LMFDB - Embedded newform 5070.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$1$
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	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} -6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{28} -1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} -1.00000 q^{40} -6.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -1.00000 q^{45} -6.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -6.00000 q^{53} +1.00000 q^{54} -2.00000 q^{56} -2.00000 q^{57} -1.00000 q^{60} +14.0000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} -6.00000 q^{69} +2.00000 q^{70} +1.00000 q^{72} +4.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -2.00000 q^{76} -16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} -2.00000 q^{84} -4.00000 q^{86} +6.00000 q^{89} -1.00000 q^{90} -6.00000 q^{92} -8.00000 q^{93} +2.00000 q^{95} +1.00000 q^{96} +4.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	−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
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−2.00000	−0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−1.00000	−0.223607
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−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	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	−6.00000	−0.884652
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	0	0
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	−2.00000	−0.264906
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−1.00000	−0.129099
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−8.00000	−1.01600
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	2.00000	0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	−1.00000	−0.111803
$81$	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
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−8.00000	−0.829561
$94$	0	0
$95$	2.00000	0.205196
$96$	1.00000	0.102062
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	1.00000	0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	−2.00000	−0.188982
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	−2.00000	−0.187317
$115$	6.00000	0.559503
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−11.0000	−1.00000
$122$	14.0000	1.26750
$123$	−6.00000	−0.541002
$124$	−8.00000	−0.718421
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	4.00000	0.345547
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−6.00000	−0.510754
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	−3.00000	−0.247436
$148$	−2.00000	−0.164399
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	1.00000	0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−16.0000	−1.27289
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	12.0000	0.945732
$162$	1.00000	0.0785674
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−2.00000	−0.154303
$169$	0	0
$170$	0	0
$171$	−2.00000	−0.152944
$172$	−4.00000	−0.304997
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	−1.00000	−0.0745356
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	−6.00000	−0.442326
$185$	2.00000	0.147043
$186$	−8.00000	−0.586588
$187$	0	0
$188$	0	0
$189$	−2.00000	−0.145479
$190$	2.00000	0.145095
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	−12.0000	−0.844317
$203$	0	0
$204$	0	0
$205$	6.00000	0.419058
$206$	−16.0000	−1.11477
$207$	−6.00000	−0.417029
$208$	0	0
$209$	0	0
$210$	2.00000	0.138013
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−12.0000	−0.820303
$215$	4.00000	0.272798
$216$	1.00000	0.0680414
$217$	16.0000	1.08615
$218$	4.00000	0.270914
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	−2.00000	−0.134231
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	−2.00000	−0.133631
$225$	1.00000	0.0666667
$226$	12.0000	0.798228
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−2.00000	−0.132453
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−1.00000	−0.0645497
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	3.00000	0.191663
$246$	−6.00000	−0.382546
$247$	0	0
$248$	−8.00000	−0.508001
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	−4.00000	−0.249029
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	6.00000	0.370681
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	4.00000	0.245256
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	−1.00000	−0.0608581
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−6.00000	−0.361158
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	8.00000	0.479808
$279$	−8.00000	−0.478947
$280$	2.00000	0.119523
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	12.0000	0.708338
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	0	0
$291$	4.00000	0.234484
$292$	4.00000	0.234082
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−18.0000	−1.04271
$299$	0	0
$300$	1.00000	0.0577350
$301$	8.00000	0.461112
$302$	−8.00000	−0.460348
$303$	−12.0000	−0.689382
$304$	−2.00000	−0.114708
$305$	−14.0000	−0.801638
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	8.00000	0.454369
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	2.00000	0.112867
$315$	2.00000	0.112687
$316$	−16.0000	−0.900070
