LMFDB - Embedded newform 2800.2.g.t.449.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2800,2,Mod(449,2800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2800.449"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-6,0,-2,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	2800.449
Dual form			2800.2.g.t.449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} +1.56155 q^{11} -0.438447i q^{13} -0.438447i q^{17} -7.12311 q^{19} -1.56155 q^{21} +3.12311i q^{23} +5.56155i q^{27} -6.68466 q^{29} +2.43845i q^{33} +6.00000i q^{37} +0.684658 q^{39} +5.12311 q^{41} +0.876894i q^{43} +8.68466i q^{47} -1.00000 q^{49} +0.684658 q^{51} +5.12311i q^{53} -11.1231i q^{57} -4.00000 q^{59} +15.3693 q^{61} +0.561553i q^{63} -10.2462i q^{67} -4.87689 q^{69} -8.00000 q^{71} +12.2462i q^{73} +1.56155i q^{77} -2.43845 q^{79} -7.00000 q^{81} +4.00000i q^{83} -10.4384i q^{87} +1.12311 q^{89} +0.438447 q^{91} +5.80776i q^{97} +0.876894 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9} - 2 q^{11} - 12 q^{19} + 2 q^{21} - 2 q^{29} - 22 q^{39} + 4 q^{41} - 4 q^{49} - 22 q^{51} - 16 q^{59} + 12 q^{61} - 36 q^{69} - 32 q^{71} - 18 q^{79} - 28 q^{81} - 12 q^{89} + 10 q^{91} + 20 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 2 * q^11 - 12 * q^19 + 2 * q^21 - 2 * q^29 - 22 * q^39 + 4 * q^41 - 4 * q^49 - 22 * q^51 - 16 * q^59 + 12 * q^61 - 36 * q^69 - 32 * q^71 - 18 * q^79 - 28 * q^81 - 12 * q^89 + 10 * q^91 + 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$.

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.56155i	0.901563i	0.892634	+	0.450781i	$0.148855\pi$
$3$	1.56155i	0.901563i	−0.892634	+	0.450781i	$0.851145\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	− 0.438447i	− 0.121603i	−0.998150	−	0.0608017i	$-0.980634\pi$
$13$	− 0.438447i	− 0.121603i	0.998150	−	0.0608017i	$-0.0193657\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.438447i	− 0.106339i	−0.998586	−	0.0531695i	$-0.983068\pi$
$17$	− 0.438447i	− 0.106339i	0.998586	−	0.0531695i	$-0.0169324\pi$
$18$	0	0
$19$	−7.12311	−1.63415	−0.817076	−	0.576530i	$-0.804407\pi$
$19$	−7.12311	−1.63415	−0.817076	+	0.576530i	$0.804407\pi$
$20$	0	0
$21$	−1.56155	−0.340759
$22$	0	0
$23$	3.12311i	0.651213i	0.945505	+	0.325606i	$0.105568\pi$
$23$	3.12311i	0.651213i	−0.945505	+	0.325606i	$0.894432\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.56155i	1.07032i
$28$	0	0
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	2.43845i	0.424479i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0.684658	0.109633
$40$	0	0
$41$	5.12311	0.800095	0.400047	−	0.916494i	$-0.368994\pi$
$41$	5.12311	0.800095	0.400047	+	0.916494i	$0.368994\pi$
$42$	0	0
$43$	0.876894i	0.133725i	0.997762	+	0.0668626i	$0.0212989\pi$
$43$	0.876894i	0.133725i	−0.997762	+	0.0668626i	$0.978701\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.68466i	1.26679i	0.773830	+	0.633394i	$0.218339\pi$
$47$	8.68466i	1.26679i	−0.773830	+	0.633394i	$0.781661\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0.684658	0.0958714
$52$	0	0
$53$	5.12311i	0.703713i	0.936054	+	0.351856i	$0.114449\pi$
$53$	5.12311i	0.703713i	−0.936054	+	0.351856i	$0.885551\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 11.1231i	− 1.47329i
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	15.3693	1.96784	0.983920	−	0.178611i	$-0.0571605\pi$
$61$	15.3693	1.96784	0.983920	+	0.178611i	$0.0571605\pi$
$62$	0	0
$63$	0.561553i	0.0707490i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.2462i	− 1.25177i	−0.779914	−	0.625887i	$-0.784737\pi$
$67$	− 10.2462i	− 1.25177i	0.779914	−	0.625887i	$-0.215263\pi$
$68$	0	0
$69$	−4.87689	−0.587109
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	12.2462i	1.43331i	0.697428	+	0.716655i	$0.254328\pi$
$73$	12.2462i	1.43331i	−0.697428	+	0.716655i	$0.745672\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.56155i	0.177955i
$78$	0	0
$79$	−2.43845	−0.274347	−0.137173	−	0.990547i	$-0.543802\pi$
$79$	−2.43845	−0.274347	−0.137173	+	0.990547i	$0.543802\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 10.4384i	− 1.11912i
$88$	0	0
$89$	1.12311	0.119049	0.0595245	−	0.998227i	$-0.481042\pi$
$89$	1.12311	0.119049	0.0595245	+	0.998227i	$0.481042\pi$
$90$	0	0
$91$	0.438447	0.0459618
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.80776i	0.589689i	0.955545	+	0.294845i	$0.0952679\pi$
$97$	5.80776i	0.589689i	−0.955545	+	0.294845i	$0.904732\pi$
$98$	0	0
$99$	0.876894	0.0881312
$100$	0	0
$101$	−16.2462	−1.61656	−0.808279	−	0.588799i	$-0.799601\pi$
$101$	−16.2462	−1.61656	−0.808279	+	0.588799i	$0.799601\pi$
