LMFDB - Embedded newform 2912.2.c.a.1457.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(1457,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2912.1457"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.20
Character	$\chi$	$=$	2912.1457
Dual form			2912.2.c.a.1457.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.302261i q^{3} -1.42871i q^{5} +1.00000 q^{7} +2.90864 q^{9} -2.11205i q^{11} +1.00000i q^{13} +0.431843 q^{15} +1.07522 q^{17} -3.52230i q^{19} +0.302261i q^{21} -5.47592 q^{23} +2.95878 q^{25} +1.78595i q^{27} -3.52776i q^{29} +3.26031 q^{31} +0.638391 q^{33} -1.42871i q^{35} -10.7444i q^{37} -0.302261 q^{39} +9.67803 q^{41} +9.02912i q^{43} -4.15560i q^{45} -4.79483 q^{47} +1.00000 q^{49} +0.324997i q^{51} +0.906733i q^{53} -3.01752 q^{55} +1.06465 q^{57} -6.88024i q^{59} +5.18552i q^{61} +2.90864 q^{63} +1.42871 q^{65} +5.46202i q^{67} -1.65516i q^{69} -0.895683 q^{71} -14.4466 q^{73} +0.894324i q^{75} -2.11205i q^{77} -15.5041 q^{79} +8.18609 q^{81} -9.02943i q^{83} -1.53618i q^{85} +1.06630 q^{87} +17.4324 q^{89} +1.00000i q^{91} +0.985464i q^{93} -5.03235 q^{95} +4.26779 q^{97} -6.14320i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) $ 34 * q + 34 * q^7 - 26 * q^9 + 8 * q^15 - 20 * q^17 + 20 * q^23 - 22 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 34 * q^49 + 32 * q^55 + 8 * q^57 - 26 * q^63 - 20 * q^65 - 64 * q^71 - 20 * q^79 - 6 * q^81 + 56 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	0.302261i	0.174510i	0.996186	+	0.0872551i	$0.0278095\pi$
$3$	0.302261i	0.174510i	−0.996186	+	0.0872551i	$0.972190\pi$
$4$	0	0
$5$	− 1.42871i	− 0.638939i	−0.947597	−	0.319470i	$-0.896495\pi$
$5$	− 1.42871i	− 0.638939i	0.947597	−	0.319470i	$-0.103505\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.90864	0.969546
$10$	0	0
$11$	− 2.11205i	− 0.636808i	−0.947955	−	0.318404i	$-0.896853\pi$
$11$	− 2.11205i	− 0.636808i	0.947955	−	0.318404i	$-0.103147\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0.431843	0.111501
$16$	0	0
$17$	1.07522	0.260780	0.130390	−	0.991463i	$-0.458377\pi$
$17$	1.07522	0.260780	0.130390	+	0.991463i	$0.458377\pi$
$18$	0	0
$19$	− 3.52230i	− 0.808071i	−0.914743	−	0.404036i	$-0.867607\pi$
$19$	− 3.52230i	− 0.808071i	0.914743	−	0.404036i	$-0.132393\pi$
$20$	0	0
$21$	0.302261i	0.0659587i
$22$	0	0
$23$	−5.47592	−1.14181	−0.570904	−	0.821017i	$-0.693407\pi$
$23$	−5.47592	−1.14181	−0.570904	+	0.821017i	$0.693407\pi$
$24$	0	0
$25$	2.95878	0.591757
$26$	0	0
$27$	1.78595i	0.343706i
$28$	0	0
$29$	− 3.52776i	− 0.655089i	−0.944836	−	0.327545i	$-0.893779\pi$
$29$	− 3.52776i	− 0.655089i	0.944836	−	0.327545i	$-0.106221\pi$
$30$	0	0
$31$	3.26031	0.585569	0.292785	−	0.956178i	$-0.405418\pi$
$31$	3.26031	0.585569	0.292785	+	0.956178i	$0.405418\pi$
$32$	0	0
$33$	0.638391	0.111130
$34$	0	0
$35$	− 1.42871i	− 0.241496i
$36$	0	0
$37$	− 10.7444i	− 1.76637i	−0.469020	−	0.883187i	$-0.655393\pi$
$37$	− 10.7444i	− 1.76637i	0.469020	−	0.883187i	$-0.344607\pi$
$38$	0	0
$39$	−0.302261	−0.0484004
$40$	0	0
$41$	9.67803	1.51145	0.755727	−	0.654887i	$-0.227284\pi$
$41$	9.67803	1.51145	0.755727	+	0.654887i	$0.227284\pi$
$42$	0	0
$43$	9.02912i	1.37693i	0.725271	+	0.688464i	$0.241715\pi$
$43$	9.02912i	1.37693i	−0.725271	+	0.688464i	$0.758285\pi$
$44$	0	0
$45$	− 4.15560i	− 0.619481i
$46$	0	0
$47$	−4.79483	−0.699398	−0.349699	−	0.936862i	$-0.613716\pi$
$47$	−4.79483	−0.699398	−0.349699	+	0.936862i	$0.613716\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.324997i	0.0455087i
$52$	0	0
$53$	0.906733i	0.124549i	0.998059	+	0.0622747i	$0.0198355\pi$
$53$	0.906733i	0.124549i	−0.998059	+	0.0622747i	$0.980165\pi$
$54$	0	0
$55$	−3.01752	−0.406882
$56$	0	0
$57$	1.06465	0.141017
$58$	0	0
$59$	− 6.88024i	− 0.895731i	−0.894101	−	0.447865i	$-0.852184\pi$
$59$	− 6.88024i	− 0.895731i	0.894101	−	0.447865i	$-0.147816\pi$
$60$	0	0
$61$	5.18552i	0.663938i	0.943290	+	0.331969i	$0.107713\pi$
$61$	5.18552i	0.663938i	−0.943290	+	0.331969i	$0.892287\pi$
$62$	0	0
$63$	2.90864	0.366454
$64$	0	0
$65$	1.42871	0.177210
$66$	0	0
$67$	5.46202i	0.667292i	0.942699	+	0.333646i	$0.108279\pi$
$67$	5.46202i	0.667292i	−0.942699	+	0.333646i	$0.891721\pi$
$68$	0	0
$69$	− 1.65516i	− 0.199257i
$70$	0	0
$71$	−0.895683	−0.106298	−0.0531490	−	0.998587i	$-0.516926\pi$
$71$	−0.895683	−0.106298	−0.0531490	+	0.998587i	$0.516926\pi$
$72$	0	0
$73$	−14.4466	−1.69085	−0.845425	−	0.534094i	$-0.820653\pi$
$73$	−14.4466	−1.69085	−0.845425	+	0.534094i	$0.820653\pi$
$74$	0	0
$75$	0.894324i	0.103268i
$76$	0	0
$77$	− 2.11205i	− 0.240691i
$78$	0	0
$79$	−15.5041	−1.74435	−0.872175	−	0.489195i	$-0.837291\pi$
$79$	−15.5041	−1.74435	−0.872175	+	0.489195i	$0.837291\pi$
