LMFDB - Embedded newform 5070.2.a.d.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} +5.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} +5.00000 q^{29} +1.00000 q^{30} -11.0000 q^{31} -1.00000 q^{32} +5.00000 q^{33} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +3.00000 q^{37} +2.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} -2.00000 q^{42} -11.0000 q^{43} -5.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +6.00000 q^{53} +1.00000 q^{54} -5.00000 q^{55} +2.00000 q^{56} +2.00000 q^{57} -5.00000 q^{58} -15.0000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +11.0000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +16.0000 q^{67} -2.00000 q^{68} +1.00000 q^{69} +2.00000 q^{70} -1.00000 q^{72} -6.00000 q^{73} -3.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} +10.0000 q^{77} -11.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +6.00000 q^{83} +2.00000 q^{84} -2.00000 q^{85} +11.0000 q^{86} -5.00000 q^{87} +5.00000 q^{88} +2.00000 q^{89} -1.00000 q^{90} -1.00000 q^{92} +11.0000 q^{93} -9.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +3.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
$5$	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
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	−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
$17$	−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
$19$	−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
$22$	5.00000	1.06600
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	1.00000	0.182574
$31$	−11.0000	−1.97566	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−11.0000	−1.97566	−0.987829	+	0.155543i	$0.950287\pi$
$32$	−1.00000	−0.176777
$33$	5.00000	0.870388
$34$	2.00000	0.342997
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−2.00000	−0.308607
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	−5.00000	−0.753778
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	2.00000	0.280056
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	−5.00000	−0.674200
$56$	2.00000	0.267261
$57$	2.00000	0.264906
$58$	−5.00000	−0.656532
$59$	−15.0000	−1.95283	−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000	−1.95283	−0.976417	+	0.215894i	$0.930733\pi$
$60$	−1.00000	−0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	11.0000	1.39700
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.00000	−0.615457
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	−2.00000	−0.242536
$69$	1.00000	0.120386
$70$	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
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−3.00000	−0.348743
$75$	−1.00000	−0.115470
$76$	−2.00000	−0.229416
$77$	10.0000	1.13961
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	2.00000	0.218218
$85$	−2.00000	−0.216930
$86$	11.0000	1.18616
$87$	−5.00000	−0.536056
$88$	5.00000	0.533002
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−1.00000	−0.104257
$93$	11.0000	1.14065
$94$	−9.00000	−0.928279
$95$	−2.00000	−0.205196
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	3.00000	0.303046
$99$	−5.00000	−0.502519
$100$	1.00000	0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−2.00000	−0.198030
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	−6.00000	−0.582772
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	−1.00000	−0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	5.00000	0.476731
$111$	−3.00000	−0.284747
$112$	−2.00000	−0.188982
$113$	11.0000	1.03479	0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000	1.03479	0.517396	+	0.855746i	$0.326901\pi$
$114$	−2.00000	−0.187317
$115$	−1.00000	−0.0932505
$116$	5.00000	0.464238
$117$	0	0
$118$	15.0000	1.38086
$119$	4.00000	0.366679
$120$	1.00000	0.0912871
$121$	14.0000	1.27273
$122$	−10.0000	−0.905357
$123$	2.00000	0.180334
