LMFDB - Embedded newform 2624.2.b.f.1313.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2624,2,Mod(1313,2624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2624, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2624.1313");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2624 = 2^{6} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2624.b (of order $2$, degree $1$, 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:	$20.9527454904$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1313.21
Character	$\chi$	$=$	2624.1313
Dual form			2624.2.b.f.1313.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70996i q^{3} -3.02571i q^{5} +0.654512 q^{7} +0.0760526 q^{9} +O(q^{10})$	$q+1.70996i q^{3} -3.02571i q^{5} +0.654512 q^{7} +0.0760526 q^{9} +1.69032i q^{11} -4.95067i q^{13} +5.17382 q^{15} +4.61075 q^{17} -4.15608i q^{19} +1.11919i q^{21} -6.69617 q^{23} -4.15490 q^{25} +5.25991i q^{27} -8.79007i q^{29} -6.18239 q^{31} -2.89037 q^{33} -1.98036i q^{35} -6.86510i q^{37} +8.46543 q^{39} -1.00000 q^{41} +9.40114i q^{43} -0.230113i q^{45} +3.05365 q^{47} -6.57161 q^{49} +7.88418i q^{51} +13.3120i q^{53} +5.11441 q^{55} +7.10672 q^{57} -5.97887i q^{59} -1.94680i q^{61} +0.0497774 q^{63} -14.9793 q^{65} -15.3096i q^{67} -11.4502i q^{69} +1.61365 q^{71} -10.9233 q^{73} -7.10470i q^{75} +1.10633i q^{77} -2.10530 q^{79} -8.76606 q^{81} -11.7257i q^{83} -13.9508i q^{85} +15.0306 q^{87} +7.43349 q^{89} -3.24028i q^{91} -10.5716i q^{93} -12.5751 q^{95} +15.4578 q^{97} +0.128553i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 28 q^{9}+O(q^{10}) $ 28 * q - 28 * q^9	$ 28 q - 28 q^{9} - 8 q^{17} - 36 q^{25} - 64 q^{33} - 28 q^{41} - 20 q^{49} - 80 q^{57} - 8 q^{73} - 20 q^{81} - 72 q^{89} - 40 q^{97}+O(q^{100}) $ 28 * q - 28 * q^9 - 8 * q^17 - 36 * q^25 - 64 * q^33 - 28 * q^41 - 20 * q^49 - 80 * q^57 - 8 * q^73 - 20 * q^81 - 72 * q^89 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2624\mathbb{Z}\right)^\times$.

$n$	$129$	$575$	$1477$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.70996i	0.987243i	0.869677	+	0.493622i	$0.164327\pi$
$3$	1.70996i	0.987243i	−0.869677	+	0.493622i	$0.835673\pi$
$4$	0	0
$5$	− 3.02571i	− 1.35314i	−0.736380	−	0.676569i	$-0.763466\pi$
$5$	− 3.02571i	− 1.35314i	0.736380	−	0.676569i	$-0.236534\pi$
$6$	0	0
$7$	0.654512	0.247382	0.123691	−	0.992321i	$-0.460527\pi$
$7$	0.654512	0.247382	0.123691	+	0.992321i	$0.460527\pi$
$8$	0	0
$9$	0.0760526	0.0253509
$10$	0	0
$11$	1.69032i	0.509650i	0.966987	+	0.254825i	$0.0820179\pi$
$11$	1.69032i	0.509650i	−0.966987	+	0.254825i	$0.917982\pi$
$12$	0	0
$13$	− 4.95067i	− 1.37307i	−0.727097	−	0.686534i	$-0.759131\pi$
$13$	− 4.95067i	− 1.37307i	0.727097	−	0.686534i	$-0.240869\pi$
$14$	0	0
$15$	5.17382	1.33588
$16$	0	0
$17$	4.61075	1.11827	0.559136	−	0.829076i	$-0.311133\pi$
$17$	4.61075	1.11827	0.559136	+	0.829076i	$0.311133\pi$
$18$	0	0
$19$	− 4.15608i	− 0.953471i	−0.879047	−	0.476735i	$-0.841820\pi$
$19$	− 4.15608i	− 0.953471i	0.879047	−	0.476735i	$-0.158180\pi$
$20$	0	0
$21$	1.11919i	0.244227i
$22$	0	0
$23$	−6.69617	−1.39625	−0.698125	−	0.715976i	$-0.745982\pi$
$23$	−6.69617	−1.39625	−0.698125	+	0.715976i	$0.745982\pi$
$24$	0	0
$25$	−4.15490	−0.830980
$26$	0	0
$27$	5.25991i	1.01227i
$28$	0	0
$29$	− 8.79007i	− 1.63227i	−0.577858	−	0.816137i	$-0.696111\pi$
$29$	− 8.79007i	− 1.63227i	0.577858	−	0.816137i	$-0.303889\pi$
$30$	0	0
$31$	−6.18239	−1.11039	−0.555195	−	0.831720i	$-0.687357\pi$
$31$	−6.18239	−1.11039	−0.555195	+	0.831720i	$0.687357\pi$
$32$	0	0
$33$	−2.89037	−0.503149
$34$	0	0
$35$	− 1.98036i	− 0.334742i
$36$	0	0
$37$	− 6.86510i	− 1.12862i	−0.825564	−	0.564308i	$-0.809143\pi$
$37$	− 6.86510i	− 1.12862i	0.825564	−	0.564308i	$-0.190857\pi$
$38$	0	0
$39$	8.46543	1.35555
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	9.40114i	1.43366i	0.697248	+	0.716830i	$0.254408\pi$
$43$	9.40114i	1.43366i	−0.697248	+	0.716830i	$0.745592\pi$
$44$	0	0
$45$	− 0.230113i	− 0.0343032i
$46$	0	0
$47$	3.05365	0.445421	0.222711	−	0.974885i	$-0.428510\pi$
$47$	3.05365	0.445421	0.222711	+	0.974885i	$0.428510\pi$
$48$	0	0
$49$	−6.57161	−0.938802
$50$	0	0
$51$	7.88418i	1.10401i
$52$	0	0
$53$	13.3120i	1.82855i	0.405098	+	0.914273i	$0.367238\pi$
$53$	13.3120i	1.82855i	−0.405098	+	0.914273i	$0.632762\pi$
$54$	0	0
$55$	5.11441	0.689627
$56$	0	0
$57$	7.10672	0.941308
$58$	0	0
$59$	− 5.97887i	− 0.778382i	−0.921157	−	0.389191i	$-0.872755\pi$
$59$	− 5.97887i	− 0.778382i	0.921157	−	0.389191i	$-0.127245\pi$
$60$	0	0
$61$	− 1.94680i	− 0.249262i	−0.992203	−	0.124631i	$-0.960225\pi$
$61$	− 1.94680i	− 0.249262i	0.992203	−	0.124631i	$-0.0397746\pi$
$62$	0	0
$63$	0.0497774	0.00627136
$64$	0	0
$65$	−14.9793	−1.85795
$66$	0	0
$67$	− 15.3096i	− 1.87036i	−0.354172	−	0.935180i	$-0.615237\pi$
