LMFDB - Embedded newform 1050.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,Mod(1,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, 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("1050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.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:	$8.38429221223$
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	$\chi$	$=$	1050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} +1.00000 q^{21} -2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -5.00000 q^{29} +7.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -1.00000 q^{39} +7.00000 q^{41} -1.00000 q^{42} +11.0000 q^{43} +2.00000 q^{44} -1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} +1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} +5.00000 q^{58} -5.00000 q^{59} -3.00000 q^{61} -7.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +12.0000 q^{67} -3.00000 q^{68} -1.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} -2.00000 q^{77} +1.00000 q^{78} +10.0000 q^{79} +1.00000 q^{81} -7.00000 q^{82} +11.0000 q^{83} +1.00000 q^{84} -11.0000 q^{86} +5.00000 q^{87} -2.00000 q^{88} -10.0000 q^{89} -1.00000 q^{91} +1.00000 q^{92} -7.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−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	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−2.00000	−0.426401
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	−1.00000	−0.154303
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−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
$49$	1.00000	0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	1.00000	0.138675
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	5.00000	0.656532
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	0	0
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	−7.00000	−0.889001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−3.00000	−0.363803
$69$	−1.00000	−0.120386
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	1.00000	0.113228
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−7.00000	−0.773021
$83$	11.0000	1.20741	0.603703	−	0.797209i	$-0.293691\pi$
$83$	11.0000	1.20741	0.603703	+	0.797209i	$0.293691\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−11.0000	−1.18616
$87$	5.00000	0.536056
$88$	−2.00000	−0.213201
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	1.00000	0.104257
$93$	−7.00000	−0.725866
$94$	8.00000	0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−1.00000	−0.101015
$99$	2.00000	0.201008
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	−3.00000	−0.297044
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	−1.00000	−0.0962250
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	−1.00000	−0.0944911
$113$	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
$116$	−5.00000	−0.464238
$117$	1.00000	0.0924500
$118$	5.00000	0.460287
$119$	3.00000	0.275010
$120$	0	0
$121$	−7.00000	−0.636364
$122$	3.00000	0.271607
$123$	−7.00000	−0.631169
$124$	7.00000	0.628619
$125$	0	0
$126$	1.00000	0.0890871
$127$	22.0000	1.95218	0.976092	−	0.217357i	$-0.0697436\pi$
$127$	22.0000	1.95218	0.976092	+	0.217357i	$0.0697436\pi$
$128$	−1.00000	−0.0883883
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	3.00000	0.257248
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	1.00000	0.0851257
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−12.0000	−1.00702
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	−1.00000	−0.0824786
$148$	2.00000	0.164399
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	2.00000	0.161165
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−10.0000	−0.795557
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	−1.00000	−0.0785674
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	7.00000	0.546608
$165$	0	0
$166$	−11.0000	−0.853766
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	−1.00000	−0.0771517
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	11.0000	0.838742
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	−5.00000	−0.379049
$175$	0	0
$176$	2.00000	0.150756
$177$	5.00000	0.375823
$178$	10.0000	0.749532
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	1.00000	0.0741249
$183$	3.00000	0.221766