$102$	0	0
$103$	5.56155i	0.547996i	0.961730	+	0.273998i	$0.0883462\pi$
$103$	5.56155i	0.547996i	−0.961730	+	0.273998i	$0.911654\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.3693i	− 1.29246i	−0.763142	−	0.646230i	$-0.776345\pi$
$107$	− 13.3693i	− 1.29246i	0.763142	−	0.646230i	$-0.223655\pi$
$108$	0	0
$109$	−5.31534	−0.509117	−0.254559	−	0.967057i	$-0.581930\pi$
$109$	−5.31534	−0.509117	−0.254559	+	0.967057i	$0.581930\pi$
$110$	0	0
$111$	−9.36932	−0.889296
$112$	0	0
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.246211i	− 0.0227622i
$118$	0	0
$119$	0.438447	0.0401924
$120$	0	0
$121$	−8.56155	−0.778323
$122$	0	0
$123$	8.00000i	0.721336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.24621i	0.554262i	0.960832	+	0.277131i	$0.0893835\pi$
$127$	6.24621i	0.554262i	−0.960832	+	0.277131i	$0.910616\pi$
$128$	0	0
$129$	−1.36932	−0.120562
$130$	0	0
$131$	0.876894	0.0766146	0.0383073	−	0.999266i	$-0.487803\pi$
$131$	0.876894	0.0766146	0.0383073	+	0.999266i	$0.487803\pi$
$132$	0	0
$133$	− 7.12311i	− 0.617652i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.1231i	− 1.46293i	−0.681881	−	0.731463i	$-0.738838\pi$
$137$	− 17.1231i	− 1.46293i	0.681881	−	0.731463i	$-0.261162\pi$
$138$	0	0
$139$	−15.1231	−1.28273	−0.641363	−	0.767238i	$-0.721631\pi$
$139$	−15.1231	−1.28273	−0.641363	+	0.767238i	$0.721631\pi$
$140$	0	0
$141$	−13.5616	−1.14209
$142$	0	0
$143$	− 0.684658i	− 0.0572540i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.56155i	− 0.128795i
$148$	0	0
$149$	−12.2462	−1.00325	−0.501624	−	0.865086i	$-0.667264\pi$
$149$	−12.2462	−1.00325	−0.501624	+	0.865086i	$0.667264\pi$
$150$	0	0
$151$	6.93087	0.564026	0.282013	−	0.959411i	$-0.408998\pi$
$151$	6.93087	0.564026	0.282013	+	0.959411i	$0.408998\pi$
$152$	0	0
$153$	− 0.246211i	− 0.0199050i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.2462i	1.61582i	0.589303	+	0.807912i	$0.299402\pi$
$157$	20.2462i	1.61582i	−0.589303	+	0.807912i	$0.700598\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	−3.12311	−0.246135
$162$	0	0
$163$	− 7.12311i	− 0.557925i	−0.960302	−	0.278962i	$-0.910010\pi$
$163$	− 7.12311i	− 0.557925i	0.960302	−	0.278962i	$-0.0899905\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.93087i	0.536327i	0.963373	+	0.268163i	$0.0864167\pi$
$167$	6.93087i	0.536327i	−0.963373	+	0.268163i	$0.913583\pi$
$168$	0	0
$169$	12.8078	0.985213
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	4.43845i	0.337449i	0.985663	+	0.168724i	$0.0539648\pi$
$173$	4.43845i	0.337449i	−0.985663	+	0.168724i	$0.946035\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.24621i	− 0.469494i
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−17.6155	−1.30935	−0.654676	−	0.755910i	$-0.727195\pi$
$181$	−17.6155	−1.30935	−0.654676	+	0.755910i	$0.727195\pi$
$182$	0	0
$183$	24.0000i	1.77413i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.684658i	− 0.0500672i
$188$	0	0
$189$	−5.56155	−0.404543
$190$	0	0
$191$	13.5616	0.981280	0.490640	−	0.871363i	$-0.336763\pi$
$191$	13.5616	0.981280	0.490640	+	0.871363i	$0.336763\pi$
$192$	0	0
$193$	− 19.3693i	− 1.39423i	−0.716957	−	0.697117i	$-0.754466\pi$
$193$	− 19.3693i	− 1.39423i	0.716957	−	0.697117i	$-0.245534\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.12311i	0.0800180i	0.999199	+	0.0400090i	$0.0127387\pi$
$197$	1.12311i	0.0800180i	−0.999199	+	0.0400090i	$0.987261\pi$
$198$	0	0
$199$	−1.75379	−0.124323	−0.0621614	−	0.998066i	$-0.519799\pi$
$199$	−1.75379	−0.124323	−0.0621614	+	0.998066i	$0.519799\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	− 6.68466i	− 0.469171i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.75379i	0.121897i
$208$	0	0
$209$	−11.1231	−0.769401
$210$	0	0
$211$	−14.0540	−0.967516	−0.483758	−	0.875202i	$-0.660728\pi$
$211$	−14.0540	−0.967516	−0.483758	+	0.875202i	$0.660728\pi$
$212$	0	0
$213$	− 12.4924i	− 0.855967i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−19.1231	−1.29222
$220$	0	0
$221$	−0.192236	−0.0129312
$222$	0	0
$223$	− 2.43845i	− 0.163291i	−0.996661	−	0.0816453i	$-0.973983\pi$
$223$	− 2.43845i	− 0.163291i	0.996661	−	0.0816453i	$-0.0260175\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.3153i	− 0.751026i	−0.926817	−	0.375513i	$-0.877467\pi$
$227$	− 11.3153i	− 0.751026i	0.926817	−	0.375513i	$-0.122533\pi$
$228$	0	0
$229$	−10.8769	−0.718765	−0.359383	−	0.933190i	$-0.617013\pi$
$229$	−10.8769	−0.718765	−0.359383	+	0.933190i	$0.617013\pi$
$230$	0	0
$231$	−2.43845	−0.160438
$232$	0	0
$233$	− 5.12311i	− 0.335626i	−0.985819	−	0.167813i	$-0.946330\pi$