$67$	− 15.3096i	− 1.87036i	0.354172	−	0.935180i	$-0.384763\pi$
$68$	0	0
$69$	− 11.4502i	− 1.37844i
$70$	0	0
$71$	1.61365	0.191505	0.0957524	−	0.995405i	$-0.469474\pi$
$71$	1.61365	0.191505	0.0957524	+	0.995405i	$0.469474\pi$
$72$	0	0
$73$	−10.9233	−1.27848	−0.639239	−	0.769008i	$-0.720751\pi$
$73$	−10.9233	−1.27848	−0.639239	+	0.769008i	$0.720751\pi$
$74$	0	0
$75$	− 7.10470i	− 0.820380i
$76$	0	0
$77$	1.10633i	0.126078i
$78$	0	0
$79$	−2.10530	−0.236865	−0.118432	−	0.992962i	$-0.537787\pi$
$79$	−2.10530	−0.236865	−0.118432	+	0.992962i	$0.537787\pi$
$80$	0	0
$81$	−8.76606	−0.974006
$82$	0	0
$83$	− 11.7257i	− 1.28706i	−0.765421	−	0.643530i	$-0.777469\pi$
$83$	− 11.7257i	− 1.28706i	0.765421	−	0.643530i	$-0.222531\pi$
$84$	0	0
$85$	− 13.9508i	− 1.51318i
$86$	0	0
$87$	15.0306	1.61145
$88$	0	0
$89$	7.43349	0.787948	0.393974	−	0.919122i	$-0.371100\pi$
$89$	7.43349	0.787948	0.393974	+	0.919122i	$0.371100\pi$
$90$	0	0
$91$	− 3.24028i	− 0.339673i
$92$	0	0
$93$	− 10.5716i	− 1.09623i
$94$	0	0
$95$	−12.5751	−1.29018
$96$	0	0
$97$	15.4578	1.56950	0.784749	−	0.619814i	$-0.212792\pi$
$97$	15.4578	1.56950	0.784749	+	0.619814i	$0.212792\pi$
$98$	0	0
$99$	0.128553i	0.0129201i
$100$	0	0
$101$	− 3.97611i	− 0.395638i	−0.980239	−	0.197819i	$-0.936614\pi$
$101$	− 3.97611i	− 0.395638i	0.980239	−	0.197819i	$-0.0633858\pi$
$102$	0	0
$103$	2.35106	0.231656	0.115828	−	0.993269i	$-0.463048\pi$
$103$	2.35106	0.231656	0.115828	+	0.993269i	$0.463048\pi$
$104$	0	0
$105$	3.38633	0.330472
$106$	0	0
$107$	− 16.6783i	− 1.61236i	−0.591674	−	0.806178i	$-0.701533\pi$
$107$	− 16.6783i	− 1.61236i	0.591674	−	0.806178i	$-0.298467\pi$
$108$	0	0
$109$	9.97114i	0.955062i	0.878615	+	0.477531i	$0.158468\pi$
$109$	9.97114i	0.955062i	−0.878615	+	0.477531i	$0.841532\pi$
$110$	0	0
$111$	11.7390	1.11422
$112$	0	0
$113$	−8.64807	−0.813542	−0.406771	−	0.913530i	$-0.633345\pi$
$113$	−8.64807	−0.813542	−0.406771	+	0.913530i	$0.633345\pi$
$114$	0	0
$115$	20.2607i	1.88932i
$116$	0	0
$117$	− 0.376512i	− 0.0348085i
$118$	0	0
$119$	3.01780	0.276641
$120$	0	0
$121$	8.14282	0.740257
$122$	0	0
$123$	− 1.70996i	− 0.154181i
$124$	0	0
$125$	− 2.55702i	− 0.228707i
$126$	0	0
$127$	−4.33403	−0.384583	−0.192292	−	0.981338i	$-0.561592\pi$
$127$	−4.33403	−0.384583	−0.192292	+	0.981338i	$0.561592\pi$
$128$	0	0
$129$	−16.0755	−1.41537
$130$	0	0
$131$	− 6.91015i	− 0.603742i	−0.953349	−	0.301871i	$-0.902389\pi$
$131$	− 6.91015i	− 0.603742i	0.953349	−	0.301871i	$-0.0976112\pi$
$132$	0	0
$133$	− 2.72021i	− 0.235872i
$134$	0	0
$135$	15.9150	1.36974
$136$	0	0
$137$	4.96922	0.424549	0.212274	−	0.977210i	$-0.431913\pi$
$137$	4.96922	0.424549	0.212274	+	0.977210i	$0.431913\pi$
$138$	0	0
$139$	4.92811i	0.417997i	0.977916	+	0.208998i	$0.0670203\pi$
$139$	4.92811i	0.417997i	−0.977916	+	0.208998i	$0.932980\pi$
$140$	0	0
$141$	5.22161i	0.439739i
$142$	0	0
$143$	8.36821	0.699785
$144$	0	0
$145$	−26.5962	−2.20869
$146$	0	0
$147$	− 11.2372i	− 0.926826i
$148$	0	0
$149$	− 14.9271i	− 1.22288i	−0.791292	−	0.611438i	$-0.790591\pi$
$149$	− 14.9271i	− 1.22288i	0.791292	−	0.611438i	$-0.209409\pi$
$150$	0	0
$151$	−9.09646	−0.740259	−0.370130	−	0.928980i	$-0.620687\pi$
$151$	−9.09646	−0.740259	−0.370130	+	0.928980i	$0.620687\pi$
$152$	0	0
$153$	0.350660	0.0283492
$154$	0	0
$155$	18.7061i	1.50251i
$156$	0	0
$157$	16.7192i	1.33434i	0.744907	+	0.667168i	$0.232494\pi$
$157$	16.7192i	1.33434i	−0.744907	+	0.667168i	$0.767506\pi$
$158$	0	0
$159$	−22.7630	−1.80522
$160$	0	0
$161$	−4.38273	−0.345408
$162$	0	0
$163$	− 15.0280i	− 1.17709i	−0.808466	−	0.588543i	$-0.799702\pi$
$163$	− 15.0280i	− 1.17709i	0.808466	−	0.588543i	$-0.200298\pi$
$164$	0	0
$165$	8.74541i	0.680829i
$166$	0	0
$167$	12.4817	0.965866	0.482933	−	0.875657i	$-0.339571\pi$
$167$	12.4817	0.965866	0.482933	+	0.875657i	$0.339571\pi$
$168$	0	0
$169$	−11.5091	−0.885318
$170$	0	0
$171$	− 0.316081i	− 0.0241713i
$172$	0	0
$173$	10.5804i	0.804410i	0.915550	+	0.402205i	$0.131756\pi$
$173$	10.5804i	0.804410i	−0.915550	+	0.402205i	$0.868244\pi$
$174$	0	0
$175$	−2.71944	−0.205570
$176$	0	0
$177$	10.2236	0.768453
$178$	0	0
$179$	− 9.39866i	− 0.702489i	−0.936284	−	0.351244i	$-0.885759\pi$
$179$	− 9.39866i	− 0.702489i	0.936284	−	0.351244i	$-0.114241\pi$
$180$	0	0
$181$	− 2.60016i	− 0.193269i	−0.995320	−	0.0966344i	$-0.969192\pi$
$181$	− 2.60016i	− 0.193269i	0.995320	−	0.0966344i	$-0.0308078\pi$
$182$	0	0
$183$	3.32893	0.246082
$184$	0	0
$185$	−20.7718	−1.52717
$186$	0	0
$187$	7.79364i	0.569927i
$188$	0	0
$189$	3.44268i	0.250418i
$190$	0	0
$191$	18.7132	1.35404	0.677019	−	0.735966i	$-0.263272\pi$
$191$	18.7132	1.35404	0.677019	+	0.735966i	$0.263272\pi$
$192$	0	0
$193$	15.3001	1.10132	0.550661	−	0.834729i	$-0.314376\pi$
$193$	15.3001	1.10132	0.550661	+	0.834729i	$0.314376\pi$
$194$	0	0
