LMFDB - Embedded newform 2912.2.c.b.1457.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(1457,2912)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.2
Character	$\chi$	$=$	2912.1457
Dual form			2912.2.c.b.1457.37

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.17321i q^{3} +3.62722i q^{5} -1.00000 q^{7} -7.06928 q^{9} +4.46405i q^{11} -1.00000i q^{13} +11.5100 q^{15} -4.52215 q^{17} -5.32637i q^{19} +3.17321i q^{21} +3.03715 q^{23} -8.15676 q^{25} +12.9127i q^{27} -5.95865i q^{29} +2.16610 q^{31} +14.1654 q^{33} -3.62722i q^{35} +5.35761i q^{37} -3.17321 q^{39} +7.13247 q^{41} -10.8225i q^{43} -25.6419i q^{45} +5.62889 q^{47} +1.00000 q^{49} +14.3498i q^{51} -4.80681i q^{53} -16.1921 q^{55} -16.9017 q^{57} -9.69248i q^{59} -1.86977i q^{61} +7.06928 q^{63} +3.62722 q^{65} -2.96132i q^{67} -9.63752i q^{69} +7.49730 q^{71} +8.99757 q^{73} +25.8831i q^{75} -4.46405i q^{77} +0.795743 q^{79} +19.7668 q^{81} -15.6044i q^{83} -16.4029i q^{85} -18.9080 q^{87} +5.98526 q^{89} +1.00000i q^{91} -6.87349i q^{93} +19.3200 q^{95} -4.04656 q^{97} -31.5576i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 38 q^{7} - 46 q^{9} + 8 q^{15} + 20 q^{17} - 12 q^{23} - 50 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 38 q^{49} + 8 q^{57} + 46 q^{63} + 20 q^{65} + 12 q^{79} + 62 q^{81} + 56 q^{87}+O(q^{100}) $ 38 * q - 38 * q^7 - 46 * q^9 + 8 * q^15 + 20 * q^17 - 12 * q^23 - 50 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 38 * q^49 + 8 * q^57 + 46 * q^63 + 20 * q^65 + 12 * q^79 + 62 * 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
$3$	− 3.17321i	− 1.83206i	−0.401115	−	0.916028i	$-0.631377\pi$
$3$	− 3.17321i	− 1.83206i	0.401115	−	0.916028i	$-0.368623\pi$
$4$	0	0
$5$	3.62722i	1.62214i	0.584946	+	0.811072i	$0.301116\pi$
$5$	3.62722i	1.62214i	−0.584946	+	0.811072i	$0.698884\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−7.06928	−2.35643
$10$	0	0
$11$	4.46405i	1.34596i	0.739659	+	0.672981i	$0.234987\pi$
$11$	4.46405i	1.34596i	−0.739659	+	0.672981i	$0.765013\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	11.5100	2.97186
$16$	0	0
$17$	−4.52215	−1.09678	−0.548392	−	0.836222i	$-0.684760\pi$
$17$	−4.52215	−1.09678	−0.548392	+	0.836222i	$0.684760\pi$
$18$	0	0
$19$	− 5.32637i	− 1.22195i	−0.791648	−	0.610977i	$-0.790777\pi$
$19$	− 5.32637i	− 1.22195i	0.791648	−	0.610977i	$-0.209223\pi$
$20$	0	0
$21$	3.17321i	0.692452i
$22$	0	0
$23$	3.03715	0.633289	0.316645	−	0.948544i	$-0.397444\pi$
$23$	3.03715	0.633289	0.316645	+	0.948544i	$0.397444\pi$
$24$	0	0
$25$	−8.15676	−1.63135
$26$	0	0
$27$	12.9127i	2.48505i
$28$	0	0
$29$	− 5.95865i	− 1.10649i	−0.833018	−	0.553246i	$-0.813389\pi$
$29$	− 5.95865i	− 1.10649i	0.833018	−	0.553246i	$-0.186611\pi$
$30$	0	0
$31$	2.16610	0.389043	0.194521	−	0.980898i	$-0.437685\pi$
$31$	2.16610	0.389043	0.194521	+	0.980898i	$0.437685\pi$
$32$	0	0
$33$	14.1654	2.46588
$34$	0	0
$35$	− 3.62722i	− 0.613113i
$36$	0	0
$37$	5.35761i	0.880786i	0.897805	+	0.440393i	$0.145161\pi$
$37$	5.35761i	0.880786i	−0.897805	+	0.440393i	$0.854839\pi$
$38$	0	0
$39$	−3.17321	−0.508121
$40$	0	0
$41$	7.13247	1.11391	0.556953	−	0.830544i	$-0.311971\pi$
$41$	7.13247	1.11391	0.556953	+	0.830544i	$0.311971\pi$
$42$	0	0
$43$	− 10.8225i	− 1.65041i	−0.564832	−	0.825206i	$-0.691059\pi$
$43$	− 10.8225i	− 1.65041i	0.564832	−	0.825206i	$-0.308941\pi$
$44$	0	0
$45$	− 25.6419i	− 3.82246i
$46$	0	0
$47$	5.62889	0.821058	0.410529	−	0.911848i	$-0.365344\pi$
$47$	5.62889	0.821058	0.410529	+	0.911848i	$0.365344\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	14.3498i	2.00937i
$52$	0	0
$53$	− 4.80681i	− 0.660266i	−0.943934	−	0.330133i	$-0.892906\pi$
$53$	− 4.80681i	− 0.660266i	0.943934	−	0.330133i	$-0.107094\pi$
$54$	0	0
$55$	−16.1921	−2.18335
$56$	0	0
$57$	−16.9017	−2.23869
$58$	0	0
$59$	− 9.69248i	− 1.26185i	−0.775843	−	0.630927i	$-0.782675\pi$
$59$	− 9.69248i	− 1.26185i	0.775843	−	0.630927i	$-0.217325\pi$
$60$	0	0
$61$	− 1.86977i	− 0.239400i	−0.992810	−	0.119700i	$-0.961807\pi$
$61$	− 1.86977i	− 0.239400i	0.992810	−	0.119700i	$-0.0381932\pi$
$62$	0	0
$63$	7.06928	0.890645
$64$	0	0
$65$	3.62722	0.449902
$66$	0	0
$67$	− 2.96132i	− 0.361783i	−0.983503	−	0.180891i	$-0.942102\pi$
$67$	− 2.96132i	− 0.361783i	0.983503	−	0.180891i	$-0.0578982\pi$
$68$	0	0
$69$	− 9.63752i	− 1.16022i
$70$	0	0
$71$	7.49730	0.889766	0.444883	−	0.895589i	$-0.353245\pi$
$71$	7.49730	0.889766	0.444883	+	0.895589i	$0.353245\pi$
$72$	0	0
$73$	8.99757	1.05309	0.526543	−	0.850148i	$-0.323488\pi$
$73$	8.99757	1.05309	0.526543	+	0.850148i	$0.323488\pi$
$74$	0	0
$75$	25.8831i	2.98873i
$76$	0	0
$77$	− 4.46405i	− 0.508726i
$78$	0	0
$79$	0.795743	0.0895280	0.0447640	−	0.998998i	$-0.485746\pi$
