94853e10	−0.708510	−0.354255	−	0.935149i	$-0.615266\pi$
$229$	−2.94853e10	−0.708510	−0.354255	+	0.935149i	$0.615266\pi$
$230$	0	0
$231$	1.03227e9	0.0238529
$232$	0	0
$233$	−6.67468e10	−1.48364	−0.741820	−	0.670599i	$-0.766037\pi$
$233$	−6.67468e10	−1.48364	−0.741820	+	0.670599i	$0.766037\pi$
$234$	0	0
$235$	−3.65396e10	−0.781553
$236$	0	0
$237$	−6.06241e9	−0.124818
$238$	0	0
$239$	7.42940e10	1.47286	0.736432	−	0.676511i	$-0.236509\pi$
$239$	7.42940e10	1.47286	0.736432	+	0.676511i	$0.236509\pi$
$240$	0	0
$241$	9.01496e10	1.72142	0.860711	−	0.509095i	$-0.170020\pi$
$241$	9.01496e10	1.72142	0.860711	+	0.509095i	$0.170020\pi$
$242$	0	0
$243$	−4.35842e10	−0.801864
$244$	0	0
$245$	4.89034e10	0.867146
$246$	0	0
$247$	1.22494e10	0.209400
$248$	0	0
$249$	−2.47323e9	−0.0407726
$250$	0	0
$251$	−3.62007e10	−0.575685	−0.287843	−	0.957678i	$-0.592938\pi$
$251$	−3.62007e10	−0.575685	−0.287843	+	0.957678i	$0.592938\pi$
$252$	0	0
$253$	−5.53053e10	−0.848641
$254$	0	0
$255$	−1.82551e9	−0.0270367
$256$	0	0
$257$	−1.97638e10	−0.282600	−0.141300	−	0.989967i	$-0.545128\pi$
$257$	−1.97638e10	−0.282600	−0.141300	+	0.989967i	$0.545128\pi$
$258$	0	0
$259$	6.76905e9	0.0934714
$260$	0	0
$261$	9.40474e10	1.25448
$262$	0	0
$263$	1.70457e10	0.219692	0.109846	−	0.993949i	$-0.464964\pi$
$263$	1.70457e10	0.219692	0.109846	+	0.993949i	$0.464964\pi$
$264$	0	0
$265$	3.36669e10	0.419369
$266$	0	0
$267$	3.66539e10	0.441386
$268$	0	0
$269$	6.56819e10	0.764822	0.382411	−	0.923992i	$-0.375094\pi$
$269$	6.56819e10	0.764822	0.382411	+	0.923992i	$0.375094\pi$
$270$	0	0
$271$	−1.52442e11	−1.71689	−0.858447	−	0.512902i	$-0.828570\pi$
$271$	−1.52442e11	−1.71689	−0.858447	+	0.512902i	$0.828570\pi$
$272$	0	0
$273$	−1.08593e9	−0.0118323
$274$	0	0
$275$	−1.15336e10	−0.121610
$276$	0	0
$277$	−1.21612e11	−1.24113	−0.620566	−	0.784154i	$-0.713097\pi$
$277$	−1.21612e11	−1.24113	−0.620566	+	0.784154i	$0.713097\pi$
$278$	0	0
$279$	−7.43071e10	−0.734195
$280$	0	0
$281$	−2.03083e11	−1.94310	−0.971548	−	0.236842i	$-0.923888\pi$
$281$	−2.03083e11	−1.94310	−0.971548	+	0.236842i	$0.923888\pi$
$282$	0	0
$283$	−3.09567e10	−0.286891	−0.143445	−	0.989658i	$-0.545818\pi$
$283$	−3.09567e10	−0.286891	−0.143445	+	0.989658i	$0.545818\pi$
$284$	0	0
$285$	−2.25998e10	−0.202909
$286$	0	0
$287$	4.39347e9	0.0382243
$288$	0	0
$289$	−1.17388e11	−0.989880
$290$	0	0
$291$	−6.26449e10	−0.512115
$292$	0	0
$293$	−7.81733e10	−0.619661	−0.309831	−	0.950792i	$-0.600272\pi$
$293$	−7.81733e10	−0.619661	−0.309831	+	0.950792i	$0.600272\pi$
$294$	0	0
$295$	−1.40441e11	−1.07968
$296$	0	0
$297$	4.34535e10	0.324057
$298$	0	0
$299$	5.81798e10	0.420971
$300$	0	0
$301$	1.53321e10	0.107659
$302$	0	0
$303$	1.76933e10	0.120592
$304$	0	0
$305$	4.65902e10	0.308280
$306$	0	0
$307$	−1.19962e11	−0.770766	−0.385383	−	0.922757i	$-0.625931\pi$
$307$	−1.19962e11	−0.770766	−0.385383	+	0.922757i	$0.625931\pi$
$308$	0	0
$309$	7.94419e10	0.495720
$310$	0	0
$311$	−1.16227e11	−0.704504	−0.352252	−	0.935905i	$-0.614584\pi$
$311$	−1.16227e11	−0.704504	−0.352252	+	0.935905i	$0.614584\pi$
$312$	0	0
$313$	9.92344e10	0.584403	0.292202	−	0.956357i	$-0.405612\pi$
$313$	9.92344e10	0.584403	0.292202	+	0.956357i	$0.405612\pi$
$314$	0	0
$315$	−1.97018e10	−0.112748
$316$	0	0
$317$	−2.98194e11	−1.65856	−0.829281	−	0.558832i	$-0.811250\pi$
$317$	−2.98194e11	−1.65856	−0.829281	+	0.558832i	$0.811250\pi$
$318$	0	0
$319$	−1.42917e11	−0.772725
$320$	0	0
$321$	−3.20014e10	−0.168227
$322$	0	0
$323$	1.48580e10	0.0759539
$324$	0	0
$325$	1.21331e10	0.0603250
$326$	0	0
$327$	−8.82367e10	−0.426761
$328$	0	0
$329$	−2.63650e10	−0.124064
$330$	0	0
$331$	−6.87707e10	−0.314904	−0.157452	−	0.987527i	$-0.550328\pi$
$331$	−6.87707e10	−0.314904	−0.157452	+	0.987527i	$0.550328\pi$