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−12.0000	−0.669775
$322$	12.0000	0.668734
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	4.00000	0.221201
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	12.0000	0.658586
$333$	−2.00000	−0.109599
$334$	−12.0000	−0.656611
$335$	−4.00000	−0.218543
$336$	−2.00000	−0.109109
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	0	0
$342$	−2.00000	−0.108148
$343$	20.0000	1.07990
$344$	−4.00000	−0.215666
$345$	6.00000	0.323029
$346$	18.0000	0.967686
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	−6.00000	−0.317110
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−1.00000	−0.0527046
$361$	−15.0000	−0.789474
$362$	2.00000	0.105118
$363$	−11.0000	−0.577350
$364$	0	0
$365$	−4.00000	−0.209370
$366$	14.0000	0.731792
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−6.00000	−0.312772
$369$	−6.00000	−0.312348
$370$	2.00000	0.103975
$371$	12.0000	0.623009
$372$	−8.00000	−0.414781
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0	0
$378$	−2.00000	−0.102869
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	2.00000	0.102598
$381$	−16.0000	−0.819705
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−8.00000	−0.407189
$387$	−4.00000	−0.203331
$388$	4.00000	0.203069
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	0	0
$392$	−3.00000	−0.151523
$393$	6.00000	0.302660
$394$	−18.0000	−0.906827
$395$	16.0000	0.805047
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	−16.0000	−0.802008
$399$	4.00000	0.200250
$400$	1.00000	0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	−12.0000	−0.597022
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	6.00000	0.296319
$411$	−6.00000	−0.295958
$412$	−16.0000	−0.788263
$413$	0	0
$414$	−6.00000	−0.294884
$415$	−12.0000	−0.589057
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	2.00000	0.0975900
$421$	−32.0000	−1.55958	−0.779792	−	0.626038i	$-0.784675\pi$
$421$	−32.0000	−1.55958	−0.779792	+	0.626038i	$0.784675\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	−28.0000	−1.35501
$428$	−12.0000	−0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	1.00000	0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	4.00000	0.191565
$437$	12.0000	0.574038
$438$	4.00000	0.191127
$439$	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
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−2.00000	−0.0949158
$445$	−6.00000	−0.284427
$446$	10.0000	0.473514
$447$	−18.0000	−0.851371
$448$	−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	$0.249646\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	12.0000	0.564433
$453$	−8.00000	−0.375873
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	4.00000	0.186908
$459$	0	0
$460$	6.00000	0.279751
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	0	0
$463$	10.0000	0.464739	0.232370	−	0.972628i	$-0.425352\pi$
$463$	10.0000	0.464739	0.232370	+	0.972628i	$0.425352\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	0	0
$474$	−16.0000	−0.734904
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−26.0000	−1.18427
$483$	12.0000	0.546019
$484$	−11.0000	−0.500000
$485$	−4.00000	−0.181631
$486$	1.00000	0.0453609
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	14.0000	0.633750
$489$	16.0000	0.723545
$490$	3.00000	0.135526
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	−6.00000	−0.270501
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	12.0000	0.537733
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	−1.00000	−0.0447214
$501$	−12.0000	−0.536120
$502$	−30.0000	−1.33897
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	−2.00000	−0.0890871
$505$	12.0000	0.533993
$506$	0	0
$507$	0	0
$508$	−16.0000	−0.709885
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	1.00000	0.0441942
$513$	−2.00000	−0.0883022
$514$	−12.0000	−0.529297
$515$	16.0000	0.705044
$516$	−4.00000	−0.176090
$517$	0	0
$518$	4.00000	0.175750
$519$	18.0000	0.790112
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	6.00000	0.262111
$525$	−2.00000	−0.0872872
$526$	30.0000	1.30806
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	6.00000	0.260623
$531$	0	0
$532$	4.00000	0.173422
$533$	0	0
$534$	6.00000	0.259645
$535$	12.0000	0.518805
$536$	4.00000	0.172774
$537$	−6.00000	−0.258919
$538$	12.0000	0.517357
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	−8.00000	−0.343629
$543$	2.00000	0.0858282
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−6.00000	−0.256307
$549$	14.0000	0.597505
$550$	0	0
$551$	0	0
$552$	−6.00000	−0.255377
$553$	32.0000	1.36078
$554$	26.0000	1.10463
$555$	2.00000	0.0848953
$556$	8.00000	0.339276
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−8.00000	−0.338667
$559$	0	0
$560$	2.00000	0.0845154
$561$	0	0
$562$	30.0000	1.26547
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	−28.0000	−1.17693
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	2.00000	0.0837708
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	0	0
$574$	12.0000	0.500870
$575$	−6.00000	−0.250217
$576$	1.00000	0.0416667
$577$	16.0000	0.666089	0.333044	−	0.942911i	$-0.391924\pi$
$577$	16.0000	0.666089	0.333044	+	0.942911i	$0.391924\pi$
$578$	−17.0000	−0.707107