$80$	0	0
$81$	8.18609	0.909566
$82$	0	0
$83$	− 9.02943i	− 0.991109i	−0.868577	−	0.495555i	$-0.834965\pi$
$83$	− 9.02943i	− 0.991109i	0.868577	−	0.495555i	$-0.165035\pi$
$84$	0	0
$85$	− 1.53618i	− 0.166622i
$86$	0	0
$87$	1.06630	0.114320
$88$	0	0
$89$	17.4324	1.84783	0.923917	−	0.382592i	$-0.124968\pi$
$89$	17.4324	1.84783	0.923917	+	0.382592i	$0.124968\pi$
$90$	0	0
$91$	1.00000i	0.104828i
$92$	0	0
$93$	0.985464i	0.102188i
$94$	0	0
$95$	−5.03235	−0.516308
$96$	0	0
$97$	4.26779	0.433328	0.216664	−	0.976246i	$-0.430482\pi$
$97$	4.26779	0.433328	0.216664	+	0.976246i	$0.430482\pi$
$98$	0	0
$99$	− 6.14320i	− 0.617415i
$100$	0	0
$101$	− 7.48731i	− 0.745015i	−0.928029	−	0.372508i	$-0.878498\pi$
$101$	− 7.48731i	− 0.745015i	0.928029	−	0.372508i	$-0.121502\pi$
$102$	0	0
$103$	2.54505	0.250771	0.125386	−	0.992108i	$-0.459983\pi$
$103$	2.54505	0.250771	0.125386	+	0.992108i	$0.459983\pi$
$104$	0	0
$105$	0.431843	0.0421436
$106$	0	0
$107$	− 6.43975i	− 0.622554i	−0.950319	−	0.311277i	$-0.899243\pi$
$107$	− 6.43975i	− 0.622554i	0.950319	−	0.311277i	$-0.100757\pi$
$108$	0	0
$109$	− 4.08818i	− 0.391577i	−0.980646	−	0.195788i	$-0.937273\pi$
$109$	− 4.08818i	− 0.391577i	0.980646	−	0.195788i	$-0.0627266\pi$
$110$	0	0
$111$	3.24762	0.308251
$112$	0	0
$113$	−10.4398	−0.982095	−0.491048	−	0.871133i	$-0.663386\pi$
$113$	−10.4398	−0.982095	−0.491048	+	0.871133i	$0.663386\pi$
$114$	0	0
$115$	7.82351i	0.729546i
$116$	0	0
$117$	2.90864i	0.268904i
$118$	0	0
$119$	1.07522	0.0985655
$120$	0	0
$121$	6.53923	0.594475
$122$	0	0
$123$	2.92529i	0.263764i
$124$	0	0
$125$	− 11.3708i	− 1.01704i
$126$	0	0
$127$	14.5900	1.29466	0.647328	−	0.762212i	$-0.275886\pi$
$127$	14.5900	1.29466	0.647328	+	0.762212i	$0.275886\pi$
$128$	0	0
$129$	−2.72915	−0.240288
$130$	0	0
$131$	− 8.54709i	− 0.746763i	−0.927678	−	0.373381i	$-0.878198\pi$
$131$	− 8.54709i	− 0.746763i	0.927678	−	0.373381i	$-0.121802\pi$
$132$	0	0
$133$	− 3.52230i	− 0.305422i
$134$	0	0
$135$	2.55161	0.219607
$136$	0	0
$137$	−4.83312	−0.412921	−0.206461	−	0.978455i	$-0.566195\pi$
$137$	−4.83312	−0.412921	−0.206461	+	0.978455i	$0.566195\pi$
$138$	0	0
$139$	− 20.7091i	− 1.75652i	−0.478180	−	0.878262i	$-0.658703\pi$
$139$	− 20.7091i	− 1.75652i	0.478180	−	0.878262i	$-0.341297\pi$
$140$	0	0
$141$	− 1.44929i	− 0.122052i
$142$	0	0
$143$	2.11205	0.176619
$144$	0	0
$145$	−5.04016	−0.418562
$146$	0	0
$147$	0.302261i	0.0249300i
$148$	0	0
$149$	5.23881i	0.429180i	0.976704	+	0.214590i	$0.0688415\pi$
$149$	5.23881i	0.429180i	−0.976704	+	0.214590i	$0.931159\pi$
$150$	0	0
$151$	17.9190	1.45823	0.729113	−	0.684393i	$-0.239933\pi$
$151$	17.9190	1.45823	0.729113	+	0.684393i	$0.239933\pi$
$152$	0	0
$153$	3.12743	0.252838
$154$	0	0
$155$	− 4.65805i	− 0.374143i
$156$	0	0
$157$	− 0.00849787i	0 0.000678204i	−1.00000		0.000339102i	$-0.999892\pi$
$157$	− 0.00849787i	0 0.000678204i	1.00000		0.000339102i	$-0.000107939\pi$
$158$	0	0
$159$	−0.274070	−0.0217351
$160$	0	0
$161$	−5.47592	−0.431563
$162$	0	0
$163$	− 2.16450i	− 0.169537i	−0.996401	−	0.0847684i	$-0.972985\pi$
$163$	− 2.16450i	− 0.169537i	0.996401	−	0.0847684i	$-0.0270150\pi$
$164$	0	0
$165$	− 0.912076i	− 0.0710050i
$166$	0	0
$167$	3.67872	0.284668	0.142334	−	0.989819i	$-0.454539\pi$
$167$	3.67872	0.284668	0.142334	+	0.989819i	$0.454539\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 10.2451i	− 0.783462i
$172$	0	0
$173$	− 22.9325i	− 1.74353i	−0.489927	−	0.871763i	$-0.662977\pi$
$173$	− 22.9325i	− 1.74353i	0.489927	−	0.871763i	$-0.337023\pi$
$174$	0	0
$175$	2.95878	0.223663
$176$	0	0
$177$	2.07962	0.156314
$178$	0	0
$179$	22.2311i	1.66163i	0.556546	+	0.830817i	$0.312126\pi$
$179$	22.2311i	1.66163i	−0.556546	+	0.830817i	$0.687874\pi$
$180$	0	0
$181$	− 3.16747i	− 0.235437i	−0.993047	−	0.117718i	$-0.962442\pi$
$181$	− 3.16747i	− 0.235437i	0.993047	−	0.117718i	$-0.0375580\pi$
$182$	0	0
$183$	−1.56738	−0.115864
$184$	0	0
$185$	−15.3507	−1.12861
$186$	0	0
$187$	− 2.27093i	− 0.166067i
$188$	0	0
$189$	1.78595i	0.129909i
$190$	0	0
$191$	−3.10833	−0.224911	−0.112455	−	0.993657i	$-0.535872\pi$
$191$	−3.10833	−0.224911	−0.112455	+	0.993657i	$0.535872\pi$
$192$	0	0
$193$	−8.51520	−0.612938	−0.306469	−	0.951881i	$-0.599148\pi$
$193$	−8.51520	−0.612938	−0.306469	+	0.951881i	$0.599148\pi$
$194$	0	0
$195$	0.431843i	0.0309249i
$196$	0	0
$197$	14.2869i	1.01790i	0.860797	+	0.508948i	$0.169966\pi$
$197$	14.2869i	1.01790i	−0.860797	+	0.508948i	$0.830034\pi$
$198$	0	0
$199$	26.2298	1.85938	0.929689	−	0.368345i	$-0.120076\pi$
$199$	26.2298	1.85938	0.929689	+	0.368345i	$0.120076\pi$
$200$	0	0
$201$	−1.65095	−0.116449
$202$	0	0
$203$	− 3.52776i	− 0.247600i
$204$	0	0
$205$	− 13.8271i	− 0.965727i