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	7.00000	0.513265
$187$	−6.00000	−0.438763
$188$	−8.00000	−0.583460
$189$	1.00000	0.0727393
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	17.0000	1.21120	0.605600	−	0.795769i	$-0.292933\pi$
$197$	17.0000	1.21120	0.605600	+	0.795769i	$0.292933\pi$
$198$	−2.00000	−0.142134
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	−12.0000	−0.844317
$203$	5.00000	0.350931
$204$	3.00000	0.210042
$205$	0	0
$206$	−11.0000	−0.766406
$207$	1.00000	0.0695048
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	1.00000	0.0686803
$213$	−12.0000	−0.822226
$214$	−2.00000	−0.136717
$215$	0	0
$216$	1.00000	0.0680414
$217$	−7.00000	−0.475191
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−3.00000	−0.201802
$222$	2.00000	0.134231
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−6.00000	−0.399114
$227$	17.0000	1.12833	0.564165	−	0.825662i	$-0.309198\pi$
$227$	17.0000	1.12833	0.564165	+	0.825662i	$0.309198\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	5.00000	0.328266
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−5.00000	−0.325472
$237$	−10.0000	−0.649570
$238$	−3.00000	−0.194461
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	−3.00000	−0.192055
$245$	0	0
$246$	7.00000	0.446304
$247$	0	0
$248$	−7.00000	−0.444500
$249$	−11.0000	−0.697097
$250$	0	0
$251$	7.00000	0.441836	0.220918	−	0.975292i	$-0.429095\pi$
$251$	7.00000	0.441836	0.220918	+	0.975292i	$0.429095\pi$
$252$	−1.00000	−0.0629941
$253$	2.00000	0.125739
$254$	−22.0000	−1.38040
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.0000	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$257$	−23.0000	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$258$	11.0000	0.684830
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−5.00000	−0.309492
$262$	8.00000	0.494242
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	0	0
$267$	10.0000	0.611990
$268$	12.0000	0.733017
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−3.00000	−0.181902
$273$	1.00000	0.0605228
$274$	−12.0000	−0.724947
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−20.0000	−1.19952
$279$	7.00000	0.419079
$280$	0	0
$281$	32.0000	1.90896	0.954480	−	0.298275i	$-0.0964112\pi$
$281$	32.0000	1.90896	0.954480	+	0.298275i	$0.0964112\pi$
$282$	−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	$0.342249\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−7.00000	−0.413197
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	6.00000	0.351123
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	−2.00000	−0.116248
$297$	−2.00000	−0.116052
$298$	−15.0000	−0.868927
$299$	1.00000	0.0578315
$300$	0	0
$301$	−11.0000	−0.634029
$302$	18.0000	1.03578
$303$	−12.0000	−0.689382
$304$	0	0
$305$	0	0
$306$	3.00000	0.171499
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	−2.00000	−0.113961
$309$	−11.0000	−0.625768
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	1.00000	0.0566139
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	10.0000	0.562544
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	1.00000	0.0560772
$319$	−10.0000	−0.559893
$320$	0	0
$321$	−2.00000	−0.111629
$322$	1.00000	0.0557278
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	−1.00000	−0.0553849
$327$	0	0
$328$	−7.00000	−0.386510
$329$	8.00000	0.441054
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	11.0000	0.603703
$333$	2.00000	0.109599
$334$	18.0000	0.984916
$335$	0	0
$336$	1.00000	0.0545545
$337$	27.0000	1.47078	0.735392	−	0.677642i	$-0.236998\pi$
$337$	27.0000	1.47078	0.735392	+	0.677642i	$0.236998\pi$
$338$	12.0000	0.652714
$339$	−6.00000	−0.325875
$340$	0	0
$341$	14.0000	0.758143
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−11.0000	−0.593080
$345$	0	0
$346$	24.0000	1.29025
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	5.00000	0.268028
$349$	15.0000	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−2.00000	−0.106600
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−5.00000	−0.265747
$355$	0	0
$356$	−10.0000	−0.529999
$357$	−3.00000	−0.158777
$358$	10.0000	0.528516
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	18.0000	0.946059
$363$	7.00000	0.367405
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	−3.00000	−0.156813