$79$	0.795743	0.0895280	0.0447640	+	0.998998i	$0.485746\pi$
$80$	0	0
$81$	19.7668	2.19632
$82$	0	0
$83$	− 15.6044i	− 1.71280i	−0.516312	−	0.856401i	$-0.672695\pi$
$83$	− 15.6044i	− 1.71280i	0.516312	−	0.856401i	$-0.327305\pi$
$84$	0	0
$85$	− 16.4029i	− 1.77914i
$86$	0	0
$87$	−18.9080	−2.02716
$88$	0	0
$89$	5.98526	0.634436	0.317218	−	0.948353i	$-0.397251\pi$
$89$	5.98526	0.634436	0.317218	+	0.948353i	$0.397251\pi$
$90$	0	0
$91$	1.00000i	0.104828i
$92$	0	0
$93$	− 6.87349i	− 0.712747i
$94$	0	0
$95$	19.3200	1.98219
$96$	0	0
$97$	−4.04656	−0.410866	−0.205433	−	0.978671i	$-0.565860\pi$
$97$	−4.04656	−0.410866	−0.205433	+	0.978671i	$0.565860\pi$
$98$	0	0
$99$	− 31.5576i	− 3.17166i
$100$	0	0
$101$	− 11.2085i	− 1.11529i	−0.830080	−	0.557644i	$-0.811705\pi$
$101$	− 11.2085i	− 1.11529i	0.830080	−	0.557644i	$-0.188295\pi$
$102$	0	0
$103$	9.91121	0.976580	0.488290	−	0.872681i	$-0.337621\pi$
$103$	9.91121	0.976580	0.488290	+	0.872681i	$0.337621\pi$
$104$	0	0
$105$	−11.5100	−1.12326
$106$	0	0
$107$	− 18.7309i	− 1.81079i	−0.424572	−	0.905394i	$-0.639575\pi$
$107$	− 18.7309i	− 1.81079i	0.424572	−	0.905394i	$-0.360425\pi$
$108$	0	0
$109$	1.34276i	0.128613i	0.997930	+	0.0643064i	$0.0204835\pi$
$109$	1.34276i	0.128613i	−0.997930	+	0.0643064i	$0.979517\pi$
$110$	0	0
$111$	17.0008	1.61365
$112$	0	0
$113$	−3.29148	−0.309637	−0.154818	−	0.987943i	$-0.549479\pi$
$113$	−3.29148	−0.309637	−0.154818	+	0.987943i	$0.549479\pi$
$114$	0	0
$115$	11.0164i	1.02729i
$116$	0	0
$117$	7.06928i	0.653555i
$118$	0	0
$119$	4.52215	0.414545
$120$	0	0
$121$	−8.92778	−0.811616
$122$	0	0
$123$	− 22.6329i	− 2.04074i
$124$	0	0
$125$	− 11.4503i	− 1.02414i
$126$	0	0
$127$	−5.40864	−0.479939	−0.239970	−	0.970780i	$-0.577137\pi$
$127$	−5.40864	−0.479939	−0.239970	+	0.970780i	$0.577137\pi$
$128$	0	0
$129$	−34.3420	−3.02364
$130$	0	0
$131$	2.37850i	0.207811i	0.994587	+	0.103905i	$0.0331339\pi$
$131$	2.37850i	0.207811i	−0.994587	+	0.103905i	$0.966866\pi$
$132$	0	0
$133$	5.32637i	0.461855i
$134$	0	0
$135$	−46.8372	−4.03110
$136$	0	0
$137$	11.6939	0.999078	0.499539	−	0.866291i	$-0.333503\pi$
$137$	11.6939	0.999078	0.499539	+	0.866291i	$0.333503\pi$
$138$	0	0
$139$	8.82105i	0.748191i	0.927390	+	0.374096i	$0.122047\pi$
$139$	8.82105i	0.748191i	−0.927390	+	0.374096i	$0.877953\pi$
$140$	0	0
$141$	− 17.8617i	− 1.50422i
$142$	0	0
$143$	4.46405	0.373303
$144$	0	0
$145$	21.6133	1.79489
$146$	0	0
$147$	− 3.17321i	− 0.261722i
$148$	0	0
$149$	21.8767i	1.79221i	0.443847	+	0.896103i	$0.353613\pi$
$149$	21.8767i	1.79221i	−0.443847	+	0.896103i	$0.646387\pi$
$150$	0	0
$151$	−14.7699	−1.20196	−0.600980	−	0.799264i	$-0.705223\pi$
$151$	−14.7699	−1.20196	−0.600980	+	0.799264i	$0.705223\pi$
$152$	0	0
$153$	31.9684	2.58449
$154$	0	0
$155$	7.85692i	0.631083i
$156$	0	0
$157$	20.9384i	1.67107i	0.549439	+	0.835534i	$0.314841\pi$
$157$	20.9384i	1.67107i	−0.549439	+	0.835534i	$0.685159\pi$
$158$	0	0
$159$	−15.2530	−1.20964
$160$	0	0
$161$	−3.03715	−0.239361
$162$	0	0
$163$	8.42682i	0.660039i	0.943974	+	0.330020i	$0.107055\pi$
$163$	8.42682i	0.660039i	−0.943974	+	0.330020i	$0.892945\pi$
$164$	0	0
$165$	51.3811i	4.00001i
$166$	0	0
$167$	6.62169	0.512402	0.256201	−	0.966624i	$-0.417529\pi$
$167$	6.62169	0.512402	0.256201	+	0.966624i	$0.417529\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	37.6536i	2.87944i
$172$	0	0
$173$	− 0.690244i	− 0.0524782i	−0.999656	−	0.0262391i	$-0.991647\pi$
$173$	− 0.690244i	− 0.0524782i	0.999656	−	0.0262391i	$-0.00835313\pi$
$174$	0	0
$175$	8.15676	0.616593
$176$	0	0
$177$	−30.7563	−2.31178
$178$	0	0
$179$	8.77446i	0.655834i	0.944707	+	0.327917i	$0.106347\pi$
$179$	8.77446i	0.655834i	−0.944707	+	0.327917i	$0.893653\pi$
$180$	0	0
$181$	− 17.0171i	− 1.26487i	−0.774612	−	0.632437i	$-0.782055\pi$
$181$	− 17.0171i	− 1.26487i	0.774612	−	0.632437i	$-0.217945\pi$
$182$	0	0
$183$	−5.93318	−0.438593
$184$	0	0
$185$	−19.4333	−1.42876
$186$	0	0
$187$	− 20.1871i	− 1.47623i
$188$	0	0
$189$	− 12.9127i	− 0.939259i
$190$	0	0
$191$	4.35994	0.315474	0.157737	−	0.987481i	$-0.449580\pi$
$191$	4.35994	0.315474	0.157737	+	0.987481i	$0.449580\pi$
$192$	0	0
$193$	−6.26746	−0.451142	−0.225571	−	0.974227i	$-0.572425\pi$
$193$	−6.26746	−0.451142	−0.225571	+	0.974227i	$0.572425\pi$
$194$	0	0
$195$	− 11.5100i	− 0.824245i
$196$	0	0
$197$	− 4.63140i	− 0.329973i	−0.986296	−	0.164987i	$-0.947242\pi$
$197$	− 4.63140i	− 0.329973i	0.986296	−	0.164987i	$-0.0527581\pi$
$198$	0	0
$199$	−4.22515	−0.299513	−0.149757	−	0.988723i	$-0.547849\pi$
$199$	−4.22515	−0.299513	−0.149757	+	0.988723i	$0.547849\pi$
$200$	0	0
$201$	−9.39689	−0.662806
$202$	0	0
$203$	5.95865i	0.418215i
$204$	0	0