$206$	0	0
$207$	−15.9275	−1.10704
$208$	0	0
$209$	−7.43929	−0.514586
$210$	0	0
$211$	5.07186i	0.349161i	0.984643	+	0.174580i	$0.0558569\pi$
$211$	5.07186i	0.349161i	−0.984643	+	0.174580i	$0.944143\pi$
$212$	0	0
$213$	− 0.270730i	− 0.0185501i
$214$	0	0
$215$	12.9000	0.879773
$216$	0	0
$217$	3.26031	0.221324
$218$	0	0
$219$	− 4.36665i	− 0.295071i
$220$	0	0
$221$	1.07522i	0.0723273i
$222$	0	0
$223$	8.10346	0.542648	0.271324	−	0.962488i	$-0.412539\pi$
$223$	8.10346	0.542648	0.271324	+	0.962488i	$0.412539\pi$
$224$	0	0
$225$	8.60603	0.573736
$226$	0	0
$227$	− 22.2426i	− 1.47629i	−0.674641	−	0.738146i	$-0.735702\pi$
$227$	− 22.2426i	− 1.47629i	0.674641	−	0.738146i	$-0.264298\pi$
$228$	0	0
$229$	0.687851i	0.0454545i	0.999742	+	0.0227272i	$0.00723493\pi$
$229$	0.687851i	0.0454545i	−0.999742	+	0.0227272i	$0.992765\pi$
$230$	0	0
$231$	0.638391	0.0420030
$232$	0	0
$233$	21.7686	1.42611	0.713054	−	0.701109i	$-0.247311\pi$
$233$	21.7686	1.42611	0.713054	+	0.701109i	$0.247311\pi$
$234$	0	0
$235$	6.85043i	0.446873i
$236$	0	0
$237$	− 4.68628i	− 0.304407i
$238$	0	0
$239$	12.4111	0.802810	0.401405	−	0.915901i	$-0.368522\pi$
$239$	12.4111	0.802810	0.401405	+	0.915901i	$0.368522\pi$
$240$	0	0
$241$	9.86867	0.635697	0.317848	−	0.948142i	$-0.397040\pi$
$241$	9.86867	0.635697	0.317848	+	0.948142i	$0.397040\pi$
$242$	0	0
$243$	7.83218i	0.502435i
$244$	0	0
$245$	− 1.42871i	− 0.0912770i
$246$	0	0
$247$	3.52230	0.224119
$248$	0	0
$249$	2.72924	0.172959
$250$	0	0
$251$	− 12.2813i	− 0.775187i	−0.921830	−	0.387593i	$-0.873306\pi$
$251$	− 12.2813i	− 0.775187i	0.921830	−	0.387593i	$-0.126694\pi$
$252$	0	0
$253$	11.5654i	0.727113i
$254$	0	0
$255$	0.464327	0.0290773
$256$	0	0
$257$	−8.06646	−0.503172	−0.251586	−	0.967835i	$-0.580952\pi$
$257$	−8.06646	−0.503172	−0.251586	+	0.967835i	$0.580952\pi$
$258$	0	0
$259$	− 10.7444i	− 0.667627i
$260$	0	0
$261$	− 10.2610i	− 0.635139i
$262$	0	0
$263$	−16.3897	−1.01063	−0.505316	−	0.862934i	$-0.668624\pi$
$263$	−16.3897	−1.01063	−0.505316	+	0.862934i	$0.668624\pi$
$264$	0	0
$265$	1.29546	0.0795795
$266$	0	0
$267$	5.26914i	0.322466i
$268$	0	0
$269$	25.7500i	1.57001i	0.619491	+	0.785004i	$0.287339\pi$
$269$	25.7500i	1.57001i	−0.619491	+	0.785004i	$0.712661\pi$
$270$	0	0
$271$	−3.70858	−0.225280	−0.112640	−	0.993636i	$-0.535931\pi$
$271$	−3.70858	−0.225280	−0.112640	+	0.993636i	$0.535931\pi$
$272$	0	0
$273$	−0.302261	−0.0182936
$274$	0	0
$275$	− 6.24911i	− 0.376836i
$276$	0	0
$277$	3.54833i	0.213199i	0.994302	+	0.106599i	$0.0339962\pi$
$277$	3.54833i	0.213199i	−0.994302	+	0.106599i	$0.966004\pi$
$278$	0	0
$279$	9.48307	0.567737
$280$	0	0
$281$	−0.0751627	−0.00448383	−0.00224192	−	0.999997i	$-0.500714\pi$
$281$	−0.0751627	−0.00448383	−0.00224192	+	0.999997i	$0.500714\pi$
$282$	0	0
$283$	22.3137i	1.32641i	0.748437	+	0.663205i	$0.230804\pi$
$283$	22.3137i	1.32641i	−0.748437	+	0.663205i	$0.769196\pi$
$284$	0	0
$285$	− 1.52108i	− 0.0901011i
$286$	0	0
$287$	9.67803	0.571276
$288$	0	0
$289$	−15.8439	−0.931994
$290$	0	0
$291$	1.28998i	0.0756202i
$292$	0	0
$293$	15.1800i	0.886825i	0.896318	+	0.443412i	$0.146232\pi$
$293$	15.1800i	0.886825i	−0.896318	+	0.443412i	$0.853768\pi$
$294$	0	0
$295$	−9.82987	−0.572317
$296$	0	0
$297$	3.77202	0.218875
$298$	0	0
$299$	− 5.47592i	− 0.316681i
$300$	0	0
$301$	9.02912i	0.520430i
$302$	0	0
$303$	2.26312	0.130013
$304$	0	0
$305$	7.40862	0.424216
$306$	0	0
$307$	10.7178i	0.611699i	0.952080	+	0.305850i	$0.0989405\pi$
$307$	10.7178i	0.611699i	−0.952080	+	0.305850i	$0.901060\pi$
$308$	0	0
$309$	0.769269i	0.0437622i
$310$	0	0
$311$	−17.1663	−0.973413	−0.486707	−	0.873565i	$-0.661802\pi$
$311$	−17.1663	−0.973413	−0.486707	+	0.873565i	$0.661802\pi$
$312$	0	0
$313$	−11.5995	−0.655645	−0.327823	−	0.944739i	$-0.606315\pi$
$313$	−11.5995	−0.655645	−0.327823	+	0.944739i	$0.606315\pi$
$314$	0	0
$315$	− 4.15560i	− 0.234142i
$316$	0	0
$317$	− 7.72578i	− 0.433923i	−0.976180	−	0.216962i	$-0.930385\pi$
$317$	− 7.72578i	− 0.433923i	0.976180	−	0.216962i	$-0.0696146\pi$
$318$	0	0
$319$	−7.45083	−0.417166
$320$	0	0
$321$	1.94648	0.108642
$322$	0	0
$323$	− 3.78726i	− 0.210729i
$324$	0	0
$325$	2.95878i	0.164124i
$326$	0	0
$327$	1.23570	0.0683342
$328$	0	0
$329$	−4.79483	−0.264348
$330$	0	0
$331$	− 7.83452i	− 0.430624i	−0.976545	−	0.215312i	$-0.930923\pi$
$331$	− 7.83452i	− 0.430624i	0.976545	−	0.215312i	$-0.0690769\pi$
$332$	0	0
$333$	− 31.2517i	− 1.71258i
$334$	0	0
$335$	7.80365	0.426359
$336$	0	0
$337$	−6.71021	−0.365528	−0.182764	−	0.983157i	$-0.558504\pi$
$337$	−6.71021	−0.365528	−0.182764	+	0.983157i	$0.558504\pi$
$338$	0	0
$339$	− 3.15554i	− 0.171386i
$340$	0	0
$341$	− 6.88596i	− 0.372896i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.36474	−0.127313
