LMFDB - Embedded newform 1950.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.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:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} +2.00000 q^{22} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} +2.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} -1.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} -4.00000 q^{57} +4.00000 q^{58} +10.0000 q^{59} -14.0000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -16.0000 q^{67} +2.00000 q^{68} -4.00000 q^{71} +1.00000 q^{72} +8.00000 q^{73} +6.00000 q^{74} -4.00000 q^{76} +4.00000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} +4.00000 q^{87} +2.00000 q^{88} +6.00000 q^{89} -2.00000 q^{91} +8.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} +12.0000 q^{97} -3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	2.00000	0.426401
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	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
$33$	2.00000	0.348155
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	2.00000	0.308607
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	−1.00000	−0.138675
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	−4.00000	−0.529813
$58$	4.00000	0.525226
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	8.00000	1.01600
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	1.00000	0.117851
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−4.00000	−0.458831
$77$	4.00000	0.455842
$78$	−1.00000	−0.113228
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	4.00000	0.431331
$87$	4.00000	0.428845
$88$	2.00000	0.213201
$89$	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
$92$	0	0
$93$	8.00000	0.829561
$94$	−8.00000	−0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	−3.00000	−0.303046
$99$	2.00000	0.201008
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	2.00000	0.198030
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	2.00000	0.194257
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	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	$0.408930\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	4.00000	0.371391
$117$	−1.00000	−0.0924500
$118$	10.0000	0.920575
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−14.0000	−1.26750
$123$	−6.00000	−0.541002
$124$	8.00000	0.718421
$125$	0	0
$126$	2.00000	0.178174
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	2.00000	0.174078
$133$	−8.00000	−0.693688
$134$	−16.0000	−1.38219
$135$	0	0
$136$	2.00000	0.171499
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	−4.00000	−0.335673
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	0	0
$146$	8.00000	0.662085
$147$	−3.00000	−0.247436
$148$	6.00000	0.493197
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−4.00000	−0.324443
$153$	2.00000	0.161690
$154$	4.00000	0.322329
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−8.00000	−0.636446
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	2.00000	0.154303
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.00000	−0.305888
$172$	4.00000	0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	2.00000	0.150756
$177$	10.0000	0.751646
$178$	6.00000	0.449719
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−2.00000	−0.148250
$183$	−14.0000	−1.03491
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	4.00000	0.292509
$188$	−8.00000	−0.583460
$189$	2.00000	0.145479
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000	0.0721688
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	−3.00000	−0.214286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	2.00000	0.142134
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	4.00000	0.281439
$203$	8.00000	0.561490
$204$	2.00000	0.140028
$205$	0	0
$206$	6.00000	0.418040
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	2.00000	0.137361
$213$	−4.00000	−0.274075
$214$	−4.00000	−0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	16.0000	1.08615
$218$	−18.0000	−1.21911
$219$	8.00000	0.540590
$220$	0	0
$221$	−2.00000	−0.134535
$222$	6.00000	0.402694
$223$	−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
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−4.00000	−0.264906
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	4.00000	0.262613
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	10.0000	0.650945
$237$	−8.00000	−0.519656
$238$	4.00000	0.259281