$205$	25.8711i	1.80691i
$206$	0	0
$207$	−21.4704	−1.49230
$208$	0	0
$209$	23.7772	1.64470
$210$	0	0
$211$	18.5552i	1.27739i	0.769458	+	0.638697i	$0.220526\pi$
$211$	18.5552i	1.27739i	−0.769458	+	0.638697i	$0.779474\pi$
$212$	0	0
$213$	− 23.7905i	− 1.63010i
$214$	0	0
$215$	39.2555	2.67721
$216$	0	0
$217$	−2.16610	−0.147044
$218$	0	0
$219$	− 28.5512i	− 1.92931i
$220$	0	0
$221$	4.52215i	0.304193i
$222$	0	0
$223$	−21.6455	−1.44949	−0.724744	−	0.689018i	$-0.758042\pi$
$223$	−21.6455	−1.44949	−0.724744	+	0.689018i	$0.758042\pi$
$224$	0	0
$225$	57.6624	3.84416
$226$	0	0
$227$	− 6.40745i	− 0.425278i	−0.977131	−	0.212639i	$-0.931794\pi$
$227$	− 6.40745i	− 0.425278i	0.977131	−	0.212639i	$-0.0682058\pi$
$228$	0	0
$229$	16.2529i	1.07402i	0.843576	+	0.537009i	$0.180446\pi$
$229$	16.2529i	1.07402i	−0.843576	+	0.537009i	$0.819554\pi$
$230$	0	0
$231$	−14.1654	−0.932014
$232$	0	0
$233$	−10.9511	−0.717433	−0.358716	−	0.933447i	$-0.616785\pi$
$233$	−10.9511	−0.717433	−0.358716	+	0.933447i	$0.616785\pi$
$234$	0	0
$235$	20.4172i	1.33187i
$236$	0	0
$237$	− 2.52506i	− 0.164020i
$238$	0	0
$239$	18.6060	1.20352	0.601762	−	0.798675i	$-0.294466\pi$
$239$	18.6060	1.20352	0.601762	+	0.798675i	$0.294466\pi$
$240$	0	0
$241$	−9.60417	−0.618659	−0.309330	−	0.950955i	$-0.600105\pi$
$241$	−9.60417	−0.618659	−0.309330	+	0.950955i	$0.600105\pi$
$242$	0	0
$243$	− 23.9864i	− 1.53873i
$244$	0	0
$245$	3.62722i	0.231735i
$246$	0	0
$247$	−5.32637	−0.338909
$248$	0	0
$249$	−49.5160	−3.13795
$250$	0	0
$251$	− 15.9028i	− 1.00377i	−0.864933	−	0.501887i	$-0.832639\pi$
$251$	− 15.9028i	− 1.00377i	0.864933	−	0.501887i	$-0.167361\pi$
$252$	0	0
$253$	13.5580i	0.852384i
$254$	0	0
$255$	−52.0498	−3.25948
$256$	0	0
$257$	−0.438800	−0.0273716	−0.0136858	−	0.999906i	$-0.504356\pi$
$257$	−0.438800	−0.0273716	−0.0136858	+	0.999906i	$0.504356\pi$
$258$	0	0
$259$	− 5.35761i	− 0.332906i
$260$	0	0
$261$	42.1233i	2.60737i
$262$	0	0
$263$	−8.10209	−0.499597	−0.249798	−	0.968298i	$-0.580364\pi$
$263$	−8.10209	−0.499597	−0.249798	+	0.968298i	$0.580364\pi$
$264$	0	0
$265$	17.4354	1.07105
$266$	0	0
$267$	− 18.9925i	− 1.16232i
$268$	0	0
$269$	7.74819i	0.472415i	0.971703	+	0.236208i	$0.0759046\pi$
$269$	7.74819i	0.472415i	−0.971703	+	0.236208i	$0.924095\pi$
$270$	0	0
$271$	−1.96747	−0.119515	−0.0597577	−	0.998213i	$-0.519033\pi$
$271$	−1.96747	−0.119515	−0.0597577	+	0.998213i	$0.519033\pi$
$272$	0	0
$273$	3.17321	0.192052
$274$	0	0
$275$	− 36.4122i	− 2.19574i
$276$	0	0
$277$	− 27.2840i	− 1.63934i	−0.572839	−	0.819668i	$-0.694158\pi$
$277$	− 27.2840i	− 1.63934i	0.572839	−	0.819668i	$-0.305842\pi$
$278$	0	0
$279$	−15.3127	−0.916750
$280$	0	0
$281$	−11.1502	−0.665168	−0.332584	−	0.943074i	$-0.607920\pi$
$281$	−11.1502	−0.665168	−0.332584	+	0.943074i	$0.607920\pi$
$282$	0	0
$283$	− 28.0765i	− 1.66897i	−0.551028	−	0.834487i	$-0.685764\pi$
$283$	− 28.0765i	− 1.66897i	0.551028	−	0.834487i	$-0.314236\pi$
$284$	0	0
$285$	− 61.3063i	− 3.63147i
$286$	0	0
$287$	−7.13247	−0.421017
$288$	0	0
$289$	3.44987	0.202933
$290$	0	0
$291$	12.8406i	0.752729i
$292$	0	0
$293$	22.9916i	1.34318i	0.740921	+	0.671592i	$0.234389\pi$
$293$	22.9916i	1.34318i	−0.740921	+	0.671592i	$0.765611\pi$
$294$	0	0
$295$	35.1568	2.04691
$296$	0	0
$297$	−57.6429	−3.34478
$298$	0	0
$299$	− 3.03715i	− 0.175643i
$300$	0	0
$301$	10.8225i	0.623797i
$302$	0	0
$303$	−35.5670	−2.04327
$304$	0	0
$305$	6.78208	0.388341
$306$	0	0
$307$	− 4.76516i	− 0.271962i	−0.990711	−	0.135981i	$-0.956581\pi$
$307$	− 4.76516i	− 0.271962i	0.990711	−	0.135981i	$-0.0434186\pi$
$308$	0	0
$309$	− 31.4504i	− 1.78915i
$310$	0	0
$311$	17.7286	1.00530	0.502649	−	0.864490i	$-0.332359\pi$
$311$	17.7286	1.00530	0.502649	+	0.864490i	$0.332359\pi$
$312$	0	0
$313$	1.92067	0.108563	0.0542813	−	0.998526i	$-0.482713\pi$
$313$	1.92067	0.108563	0.0542813	+	0.998526i	$0.482713\pi$
$314$	0	0
$315$	25.6419i	1.44475i
$316$	0	0
$317$	− 14.6051i	− 0.820305i	−0.912017	−	0.410152i	$-0.865476\pi$
$317$	− 14.6051i	− 0.820305i	0.912017	−	0.410152i	$-0.134524\pi$
$318$	0	0
$319$	26.5997	1.48930
$320$	0	0
$321$	−59.4373	−3.31746
$322$	0	0
$323$	24.0867i	1.34022i
$324$	0	0
$325$	8.15676i	0.452456i
$326$	0	0
$327$	4.26085	0.235626
$328$	0	0
$329$	−5.62889	−0.310331
$330$	0	0
$331$	− 23.8200i	− 1.30927i	−0.755947	−	0.654633i	$-0.772823\pi$
$331$	− 23.8200i	− 1.30927i	0.755947	−	0.654633i	$-0.227177\pi$
$332$	0	0
$333$	− 37.8745i	− 2.07551i
$334$	0	0
$335$	10.7414	0.586864
$336$	0	0
$337$	12.2389	0.666698	0.333349	−	0.942804i	$-0.391821\pi$
$337$	12.2389	0.666698	0.333349	+	0.942804i	$0.391821\pi$
$338$	0	0
$339$	10.4446i	0.567271i
$340$	0	0
$341$	9.66958i	0.523637i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	34.9574	1.88205
