LMFDB - Embedded newform 5070.2.a.u.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:	$0$
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	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} +8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{42} +12.0000 q^{43} +1.00000 q^{45} +8.00000 q^{46} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +1.00000 q^{60} -10.0000 q^{61} +8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} -2.00000 q^{68} +8.00000 q^{69} -4.00000 q^{70} +16.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} -4.00000 q^{84} -2.00000 q^{85} +12.0000 q^{86} +2.00000 q^{87} +14.0000 q^{89} +1.00000 q^{90} +8.00000 q^{92} +8.00000 q^{93} -4.00000 q^{95} +1.00000 q^{96} +6.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	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
$14$	−4.00000	−1.06904
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	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	$0.651732\pi$
$20$	1.00000	0.223607
$21$	−4.00000	−0.872872
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	1.00000	0.182574
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−4.00000	−0.676123
$36$	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	$0.552568\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	1.00000	0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−4.00000	−0.617213
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	8.00000	1.17954
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	−4.00000	−0.529813
$58$	2.00000	0.262613
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000	0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	8.00000	1.01600
$63$	−4.00000	−0.503953
$64$	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	$0.421429\pi$
$68$	−2.00000	−0.242536
$69$	8.00000	0.963087
$70$	−4.00000	−0.478091
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	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
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−4.00000	−0.436436
$85$	−2.00000	−0.216930
$86$	12.0000	1.29399
$87$	2.00000	0.214423
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	8.00000	0.834058
$93$	8.00000	0.829561
$94$	0	0
$95$	−4.00000	−0.410391
$96$	1.00000	0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	1.00000	0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−2.00000	−0.198030
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	10.0000	0.971286
$107$	−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
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	−4.00000	−0.377964
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	−4.00000	−0.374634
$115$	8.00000	0.746004
$116$	2.00000	0.185695
$117$	0	0
$118$	0	0
$119$	8.00000	0.733359
$120$	1.00000	0.0912871
$121$	−11.0000	−1.00000
$122$	−10.0000	−0.905357
$123$	6.00000	0.541002
$124$	8.00000	0.718421
$125$	1.00000	0.0894427
$126$	−4.00000	−0.356348
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	1.00000	0.0883883
$129$	12.0000	1.05654
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	4.00000	0.345547
$135$	1.00000	0.0860663
$136$	−2.00000	−0.171499
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	8.00000	0.681005
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	16.0000	1.34269
$143$	0	0
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	6.00000	0.496564
$147$	9.00000	0.742307
$148$	−2.00000	−0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	1.00000	0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−4.00000	−0.324443
$153$	−2.00000	−0.161690
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−8.00000	−0.636446
$159$	10.0000	0.793052
$160$	1.00000	0.0790569
$161$	−32.0000	−2.52195
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	−4.00000	−0.308607
$169$	0	0
$170$	−2.00000	−0.153393
$171$	−4.00000	−0.305888
$172$	12.0000	0.914991
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	2.00000	0.151620
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	14.0000	1.04934
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	1.00000	0.0745356
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	8.00000	0.589768
$185$	−2.00000	−0.147043
$186$	8.00000	0.586588
$187$	0	0
$188$	0	0
$189$	−4.00000	−0.290957
$190$	−4.00000	−0.290191
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	1.00000	0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	9.00000	0.642857
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	10.0000	0.703598
$203$	−8.00000	−0.561490
$204$	−2.00000	−0.140028
$205$	6.00000	0.419058