$233$	− 5.12311i	− 0.335626i	0.985819	−	0.167813i	$-0.0536704\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3.80776i	− 0.247341i
$238$	0	0
$239$	19.8078	1.28126	0.640629	−	0.767851i	$-0.278674\pi$
$239$	19.8078	1.28126	0.640629	+	0.767851i	$0.278674\pi$
$240$	0	0
$241$	−4.24621	−0.273523	−0.136761	−	0.990604i	$-0.543669\pi$
$241$	−4.24621	−0.273523	−0.136761	+	0.990604i	$0.543669\pi$
$242$	0	0
$243$	5.75379i	0.369106i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.12311i	0.198718i
$248$	0	0
$249$	−6.24621	−0.395838
$250$	0	0
$251$	8.87689	0.560305	0.280152	−	0.959956i	$-0.409615\pi$
$251$	8.87689	0.560305	0.280152	+	0.959956i	$0.409615\pi$
$252$	0	0
$253$	4.87689i	0.306608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 10.4924i	− 0.654499i	−0.944938	−	0.327250i	$-0.893878\pi$
$257$	− 10.4924i	− 0.654499i	0.944938	−	0.327250i	$-0.106122\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−3.75379	−0.232354
$262$	0	0
$263$	− 12.8769i	− 0.794023i	−0.917814	−	0.397012i	$-0.870047\pi$
$263$	− 12.8769i	− 0.794023i	0.917814	−	0.397012i	$-0.129953\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.75379i	0.107330i
$268$	0	0
$269$	20.7386	1.26446	0.632228	−	0.774782i	$-0.282140\pi$
$269$	20.7386	1.26446	0.632228	+	0.774782i	$0.282140\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	0.684658i	0.0414374i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 0.246211i	− 0.0147934i	−0.999973	−	0.00739670i	$-0.997646\pi$
$277$	− 0.246211i	− 0.0147934i	0.999973	−	0.00739670i	$-0.00235446\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.4384	0.742016	0.371008	−	0.928630i	$-0.379012\pi$
$281$	12.4384	0.742016	0.371008	+	0.928630i	$0.379012\pi$
$282$	0	0
$283$	− 11.3153i	− 0.672627i	−0.941750	−	0.336314i	$-0.890820\pi$
$283$	− 11.3153i	− 0.672627i	0.941750	−	0.336314i	$-0.109180\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.12311i	0.302407i
$288$	0	0
$289$	16.8078	0.988692
$290$	0	0
$291$	−9.06913	−0.531642
$292$	0	0
$293$	2.68466i	0.156839i	0.996920	+	0.0784197i	$0.0249874\pi$
$293$	2.68466i	0.156839i	−0.996920	+	0.0784197i	$0.975013\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8.68466i	0.503935i
$298$	0	0
$299$	1.36932	0.0791896
$300$	0	0
$301$	−0.876894	−0.0505434
$302$	0	0
$303$	− 25.3693i	− 1.45743i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.3153i	1.10238i	0.834378	+	0.551192i	$0.185827\pi$
$307$	19.3153i	1.10238i	−0.834378	+	0.551192i	$0.814173\pi$
$308$	0	0
$309$	−8.68466	−0.494053
$310$	0	0
$311$	−31.6155	−1.79275	−0.896376	−	0.443294i	$-0.853810\pi$
$311$	−31.6155	−1.79275	−0.896376	+	0.443294i	$0.853810\pi$
$312$	0	0
$313$	22.3002i	1.26048i	0.776400	+	0.630241i	$0.217044\pi$
$313$	22.3002i	1.26048i	−0.776400	+	0.630241i	$0.782956\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.4924i	0.589313i	0.955603	+	0.294657i	$0.0952053\pi$
$317$	10.4924i	0.589313i	−0.955603	+	0.294657i	$0.904795\pi$
$318$	0	0
$319$	−10.4384	−0.584441
$320$	0	0
$321$	20.8769	1.16523
$322$	0	0
$323$	3.12311i	0.173774i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 8.30019i	− 0.459001i
$328$	0	0
$329$	−8.68466	−0.478801
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	3.36932i	0.184637i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.50758i	− 0.0821230i	−0.999157	−	0.0410615i	$-0.986926\pi$
$337$	− 1.50758i	− 0.0821230i	0.999157	−	0.0410615i	$-0.0130740\pi$
$338$	0	0
$339$	−21.8617	−1.18737
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.12311i	− 0.382388i	−0.981552	−	0.191194i	$-0.938764\pi$
$347$	− 7.12311i	− 0.382388i	0.981552	−	0.191194i	$-0.0612360\pi$
$348$	0	0
$349$	−10.4924	−0.561646	−0.280823	−	0.959760i	$-0.590607\pi$
$349$	−10.4924	−0.561646	−0.280823	+	0.959760i	$0.590607\pi$
$350$	0	0
$351$	2.43845	0.130155
$352$	0	0
$353$	− 5.80776i	− 0.309116i	−0.987984	−	0.154558i	$-0.950605\pi$
$353$	− 5.80776i	− 0.309116i	0.987984	−	0.154558i	$-0.0493954\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.684658i	0.0362360i
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	0	0
$363$	− 13.3693i	− 0.701707i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.68466i	0.453335i	0.973972	+	0.226668i	$0.0727831\pi$
$367$	8.68466i	0.453335i	−0.973972	+	0.226668i	$0.927217\pi$
$368$	0	0
$369$	2.87689	0.149765
$370$	0	0
$371$	−5.12311	−0.265978
$372$	0	0
$373$	− 4.63068i	− 0.239768i	−0.992788	−	0.119884i	$-0.961748\pi$
$373$	− 4.63068i	− 0.239768i	0.992788	−	0.119884i	$-0.0382522\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.93087i	0.150947i
$378$	0	0