$332$	0	0
$333$	1.35578e11	0.604214
$334$	0	0
$335$	2.35003e11	1.01946
$336$	0	0
$337$	−1.56091e11	−0.659239	−0.329619	−	0.944114i	$-0.606920\pi$
$337$	−1.56091e11	−0.659239	−0.329619	+	0.944114i	$0.606920\pi$
$338$	0	0
$339$	8.95772e10	0.368382
$340$	0	0
$341$	1.12919e11	0.452243
$342$	0	0
$343$	7.12819e10	0.278071
$344$	0	0
$345$	−1.07340e11	−0.407921
$346$	0	0
$347$	1.65467e11	0.612673	0.306337	−	0.951923i	$-0.400897\pi$
$347$	1.65467e11	0.612673	0.306337	+	0.951923i	$0.400897\pi$
$348$	0	0
$349$	−4.02009e11	−1.45051	−0.725257	−	0.688478i	$-0.758279\pi$
$349$	−4.02009e11	−1.45051	−0.725257	+	0.688478i	$0.758279\pi$
$350$	0	0
$351$	−4.57121e10	−0.160749
$352$	0	0
$353$	2.74001e11	0.939217	0.469609	−	0.882875i	$-0.344395\pi$
$353$	2.74001e11	0.939217	0.469609	+	0.882875i	$0.344395\pi$
$354$	0	0
$355$	8.49464e10	0.283869
$356$	0	0
$357$	−1.31719e9	−0.00429182
$358$	0	0
$359$	−8.26405e10	−0.262584	−0.131292	−	0.991344i	$-0.541913\pi$
$359$	−8.26405e10	−0.262584	−0.131292	+	0.991344i	$0.541913\pi$
$360$	0	0
$361$	−1.38746e11	−0.429969
$362$	0	0
$363$	6.90869e10	0.208841
$364$	0	0
$365$	−4.46898e11	−1.31792
$366$	0	0
$367$	−4.83059e10	−0.138996	−0.0694980	−	0.997582i	$-0.522140\pi$
$367$	−4.83059e10	−0.138996	−0.0694980	+	0.997582i	$0.522140\pi$
$368$	0	0
$369$	8.79974e10	0.247088
$370$	0	0
$371$	2.42922e10	0.0665708
$372$	0	0
$373$	8.53749e10	0.228371	0.114185	−	0.993459i	$-0.463574\pi$
$373$	8.53749e10	0.228371	0.114185	+	0.993459i	$0.463574\pi$
$374$	0	0
$375$	−1.25304e11	−0.327207
$376$	0	0
$377$	1.50345e11	0.383312
$378$	0	0
$379$	2.25085e11	0.560365	0.280182	−	0.959947i	$-0.409605\pi$
$379$	2.25085e11	0.560365	0.280182	+	0.959947i	$0.409605\pi$
$380$	0	0
$381$	−2.03614e11	−0.495047
$382$	0	0
$383$	−6.84152e11	−1.62464	−0.812322	−	0.583210i	$-0.801797\pi$
$383$	−6.84152e11	−1.62464	−0.812322	+	0.583210i	$0.801797\pi$
$384$	0	0
$385$	2.99394e10	0.0694495
$386$	0	0
$387$	3.07088e11	0.695926
$388$	0	0
$389$	7.47317e11	1.65475	0.827374	−	0.561652i	$-0.189834\pi$
$389$	7.47317e11	1.65475	0.827374	+	0.561652i	$0.189834\pi$
$390$	0	0
$391$	7.05700e10	0.152695
$392$	0	0
$393$	−1.97301e11	−0.417218
$394$	0	0
$395$	−1.75830e11	−0.363419
$396$	0	0
$397$	1.63235e11	0.329804	0.164902	−	0.986310i	$-0.447269\pi$
$397$	1.63235e11	0.329804	0.164902	+	0.986310i	$0.447269\pi$
$398$	0	0
$399$	−1.63067e10	−0.0322099
$400$	0	0
$401$	−7.33836e11	−1.41726	−0.708629	−	0.705581i	$-0.750686\pi$
$401$	−7.33836e11	−1.41726	−0.708629	+	0.705581i	$0.750686\pi$
$402$	0	0
$403$	−1.18788e11	−0.224336
$404$	0	0
$405$	−3.50401e11	−0.647170
$406$	0	0
$407$	−2.06028e11	−0.372178
$408$	0	0
$409$	7.45800e11	1.31785	0.658927	−	0.752207i	$-0.271010\pi$
$409$	7.45800e11	1.31785	0.658927	+	0.752207i	$0.271010\pi$
$410$	0	0
$411$	−1.64518e11	−0.284397
$412$	0	0
$413$	−1.01334e11	−0.171388
$414$	0	0
$415$	−7.17321e10	−0.118713
$416$	0	0
$417$	−3.57221e11	−0.578528
$418$	0	0
$419$	5.46254e11	0.865828	0.432914	−	0.901435i	$-0.357485\pi$
$419$	5.46254e11	0.865828	0.432914	+	0.901435i	$0.357485\pi$
$420$	0	0
$421$	−2.95449e11	−0.458366	−0.229183	−	0.973383i	$-0.573606\pi$
$421$	−2.95449e11	−0.458366	−0.229183	+	0.973383i	$0.573606\pi$
$422$	0	0
$423$	−5.28067e11	−0.801969
$424$	0	0
$425$	1.47170e10	0.0218811
$426$	0	0
$427$	3.36170e10	0.0489365
$428$	0	0
$429$	3.30520e10	0.0471129
$430$	0	0
$431$	5.49479e11	0.767014	0.383507	−	0.923538i	$-0.374716\pi$
$431$	5.49479e11	0.767014	0.383507	+	0.923538i	$0.374716\pi$
$432$	0	0
$433$	−7.54093e11	−1.03093	−0.515465	−	0.856910i	$-0.672381\pi$
$433$	−7.54093e11	−1.03093	−0.515465	+	0.856910i	$0.672381\pi$
$434$	0	0
$435$	−2.77382e11	−0.371430
$436$	0	0
$437$	8.73654e11	1.14597
$438$	0	0