$206$	−4.00000	−0.278693
$207$	8.00000	0.556038
$208$	0	0
$209$	0	0
$210$	−4.00000	−0.276026
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	10.0000	0.686803
$213$	16.0000	1.09630
$214$	−12.0000	−0.820303
$215$	12.0000	0.818393
$216$	1.00000	0.0680414
$217$	−32.0000	−2.17230
$218$	−14.0000	−0.948200
$219$	6.00000	0.405442
$220$	0	0
$221$	0	0
$222$	−2.00000	−0.134231
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	−4.00000	−0.267261
$225$	1.00000	0.0666667
$226$	−10.0000	−0.665190
$227$	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
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	2.00000	0.131306
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.00000	−0.519656
$238$	8.00000	0.518563
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	1.00000	0.0645497
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	9.00000	0.574989
$246$	6.00000	0.382546
$247$	0	0
$248$	8.00000	0.508001
$249$	4.00000	0.253490
$250$	1.00000	0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−4.00000	−0.251976
$253$	0	0
$254$	−12.0000	−0.752947
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	12.0000	0.747087
$259$	8.00000	0.497096
$260$	0	0
$261$	2.00000	0.123797
$262$	8.00000	0.494242
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	16.0000	0.981023
$267$	14.0000	0.856786
$268$	4.00000	0.244339
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	1.00000	0.0608581
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	8.00000	0.481543
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−20.0000	−1.19952
$279$	8.00000	0.478947
$280$	−4.00000	−0.239046
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	16.0000	0.949425
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−24.0000	−1.41668
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	6.00000	0.351726
$292$	6.00000	0.351123
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−10.0000	−0.579284
$299$	0	0
$300$	1.00000	0.0577350
$301$	−48.0000	−2.76667
$302$	8.00000	0.460348
$303$	10.0000	0.574485
$304$	−4.00000	−0.229416
$305$	−10.0000	−0.572598
$306$	−2.00000	−0.114332
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	8.00000	0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	18.0000	1.01580
$315$	−4.00000	−0.225374
$316$	−8.00000	−0.450035
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	10.0000	0.560772
$319$	0	0
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	−32.0000	−1.78329
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	−14.0000	−0.774202
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	4.00000	0.219529
$333$	−2.00000	−0.109599
$334$	−16.0000	−0.875481
$335$	4.00000	0.218543
$336$	−4.00000	−0.218218
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	−2.00000	−0.108465
$341$	0	0
$342$	−4.00000	−0.216295
$343$	−8.00000	−0.431959
$344$	12.0000	0.646997
$345$	8.00000	0.430706
$346$	2.00000	0.107521
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	2.00000	0.107211
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	14.0000	0.741999
$357$	8.00000	0.423405
$358$	16.0000	0.845626
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	−11.0000	−0.577350
$364$	0	0
$365$	6.00000	0.314054
$366$	−10.0000	−0.522708
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	8.00000	0.417029
$369$	6.00000	0.312348
$370$	−2.00000	−0.103975
$371$	−40.0000	−2.07670
$372$	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	$0.692886\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	−4.00000	−0.205738
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−4.00000	−0.205196
$381$	−12.0000	−0.614779
$382$	8.00000	0.409316
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−18.0000	−0.916176
$387$	12.0000	0.609994
$388$	6.00000	0.304604
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	9.00000	0.454569
$393$	8.00000	0.403547
$394$	6.00000	0.302276
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	−8.00000	−0.401004
$399$	16.0000	0.801002
$400$	1.00000	0.0500000
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	10.0000	0.497519
$405$	1.00000	0.0496904
$406$	−8.00000	−0.397033
$407$	0	0
$408$	−2.00000	−0.0990148
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	6.00000	0.296319
$411$	−6.00000	−0.295958
$412$	−4.00000	−0.197066
$413$	0	0
$414$	8.00000	0.393179
$415$	4.00000	0.196352
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	−4.00000	−0.195180
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	10.0000	0.485643
$425$	−2.00000	−0.0970143
$426$	16.0000	0.775203
$427$	40.0000	1.93574