$379$	−16.4924	−0.847159	−0.423579	−	0.905859i	$-0.639227\pi$
$379$	−16.4924	−0.847159	−0.423579	+	0.905859i	$0.639227\pi$
$380$	0	0
$381$	−9.75379	−0.499702
$382$	0	0
$383$	6.24621i	0.319166i	0.987184	+	0.159583i	$0.0510150\pi$
$383$	6.24621i	0.319166i	−0.987184	+	0.159583i	$0.948985\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.492423i	0.0250312i
$388$	0	0
$389$	24.9309	1.26405	0.632023	−	0.774950i	$-0.282225\pi$
$389$	24.9309	1.26405	0.632023	+	0.774950i	$0.282225\pi$
$390$	0	0
$391$	1.36932	0.0692493
$392$	0	0
$393$	1.36932i	0.0690729i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.5616i	1.38327i	0.722245	+	0.691637i	$0.243110\pi$
$397$	27.5616i	1.38327i	−0.722245	+	0.691637i	$0.756890\pi$
$398$	0	0
$399$	11.1231	0.556852
$400$	0	0
$401$	31.5616	1.57611	0.788054	−	0.615606i	$-0.211089\pi$
$401$	31.5616	1.57611	0.788054	+	0.615606i	$0.211089\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.36932i	0.464420i
$408$	0	0
$409$	−6.49242	−0.321030	−0.160515	−	0.987033i	$-0.551315\pi$
$409$	−6.49242	−0.321030	−0.160515	+	0.987033i	$0.551315\pi$
$410$	0	0
$411$	26.7386	1.31892
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 23.6155i	− 1.15646i
$418$	0	0
$419$	26.2462	1.28221	0.641106	−	0.767453i	$-0.278476\pi$
$419$	26.2462	1.28221	0.641106	+	0.767453i	$0.278476\pi$
$420$	0	0
$421$	−2.68466	−0.130842	−0.0654211	−	0.997858i	$-0.520839\pi$
$421$	−2.68466	−0.130842	−0.0654211	+	0.997858i	$0.520839\pi$
$422$	0	0
$423$	4.87689i	0.237123i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15.3693i	0.743773i
$428$	0	0
$429$	1.06913	0.0516181
$430$	0	0
$431$	19.8078	0.954106	0.477053	−	0.878874i	$-0.341705\pi$
$431$	19.8078	0.954106	0.477053	+	0.878874i	$0.341705\pi$
$432$	0	0
$433$	− 8.24621i	− 0.396288i	−0.980173	−	0.198144i	$-0.936509\pi$
$433$	− 8.24621i	− 0.396288i	0.980173	−	0.198144i	$-0.0634913\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 22.2462i	− 1.06418i
$438$	0	0
$439$	9.36932	0.447173	0.223587	−	0.974684i	$-0.428223\pi$
$439$	9.36932	0.447173	0.223587	+	0.974684i	$0.428223\pi$
$440$	0	0
$441$	−0.561553	−0.0267406
$442$	0	0
$443$	− 2.63068i	− 0.124988i	−0.998045	−	0.0624938i	$-0.980095\pi$
$443$	− 2.63068i	− 0.124988i	0.998045	−	0.0624938i	$-0.0199054\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 19.1231i	− 0.904492i
$448$	0	0
$449$	1.80776	0.0853137	0.0426568	−	0.999090i	$-0.486418\pi$
$449$	1.80776	0.0853137	0.0426568	+	0.999090i	$0.486418\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	10.8229i	0.508505i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17.1231i	− 0.800985i	−0.916300	−	0.400493i	$-0.868839\pi$
$457$	− 17.1231i	− 0.800985i	0.916300	−	0.400493i	$-0.131161\pi$
$458$	0	0
$459$	2.43845	0.113817
$460$	0	0
$461$	−13.1231	−0.611204	−0.305602	−	0.952159i	$-0.598858\pi$
$461$	−13.1231	−0.611204	−0.305602	+	0.952159i	$0.598858\pi$
$462$	0	0
$463$	12.4924i	0.580572i	0.956940	+	0.290286i	$0.0937505\pi$
$463$	12.4924i	0.580572i	−0.956940	+	0.290286i	$0.906250\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 22.4384i	− 1.03833i	−0.854675	−	0.519164i	$-0.826243\pi$
$467$	− 22.4384i	− 1.03833i	0.854675	−	0.519164i	$-0.173757\pi$
$468$	0	0
$469$	10.2462	0.473126
$470$	0	0
$471$	−31.6155	−1.45677
$472$	0	0
$473$	1.36932i	0.0629613i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.87689i	0.131724i
$478$	0	0
$479$	4.87689	0.222831	0.111415	−	0.993774i	$-0.464462\pi$
$479$	4.87689	0.222831	0.111415	+	0.993774i	$0.464462\pi$
$480$	0	0
$481$	2.63068	0.119949
$482$	0	0
$483$	− 4.87689i	− 0.221906i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.12311i	0.141521i	0.997493	+	0.0707607i	$0.0225427\pi$
$487$	3.12311i	0.141521i	−0.997493	+	0.0707607i	$0.977457\pi$
$488$	0	0
$489$	11.1231	0.503004
$490$	0	0
$491$	41.1771	1.85830	0.929148	−	0.369708i	$-0.120542\pi$
$491$	41.1771	1.85830	0.929148	+	0.369708i	$0.120542\pi$
$492$	0	0
$493$	2.93087i	0.132000i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	41.1771	1.84334	0.921670	−	0.387976i	$-0.126826\pi$
$499$	41.1771	1.84334	0.921670	+	0.387976i	$0.126826\pi$
$500$	0	0
$501$	−10.8229	−0.483532
$502$	0	0
$503$	38.9309i	1.73584i	0.496703	+	0.867921i	$0.334544\pi$
$503$	38.9309i	1.73584i	−0.496703	+	0.867921i	$0.665456\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	20.0000i	0.888231i
$508$	0	0
$509$	11.7538	0.520978	0.260489	−	0.965477i	$-0.416116\pi$
$509$	11.7538	0.520978	0.260489	+	0.965477i	$0.416116\pi$
$510$	0	0
$511$	−12.2462	−0.541740
$512$	0	0
$513$	− 39.6155i	− 1.74907i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13.5616i	0.596436i