$367$	−23.0000	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$367$	−23.0000	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$368$	1.00000	0.0521286
$369$	7.00000	0.364405
$370$	0	0
$371$	−1.00000	−0.0519174
$372$	−7.00000	−0.362933
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	8.00000	0.412568
$377$	−5.00000	−0.257513
$378$	−1.00000	−0.0514344
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	0	0
$381$	−22.0000	−1.12709
$382$	3.00000	0.153493
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	11.0000	0.559161
$388$	2.00000	0.101535
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	−1.00000	−0.0505076
$393$	8.00000	0.403547
$394$	−17.0000	−0.856448
$395$	0	0
$396$	2.00000	0.100504
$397$	−13.0000	−0.652451	−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000	−0.652451	−0.326226	+	0.945292i	$0.605777\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	12.0000	0.598506
$403$	7.00000	0.348695
$404$	12.0000	0.597022
$405$	0	0
$406$	−5.00000	−0.248146
$407$	4.00000	0.198273
$408$	−3.00000	−0.148522
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	11.0000	0.541931
$413$	5.00000	0.246034
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−20.0000	−0.979404
$418$	0	0
$419$	25.0000	1.22133	0.610665	−	0.791889i	$-0.290902\pi$
$419$	25.0000	1.22133	0.610665	+	0.791889i	$0.290902\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	13.0000	0.632830
$423$	−8.00000	−0.388973
$424$	−1.00000	−0.0485643
$425$	0	0
$426$	12.0000	0.581402
$427$	3.00000	0.145180
$428$	2.00000	0.0966736
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	−1.00000	−0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	7.00000	0.336011
$435$	0	0
$436$	0	0
$437$	0	0
$438$	6.00000	0.286691
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	3.00000	0.142695
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	9.00000	0.426162
$447$	−15.0000	−0.709476
$448$	−1.00000	−0.0472456
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	14.0000	0.659234
$452$	6.00000	0.282216
$453$	18.0000	0.845714
$454$	−17.0000	−0.797850
$455$	0	0
$456$	0	0
$457$	−33.0000	−1.54367	−0.771837	−	0.635820i	$-0.780662\pi$
$457$	−33.0000	−1.54367	−0.771837	+	0.635820i	$0.780662\pi$
$458$	10.0000	0.467269
$459$	3.00000	0.140028
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	−2.00000	−0.0930484
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	24.0000	1.11178
$467$	−33.0000	−1.52706	−0.763529	−	0.645774i	$-0.776535\pi$
$467$	−33.0000	−1.52706	−0.763529	+	0.645774i	$0.776535\pi$
$468$	1.00000	0.0462250
$469$	−12.0000	−0.554109
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	5.00000	0.230144
$473$	22.0000	1.01156
$474$	10.0000	0.459315
$475$	0	0
$476$	3.00000	0.137505
$477$	1.00000	0.0457869
$478$	0	0
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−22.0000	−1.00207
$483$	1.00000	0.0455016
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	3.00000	0.135804
$489$	−1.00000	−0.0452216
$490$	0	0
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	−7.00000	−0.315584
$493$	15.0000	0.675566
$494$	0	0
$495$	0	0
$496$	7.00000	0.314309
$497$	−12.0000	−0.538274
$498$	11.0000	0.492922
$499$	25.0000	1.11915	0.559577	−	0.828778i	$-0.310964\pi$
$499$	25.0000	1.11915	0.559577	+	0.828778i	$0.310964\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	−7.00000	−0.312425
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	−2.00000	−0.0889108
$507$	12.0000	0.532939
$508$	22.0000	0.976092
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	23.0000	1.01449
$515$	0	0
$516$	−11.0000	−0.484248
$517$	−16.0000	−0.703679
$518$	2.00000	0.0878750
$519$	24.0000	1.05348
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	5.00000	0.218844
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	−21.0000	−0.915644
$527$	−21.0000	−0.914774
$528$	−2.00000	−0.0870388
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−5.00000	−0.216982
$532$	0	0
$533$	7.00000	0.303204
$534$	−10.0000	−0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	10.0000	0.431532
$538$	10.0000	0.431131
$539$	2.00000	0.0861461
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	28.0000	1.20270
$543$	18.0000	0.772454
$544$	3.00000	0.128624
$545$	0	0
$546$	−1.00000	−0.0427960
$547$	37.0000	1.58201	0.791003	−	0.611812i	$-0.209559\pi$