$579$	−8.00000	−0.332469
$580$	0	0
$581$	−24.0000	−0.995688
$582$	4.00000	0.165805
$583$	0	0
$584$	4.00000	0.165521
$585$	0	0
$586$	6.00000	0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−3.00000	−0.123718
$589$	16.0000	0.659269
$590$	0	0
$591$	−18.0000	−0.740421
$592$	−2.00000	−0.0821995
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	−16.0000	−0.654836
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	1.00000	0.0408248
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	8.00000	0.326056
$603$	4.00000	0.162893
$604$	−8.00000	−0.325515
$605$	11.0000	0.447214
$606$	−12.0000	−0.487467
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	−14.0000	−0.566843
$611$	0	0
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	4.00000	0.161427
$615$	6.00000	0.241943
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−16.0000	−0.643614
$619$	−38.0000	−1.52735	−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000	−1.52735	−0.763674	+	0.645601i	$0.776607\pi$
$620$	8.00000	0.321288
$621$	−6.00000	−0.240772
$622$	24.0000	0.962312
$623$	−12.0000	−0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	2.00000	0.0798087
$629$	0	0
$630$	2.00000	0.0796819
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	−16.0000	−0.636446
$633$	−4.00000	−0.158986
$634$	18.0000	0.714871
$635$	16.0000	0.634941
$636$	−6.00000	−0.237915
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	−12.0000	−0.473602
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	12.0000	0.472866
$645$	4.00000	0.157500
$646$	0	0
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	16.0000	0.627089
$652$	16.0000	0.626608
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	4.00000	0.156412
$655$	−6.00000	−0.234439
$656$	−6.00000	−0.234261
$657$	4.00000	0.156055
$658$	0	0
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	0	0
$661$	16.0000	0.622328	0.311164	−	0.950356i	$-0.399281\pi$
$661$	16.0000	0.622328	0.311164	+	0.950356i	$0.399281\pi$
$662$	−2.00000	−0.0777322
$663$	0	0
$664$	12.0000	0.465690
$665$	−4.00000	−0.155113
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	−12.0000	−0.464294
$669$	10.0000	0.386622
$670$	−4.00000	−0.154533
$671$	0	0
$672$	−2.00000	−0.0771517
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	14.0000	0.539260
$675$	1.00000	0.0384900
$676$	0	0
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	12.0000	0.460857
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	−2.00000	−0.0764719
$685$	6.00000	0.229248
$686$	20.0000	0.763604
$687$	4.00000	0.152610
$688$	−4.00000	−0.152499
$689$	0	0
$690$	6.00000	0.228416
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	24.0000	0.911028
$695$	−8.00000	−0.303457
$696$	0	0
$697$	0	0
$698$	−20.0000	−0.757011
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	−6.00000	−0.225813
$707$	24.0000	0.902613
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	6.00000	0.224860
$713$	48.0000	1.79761
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	0	0
$718$	24.0000	0.895672
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	−1.00000	−0.0372678
$721$	32.0000	1.19174
$722$	−15.0000	−0.558242
$723$	−26.0000	−0.966950
$724$	2.00000	0.0743294
$725$	0	0
$726$	−11.0000	−0.408248
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	0	0
$732$	14.0000	0.517455
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	8.00000	0.295285
$735$	3.00000	0.110657
$736$	−6.00000	−0.221163
$737$	0	0
$738$	−6.00000	−0.220863
$739$	−38.0000	−1.39785	−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000	−1.39785	−0.698926	+	0.715194i	$0.746338\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	12.0000	0.440534
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	−8.00000	−0.293294
$745$	18.0000	0.659469
$746$	26.0000	0.951928
$747$	12.0000	0.439057
$748$	0	0
$749$	24.0000	0.876941
$750$	−1.00000	−0.0365148
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	0	0
$755$	8.00000	0.291150
$756$	−2.00000	−0.0727393
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	34.0000	1.23494
$759$	0	0
$760$	2.00000	0.0725476
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	−16.0000	−0.579619
$763$	−8.00000	−0.289619
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	0	0
$768$	1.00000	0.0360844
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	−8.00000	−0.287926
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−4.00000	−0.143777
$775$	−8.00000	−0.287368
$776$	4.00000	0.143592
$777$	4.00000	0.143499
$778$	36.0000	1.29066
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−3.00000	−0.107143
$785$	−2.00000	−0.0713831
$786$	6.00000	0.214013
$787$	16.0000	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$787$	16.0000	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$788$	−18.0000	−0.641223
$789$	30.0000	1.06803
$790$	16.0000	0.569254
$791$	−24.0000	−0.853342
$792$	0	0
$793$	0	0
$794$	−26.0000	−0.922705
$795$	6.00000	0.212798
$796$	−16.0000	−0.567105
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	4.00000	0.141598
$799$	0	0
$800$	1.00000	0.0353553
$801$	6.00000	0.212000
$802$	18.0000	0.635602
$803$	0	0
$804$	4.00000	0.141069
$805$	−12.0000	−0.422944