$518$	0	0
$519$	−6.93087	−0.304231
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	40.4924i	1.77061i	0.465011	+	0.885305i	$0.346050\pi$
$523$	40.4924i	1.77061i	−0.465011	+	0.885305i	$0.653950\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.2462	0.575922
$530$	0	0
$531$	−2.24621	−0.0974773
$532$	0	0
$533$	− 2.24621i	− 0.0972942i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	31.2311i	1.34772i
$538$	0	0
$539$	−1.56155	−0.0672608
$540$	0	0
$541$	−37.8078	−1.62548	−0.812741	−	0.582625i	$-0.802026\pi$
$541$	−37.8078	−1.62548	−0.812741	+	0.582625i	$0.802026\pi$
$542$	0	0
$543$	− 27.5076i	− 1.18046i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.24621i	0.0960411i	0.998846	+	0.0480205i	$0.0152913\pi$
$547$	2.24621i	0.0960411i	−0.998846	+	0.0480205i	$0.984709\pi$
$548$	0	0
$549$	8.63068	0.368349
$550$	0	0
$551$	47.6155	2.02849
$552$	0	0
$553$	− 2.43845i	− 0.103693i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 13.1231i	− 0.556044i	−0.960575	−	0.278022i	$-0.910321\pi$
$557$	− 13.1231i	− 0.556044i	0.960575	−	0.278022i	$-0.0896788\pi$
$558$	0	0
$559$	0.384472	0.0162614
$560$	0	0
$561$	1.06913	0.0451387
$562$	0	0
$563$	− 28.0000i	− 1.18006i	−0.807382	−	0.590030i	$-0.799116\pi$
$563$	− 28.0000i	− 1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 7.00000i	− 0.293972i
$568$	0	0
$569$	30.9848	1.29895	0.649476	−	0.760382i	$-0.274988\pi$
$569$	30.9848	1.29895	0.649476	+	0.760382i	$0.274988\pi$
$570$	0	0
$571$	−40.4924	−1.69456	−0.847278	−	0.531150i	$-0.821760\pi$
$571$	−40.4924	−1.69456	−0.847278	+	0.531150i	$0.821760\pi$
$572$	0	0
$573$	21.1771i	0.884685i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 24.0540i	− 1.00138i	−0.865627	−	0.500690i	$-0.833080\pi$
$577$	− 24.0540i	− 1.00138i	0.865627	−	0.500690i	$-0.166920\pi$
$578$	0	0
$579$	30.2462	1.25699
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	8.00000i	0.331326i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.2462i	1.08330i	0.840605	+	0.541649i	$0.182200\pi$
$587$	26.2462i	1.08330i	−0.840605	+	0.541649i	$0.817800\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−1.75379	−0.0721412
$592$	0	0
$593$	27.5616i	1.13182i	0.824468	+	0.565909i	$0.191475\pi$
$593$	27.5616i	1.13182i	−0.824468	+	0.565909i	$0.808525\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2.73863i	− 0.112085i
$598$	0	0
$599$	−11.8078	−0.482452	−0.241226	−	0.970469i	$-0.577550\pi$
$599$	−11.8078	−0.482452	−0.241226	+	0.970469i	$0.577550\pi$
$600$	0	0
$601$	6.49242	0.264831	0.132416	−	0.991194i	$-0.457727\pi$
$601$	6.49242	0.264831	0.132416	+	0.991194i	$0.457727\pi$
$602$	0	0
$603$	− 5.75379i	− 0.234312i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	42.0540i	1.70692i	0.521160	+	0.853459i	$0.325500\pi$
$607$	42.0540i	1.70692i	−0.521160	+	0.853459i	$0.674500\pi$
$608$	0	0
$609$	10.4384	0.422987
$610$	0	0
$611$	3.80776	0.154046
$612$	0	0
$613$	− 40.7386i	− 1.64542i	−0.568463	−	0.822709i	$-0.692462\pi$
$613$	− 40.7386i	− 1.64542i	0.568463	−	0.822709i	$-0.307538\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.2462i	1.29818i	0.760710	+	0.649092i	$0.224851\pi$
$617$	32.2462i	1.29818i	−0.760710	+	0.649092i	$0.775149\pi$
$618$	0	0
$619$	32.1080	1.29053	0.645264	−	0.763960i	$-0.276747\pi$
$619$	32.1080	1.29053	0.645264	+	0.763960i	$0.276747\pi$
$620$	0	0
$621$	−17.3693	−0.697007
$622$	0	0
$623$	1.12311i	0.0449963i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 17.3693i	− 0.693664i
$628$	0	0
$629$	2.63068	0.104892
$630$	0	0
$631$	11.8078	0.470060	0.235030	−	0.971988i	$-0.424481\pi$
$631$	11.8078	0.470060	0.235030	+	0.971988i	$0.424481\pi$
$632$	0	0
$633$	− 21.9460i	− 0.872276i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.438447i	0.0173719i
$638$	0	0
$639$	−4.49242	−0.177717
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	1.56155i	0.0615816i	0.999526	+	0.0307908i	$0.00980257\pi$
$643$	1.56155i	0.0615816i	−0.999526	+	0.0307908i	$0.990197\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 36.4924i	− 1.43467i	−0.696731	−	0.717333i	$-0.745363\pi$
$647$	− 36.4924i	− 1.43467i	0.696731	−	0.717333i	$-0.254637\pi$
$648$	0	0
$649$	−6.24621	−0.245185
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.2311i	1.30043i	0.759750	+	0.650216i	$0.225322\pi$
$653$	33.2311i	1.30043i	−0.759750	+	0.650216i	$0.774678\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.87689i	0.268293i
$658$	0	0
$659$	9.17708	0.357488	0.178744	−	0.983896i	$-0.442797\pi$
$659$	9.17708	0.357488	0.178744	+	0.983896i	$0.442797\pi$
