LMFDB - Embedded newform 5850.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(1,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{8} -6.00000 q^{11} -1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000 q^{19} -6.00000 q^{22} +6.00000 q^{23} -1.00000 q^{26} -2.00000 q^{28} +3.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{37} -1.00000 q^{38} +9.00000 q^{41} -8.00000 q^{43} -6.00000 q^{44} +6.00000 q^{46} -3.00000 q^{47} -3.00000 q^{49} -1.00000 q^{52} +3.00000 q^{53} -2.00000 q^{56} +3.00000 q^{58} +6.00000 q^{59} -10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +13.0000 q^{67} -9.00000 q^{71} +4.00000 q^{73} +1.00000 q^{74} -1.00000 q^{76} +12.0000 q^{77} +11.0000 q^{79} +9.00000 q^{82} +12.0000 q^{83} -8.00000 q^{86} -6.00000 q^{88} +6.00000 q^{89} +2.00000 q^{91} +6.00000 q^{92} -3.00000 q^{94} +10.0000 q^{97} -3.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	−2.00000	−0.377964
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	6.00000	0.884652
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	−1.00000	−0.138675
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	3.00000	0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−1.00000	−0.114708
$77$	12.0000	1.36753
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	0	0
$82$	9.00000	0.993884
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	−6.00000	−0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	6.00000	0.625543
$93$	0	0
$94$	−3.00000	−0.309426
$95$	0	0
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	3.00000	0.291386
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	0	0
$112$	−2.00000	−0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	3.00000	0.278543
$117$	0	0
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	−10.0000	−0.905357
$123$	0	0
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	13.0000	1.12303
$135$	0	0
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	−9.00000	−0.755263
$143$	6.00000	0.501745
$144$	0	0
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	1.00000	0.0821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	11.0000	0.875113
$159$	0	0
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	12.0000	0.931381
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	0	0
$175$	0	0
$176$	−6.00000	−0.452267
$177$	0	0
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	6.00000	0.442326
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−3.00000	−0.218797
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	0	0
$201$	0	0
$202$	6.00000	0.422159
$203$	−6.00000	−0.421117
$204$	0	0
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	6.00000	0.415029
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	3.00000	0.206041
$213$	0	0
$214$	15.0000	1.02538
$215$	0	0
$216$	0	0
$217$	−16.0000	−1.08615
$218$	5.00000	0.338643
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	6.00000	0.399114
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	−25.0000	−1.65205	−0.826023	−	0.563636i	$-0.809402\pi$
$229$	−25.0000	−1.65205	−0.826023	+	0.563636i	$0.809402\pi$
$230$	0	0
$231$	0	0
$232$	3.00000	0.196960
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	25.0000	1.60706
$243$	0	0
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	1.00000	0.0636285
$248$	8.00000	0.508001
$249$	0	0
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	−36.0000	−2.26330
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	−9.00000	−0.556022
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	2.00000	0.122628
$267$	0	0
$268$	13.0000	0.794101
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	0	0
$274$	3.00000	0.181237
$275$	0	0
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	−10.0000	−0.599760
$279$	0	0
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	6.00000	0.354787
$287$	−18.0000	−1.06251
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	4.00000	0.234082
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	0	0
$296$	1.00000	0.0581238
$297$	0	0
$298$	6.00000	0.347571
$299$	−6.00000	−0.346989
$300$	0	0
$301$	16.0000	0.922225
$302$	−22.0000	−1.26596
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−23.0000	−1.31268	−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000	−1.31268	−0.656340	+	0.754466i	$0.727896\pi$
$308$	12.0000	0.683763
$309$	0	0
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	11.0000	0.618798
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	0	0
$322$	−12.0000	−0.668734
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−20.0000	−1.10770