$806$	0	0
$807$	12.0000	0.422420
$808$	−12.0000	−0.422159
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−1.00000	−0.0351364
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	0	0
$815$	−16.0000	−0.560456
$816$	0	0
$817$	8.00000	0.279885
$818$	−14.0000	−0.489499
$819$	0	0
$820$	6.00000	0.209529
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−6.00000	−0.209274
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	−6.00000	−0.208514
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	−12.0000	−0.416526
$831$	26.0000	0.901930
$832$	0	0
$833$	0	0
$834$	8.00000	0.277017
$835$	12.0000	0.415277
$836$	0	0
$837$	−8.00000	−0.276520
$838$	6.00000	0.207267
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	2.00000	0.0690066
$841$	−29.0000	−1.00000
$842$	−32.0000	−1.10279
$843$	30.0000	1.03325
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	22.0000	0.755929
$848$	−6.00000	−0.206041
$849$	−28.0000	−0.960958
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−28.0000	−0.958140
$855$	2.00000	0.0683986
$856$	−12.0000	−0.410152
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	4.00000	0.136399
$861$	12.0000	0.408959
$862$	−24.0000	−0.817443
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	1.00000	0.0340207
$865$	−18.0000	−0.612018
$866$	14.0000	0.475739
$867$	−17.0000	−0.577350
$868$	16.0000	0.543075
$869$	0	0
$870$	0	0
$871$	0	0
$872$	4.00000	0.135457
$873$	4.00000	0.135379
$874$	12.0000	0.405906
$875$	2.00000	0.0676123
$876$	4.00000	0.135147
$877$	58.0000	1.95852	0.979260	−	0.202606i	$-0.0649409\pi$
$877$	58.0000	1.95852	0.979260	+	0.202606i	$0.0649409\pi$
$878$	32.0000	1.07995
$879$	6.00000	0.202375
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	−3.00000	−0.101015
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	−2.00000	−0.0671156
$889$	32.0000	1.07325
$890$	−6.00000	−0.201120
$891$	0	0
$892$	10.0000	0.334825
$893$	0	0
$894$	−18.0000	−0.602010
$895$	6.00000	0.200558
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	30.0000	1.00111
$899$	0	0
$900$	1.00000	0.0333333
$901$	0	0
$902$	0	0
$903$	8.00000	0.266223
$904$	12.0000	0.399114
$905$	−2.00000	−0.0664822
$906$	−8.00000	−0.265782
$907$	−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
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	−2.00000	−0.0662266
$913$	0	0
$914$	28.0000	0.926158
$915$	−14.0000	−0.462826
$916$	4.00000	0.132164
$917$	−12.0000	−0.396275
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	6.00000	0.197814
$921$	4.00000	0.131804
$922$	42.0000	1.38320
$923$	0	0
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	10.0000	0.328620
$927$	−16.0000	−0.525509
$928$	0	0
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	8.00000	0.262330
$931$	6.00000	0.196642
$932$	0	0
$933$	24.0000	0.785725
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−8.00000	−0.261209
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	2.00000	0.0651635
$943$	36.0000	1.17232
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	−16.0000	−0.519656
$949$	0	0
$950$	−2.00000	−0.0648886
$951$	18.0000	0.583690
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	−1.00000	−0.0322749
$961$	33.0000	1.06452
$962$	0	0
$963$	−12.0000	−0.386695
$964$	−26.0000	−0.837404
$965$	8.00000	0.257529
$966$	12.0000	0.386094
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	−4.00000	−0.128432
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	1.00000	0.0320750
$973$	−16.0000	−0.512936
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	14.0000	0.448129
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	16.0000	0.511624
$979$	0	0
$980$	3.00000	0.0958315
$981$	4.00000	0.127710
$982$	6.00000	0.191468
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−6.00000	−0.191273
$985$	18.0000	0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.0000	0.763156
$990$	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
$993$	−2.00000	−0.0634681
$994$	0	0
$995$	16.0000	0.507234
$996$	12.0000	0.380235
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	−14.0000	−0.443162
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.t.1.1		1
13.5	odd	4		5070.2.b.m.1351.1		2
13.8	odd	4		5070.2.b.m.1351.2		2
13.12	even	2		390.2.a.d.1.1	✓	1
39.38	odd	2		1170.2.a.k.1.1		1
52.51	odd	2		3120.2.a.j.1.1		1
65.12	odd	4		1950.2.e.d.1249.1		2
65.38	odd	4		1950.2.e.d.1249.2		2
65.64	even	2		1950.2.a.o.1.1		1
156.155	even	2		9360.2.a.g.1.1		1
195.38	even	4		5850.2.e.o.5149.1		2
195.77	even	4		5850.2.e.o.5149.2		2
195.194	odd	2		5850.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.d.1.1	✓	1	13.12	even	2
1170.2.a.k.1.1		1	39.38	odd	2
1950.2.a.o.1.1		1	65.64	even	2
1950.2.e.d.1249.1		2	65.12	odd	4
1950.2.e.d.1249.2		2	65.38	odd	4
3120.2.a.j.1.1		1	52.51	odd	2
5070.2.a.t.1.1		1	1.1	even	1	trivial
5070.2.b.m.1351.1		2	13.5	odd	4
5070.2.b.m.1351.2		2	13.8	odd	4
5850.2.a.g.1.1		1	195.194	odd	2
5850.2.e.o.5149.1		2	195.38	even	4
5850.2.e.o.5149.2		2	195.77	even	4
9360.2.a.g.1.1		1	156.155	even	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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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