$660$	0	0
$661$	−5.12311	−0.199266	−0.0996329	−	0.995024i	$-0.531767\pi$
$661$	−5.12311	−0.199266	−0.0996329	+	0.995024i	$0.531767\pi$
$662$	0	0
$663$	− 0.300187i	− 0.0116583i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 20.8769i	− 0.808357i
$668$	0	0
$669$	3.80776	0.147217
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	− 31.8617i	− 1.22818i	−0.789236	−	0.614090i	$-0.789523\pi$
$673$	− 31.8617i	− 1.22818i	0.789236	−	0.614090i	$-0.210477\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.93087i	0.189509i	0.995501	+	0.0947544i	$0.0302066\pi$
$677$	4.93087i	0.189509i	−0.995501	+	0.0947544i	$0.969793\pi$
$678$	0	0
$679$	−5.80776	−0.222882
$680$	0	0
$681$	17.6695	0.677097
$682$	0	0
$683$	− 6.73863i	− 0.257847i	−0.991655	−	0.128923i	$-0.958848\pi$
$683$	− 6.73863i	− 0.257847i	0.991655	−	0.128923i	$-0.0411521\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 16.9848i	− 0.648012i
$688$	0	0
$689$	2.24621	0.0855738
$690$	0	0
$691$	24.4924	0.931736	0.465868	−	0.884854i	$-0.345742\pi$
$691$	24.4924	0.931736	0.465868	+	0.884854i	$0.345742\pi$
$692$	0	0
$693$	0.876894i	0.0333105i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 2.24621i	− 0.0850813i
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	28.9309	1.09270	0.546352	−	0.837556i	$-0.316016\pi$
$701$	28.9309	1.09270	0.546352	+	0.837556i	$0.316016\pi$
$702$	0	0
$703$	− 42.7386i	− 1.61192i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 16.2462i	− 0.611002i
$708$	0	0
$709$	−27.1771	−1.02066	−0.510328	−	0.859980i	$-0.670476\pi$
$709$	−27.1771	−1.02066	−0.510328	+	0.859980i	$0.670476\pi$
$710$	0	0
$711$	−1.36932	−0.0513534
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	30.9309i	1.15513i
$718$	0	0
$719$	8.38447	0.312688	0.156344	−	0.987703i	$-0.450029\pi$
$719$	8.38447	0.312688	0.156344	+	0.987703i	$0.450029\pi$
$720$	0	0
$721$	−5.56155	−0.207123
$722$	0	0
$723$	− 6.63068i	− 0.246598i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 52.4924i	− 1.94684i	−0.229035	−	0.973418i	$-0.573557\pi$
$727$	− 52.4924i	− 1.94684i	0.229035	−	0.973418i	$-0.426443\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	0.384472	0.0142202
$732$	0	0
$733$	− 6.68466i	− 0.246903i	−0.992351	−	0.123452i	$-0.960604\pi$
$733$	− 6.68466i	− 0.246903i	0.992351	−	0.123452i	$-0.0393964\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	34.9309	1.28495	0.642476	−	0.766305i	$-0.277907\pi$
$739$	34.9309	1.28495	0.642476	+	0.766305i	$0.277907\pi$
$740$	0	0
$741$	−4.87689	−0.179157
$742$	0	0
$743$	− 32.9848i	− 1.21010i	−0.796189	−	0.605048i	$-0.793154\pi$
$743$	− 32.9848i	− 1.21010i	0.796189	−	0.605048i	$-0.206846\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.24621i	0.0821846i
$748$	0	0
$749$	13.3693	0.488504
$750$	0	0
$751$	−17.0691	−0.622861	−0.311431	−	0.950269i	$-0.600808\pi$
$751$	−17.0691	−0.622861	−0.311431	+	0.950269i	$0.600808\pi$
$752$	0	0
$753$	13.8617i	0.505150i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	39.3693i	1.43090i	0.698663	+	0.715451i	$0.253779\pi$
$757$	39.3693i	1.43090i	−0.698663	+	0.715451i	$0.746221\pi$
$758$	0	0
$759$	−7.61553	−0.276426
$760$	0	0
$761$	48.2462	1.74892	0.874462	−	0.485094i	$-0.161215\pi$
$761$	48.2462	1.74892	0.874462	+	0.485094i	$0.161215\pi$
$762$	0	0
$763$	− 5.31534i	− 0.192428i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.75379i	0.0633256i
$768$	0	0
$769$	42.4924	1.53232	0.766158	−	0.642652i	$-0.222166\pi$
$769$	42.4924	1.53232	0.766158	+	0.642652i	$0.222166\pi$
$770$	0	0
$771$	16.3845	0.590072
$772$	0	0
$773$	− 36.9309i	− 1.32831i	−0.747594	−	0.664156i	$-0.768791\pi$
$773$	− 36.9309i	− 1.32831i	0.747594	−	0.664156i	$-0.231209\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 9.36932i	− 0.336122i
$778$	0	0
$779$	−36.4924	−1.30748
$780$	0	0
$781$	−12.4924	−0.447014
$782$	0	0
$783$	− 37.1771i	− 1.32860i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	49.1771i	1.75297i	0.481426	+	0.876487i	$0.340119\pi$
$787$	49.1771i	1.75297i	−0.481426	+	0.876487i	$0.659881\pi$
$788$	0	0
$789$	20.1080	0.715862
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	− 6.73863i	− 0.239296i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.0540i	0.852036i	0.904715	+	0.426018i	$0.140084\pi$
$797$	24.0540i	0.852036i	−0.904715	+	0.426018i	$0.859916\pi$
$798$	0	0
$799$	3.80776	0.134709
$800$	0	0
$801$	0.630683	0.0222841
$802$	0	0
$803$	19.1231i	0.674840i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32.3845i	1.13999i
$808$	0	0
$809$	−16.5464	−0.581740	−0.290870	−	0.956763i	$-0.593945\pi$