$346$	0	0
$347$	− 17.3706i	− 0.932505i	−0.884652	−	0.466253i	$-0.845604\pi$
$347$	− 17.3706i	− 0.932505i	0.884652	−	0.466253i	$-0.154396\pi$
$348$	0	0
$349$	− 20.6803i	− 1.10699i	−0.832852	−	0.553497i	$-0.813293\pi$
$349$	− 20.6803i	− 1.10699i	0.832852	−	0.553497i	$-0.186707\pi$
$350$	0	0
$351$	12.9127	0.689228
$352$	0	0
$353$	25.0345	1.33245	0.666227	−	0.745749i	$-0.267908\pi$
$353$	25.0345	1.33245	0.666227	+	0.745749i	$0.267908\pi$
$354$	0	0
$355$	27.1944i	1.44333i
$356$	0	0
$357$	− 14.3498i	− 0.759469i
$358$	0	0
$359$	−34.6180	−1.82707	−0.913535	−	0.406761i	$-0.866658\pi$
$359$	−34.6180	−1.82707	−0.913535	+	0.406761i	$0.866658\pi$
$360$	0	0
$361$	−9.37027	−0.493172
$362$	0	0
$363$	28.3297i	1.48693i
$364$	0	0
$365$	32.6362i	1.70826i
$366$	0	0
$367$	29.6400	1.54719	0.773597	−	0.633677i	$-0.218455\pi$
$367$	29.6400	1.54719	0.773597	+	0.633677i	$0.218455\pi$
$368$	0	0
$369$	−50.4214	−2.62483
$370$	0	0
$371$	4.80681i	0.249557i
$372$	0	0
$373$	− 2.18046i	− 0.112900i	−0.998405	−	0.0564499i	$-0.982022\pi$
$373$	− 2.18046i	− 0.112900i	0.998405	−	0.0564499i	$-0.0179781\pi$
$374$	0	0
$375$	−36.3342	−1.87629
$376$	0	0
$377$	−5.95865	−0.306886
$378$	0	0
$379$	6.07608i	0.312107i	0.987749	+	0.156054i	$0.0498773\pi$
$379$	6.07608i	0.312107i	−0.987749	+	0.156054i	$0.950123\pi$
$380$	0	0
$381$	17.1628i	0.879275i
$382$	0	0
$383$	17.9109	0.915204	0.457602	−	0.889157i	$-0.348708\pi$
$383$	17.9109	0.915204	0.457602	+	0.889157i	$0.348708\pi$
$384$	0	0
$385$	16.1921	0.825227
$386$	0	0
$387$	76.5071i	3.88907i
$388$	0	0
$389$	− 2.17676i	− 0.110366i	−0.998476	−	0.0551830i	$-0.982426\pi$
$389$	− 2.17676i	− 0.110366i	0.998476	−	0.0551830i	$-0.0175742\pi$
$390$	0	0
$391$	−13.7344	−0.694581
$392$	0	0
$393$	7.54749	0.380720
$394$	0	0
$395$	2.88634i	0.145227i
$396$	0	0
$397$	2.95214i	0.148164i	0.997252	+	0.0740818i	$0.0236026\pi$
$397$	2.95214i	0.148164i	−0.997252	+	0.0740818i	$0.976397\pi$
$398$	0	0
$399$	16.9017	0.846144
$400$	0	0
$401$	−2.03003	−0.101375	−0.0506875	−	0.998715i	$-0.516141\pi$
$401$	−2.03003	−0.101375	−0.0506875	+	0.998715i	$0.516141\pi$
$402$	0	0
$403$	− 2.16610i	− 0.107901i
$404$	0	0
$405$	71.6988i	3.56274i
$406$	0	0
$407$	−23.9167	−1.18551
$408$	0	0
$409$	−35.9771	−1.77895	−0.889477	−	0.456981i	$-0.848931\pi$
$409$	−35.9771	−1.77895	−0.889477	+	0.456981i	$0.848931\pi$
$410$	0	0
$411$	− 37.1073i	− 1.83037i
$412$	0	0
$413$	9.69248i	0.476936i
$414$	0	0
$415$	56.6005	2.77841
$416$	0	0
$417$	27.9910	1.37073
$418$	0	0
$419$	4.64982i	0.227158i	0.993529	+	0.113579i	$0.0362316\pi$
$419$	4.64982i	0.227158i	−0.993529	+	0.113579i	$0.963768\pi$
$420$	0	0
$421$	− 2.96094i	− 0.144307i	−0.997394	−	0.0721537i	$-0.977013\pi$
$421$	− 2.96094i	− 0.144307i	0.997394	−	0.0721537i	$-0.0229872\pi$
$422$	0	0
$423$	−39.7922	−1.93476
$424$	0	0
$425$	36.8861	1.78924
$426$	0	0
$427$	1.86977i	0.0904846i
$428$	0	0
$429$	− 14.1654i	− 0.683912i
$430$	0	0
$431$	−18.2306	−0.878136	−0.439068	−	0.898454i	$-0.644691\pi$
$431$	−18.2306	−0.878136	−0.439068	+	0.898454i	$0.644691\pi$
$432$	0	0
$433$	32.8482	1.57858	0.789291	−	0.614019i	$-0.210448\pi$
$433$	32.8482	1.57858	0.789291	+	0.614019i	$0.210448\pi$
$434$	0	0
$435$	− 68.5837i	− 3.28834i
$436$	0	0
$437$	− 16.1770i	− 0.773850i
$438$	0	0
$439$	−22.7606	−1.08630	−0.543151	−	0.839635i	$-0.682769\pi$
$439$	−22.7606	−1.08630	−0.543151	+	0.839635i	$0.682769\pi$
$440$	0	0
$441$	−7.06928	−0.336632
$442$	0	0
$443$	3.26039i	0.154906i	0.996996	+	0.0774529i	$0.0246787\pi$
$443$	3.26039i	0.154906i	−0.996996	+	0.0774529i	$0.975321\pi$
$444$	0	0
$445$	21.7099i	1.02915i
$446$	0	0
$447$	69.4193	3.28342
$448$	0	0
$449$	−2.78848	−0.131597	−0.0657983	−	0.997833i	$-0.520959\pi$
$449$	−2.78848	−0.131597	−0.0657983	+	0.997833i	$0.520959\pi$
$450$	0	0
$451$	31.8397i	1.49927i
$452$	0	0
$453$	46.8682i	2.20206i
$454$	0	0
$455$	−3.62722	−0.170047
$456$	0	0
$457$	14.7310	0.689088	0.344544	−	0.938770i	$-0.388033\pi$
$457$	14.7310	0.689088	0.344544	+	0.938770i	$0.388033\pi$
$458$	0	0
$459$	− 58.3931i	− 2.72556i
$460$	0	0
$461$	− 20.4768i	− 0.953698i	−0.878985	−	0.476849i	$-0.841779\pi$
$461$	− 20.4768i	− 0.953698i	0.878985	−	0.476849i	$-0.158221\pi$
$462$	0	0
$463$	10.3546	0.481220	0.240610	−	0.970622i	$-0.422652\pi$
$463$	10.3546	0.481220	0.240610	+	0.970622i	$0.422652\pi$
$464$	0	0
$465$	24.9317	1.15618
$466$	0	0
$467$	− 31.9235i	− 1.47724i	−0.674120	−	0.738622i	$-0.735477\pi$
$467$	− 31.9235i	− 1.47724i	0.674120	−	0.738622i	$-0.264523\pi$
$468$	0	0
$469$	2.96132i	0.136741i
$470$	0	0
$471$	66.4420	3.06149
$472$	0	0
$473$	48.3121	2.22139
$474$	0	0
$475$	43.4460i	1.99344i
$476$	0	0
$477$	33.9806i	1.55587i
$478$	0	0