$195$	− 25.6139i	− 1.83425i
$196$	0	0
$197$	7.73192i	0.550876i	0.961319	+	0.275438i	$0.0888229\pi$
$197$	7.73192i	0.550876i	−0.961319	+	0.275438i	$0.911177\pi$
$198$	0	0
$199$	19.2053	1.36143	0.680715	−	0.732548i	$-0.261669\pi$
$199$	19.2053	1.36143	0.680715	+	0.732548i	$0.261669\pi$
$200$	0	0
$201$	26.1787	1.84650
$202$	0	0
$203$	− 5.75321i	− 0.403796i
$204$	0	0
$205$	3.02571i	0.211325i
$206$	0	0
$207$	−0.509262	−0.0353961
$208$	0	0
$209$	7.02510	0.485937
$210$	0	0
$211$	9.92488i	0.683257i	0.939835	+	0.341629i	$0.110978\pi$
$211$	9.92488i	0.683257i	−0.939835	+	0.341629i	$0.889022\pi$
$212$	0	0
$213$	2.75927i	0.189062i
$214$	0	0
$215$	28.4451	1.93994
$216$	0	0
$217$	−4.04645	−0.274691
$218$	0	0
$219$	− 18.6784i	− 1.26217i
$220$	0	0
$221$	− 22.8263i	− 1.53546i
$222$	0	0
$223$	22.5179	1.50791	0.753957	−	0.656924i	$-0.228143\pi$
$223$	22.5179	1.50791	0.753957	+	0.656924i	$0.228143\pi$
$224$	0	0
$225$	−0.315991	−0.0210661
$226$	0	0
$227$	− 15.0419i	− 0.998368i	−0.866496	−	0.499184i	$-0.833633\pi$
$227$	− 15.0419i	− 0.998368i	0.866496	−	0.499184i	$-0.166367\pi$
$228$	0	0
$229$	− 8.16360i	− 0.539466i	−0.962935	−	0.269733i	$-0.913065\pi$
$229$	− 8.16360i	− 0.539466i	0.962935	−	0.269733i	$-0.0869355\pi$
$230$	0	0
$231$	−1.89178	−0.124470
$232$	0	0
$233$	−2.26571	−0.148432	−0.0742159	−	0.997242i	$-0.523645\pi$
$233$	−2.26571	−0.148432	−0.0742159	+	0.997242i	$0.523645\pi$
$234$	0	0
$235$	− 9.23946i	− 0.602716i
$236$	0	0
$237$	− 3.59997i	− 0.233843i
$238$	0	0
$239$	−8.97711	−0.580681	−0.290341	−	0.956923i	$-0.593769\pi$
$239$	−8.97711	−0.580681	−0.290341	+	0.956923i	$0.593769\pi$
$240$	0	0
$241$	9.42139	0.606885	0.303443	−	0.952850i	$-0.401864\pi$
$241$	9.42139	0.606885	0.303443	+	0.952850i	$0.401864\pi$
$242$	0	0
$243$	0.790170i	0.0506894i
$244$	0	0
$245$	19.8838i	1.27033i
$246$	0	0
$247$	−20.5754	−1.30918
$248$	0	0
$249$	20.0504	1.27064
$250$	0	0
$251$	25.8271i	1.63019i	0.579327	+	0.815096i	$0.303316\pi$
$251$	25.8271i	1.63019i	−0.579327	+	0.815096i	$0.696684\pi$
$252$	0	0
$253$	− 11.3187i	− 0.711598i
$254$	0	0
$255$	23.8552	1.49387
$256$	0	0
$257$	−19.3650	−1.20796	−0.603978	−	0.797001i	$-0.706418\pi$
$257$	−19.3650	−1.20796	−0.603978	+	0.797001i	$0.706418\pi$
$258$	0	0
$259$	− 4.49329i	− 0.279200i
$260$	0	0
$261$	− 0.668508i	− 0.0413796i
$262$	0	0
$263$	−10.8885	−0.671415	−0.335708	−	0.941966i	$-0.608975\pi$
$263$	−10.8885	−0.671415	−0.335708	+	0.941966i	$0.608975\pi$
$264$	0	0
$265$	40.2783	2.47428
$266$	0	0
$267$	12.7109i	0.777897i
$268$	0	0
$269$	− 18.3355i	− 1.11793i	−0.829190	−	0.558967i	$-0.811198\pi$
$269$	− 18.3355i	− 1.11793i	0.829190	−	0.558967i	$-0.188802\pi$
$270$	0	0
$271$	1.81814	0.110444	0.0552219	−	0.998474i	$-0.482413\pi$
$271$	1.81814	0.110444	0.0552219	+	0.998474i	$0.482413\pi$
$272$	0	0
$273$	5.54073	0.335340
$274$	0	0
$275$	− 7.02311i	− 0.423509i
$276$	0	0
$277$	16.6456i	1.00014i	0.865986	+	0.500068i	$0.166692\pi$
$277$	16.6456i	1.00014i	−0.865986	+	0.500068i	$0.833308\pi$
$278$	0	0
$279$	−0.470187	−0.0281494
$280$	0	0
$281$	−2.86255	−0.170765	−0.0853826	−	0.996348i	$-0.527211\pi$
$281$	−2.86255	−0.170765	−0.0853826	+	0.996348i	$0.527211\pi$
$282$	0	0
$283$	26.6123i	1.58194i	0.611856	+	0.790969i	$0.290423\pi$
$283$	26.6123i	1.58194i	−0.611856	+	0.790969i	$0.709577\pi$
$284$	0	0
$285$	− 21.5028i	− 1.27372i
$286$	0	0
$287$	−0.654512	−0.0386346
$288$	0	0
$289$	4.25904	0.250532
$290$	0	0
$291$	26.4321i	1.54948i
$292$	0	0
$293$	− 19.9761i	− 1.16701i	−0.812108	−	0.583507i	$-0.801680\pi$
$293$	− 19.9761i	− 1.16701i	0.812108	−	0.583507i	$-0.198320\pi$
$294$	0	0
$295$	−18.0903	−1.05326
$296$	0	0
$297$	−8.89093	−0.515904
$298$	0	0
$299$	33.1506i	1.91715i
$300$	0	0
$301$	6.15316i	0.354662i
$302$	0	0
$303$	6.79898	0.390591
$304$	0	0
$305$	−5.89043	−0.337285
$306$	0	0
$307$	8.00770i	0.457024i	0.973541	+	0.228512i	$0.0733859\pi$
$307$	8.00770i	0.457024i	−0.973541	+	0.228512i	$0.926614\pi$
$308$	0	0
$309$	4.02020i	0.228701i
$310$	0	0
$311$	−6.52033	−0.369734	−0.184867	−	0.982764i	$-0.559185\pi$
$311$	−6.52033	−0.369734	−0.184867	+	0.982764i	$0.559185\pi$
$312$	0	0
$313$	30.8918	1.74611	0.873053	−	0.487626i	$-0.162137\pi$
$313$	30.8918	1.74611	0.873053	+	0.487626i	$0.162137\pi$
$314$	0	0
$315$	− 0.150612i	− 0.00848601i
$316$	0	0
$317$	− 25.3428i	− 1.42339i	−0.702487	−	0.711696i	$-0.747927\pi$
$317$	− 25.3428i	− 1.42339i	0.702487	−	0.711696i	$-0.252073\pi$
$318$	0	0
$319$	14.8580	0.831889
$320$	0	0
$321$	28.5192	1.59179
$322$	0	0
$323$	− 19.1627i	− 1.06624i
$324$	0	0
$325$	20.5696i	1.14099i
$326$	0	0
$327$	−17.0502	−0.942878
$328$	0	0
$329$	1.99865	0.110189
$330$	0	0
$331$	25.9202i	1.42470i	0.701824	+	0.712350i	$0.252369\pi$
$331$	25.9202i	1.42470i	−0.701824	+	0.712350i	$0.747631\pi$
$332$	0	0
$333$	− 0.522109i	− 0.0286114i
$334$	0	0