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	0	0
$246$	−6.00000	−0.382546
$247$	4.00000	0.254514
$248$	8.00000	0.508001
$249$	12.0000	0.760469
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	6.00000	0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	4.00000	0.249029
$259$	12.0000	0.745644
$260$	0	0
$261$	4.00000	0.247594
$262$	−6.00000	−0.370681
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	−8.00000	−0.490511
$267$	6.00000	0.367194
$268$	−16.0000	−0.977356
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	2.00000	0.121268
$273$	−2.00000	−0.121046
$274$	−14.0000	−0.845771
$275$	0	0
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	4.00000	0.239904
$279$	8.00000	0.478947
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−8.00000	−0.476393
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−12.0000	−0.708338
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	12.0000	0.703452
$292$	8.00000	0.468165
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	6.00000	0.348743
$297$	2.00000	0.116052
$298$	12.0000	0.695141
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	−16.0000	−0.920697
$303$	4.00000	0.229794
$304$	−4.00000	−0.229416
$305$	0	0
$306$	2.00000	0.114332
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	4.00000	0.227921
$309$	6.00000	0.341328
$310$	0	0
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	−1.00000	−0.0566139
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	2.00000	0.112154
$319$	8.00000	0.447914
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−8.00000	−0.445132
$324$	1.00000	0.0555556
$325$	0	0
$326$	−8.00000	−0.443079
$327$	−18.0000	−0.995402
$328$	−6.00000	−0.331295
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	12.0000	0.658586
$333$	6.00000	0.328798
$334$	12.0000	0.656611
$335$	0	0
$336$	2.00000	0.109109
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	1.00000	0.0543928
$339$	6.00000	0.325875
$340$	0	0
$341$	16.0000	0.866449
$342$	−4.00000	−0.216295
$343$	−20.0000	−1.07990
$344$	4.00000	0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	4.00000	0.214423
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	2.00000	0.106600
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	10.0000	0.531494
$355$	0	0
$356$	6.00000	0.317999
$357$	4.00000	0.211702
$358$	−18.0000	−0.951330
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−6.00000	−0.315353
$363$	−7.00000	−0.367405
$364$	−2.00000	−0.104828
$365$	0	0
$366$	−14.0000	−0.731792
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	4.00000	0.207670
$372$	8.00000	0.414781
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−4.00000	−0.206010
$378$	2.00000	0.102869
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	−12.0000	−0.613973
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	12.0000	0.609208
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	0	0
$392$	−3.00000	−0.151523
$393$	−6.00000	−0.302660
$394$	10.0000	0.503793
$395$	0	0
$396$	2.00000	0.100504
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−24.0000	−1.20301
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−16.0000	−0.798007
$403$	−8.00000	−0.398508
$404$	4.00000	0.199007
$405$	0	0
$406$	8.00000	0.397033
$407$	12.0000	0.594818
$408$	2.00000	0.0990148
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	6.00000	0.295599
$413$	20.0000	0.984136
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	4.00000	0.195881
$418$	−8.00000	−0.391293
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	−4.00000	−0.194717
$423$	−8.00000	−0.388973
$424$	2.00000	0.0971286
$425$	0	0
$426$	−4.00000	−0.193801
$427$	−28.0000	−1.35501
$428$	−4.00000	−0.193347
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	1.00000	0.0481125
$433$	12.0000	0.576683	0.288342	−	0.957528i	$-0.406896\pi$
$433$	12.0000	0.576683	0.288342	+	0.957528i	$0.406896\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	−18.0000	−0.862044
$437$	0	0
$438$	8.00000	0.382255
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−2.00000	−0.0951303
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−14.0000	−0.662919
$447$	12.0000	0.567581
$448$	2.00000	0.0944911
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	6.00000	0.282216
$453$	−16.0000	−0.751746
$454$	12.0000	0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	−22.0000	−1.02799
$459$	2.00000	0.0933520
$460$	0	0
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	4.00000	0.186097