$346$	0	0
$347$	− 28.7053i	− 1.54098i	−0.637450	−	0.770492i	$-0.720011\pi$
$347$	− 28.7053i	− 1.54098i	0.637450	−	0.770492i	$-0.279989\pi$
$348$	0	0
$349$	31.5592i	1.68933i	0.535298	+	0.844663i	$0.320199\pi$
$349$	31.5592i	1.68933i	−0.535298	+	0.844663i	$0.679801\pi$
$350$	0	0
$351$	−1.78595	−0.0953269
$352$	0	0
$353$	−32.1482	−1.71108	−0.855538	−	0.517741i	$-0.826773\pi$
$353$	−32.1482	−1.71108	−0.855538	+	0.517741i	$0.826773\pi$
$354$	0	0
$355$	1.27967i	0.0679180i
$356$	0	0
$357$	0.324997i	0.0172007i
$358$	0	0
$359$	0.201990	0.0106606	0.00533032	−	0.999986i	$-0.498303\pi$
$359$	0.201990	0.0106606	0.00533032	+	0.999986i	$0.498303\pi$
$360$	0	0
$361$	6.59340	0.347021
$362$	0	0
$363$	1.97655i	0.103742i
$364$	0	0
$365$	20.6401i	1.08035i
$366$	0	0
$367$	−12.2350	−0.638659	−0.319330	−	0.947644i	$-0.603458\pi$
$367$	−12.2350	−0.638659	−0.319330	+	0.947644i	$0.603458\pi$
$368$	0	0
$369$	28.1499	1.46542
$370$	0	0
$371$	0.906733i	0.0470753i
$372$	0	0
$373$	30.5414i	1.58138i	0.612220	+	0.790688i	$0.290277\pi$
$373$	30.5414i	1.58138i	−0.612220	+	0.790688i	$0.709723\pi$
$374$	0	0
$375$	3.43695	0.177483
$376$	0	0
$377$	3.52776	0.181689
$378$	0	0
$379$	23.3843i	1.20117i	0.799560	+	0.600586i	$0.205066\pi$
$379$	23.3843i	1.20117i	−0.799560	+	0.600586i	$0.794934\pi$
$380$	0	0
$381$	4.40999i	0.225931i
$382$	0	0
$383$	25.5623	1.30617	0.653085	−	0.757285i	$-0.273474\pi$
$383$	25.5623	1.30617	0.653085	+	0.757285i	$0.273474\pi$
$384$	0	0
$385$	−3.01752	−0.153787
$386$	0	0
$387$	26.2624i	1.33500i
$388$	0	0
$389$	18.4359i	0.934735i	0.884063	+	0.467367i	$0.154797\pi$
$389$	18.4359i	0.934735i	−0.884063	+	0.467367i	$0.845203\pi$
$390$	0	0
$391$	−5.88783	−0.297760
$392$	0	0
$393$	2.58345	0.130318
$394$	0	0
$395$	22.1509i	1.11453i
$396$	0	0
$397$	− 24.1909i	− 1.21411i	−0.794662	−	0.607053i	$-0.792352\pi$
$397$	− 24.1909i	− 1.21411i	0.794662	−	0.607053i	$-0.207648\pi$
$398$	0	0
$399$	1.06465	0.0532993
$400$	0	0
$401$	27.3572	1.36615	0.683076	−	0.730347i	$-0.260642\pi$
$401$	27.3572	1.36615	0.683076	+	0.730347i	$0.260642\pi$
$402$	0	0
$403$	3.26031i	0.162408i
$404$	0	0
$405$	− 11.6956i	− 0.581157i
$406$	0	0
$407$	−22.6928	−1.12484
$408$	0	0
$409$	−11.9361	−0.590204	−0.295102	−	0.955466i	$-0.595354\pi$
$409$	−11.9361	−0.590204	−0.295102	+	0.955466i	$0.595354\pi$
$410$	0	0
$411$	− 1.46086i	− 0.0720590i
$412$	0	0
$413$	− 6.88024i	− 0.338554i
$414$	0	0
$415$	−12.9005	−0.633258
$416$	0	0
$417$	6.25955	0.306531
$418$	0	0
$419$	2.55111i	0.124630i	0.998057	+	0.0623150i	$0.0198484\pi$
$419$	2.55111i	0.124630i	−0.998057	+	0.0623150i	$0.980152\pi$
$420$	0	0
$421$	37.9289i	1.84854i	0.381740	+	0.924270i	$0.375325\pi$
$421$	37.9289i	1.84854i	−0.381740	+	0.924270i	$0.624675\pi$
$422$	0	0
$423$	−13.9464	−0.678099
$424$	0	0
$425$	3.18135	0.154318
$426$	0	0
$427$	5.18552i	0.250945i
$428$	0	0
$429$	0.638391i	0.0308218i
$430$	0	0
$431$	3.23951	0.156041	0.0780207	−	0.996952i	$-0.475140\pi$
$431$	3.23951	0.156041	0.0780207	+	0.996952i	$0.475140\pi$
$432$	0	0
$433$	−10.6772	−0.513114	−0.256557	−	0.966529i	$-0.582588\pi$
$433$	−10.6772	−0.513114	−0.256557	+	0.966529i	$0.582588\pi$
$434$	0	0
$435$	− 1.52344i	− 0.0730434i
$436$	0	0
$437$	19.2878i	0.922662i
$438$	0	0
$439$	7.91420	0.377724	0.188862	−	0.982004i	$-0.439520\pi$
$439$	7.91420	0.377724	0.188862	+	0.982004i	$0.439520\pi$
$440$	0	0
$441$	2.90864	0.138507
$442$	0	0
$443$	− 0.488220i	− 0.0231960i	−0.999933	−	0.0115980i	$-0.996308\pi$
$443$	− 0.488220i	− 0.0231960i	0.999933	−	0.0115980i	$-0.00369185\pi$
$444$	0	0
$445$	− 24.9059i	− 1.18065i
$446$	0	0
$447$	−1.58348	−0.0748962
$448$	0	0
$449$	−32.7761	−1.54680	−0.773399	−	0.633919i	$-0.781445\pi$
$449$	−32.7761	−1.54680	−0.773399	+	0.633919i	$0.781445\pi$
$450$	0	0
$451$	− 20.4405i	− 0.962506i
$452$	0	0
$453$	5.41620i	0.254475i
$454$	0	0
$455$	1.42871	0.0669790
$456$	0	0
$457$	10.4564	0.489131	0.244566	−	0.969633i	$-0.421355\pi$
$457$	10.4564	0.489131	0.244566	+	0.969633i	$0.421355\pi$
$458$	0	0
$459$	1.92029i	0.0896315i
$460$	0	0
$461$	− 18.2242i	− 0.848786i	−0.905478	−	0.424393i	$-0.860487\pi$
$461$	− 18.2242i	− 0.848786i	0.905478	−	0.424393i	$-0.139513\pi$
$462$	0	0
$463$	−12.6199	−0.586498	−0.293249	−	0.956036i	$-0.594736\pi$
$463$	−12.6199	−0.586498	−0.293249	+	0.956036i	$0.594736\pi$
$464$	0	0
$465$	1.40794	0.0652918
$466$	0	0
$467$	− 4.87907i	− 0.225777i	−0.993608	−	0.112888i	$-0.963990\pi$
$467$	− 4.87907i	− 0.225777i	0.993608	−	0.112888i	$-0.0360102\pi$
$468$	0	0
$469$	5.46202i	0.252213i
$470$	0	0
$471$	0.00256857	0.000118353 0
$472$	0	0
$473$	19.0700	0.876839
$474$	0	0
$475$	− 10.4217i	− 0.478182i
$476$	0	0
$477$	2.63736i	0.120756i