$479$	−10.5582	−0.482414	−0.241207	−	0.970474i	$-0.577543\pi$
$479$	−10.5582	−0.482414	−0.241207	+	0.970474i	$0.577543\pi$
$480$	0	0
$481$	5.35761	0.244286
$482$	0	0
$483$	9.63752i	0.438522i
$484$	0	0
$485$	− 14.6778i	− 0.666484i
$486$	0	0
$487$	−42.8480	−1.94163	−0.970815	−	0.239831i	$-0.922908\pi$
$487$	−42.8480	−1.94163	−0.970815	+	0.239831i	$0.922908\pi$
$488$	0	0
$489$	26.7401	1.20923
$490$	0	0
$491$	5.44721i	0.245829i	0.992417	+	0.122915i	$0.0392242\pi$
$491$	5.44721i	0.245829i	−0.992417	+	0.122915i	$0.960776\pi$
$492$	0	0
$493$	26.9459i	1.21358i
$494$	0	0
$495$	114.467	5.14489
$496$	0	0
$497$	−7.49730	−0.336300
$498$	0	0
$499$	− 7.15957i	− 0.320506i	−0.987076	−	0.160253i	$-0.948769\pi$
$499$	− 7.15957i	− 0.320506i	0.987076	−	0.160253i	$-0.0512311\pi$
$500$	0	0
$501$	− 21.0120i	− 0.938748i
$502$	0	0
$503$	8.08761	0.360609	0.180304	−	0.983611i	$-0.442292\pi$
$503$	8.08761	0.360609	0.180304	+	0.983611i	$0.442292\pi$
$504$	0	0
$505$	40.6558	1.80916
$506$	0	0
$507$	3.17321i	0.140927i
$508$	0	0
$509$	17.5326i	0.777120i	0.921423	+	0.388560i	$0.127027\pi$
$509$	17.5326i	0.777120i	−0.921423	+	0.388560i	$0.872973\pi$
$510$	0	0
$511$	−8.99757	−0.398029
$512$	0	0
$513$	68.7778	3.03661
$514$	0	0
$515$	35.9502i	1.58415i
$516$	0	0
$517$	25.1277i	1.10511i
$518$	0	0
$519$	−2.19029	−0.0961430
$520$	0	0
$521$	38.2311	1.67494	0.837468	−	0.546487i	$-0.184035\pi$
$521$	38.2311	1.67494	0.837468	+	0.546487i	$0.184035\pi$
$522$	0	0
$523$	− 3.18625i	− 0.139325i	−0.997571	−	0.0696624i	$-0.977808\pi$
$523$	− 3.18625i	− 0.139325i	0.997571	−	0.0696624i	$-0.0221922\pi$
$524$	0	0
$525$	− 25.8831i	− 1.12963i
$526$	0	0
$527$	−9.79542	−0.426695
$528$	0	0
$529$	−13.7757	−0.598945
$530$	0	0
$531$	68.5188i	2.97346i
$532$	0	0
$533$	− 7.13247i	− 0.308942i
$534$	0	0
$535$	67.9413	2.93736
$536$	0	0
$537$	27.8432	1.20152
$538$	0	0
$539$	4.46405i	0.192280i
$540$	0	0
$541$	− 22.5640i	− 0.970103i	−0.874486	−	0.485051i	$-0.838801\pi$
$541$	− 22.5640i	− 0.970103i	0.874486	−	0.485051i	$-0.161199\pi$
$542$	0	0
$543$	−53.9990	−2.31732
$544$	0	0
$545$	−4.87048	−0.208628
$546$	0	0
$547$	1.52020i	0.0649990i	0.999472	+	0.0324995i	$0.0103467\pi$
$547$	1.52020i	0.0649990i	−0.999472	+	0.0324995i	$0.989653\pi$
$548$	0	0
$549$	13.2179i	0.564127i
$550$	0	0
$551$	−31.7380	−1.35208
$552$	0	0
$553$	−0.795743	−0.0338384
$554$	0	0
$555$	61.6659i	2.61757i
$556$	0	0
$557$	− 15.1703i	− 0.642785i	−0.946946	−	0.321393i	$-0.895849\pi$
$557$	− 15.1703i	− 0.642785i	0.946946	−	0.321393i	$-0.104151\pi$
$558$	0	0
$559$	−10.8225	−0.457742
$560$	0	0
$561$	−64.0581	−2.70453
$562$	0	0
$563$	22.5767i	0.951493i	0.879583	+	0.475746i	$0.157822\pi$
$563$	22.5767i	0.951493i	−0.879583	+	0.475746i	$0.842178\pi$
$564$	0	0
$565$	− 11.9389i	− 0.502275i
$566$	0	0
$567$	−19.7668	−0.830129
$568$	0	0
$569$	−30.2677	−1.26889	−0.634445	−	0.772968i	$-0.718771\pi$
$569$	−30.2677	−1.26889	−0.634445	+	0.772968i	$0.718771\pi$
$570$	0	0
$571$	− 32.5548i	− 1.36238i	−0.732108	−	0.681189i	$-0.761463\pi$
$571$	− 32.5548i	− 1.36238i	0.732108	−	0.681189i	$-0.238537\pi$
$572$	0	0
$573$	− 13.8350i	− 0.577966i
$574$	0	0
$575$	−24.7733	−1.03312
$576$	0	0
$577$	28.9127	1.20365	0.601826	−	0.798627i	$-0.294440\pi$
$577$	28.9127	1.20365	0.601826	+	0.798627i	$0.294440\pi$
$578$	0	0
$579$	19.8880i	0.826516i
$580$	0	0
$581$	15.6044i	0.647378i
$582$	0	0
$583$	21.4578	0.888693
$584$	0	0
$585$	−25.6419	−1.06016
$586$	0	0
$587$	29.5264i	1.21868i	0.792908	+	0.609342i	$0.208566\pi$
$587$	29.5264i	1.21868i	−0.792908	+	0.609342i	$0.791434\pi$
$588$	0	0
$589$	− 11.5374i	− 0.475392i
$590$	0	0
$591$	−14.6964	−0.604529
$592$	0	0
$593$	6.44558	0.264688	0.132344	−	0.991204i	$-0.457750\pi$
$593$	6.44558	0.264688	0.132344	+	0.991204i	$0.457750\pi$
$594$	0	0
$595$	16.4029i	0.672452i
$596$	0	0
$597$	13.4073i	0.548725i
$598$	0	0
$599$	9.42470	0.385083	0.192541	−	0.981289i	$-0.438327\pi$
$599$	9.42470	0.385083	0.192541	+	0.981289i	$0.438327\pi$
$600$	0	0
$601$	−7.96744	−0.324999	−0.162499	−	0.986709i	$-0.551956\pi$
$601$	−7.96744	−0.324999	−0.162499	+	0.986709i	$0.551956\pi$
$602$	0	0
$603$	20.9344i	0.852514i
$604$	0	0
$605$	− 32.3831i	− 1.31656i
$606$	0	0
$607$	24.9085	1.01100	0.505502	−	0.862826i	$-0.331307\pi$
$607$	24.9085	1.01100	0.505502	+	0.862826i	$0.331307\pi$
$608$	0	0
$609$	18.9080	0.766193
$610$	0	0
$611$	− 5.62889i	− 0.227720i
$612$	0	0
$613$	− 29.7739i	− 1.20256i	−0.799040	−	0.601278i	$-0.794659\pi$
$613$	− 29.7739i	− 1.20256i	0.799040	−	0.601278i	$-0.205341\pi$
$614$	0	0
$615$	82.0944	3.31037
$616$	0	0
$617$	33.7904	1.36035	0.680175	−	0.733050i	$-0.261904\pi$
$617$	33.7904	1.36035	0.680175	+	0.733050i	$0.261904\pi$
$618$	0	0
$619$	− 10.9841i	− 0.441488i	−0.975332	−	0.220744i	$-0.929151\pi$