$439$	−9.98220e10	−0.128273	−0.0641366	−	0.997941i	$-0.520429\pi$
$439$	−9.98220e10	−0.128273	−0.0641366	+	0.997941i	$0.520429\pi$
$440$	0	0
$441$	7.06749e11	0.889798
$442$	0	0
$443$	1.08040e12	1.33281	0.666404	−	0.745591i	$-0.267832\pi$
$443$	1.08040e12	1.33281	0.666404	+	0.745591i	$0.267832\pi$
$444$	0	0
$445$	1.06309e12	1.28513
$446$	0	0
$447$	−1.68830e11	−0.200017
$448$	0	0
$449$	1.02947e12	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$449$	1.02947e12	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$450$	0	0
$451$	−1.33723e11	−0.152199
$452$	0	0
$453$	4.74917e11	0.529878
$454$	0	0
$455$	−3.14955e10	−0.0344507
$456$	0	0
$457$	1.06106e12	1.13793	0.568966	−	0.822361i	$-0.307343\pi$
$457$	1.06106e12	1.13793	0.568966	+	0.822361i	$0.307343\pi$
$458$	0	0
$459$	−5.54471e10	−0.0583071
$460$	0	0
$461$	−7.20760e11	−0.743253	−0.371626	−	0.928382i	$-0.621200\pi$
$461$	−7.20760e11	−0.743253	−0.371626	+	0.928382i	$0.621200\pi$
$462$	0	0
$463$	−1.41179e11	−0.142776	−0.0713880	−	0.997449i	$-0.522743\pi$
$463$	−1.41179e11	−0.142776	−0.0713880	+	0.997449i	$0.522743\pi$
$464$	0	0
$465$	2.19161e11	0.217382
$466$	0	0
$467$	5.23104e11	0.508935	0.254467	−	0.967081i	$-0.418100\pi$
$467$	5.23104e11	0.508935	0.254467	+	0.967081i	$0.418100\pi$
$468$	0	0
$469$	1.69565e11	0.161830
$470$	0	0
$471$	3.15038e11	0.294964
$472$	0	0
$473$	−4.66658e11	−0.428670
$474$	0	0
$475$	1.82196e11	0.164217
$476$	0	0
$477$	4.86551e11	0.430324
$478$	0	0
$479$	−1.22387e12	−1.06224	−0.531122	−	0.847295i	$-0.678230\pi$
$479$	−1.22387e12	−1.06224	−0.531122	+	0.847295i	$0.678230\pi$
$480$	0	0
$481$	2.16736e11	0.184620
$482$	0	0
$483$	−7.74508e10	−0.0647536
$484$	0	0
$485$	−1.81691e12	−1.49107
$486$	0	0
$487$	3.77344e11	0.303988	0.151994	−	0.988381i	$-0.451431\pi$
$487$	3.77344e11	0.303988	0.151994	+	0.988381i	$0.451431\pi$
$488$	0	0
$489$	−3.12398e10	−0.0247069
$490$	0	0
$491$	2.35736e12	1.83046	0.915229	−	0.402934i	$-0.132010\pi$
$491$	2.35736e12	1.83046	0.915229	+	0.402934i	$0.132010\pi$
$492$	0	0
$493$	1.82363e11	0.139035
$494$	0	0
$495$	5.99659e11	0.448933
$496$	0	0
$497$	6.12927e10	0.0450614
$498$	0	0
$499$	2.64345e12	1.90862	0.954309	−	0.298822i	$-0.0965938\pi$
$499$	2.64345e12	1.90862	0.954309	+	0.298822i	$0.0965938\pi$
$500$	0	0
$501$	−5.26480e11	−0.373346
$502$	0	0
$503$	−1.53328e11	−0.106799	−0.0533994	−	0.998573i	$-0.517006\pi$
$503$	−1.53328e11	−0.106799	−0.0533994	+	0.998573i	$0.517006\pi$
$504$	0	0
$505$	5.13167e11	0.351113
$506$	0	0
$507$	−3.47700e10	−0.0233705
$508$	0	0
$509$	−6.36840e11	−0.420533	−0.210267	−	0.977644i	$-0.567433\pi$
$509$	−6.36840e11	−0.420533	−0.210267	+	0.977644i	$0.567433\pi$
$510$	0	0
$511$	−3.22457e11	−0.209208
$512$	0	0
$513$	−6.86433e11	−0.437593
$514$	0	0
$515$	2.30408e12	1.44333
$516$	0	0
$517$	8.02463e11	0.493989
$518$	0	0
$519$	−4.80489e11	−0.290690
$520$	0	0
$521$	1.84033e12	1.09427	0.547137	−	0.837043i	$-0.315718\pi$
$521$	1.84033e12	1.09427	0.547137	+	0.837043i	$0.315718\pi$
$522$	0	0
$523$	1.13176e10	0.00661447	0.00330724	−	0.999995i	$-0.498947\pi$
$523$	1.13176e10	0.00661447	0.00330724	+	0.999995i	$0.498947\pi$
$524$	0	0
$525$	−1.61520e10	−0.00927917
$526$	0	0
$527$	−1.44085e11	−0.0813715
$528$	0	0
$529$	2.34837e12	1.30382
$530$	0	0
$531$	−2.02964e12	−1.10788
$532$	0	0
$533$	1.40673e11	0.0754987
$534$	0	0
$535$	−9.28148e11	−0.489807
$536$	0	0
$537$	−1.41838e11	−0.0736053
$538$	0	0
$539$	−1.07399e12	−0.548089
$540$	0	0
$541$	−2.25189e12	−1.13021	−0.565107	−	0.825018i	$-0.691165\pi$
$541$	−2.25189e12	−1.13021	−0.565107	+	0.825018i	$0.691165\pi$
$542$	0	0
$543$	−6.65356e11	−0.328439
$544$	0	0
$545$	−2.55916e12	−1.24255
$546$	0	0
$547$	−1.92777e11	−0.0920686	−0.0460343	−	0.998940i	$-0.514658\pi$
$547$	−1.92777e11	−0.0920686	−0.0460343	+	0.998940i	$0.514658\pi$