$124$	−11.0000	−0.987829
$125$	1.00000	0.0894427
$126$	2.00000	0.178174
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−1.00000	−0.0883883
$129$	11.0000	0.968496
$130$	0	0
$131$	−1.00000	−0.0873704	−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000	−0.0873704	−0.0436852	+	0.999045i	$0.513910\pi$
$132$	5.00000	0.435194
$133$	4.00000	0.346844
$134$	−16.0000	−1.38219
$135$	−1.00000	−0.0860663
$136$	2.00000	0.171499
$137$	−11.0000	−0.939793	−0.469897	−	0.882721i	$-0.655709\pi$
$137$	−11.0000	−0.939793	−0.469897	+	0.882721i	$0.655709\pi$
$138$	−1.00000	−0.0851257
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	−2.00000	−0.169031
$141$	−9.00000	−0.757937
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	5.00000	0.415227
$146$	6.00000	0.496564
$147$	3.00000	0.247436
$148$	3.00000	0.246598
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	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	$0.394462\pi$
$152$	2.00000	0.162221
$153$	−2.00000	−0.161690
$154$	−10.0000	−0.805823
$155$	−11.0000	−0.883541
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	11.0000	0.875113
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	2.00000	0.157622
$162$	−1.00000	−0.0785674
$163$	−15.0000	−1.17489	−0.587445	−	0.809264i	$-0.699866\pi$
$163$	−15.0000	−1.17489	−0.587445	+	0.809264i	$0.699866\pi$
$164$	−2.00000	−0.156174
$165$	5.00000	0.389249
$166$	−6.00000	−0.465690
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	−2.00000	−0.154303
$169$	0	0
$170$	2.00000	0.153393
$171$	−2.00000	−0.152944
$172$	−11.0000	−0.838742
$173$	20.0000	1.52057	0.760286	−	0.649589i	$-0.225059\pi$
$173$	20.0000	1.52057	0.760286	+	0.649589i	$0.225059\pi$
$174$	5.00000	0.379049
$175$	−2.00000	−0.151186
$176$	−5.00000	−0.376889
$177$	15.0000	1.12747
$178$	−2.00000	−0.149906
$179$	−13.0000	−0.971666	−0.485833	−	0.874052i	$-0.661484\pi$
$179$	−13.0000	−0.971666	−0.485833	+	0.874052i	$0.661484\pi$
$180$	1.00000	0.0745356
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	1.00000	0.0737210
$185$	3.00000	0.220564
$186$	−11.0000	−0.806559
$187$	10.0000	0.731272
$188$	9.00000	0.656392
$189$	2.00000	0.145479
$190$	2.00000	0.145095
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	−1.00000	−0.0721688
$193$	−24.0000	−1.72756	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000	−1.72756	−0.863779	+	0.503871i	$0.831909\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	5.00000	0.355335
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	−1.00000	−0.0707107
$201$	−16.0000	−1.12855
$202$	−2.00000	−0.140720
$203$	−10.0000	−0.701862
$204$	2.00000	0.140028
$205$	−2.00000	−0.139686
$206$	−10.0000	−0.696733
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	10.0000	0.691714
$210$	−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
$214$	−10.0000	−0.683586
$215$	−11.0000	−0.750194
$216$	1.00000	0.0680414
$217$	22.0000	1.49346
$218$	2.00000	0.135457
$219$	6.00000	0.405442
$220$	−5.00000	−0.337100
$221$	0	0
$222$	3.00000	0.201347
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	−11.0000	−0.731709
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	2.00000	0.132453
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	1.00000	0.0659380
$231$	−10.0000	−0.657952
$232$	−5.00000	−0.328266
$233$	1.00000	0.0655122	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	1.00000	0.0655122	0.0327561	+	0.999463i	$0.489572\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	−15.0000	−0.976417
$237$	11.0000	0.714527
$238$	−4.00000	−0.259281
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	−1.00000	−0.0645497
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	−14.0000	−0.899954
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	−3.00000	−0.191663
$246$	−2.00000	−0.127515
$247$	0	0
$248$	11.0000	0.698501