$547$	37.0000	1.58201	0.791003	+	0.611812i	$0.209559\pi$
$548$	12.0000	0.512615
$549$	−3.00000	−0.128037
$550$	0	0
$551$	0	0
$552$	1.00000	0.0425628
$553$	−10.0000	−0.425243
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	20.0000	0.848189
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−7.00000	−0.296334
$559$	11.0000	0.465250
$560$	0	0
$561$	6.00000	0.253320
$562$	−32.0000	−1.34984
$563$	31.0000	1.30649	0.653247	−	0.757145i	$-0.273406\pi$
$563$	31.0000	1.30649	0.653247	+	0.757145i	$0.273406\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	−16.0000	−0.672530
$567$	−1.00000	−0.0419961
$568$	−12.0000	−0.503509
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	2.00000	0.0836242
$573$	3.00000	0.125327
$574$	7.00000	0.292174
$575$	0	0
$576$	1.00000	0.0416667
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	8.00000	0.332756
$579$	14.0000	0.581820
$580$	0	0
$581$	−11.0000	−0.456357
$582$	2.00000	0.0829027
$583$	2.00000	0.0828315
$584$	−6.00000	−0.248282
$585$	0	0
$586$	24.0000	0.991431
$587$	27.0000	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$587$	27.0000	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$588$	−1.00000	−0.0412393
$589$	0	0
$590$	0	0
$591$	−17.0000	−0.699287
$592$	2.00000	0.0821995
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	15.0000	0.614424
$597$	0	0
$598$	−1.00000	−0.0408930
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	11.0000	0.448327
$603$	12.0000	0.488678
$604$	−18.0000	−0.732410
$605$	0	0
$606$	12.0000	0.487467
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	−8.00000	−0.323645
$612$	−3.00000	−0.121268
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	2.00000	0.0805823
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	11.0000	0.442485
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	18.0000	0.721734
$623$	10.0000	0.400642
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	2.00000	0.0798087
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−38.0000	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	−10.0000	−0.397779
$633$	13.0000	0.516704
$634$	3.00000	0.119145
$635$	0	0
$636$	−1.00000	−0.0396526
$637$	1.00000	0.0396214
$638$	10.0000	0.395904
$639$	12.0000	0.474713
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	2.00000	0.0789337
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	0	0
$647$	22.0000	0.864909	0.432455	−	0.901656i	$-0.357648\pi$
$647$	22.0000	0.864909	0.432455	+	0.901656i	$0.357648\pi$
$648$	−1.00000	−0.0392837
$649$	−10.0000	−0.392534
$650$	0	0
$651$	7.00000	0.274352
$652$	1.00000	0.0391630
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	0	0
$656$	7.00000	0.273304
$657$	6.00000	0.234082
$658$	−8.00000	−0.311872
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	−17.0000	−0.660724
$663$	3.00000	0.116510
$664$	−11.0000	−0.426883
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−5.00000	−0.193601
$668$	−18.0000	−0.696441
$669$	9.00000	0.347960
$670$	0	0
$671$	−6.00000	−0.231627
$672$	−1.00000	−0.0385758
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	−27.0000	−1.04000
$675$	0	0
$676$	−12.0000	−0.461538
$677$	32.0000	1.22986	0.614930	−	0.788582i	$-0.289184\pi$
$677$	32.0000	1.22986	0.614930	+	0.788582i	$0.289184\pi$
$678$	6.00000	0.230429
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−17.0000	−0.651441
$682$	−14.0000	−0.536088
$683$	46.0000	1.76014	0.880071	−	0.474843i	$-0.157495\pi$
$683$	46.0000	1.76014	0.880071	+	0.474843i	$0.157495\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	10.0000	0.381524
$688$	11.0000	0.419371
$689$	1.00000	0.0380970
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	−24.0000	−0.912343
$693$	−2.00000	−0.0759737
$694$	−2.00000	−0.0759190
$695$	0	0
$696$	−5.00000	−0.189525
$697$	−21.0000	−0.795432
$698$	−15.0000	−0.567758
$699$	24.0000	0.907763
$700$	0	0
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	1.00000	0.0377426
$703$	0	0
$704$	2.00000	0.0753778
$705$	0	0
$706$	−6.00000	−0.225813
$707$	−12.0000	−0.451306
$708$	5.00000	0.187912
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	10.0000	0.374766
$713$	7.00000	0.262152
$714$	3.00000	0.112272
$715$	0	0
$716$	−10.0000	−0.373718
$717$	0	0
$718$	−25.0000	−0.932992
$719$	−10.0000	−0.372937	−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000	−0.372937	−0.186469	+	0.982461i	$0.559704\pi$