$478$	0	0
$479$	18.0377	0.824165	0.412082	−	0.911147i	$-0.364802\pi$
$479$	18.0377	0.824165	0.412082	+	0.911147i	$0.364802\pi$
$480$	0	0
$481$	10.7444	0.489904
$482$	0	0
$483$	− 1.65516i	− 0.0753122i
$484$	0	0
$485$	− 6.09743i	− 0.276870i
$486$	0	0
$487$	−22.7622	−1.03145	−0.515726	−	0.856753i	$-0.672478\pi$
$487$	−22.7622	−1.03145	−0.515726	+	0.856753i	$0.672478\pi$
$488$	0	0
$489$	0.654243	0.0295859
$490$	0	0
$491$	22.4017i	1.01098i	0.862833	+	0.505488i	$0.168688\pi$
$491$	22.4017i	1.01098i	−0.862833	+	0.505488i	$0.831312\pi$
$492$	0	0
$493$	− 3.79313i	− 0.170834i
$494$	0	0
$495$	−8.77686	−0.394491
$496$	0	0
$497$	−0.895683	−0.0401769
$498$	0	0
$499$	− 6.56942i	− 0.294088i	−0.989130	−	0.147044i	$-0.953024\pi$
$499$	− 6.56942i	− 0.294088i	0.989130	−	0.147044i	$-0.0469758\pi$
$500$	0	0
$501$	1.11193i	0.0496775i
$502$	0	0
$503$	28.0909	1.25251	0.626256	−	0.779617i	$-0.284586\pi$
$503$	28.0909	1.25251	0.626256	+	0.779617i	$0.284586\pi$
$504$	0	0
$505$	−10.6972	−0.476019
$506$	0	0
$507$	− 0.302261i	− 0.0134239i
$508$	0	0
$509$	− 30.4909i	− 1.35149i	−0.737137	−	0.675744i	$-0.763823\pi$
$509$	− 30.4909i	− 1.35149i	0.737137	−	0.675744i	$-0.236177\pi$
$510$	0	0
$511$	−14.4466	−0.639081
$512$	0	0
$513$	6.29065	0.277739
$514$	0	0
$515$	− 3.63614i	− 0.160228i
$516$	0	0
$517$	10.1269i	0.445383i
$518$	0	0
$519$	6.93160	0.304263
$520$	0	0
$521$	−45.5582	−1.99594	−0.997970	−	0.0636820i	$-0.979716\pi$
$521$	−45.5582	−1.99594	−0.997970	+	0.0636820i	$0.979716\pi$
$522$	0	0
$523$	6.33191i	0.276875i	0.990371	+	0.138438i	$0.0442080\pi$
$523$	6.33191i	0.276875i	−0.990371	+	0.138438i	$0.955792\pi$
$524$	0	0
$525$	0.894324i	0.0390315i
$526$	0	0
$527$	3.50556	0.152705
$528$	0	0
$529$	6.98571	0.303727
$530$	0	0
$531$	− 20.0121i	− 0.868452i
$532$	0	0
$533$	9.67803i	0.419202i
$534$	0	0
$535$	−9.20055	−0.397774
$536$	0	0
$537$	−6.71960	−0.289972
$538$	0	0
$539$	− 2.11205i	− 0.0909726i
$540$	0	0
$541$	− 22.2376i	− 0.956068i	−0.878341	−	0.478034i	$-0.841350\pi$
$541$	− 22.2376i	− 0.956068i	0.878341	−	0.478034i	$-0.158650\pi$
$542$	0	0
$543$	0.957403	0.0410861
$544$	0	0
$545$	−5.84083	−0.250194
$546$	0	0
$547$	21.1582i	0.904658i	0.891851	+	0.452329i	$0.149407\pi$
$547$	21.1582i	0.904658i	−0.891851	+	0.452329i	$0.850593\pi$
$548$	0	0
$549$	15.0828i	0.643719i
$550$	0	0
$551$	−12.4258	−0.529359
$552$	0	0
$553$	−15.5041	−0.659302
$554$	0	0
$555$	− 4.63991i	− 0.196953i
$556$	0	0
$557$	− 0.896820i	− 0.0379995i	−0.999819	−	0.0189997i	$-0.993952\pi$
$557$	− 0.896820i	− 0.0379995i	0.999819	−	0.0189997i	$-0.00604817\pi$
$558$	0	0
$559$	−9.02912	−0.381891
$560$	0	0
$561$	0.686412	0.0289803
$562$	0	0
$563$	− 5.18352i	− 0.218459i	−0.994017	−	0.109230i	$-0.965162\pi$
$563$	− 5.18352i	− 0.218459i	0.994017	−	0.109230i	$-0.0348384\pi$
$564$	0	0
$565$	14.9155i	0.627499i
$566$	0	0
$567$	8.18609	0.343784
$568$	0	0
$569$	−16.6719	−0.698924	−0.349462	−	0.936951i	$-0.613636\pi$
$569$	−16.6719	−0.698924	−0.349462	+	0.936951i	$0.613636\pi$
$570$	0	0
$571$	16.4548i	0.688611i	0.938858	+	0.344306i	$0.111886\pi$
$571$	16.4548i	0.688611i	−0.938858	+	0.344306i	$0.888114\pi$
$572$	0	0
$573$	− 0.939526i	− 0.0392492i
$574$	0	0
$575$	−16.2021	−0.675673
$576$	0	0
$577$	0.0663045	0.00276029	0.00138015	−	0.999999i	$-0.499561\pi$
$577$	0.0663045	0.00276029	0.00138015	+	0.999999i	$0.499561\pi$
$578$	0	0
$579$	− 2.57381i	− 0.106964i
$580$	0	0
$581$	− 9.02943i	− 0.374604i
$582$	0	0
$583$	1.91507	0.0793141
$584$	0	0
$585$	4.15560	0.171813
$586$	0	0
$587$	23.6255i	0.975127i	0.873087	+	0.487564i	$0.162114\pi$
$587$	23.6255i	0.975127i	−0.873087	+	0.487564i	$0.837886\pi$
$588$	0	0
$589$	− 11.4838i	− 0.473182i
$590$	0	0
$591$	−4.31836	−0.177633
$592$	0	0
$593$	−1.66405	−0.0683342	−0.0341671	−	0.999416i	$-0.510878\pi$
$593$	−1.66405	−0.0683342	−0.0341671	+	0.999416i	$0.510878\pi$
$594$	0	0
$595$	− 1.53618i	− 0.0629773i
$596$	0	0
$597$	7.92822i	0.324481i
$598$	0	0
$599$	4.60064	0.187977	0.0939885	−	0.995573i	$-0.470038\pi$
$599$	4.60064	0.187977	0.0939885	+	0.995573i	$0.470038\pi$
$600$	0	0
$601$	−13.2087	−0.538794	−0.269397	−	0.963029i	$-0.586824\pi$
$601$	−13.2087	−0.538794	−0.269397	+	0.963029i	$0.586824\pi$
$602$	0	0
$603$	15.8870i	0.646970i
$604$	0	0
$605$	− 9.34267i	− 0.379833i
$606$	0	0
$607$	−12.3140	−0.499811	−0.249905	−	0.968270i	$-0.580399\pi$
$607$	−12.3140	−0.499811	−0.249905	+	0.968270i	$0.580399\pi$
$608$	0	0
$609$	1.06630	0.0432088
$610$	0	0
$611$	− 4.79483i	− 0.193978i
$612$	0	0
$613$	35.6977i	1.44182i	0.693030	+	0.720909i	$0.256276\pi$
$613$	35.6977i	1.44182i	−0.693030	+	0.720909i	$0.743724\pi$
$614$	0	0
$615$	4.17939	0.168529
$616$	0	0
$617$	−11.9160	−0.479720	−0.239860	−	0.970808i	$-0.577102\pi$