$249$	−6.00000	−0.380235
$250$	−1.00000	−0.0632456
$251$	25.0000	1.57799	0.788993	−	0.614402i	$-0.210603\pi$
$251$	25.0000	1.57799	0.788993	+	0.614402i	$0.210603\pi$
$252$	−2.00000	−0.125988
$253$	5.00000	0.314347
$254$	−2.00000	−0.125491
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−17.0000	−1.06043	−0.530215	−	0.847863i	$-0.677889\pi$
$257$	−17.0000	−1.06043	−0.530215	+	0.847863i	$0.677889\pi$
$258$	−11.0000	−0.684830
$259$	−6.00000	−0.372822
$260$	0	0
$261$	5.00000	0.309492
$262$	1.00000	0.0617802
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	−5.00000	−0.307729
$265$	6.00000	0.368577
$266$	−4.00000	−0.245256
$267$	−2.00000	−0.122398
$268$	16.0000	0.977356
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	1.00000	0.0608581
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	11.0000	0.664534
$275$	−5.00000	−0.301511
$276$	1.00000	0.0601929
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	−2.00000	−0.119952
$279$	−11.0000	−0.658553
$280$	2.00000	0.119523
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	9.00000	0.535942
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−5.00000	−0.293610
$291$	2.00000	0.117242
$292$	−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	$0.443923\pi$
$294$	−3.00000	−0.174964
$295$	−15.0000	−0.873334
$296$	−3.00000	−0.174371
$297$	5.00000	0.290129
$298$	−17.0000	−0.984784
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	22.0000	1.26806
$302$	−8.00000	−0.460348
$303$	−2.00000	−0.114897
$304$	−2.00000	−0.114708
$305$	10.0000	0.572598
$306$	2.00000	0.114332
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	10.0000	0.569803
$309$	−10.0000	−0.568880
$310$	11.0000	0.624758
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	7.00000	0.395033
$315$	−2.00000	−0.112687
$316$	−11.0000	−0.618798
$317$	16.0000	0.898650	0.449325	−	0.893368i	$-0.351665\pi$
$317$	16.0000	0.898650	0.449325	+	0.893368i	$0.351665\pi$
$318$	6.00000	0.336463
$319$	−25.0000	−1.39973
$320$	1.00000	0.0559017
$321$	−10.0000	−0.558146
$322$	−2.00000	−0.111456
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	15.0000	0.830773
$327$	2.00000	0.110600
$328$	2.00000	0.110432
$329$	−18.0000	−0.992372
$330$	−5.00000	−0.275241
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	6.00000	0.329293
$333$	3.00000	0.164399
$334$	−3.00000	−0.164153
$335$	16.0000	0.874173
$336$	2.00000	0.109109
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	−11.0000	−0.597438
$340$	−2.00000	−0.108465
$341$	55.0000	2.97842
$342$	2.00000	0.108148
$343$	20.0000	1.07990
$344$	11.0000	0.593080
$345$	1.00000	0.0538382
$346$	−20.0000	−1.07521
$347$	34.0000	1.82522	0.912608	−	0.408836i	$-0.134065\pi$
$347$	34.0000	1.82522	0.912608	+	0.408836i	$0.134065\pi$
$348$	−5.00000	−0.268028
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	5.00000	0.266501
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	−15.0000	−0.797241
$355$	0	0
$356$	2.00000	0.106000
$357$	−4.00000	−0.211702
$358$	13.0000	0.687071
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	−1.00000	−0.0527046
$361$	−15.0000	−0.789474
$362$	16.0000	0.840941
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−6.00000	−0.314054
$366$	10.0000	0.522708
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−1.00000	−0.0521286
$369$	−2.00000	−0.104116
$370$	−3.00000	−0.155963
$371$	−12.0000	−0.623009
$372$	11.0000	0.570323
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	−10.0000	−0.517088
$375$	−1.00000	−0.0516398
$376$	−9.00000	−0.464140
$377$	0	0
$378$	−2.00000	−0.102869
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	−2.00000	−0.102598
$381$	−2.00000	−0.102463
$382$	−4.00000	−0.204658
$383$	31.0000	1.58403	0.792013	−	0.610504i	$-0.209033\pi$
$383$	31.0000	1.58403	0.792013	+	0.610504i	$0.209033\pi$
$384$	1.00000	0.0510310