$619$	− 10.9841i	− 0.441488i	0.975332	−	0.220744i	$-0.0708486\pi$
$620$	0	0
$621$	39.2177i	1.57375i
$622$	0	0
$623$	−5.98526	−0.239794
$624$	0	0
$625$	0.748927	0.0299571
$626$	0	0
$627$	− 75.4502i	− 3.01319i
$628$	0	0
$629$	− 24.2280i	− 0.966032i
$630$	0	0
$631$	−20.7918	−0.827708	−0.413854	−	0.910343i	$-0.635818\pi$
$631$	−20.7918	−0.827708	−0.413854	+	0.910343i	$0.635818\pi$
$632$	0	0
$633$	58.8797	2.34026
$634$	0	0
$635$	− 19.6183i	− 0.778530i
$636$	0	0
$637$	− 1.00000i	− 0.0396214i
$638$	0	0
$639$	−53.0005	−2.09667
$640$	0	0
$641$	20.7776	0.820665	0.410332	−	0.911936i	$-0.365413\pi$
$641$	20.7776	0.820665	0.410332	+	0.911936i	$0.365413\pi$
$642$	0	0
$643$	− 4.80576i	− 0.189521i	−0.995500	−	0.0947604i	$-0.969791\pi$
$643$	− 4.80576i	− 0.189521i	0.995500	−	0.0947604i	$-0.0302085\pi$
$644$	0	0
$645$	− 124.566i	− 4.90479i
$646$	0	0
$647$	24.8286	0.976114	0.488057	−	0.872812i	$-0.337706\pi$
$647$	24.8286	0.976114	0.488057	+	0.872812i	$0.337706\pi$
$648$	0	0
$649$	43.2677	1.69841
$650$	0	0
$651$	6.87349i	0.269393i
$652$	0	0
$653$	− 28.1755i	− 1.10259i	−0.834309	−	0.551296i	$-0.814133\pi$
$653$	− 28.1755i	− 1.10259i	0.834309	−	0.551296i	$-0.185867\pi$
$654$	0	0
$655$	−8.62736	−0.337099
$656$	0	0
$657$	−63.6063	−2.48152
$658$	0	0
$659$	− 43.4049i	− 1.69082i	−0.534121	−	0.845408i	$-0.679357\pi$
$659$	− 43.4049i	− 1.69082i	0.534121	−	0.845408i	$-0.320643\pi$
$660$	0	0
$661$	12.4759i	0.485256i	0.970119	+	0.242628i	$0.0780094\pi$
$661$	12.4759i	0.485256i	−0.970119	+	0.242628i	$0.921991\pi$
$662$	0	0
$663$	14.3498	0.557298
$664$	0	0
$665$	−19.3200	−0.749196
$666$	0	0
$667$	− 18.0973i	− 0.700730i
$668$	0	0
$669$	68.6857i	2.65554i
$670$	0	0
$671$	8.34676	0.322223
$672$	0	0
$673$	−16.7979	−0.647513	−0.323756	−	0.946141i	$-0.604946\pi$
$673$	−16.7979	−0.647513	−0.323756	+	0.946141i	$0.604946\pi$
$674$	0	0
$675$	− 105.326i	− 4.05398i
$676$	0	0
$677$	32.4535i	1.24729i	0.781709	+	0.623644i	$0.214348\pi$
$677$	32.4535i	1.24729i	−0.781709	+	0.623644i	$0.785652\pi$
$678$	0	0
$679$	4.04656	0.155293
$680$	0	0
$681$	−20.3322	−0.779132
$682$	0	0
$683$	− 47.5144i	− 1.81809i	−0.416698	−	0.909045i	$-0.636813\pi$
$683$	− 47.5144i	− 1.81809i	0.416698	−	0.909045i	$-0.363187\pi$
$684$	0	0
$685$	42.4164i	1.62065i
$686$	0	0
$687$	51.5738	1.96766
$688$	0	0
$689$	−4.80681	−0.183125
$690$	0	0
$691$	41.8900i	1.59357i	0.604263	+	0.796785i	$0.293468\pi$
$691$	41.8900i	1.59357i	−0.604263	+	0.796785i	$0.706532\pi$
$692$	0	0
$693$	31.5576i	1.19878i
$694$	0	0
$695$	−31.9959	−1.21367
$696$	0	0
$697$	−32.2541	−1.22171
$698$	0	0
$699$	34.7503i	1.31438i
$700$	0	0
$701$	− 27.9376i	− 1.05519i	−0.849496	−	0.527595i	$-0.823094\pi$
$701$	− 27.9376i	− 1.05519i	0.849496	−	0.527595i	$-0.176906\pi$
$702$	0	0
$703$	28.5367	1.07628
$704$	0	0
$705$	64.7882	2.44007
$706$	0	0
$707$	11.2085i	0.421540i
$708$	0	0
$709$	− 33.9800i	− 1.27615i	−0.769975	−	0.638074i	$-0.779731\pi$
$709$	− 33.9800i	− 1.27615i	0.769975	−	0.638074i	$-0.220269\pi$
$710$	0	0
$711$	−5.62532	−0.210966
$712$	0	0
$713$	6.57876	0.246376
$714$	0	0
$715$	16.1921i	0.605551i
$716$	0	0
$717$	− 59.0409i	− 2.20492i
$718$	0	0
$719$	−22.1904	−0.827563	−0.413781	−	0.910376i	$-0.635792\pi$
$719$	−22.1904	−0.827563	−0.413781	+	0.910376i	$0.635792\pi$
$720$	0	0
$721$	−9.91121	−0.369113
$722$	0	0
$723$	30.4761i	1.13342i
$724$	0	0
$725$	48.6032i	1.80508i
$726$	0	0
$727$	21.1722	0.785235	0.392618	−	0.919702i	$-0.371570\pi$
$727$	21.1722	0.785235	0.392618	+	0.919702i	$0.371570\pi$
$728$	0	0
$729$	−16.8133	−0.622713
$730$	0	0
$731$	48.9409i	1.81014i
$732$	0	0
$733$	31.5545i	1.16549i	0.812654	+	0.582746i	$0.198022\pi$
$733$	31.5545i	1.16549i	−0.812654	+	0.582746i	$0.801978\pi$
$734$	0	0
$735$	11.5100	0.424551
$736$	0	0
$737$	13.2195	0.486946
$738$	0	0
$739$	21.3923i	0.786930i	0.919340	+	0.393465i	$0.128724\pi$
$739$	21.3923i	0.786930i	−0.919340	+	0.393465i	$0.871276\pi$
$740$	0	0
$741$	16.9017i	0.620900i
$742$	0	0
$743$	−36.1540	−1.32636	−0.663180	−	0.748460i	$-0.730794\pi$
$743$	−36.1540	−1.32636	−0.663180	+	0.748460i	$0.730794\pi$
$744$	0	0
$745$	−79.3515	−2.90722
$746$	0	0
$747$	110.312i	4.03609i
$748$	0	0
$749$	18.7309i	0.684414i
$750$	0	0
$751$	−25.1618	−0.918167	−0.459083	−	0.888393i	$-0.651822\pi$
$751$	−25.1618	−0.918167	−0.459083	+	0.888393i	$0.651822\pi$
$752$	0	0
$753$	−50.4629	−1.83897
$754$	0	0
$755$	− 53.5739i	− 1.94975i
$756$	0	0
$757$	− 1.96348i	− 0.0713639i	−0.999363	−	0.0356819i	$-0.988640\pi$
$757$	− 1.96348i	− 0.0713639i	0.999363	−	0.0356819i	$-0.0113603\pi$
$758$	0	0
$759$	43.0224	1.56161
$760$	0	0
$761$	−13.8287	−0.501292	−0.250646	−	0.968079i	$-0.580643\pi$
$761$	−13.8287	−0.501292	−0.250646	+	0.968079i	$0.580643\pi$