$548$	0	0
$549$	6.73318e11	0.316333
$550$	0	0
$551$	2.25765e12	1.04346
$552$	0	0
$553$	−1.26870e11	−0.0576892
$554$	0	0
$555$	−3.99873e11	−0.178897
$556$	0	0
$557$	3.00574e12	1.32313	0.661566	−	0.749887i	$-0.269892\pi$
$557$	3.00574e12	1.32313	0.661566	+	0.749887i	$0.269892\pi$
$558$	0	0
$559$	4.90913e11	0.212643
$560$	0	0
$561$	4.00909e10	0.0170888
$562$	0	0
$563$	1.64962e11	0.0691984	0.0345992	−	0.999401i	$-0.488985\pi$
$563$	1.64962e11	0.0691984	0.0345992	+	0.999401i	$0.488985\pi$
$564$	0	0
$565$	2.59804e12	1.07258
$566$	0	0
$567$	−2.52831e11	−0.102732
$568$	0	0
$569$	−2.35127e11	−0.0940368	−0.0470184	−	0.998894i	$-0.514972\pi$
$569$	−2.35127e11	−0.0940368	−0.0470184	+	0.998894i	$0.514972\pi$
$570$	0	0
$571$	2.71697e12	1.06960	0.534801	−	0.844978i	$-0.320387\pi$
$571$	2.71697e12	1.06960	0.534801	+	0.844978i	$0.320387\pi$
$572$	0	0
$573$	1.12851e12	0.437329
$574$	0	0
$575$	8.65362e11	0.330136
$576$	0	0
$577$	−3.01507e12	−1.13242	−0.566209	−	0.824262i	$-0.691590\pi$
$577$	−3.01507e12	−1.13242	−0.566209	+	0.824262i	$0.691590\pi$
$578$	0	0
$579$	1.13894e11	0.0421159
$580$	0	0
$581$	−5.17580e10	−0.0188445
$582$	0	0
$583$	−7.39374e11	−0.265067
$584$	0	0
$585$	−6.30828e11	−0.222694
$586$	0	0
$587$	9.84708e11	0.342323	0.171161	−	0.985243i	$-0.445248\pi$
$587$	9.84708e11	0.342323	0.171161	+	0.985243i	$0.445248\pi$
$588$	0	0
$589$	−1.78377e12	−0.610690
$590$	0	0
$591$	−5.89208e11	−0.198667
$592$	0	0
$593$	1.01065e12	0.335625	0.167812	−	0.985819i	$-0.446330\pi$
$593$	1.01065e12	0.335625	0.167812	+	0.985819i	$0.446330\pi$
$594$	0	0
$595$	−3.82029e10	−0.0124960
$596$	0	0
$597$	−5.19213e11	−0.167286
$598$	0	0
$599$	−5.47507e12	−1.73768	−0.868839	−	0.495095i	$-0.835133\pi$
$599$	−5.47507e12	−1.73768	−0.868839	+	0.495095i	$0.835133\pi$
$600$	0	0
$601$	9.94557e11	0.310953	0.155476	−	0.987840i	$-0.450309\pi$
$601$	9.94557e11	0.310953	0.155476	+	0.987840i	$0.450309\pi$
$602$	0	0
$603$	3.39624e12	1.04609
$604$	0	0
$605$	2.00375e12	0.608058
$606$	0	0
$607$	−4.57629e12	−1.36825	−0.684124	−	0.729366i	$-0.739815\pi$
$607$	−4.57629e12	−1.36825	−0.684124	+	0.729366i	$0.739815\pi$
$608$	0	0
$609$	−2.00144e11	−0.0589610
$610$	0	0
$611$	−8.44173e11	−0.245045
$612$	0	0
$613$	2.27107e12	0.649617	0.324809	−	0.945780i	$-0.394700\pi$
$613$	2.27107e12	0.649617	0.324809	+	0.945780i	$0.394700\pi$
$614$	0	0
$615$	−2.59538e11	−0.0731583
$616$	0	0
$617$	3.53862e12	0.982994	0.491497	−	0.870879i	$-0.336450\pi$
$617$	3.53862e12	0.982994	0.491497	+	0.870879i	$0.336450\pi$
$618$	0	0
$619$	−1.97957e12	−0.541956	−0.270978	−	0.962586i	$-0.587347\pi$
$619$	−1.97957e12	−0.541956	−0.270978	+	0.962586i	$0.587347\pi$
$620$	0	0
$621$	−3.26029e12	−0.879720
$622$	0	0
$623$	7.67065e11	0.204003
$624$	0	0
$625$	−2.80452e12	−0.735187
$626$	0	0
$627$	4.96324e11	0.128251
$628$	0	0
$629$	2.62893e11	0.0669656
$630$	0	0
$631$	−3.71649e11	−0.0933256	−0.0466628	−	0.998911i	$-0.514859\pi$
$631$	−3.71649e11	−0.0933256	−0.0466628	+	0.998911i	$0.514859\pi$
$632$	0	0
$633$	−1.35068e11	−0.0334376
$634$	0	0
$635$	−5.90551e12	−1.44137
$636$	0	0
$637$	1.12981e12	0.271881
$638$	0	0
$639$	1.22764e12	0.291284
$640$	0	0
$641$	1.07772e12	0.252143	0.126071	−	0.992021i	$-0.459763\pi$
$641$	1.07772e12	0.252143	0.126071	+	0.992021i	$0.459763\pi$
$642$	0	0
$643$	6.69209e12	1.54388	0.771938	−	0.635698i	$-0.219288\pi$
$643$	6.69209e12	1.54388	0.771938	+	0.635698i	$0.219288\pi$
$644$	0	0
$645$	−9.05721e11	−0.206051
$646$	0	0
$647$	−1.58975e12	−0.356664	−0.178332	−	0.983970i	$-0.557070\pi$
$647$	−1.58975e12	−0.356664	−0.178332	+	0.983970i	$0.557070\pi$
$648$	0	0
$649$	3.08429e12	0.682423
$650$	0	0
$651$	1.58134e11	0.0345073
$652$	0	0
$653$	−1.96565e12	−0.423055	−0.211527	−	0.977372i	$-0.567844\pi$