$327$	0	0
$328$	9.00000	0.496942
$329$	6.00000	0.330791
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	15.0000	0.820763
$335$	0	0
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	0	0
$341$	−48.0000	−2.59935
$342$	0	0
$343$	20.0000	1.07990
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−21.0000	−1.12897
$347$	9.00000	0.483145	0.241573	−	0.970383i	$-0.422337\pi$
$347$	9.00000	0.483145	0.241573	+	0.970383i	$0.422337\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	0	0
$352$	−6.00000	−0.319801
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	2.00000	0.105118
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	−3.00000	−0.154713
$377$	−3.00000	−0.154508
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	−6.00000	−0.306987
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	0	0
$386$	−8.00000	−0.407189
$387$	0	0
$388$	10.0000	0.507673
$389$	−33.0000	−1.67317	−0.836583	−	0.547840i	$-0.815450\pi$
$389$	−33.0000	−1.67317	−0.836583	+	0.547840i	$0.815450\pi$
$390$	0	0
$391$	0	0
$392$	−3.00000	−0.151523
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	25.0000	1.25471	0.627357	−	0.778732i	$-0.284137\pi$
$397$	25.0000	1.25471	0.627357	+	0.778732i	$0.284137\pi$
$398$	−7.00000	−0.350878
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	6.00000	0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	−6.00000	−0.297409
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	4.00000	0.197066
$413$	−12.0000	−0.590481
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	6.00000	0.293470
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−22.0000	−1.07094
$423$	0	0
$424$	3.00000	0.145693
$425$	0	0
$426$	0	0
$427$	20.0000	0.967868
$428$	15.0000	0.725052
$429$	0	0
$430$	0	0
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	0	0
$433$	13.0000	0.624740	0.312370	−	0.949960i	$-0.398877\pi$
$433$	13.0000	0.624740	0.312370	+	0.949960i	$0.398877\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	5.00000	0.239457
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.0000	−0.997740	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000	−0.997740	−0.498870	+	0.866677i	$0.666252\pi$
$444$	0	0
$445$	0	0
$446$	−14.0000	−0.662919
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	39.0000	1.84052	0.920262	−	0.391303i	$-0.127976\pi$
$449$	39.0000	1.84052	0.920262	+	0.391303i	$0.127976\pi$
$450$	0	0
$451$	−54.0000	−2.54276
$452$	6.00000	0.282216
$453$	0	0
$454$	18.0000	0.844782
$455$	0	0
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	−25.0000	−1.16817
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	0	0
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	0	0
$469$	−26.0000	−1.20057
$470$	0	0
$471$	0	0
$472$	6.00000	0.276172
$473$	48.0000	2.20704
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	24.0000	1.09773
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	26.0000	1.18427
$483$	0	0
$484$	25.0000	1.13636
$485$	0	0
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−10.0000	−0.452679
$489$	0	0
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	0	0
$494$	1.00000	0.0449921
$495$	0	0
$496$	8.00000	0.359211
$497$	18.0000	0.807410
$498$	0	0
$499$	−43.0000	−1.92494	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	−43.0000	−1.92494	−0.962472	+	0.271380i	$0.912520\pi$
$500$	0	0
$501$	0	0
$502$	−15.0000	−0.669483
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	0	0
$505$	0	0
$506$	−36.0000	−1.60040
$507$	0	0
$508$	7.00000	0.310575
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	0	0
$516$	0	0
$517$	18.0000	0.791639
$518$	−2.00000	−0.0878750
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	−9.00000	−0.393167
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	2.00000	0.0867110
$533$	−9.00000	−0.389833
$534$	0	0
$535$	0	0
$536$	13.0000	0.561514
$537$	0	0
$538$	15.0000	0.646696
$539$	18.0000	0.775315
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	3.00000	0.128154
$549$	0	0
$550$	0	0
$551$	−3.00000	−0.127804
$552$	0	0
$553$	−22.0000	−0.935535
$554$	4.00000	0.169944
$555$	0	0
$556$	−10.0000	−0.424094
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	15.0000	0.632737
$563$	45.0000	1.89652	0.948262	−	0.317489i	$-0.102840\pi$
$563$	45.0000	1.89652	0.948262	+	0.317489i	$0.102840\pi$
$564$	0	0
$565$	0	0
$566$	10.0000	0.420331
$567$	0	0
$568$	−9.00000	−0.377632
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	6.00000	0.250873
$573$	0	0
$574$	−18.0000	−0.751305
$575$	0	0
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	−18.0000	−0.745484