$762$	0	0
$763$	− 1.34276i	− 0.0486111i
$764$	0	0
$765$	115.956i	4.19241i
$766$	0	0
$767$	−9.69248	−0.349975
$768$	0	0
$769$	26.0028	0.937684	0.468842	−	0.883282i	$-0.344671\pi$
$769$	26.0028	0.937684	0.468842	+	0.883282i	$0.344671\pi$
$770$	0	0
$771$	1.39241i	0.0501463i
$772$	0	0
$773$	4.06853i	0.146335i	0.997320	+	0.0731675i	$0.0233108\pi$
$773$	4.06853i	0.146335i	−0.997320	+	0.0731675i	$0.976689\pi$
$774$	0	0
$775$	−17.6683	−0.634665
$776$	0	0
$777$	−17.0008	−0.609902
$778$	0	0
$779$	− 37.9902i	− 1.36114i
$780$	0	0
$781$	33.4684i	1.19759i
$782$	0	0
$783$	76.9421	2.74969
$784$	0	0
$785$	−75.9483	−2.71071
$786$	0	0
$787$	− 38.3646i	− 1.36755i	−0.729692	−	0.683776i	$-0.760337\pi$
$787$	− 38.3646i	− 1.36755i	0.729692	−	0.683776i	$-0.239663\pi$
$788$	0	0
$789$	25.7097i	0.915288i
$790$	0	0
$791$	3.29148	0.117032
$792$	0	0
$793$	−1.86977	−0.0663975
$794$	0	0
$795$	− 55.3261i	− 1.96222i
$796$	0	0
$797$	− 33.8141i	− 1.19776i	−0.800840	−	0.598879i	$-0.795613\pi$
$797$	− 33.8141i	− 1.19776i	0.800840	−	0.598879i	$-0.204387\pi$
$798$	0	0
$799$	−25.4547	−0.900522
$800$	0	0
$801$	−42.3114	−1.49500
$802$	0	0
$803$	40.1656i	1.41741i
$804$	0	0
$805$	− 11.0164i	− 0.388278i
$806$	0	0
$807$	24.5866	0.865491
$808$	0	0
$809$	29.0912	1.02279	0.511396	−	0.859345i	$-0.329129\pi$
$809$	29.0912	1.02279	0.511396	+	0.859345i	$0.329129\pi$
$810$	0	0
$811$	26.0470i	0.914635i	0.889303	+	0.457318i	$0.151190\pi$
$811$	26.0470i	0.914635i	−0.889303	+	0.457318i	$0.848810\pi$
$812$	0	0
$813$	6.24320i	0.218959i
$814$	0	0
$815$	−30.5660	−1.07068
$816$	0	0
$817$	−57.6445	−2.01673
$818$	0	0
$819$	− 7.06928i	− 0.247021i
$820$	0	0
$821$	49.1118i	1.71401i	0.515306	+	0.857006i	$0.327678\pi$
$821$	49.1118i	1.71401i	−0.515306	+	0.857006i	$0.672322\pi$
$822$	0	0
$823$	32.7800	1.14264	0.571320	−	0.820728i	$-0.306432\pi$
$823$	32.7800	1.14264	0.571320	+	0.820728i	$0.306432\pi$
$824$	0	0
$825$	−115.544	−4.02271
$826$	0	0
$827$	− 13.9227i	− 0.484139i	−0.970259	−	0.242070i	$-0.922174\pi$
$827$	− 13.9227i	− 0.484139i	0.970259	−	0.242070i	$-0.0778263\pi$
$828$	0	0
$829$	29.9759i	1.04110i	0.853830	+	0.520552i	$0.174274\pi$
$829$	29.9759i	1.04110i	−0.853830	+	0.520552i	$0.825726\pi$
$830$	0	0
$831$	−86.5779	−3.00335
$832$	0	0
$833$	−4.52215	−0.156683
$834$	0	0
$835$	24.0184i	0.831190i
$836$	0	0
$837$	27.9701i	0.966789i
$838$	0	0
$839$	−22.6147	−0.780746	−0.390373	−	0.920657i	$-0.627654\pi$
$839$	−22.6147	−0.780746	−0.390373	+	0.920657i	$0.627654\pi$
$840$	0	0
$841$	−6.50546	−0.224326
$842$	0	0
$843$	35.3821i	1.21862i
$844$	0	0
$845$	− 3.62722i	− 0.124780i
$846$	0	0
$847$	8.92778	0.306762
$848$	0	0
$849$	−89.0926	−3.05765
$850$	0	0
$851$	16.2719i	0.557793i
$852$	0	0
$853$	29.5133i	1.01052i	0.862968	+	0.505259i	$0.168603\pi$
$853$	29.5133i	1.01052i	−0.862968	+	0.505259i	$0.831397\pi$
$854$	0	0
$855$	−136.578	−4.67087
$856$	0	0
$857$	29.6904	1.01420	0.507102	−	0.861886i	$-0.330717\pi$
$857$	29.6904	1.01420	0.507102	+	0.861886i	$0.330717\pi$
$858$	0	0
$859$	41.2118i	1.40613i	0.711126	+	0.703065i	$0.248186\pi$
$859$	41.2118i	1.40613i	−0.711126	+	0.703065i	$0.751814\pi$
$860$	0	0
$861$	22.6329i	0.771326i
$862$	0	0
$863$	−11.3243	−0.385484	−0.192742	−	0.981249i	$-0.561738\pi$
$863$	−11.3243	−0.385484	−0.192742	+	0.981249i	$0.561738\pi$
$864$	0	0
$865$	2.50367	0.0851273
$866$	0	0
$867$	− 10.9472i	− 0.371785i
$868$	0	0
$869$	3.55224i	0.120501i
$870$	0	0
$871$	−2.96132	−0.100340
$872$	0	0
$873$	28.6062	0.968175
$874$	0	0
$875$	11.4503i	0.387090i
$876$	0	0
$877$	− 35.1561i	− 1.18714i	−0.804783	−	0.593569i	$-0.797718\pi$
$877$	− 35.1561i	− 1.18714i	0.804783	−	0.593569i	$-0.202282\pi$
$878$	0	0
$879$	72.9572	2.46079
$880$	0	0
$881$	26.8598	0.904929	0.452464	−	0.891782i	$-0.350545\pi$
$881$	26.8598	0.904929	0.452464	+	0.891782i	$0.350545\pi$
$882$	0	0
$883$	− 4.48500i	− 0.150932i	−0.997148	−	0.0754661i	$-0.975956\pi$
$883$	− 4.48500i	− 0.150932i	0.997148	−	0.0754661i	$-0.0240445\pi$
$884$	0	0
$885$	− 111.560i	− 3.75005i
$886$	0	0
$887$	−22.5853	−0.758340	−0.379170	−	0.925327i	$-0.623790\pi$
$887$	−22.5853	−0.758340	−0.379170	+	0.925327i	$0.623790\pi$
$888$	0	0
$889$	5.40864	0.181400
$890$	0	0
$891$	88.2403i	2.95616i
$892$	0	0
$893$	− 29.9816i	− 1.00329i
$894$	0	0
$895$	−31.8270	−1.06386
$896$	0	0
$897$	−9.63752	−0.321787
$898$	0	0
$899$	− 12.9070i	− 0.430473i
$900$	0	0
$901$	21.7371i	0.724168i
$902$	0	0
$903$	34.3420	1.14283
$904$	0	0
$905$	61.7250	2.05181
$906$	0	0
$907$	4.92450i	0.163515i	0.996652	+	0.0817577i	$0.0260534\pi$
$907$	4.92450i	0.163515i	−0.996652	+	0.0817577i	$0.973947\pi$
$908$	0	0
$909$	79.2361i	2.62810i
$910$	0	0
$911$	2.34456	0.0776787	0.0388393	−	0.999245i	$-0.487634\pi$