$653$	−1.96565e12	−0.423055	−0.211527	+	0.977372i	$0.567844\pi$
$654$	0	0
$655$	−5.72240e12	−1.21476
$656$	0	0
$657$	−6.45853e12	−1.35235
$658$	0	0
$659$	−1.09426e11	−0.0226014	−0.0113007	−	0.999936i	$-0.503597\pi$
$659$	−1.09426e11	−0.0226014	−0.0113007	+	0.999936i	$0.503597\pi$
$660$	0	0
$661$	−2.69590e12	−0.549285	−0.274643	−	0.961546i	$-0.588560\pi$
$661$	−2.69590e12	−0.549285	−0.274643	+	0.961546i	$0.588560\pi$
$662$	0	0
$663$	−4.21747e10	−0.00847698
$664$	0	0
$665$	−4.72951e11	−0.0937818
$666$	0	0
$667$	1.07230e13	2.09772
$668$	0	0
$669$	−4.27026e11	−0.0824208
$670$	0	0
$671$	−1.02319e12	−0.194852
$672$	0	0
$673$	−6.37187e12	−1.19729	−0.598644	−	0.801015i	$-0.704294\pi$
$673$	−6.37187e12	−1.19729	−0.598644	+	0.801015i	$0.704294\pi$
$674$	0	0
$675$	−6.79918e11	−0.126064
$676$	0	0
$677$	−6.61270e12	−1.20985	−0.604923	−	0.796284i	$-0.706796\pi$
$677$	−6.61270e12	−1.20985	−0.604923	+	0.796284i	$0.706796\pi$
$678$	0	0
$679$	−1.31099e12	−0.236692
$680$	0	0
$681$	−2.23336e12	−0.397921
$682$	0	0
$683$	−3.52480e12	−0.619785	−0.309893	−	0.950772i	$-0.600293\pi$
$683$	−3.52480e12	−0.619785	−0.309893	+	0.950772i	$0.600293\pi$
$684$	0	0
$685$	−4.77157e12	−0.828045
$686$	0	0
$687$	1.25679e12	0.215257
$688$	0	0
$689$	7.77804e11	0.131487
$690$	0	0
$691$	−1.21020e12	−0.201932	−0.100966	−	0.994890i	$-0.532193\pi$
$691$	−1.21020e12	−0.201932	−0.100966	+	0.994890i	$0.532193\pi$
$692$	0	0
$693$	4.32681e11	0.0712638
$694$	0	0
$695$	−1.03606e13	−1.68443
$696$	0	0
$697$	1.70632e11	0.0273849
$698$	0	0
$699$	2.84503e12	0.450755
$700$	0	0
$701$	3.34752e12	0.523591	0.261796	−	0.965123i	$-0.415685\pi$
$701$	3.34752e12	0.523591	0.261796	+	0.965123i	$0.415685\pi$
$702$	0	0
$703$	3.25461e12	0.502574
$704$	0	0
$705$	1.55748e12	0.237449
$706$	0	0
$707$	3.70273e11	0.0557359
$708$	0	0
$709$	1.09345e13	1.62514	0.812568	−	0.582866i	$-0.198069\pi$
$709$	1.09345e13	1.62514	0.812568	+	0.582866i	$0.198069\pi$
$710$	0	0
$711$	−2.54109e12	−0.372912
$712$	0	0
$713$	−8.47224e12	−1.22771
$714$	0	0
$715$	9.58620e11	0.137173
$716$	0	0
$717$	−3.16673e12	−0.447481
$718$	0	0
$719$	−4.73071e12	−0.660156	−0.330078	−	0.943954i	$-0.607075\pi$
$719$	−4.73071e12	−0.660156	−0.330078	+	0.943954i	$0.607075\pi$
$720$	0	0
$721$	1.66250e12	0.229115
$722$	0	0
$723$	−3.84256e12	−0.522997
$724$	0	0
$725$	2.23622e12	0.300603
$726$	0	0
$727$	−1.40402e13	−1.86410	−0.932049	−	0.362333i	$-0.881980\pi$
$727$	−1.40402e13	−1.86410	−0.932049	+	0.362333i	$0.881980\pi$
$728$	0	0
$729$	−3.72119e12	−0.487987
$730$	0	0
$731$	5.95459e11	0.0771301
$732$	0	0
$733$	−1.78493e12	−0.228377	−0.114188	−	0.993459i	$-0.536427\pi$
$733$	−1.78493e12	−0.228377	−0.114188	+	0.993459i	$0.536427\pi$
$734$	0	0
$735$	−2.08448e12	−0.263453
$736$	0	0
$737$	−5.16101e12	−0.644364
$738$	0	0
$739$	1.24956e13	1.54119	0.770595	−	0.637325i	$-0.219959\pi$
$739$	1.24956e13	1.54119	0.770595	+	0.637325i	$0.219959\pi$
$740$	0	0
$741$	−5.22121e11	−0.0636194
$742$	0	0
$743$	−4.21591e12	−0.507506	−0.253753	−	0.967269i	$-0.581665\pi$
$743$	−4.21591e12	−0.507506	−0.253753	+	0.967269i	$0.581665\pi$
$744$	0	0
$745$	−4.89665e12	−0.582366
$746$	0	0
$747$	−1.03667e12	−0.121814
$748$	0	0
$749$	−6.69701e11	−0.0777522
$750$	0	0
$751$	−9.94988e12	−1.14140	−0.570701	−	0.821158i	$-0.693328\pi$
$751$	−9.94988e12	−1.14140	−0.570701	+	0.821158i	$0.693328\pi$
$752$	0	0
$753$	1.54303e12	0.174903
$754$	0	0
$755$	1.37742e13	1.54278
$756$	0	0
$757$	−1.61992e13	−1.79293	−0.896465	−	0.443115i	$-0.853873\pi$
$757$	−1.61992e13	−1.79293	−0.896465	+	0.443115i	$0.853873\pi$
$758$	0	0
$759$	2.35735e12	0.257831
$760$	0	0
$761$	4.79250e12	0.518001	0.259001	−	0.965877i	$-0.416607\pi$
$761$	4.79250e12	0.518001	0.259001	+	0.965877i	$0.416607\pi$