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−14.0000	−0.648537
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	−1.00000	−0.0462250
$469$	−32.0000	−1.47762
$470$	0	0
$471$	10.0000	0.460776
$472$	10.0000	0.460287
$473$	8.00000	0.367840
$474$	−8.00000	−0.367452
$475$	0	0
$476$	4.00000	0.183340
$477$	2.00000	0.0915737
$478$	−12.0000	−0.548867
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	−18.0000	−0.819878
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	−14.0000	−0.633750
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−22.0000	−0.992846	−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000	−0.992846	−0.496423	+	0.868081i	$0.665354\pi$
$492$	−6.00000	−0.270501
$493$	8.00000	0.360302
$494$	4.00000	0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	−8.00000	−0.358849
$498$	12.0000	0.537733
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	18.0000	0.803379
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	6.00000	0.266207
$509$	−40.0000	−1.77297	−0.886484	−	0.462758i	$-0.846860\pi$
$509$	−40.0000	−1.77297	−0.886484	+	0.462758i	$0.846860\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−18.0000	−0.793946
$515$	0	0
$516$	4.00000	0.176090
$517$	−16.0000	−0.703679
$518$	12.0000	0.527250
$519$	14.0000	0.614532
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	4.00000	0.175075
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−8.00000	−0.348817
$527$	16.0000	0.696971
$528$	2.00000	0.0870388
$529$	−23.0000	−1.00000
$530$	0	0
$531$	10.0000	0.433963
$532$	−8.00000	−0.346844
$533$	6.00000	0.259889
$534$	6.00000	0.259645
$535$	0	0
$536$	−16.0000	−0.691095
$537$	−18.0000	−0.776757
$538$	−20.0000	−0.862261
$539$	−6.00000	−0.258438
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	16.0000	0.687259
$543$	−6.00000	−0.257485
$544$	2.00000	0.0857493
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−14.0000	−0.598050
$549$	−14.0000	−0.597505
$550$	0	0
$551$	−16.0000	−0.681623
$552$	0	0
$553$	−16.0000	−0.680389
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	4.00000	0.169638
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	8.00000	0.338667
$559$	−4.00000	−0.169182
$560$	0	0
$561$	4.00000	0.168880
$562$	−18.0000	−0.759284
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	16.0000	0.672530
$567$	2.00000	0.0839921
$568$	−4.00000	−0.167836
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	−2.00000	−0.0836242
$573$	−12.0000	−0.501307
$574$	−12.0000	−0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	12.0000	0.497416
$583$	4.00000	0.165663
$584$	8.00000	0.331042
$585$	0	0
$586$	26.0000	1.07405
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	−3.00000	−0.123718
$589$	−32.0000	−1.31854
$590$	0	0
$591$	10.0000	0.411345
$592$	6.00000	0.246598
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	12.0000	0.491539
$597$	−24.0000	−0.982255
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	8.00000	0.326056
$603$	−16.0000	−0.651570
$604$	−16.0000	−0.651031
$605$	0	0
$606$	4.00000	0.162489
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	−4.00000	−0.162221
$609$	8.00000	0.324176
$610$	0	0
$611$	8.00000	0.323645
$612$	2.00000	0.0808452
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	4.00000	0.161165
$617$	46.0000	1.85189	0.925945	−	0.377658i	$-0.123271\pi$
$617$	46.0000	1.85189	0.925945	+	0.377658i	$0.123271\pi$
$618$	6.00000	0.241355
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0	0
$621$	0	0
$622$	20.0000	0.801927
$623$	12.0000	0.480770
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−20.0000	−0.799361
$627$	−8.00000	−0.319489
$628$	10.0000	0.399043
$629$	12.0000	0.478471
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−8.00000	−0.318223
$633$	−4.00000	−0.158986
$634$	−18.0000	−0.714871
$635$	0	0
$636$	2.00000	0.0793052
$637$	3.00000	0.118864
$638$	8.00000	0.316723
$639$	−4.00000	−0.158238
$640$	0	0
$641$	−38.0000	−1.50091	−0.750455	−	0.660922i	$-0.770166\pi$
$641$	−38.0000	−1.50091	−0.750455	+	0.660922i	$0.770166\pi$
$642$	−4.00000	−0.157867
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	0	0
$646$	−8.00000	−0.314756
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	1.00000	0.0392837
$649$	20.0000	0.785069
$650$	0	0
$651$	16.0000	0.627089
$652$	−8.00000	−0.313304
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−18.0000	−0.703856
$655$	0	0
$656$	−6.00000	−0.234261
$657$	8.00000	0.312110