$335$	−46.3223	−2.53086
$336$	0	0
$337$	0.358868	0.0195488	0.00977439	−	0.999952i	$-0.496889\pi$
$337$	0.358868	0.0195488	0.00977439	+	0.999952i	$0.496889\pi$
$338$	0	0
$339$	− 14.7878i	− 0.803164i
$340$	0	0
$341$	− 10.4502i	− 0.565911i
$342$	0	0
$343$	−8.88279	−0.479626
$344$	0	0
$345$	−34.6448	−1.86521
$346$	0	0
$347$	− 13.3362i	− 0.715926i	−0.933736	−	0.357963i	$-0.883471\pi$
$347$	− 13.3362i	− 0.715926i	0.933736	−	0.357963i	$-0.116529\pi$
$348$	0	0
$349$	28.7714i	1.54010i	0.637986	+	0.770048i	$0.279768\pi$
$349$	28.7714i	1.54010i	−0.637986	+	0.770048i	$0.720232\pi$
$350$	0	0
$351$	26.0401	1.38992
$352$	0	0
$353$	−27.9780	−1.48912	−0.744559	−	0.667557i	$-0.767340\pi$
$353$	−27.9780	−1.48912	−0.744559	+	0.667557i	$0.767340\pi$
$354$	0	0
$355$	− 4.88242i	− 0.259132i
$356$	0	0
$357$	5.16030i	0.273112i
$358$	0	0
$359$	3.06436	0.161731	0.0808653	−	0.996725i	$-0.474232\pi$
$359$	3.06436	0.161731	0.0808653	+	0.996725i	$0.474232\pi$
$360$	0	0
$361$	1.72697	0.0908933
$362$	0	0
$363$	13.9239i	0.730813i
$364$	0	0
$365$	33.0508i	1.72996i
$366$	0	0
$367$	19.5762	1.02187	0.510936	−	0.859619i	$-0.329299\pi$
$367$	19.5762	1.02187	0.510936	+	0.859619i	$0.329299\pi$
$368$	0	0
$369$	−0.0760526	−0.00395914
$370$	0	0
$371$	8.71288i	0.452350i
$372$	0	0
$373$	1.71844i	0.0889776i	0.999010	+	0.0444888i	$0.0141659\pi$
$373$	1.71844i	0.0889776i	−0.999010	+	0.0444888i	$0.985834\pi$
$374$	0	0
$375$	4.37239	0.225789
$376$	0	0
$377$	−43.5167	−2.24123
$378$	0	0
$379$	6.41887i	0.329715i	0.986317	+	0.164858i	$0.0527165\pi$
$379$	6.41887i	0.329715i	−0.986317	+	0.164858i	$0.947284\pi$
$380$	0	0
$381$	− 7.41101i	− 0.379677i
$382$	0	0
$383$	−26.1636	−1.33690	−0.668449	−	0.743758i	$-0.733042\pi$
$383$	−26.1636	−1.33690	−0.668449	+	0.743758i	$0.733042\pi$
$384$	0	0
$385$	3.34744	0.170602
$386$	0	0
$387$	0.714982i	0.0363446i
$388$	0	0
$389$	0.442767i	0.0224492i	0.999937	+	0.0112246i	$0.00357297\pi$
$389$	0.442767i	0.0224492i	−0.999937	+	0.0112246i	$0.996427\pi$
$390$	0	0
$391$	−30.8744	−1.56139
$392$	0	0
$393$	11.8160	0.596040
$394$	0	0
$395$	6.37002i	0.320510i
$396$	0	0
$397$	6.01454i	0.301861i	0.988544	+	0.150931i	$0.0482270\pi$
$397$	6.01454i	0.301861i	−0.988544	+	0.150931i	$0.951773\pi$
$398$	0	0
$399$	4.65143	0.232863
$400$	0	0
$401$	9.32405	0.465621	0.232811	−	0.972522i	$-0.425208\pi$
$401$	9.32405	0.465621	0.232811	+	0.972522i	$0.425208\pi$
$402$	0	0
$403$	30.6070i	1.52464i
$404$	0	0
$405$	26.5235i	1.31796i
$406$	0	0
$407$	11.6042	0.575199
$408$	0	0
$409$	−6.93568	−0.342947	−0.171474	−	0.985189i	$-0.554853\pi$
$409$	−6.93568	−0.342947	−0.171474	+	0.985189i	$0.554853\pi$
$410$	0	0
$411$	8.49714i	0.419133i
$412$	0	0
$413$	− 3.91324i	− 0.192558i
$414$	0	0
$415$	−35.4785	−1.74157
$416$	0	0
$417$	−8.42685	−0.412664
$418$	0	0
$419$	− 23.7753i	− 1.16150i	−0.814083	−	0.580749i	$-0.802760\pi$
$419$	− 23.7753i	− 1.16150i	0.814083	−	0.580749i	$-0.197240\pi$
$420$	0	0
$421$	− 18.5897i	− 0.906004i	−0.891509	−	0.453002i	$-0.850353\pi$
$421$	− 18.5897i	− 0.906004i	0.891509	−	0.453002i	$-0.149647\pi$
$422$	0	0
$423$	0.232238	0.0112918
$424$	0	0
$425$	−19.1572	−0.929262
$426$	0	0
$427$	− 1.27420i	− 0.0616629i
$428$	0	0
$429$	14.3093i	0.690858i
$430$	0	0
$431$	0.384345	0.0185133	0.00925664	−	0.999957i	$-0.497053\pi$
$431$	0.384345	0.0185133	0.00925664	+	0.999957i	$0.497053\pi$
$432$	0	0
$433$	14.4281	0.693371	0.346686	−	0.937981i	$-0.387307\pi$
$433$	14.4281	0.693371	0.346686	+	0.937981i	$0.387307\pi$
$434$	0	0
$435$	− 45.4783i	− 2.18052i
$436$	0	0
$437$	27.8299i	1.33128i
$438$	0	0
$439$	−26.0914	−1.24527	−0.622636	−	0.782511i	$-0.713938\pi$
$439$	−26.0914	−1.24527	−0.622636	+	0.782511i	$0.713938\pi$
$440$	0	0
$441$	−0.499789	−0.0237995
$442$	0	0
$443$	− 17.7432i	− 0.843005i	−0.906827	−	0.421502i	$-0.861503\pi$
$443$	− 17.7432i	− 0.843005i	0.906827	−	0.421502i	$-0.138497\pi$
$444$	0	0
$445$	− 22.4916i	− 1.06620i
$446$	0	0
$447$	25.5247	1.20728
$448$	0	0
$449$	19.9724	0.942557	0.471279	−	0.881984i	$-0.343793\pi$
$449$	19.9724	0.942557	0.471279	+	0.881984i	$0.343793\pi$
$450$	0	0
$451$	− 1.69032i	− 0.0795940i
$452$	0	0
$453$	− 15.5545i	− 0.730816i
$454$	0	0
$455$	−9.80412	−0.459624
$456$	0	0
$457$	3.14194	0.146974	0.0734868	−	0.997296i	$-0.476587\pi$
$457$	3.14194	0.146974	0.0734868	+	0.997296i	$0.476587\pi$
$458$	0	0
$459$	24.2522i	1.13199i
$460$	0	0
$461$	26.8341i	1.24979i	0.780709	+	0.624894i	$0.214858\pi$
$461$	26.8341i	1.24979i	−0.780709	+	0.624894i	$0.785142\pi$
$462$	0	0
$463$	−1.69928	−0.0789724	−0.0394862	−	0.999220i	$-0.512572\pi$
$463$	−1.69928	−0.0789724	−0.0394862	+	0.999220i	$0.512572\pi$
$464$	0	0
$465$	−31.9866	−1.48334
$466$	0	0
$467$	17.8505i	0.826023i	0.910726	+	0.413012i	$0.135523\pi$
$467$	17.8505i	0.826023i	−0.910726	+	0.413012i	$0.864477\pi$
$468$	0	0