$584$	4.00000	0.165521
$585$	0	0
$586$	−12.0000	−0.495715
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	0	0
$592$	1.00000	0.0410997
$593$	3.00000	0.123195	0.0615976	−	0.998101i	$-0.480380\pi$
$593$	3.00000	0.123195	0.0615976	+	0.998101i	$0.480380\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	−6.00000	−0.245358
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	16.0000	0.652111
$603$	0	0
$604$	−22.0000	−0.895167
$605$	0	0
$606$	0	0
$607$	−11.0000	−0.446476	−0.223238	−	0.974764i	$-0.571663\pi$
$607$	−11.0000	−0.446476	−0.223238	+	0.974764i	$0.571663\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	−23.0000	−0.928204
$615$	0	0
$616$	12.0000	0.483494
$617$	21.0000	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$617$	21.0000	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$618$	0	0
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	0	0
$621$	0	0
$622$	−18.0000	−0.721734
$623$	−12.0000	−0.480770
$624$	0	0
$625$	0	0
$626$	1.00000	0.0399680
$627$	0	0
$628$	22.0000	0.877896
$629$	0	0
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	11.0000	0.437557
$633$	0	0
$634$	24.0000	0.953162
$635$	0	0
$636$	0	0
$637$	3.00000	0.118864
$638$	−18.0000	−0.712627
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−5.00000	−0.197181	−0.0985904	−	0.995128i	$-0.531433\pi$
$643$	−5.00000	−0.197181	−0.0985904	+	0.995128i	$0.531433\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	0	0
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	−20.0000	−0.783260
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	0	0
$656$	9.00000	0.351391
$657$	0	0
$658$	6.00000	0.233904
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	−31.0000	−1.20576	−0.602880	−	0.797832i	$-0.705980\pi$
$661$	−31.0000	−1.20576	−0.602880	+	0.797832i	$0.705980\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	18.0000	0.696963
$668$	15.0000	0.580367
$669$	0	0
$670$	0	0
$671$	60.0000	2.31627
$672$	0	0
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	1.00000	0.0384615
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	0	0
$682$	−48.0000	−1.83801
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−3.00000	−0.114291
$690$	0	0
$691$	−1.00000	−0.0380418	−0.0190209	−	0.999819i	$-0.506055\pi$
$691$	−1.00000	−0.0380418	−0.0190209	+	0.999819i	$0.506055\pi$
$692$	−21.0000	−0.798300
$693$	0	0
$694$	9.00000	0.341635
$695$	0	0
$696$	0	0
$697$	0	0
$698$	2.00000	0.0757011
$699$	0	0
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−1.00000	−0.0377157
$704$	−6.00000	−0.226134
$705$	0	0
$706$	21.0000	0.790345
$707$	−12.0000	−0.451306
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	6.00000	0.224860
$713$	48.0000	1.79761
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	15.0000	0.559795
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	−18.0000	−0.669891
$723$	0	0
$724$	2.00000	0.0743294
$725$	0	0
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	13.0000	0.480166	0.240083	−	0.970752i	$-0.422825\pi$
$733$	13.0000	0.480166	0.240083	+	0.970752i	$0.422825\pi$
$734$	−17.0000	−0.627481
$735$	0	0
$736$	6.00000	0.221163
$737$	−78.0000	−2.87317
$738$	0	0
$739$	11.0000	0.404642	0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000	0.404642	0.202321	+	0.979319i	$0.435152\pi$
$740$	0	0
$741$	0	0
$742$	−6.00000	−0.220267
$743$	−15.0000	−0.550297	−0.275148	−	0.961402i	$-0.588727\pi$
$743$	−15.0000	−0.550297	−0.275148	+	0.961402i	$0.588727\pi$
$744$	0	0
$745$	0	0
$746$	10.0000	0.366126
$747$	0	0
$748$	0	0
$749$	−30.0000	−1.09618
$750$	0	0
$751$	−7.00000	−0.255434	−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000	−0.255434	−0.127717	+	0.991811i	$0.540765\pi$
$752$	−3.00000	−0.109399
$753$	0	0
$754$	−3.00000	−0.109254
$755$	0	0
$756$	0	0
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	−6.00000	−0.217072
$765$	0	0
$766$	15.0000	0.541972
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	0	0
$772$	−8.00000	−0.287926
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	−33.0000	−1.18311
$779$	−9.00000	−0.322458
$780$	0	0
$781$	54.0000	1.93227
$782$	0	0
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	10.0000	0.355110
$794$	25.0000	0.887217
$795$	0	0
$796$	−7.00000	−0.248108
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	6.00000	0.211867
$803$	−24.0000	−0.846942
$804$	0	0
$805$	0	0
$806$	−8.00000	−0.281788
$807$	0	0
$808$	6.00000	0.211079
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	−6.00000	−0.210300
$815$	0	0
$816$	0	0
$817$	8.00000	0.279885
$818$	14.0000	0.489499
$819$	0	0