$617$	−11.9160	−0.479720	−0.239860	+	0.970808i	$0.577102\pi$
$618$	0	0
$619$	36.8286i	1.48027i	0.672460	+	0.740134i	$0.265238\pi$
$619$	36.8286i	1.48027i	−0.672460	+	0.740134i	$0.734762\pi$
$620$	0	0
$621$	− 9.77971i	− 0.392446i
$622$	0	0
$623$	17.4324	0.698416
$624$	0	0
$625$	−1.45168	−0.0580671
$626$	0	0
$627$	− 2.24860i	− 0.0898006i
$628$	0	0
$629$	− 11.5527i	− 0.460635i
$630$	0	0
$631$	−37.1252	−1.47793	−0.738966	−	0.673743i	$-0.764686\pi$
$631$	−37.1252	−1.47793	−0.738966	+	0.673743i	$0.764686\pi$
$632$	0	0
$633$	−1.53302	−0.0609322
$634$	0	0
$635$	− 20.8449i	− 0.827206i
$636$	0	0
$637$	1.00000i	0.0396214i
$638$	0	0
$639$	−2.60522	−0.103061
$640$	0	0
$641$	6.56797	0.259419	0.129710	−	0.991552i	$-0.458595\pi$
$641$	6.56797	0.259419	0.129710	+	0.991552i	$0.458595\pi$
$642$	0	0
$643$	− 1.39342i	− 0.0549511i	−0.999622	−	0.0274755i	$-0.991253\pi$
$643$	− 1.39342i	− 0.0549511i	0.999622	−	0.0274755i	$-0.00874684\pi$
$644$	0	0
$645$	3.89916i	0.153529i
$646$	0	0
$647$	35.5340	1.39699	0.698493	−	0.715617i	$-0.253854\pi$
$647$	35.5340	1.39699	0.698493	+	0.715617i	$0.253854\pi$
$648$	0	0
$649$	−14.5314	−0.570409
$650$	0	0
$651$	0.985464i	0.0386234i
$652$	0	0
$653$	− 13.3350i	− 0.521839i	−0.965361	−	0.260920i	$-0.915974\pi$
$653$	− 13.3350i	− 0.521839i	0.965361	−	0.260920i	$-0.0840258\pi$
$654$	0	0
$655$	−12.2113	−0.477136
$656$	0	0
$657$	−42.0200	−1.63936
$658$	0	0
$659$	32.7598i	1.27614i	0.769978	+	0.638070i	$0.220267\pi$
$659$	32.7598i	1.27614i	−0.769978	+	0.638070i	$0.779733\pi$
$660$	0	0
$661$	42.2267i	1.64243i	0.570620	+	0.821214i	$0.306703\pi$
$661$	42.2267i	1.64243i	−0.570620	+	0.821214i	$0.693297\pi$
$662$	0	0
$663$	−0.324997	−0.0126218
$664$	0	0
$665$	−5.03235	−0.195146
$666$	0	0
$667$	19.3178i	0.747986i
$668$	0	0
$669$	2.44936i	0.0946976i
$670$	0	0
$671$	10.9521	0.422801
$672$	0	0
$673$	−9.96162	−0.383992	−0.191996	−	0.981396i	$-0.561496\pi$
$673$	−9.96162	−0.383992	−0.191996	+	0.981396i	$0.561496\pi$
$674$	0	0
$675$	5.28424i	0.203390i
$676$	0	0
$677$	− 5.80056i	− 0.222934i	−0.993768	−	0.111467i	$-0.964445\pi$
$677$	− 5.80056i	− 0.222934i	0.993768	−	0.111467i	$-0.0355549\pi$
$678$	0	0
$679$	4.26779	0.163783
$680$	0	0
$681$	6.72306	0.257628
$682$	0	0
$683$	11.3291i	0.433494i	0.976228	+	0.216747i	$0.0695447\pi$
$683$	11.3291i	0.433494i	−0.976228	+	0.216747i	$0.930455\pi$
$684$	0	0
$685$	6.90513i	0.263832i
$686$	0	0
$687$	−0.207910	−0.00793227
$688$	0	0
$689$	−0.906733	−0.0345438
$690$	0	0
$691$	28.0266i	1.06618i	0.846058	+	0.533092i	$0.178970\pi$
$691$	28.0266i	1.06618i	−0.846058	+	0.533092i	$0.821030\pi$
$692$	0	0
$693$	− 6.14320i	− 0.233361i
$694$	0	0
$695$	−29.5873	−1.12231
$696$	0	0
$697$	10.4060	0.394156
$698$	0	0
$699$	6.57979i	0.248871i
$700$	0	0
$701$	36.2401i	1.36877i	0.729121	+	0.684384i	$0.239929\pi$
$701$	36.2401i	1.36877i	−0.729121	+	0.684384i	$0.760071\pi$
$702$	0	0
$703$	−37.8451	−1.42736
$704$	0	0
$705$	−2.07062	−0.0779839
$706$	0	0
$707$	− 7.48731i	− 0.281589i
$708$	0	0
$709$	− 22.1194i	− 0.830710i	−0.909659	−	0.415355i	$-0.863657\pi$
$709$	− 22.1194i	− 0.830710i	0.909659	−	0.415355i	$-0.136343\pi$
$710$	0	0
$711$	−45.0959	−1.69123
$712$	0	0
$713$	−17.8532	−0.668608
$714$	0	0
$715$	− 3.01752i	− 0.112849i
$716$	0	0
$717$	3.75140i	0.140099i
$718$	0	0
$719$	19.9640	0.744531	0.372265	−	0.928126i	$-0.378581\pi$
$719$	19.9640	0.744531	0.372265	+	0.928126i	$0.378581\pi$
$720$	0	0
$721$	2.54505	0.0947826
$722$	0	0
$723$	2.98291i	0.110936i
$724$	0	0
$725$	− 10.4379i	− 0.387653i
$726$	0	0
$727$	−4.84254	−0.179600	−0.0897999	−	0.995960i	$-0.528623\pi$
$727$	−4.84254	−0.179600	−0.0897999	+	0.995960i	$0.528623\pi$
$728$	0	0
$729$	22.1909	0.821886
$730$	0	0
$731$	9.70831i	0.359075i
$732$	0	0
$733$	− 8.88437i	− 0.328152i	−0.986448	−	0.164076i	$-0.947536\pi$
$733$	− 8.88437i	− 0.328152i	0.986448	−	0.164076i	$-0.0524642\pi$
$734$	0	0
$735$	0.431843	0.0159288
$736$	0	0
$737$	11.5361	0.424937
$738$	0	0
$739$	10.4639i	0.384922i	0.981305	+	0.192461i	$0.0616469\pi$
$739$	10.4639i	0.384922i	−0.981305	+	0.192461i	$0.938353\pi$
$740$	0	0
$741$	1.06465i	0.0391110i
$742$	0	0
$743$	39.6855	1.45592	0.727960	−	0.685620i	$-0.240469\pi$
$743$	39.6855	1.45592	0.727960	+	0.685620i	$0.240469\pi$
$744$	0	0
$745$	7.48474	0.274220
$746$	0	0
$747$	− 26.2634i	− 0.960926i
$748$	0	0
$749$	− 6.43975i	− 0.235303i
$750$	0	0
$751$	33.4376	1.22016	0.610078	−	0.792342i	$-0.291138\pi$
$751$	33.4376	1.22016	0.610078	+	0.792342i	$0.291138\pi$
$752$	0	0
$753$	3.71214	0.135278
$754$	0	0
$755$	− 25.6011i	− 0.931718i
$756$	0	0
$757$	45.0370i	1.63690i	0.574581	+	0.818448i	$0.305165\pi$
$757$	45.0370i	1.63690i	−0.574581	+	0.818448i	$0.694835\pi$
$758$	0	0