$469$	− 10.0203i	− 0.462694i
$470$	0	0
$471$	−28.5891	−1.31731
$472$	0	0
$473$	−15.8909	−0.730665
$474$	0	0
$475$	17.2681i	0.792316i
$476$	0	0
$477$	1.01241i	0.0463553i
$478$	0	0
$479$	−25.5933	−1.16939	−0.584695	−	0.811254i	$-0.698786\pi$
$479$	−25.5933	−1.16939	−0.584695	+	0.811254i	$0.698786\pi$
$480$	0	0
$481$	−33.9869	−1.54967
$482$	0	0
$483$	− 7.49427i	− 0.341001i
$484$	0	0
$485$	− 46.7706i	− 2.12375i
$486$	0	0
$487$	−21.2988	−0.965140	−0.482570	−	0.875857i	$-0.660297\pi$
$487$	−21.2988	−0.965140	−0.482570	+	0.875857i	$0.660297\pi$
$488$	0	0
$489$	25.6973	1.16207
$490$	0	0
$491$	− 35.3327i	− 1.59454i	−0.603622	−	0.797271i	$-0.706276\pi$
$491$	− 35.3327i	− 1.59454i	0.603622	−	0.797271i	$-0.293724\pi$
$492$	0	0
$493$	− 40.5288i	− 1.82533i
$494$	0	0
$495$	0.388964	0.0174826
$496$	0	0
$497$	1.05615	0.0473749
$498$	0	0
$499$	− 28.9892i	− 1.29774i	−0.760901	−	0.648868i	$-0.775243\pi$
$499$	− 28.9892i	− 1.29774i	0.760901	−	0.648868i	$-0.224757\pi$
$500$	0	0
$501$	21.3432i	0.953545i
$502$	0	0
$503$	1.51749	0.0676617	0.0338309	−	0.999428i	$-0.489229\pi$
$503$	1.51749	0.0676617	0.0338309	+	0.999428i	$0.489229\pi$
$504$	0	0
$505$	−12.0306	−0.535353
$506$	0	0
$507$	− 19.6801i	− 0.874025i
$508$	0	0
$509$	28.5260i	1.26439i	0.774809	+	0.632196i	$0.217846\pi$
$509$	28.5260i	1.26439i	−0.774809	+	0.632196i	$0.782154\pi$
$510$	0	0
$511$	−7.14945	−0.316273
$512$	0	0
$513$	21.8606	0.965171
$514$	0	0
$515$	− 7.11361i	− 0.313463i
$516$	0	0
$517$	5.16165i	0.227009i
$518$	0	0
$519$	−18.0920	−0.794148
$520$	0	0
$521$	11.5395	0.505555	0.252777	−	0.967524i	$-0.418656\pi$
$521$	11.5395	0.505555	0.252777	+	0.967524i	$0.418656\pi$
$522$	0	0
$523$	30.7082i	1.34278i	0.741106	+	0.671388i	$0.234302\pi$
$523$	30.7082i	1.34278i	−0.741106	+	0.671388i	$0.765698\pi$
$524$	0	0
$525$	− 4.65011i	− 0.202948i
$526$	0	0
$527$	−28.5055	−1.24172
$528$	0	0
$529$	21.8388	0.949511
$530$	0	0
$531$	− 0.454709i	− 0.0197327i
$532$	0	0
$533$	4.95067i	0.214437i
$534$	0	0
$535$	−50.4637	−2.18174
$536$	0	0
$537$	16.0713	0.693527
$538$	0	0
$539$	− 11.1081i	− 0.478461i
$540$	0	0
$541$	33.4617i	1.43863i	0.694684	+	0.719315i	$0.255544\pi$
$541$	33.4617i	1.43863i	−0.694684	+	0.719315i	$0.744456\pi$
$542$	0	0
$543$	4.44617	0.190803
$544$	0	0
$545$	30.1697	1.29233
$546$	0	0
$547$	18.7475i	0.801587i	0.916168	+	0.400793i	$0.131266\pi$
$547$	18.7475i	0.801587i	−0.916168	+	0.400793i	$0.868734\pi$
$548$	0	0
$549$	− 0.148059i	− 0.00631900i
$550$	0	0
$551$	−36.5322	−1.55633
$552$	0	0
$553$	−1.37794	−0.0585962
$554$	0	0
$555$	− 35.5188i	− 1.50769i
$556$	0	0
$557$	23.1905i	0.982615i	0.870986	+	0.491308i	$0.163481\pi$
$557$	23.1905i	0.982615i	−0.870986	+	0.491308i	$0.836519\pi$
$558$	0	0
$559$	46.5420	1.96851
$560$	0	0
$561$	−13.3268	−0.562657
$562$	0	0
$563$	25.7192i	1.08393i	0.840400	+	0.541967i	$0.182320\pi$
$563$	25.7192i	1.08393i	−0.840400	+	0.541967i	$0.817680\pi$
$564$	0	0
$565$	26.1665i	1.10083i
$566$	0	0
$567$	−5.73749	−0.240952
$568$	0	0
$569$	−30.4354	−1.27592	−0.637959	−	0.770070i	$-0.720221\pi$
$569$	−30.4354	−1.27592	−0.637959	+	0.770070i	$0.720221\pi$
$570$	0	0
$571$	31.8308i	1.33208i	0.745917	+	0.666039i	$0.232012\pi$
$571$	31.8308i	1.33208i	−0.745917	+	0.666039i	$0.767988\pi$
$572$	0	0
$573$	31.9987i	1.33676i
$574$	0	0
$575$	27.8220	1.16026
$576$	0	0
$577$	37.6528	1.56751	0.783754	−	0.621072i	$-0.213302\pi$
$577$	37.6528	1.56751	0.783754	+	0.621072i	$0.213302\pi$
$578$	0	0
$579$	26.1624i	1.08727i
$580$	0	0
$581$	− 7.67460i	− 0.318396i
$582$	0	0
$583$	−22.5016	−0.931919
$584$	0	0
$585$	−1.13921	−0.0471007
$586$	0	0
$587$	40.9704i	1.69103i	0.533951	+	0.845515i	$0.320707\pi$
$587$	40.9704i	1.69103i	−0.533951	+	0.845515i	$0.679293\pi$
$588$	0	0
$589$	25.6945i	1.05873i
$590$	0	0
$591$	−13.2212	−0.543849
$592$	0	0
$593$	−22.0504	−0.905501	−0.452751	−	0.891637i	$-0.649557\pi$
$593$	−22.0504	−0.905501	−0.452751	+	0.891637i	$0.649557\pi$
$594$	0	0
$595$	− 9.13096i	− 0.374333i
$596$	0	0
$597$	32.8403i	1.34406i
$598$	0	0
$599$	24.4091	0.997327	0.498664	−	0.866796i	$-0.333824\pi$
$599$	24.4091	0.997327	0.498664	+	0.866796i	$0.333824\pi$
$600$	0	0
$601$	−9.00942	−0.367502	−0.183751	−	0.982973i	$-0.558824\pi$
$601$	−9.00942	−0.367502	−0.183751	+	0.982973i	$0.558824\pi$
$602$	0	0
$603$	− 1.16433i	− 0.0474153i
$604$	0	0
$605$	− 24.6378i	− 1.00167i
$606$	0	0
$607$	7.60037	0.308489	0.154245	−	0.988033i	$-0.450706\pi$
$607$	7.60037	0.308489	0.154245	+	0.988033i	$0.450706\pi$
$608$	0	0
$609$	9.83773	0.398645
$610$	0	0
$611$	− 15.1176i	− 0.611594i
$612$	0	0
$613$	− 0.401605i	− 0.0162207i	−0.999967	−	0.00811034i	$-0.997418\pi$
$613$	− 0.401605i	− 0.0162207i	0.999967	−	0.00811034i	$-0.00258163\pi$
$614$	0	0
$615$	−5.17382	−0.208629
$616$	0	0
$617$	28.0114	1.12770	0.563848	−	0.825879i	$-0.309321\pi$