$658$	−16.0000	−0.623745
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	−12.0000	−0.466393
$663$	−2.00000	−0.0776736
$664$	12.0000	0.465690
$665$	0	0
$666$	6.00000	0.232495
$667$	0	0
$668$	12.0000	0.464294
$669$	−14.0000	−0.541271
$670$	0	0
$671$	−28.0000	−1.08093
$672$	2.00000	0.0771517
$673$	−36.0000	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$674$	20.0000	0.770371
$675$	0	0
$676$	1.00000	0.0384615
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	6.00000	0.230429
$679$	24.0000	0.921035
$680$	0	0
$681$	12.0000	0.459841
$682$	16.0000	0.612672
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−20.0000	−0.763604
$687$	−22.0000	−0.839352
$688$	4.00000	0.152499
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	14.0000	0.532200
$693$	4.00000	0.151947
$694$	12.0000	0.455514
$695$	0	0
$696$	4.00000	0.151620
$697$	−12.0000	−0.454532
$698$	14.0000	0.529908
$699$	−14.0000	−0.529529
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	−1.00000	−0.0377426
$703$	−24.0000	−0.905177
$704$	2.00000	0.0753778
$705$	0	0
$706$	−34.0000	−1.27961
$707$	8.00000	0.300871
$708$	10.0000	0.375823
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	4.00000	0.149696
$715$	0	0
$716$	−18.0000	−0.672692
$717$	−12.0000	−0.448148
$718$	32.0000	1.19423
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	−3.00000	−0.111648
$723$	−18.0000	−0.669427
$724$	−6.00000	−0.222988
$725$	0	0
$726$	−7.00000	−0.259794
$727$	34.0000	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$727$	34.0000	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	−14.0000	−0.517455
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	0	0
$737$	−32.0000	−1.17874
$738$	−6.00000	−0.220863
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	4.00000	0.146845
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−22.0000	−0.805477
$747$	12.0000	0.439057
$748$	4.00000	0.146254
$749$	−8.00000	−0.292314
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−8.00000	−0.291730
$753$	18.0000	0.655956
$754$	−4.00000	−0.145671
$755$	0	0
$756$	2.00000	0.0727393
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	0	0
$761$	38.0000	1.37750	0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000	1.37750	0.688749	+	0.724999i	$0.258160\pi$
$762$	6.00000	0.217357
$763$	−36.0000	−1.30329
$764$	−12.0000	−0.434145
$765$	0	0
$766$	24.0000	0.867155
$767$	−10.0000	−0.361079
$768$	1.00000	0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	22.0000	0.791285	0.395643	−	0.918405i	$-0.370522\pi$
$773$	22.0000	0.791285	0.395643	+	0.918405i	$0.370522\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	12.0000	0.430775
$777$	12.0000	0.430498
$778$	16.0000	0.573628
$779$	24.0000	0.859889
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	4.00000	0.142948
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−6.00000	−0.214013
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	10.0000	0.356235
$789$	−8.00000	−0.284808
$790$	0	0
$791$	12.0000	0.426671
$792$	2.00000	0.0710669
$793$	14.0000	0.497155
$794$	−18.0000	−0.638796
$795$	0	0
$796$	−24.0000	−0.850657
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	−8.00000	−0.283197
$799$	−16.0000	−0.566039
$800$	0	0
$801$	6.00000	0.212000
$802$	−18.0000	−0.635602
$803$	16.0000	0.564628
$804$	−16.0000	−0.564276
$805$	0	0
$806$	−8.00000	−0.281788
$807$	−20.0000	−0.704033
$808$	4.00000	0.140720
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	8.00000	0.280745
$813$	16.0000	0.561144
$814$	12.0000	0.420600
$815$	0	0
$816$	2.00000	0.0700140
$817$	−16.0000	−0.559769
$818$	−26.0000	−0.909069
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	−14.0000	−0.488306
$823$	34.0000	1.18517	0.592583	−	0.805510i	$-0.298108\pi$
$823$	34.0000	1.18517	0.592583	+	0.805510i	$0.298108\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	20.0000	0.695889
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	−1.00000	−0.0346688
$833$	−6.00000	−0.207888
$834$	4.00000	0.138509
$835$	0	0
$836$	−8.00000	−0.276686
$837$	8.00000	0.276520
$838$	−14.0000	−0.483622
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	34.0000	1.17172
$843$	−18.0000	−0.619953
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−8.00000	−0.275046
$847$	−14.0000	−0.481046
$848$	2.00000	0.0686803
$849$	16.0000	0.549119
$850$	0	0
$851$	0	0
$852$	−4.00000	−0.137038
$853$	−18.0000	−0.616308	−0.308154	−	0.951336i	$-0.599711\pi$