$809$	−16.5464	−0.581740	−0.290870	+	0.956763i	$0.593945\pi$
$810$	0	0
$811$	−19.6155	−0.688794	−0.344397	−	0.938824i	$-0.611917\pi$
$811$	−19.6155	−0.688794	−0.344397	+	0.938824i	$0.611917\pi$
$812$	0	0
$813$	24.9848i	0.876257i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 6.24621i	− 0.218527i
$818$	0	0
$819$	0.246211	0.00860332
$820$	0	0
$821$	−21.4233	−0.747678	−0.373839	−	0.927494i	$-0.621959\pi$
$821$	−21.4233	−0.747678	−0.373839	+	0.927494i	$0.621959\pi$
$822$	0	0
$823$	− 36.4924i	− 1.27205i	−0.771670	−	0.636023i	$-0.780578\pi$
$823$	− 36.4924i	− 1.27205i	0.771670	−	0.636023i	$-0.219422\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.36932i	0.186709i	0.995633	+	0.0933547i	$0.0297591\pi$
$827$	5.36932i	0.186709i	−0.995633	+	0.0933547i	$0.970241\pi$
$828$	0	0
$829$	−34.8769	−1.21132	−0.605662	−	0.795722i	$-0.707092\pi$
$829$	−34.8769	−1.21132	−0.605662	+	0.795722i	$0.707092\pi$
$830$	0	0
$831$	0.384472	0.0133372
$832$	0	0
$833$	0.438447i	0.0151913i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−28.8769	−0.996941	−0.498471	−	0.866907i	$-0.666105\pi$
$839$	−28.8769	−0.996941	−0.498471	+	0.866907i	$0.666105\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	19.4233i	0.668974i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 8.56155i	− 0.294178i
$848$	0	0
$849$	17.6695	0.606416
$850$	0	0
$851$	−18.7386	−0.642352
$852$	0	0
$853$	7.26137i	0.248624i	0.992243	+	0.124312i	$0.0396724\pi$
$853$	7.26137i	0.248624i	−0.992243	+	0.124312i	$0.960328\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 15.7538i	− 0.538139i	−0.963121	−	0.269070i	$-0.913284\pi$
$857$	− 15.7538i	− 0.538139i	0.963121	−	0.269070i	$-0.0867162\pi$
$858$	0	0
$859$	−16.4924	−0.562714	−0.281357	−	0.959603i	$-0.590785\pi$
$859$	−16.4924	−0.562714	−0.281357	+	0.959603i	$0.590785\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	− 25.7538i	− 0.876669i	−0.898812	−	0.438335i	$-0.855569\pi$
$863$	− 25.7538i	− 0.876669i	0.898812	−	0.438335i	$-0.144431\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.2462i	0.891368i
$868$	0	0
$869$	−3.80776	−0.129170
$870$	0	0
$871$	−4.49242	−0.152220
$872$	0	0
$873$	3.26137i	0.110381i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 40.2462i	− 1.35902i	−0.733667	−	0.679509i	$-0.762193\pi$
$877$	− 40.2462i	− 1.35902i	0.733667	−	0.679509i	$-0.237807\pi$
$878$	0	0
$879$	−4.19224	−0.141401
$880$	0	0
$881$	−11.8617	−0.399632	−0.199816	−	0.979833i	$-0.564034\pi$
$881$	−11.8617	−0.399632	−0.199816	+	0.979833i	$0.564034\pi$
$882$	0	0
$883$	− 8.49242i	− 0.285793i	−0.989738	−	0.142896i	$-0.954358\pi$
$883$	− 8.49242i	− 0.285793i	0.989738	−	0.142896i	$-0.0456416\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 20.4924i	− 0.688068i	−0.938957	−	0.344034i	$-0.888206\pi$
$887$	− 20.4924i	− 0.688068i	0.938957	−	0.344034i	$-0.111794\pi$
$888$	0	0
$889$	−6.24621	−0.209491
$890$	0	0
$891$	−10.9309	−0.366198
$892$	0	0
$893$	− 61.8617i	− 2.07012i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.13826i	0.0713944i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	2.24621	0.0748321
$902$	0	0
$903$	− 1.36932i	− 0.0455680i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	24.1080i	0.800491i	0.916408	+	0.400246i	$0.131075\pi$
$907$	24.1080i	0.800491i	−0.916408	+	0.400246i	$0.868925\pi$
$908$	0	0
$909$	−9.12311	−0.302594
$910$	0	0
$911$	28.4924	0.943996	0.471998	−	0.881600i	$-0.343533\pi$
$911$	28.4924	0.943996	0.471998	+	0.881600i	$0.343533\pi$
$912$	0	0
$913$	6.24621i	0.206719i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.876894i	0.0289576i
$918$	0	0
$919$	40.3002	1.32938	0.664690	−	0.747119i	$-0.268564\pi$
$919$	40.3002	1.32938	0.664690	+	0.747119i	$0.268564\pi$
$920$	0	0
$921$	−30.1619	−0.993869
$922$	0	0
$923$	3.50758i	0.115453i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.12311i	0.102576i
$928$	0	0
$929$	−22.1080	−0.725338	−0.362669	−	0.931918i	$-0.618134\pi$
$929$	−22.1080	−0.725338	−0.362669	+	0.931918i	$0.618134\pi$
$930$	0	0
$931$	7.12311	0.233450
$932$	0	0
$933$	− 49.3693i	− 1.61628i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 55.6695i	− 1.81864i	−0.416094	−	0.909322i	$-0.636601\pi$
$937$	− 55.6695i	− 1.81864i	0.416094	−	0.909322i	$-0.363399\pi$
$938$	0	0
$939$	−34.8229	−1.13640
$940$	0	0
$941$	43.8617	1.42985	0.714926	−	0.699200i	$-0.246460\pi$
$941$	43.8617	1.42985	0.714926	+	0.699200i	$0.246460\pi$
$942$	0	0
$943$	16.0000i	0.521032i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	0	0
$949$	5.36932	0.174295
$950$	0	0
$951$	−16.3845	−0.531303
$952$	0	0