$617$	28.0114	1.12770	0.563848	+	0.825879i	$0.309321\pi$
$618$	0	0
$619$	21.3586i	0.858476i	0.903191	+	0.429238i	$0.141218\pi$
$619$	21.3586i	0.858476i	−0.903191	+	0.429238i	$0.858782\pi$
$620$	0	0
$621$	− 35.2213i	− 1.41338i
$622$	0	0
$623$	4.86531	0.194925
$624$	0	0
$625$	−28.5113	−1.14045
$626$	0	0
$627$	12.0126i	0.479738i
$628$	0	0
$629$	− 31.6533i	− 1.26210i
$630$	0	0
$631$	−24.2901	−0.966973	−0.483487	−	0.875352i	$-0.660630\pi$
$631$	−24.2901	−0.966973	−0.483487	+	0.875352i	$0.660630\pi$
$632$	0	0
$633$	−16.9711	−0.674541
$634$	0	0
$635$	13.1135i	0.520394i
$636$	0	0
$637$	32.5339i	1.28904i
$638$	0	0
$639$	0.122722	0.00485481
$640$	0	0
$641$	11.4956	0.454048	0.227024	−	0.973889i	$-0.427100\pi$
$641$	11.4956	0.454048	0.227024	+	0.973889i	$0.427100\pi$
$642$	0	0
$643$	− 25.2706i	− 0.996576i	−0.867012	−	0.498288i	$-0.833962\pi$
$643$	− 25.2706i	− 0.996576i	0.867012	−	0.498288i	$-0.166038\pi$
$644$	0	0
$645$	48.6398i	1.91519i
$646$	0	0
$647$	16.3017	0.640886	0.320443	−	0.947268i	$-0.396168\pi$
$647$	16.3017	0.640886	0.320443	+	0.947268i	$0.396168\pi$
$648$	0	0
$649$	10.1062	0.396703
$650$	0	0
$651$	− 6.91926i	− 0.271187i
$652$	0	0
$653$	− 32.1298i	− 1.25733i	−0.777674	−	0.628667i	$-0.783601\pi$
$653$	− 32.1298i	− 1.25733i	0.777674	−	0.628667i	$-0.216399\pi$
$654$	0	0
$655$	−20.9081	−0.816946
$656$	0	0
$657$	−0.830747	−0.0324105
$658$	0	0
$659$	16.5842i	0.646029i	0.946394	+	0.323014i	$0.104696\pi$
$659$	16.5842i	0.646029i	−0.946394	+	0.323014i	$0.895304\pi$
$660$	0	0
$661$	− 15.5250i	− 0.603853i	−0.953331	−	0.301926i	$-0.902370\pi$
$661$	− 15.5250i	− 0.603853i	0.953331	−	0.301926i	$-0.0976296\pi$
$662$	0	0
$663$	39.0320	1.51588
$664$	0	0
$665$	−8.23055	−0.319167
$666$	0	0
$667$	58.8598i	2.27906i
$668$	0	0
$669$	38.5047i	1.48868i
$670$	0	0
$671$	3.29070	0.127036
$672$	0	0
$673$	12.3277	0.475197	0.237598	−	0.971364i	$-0.423640\pi$
$673$	12.3277	0.475197	0.237598	+	0.971364i	$0.423640\pi$
$674$	0	0
$675$	− 21.8544i	− 0.841177i
$676$	0	0
$677$	− 34.6533i	− 1.33183i	−0.746026	−	0.665917i	$-0.768040\pi$
$677$	− 34.6533i	− 1.33183i	0.746026	−	0.665917i	$-0.231960\pi$
$678$	0	0
$679$	10.1173	0.388266
$680$	0	0
$681$	25.7210	0.985632
$682$	0	0
$683$	− 23.1508i	− 0.885839i	−0.896561	−	0.442920i	$-0.853943\pi$
$683$	− 23.1508i	− 0.885839i	0.896561	−	0.442920i	$-0.146057\pi$
$684$	0	0
$685$	− 15.0354i	− 0.574473i
$686$	0	0
$687$	13.9594	0.532584
$688$	0	0
$689$	65.9034	2.51072
$690$	0	0
$691$	− 2.63453i	− 0.100222i	−0.998744	−	0.0501111i	$-0.984042\pi$
$691$	− 2.63453i	− 0.100222i	0.998744	−	0.0501111i	$-0.0159575\pi$
$692$	0	0
$693$	0.0841396i	0.00319620i
$694$	0	0
$695$	14.9110	0.565607
$696$	0	0
$697$	−4.61075	−0.174645
$698$	0	0
$699$	− 3.87427i	− 0.146538i
$700$	0	0
$701$	11.5063i	0.434588i	0.976106	+	0.217294i	$0.0697231\pi$
$701$	11.5063i	0.434588i	−0.976106	+	0.217294i	$0.930277\pi$
$702$	0	0
$703$	−28.5319	−1.07610
$704$	0	0
$705$	15.7991	0.595027
$706$	0	0
$707$	− 2.60242i	− 0.0978739i
$708$	0	0
$709$	43.0870i	1.61816i	0.587695	+	0.809082i	$0.300035\pi$
$709$	43.0870i	1.61816i	−0.587695	+	0.809082i	$0.699965\pi$
$710$	0	0
$711$	−0.160114	−0.00600473
$712$	0	0
$713$	41.3984	1.55038
$714$	0	0
$715$	− 25.3197i	− 0.946905i
$716$	0	0
$717$	− 15.3505i	− 0.573273i
$718$	0	0
$719$	−9.99334	−0.372689	−0.186344	−	0.982484i	$-0.559664\pi$
$719$	−9.99334	−0.372689	−0.186344	+	0.982484i	$0.559664\pi$
$720$	0	0
$721$	1.53880	0.0573077
$722$	0	0
$723$	16.1102i	0.599143i
$724$	0	0
$725$	36.5219i	1.35639i
$726$	0	0
$727$	3.21740	0.119327	0.0596634	−	0.998219i	$-0.480997\pi$
$727$	3.21740	0.119327	0.0596634	+	0.998219i	$0.480997\pi$
$728$	0	0
$729$	−27.6493	−1.02405
$730$	0	0
$731$	43.3463i	1.60322i
$732$	0	0
$733$	− 22.8606i	− 0.844375i	−0.906508	−	0.422188i	$-0.861262\pi$
$733$	− 22.8606i	− 0.844375i	0.906508	−	0.422188i	$-0.138738\pi$
$734$	0	0
$735$	−34.0004	−1.25412
$736$	0	0
$737$	25.8780	0.953230
$738$	0	0
$739$	0.369645i	0.0135976i	0.999977	+	0.00679881i	$0.00216414\pi$
$739$	0.369645i	0.0135976i	−0.999977	+	0.00679881i	$0.997836\pi$
$740$	0	0
$741$	− 35.1830i	− 1.29248i
$742$	0	0
$743$	41.6693	1.52870	0.764349	−	0.644803i	$-0.223060\pi$
$743$	41.6693	1.52870	0.764349	+	0.644803i	$0.223060\pi$
$744$	0	0
$745$	−45.1651	−1.65472
$746$	0	0
$747$	− 0.891769i	− 0.0326281i
$748$	0	0
$749$	− 10.9162i	− 0.398868i
$750$	0	0
$751$	−35.5518	−1.29730	−0.648651	−	0.761086i	$-0.724666\pi$
$751$	−35.5518	−1.29730	−0.648651	+	0.761086i	$0.724666\pi$
$752$	0	0
$753$	−44.1632	−1.60940
$754$	0	0
$755$	27.5232i	1.00167i
$756$	0	0
$757$	40.6113i	1.47604i	0.674778	+	0.738021i	$0.264240\pi$
$757$	40.6113i	1.47604i	−0.674778	+	0.738021i	$0.735760\pi$
$758$	0	0
$759$	19.3544	0.702521
$760$	0	0
$761$	−18.3027	−0.663471	−0.331735	−	0.943372i	$-0.607634\pi$