$911$	2.34456	0.0776787	0.0388393	+	0.999245i	$0.487634\pi$
$912$	0	0
$913$	69.6587	2.30537
$914$	0	0
$915$	− 21.5210i	− 0.711462i
$916$	0	0
$917$	− 2.37850i	− 0.0785450i
$918$	0	0
$919$	46.5700	1.53620	0.768102	−	0.640328i	$-0.221201\pi$
$919$	46.5700	1.53620	0.768102	+	0.640328i	$0.221201\pi$
$920$	0	0
$921$	−15.1209	−0.498249
$922$	0	0
$923$	− 7.49730i	− 0.246777i
$924$	0	0
$925$	− 43.7008i	− 1.43687i
$926$	0	0
$927$	−70.0651	−2.30124
$928$	0	0
$929$	21.9902	0.721475	0.360737	−	0.932667i	$-0.382525\pi$
$929$	21.9902	0.721475	0.360737	+	0.932667i	$0.382525\pi$
$930$	0	0
$931$	− 5.32637i	− 0.174565i
$932$	0	0
$933$	− 56.2567i	− 1.84176i
$934$	0	0
$935$	73.2233	2.39466
$936$	0	0
$937$	−39.7320	−1.29799	−0.648994	−	0.760794i	$-0.724810\pi$
$937$	−39.7320	−1.29799	−0.648994	+	0.760794i	$0.724810\pi$
$938$	0	0
$939$	− 6.09469i	− 0.198893i
$940$	0	0
$941$	− 35.5095i	− 1.15758i	−0.815477	−	0.578789i	$-0.803525\pi$
$941$	− 35.5095i	− 1.15758i	0.815477	−	0.578789i	$-0.196475\pi$
$942$	0	0
$943$	21.6624	0.705424
$944$	0	0
$945$	46.8372	1.52361
$946$	0	0
$947$	36.8412i	1.19718i	0.801056	+	0.598590i	$0.204272\pi$
$947$	36.8412i	1.19718i	−0.801056	+	0.598590i	$0.795728\pi$
$948$	0	0
$949$	− 8.99757i	− 0.292074i
$950$	0	0
$951$	−46.3451	−1.50284
$952$	0	0
$953$	−54.3777	−1.76147	−0.880733	−	0.473612i	$-0.842950\pi$
$953$	−54.3777	−1.76147	−0.880733	+	0.473612i	$0.842950\pi$
$954$	0	0
$955$	15.8145i	0.511744i
$956$	0	0
$957$	− 84.4066i	− 2.72848i
$958$	0	0
$959$	−11.6939	−0.377616
$960$	0	0
$961$	−26.3080	−0.848646
$962$	0	0
$963$	132.414i	4.26699i
$964$	0	0
$965$	− 22.7335i	− 0.731817i
$966$	0	0
$967$	−39.4871	−1.26982	−0.634910	−	0.772586i	$-0.718963\pi$
$967$	−39.4871	−1.26982	−0.634910	+	0.772586i	$0.718963\pi$
$968$	0	0
$969$	76.4322	2.45535
$970$	0	0
$971$	− 31.6915i	− 1.01703i	−0.861054	−	0.508514i	$-0.830195\pi$
$971$	− 31.6915i	− 1.01703i	0.861054	−	0.508514i	$-0.169805\pi$
$972$	0	0
$973$	− 8.82105i	− 0.282790i
$974$	0	0
$975$	25.8831	0.828924
$976$	0	0
$977$	−22.8224	−0.730154	−0.365077	−	0.930977i	$-0.618957\pi$
$977$	−22.8224	−0.730154	−0.365077	+	0.930977i	$0.618957\pi$
$978$	0	0
$979$	26.7185i	0.853927i
$980$	0	0
$981$	− 9.49232i	− 0.303066i
$982$	0	0
$983$	22.5531	0.719331	0.359665	−	0.933081i	$-0.382891\pi$
$983$	22.5531	0.719331	0.359665	+	0.933081i	$0.382891\pi$
$984$	0	0
$985$	16.7991	0.535264
$986$	0	0
$987$	17.8617i	0.568543i
$988$	0	0
$989$	− 32.8695i	− 1.04519i
$990$	0	0
$991$	−10.4713	−0.332631	−0.166315	−	0.986073i	$-0.553187\pi$
$991$	−10.4713	−0.332631	−0.166315	+	0.986073i	$0.553187\pi$
$992$	0	0
$993$	−75.5859	−2.39865
$994$	0	0
$995$	− 15.3256i	− 0.485854i
$996$	0	0
$997$	15.9240i	0.504319i	0.967686	+	0.252159i	$0.0811408\pi$
$997$	15.9240i	0.504319i	−0.967686	+	0.252159i	$0.918859\pi$
$998$	0	0
$999$	−69.1812	−2.18880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.c.b.1457.2		38
4.3	odd	2		728.2.c.b.365.3	✓	38
8.3	odd	2		728.2.c.b.365.4	yes	38
8.5	even	2	inner	2912.2.c.b.1457.37		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.c.b.365.3	✓	38	4.3	odd	2
728.2.c.b.365.4	yes	38	8.3	odd	2
2912.2.c.b.1457.2		38	1.1	even	1	trivial
2912.2.c.b.1457.37		38	8.5	even	2	inner

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

\(f(q)\)	\(=\)	\(q-3.17321i q^{3} +3.62722i q^{5} -1.00000 q^{7} -7.06928 q^{9} +4.46405i q^{11} -1.00000i q^{13} +11.5100 q^{15} -4.52215 q^{17} -5.32637i q^{19} +3.17321i q^{21} +3.03715 q^{23} -8.15676 q^{25} +12.9127i q^{27} -5.95865i q^{29} +2.16610 q^{31} +14.1654 q^{33} -3.62722i q^{35} +5.35761i q^{37} -3.17321 q^{39} +7.13247 q^{41} -10.8225i q^{43} -25.6419i q^{45} +5.62889 q^{47} +1.00000 q^{49} +14.3498i q^{51} -4.80681i q^{53} -16.1921 q^{55} -16.9017 q^{57} -9.69248i q^{59} -1.86977i q^{61} +7.06928 q^{63} +3.62722 q^{65} -2.96132i q^{67} -9.63752i q^{69} +7.49730 q^{71} +8.99757 q^{73} +25.8831i q^{75} -4.46405i q^{77} +0.795743 q^{79} +19.7668 q^{81} -15.6044i q^{83} -16.4029i q^{85} -18.9080 q^{87} +5.98526 q^{89} +1.00000i q^{91} -6.87349i q^{93} +19.3200 q^{95} -4.04656 q^{97} -31.5576i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 38 q^{7} - 46 q^{9} + 8 q^{15} + 20 q^{17} - 12 q^{23} - 50 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 38 q^{49} + 8 q^{57} + 46 q^{63} + 20 q^{65} + 12 q^{79} + 62 q^{81} + 56 q^{87}+O(q^{100}) \) 38 * q - 38 * q^7 - 46 * q^9 + 8 * q^15 + 20 * q^17 - 12 * q^23 - 50 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 38 * q^49 + 8 * q^57 + 46 * q^63 + 20 * q^65 + 12 * q^79 + 62 * q^81 + 56 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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