$759$	−3.49578	−0.126889
$760$	0	0
$761$	13.7077	0.496905	0.248452	−	0.968644i	$-0.420078\pi$
$761$	13.7077	0.496905	0.248452	+	0.968644i	$0.420078\pi$
$762$	0	0
$763$	− 4.08818i	− 0.148002i
$764$	0	0
$765$	− 4.46820i	− 0.161548i
$766$	0	0
$767$	6.88024	0.248431
$768$	0	0
$769$	23.3751	0.842929	0.421464	−	0.906845i	$-0.361516\pi$
$769$	23.3751	0.842929	0.421464	+	0.906845i	$0.361516\pi$
$770$	0	0
$771$	− 2.43817i	− 0.0878087i
$772$	0	0
$773$	23.7114i	0.852838i	0.904526	+	0.426419i	$0.140225\pi$
$773$	23.7114i	0.852838i	−0.904526	+	0.426419i	$0.859775\pi$
$774$	0	0
$775$	9.64656	0.346515
$776$	0	0
$777$	3.24762	0.116508
$778$	0	0
$779$	− 34.0889i	− 1.22136i
$780$	0	0
$781$	1.89173i	0.0676915i
$782$	0	0
$783$	6.30040	0.225158
$784$	0	0
$785$	−0.0121410	−0.000433331 0
$786$	0	0
$787$	46.6267i	1.66206i	0.556227	+	0.831030i	$0.312249\pi$
$787$	46.6267i	1.66206i	−0.556227	+	0.831030i	$0.687751\pi$
$788$	0	0
$789$	− 4.95396i	− 0.176366i
$790$	0	0
$791$	−10.4398	−0.371197
$792$	0	0
$793$	−5.18552	−0.184143
$794$	0	0
$795$	0.391567i	0.0138874i
$796$	0	0
$797$	− 25.0129i	− 0.886003i	−0.896521	−	0.443001i	$-0.853914\pi$
$797$	− 25.0129i	− 0.886003i	0.896521	−	0.443001i	$-0.146086\pi$
$798$	0	0
$799$	−5.15551	−0.182389
$800$	0	0
$801$	50.7047	1.79156
$802$	0	0
$803$	30.5121i	1.07675i
$804$	0	0
$805$	7.82351i	0.275743i
$806$	0	0
$807$	−7.78322	−0.273982
$808$	0	0
$809$	−6.23755	−0.219300	−0.109650	−	0.993970i	$-0.534973\pi$
$809$	−6.23755	−0.219300	−0.109650	+	0.993970i	$0.534973\pi$
$810$	0	0
$811$	30.5842i	1.07396i	0.843596	+	0.536978i	$0.180434\pi$
$811$	30.5842i	1.07396i	−0.843596	+	0.536978i	$0.819566\pi$
$812$	0	0
$813$	− 1.12096i	− 0.0393137i
$814$	0	0
$815$	−3.09245	−0.108324
$816$	0	0
$817$	31.8033	1.11266
$818$	0	0
$819$	2.90864i	0.101636i
$820$	0	0
$821$	31.2474i	1.09054i	0.838260	+	0.545271i	$0.183573\pi$
$821$	31.2474i	1.09054i	−0.838260	+	0.545271i	$0.816427\pi$
$822$	0	0
$823$	−57.0364	−1.98816	−0.994082	−	0.108628i	$-0.965354\pi$
$823$	−57.0364	−1.98816	−0.994082	+	0.108628i	$0.965354\pi$
$824$	0	0
$825$	1.88886	0.0657617
$826$	0	0
$827$	22.0341i	0.766200i	0.923707	+	0.383100i	$0.125143\pi$
$827$	22.0341i	0.766200i	−0.923707	+	0.383100i	$0.874857\pi$
$828$	0	0
$829$	− 1.38481i	− 0.0480963i	−0.999711	−	0.0240482i	$-0.992344\pi$
$829$	− 1.38481i	− 0.0480963i	0.999711	−	0.0240482i	$-0.00765551\pi$
$830$	0	0
$831$	−1.07252	−0.0372053
$832$	0	0
$833$	1.07522	0.0372542
$834$	0	0
$835$	− 5.25584i	− 0.181886i
$836$	0	0
$837$	5.82275i	0.201264i
$838$	0	0
$839$	−1.94568	−0.0671723	−0.0335861	−	0.999436i	$-0.510693\pi$
$839$	−1.94568	−0.0671723	−0.0335861	+	0.999436i	$0.510693\pi$
$840$	0	0
$841$	16.5549	0.570858
$842$	0	0
$843$	− 0.0227187i	0 0.000782475i
$844$	0	0
$845$	1.42871i	0.0491492i
$846$	0	0
$847$	6.53923	0.224690
$848$	0	0
$849$	−6.74455	−0.231472
$850$	0	0
$851$	58.8357i	2.01686i
$852$	0	0
$853$	− 3.26963i	− 0.111950i	−0.998432	−	0.0559749i	$-0.982173\pi$
$853$	− 3.26963i	− 0.111950i	0.998432	−	0.0559749i	$-0.0178267\pi$
$854$	0	0
$855$	−14.6373	−0.500585
$856$	0	0
$857$	49.4615	1.68957	0.844786	−	0.535104i	$-0.179728\pi$
$857$	49.4615	1.68957	0.844786	+	0.535104i	$0.179728\pi$
$858$	0	0
$859$	− 39.4168i	− 1.34489i	−0.740149	−	0.672443i	$-0.765245\pi$
$859$	− 39.4168i	− 1.34489i	0.740149	−	0.672443i	$-0.234755\pi$
$860$	0	0
$861$	2.92529i	0.0996935i
$862$	0	0
$863$	−1.05337	−0.0358572	−0.0179286	−	0.999839i	$-0.505707\pi$
$863$	−1.05337	−0.0358572	−0.0179286	+	0.999839i	$0.505707\pi$
$864$	0	0
$865$	−32.7639	−1.11401
$866$	0	0
$867$	− 4.78899i	− 0.162642i
$868$	0	0
$869$	32.7455i	1.11082i
$870$	0	0
$871$	−5.46202	−0.185073
$872$	0	0
$873$	12.4134	0.420131
$874$	0	0
$875$	− 11.3708i	− 0.384403i
$876$	0	0
$877$	13.9600i	0.471394i	0.971827	+	0.235697i	$0.0757373\pi$
$877$	13.9600i	0.471394i	−0.971827	+	0.235697i	$0.924263\pi$
$878$	0	0
$879$	−4.58831	−0.154760
$880$	0	0
$881$	23.8927	0.804967	0.402483	−	0.915427i	$-0.368147\pi$
$881$	23.8927	0.804967	0.402483	+	0.915427i	$0.368147\pi$
$882$	0	0
$883$	− 36.0747i	− 1.21401i	−0.794698	−	0.607005i	$-0.792371\pi$
$883$	− 36.0747i	− 1.21401i	0.794698	−	0.607005i	$-0.207629\pi$
$884$	0	0
$885$	− 2.97118i	− 0.0998752i
$886$	0	0
$887$	−16.9060	−0.567647	−0.283824	−	0.958877i	$-0.591603\pi$
$887$	−16.9060	−0.567647	−0.283824	+	0.958877i	$0.591603\pi$
$888$	0	0
$889$	14.5900	0.489334
$890$	0	0
$891$	− 17.2895i	− 0.579219i
$892$	0	0
$893$	16.8888i	0.565163i
$894$	0	0
$895$	31.7619	1.06168
$896$	0	0
$897$	1.65516	0.0552640
$898$	0	0
$899$	− 11.5016i	− 0.383600i
$900$	0	0
$901$	0.974940i	0.0324800i
$902$	0	0
$903$	−2.72915	−0.0908203
$904$	0	0
$905$	−4.52541	−0.150430
$906$	0	0