$761$	−18.3027	−0.663471	−0.331735	+	0.943372i	$0.607634\pi$
$762$	0	0
$763$	6.52623i	0.236265i
$764$	0	0
$765$	− 1.06099i	− 0.0383603i
$766$	0	0
$767$	−29.5994	−1.06877
$768$	0	0
$769$	29.7199	1.07173	0.535864	−	0.844304i	$-0.319986\pi$
$769$	29.7199	1.07173	0.535864	+	0.844304i	$0.319986\pi$
$770$	0	0
$771$	− 33.1133i	− 1.19255i
$772$	0	0
$773$	16.7108i	0.601047i	0.953774	+	0.300524i	$0.0971614\pi$
$773$	16.7108i	0.601047i	−0.953774	+	0.300524i	$0.902839\pi$
$774$	0	0
$775$	25.6872	0.922713
$776$	0	0
$777$	7.68333	0.275638
$778$	0	0
$779$	4.15608i	0.148907i
$780$	0	0
$781$	2.72758i	0.0976004i
$782$	0	0
$783$	46.2350	1.65230
$784$	0	0
$785$	50.5873	1.80554
$786$	0	0
$787$	− 45.7572i	− 1.63107i	−0.578710	−	0.815533i	$-0.696444\pi$
$787$	− 45.7572i	− 1.63107i	0.578710	−	0.815533i	$-0.303556\pi$
$788$	0	0
$789$	− 18.6189i	− 0.662850i
$790$	0	0
$791$	−5.66027	−0.201256
$792$	0	0
$793$	−9.63794	−0.342253
$794$	0	0
$795$	68.8741i	2.44271i
$796$	0	0
$797$	27.0694i	0.958849i	0.877583	+	0.479424i	$0.159155\pi$
$797$	27.0694i	0.958849i	−0.877583	+	0.479424i	$0.840845\pi$
$798$	0	0
$799$	14.0796	0.498102
$800$	0	0
$801$	0.565337	0.0199752
$802$	0	0
$803$	− 18.4639i	− 0.651577i
$804$	0	0
$805$	13.2609i	0.467384i
$806$	0	0
$807$	31.3528	1.10367
$808$	0	0
$809$	0.711657	0.0250205	0.0125103	−	0.999922i	$-0.496018\pi$
$809$	0.711657	0.0250205	0.0125103	+	0.999922i	$0.496018\pi$
$810$	0	0
$811$	− 47.7972i	− 1.67839i	−0.543833	−	0.839194i	$-0.683027\pi$
$811$	− 47.7972i	− 1.67839i	0.543833	−	0.839194i	$-0.316973\pi$
$812$	0	0
$813$	3.10893i	0.109035i
$814$	0	0
$815$	−45.4704	−1.59276
$816$	0	0
$817$	39.0719	1.36695
$818$	0	0
$819$	− 0.246432i	− 0.00861101i
$820$	0	0
$821$	− 6.53158i	− 0.227954i	−0.993483	−	0.113977i	$-0.963641\pi$
$821$	− 6.53158i	− 0.227954i	0.993483	−	0.113977i	$-0.0363590\pi$
$822$	0	0
$823$	32.4825	1.13227	0.566134	−	0.824313i	$-0.308439\pi$
$823$	32.4825	1.13227	0.566134	+	0.824313i	$0.308439\pi$
$824$	0	0
$825$	12.0092	0.418107
$826$	0	0
$827$	− 4.43267i	− 0.154139i	−0.997026	−	0.0770694i	$-0.975444\pi$
$827$	− 4.43267i	− 0.154139i	0.997026	−	0.0770694i	$-0.0245563\pi$
$828$	0	0
$829$	38.8666i	1.34989i	0.737866	+	0.674947i	$0.235833\pi$
$829$	38.8666i	1.34989i	−0.737866	+	0.674947i	$0.764167\pi$
$830$	0	0
$831$	−28.4632	−0.987377
$832$	0	0
$833$	−30.3001	−1.04984
$834$	0	0
$835$	− 37.7661i	− 1.30695i
$836$	0	0
$837$	− 32.5189i	− 1.12402i
$838$	0	0
$839$	36.4360	1.25791	0.628955	−	0.777442i	$-0.283483\pi$
$839$	36.4360	1.25791	0.628955	+	0.777442i	$0.283483\pi$
$840$	0	0
$841$	−48.2653	−1.66432
$842$	0	0
$843$	− 4.89483i	− 0.168587i
$844$	0	0
$845$	34.8233i	1.19796i
$846$	0	0
$847$	5.32958	0.183127
$848$	0	0
$849$	−45.5059	−1.56176
$850$	0	0
$851$	45.9699i	1.57583i
$852$	0	0
$853$	− 49.7084i	− 1.70198i	−0.525181	−	0.850991i	$-0.676002\pi$
$853$	− 49.7084i	− 1.70198i	0.525181	−	0.850991i	$-0.323998\pi$
$854$	0	0
$855$	−0.956369	−0.0327071
$856$	0	0
$857$	40.2498	1.37491	0.687454	−	0.726228i	$-0.258728\pi$
$857$	40.2498	1.37491	0.687454	+	0.726228i	$0.258728\pi$
$858$	0	0
$859$	18.7277i	0.638979i	0.947590	+	0.319490i	$0.103511\pi$
$859$	18.7277i	0.638979i	−0.947590	+	0.319490i	$0.896489\pi$
$860$	0	0
$861$	− 1.11919i	− 0.0381418i
$862$	0	0
$863$	41.5149	1.41318	0.706591	−	0.707622i	$-0.250232\pi$
$863$	41.5149	1.41318	0.706591	+	0.707622i	$0.250232\pi$
$864$	0	0
$865$	32.0131	1.08848
$866$	0	0
$867$	7.28278i	0.247336i
$868$	0	0
$869$	− 3.55863i	− 0.120718i
$870$	0	0
$871$	−75.7926	−2.56813
$872$	0	0
$873$	1.17560	0.0397881
$874$	0	0
$875$	− 1.67360i	− 0.0565780i
$876$	0	0
$877$	28.0656i	0.947707i	0.880604	+	0.473854i	$0.157137\pi$
$877$	28.0656i	0.947707i	−0.880604	+	0.473854i	$0.842863\pi$
$878$	0	0
$879$	34.1582	1.15213
$880$	0	0
$881$	1.09142	0.0367708	0.0183854	−	0.999831i	$-0.494147\pi$
$881$	1.09142	0.0367708	0.0183854	+	0.999831i	$0.494147\pi$
$882$	0	0
$883$	− 11.4774i	− 0.386244i	−0.981175	−	0.193122i	$-0.938139\pi$
$883$	− 11.4774i	− 0.386244i	0.981175	−	0.193122i	$-0.0618613\pi$
$884$	0	0
$885$	− 30.9336i	− 1.03982i
$886$	0	0
$887$	38.8690	1.30509	0.652547	−	0.757748i	$-0.273701\pi$
$887$	38.8690	1.30509	0.652547	+	0.757748i	$0.273701\pi$
$888$	0	0
$889$	−2.83668	−0.0951392
$890$	0	0
$891$	− 14.8174i	− 0.496403i
$892$	0	0
$893$	− 12.6912i	− 0.424696i
$894$	0	0
$895$	−28.4376	−0.950564
$896$	0	0
$897$	−56.6860	−1.89269
$898$	0	0
$899$	54.3437i	1.81246i
$900$	0	0
$901$	61.3785i	2.04481i
$902$	0	0
$903$	−10.5216	−0.350138
$904$	0	0
$905$	−7.86734	−0.261519
$906$	0	0
$907$	− 44.3257i	− 1.47181i	−0.677085	−	0.735905i	$-0.736757\pi$
$907$	− 44.3257i	− 1.47181i	0.677085	−	0.735905i	$-0.263243\pi$
$908$	0	0
$909$	− 0.302394i	− 0.0100298i
$910$	0	0
$911$	−15.8440	−0.524936	−0.262468	−	0.964941i	$-0.584536\pi$