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	0	0
$823$	13.0000	0.453152	0.226576	−	0.973994i	$-0.427247\pi$
$823$	13.0000	0.453152	0.226576	+	0.973994i	$0.427247\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−12.0000	−0.417533
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	0	0
$835$	0	0
$836$	6.00000	0.207514
$837$	0	0
$838$	9.00000	0.310900
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−10.0000	−0.344623
$843$	0	0
$844$	−22.0000	−0.757271
$845$	0	0
$846$	0	0
$847$	−50.0000	−1.71802
$848$	3.00000	0.103020
$849$	0	0
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	19.0000	0.650548	0.325274	−	0.945620i	$-0.394544\pi$
$853$	19.0000	0.650548	0.325274	+	0.945620i	$0.394544\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	15.0000	0.512689
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	0	0
$862$	−15.0000	−0.510902
$863$	−3.00000	−0.102121	−0.0510606	−	0.998696i	$-0.516260\pi$
$863$	−3.00000	−0.102121	−0.0510606	+	0.998696i	$0.516260\pi$
$864$	0	0
$865$	0	0
$866$	13.0000	0.441758
$867$	0	0
$868$	−16.0000	−0.543075
$869$	−66.0000	−2.23890
$870$	0	0
$871$	−13.0000	−0.440488
$872$	5.00000	0.169321
$873$	0	0
$874$	−6.00000	−0.202953
$875$	0	0
$876$	0	0
$877$	−29.0000	−0.979260	−0.489630	−	0.871930i	$-0.662868\pi$
$877$	−29.0000	−0.979260	−0.489630	+	0.871930i	$0.662868\pi$
$878$	−19.0000	−0.641219
$879$	0	0
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	−21.0000	−0.705509
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	0	0
$892$	−14.0000	−0.468755
$893$	3.00000	0.100391
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	39.0000	1.30145
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	−54.0000	−1.79800
$903$	0	0
$904$	6.00000	0.199557
$905$	0	0
$906$	0	0
$907$	22.0000	0.730498	0.365249	−	0.930910i	$-0.380984\pi$
$907$	22.0000	0.730498	0.365249	+	0.930910i	$0.380984\pi$
$908$	18.0000	0.597351
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−72.0000	−2.38285
$914$	28.0000	0.926158
$915$	0	0
$916$	−25.0000	−0.826023
$917$	18.0000	0.594412
$918$	0	0
$919$	17.0000	0.560778	0.280389	−	0.959886i	$-0.409536\pi$
$919$	17.0000	0.560778	0.280389	+	0.959886i	$0.409536\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.00000	0.296239
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	0	0
$928$	3.00000	0.0984798
$929$	−15.0000	−0.492134	−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000	−0.492134	−0.246067	+	0.969253i	$0.579138\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	0	0
$933$	0	0
$934$	−27.0000	−0.883467
$935$	0	0
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	−26.0000	−0.848930
$939$	0	0
$940$	0	0
$941$	−36.0000	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$942$	0	0
$943$	54.0000	1.75848
$944$	6.00000	0.195283
$945$	0	0
$946$	48.0000	1.56061
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−48.0000	−1.55487	−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000	−1.55487	−0.777436	+	0.628962i	$0.783480\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	−15.0000	−0.484628
$959$	−6.00000	−0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	−1.00000	−0.0322413
$963$	0	0
$964$	26.0000	0.837404
$965$	0	0
$966$	0	0
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	0	0
$971$	57.0000	1.82922	0.914609	−	0.404341i	$-0.132499\pi$
$971$	57.0000	1.82922	0.914609	+	0.404341i	$0.132499\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	0	0
$982$	36.0000	1.14881
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	1.00000	0.0318142
$989$	−48.0000	−1.52631
$990$	0	0
$991$	17.0000	0.540023	0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000	0.540023	0.270011	+	0.962857i	$0.412973\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	18.0000	0.570925
$995$	0	0
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−43.0000	−1.36114
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.a.bg.1.1	yes	1
3.2	odd	2		5850.2.a.i.1.1	✓	1
5.2	odd	4		5850.2.e.b.5149.2		2
5.3	odd	4		5850.2.e.b.5149.1		2
5.4	even	2		5850.2.a.r.1.1	yes	1
15.2	even	4		5850.2.e.bf.5149.1		2
15.8	even	4		5850.2.e.bf.5149.2		2
15.14	odd	2		5850.2.a.bx.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5850.2.a.i.1.1	✓	1	3.2	odd	2
5850.2.a.r.1.1	yes	1	5.4	even	2
5850.2.a.bg.1.1	yes	1	1.1	even	1	trivial
5850.2.a.bx.1.1	yes	1	15.14	odd	2
5850.2.e.b.5149.1		2	5.3	odd	4
5850.2.e.b.5149.2		2	5.2	odd	4
5850.2.e.bf.5149.1		2	15.2	even	4
5850.2.e.bf.5149.2		2	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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