$762$	0	0
$763$	−1.84655e12	−0.197243
$764$	0	0
$765$	−7.65171e11	−0.0807759
$766$	0	0
$767$	−3.24460e12	−0.338518
$768$	0	0
$769$	−8.41625e11	−0.0867861	−0.0433930	−	0.999058i	$-0.513817\pi$
$769$	−8.41625e11	−0.0867861	−0.0433930	+	0.999058i	$0.513817\pi$
$770$	0	0
$771$	8.42419e11	0.0858585
$772$	0	0
$773$	3.90333e12	0.393213	0.196607	−	0.980482i	$-0.437008\pi$
$773$	3.90333e12	0.393213	0.196607	+	0.980482i	$0.437008\pi$
$774$	0	0
$775$	−1.76684e12	−0.175930
$776$	0	0
$777$	−2.88526e11	−0.0283982
$778$	0	0
$779$	2.11241e12	0.205523
$780$	0	0
$781$	−1.86555e12	−0.179422
$782$	0	0
$783$	−8.42507e12	−0.801024
$784$	0	0
$785$	9.13716e12	0.858812
$786$	0	0
$787$	1.42677e13	1.32577	0.662884	−	0.748723i	$-0.269332\pi$
$787$	1.42677e13	1.32577	0.662884	+	0.748723i	$0.269332\pi$
$788$	0	0
$789$	−7.26562e11	−0.0667461
$790$	0	0
$791$	1.87460e12	0.170261
$792$	0	0
$793$	1.07637e12	0.0966570
$794$	0	0
$795$	−1.43503e12	−0.127411
$796$	0	0
$797$	1.77300e13	1.55649	0.778245	−	0.627960i	$-0.216110\pi$
$797$	1.77300e13	1.55649	0.778245	+	0.627960i	$0.216110\pi$
$798$	0	0
$799$	−1.02395e12	−0.0888829
$800$	0	0
$801$	1.53636e13	1.31870
$802$	0	0
$803$	9.81453e12	0.833009
$804$	0	0
$805$	−2.24634e12	−0.188535
$806$	0	0
$807$	−2.79964e12	−0.232366
$808$	0	0
$809$	6.57374e12	0.539565	0.269783	−	0.962921i	$-0.413048\pi$
$809$	6.57374e12	0.539565	0.269783	+	0.962921i	$0.413048\pi$
$810$	0	0
$811$	−1.30848e13	−1.06212	−0.531060	−	0.847334i	$-0.678206\pi$
$811$	−1.30848e13	−1.06212	−0.531060	+	0.847334i	$0.678206\pi$
$812$	0	0
$813$	6.49774e12	0.521621
$814$	0	0
$815$	−9.06060e11	−0.0719363
$816$	0	0
$817$	7.37177e12	0.578858
$818$	0	0
$819$	−4.55171e11	−0.0353506
$820$	0	0
$821$	4.25638e12	0.326961	0.163480	−	0.986547i	$-0.447728\pi$
$821$	4.25638e12	0.326961	0.163480	+	0.986547i	$0.447728\pi$
$822$	0	0
$823$	4.41691e12	0.335598	0.167799	−	0.985821i	$-0.446334\pi$
$823$	4.41691e12	0.335598	0.167799	+	0.985821i	$0.446334\pi$
$824$	0	0
$825$	4.91613e11	0.0369471
$826$	0	0
$827$	1.65971e13	1.23384	0.616919	−	0.787026i	$-0.288380\pi$
$827$	1.65971e13	1.23384	0.616919	+	0.787026i	$0.288380\pi$
$828$	0	0
$829$	−1.79742e13	−1.32176	−0.660882	−	0.750490i	$-0.729817\pi$
$829$	−1.79742e13	−1.32176	−0.660882	+	0.750490i	$0.729817\pi$
$830$	0	0
$831$	5.18363e12	0.377077
$832$	0	0
$833$	1.37042e12	0.0986171
$834$	0	0
$835$	−1.52697e13	−1.08703
$836$	0	0
$837$	6.65667e12	0.468805
$838$	0	0
$839$	−3.10688e12	−0.216469	−0.108234	−	0.994125i	$-0.534520\pi$
$839$	−3.10688e12	−0.216469	−0.108234	+	0.994125i	$0.534520\pi$
$840$	0	0
$841$	1.32025e13	0.910070
$842$	0	0
$843$	8.65626e12	0.590345
$844$	0	0
$845$	−1.00845e12	−0.0680452
$846$	0	0
$847$	1.44580e12	0.0965234
$848$	0	0
$849$	1.31951e12	0.0871622
$850$	0	0
$851$	1.54581e13	1.01036
$852$	0	0
$853$	1.20700e13	0.780613	0.390306	−	0.920685i	$-0.372369\pi$
$853$	1.20700e13	0.780613	0.390306	+	0.920685i	$0.372369\pi$
$854$	0	0
$855$	−9.47279e12	−0.606220
$856$	0	0
$857$	−6.31368e12	−0.399824	−0.199912	−	0.979814i	$-0.564066\pi$
$857$	−6.31368e12	−0.399824	−0.199912	+	0.979814i	$0.564066\pi$
$858$	0	0
$859$	−1.86663e13	−1.16974	−0.584870	−	0.811127i	$-0.698854\pi$
$859$	−1.86663e13	−1.16974	−0.584870	+	0.811127i	$0.698854\pi$
$860$	0	0
$861$	−1.87269e11	−0.0116132
$862$	0	0
$863$	1.78712e13	1.09675	0.548373	−	0.836234i	$-0.315247\pi$
$863$	1.78712e13	1.09675	0.548373	+	0.836234i	$0.315247\pi$
$864$	0	0
$865$	−1.39358e13	−0.846368
$866$	0	0
$867$	5.00357e12	0.300742
$868$	0	0
$869$	3.86149e12	0.229703
$870$	0	0
$871$	5.42926e12	0.319639
$872$	0	0
$873$	−2.62579e13	−1.53002
$874$	0	0
$875$	−2.62226e12	−0.151231
$876$	0	0
$877$	−3.76100e12	−0.214687	−0.107343	−	0.994222i	$-0.534234\pi$