$911$	−15.8440	−0.524936	−0.262468	+	0.964941i	$0.584536\pi$
$912$	0	0
$913$	19.8201	0.655950
$914$	0	0
$915$	− 10.0724i	− 0.332982i
$916$	0	0
$917$	− 4.52278i	− 0.149355i
$918$	0	0
$919$	−3.32293	−0.109613	−0.0548066	−	0.998497i	$-0.517454\pi$
$919$	−3.32293	−0.109613	−0.0548066	+	0.998497i	$0.517454\pi$
$920$	0	0
$921$	−13.6928	−0.451193
$922$	0	0
$923$	− 7.98864i	− 0.262949i
$924$	0	0
$925$	28.5238i	0.937858i
$926$	0	0
$927$	0.178804	0.00587270
$928$	0	0
$929$	59.6827	1.95813	0.979063	−	0.203555i	$-0.0652497\pi$
$929$	59.6827	1.95813	0.979063	+	0.203555i	$0.0652497\pi$
$930$	0	0
$931$	27.3122i	0.895120i
$932$	0	0
$933$	− 11.1495i	− 0.365017i
$934$	0	0
$935$	23.5813	0.771190
$936$	0	0
$937$	51.6947	1.68879	0.844396	−	0.535720i	$-0.179960\pi$
$937$	51.6947	1.68879	0.844396	+	0.535720i	$0.179960\pi$
$938$	0	0
$939$	52.8235i	1.72383i
$940$	0	0
$941$	− 31.2160i	− 1.01761i	−0.860881	−	0.508806i	$-0.830087\pi$
$941$	− 31.2160i	− 1.01761i	0.860881	−	0.508806i	$-0.169913\pi$
$942$	0	0
$943$	6.69617	0.218057
$944$	0	0
$945$	10.4165	0.338850
$946$	0	0
$947$	− 15.4664i	− 0.502590i	−0.967911	−	0.251295i	$-0.919144\pi$
$947$	− 15.4664i	− 0.502590i	0.967911	−	0.251295i	$-0.0808564\pi$
$948$	0	0
$949$	54.0778i	1.75544i
$950$	0	0
$951$	43.3350	1.40523
$952$	0	0
$953$	26.0920	0.845202	0.422601	−	0.906316i	$-0.361117\pi$
$953$	26.0920	0.845202	0.422601	+	0.906316i	$0.361117\pi$
$954$	0	0
$955$	− 56.6206i	− 1.83220i
$956$	0	0
$957$	25.4065i	0.821277i
$958$	0	0
$959$	3.25242	0.105026
$960$	0	0
$961$	7.22200	0.232968
$962$	0	0
$963$	− 1.26843i	− 0.0408746i
$964$	0	0
$965$	− 46.2935i	− 1.49024i
$966$	0	0
$967$	−43.4016	−1.39570	−0.697851	−	0.716243i	$-0.745860\pi$
$967$	−43.4016	−1.39570	−0.697851	+	0.716243i	$0.745860\pi$
$968$	0	0
$969$	32.7673	1.05264
$970$	0	0
$971$	17.2893i	0.554840i	0.960749	+	0.277420i	$0.0894794\pi$
$971$	17.2893i	0.554840i	−0.960749	+	0.277420i	$0.910521\pi$
$972$	0	0
$973$	3.22551i	0.103405i
$974$	0	0
$975$	−35.1730	−1.12644
$976$	0	0
$977$	−22.5021	−0.719906	−0.359953	−	0.932971i	$-0.617207\pi$
$977$	−22.5021	−0.719906	−0.359953	+	0.932971i	$0.617207\pi$
$978$	0	0
$979$	12.5650i	0.401578i
$980$	0	0
$981$	0.758331i	0.0242117i
$982$	0	0
$983$	50.4448	1.60894	0.804469	−	0.593994i	$-0.202450\pi$
$983$	50.4448	1.60894	0.804469	+	0.593994i	$0.202450\pi$
$984$	0	0
$985$	23.3945	0.745411
$986$	0	0
$987$	3.41761i	0.108784i
$988$	0	0
$989$	− 62.9517i	− 2.00175i
$990$	0	0
$991$	19.9008	0.632169	0.316085	−	0.948731i	$-0.397632\pi$
$991$	19.9008	0.632169	0.316085	+	0.948731i	$0.397632\pi$
$992$	0	0
$993$	−44.3223	−1.40653
$994$	0	0
$995$	− 58.1097i	− 1.84220i
$996$	0	0
$997$	− 25.7252i	− 0.814725i	−0.913267	−	0.407362i	$-0.866449\pi$
$997$	− 25.7252i	− 0.814725i	0.913267	−	0.407362i	$-0.133551\pi$
$998$	0	0
$999$	36.1098	1.14246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2624.2.b.f.1313.21	yes	28
4.3	odd	2	inner	2624.2.b.f.1313.7	✓	28
8.3	odd	2	inner	2624.2.b.f.1313.22	yes	28
8.5	even	2	inner	2624.2.b.f.1313.8	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2624.2.b.f.1313.7	✓	28	4.3	odd	2	inner
2624.2.b.f.1313.8	yes	28	8.5	even	2	inner
2624.2.b.f.1313.21	yes	28	1.1	even	1	trivial
2624.2.b.f.1313.22	yes	28	8.3	odd	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:	\( N \)	\(=\)	\( 2624 = 2^{6} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2624.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.70996i q^{3} -3.02571i q^{5} +0.654512 q^{7} +0.0760526 q^{9} +O(q^{10})\)	\(q+1.70996i q^{3} -3.02571i q^{5} +0.654512 q^{7} +0.0760526 q^{9} +1.69032i q^{11} -4.95067i q^{13} +5.17382 q^{15} +4.61075 q^{17} -4.15608i q^{19} +1.11919i q^{21} -6.69617 q^{23} -4.15490 q^{25} +5.25991i q^{27} -8.79007i q^{29} -6.18239 q^{31} -2.89037 q^{33} -1.98036i q^{35} -6.86510i q^{37} +8.46543 q^{39} -1.00000 q^{41} +9.40114i q^{43} -0.230113i q^{45} +3.05365 q^{47} -6.57161 q^{49} +7.88418i q^{51} +13.3120i q^{53} +5.11441 q^{55} +7.10672 q^{57} -5.97887i q^{59} -1.94680i q^{61} +0.0497774 q^{63} -14.9793 q^{65} -15.3096i q^{67} -11.4502i q^{69} +1.61365 q^{71} -10.9233 q^{73} -7.10470i q^{75} +1.10633i q^{77} -2.10530 q^{79} -8.76606 q^{81} -11.7257i q^{83} -13.9508i q^{85} +15.0306 q^{87} +7.43349 q^{89} -3.24028i q^{91} -10.5716i q^{93} -12.5751 q^{95} +15.4578 q^{97} +0.128553i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 28 q^{9}+O(q^{10}) \) 28 * q - 28 * q^9	\( 28 q - 28 q^{9} - 8 q^{17} - 36 q^{25} - 64 q^{33} - 28 q^{41} - 20 q^{49} - 80 q^{57} - 8 q^{73} - 20 q^{81} - 72 q^{89} - 40 q^{97}+O(q^{100}) \) 28 * q - 28 * q^9 - 8 * q^17 - 36 * q^25 - 64 * q^33 - 28 * q^41 - 20 * q^49 - 80 * q^57 - 8 * q^73 - 20 * q^81 - 72 * q^89 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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