$853$	−18.0000	−0.616308	−0.308154	+	0.951336i	$0.599711\pi$
$854$	−28.0000	−0.958140
$855$	0	0
$856$	−4.00000	−0.136717
$857$	34.0000	1.16142	0.580709	−	0.814111i	$-0.302775\pi$
$857$	34.0000	1.16142	0.580709	+	0.814111i	$0.302775\pi$
$858$	−2.00000	−0.0682789
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	16.0000	0.544962
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	12.0000	0.407777
$867$	−13.0000	−0.441503
$868$	16.0000	0.543075
$869$	−16.0000	−0.542763
$870$	0	0
$871$	16.0000	0.542139
$872$	−18.0000	−0.609557
$873$	12.0000	0.406138
$874$	0	0
$875$	0	0
$876$	8.00000	0.270295
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	32.0000	1.07995
$879$	26.0000	0.876958
$880$	0	0
$881$	−10.0000	−0.336909	−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000	−0.336909	−0.168454	+	0.985709i	$0.553878\pi$
$882$	−3.00000	−0.101015
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	4.00000	0.134383
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	6.00000	0.201347
$889$	12.0000	0.402467
$890$	0	0
$891$	2.00000	0.0670025
$892$	−14.0000	−0.468755
$893$	32.0000	1.07084
$894$	12.0000	0.401340
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	30.0000	1.00111
$899$	32.0000	1.06726
$900$	0	0
$901$	4.00000	0.133259
$902$	−12.0000	−0.399556
$903$	8.00000	0.266223
$904$	6.00000	0.199557
$905$	0	0
$906$	−16.0000	−0.531564
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	12.0000	0.398234
$909$	4.00000	0.132672
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	−4.00000	−0.132453
$913$	24.0000	0.794284
$914$	−28.0000	−0.926158
$915$	0	0
$916$	−22.0000	−0.726900
$917$	−12.0000	−0.396275
$918$	2.00000	0.0660098
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	20.0000	0.658665
$923$	4.00000	0.131662
$924$	4.00000	0.131590
$925$	0	0
$926$	6.00000	0.197172
$927$	6.00000	0.197066
$928$	4.00000	0.131306
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	−14.0000	−0.458585
$933$	20.0000	0.654771
$934$	28.0000	0.916188
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	56.0000	1.82944	0.914720	−	0.404088i	$-0.132411\pi$
$937$	56.0000	1.82944	0.914720	+	0.404088i	$0.132411\pi$
$938$	−32.0000	−1.04484
$939$	−20.0000	−0.652675
$940$	0	0
$941$	−44.0000	−1.43436	−0.717180	−	0.696888i	$-0.754567\pi$
$941$	−44.0000	−1.43436	−0.717180	+	0.696888i	$0.754567\pi$
$942$	10.0000	0.325818
$943$	0	0
$944$	10.0000	0.325472
$945$	0	0
$946$	8.00000	0.260102
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	−8.00000	−0.259828
$949$	−8.00000	−0.259691
$950$	0	0
$951$	−18.0000	−0.583690
$952$	4.00000	0.129641
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−12.0000	−0.388108
$957$	8.00000	0.258603
$958$	36.0000	1.16311
$959$	−28.0000	−0.904167
$960$	0	0
$961$	33.0000	1.06452
$962$	−6.00000	−0.193448
$963$	−4.00000	−0.128898
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	−7.00000	−0.224989
$969$	−8.00000	−0.256997
$970$	0	0
$971$	50.0000	1.60458	0.802288	−	0.596937i	$-0.203616\pi$
$971$	50.0000	1.60458	0.802288	+	0.596937i	$0.203616\pi$
$972$	1.00000	0.0320750
$973$	8.00000	0.256468
$974$	−26.0000	−0.833094
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−8.00000	−0.255812
$979$	12.0000	0.383522
$980$	0	0
$981$	−18.0000	−0.574696
$982$	−22.0000	−0.702048
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	8.00000	0.254772
$987$	−16.0000	−0.509286
$988$	4.00000	0.127257
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.00000	0.254000
$993$	−12.0000	−0.380808
$994$	−8.00000	−0.253745
$995$	0	0
$996$	12.0000	0.380235
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	36.0000	1.13956
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.a.z.1.1		1
3.2	odd	2		5850.2.a.t.1.1		1
5.2	odd	4		390.2.e.c.79.2	yes	2
5.3	odd	4		390.2.e.c.79.1	✓	2
5.4	even	2		1950.2.a.c.1.1		1
15.2	even	4		1170.2.e.a.469.1		2
15.8	even	4		1170.2.e.a.469.2		2
15.14	odd	2		5850.2.a.bj.1.1		1
20.3	even	4		3120.2.l.g.1249.1		2
20.7	even	4		3120.2.l.g.1249.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.c.79.1	✓	2	5.3	odd	4
390.2.e.c.79.2	yes	2	5.2	odd	4
1170.2.e.a.469.1		2	15.2	even	4
1170.2.e.a.469.2		2	15.8	even	4
1950.2.a.c.1.1		1	5.4	even	2
1950.2.a.z.1.1		1	1.1	even	1	trivial
3120.2.l.g.1249.1		2	20.3	even	4
3120.2.l.g.1249.2		2	20.7	even	4
5850.2.a.t.1.1		1	3.2	odd	2
5850.2.a.bj.1.1		1	15.14	odd	2

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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more