$877$	−3.76100e12	−0.214687	−0.107343	+	0.994222i	$0.534234\pi$
$878$	0	0
$879$	3.33208e12	0.188263
$880$	0	0
$881$	2.05824e13	1.15108	0.575539	−	0.817775i	$-0.304792\pi$
$881$	2.05824e13	1.15108	0.575539	+	0.817775i	$0.304792\pi$
$882$	0	0
$883$	3.01536e13	1.66923	0.834616	−	0.550833i	$-0.185690\pi$
$883$	3.01536e13	1.66923	0.834616	+	0.550833i	$0.185690\pi$
$884$	0	0
$885$	5.98619e12	0.328024
$886$	0	0
$887$	3.34317e13	1.81343	0.906717	−	0.421739i	$-0.138580\pi$
$887$	3.34317e13	1.81343	0.906717	+	0.421739i	$0.138580\pi$
$888$	0	0
$889$	−4.26109e12	−0.228804
$890$	0	0
$891$	7.69533e12	0.409051
$892$	0	0
$893$	−1.26765e13	−0.667063
$894$	0	0
$895$	−4.11379e12	−0.214308
$896$	0	0
$897$	−2.47988e12	−0.127898
$898$	0	0
$899$	−2.18935e13	−1.11788
$900$	0	0
$901$	9.43448e11	0.0476932
$902$	0	0
$903$	−6.53519e11	−0.0327087
$904$	0	0
$905$	−1.92976e13	−0.956277
$906$	0	0
$907$	−2.03004e13	−0.996030	−0.498015	−	0.867169i	$-0.665937\pi$
$907$	−2.03004e13	−0.996030	−0.498015	+	0.867169i	$0.665937\pi$
$908$	0	0
$909$	7.41625e12	0.360285
$910$	0	0
$911$	−1.74407e13	−0.838940	−0.419470	−	0.907769i	$-0.637784\pi$
$911$	−1.74407e13	−0.838940	−0.419470	+	0.907769i	$0.637784\pi$
$912$	0	0
$913$	1.57534e12	0.0750337
$914$	0	0
$915$	−1.98588e12	−0.0936607
$916$	0	0
$917$	−4.12897e12	−0.192832
$918$	0	0
$919$	−1.69619e13	−0.784430	−0.392215	−	0.919874i	$-0.628291\pi$
$919$	−1.69619e13	−0.784430	−0.392215	+	0.919874i	$0.628291\pi$
$920$	0	0
$921$	5.11332e12	0.234172
$922$	0	0
$923$	1.96251e12	0.0890031
$924$	0	0
$925$	3.22372e12	0.144784
$926$	0	0
$927$	3.32984e13	1.48103
$928$	0	0
$929$	4.90922e12	0.216243	0.108121	−	0.994138i	$-0.465516\pi$
$929$	4.90922e12	0.216243	0.108121	+	0.994138i	$0.465516\pi$
$930$	0	0
$931$	1.69658e13	0.740117
$932$	0	0
$933$	4.95408e12	0.214040
$934$	0	0
$935$	1.16277e12	0.0497556
$936$	0	0
$937$	1.13139e13	0.479497	0.239749	−	0.970835i	$-0.422935\pi$
$937$	1.13139e13	0.479497	0.239749	+	0.970835i	$0.422935\pi$
$938$	0	0
$939$	−4.22980e12	−0.177552
$940$	0	0
$941$	7.23052e11	0.0300619	0.0150310	−	0.999887i	$-0.495215\pi$
$941$	7.23052e11	0.0300619	0.0150310	+	0.999887i	$0.495215\pi$
$942$	0	0
$943$	1.00332e13	0.413176
$944$	0	0
$945$	1.76495e12	0.0719930
$946$	0	0
$947$	−4.04364e13	−1.63380	−0.816898	−	0.576782i	$-0.804308\pi$
$947$	−4.04364e13	−1.63380	−0.816898	+	0.576782i	$0.804308\pi$
$948$	0	0
$949$	−1.03247e13	−0.413216
$950$	0	0
$951$	1.27103e13	0.503899
$952$	0	0
$953$	4.03093e13	1.58302	0.791511	−	0.611155i	$-0.209295\pi$
$953$	4.03093e13	1.58302	0.791511	+	0.611155i	$0.209295\pi$
$954$	0	0
$955$	3.27305e13	1.27332
$956$	0	0
$957$	6.09172e12	0.234767
$958$	0	0
$959$	−3.44291e12	−0.131444
$960$	0	0
$961$	−9.14153e12	−0.345751
$962$	0	0
$963$	−1.34135e13	−0.502602
$964$	0	0
$965$	3.30330e12	0.122624
$966$	0	0
$967$	−1.74415e13	−0.641451	−0.320726	−	0.947172i	$-0.603927\pi$
$967$	−1.74415e13	−0.641451	−0.320726	+	0.947172i	$0.603927\pi$
$968$	0	0
$969$	−6.33314e11	−0.0230761
$970$	0	0
$971$	5.09597e13	1.83967	0.919836	−	0.392303i	$-0.128322\pi$
$971$	5.09597e13	1.83967	0.919836	+	0.392303i	$0.128322\pi$
$972$	0	0
$973$	−7.47565e12	−0.267388
$974$	0	0
$975$	−5.17166e11	−0.0183277
$976$	0	0
$977$	3.19652e13	1.12241	0.561205	−	0.827677i	$-0.310338\pi$
$977$	3.19652e13	1.12241	0.561205	+	0.827677i	$0.310338\pi$
$978$	0	0
$979$	−2.33469e13	−0.812283
$980$	0	0
$981$	−3.69848e13	−1.27501
$982$	0	0
$983$	5.12376e13	1.75024	0.875121	−	0.483904i	$-0.160782\pi$
$983$	5.12376e13	1.75024	0.875121	+	0.483904i	$0.160782\pi$
$984$	0	0
$985$	−1.70890e13	−0.578434
$986$	0	0
$987$	1.12379e12	0.0376927
$988$	0	0
$989$	3.50131e13	1.16372
$990$	0	0
$991$	1.90444e13	0.627243	0.313622	−	0.949548i	$-0.398458\pi$
$991$	1.90444e13	0.627243	0.313622	+	0.949548i	$0.398458\pi$