$428$	−12.0000	−0.580042
$429$	0	0
$430$	12.0000	0.578691
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	−32.0000	−1.53605
$435$	2.00000	0.0958927
$436$	−14.0000	−0.670478
$437$	−32.0000	−1.53077
$438$	6.00000	0.286691
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−2.00000	−0.0949158
$445$	14.0000	0.663664
$446$	−12.0000	−0.568216
$447$	−10.0000	−0.472984
$448$	−4.00000	−0.188982
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−10.0000	−0.470360
$453$	8.00000	0.375873
$454$	12.0000	0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	−14.0000	−0.654177
$459$	−2.00000	−0.0933520
$460$	8.00000	0.373002
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	2.00000	0.0928477
$465$	8.00000	0.370991
$466$	−26.0000	−1.20443
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	0	0
$474$	−8.00000	−0.367452
$475$	−4.00000	−0.183533
$476$	8.00000	0.366679
$477$	10.0000	0.457869
$478$	−24.0000	−1.09773
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−2.00000	−0.0910975
$483$	−32.0000	−1.45605
$484$	−11.0000	−0.500000
$485$	6.00000	0.272446
$486$	1.00000	0.0453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−10.0000	−0.452679
$489$	20.0000	0.904431
$490$	9.00000	0.406579
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	6.00000	0.270501
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	−64.0000	−2.87079
$498$	4.00000	0.179244
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	1.00000	0.0447214
$501$	−16.0000	−0.714827
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	−4.00000	−0.178174
$505$	10.0000	0.444994
$506$	0	0
$507$	0	0
$508$	−12.0000	−0.532414
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	−2.00000	−0.0885615
$511$	−24.0000	−1.06170
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	22.0000	0.970378
$515$	−4.00000	−0.176261
$516$	12.0000	0.528271
$517$	0	0
$518$	8.00000	0.351500
$519$	2.00000	0.0877903
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	2.00000	0.0875376
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	8.00000	0.349482
$525$	−4.00000	−0.174574
$526$	24.0000	1.04645
$527$	−16.0000	−0.696971
$528$	0	0
$529$	41.0000	1.78261
$530$	10.0000	0.434372
$531$	0	0
$532$	16.0000	0.693688
$533$	0	0
$534$	14.0000	0.605839
$535$	−12.0000	−0.518805
$536$	4.00000	0.172774
$537$	16.0000	0.690451
$538$	−14.0000	−0.603583
$539$	0	0
$540$	1.00000	0.0430331
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	32.0000	1.37452
$543$	−2.00000	−0.0858282
$544$	−2.00000	−0.0857493
$545$	−14.0000	−0.599694
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	−6.00000	−0.256307
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−8.00000	−0.340811
$552$	8.00000	0.340503
$553$	32.0000	1.36078
$554$	18.0000	0.764747
$555$	−2.00000	−0.0848953
$556$	−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	$0.624538\pi$
$558$	8.00000	0.338667
$559$	0	0
$560$	−4.00000	−0.169031
$561$	0	0
$562$	−18.0000	−0.759284
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	20.0000	0.840663
$567$	−4.00000	−0.167984
$568$	16.0000	0.671345
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	−4.00000	−0.167542
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	−24.0000	−1.00174
$575$	8.00000	0.333623
$576$	1.00000	0.0416667
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	−13.0000	−0.540729
$579$	−18.0000	−0.748054
$580$	2.00000	0.0830455
$581$	−16.0000	−0.663792
$582$	6.00000	0.248708
$583$	0	0
$584$	6.00000	0.248282
$585$	0	0
$586$	6.00000	0.247858
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	9.00000	0.371154
$589$	−32.0000	−1.31854
$590$	0	0
$591$	6.00000	0.246807
$592$	−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	$0.592809\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	−10.0000	−0.409616
$597$	−8.00000	−0.327418
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	1.00000	0.0408248
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−48.0000	−1.95633
$603$	4.00000	0.162893
$604$	8.00000	0.325515
$605$	−11.0000	−0.447214
$606$	10.0000	0.406222
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	−4.00000	−0.162221
$609$	−8.00000	−0.324176
$610$	−10.0000	−0.404888
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	12.0000	0.484281
$615$	6.00000	0.241943
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	−4.00000	−0.160904
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	8.00000	0.321288
$621$	8.00000	0.321029
$622$	0	0
$623$	−56.0000	−2.24359
$624$	0	0
$625$	1.00000	0.0400000
$626$	18.0000	0.719425
$627$	0	0
$628$	18.0000	0.718278
$629$	4.00000	0.159490