$385$	10.0000	0.509647
$386$	24.0000	1.22157
$387$	−11.0000	−0.559161
$388$	−2.00000	−0.101535
$389$	5.00000	0.253510	0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000	0.253510	0.126755	+	0.991934i	$0.459544\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	3.00000	0.151523
$393$	1.00000	0.0504433
$394$	8.00000	0.403034
$395$	−11.0000	−0.553470
$396$	−5.00000	−0.251259
$397$	3.00000	0.150566	0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000	0.150566	0.0752828	+	0.997162i	$0.476014\pi$
$398$	−24.0000	−1.20301
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	16.0000	0.798007
$403$	0	0
$404$	2.00000	0.0995037
$405$	1.00000	0.0496904
$406$	10.0000	0.496292
$407$	−15.0000	−0.743522
$408$	−2.00000	−0.0990148
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	2.00000	0.0987730
$411$	11.0000	0.542590
$412$	10.0000	0.492665
$413$	30.0000	1.47620
$414$	1.00000	0.0491473
$415$	6.00000	0.294528
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	−10.0000	−0.489116
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	2.00000	0.0975900
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	4.00000	0.194717
$423$	9.00000	0.437595
$424$	−6.00000	−0.291386
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−20.0000	−0.967868
$428$	10.0000	0.483368
$429$	0	0
$430$	11.0000	0.530467
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	−1.00000	−0.0481125
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	−22.0000	−1.05603
$435$	−5.00000	−0.239732
$436$	−2.00000	−0.0957826
$437$	2.00000	0.0956730
$438$	−6.00000	−0.286691
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	5.00000	0.238366
$441$	−3.00000	−0.142857
$442$	0	0
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	−3.00000	−0.142374
$445$	2.00000	0.0948091
$446$	−26.0000	−1.23114
$447$	−17.0000	−0.804072
$448$	−2.00000	−0.0944911
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	−1.00000	−0.0471405
$451$	10.0000	0.470882
$452$	11.0000	0.517396
$453$	−8.00000	−0.375873
$454$	0	0
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	−10.0000	−0.467269
$459$	2.00000	0.0933520
$460$	−1.00000	−0.0466252
$461$	−39.0000	−1.81641	−0.908206	−	0.418524i	$-0.862547\pi$
$461$	−39.0000	−1.81641	−0.908206	+	0.418524i	$0.862547\pi$
$462$	10.0000	0.465242
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	5.00000	0.232119
$465$	11.0000	0.510113
$466$	−1.00000	−0.0463241
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	−9.00000	−0.415139
$471$	7.00000	0.322543
$472$	15.0000	0.690431
$473$	55.0000	2.52890
$474$	−11.0000	−0.505247
$475$	−2.00000	−0.0917663
$476$	4.00000	0.183340
$477$	6.00000	0.274721
$478$	20.0000	0.914779
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	7.00000	0.318841
$483$	−2.00000	−0.0910032
$484$	14.0000	0.636364
$485$	−2.00000	−0.0908153
$486$	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	$0.485571\pi$
$488$	−10.0000	−0.452679
$489$	15.0000	0.678323
$490$	3.00000	0.135526
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	2.00000	0.0901670
$493$	−10.0000	−0.450377
$494$	0	0
$495$	−5.00000	−0.224733
$496$	−11.0000	−0.493915
$497$	0	0
$498$	6.00000	0.268866
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	1.00000	0.0447214
$501$	−3.00000	−0.134030
$502$	−25.0000	−1.11580
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	2.00000	0.0890871
$505$	2.00000	0.0889988
$506$	−5.00000	−0.222277
$507$	0	0
$508$	2.00000	0.0887357
$509$	29.0000	1.28540	0.642701	−	0.766117i	$-0.277814\pi$
$509$	29.0000	1.28540	0.642701	+	0.766117i	$0.277814\pi$
$510$	−2.00000	−0.0885615
$511$	12.0000	0.530849
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	17.0000	0.749838
$515$	10.0000	0.440653
$516$	11.0000	0.484248
$517$	−45.0000	−1.97910
$518$	6.00000	0.263625
$519$	−20.0000	−0.877903