$992$	0	0
$993$	2.93130e12	0.0956730
$994$	0	0
$995$	−1.50589e13	−0.487068
$996$	0	0
$997$	4.27709e13	1.37095	0.685473	−	0.728098i	$-0.259595\pi$
$997$	4.27709e13	1.37095	0.685473	+	0.728098i	$0.259595\pi$
$998$	0	0
$999$	−1.21455e13	−0.385808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.g.1.2		4
4.3	odd	2		13.10.a.a.1.3	✓	4
12.11	even	2		117.10.a.c.1.2		4
20.19	odd	2		325.10.a.a.1.2		4
52.51	odd	2		169.10.a.a.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.a.a.1.3	✓	4	4.3	odd	2
117.10.a.c.1.2		4	12.11	even	2
169.10.a.a.1.2		4	52.51	odd	2
208.10.a.g.1.2		4	1.1	even	1	trivial
325.10.a.a.1.2		4	20.19	odd	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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q-42.6243 q^{3} -1236.25 q^{5} -892.010 q^{7} -17866.2 q^{9} +O(q^{10})\)	\(q-42.6243 q^{3} -1236.25 q^{5} -892.010 q^{7} -17866.2 q^{9} +27149.8 q^{11} -28561.0 q^{13} +52694.2 q^{15} -34643.4 q^{17} -428885. q^{19} +38021.3 q^{21} -2.03704e6 q^{23} -424814. q^{25} +1.60051e6 q^{27} -5.26400e6 q^{29} +4.15910e6 q^{31} -1.15724e6 q^{33} +1.10275e6 q^{35} -7.58854e6 q^{37} +1.21739e6 q^{39} -4.92536e6 q^{41} -1.71882e7 q^{43} +2.20870e7 q^{45} +2.95568e7 q^{47} -3.95579e7 q^{49} +1.47665e6 q^{51} -2.72331e7 q^{53} -3.35640e7 q^{55} +1.82809e7 q^{57} +1.13602e8 q^{59} -3.76868e7 q^{61} +1.59368e7 q^{63} +3.53085e7 q^{65} -1.90094e8 q^{67} +8.68273e7 q^{69} -6.87130e7 q^{71} +3.61495e8 q^{73} +1.81074e7 q^{75} -2.42179e7 q^{77} +1.42229e8 q^{79} +2.83439e8 q^{81} +5.80240e7 q^{83} +4.28279e7 q^{85} +2.24374e8 q^{87} -8.59928e8 q^{89} +2.54767e7 q^{91} -1.77279e8 q^{93} +5.30208e8 q^{95} +1.46970e9 q^{97} -4.85064e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 163 q^{3} + 471 q^{5} + 11241 q^{7} - 29953 q^{9}+O(q^{10}) \) 4 * q + 163 * q^3 + 471 * q^5 + 11241 * q^7 - 29953 * q^9	\( 4 q + 163 q^{3} + 471 q^{5} + 11241 q^{7} - 29953 q^{9} + 40140 q^{11} - 114244 q^{13} - 83307 q^{15} + 78717 q^{17} - 209664 q^{19} + 1138431 q^{21} + 4257444 q^{23} - 2900157 q^{25} + 2077801 q^{27} - 1647936 q^{29} + 11366002 q^{31} - 14413222 q^{33} + 13789797 q^{35} + 4636891 q^{37} - 4655443 q^{39} + 13859538 q^{41} + 33368081 q^{43} - 17423928 q^{45} + 3943005 q^{47} + 23294923 q^{49} + 19664471 q^{51} - 171019326 q^{53} + 121160538 q^{55} - 47829030 q^{57} + 63389388 q^{59} + 77050190 q^{61} + 155695476 q^{63} - 13452231 q^{65} + 41174072 q^{67} + 546642556 q^{69} - 252460989 q^{71} + 594415068 q^{73} - 533318748 q^{75} + 561950454 q^{77} - 115998984 q^{79} + 437803700 q^{81} + 79577862 q^{83} + 549463469 q^{85} + 1087526510 q^{87} - 1152240276 q^{89} - 321054201 q^{91} + 1618266556 q^{93} + 1273705170 q^{95} + 1049098084 q^{97} - 2132181050 q^{99}+O(q^{100}) \) 4 * q + 163 * q^3 + 471 * q^5 + 11241 * q^7 - 29953 * q^9 + 40140 * q^11 - 114244 * q^13 - 83307 * q^15 + 78717 * q^17 - 209664 * q^19 + 1138431 * q^21 + 4257444 * q^23 - 2900157 * q^25 + 2077801 * q^27 - 1647936 * q^29 + 11366002 * q^31 - 14413222 * q^33 + 13789797 * q^35 + 4636891 * q^37 - 4655443 * q^39 + 13859538 * q^41 + 33368081 * q^43 - 17423928 * q^45 + 3943005 * q^47 + 23294923 * q^49 + 19664471 * q^51 - 171019326 * q^53 + 121160538 * q^55 - 47829030 * q^57 + 63389388 * q^59 + 77050190 * q^61 + 155695476 * q^63 - 13452231 * q^65 + 41174072 * q^67 + 546642556 * q^69 - 252460989 * q^71 + 594415068 * q^73 - 533318748 * q^75 + 561950454 * q^77 - 115998984 * q^79 + 437803700 * q^81 + 79577862 * q^83 + 549463469 * q^85 + 1087526510 * q^87 - 1152240276 * q^89 - 321054201 * q^91 + 1618266556 * q^93 + 1273705170 * q^95 + 1049098084 * q^97 - 2132181050 * q^99

Introduction

L-functions

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