$630$	−4.00000	−0.159364
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−8.00000	−0.318223
$633$	−12.0000	−0.476957
$634$	14.0000	0.556011
$635$	−12.0000	−0.476205
$636$	10.0000	0.396526
$637$	0	0
$638$	0	0
$639$	16.0000	0.632950
$640$	1.00000	0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−12.0000	−0.473602
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	−32.0000	−1.26098
$645$	12.0000	0.472500
$646$	8.00000	0.314756
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	−32.0000	−1.25418
$652$	20.0000	0.783260
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−14.0000	−0.547443
$655$	8.00000	0.312586
$656$	6.00000	0.234261
$657$	6.00000	0.234082
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	4.00000	0.155230
$665$	16.0000	0.620453
$666$	−2.00000	−0.0774984
$667$	16.0000	0.619522
$668$	−16.0000	−0.619059
$669$	−12.0000	−0.463947
$670$	4.00000	0.154533
$671$	0	0
$672$	−4.00000	−0.154303
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	−22.0000	−0.847408
$675$	1.00000	0.0384900
$676$	0	0
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	−10.0000	−0.384048
$679$	−24.0000	−0.921035
$680$	−2.00000	−0.0766965
$681$	12.0000	0.459841
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−4.00000	−0.152944
$685$	−6.00000	−0.229248
$686$	−8.00000	−0.305441
$687$	−14.0000	−0.534133
$688$	12.0000	0.457496
$689$	0	0
$690$	8.00000	0.304555
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−20.0000	−0.758643
$696$	2.00000	0.0758098
$697$	−12.0000	−0.454532
$698$	−14.0000	−0.529908
$699$	−26.0000	−0.983410
$700$	−4.00000	−0.151186
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	26.0000	0.978523
$707$	−40.0000	−1.50435
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	16.0000	0.600469
$711$	−8.00000	−0.300023
$712$	14.0000	0.524672
$713$	64.0000	2.39682
$714$	8.00000	0.299392
$715$	0	0
$716$	16.0000	0.597948
$717$	−24.0000	−0.896296
$718$	−24.0000	−0.895672
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	1.00000	0.0372678
$721$	16.0000	0.595871
$722$	−3.00000	−0.111648
$723$	−2.00000	−0.0743808
$724$	−2.00000	−0.0743294
$725$	2.00000	0.0742781
$726$	−11.0000	−0.408248
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	6.00000	0.222070
$731$	−24.0000	−0.887672
$732$	−10.0000	−0.369611
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	4.00000	0.147643
$735$	9.00000	0.331970
$736$	8.00000	0.294884
$737$	0	0
$738$	6.00000	0.220863
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	−40.0000	−1.46845
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	8.00000	0.293294
$745$	−10.0000	−0.366372
$746$	−22.0000	−0.805477
$747$	4.00000	0.146352
$748$	0	0
$749$	48.0000	1.75388
$750$	1.00000	0.0365148
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.00000	0.291150
$756$	−4.00000	−0.145479
$757$	50.0000	1.81728	0.908640	−	0.417579i	$-0.137121\pi$
$757$	50.0000	1.81728	0.908640	+	0.417579i	$0.137121\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	−4.00000	−0.145095
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−12.0000	−0.434714
$763$	56.0000	2.02734
$764$	8.00000	0.289430
$765$	−2.00000	−0.0723102
$766$	16.0000	0.578103
$767$	0	0
$768$	1.00000	0.0360844
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	−18.0000	−0.647834
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	12.0000	0.431331
$775$	8.00000	0.287368
$776$	6.00000	0.215387
$777$	8.00000	0.286998
$778$	−14.0000	−0.501924
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	−16.0000	−0.572159
$783$	2.00000	0.0714742
$784$	9.00000	0.321429
$785$	18.0000	0.642448
$786$	8.00000	0.285351
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	6.00000	0.213741
$789$	24.0000	0.854423
$790$	−8.00000	−0.284627
$791$	40.0000	1.42224
$792$	0	0
$793$	0	0
$794$	−10.0000	−0.354887
$795$	10.0000	0.354663
$796$	−8.00000	−0.283552
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	16.0000	0.566394
$799$	0	0
$800$	1.00000	0.0353553
$801$	14.0000	0.494666
$802$	−26.0000	−0.918092
$803$	0	0
$804$	4.00000	0.141069
$805$	−32.0000	−1.12785
$806$	0	0
$807$	−14.0000	−0.492823
$808$	10.0000	0.351799
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	1.00000	0.0351364
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	−8.00000	−0.280745
$813$	32.0000	1.12229
$814$	0	0
$815$	20.0000	0.700569
$816$	−2.00000	−0.0700140
$817$	−48.0000	−1.67931
$818$	−10.0000	−0.349642
$819$	0	0
$820$	6.00000	0.209529
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	−6.00000	−0.209274
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	8.00000	0.278019