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−5.00000	−0.218844
$523$	−31.0000	−1.35554	−0.677768	−	0.735276i	$-0.737052\pi$
$523$	−31.0000	−1.35554	−0.677768	+	0.735276i	$0.737052\pi$
$524$	−1.00000	−0.0436852
$525$	2.00000	0.0872872
$526$	−21.0000	−0.915644
$527$	22.0000	0.958335
$528$	5.00000	0.217597
$529$	−22.0000	−0.956522
$530$	−6.00000	−0.260623
$531$	−15.0000	−0.650945
$532$	4.00000	0.173422
$533$	0	0
$534$	2.00000	0.0865485
$535$	10.0000	0.432338
$536$	−16.0000	−0.691095
$537$	13.0000	0.560991
$538$	−14.0000	−0.603583
$539$	15.0000	0.646096
$540$	−1.00000	−0.0430331
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	13.0000	0.558398
$543$	16.0000	0.686626
$544$	2.00000	0.0857493
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−11.0000	−0.469897
$549$	10.0000	0.426790
$550$	5.00000	0.213201
$551$	−10.0000	−0.426014
$552$	−1.00000	−0.0425628
$553$	22.0000	0.935535
$554$	11.0000	0.467345
$555$	−3.00000	−0.127343
$556$	2.00000	0.0848189
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	11.0000	0.465667
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	−10.0000	−0.422200
$562$	10.0000	0.421825
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−9.00000	−0.378968
$565$	11.0000	0.462773
$566$	−19.0000	−0.798630
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	−2.00000	−0.0837708
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	−4.00000	−0.167102
$574$	−4.00000	−0.166957
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	13.0000	0.540729
$579$	24.0000	0.997406
$580$	5.00000	0.207614
$581$	−12.0000	−0.497844
$582$	−2.00000	−0.0829027
$583$	−30.0000	−1.24247
$584$	6.00000	0.248282
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	3.00000	0.123718
$589$	22.0000	0.906494
$590$	15.0000	0.617540
$591$	8.00000	0.329076
$592$	3.00000	0.123299
$593$	31.0000	1.27302	0.636509	−	0.771270i	$-0.280378\pi$
$593$	31.0000	1.27302	0.636509	+	0.771270i	$0.280378\pi$
$594$	−5.00000	−0.205152
$595$	4.00000	0.163984
$596$	17.0000	0.696347
$597$	−24.0000	−0.982255
$598$	0	0
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	1.00000	0.0408248
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	−22.0000	−0.896653
$603$	16.0000	0.651570
$604$	8.00000	0.325515
$605$	14.0000	0.569181
$606$	2.00000	0.0812444
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	2.00000	0.0811107
$609$	10.0000	0.405220
$610$	−10.0000	−0.404888
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	−10.0000	−0.402911
$617$	41.0000	1.65060	0.825299	−	0.564696i	$-0.191007\pi$
$617$	41.0000	1.65060	0.825299	+	0.564696i	$0.191007\pi$
$618$	10.0000	0.402259
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	−11.0000	−0.441771
$621$	1.00000	0.0401286
$622$	−20.0000	−0.801927
$623$	−4.00000	−0.160257
$624$	0	0
$625$	1.00000	0.0400000
$626$	−20.0000	−0.799361
$627$	−10.0000	−0.399362
$628$	−7.00000	−0.279330
$629$	−6.00000	−0.239236
$630$	2.00000	0.0796819
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	11.0000	0.437557
$633$	4.00000	0.158986
$634$	−16.0000	−0.635441
$635$	2.00000	0.0793676
$636$	−6.00000	−0.237915
$637$	0	0
$638$	25.0000	0.989759
$639$	0	0
$640$	−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	$0.701845\pi$
$642$	10.0000	0.394669
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	2.00000	0.0788110
$645$	11.0000	0.433125
$646$	−4.00000	−0.157378
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−1.00000	−0.0392837
$649$	75.0000	2.94401
$650$	0	0
$651$	−22.0000	−0.862248
$652$	−15.0000	−0.587445
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	−2.00000	−0.0782062
$655$	−1.00000	−0.0390732
$656$	−2.00000	−0.0780869
$657$	−6.00000	−0.234082
$658$	18.0000	0.701713
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	5.00000	0.194625
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	−6.00000	−0.232845