$720$	0	0
$721$	−11.0000	−0.409661
$722$	19.0000	0.707107
$723$	−22.0000	−0.818189
$724$	−18.0000	−0.668965
$725$	0	0
$726$	−7.00000	−0.259794
$727$	−13.0000	−0.482143	−0.241072	−	0.970507i	$-0.577499\pi$
$727$	−13.0000	−0.482143	−0.241072	+	0.970507i	$0.577499\pi$
$728$	1.00000	0.0370625
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.0000	−1.22055
$732$	3.00000	0.110883
$733$	11.0000	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$733$	11.0000	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$734$	23.0000	0.848945
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	24.0000	0.884051
$738$	−7.00000	−0.257674
$739$	−25.0000	−0.919640	−0.459820	−	0.888012i	$-0.652086\pi$
$739$	−25.0000	−0.919640	−0.459820	+	0.888012i	$0.652086\pi$
$740$	0	0
$741$	0	0
$742$	1.00000	0.0367112
$743$	−29.0000	−1.06391	−0.531953	−	0.846774i	$-0.678542\pi$
$743$	−29.0000	−1.06391	−0.531953	+	0.846774i	$0.678542\pi$
$744$	7.00000	0.256632
$745$	0	0
$746$	24.0000	0.878702
$747$	11.0000	0.402469
$748$	−6.00000	−0.219382
$749$	−2.00000	−0.0730784
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−8.00000	−0.291730
$753$	−7.00000	−0.255094
$754$	5.00000	0.182089
$755$	0	0
$756$	1.00000	0.0363696
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	25.0000	0.908041
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	22.0000	0.796976
$763$	0	0
$764$	−3.00000	−0.108536
$765$	0	0
$766$	4.00000	0.144526
$767$	−5.00000	−0.180540
$768$	−1.00000	−0.0360844
$769$	−20.0000	−0.721218	−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000	−0.721218	−0.360609	+	0.932717i	$0.617431\pi$
$770$	0	0
$771$	23.0000	0.828325
$772$	−14.0000	−0.503871
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	−11.0000	−0.395387
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	2.00000	0.0717496
$778$	−10.0000	−0.358517
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	3.00000	0.107280
$783$	5.00000	0.178685
$784$	1.00000	0.0357143
$785$	0	0
$786$	−8.00000	−0.285351
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	17.0000	0.605600
$789$	−21.0000	−0.747620
$790$	0	0
$791$	−6.00000	−0.213335
$792$	−2.00000	−0.0710669
$793$	−3.00000	−0.106533
$794$	13.0000	0.461353
$795$	0	0
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	−10.0000	−0.353333
$802$	8.00000	0.282490
$803$	12.0000	0.423471
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−7.00000	−0.246564
$807$	10.0000	0.352017
$808$	−12.0000	−0.422159
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−18.0000	−0.632065	−0.316033	−	0.948748i	$-0.602351\pi$
$811$	−18.0000	−0.632065	−0.316033	+	0.948748i	$0.602351\pi$
$812$	5.00000	0.175466
$813$	28.0000	0.982003
$814$	−4.00000	−0.140200
$815$	0	0
$816$	3.00000	0.105021
$817$	0	0
$818$	10.0000	0.349642
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	12.0000	0.418548
$823$	−54.0000	−1.88232	−0.941161	−	0.337959i	$-0.890263\pi$
$823$	−54.0000	−1.88232	−0.941161	+	0.337959i	$0.890263\pi$
$824$	−11.0000	−0.383203
$825$	0	0
$826$	−5.00000	−0.173972
$827$	2.00000	0.0695468	0.0347734	−	0.999395i	$-0.488929\pi$
$827$	2.00000	0.0695468	0.0347734	+	0.999395i	$0.488929\pi$
$828$	1.00000	0.0347524
$829$	−5.00000	−0.173657	−0.0868286	−	0.996223i	$-0.527673\pi$
$829$	−5.00000	−0.173657	−0.0868286	+	0.996223i	$0.527673\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	1.00000	0.0346688
$833$	−3.00000	−0.103944
$834$	20.0000	0.692543
$835$	0	0
$836$	0	0
$837$	−7.00000	−0.241955
$838$	−25.0000	−0.863611
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−22.0000	−0.758170
$843$	−32.0000	−1.10214
$844$	−13.0000	−0.447478
$845$	0	0
$846$	8.00000	0.275046
$847$	7.00000	0.240523
$848$	1.00000	0.0343401
$849$	−16.0000	−0.549119
$850$	0	0
$851$	2.00000	0.0685591
$852$	−12.0000	−0.411113
$853$	−29.0000	−0.992941	−0.496471	−	0.868054i	$-0.665371\pi$
$853$	−29.0000	−0.992941	−0.496471	+	0.868054i	$0.665371\pi$
$854$	−3.00000	−0.102658
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	2.00000	0.0682789
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	7.00000	0.238559
$862$	3.00000	0.102180
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	14.0000	0.475739
$867$	8.00000	0.271694
$868$	−7.00000	−0.237595