$953$	33.1231i	1.07296i	0.843912	+	0.536481i	$0.180247\pi$
$953$	33.1231i	1.07296i	−0.843912	+	0.536481i	$0.819753\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 16.3002i	− 0.526910i
$958$	0	0
$959$	17.1231	0.552934
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 7.50758i	− 0.241928i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	35.1231i	1.12948i	0.825268	+	0.564741i	$0.191024\pi$
$967$	35.1231i	1.12948i	−0.825268	+	0.564741i	$0.808976\pi$
$968$	0	0
$969$	−4.87689	−0.156668
$970$	0	0
$971$	−49.4773	−1.58780	−0.793901	−	0.608048i	$-0.791953\pi$
$971$	−49.4773	−1.58780	−0.793901	+	0.608048i	$0.791953\pi$
$972$	0	0
$973$	− 15.1231i	− 0.484825i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.2311i	1.06316i	0.847009	+	0.531578i	$0.178401\pi$
$977$	33.2311i	1.06316i	−0.847009	+	0.531578i	$0.821599\pi$
$978$	0	0
$979$	1.75379	0.0560513
$980$	0	0
$981$	−2.98485	−0.0952988
$982$	0	0
$983$	51.4233i	1.64015i	0.572257	+	0.820074i	$0.306068\pi$
$983$	51.4233i	1.64015i	−0.572257	+	0.820074i	$0.693932\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 13.5616i	− 0.431669i
$988$	0	0
$989$	−2.73863	−0.0870835
$990$	0	0
$991$	−12.4924	−0.396835	−0.198417	−	0.980118i	$-0.563580\pi$
$991$	−12.4924	−0.396835	−0.198417	+	0.980118i	$0.563580\pi$
$992$	0	0
$993$	− 18.7386i	− 0.594653i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.68466i	− 0.0850240i	−0.999096	−	0.0425120i	$-0.986464\pi$
$997$	− 2.68466i	− 0.0850240i	0.999096	−	0.0425120i	$-0.0135361\pi$
$998$	0	0
$999$	−33.3693	−1.05576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.g.t.449.3		4
4.3	odd	2		175.2.b.b.99.1		4
5.2	odd	4		2800.2.a.bi.1.2		2
5.3	odd	4		560.2.a.i.1.1		2
5.4	even	2	inner	2800.2.g.t.449.2		4
12.11	even	2		1575.2.d.e.1324.4		4
15.8	even	4		5040.2.a.bt.1.1		2
20.3	even	4		35.2.a.b.1.1	✓	2
20.7	even	4		175.2.a.f.1.2		2
20.19	odd	2		175.2.b.b.99.4		4
28.27	even	2		1225.2.b.f.99.1		4
35.13	even	4		3920.2.a.bs.1.2		2
40.3	even	4		2240.2.a.bh.1.1		2
40.13	odd	4		2240.2.a.bd.1.2		2
60.23	odd	4		315.2.a.e.1.2		2
60.47	odd	4		1575.2.a.p.1.1		2
60.59	even	2		1575.2.d.e.1324.1		4
140.3	odd	12		245.2.e.h.226.2		4
140.23	even	12		245.2.e.i.116.2		4
140.27	odd	4		1225.2.a.s.1.2		2
140.83	odd	4		245.2.a.d.1.1		2
140.103	odd	12		245.2.e.h.116.2		4
140.123	even	12		245.2.e.i.226.2		4
140.139	even	2		1225.2.b.f.99.4		4
220.43	odd	4		4235.2.a.m.1.2		2
260.103	even	4		5915.2.a.l.1.2		2
420.83	even	4		2205.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.a.b.1.1	✓	2	20.3	even	4
175.2.a.f.1.2		2	20.7	even	4
175.2.b.b.99.1		4	4.3	odd	2
175.2.b.b.99.4		4	20.19	odd	2
245.2.a.d.1.1		2	140.83	odd	4
245.2.e.h.116.2		4	140.103	odd	12
245.2.e.h.226.2		4	140.3	odd	12
245.2.e.i.116.2		4	140.23	even	12
245.2.e.i.226.2		4	140.123	even	12
315.2.a.e.1.2		2	60.23	odd	4
560.2.a.i.1.1		2	5.3	odd	4
1225.2.a.s.1.2		2	140.27	odd	4
1225.2.b.f.99.1		4	28.27	even	2
1225.2.b.f.99.4		4	140.139	even	2
1575.2.a.p.1.1		2	60.47	odd	4
1575.2.d.e.1324.1		4	60.59	even	2
1575.2.d.e.1324.4		4	12.11	even	2
2205.2.a.x.1.2		2	420.83	even	4
2240.2.a.bd.1.2		2	40.13	odd	4
2240.2.a.bh.1.1		2	40.3	even	4
2800.2.a.bi.1.2		2	5.2	odd	4
2800.2.g.t.449.2		4	5.4	even	2	inner
2800.2.g.t.449.3		4	1.1	even	1	trivial
3920.2.a.bs.1.2		2	35.13	even	4
4235.2.a.m.1.2		2	220.43	odd	4
5040.2.a.bt.1.1		2	15.8	even	4
5915.2.a.l.1.2		2	260.103	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} +1.56155 q^{11} -0.438447i q^{13} -0.438447i q^{17} -7.12311 q^{19} -1.56155 q^{21} +3.12311i q^{23} +5.56155i q^{27} -6.68466 q^{29} +2.43845i q^{33} +6.00000i q^{37} +0.684658 q^{39} +5.12311 q^{41} +0.876894i q^{43} +8.68466i q^{47} -1.00000 q^{49} +0.684658 q^{51} +5.12311i q^{53} -11.1231i q^{57} -4.00000 q^{59} +15.3693 q^{61} +0.561553i q^{63} -10.2462i q^{67} -4.87689 q^{69} -8.00000 q^{71} +12.2462i q^{73} +1.56155i q^{77} -2.43845 q^{79} -7.00000 q^{81} +4.00000i q^{83} -10.4384i q^{87} +1.12311 q^{89} +0.438447 q^{91} +5.80776i q^{97} +0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9} - 2 q^{11} - 12 q^{19} + 2 q^{21} - 2 q^{29} - 22 q^{39} + 4 q^{41} - 4 q^{49} - 22 q^{51} - 16 q^{59} + 12 q^{61} - 36 q^{69} - 32 q^{71} - 18 q^{79} - 28 q^{81} - 12 q^{89} + 10 q^{91} + 20 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 2 * q^11 - 12 * q^19 + 2 * q^21 - 2 * q^29 - 22 * q^39 + 4 * q^41 - 4 * q^49 - 22 * q^51 - 16 * q^59 + 12 * q^61 - 36 * q^69 - 32 * q^71 - 18 * q^79 - 28 * q^81 - 12 * q^89 + 10 * q^91 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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