$907$	22.2018i	0.737200i	0.929588	+	0.368600i	$0.120163\pi$
$907$	22.2018i	0.737200i	−0.929588	+	0.368600i	$0.879837\pi$
$908$	0	0
$909$	− 21.7779i	− 0.722327i
$910$	0	0
$911$	−43.7053	−1.44802	−0.724011	−	0.689788i	$-0.757704\pi$
$911$	−43.7053	−1.44802	−0.724011	+	0.689788i	$0.757704\pi$
$912$	0	0
$913$	−19.0707	−0.631147
$914$	0	0
$915$	2.23933i	0.0740300i
$916$	0	0
$917$	− 8.54709i	− 0.282250i
$918$	0	0
$919$	44.9547	1.48292	0.741459	−	0.670998i	$-0.234134\pi$
$919$	44.9547	1.48292	0.741459	+	0.670998i	$0.234134\pi$
$920$	0	0
$921$	−3.23958	−0.106748
$922$	0	0
$923$	− 0.895683i	− 0.0294818i
$924$	0	0
$925$	− 31.7905i	− 1.04526i
$926$	0	0
$927$	7.40263	0.243134
$928$	0	0
$929$	11.1318	0.365222	0.182611	−	0.983185i	$-0.441545\pi$
$929$	11.1318	0.365222	0.182611	+	0.983185i	$0.441545\pi$
$930$	0	0
$931$	− 3.52230i	− 0.115439i
$932$	0	0
$933$	− 5.18871i	− 0.169871i
$934$	0	0
$935$	−3.24450	−0.106107
$936$	0	0
$937$	20.9629	0.684827	0.342413	−	0.939549i	$-0.388756\pi$
$937$	20.9629	0.684827	0.342413	+	0.939549i	$0.388756\pi$
$938$	0	0
$939$	− 3.50609i	− 0.114417i
$940$	0	0
$941$	0.215532i	0.00702615i	0.999994	+	0.00351308i	$0.00111825\pi$
$941$	0.215532i	0.00702615i	−0.999994	+	0.00351308i	$0.998882\pi$
$942$	0	0
$943$	−52.9961	−1.72579
$944$	0	0
$945$	2.55161	0.0830037
$946$	0	0
$947$	53.2972i	1.73193i	0.500108	+	0.865963i	$0.333294\pi$
$947$	53.2972i	1.73193i	−0.500108	+	0.865963i	$0.666706\pi$
$948$	0	0
$949$	− 14.4466i	− 0.468957i
$950$	0	0
$951$	2.33520	0.0757240
$952$	0	0
$953$	−56.7114	−1.83706	−0.918531	−	0.395349i	$-0.870624\pi$
$953$	−56.7114	−1.83706	−0.918531	+	0.395349i	$0.870624\pi$
$954$	0	0
$955$	4.44091i	0.143704i
$956$	0	0
$957$	− 2.25209i	− 0.0727998i
$958$	0	0
$959$	−4.83312	−0.156070
$960$	0	0
$961$	−20.3704	−0.657108
$962$	0	0
$963$	− 18.7309i	− 0.603595i
$964$	0	0
$965$	12.1658i	0.391630i
$966$	0	0
$967$	−23.7485	−0.763699	−0.381850	−	0.924224i	$-0.624713\pi$
$967$	−23.7485	−0.763699	−0.381850	+	0.924224i	$0.624713\pi$
$968$	0	0
$969$	1.14474	0.0367743
$970$	0	0
$971$	28.6064i	0.918022i	0.888431	+	0.459011i	$0.151796\pi$
$971$	28.6064i	0.918022i	−0.888431	+	0.459011i	$0.848204\pi$
$972$	0	0
$973$	− 20.7091i	− 0.663903i
$974$	0	0
$975$	−0.894324	−0.0286413
$976$	0	0
$977$	18.5455	0.593323	0.296661	−	0.954983i	$-0.404127\pi$
$977$	18.5455	0.593323	0.296661	+	0.954983i	$0.404127\pi$
$978$	0	0
$979$	− 36.8183i	− 1.17672i
$980$	0	0
$981$	− 11.8910i	− 0.379652i
$982$	0	0
$983$	57.1876	1.82400	0.912001	−	0.410188i	$-0.134537\pi$
$983$	57.1876	1.82400	0.912001	+	0.410188i	$0.134537\pi$
$984$	0	0
$985$	20.4118	0.650374
$986$	0	0
$987$	− 1.44929i	− 0.0461314i
$988$	0	0
$989$	− 49.4427i	− 1.57219i
$990$	0	0
$991$	0.0362040	0.00115006	0.000575029	−	1.00000i	$-0.499817\pi$
$991$	0.0362040	0.00115006	0.000575029		1.00000i	$0.499817\pi$
$992$	0	0
$993$	2.36807	0.0751484
$994$	0	0
$995$	− 37.4748i	− 1.18803i
$996$	0	0
$997$	22.5909i	0.715462i	0.933825	+	0.357731i	$0.116450\pi$
$997$	22.5909i	0.715462i	−0.933825	+	0.357731i	$0.883550\pi$
$998$	0	0
$999$	19.1890	0.607114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.c.a.1457.20		34
4.3	odd	2		728.2.c.a.365.4	yes	34
8.3	odd	2		728.2.c.a.365.3	✓	34
8.5	even	2	inner	2912.2.c.a.1457.15		34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.c.a.365.3	✓	34	8.3	odd	2
728.2.c.a.365.4	yes	34	4.3	odd	2
2912.2.c.a.1457.15		34	8.5	even	2	inner
2912.2.c.a.1457.20		34	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.302261i q^{3} -1.42871i q^{5} +1.00000 q^{7} +2.90864 q^{9} -2.11205i q^{11} +1.00000i q^{13} +0.431843 q^{15} +1.07522 q^{17} -3.52230i q^{19} +0.302261i q^{21} -5.47592 q^{23} +2.95878 q^{25} +1.78595i q^{27} -3.52776i q^{29} +3.26031 q^{31} +0.638391 q^{33} -1.42871i q^{35} -10.7444i q^{37} -0.302261 q^{39} +9.67803 q^{41} +9.02912i q^{43} -4.15560i q^{45} -4.79483 q^{47} +1.00000 q^{49} +0.324997i q^{51} +0.906733i q^{53} -3.01752 q^{55} +1.06465 q^{57} -6.88024i q^{59} +5.18552i q^{61} +2.90864 q^{63} +1.42871 q^{65} +5.46202i q^{67} -1.65516i q^{69} -0.895683 q^{71} -14.4466 q^{73} +0.894324i q^{75} -2.11205i q^{77} -15.5041 q^{79} +8.18609 q^{81} -9.02943i q^{83} -1.53618i q^{85} +1.06630 q^{87} +17.4324 q^{89} +1.00000i q^{91} +0.985464i q^{93} -5.03235 q^{95} +4.26779 q^{97} -6.14320i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) \) 34 * q + 34 * q^7 - 26 * q^9 + 8 * q^15 - 20 * q^17 + 20 * q^23 - 22 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 34 * q^49 + 32 * q^55 + 8 * q^57 - 26 * q^63 - 20 * q^65 - 64 * q^71 - 20 * q^79 - 6 * q^81 + 56 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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