$869$	20.0000	0.678454
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	−6.00000	−0.202721
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−35.0000	−1.18119
$879$	24.0000	0.809500
$880$	0	0
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	−1.00000	−0.0336718
$883$	−39.0000	−1.31245	−0.656227	−	0.754563i	$-0.727849\pi$
$883$	−39.0000	−1.31245	−0.656227	+	0.754563i	$0.727849\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	−16.0000	−0.537531
$887$	−58.0000	−1.94745	−0.973725	−	0.227728i	$-0.926870\pi$
$887$	−58.0000	−1.94745	−0.973725	+	0.227728i	$0.926870\pi$
$888$	2.00000	0.0671156
$889$	−22.0000	−0.737856
$890$	0	0
$891$	2.00000	0.0670025
$892$	−9.00000	−0.301342
$893$	0	0
$894$	15.0000	0.501675
$895$	0	0
$896$	1.00000	0.0334077
$897$	−1.00000	−0.0333890
$898$	−20.0000	−0.667409
$899$	−35.0000	−1.16732
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	−14.0000	−0.466149
$903$	11.0000	0.366057
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−18.0000	−0.598010
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	17.0000	0.564165
$909$	12.0000	0.398015
$910$	0	0
$911$	−53.0000	−1.75597	−0.877984	−	0.478690i	$-0.841112\pi$
$911$	−53.0000	−1.75597	−0.877984	+	0.478690i	$0.841112\pi$
$912$	0	0
$913$	22.0000	0.728094
$914$	33.0000	1.09154
$915$	0	0
$916$	−10.0000	−0.330409
$917$	8.00000	0.264183
$918$	−3.00000	−0.0990148
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	−12.0000	−0.395199
$923$	12.0000	0.394985
$924$	2.00000	0.0657952
$925$	0	0
$926$	4.00000	0.131448
$927$	11.0000	0.361287
$928$	5.00000	0.164133
$929$	−45.0000	−1.47640	−0.738201	−	0.674581i	$-0.764324\pi$
$929$	−45.0000	−1.47640	−0.738201	+	0.674581i	$0.764324\pi$
$930$	0	0
$931$	0	0
$932$	−24.0000	−0.786146
$933$	18.0000	0.589294
$934$	33.0000	1.07979
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	12.0000	0.391814
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	2.00000	0.0651635
$943$	7.00000	0.227951
$944$	−5.00000	−0.162736
$945$	0	0
$946$	−22.0000	−0.715282
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	−10.0000	−0.324785
$949$	6.00000	0.194768
$950$	0	0
$951$	3.00000	0.0972817
$952$	−3.00000	−0.0972306
$953$	16.0000	0.518291	0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000	0.518291	0.259145	+	0.965838i	$0.416559\pi$
$954$	−1.00000	−0.0323762
$955$	0	0
$956$	0	0
$957$	10.0000	0.323254
$958$	20.0000	0.646171
$959$	−12.0000	−0.387500
$960$	0	0
$961$	18.0000	0.580645
$962$	−2.00000	−0.0644826
$963$	2.00000	0.0644491
$964$	22.0000	0.708572
$965$	0	0
$966$	−1.00000	−0.0321745
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−1.00000	−0.0320750
$973$	−20.0000	−0.641171
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−3.00000	−0.0960277
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	1.00000	0.0319765
$979$	−20.0000	−0.639203
$980$	0	0
$981$	0	0
$982$	−2.00000	−0.0638226
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	7.00000	0.223152
$985$	0	0
$986$	−15.0000	−0.477697
$987$	−8.00000	−0.254643
$988$	0	0
$989$	11.0000	0.349780
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−7.00000	−0.222250
$993$	−17.0000	−0.539479
$994$	12.0000	0.380617
$995$	0	0
$996$	−11.0000	−0.348548
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	−25.0000	−0.791361
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.a.b.1.1	✓	1
3.2	odd	2		3150.2.a.y.1.1		1
4.3	odd	2		8400.2.a.ck.1.1		1
5.2	odd	4		1050.2.g.i.799.1		2
5.3	odd	4		1050.2.g.i.799.2		2
5.4	even	2		1050.2.a.r.1.1	yes	1
7.6	odd	2		7350.2.a.bi.1.1		1
15.2	even	4		3150.2.g.h.2899.2		2
15.8	even	4		3150.2.g.h.2899.1		2
15.14	odd	2		3150.2.a.n.1.1		1
20.19	odd	2		8400.2.a.d.1.1		1
35.34	odd	2		7350.2.a.cb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.2.a.b.1.1	✓	1	1.1	even	1	trivial
1050.2.a.r.1.1	yes	1	5.4	even	2
1050.2.g.i.799.1		2	5.2	odd	4
1050.2.g.i.799.2		2	5.3	odd	4
3150.2.a.n.1.1		1	15.14	odd	2
3150.2.a.y.1.1		1	3.2	odd	2
3150.2.g.h.2899.1		2	15.8	even	4
3150.2.g.h.2899.2		2	15.2	even	4
7350.2.a.bi.1.1		1	7.6	odd	2
7350.2.a.cb.1.1		1	35.34	odd	2
8400.2.a.d.1.1		1	20.19	odd	2
8400.2.a.ck.1.1		1	4.3	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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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