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	4.00000	0.138842
$831$	18.0000	0.624413
$832$	0	0
$833$	−18.0000	−0.623663
$834$	−20.0000	−0.692543
$835$	−16.0000	−0.553703
$836$	0	0
$837$	8.00000	0.276520
$838$	−24.0000	−0.829066
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	−4.00000	−0.138013
$841$	−25.0000	−0.862069
$842$	−30.0000	−1.03387
$843$	−18.0000	−0.619953
$844$	−12.0000	−0.413057
$845$	0	0
$846$	0	0
$847$	44.0000	1.51186
$848$	10.0000	0.343401
$849$	20.0000	0.686398
$850$	−2.00000	−0.0685994
$851$	−16.0000	−0.548473
$852$	16.0000	0.548151
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	40.0000	1.36877
$855$	−4.00000	−0.136797
$856$	−12.0000	−0.410152
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	12.0000	0.409197
$861$	−24.0000	−0.817918
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	1.00000	0.0340207
$865$	2.00000	0.0680020
$866$	10.0000	0.339814
$867$	−13.0000	−0.441503
$868$	−32.0000	−1.08615
$869$	0	0
$870$	2.00000	0.0678064
$871$	0	0
$872$	−14.0000	−0.474100
$873$	6.00000	0.203069
$874$	−32.0000	−1.08242
$875$	−4.00000	−0.135225
$876$	6.00000	0.202721
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	40.0000	1.34993
$879$	6.00000	0.202375
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	9.00000	0.303046
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	0	0
$885$	0	0
$886$	−36.0000	−1.20944
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	−2.00000	−0.0671156
$889$	48.0000	1.60987
$890$	14.0000	0.469281
$891$	0	0
$892$	−12.0000	−0.401790
$893$	0	0
$894$	−10.0000	−0.334450
$895$	16.0000	0.534821
$896$	−4.00000	−0.133631
$897$	0	0
$898$	6.00000	0.200223
$899$	16.0000	0.533630
$900$	1.00000	0.0333333
$901$	−20.0000	−0.666297
$902$	0	0
$903$	−48.0000	−1.59734
$904$	−10.0000	−0.332595
$905$	−2.00000	−0.0664822
$906$	8.00000	0.265782
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	12.0000	0.398234
$909$	10.0000	0.331679
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	−18.0000	−0.595387
$915$	−10.0000	−0.330590
$916$	−14.0000	−0.462573
$917$	−32.0000	−1.05673
$918$	−2.00000	−0.0660098
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	8.00000	0.263752
$921$	12.0000	0.395413
$922$	−2.00000	−0.0658665
$923$	0	0
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−20.0000	−0.657241
$927$	−4.00000	−0.131377
$928$	2.00000	0.0656532
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	8.00000	0.262330
$931$	−36.0000	−1.17985
$932$	−26.0000	−0.851658
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	−16.0000	−0.522419
$939$	18.0000	0.587408
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	18.0000	0.586472
$943$	48.0000	1.56310
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	−4.00000	−0.129777
$951$	14.0000	0.453981
$952$	8.00000	0.259281
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	10.0000	0.323762
$955$	8.00000	0.258874
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−8.00000	−0.258468
$959$	24.0000	0.775000
$960$	1.00000	0.0322749
$961$	33.0000	1.06452
$962$	0	0
$963$	−12.0000	−0.386695
$964$	−2.00000	−0.0644157
$965$	−18.0000	−0.579441
$966$	−32.0000	−1.02958
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	−11.0000	−0.353553
$969$	8.00000	0.256997
$970$	6.00000	0.192648
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	1.00000	0.0320750
$973$	80.0000	2.56468
$974$	20.0000	0.640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	20.0000	0.639529
$979$	0	0
$980$	9.00000	0.287494
$981$	−14.0000	−0.446986
$982$	0	0
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	6.00000	0.191273
$985$	6.00000	0.191176
$986$	−4.00000	−0.127386
$987$	0	0
$988$	0	0
$989$	96.0000	3.05262
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	8.00000	0.254000
$993$	20.0000	0.634681
$994$	−64.0000	−2.02996
$995$	−8.00000	−0.253617
$996$	4.00000	0.126745
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	44.0000	1.39280
$999$	−2.00000	−0.0632772

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.c.1.1	✓	1	13.12	even	2
1170.2.a.n.1.1		1	39.38	odd	2
1950.2.a.n.1.1		1	65.64	even	2
1950.2.e.e.1249.1		2	65.12	odd	4
1950.2.e.e.1249.2		2	65.38	odd	4
3120.2.a.a.1.1		1	52.51	odd	2
5070.2.a.u.1.1		1	1.1	even	1	trivial
5070.2.b.i.1351.1		2	13.5	odd	4
5070.2.b.i.1351.2		2	13.8	odd	4
5850.2.a.c.1.1		1	195.194	odd	2
5850.2.e.m.5149.1		2	195.38	even	4
5850.2.e.m.5149.2		2	195.77	even	4
9360.2.a.bc.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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