$665$	4.00000	0.155113
$666$	−3.00000	−0.116248
$667$	−5.00000	−0.193601
$668$	3.00000	0.116073
$669$	−26.0000	−1.00522
$670$	−16.0000	−0.618134
$671$	−50.0000	−1.93023
$672$	−2.00000	−0.0771517
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	2.00000	0.0770371
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−4.00000	−0.153732	−0.0768662	−	0.997041i	$-0.524491\pi$
$677$	−4.00000	−0.153732	−0.0768662	+	0.997041i	$0.524491\pi$
$678$	11.0000	0.422452
$679$	4.00000	0.153506
$680$	2.00000	0.0766965
$681$	0	0
$682$	−55.0000	−2.10606
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−2.00000	−0.0764719
$685$	−11.0000	−0.420288
$686$	−20.0000	−0.763604
$687$	−10.0000	−0.381524
$688$	−11.0000	−0.419371
$689$	0	0
$690$	−1.00000	−0.0380693
$691$	−30.0000	−1.14125	−0.570627	−	0.821209i	$-0.693300\pi$
$691$	−30.0000	−1.14125	−0.570627	+	0.821209i	$0.693300\pi$
$692$	20.0000	0.760286
$693$	10.0000	0.379869
$694$	−34.0000	−1.29062
$695$	2.00000	0.0758643
$696$	5.00000	0.189525
$697$	4.00000	0.151511
$698$	−20.0000	−0.757011
$699$	−1.00000	−0.0378235
$700$	−2.00000	−0.0755929
$701$	13.0000	0.491003	0.245502	−	0.969396i	$-0.421047\pi$
$701$	13.0000	0.491003	0.245502	+	0.969396i	$0.421047\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	−5.00000	−0.188445
$705$	−9.00000	−0.338960
$706$	18.0000	0.677439
$707$	−4.00000	−0.150435
$708$	15.0000	0.563735
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	−11.0000	−0.412532
$712$	−2.00000	−0.0749532
$713$	11.0000	0.411953
$714$	4.00000	0.149696
$715$	0	0
$716$	−13.0000	−0.485833
$717$	20.0000	0.746914
$718$	−12.0000	−0.447836
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	1.00000	0.0372678
$721$	−20.0000	−0.744839
$722$	15.0000	0.558242
$723$	7.00000	0.260333
$724$	−16.0000	−0.594635
$725$	5.00000	0.185695
$726$	14.0000	0.519589
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	6.00000	0.222070
$731$	22.0000	0.813699
$732$	−10.0000	−0.369611
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	16.0000	0.590571
$735$	3.00000	0.110657
$736$	1.00000	0.0368605
$737$	−80.0000	−2.94684
$738$	2.00000	0.0736210
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	3.00000	0.110282
$741$	0	0
$742$	12.0000	0.440534
$743$	51.0000	1.87101	0.935504	−	0.353315i	$-0.114946\pi$
$743$	51.0000	1.87101	0.935504	+	0.353315i	$0.114946\pi$
$744$	−11.0000	−0.403280
$745$	17.0000	0.622832
$746$	19.0000	0.695639
$747$	6.00000	0.219529
$748$	10.0000	0.365636
$749$	−20.0000	−0.730784
$750$	1.00000	0.0365148
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	9.00000	0.328196
$753$	−25.0000	−0.911051
$754$	0	0
$755$	8.00000	0.291150
$756$	2.00000	0.0727393
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	2.00000	0.0726433
$759$	−5.00000	−0.181489
$760$	2.00000	0.0725476
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	2.00000	0.0724524
$763$	4.00000	0.144810
$764$	4.00000	0.144715
$765$	−2.00000	−0.0723102
$766$	−31.0000	−1.12008
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	43.0000	1.55062	0.775310	−	0.631581i	$-0.217594\pi$
$769$	43.0000	1.55062	0.775310	+	0.631581i	$0.217594\pi$
$770$	−10.0000	−0.360375
$771$	17.0000	0.612240
$772$	−24.0000	−0.863779
$773$	32.0000	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$774$	11.0000	0.395387
$775$	−11.0000	−0.395132
$776$	2.00000	0.0717958
$777$	6.00000	0.215249
$778$	−5.00000	−0.179259
$779$	4.00000	0.143315
$780$	0	0
$781$	0	0
$782$	−2.00000	−0.0715199
$783$	−5.00000	−0.178685
$784$	−3.00000	−0.107143
$785$	−7.00000	−0.249841
$786$	−1.00000	−0.0356688
$787$	−31.0000	−1.10503	−0.552515	−	0.833503i	$-0.686332\pi$
$787$	−31.0000	−1.10503	−0.552515	+	0.833503i	$0.686332\pi$
$788$	−8.00000	−0.284988
$789$	−21.0000	−0.747620
$790$	11.0000	0.391362
$791$	−22.0000	−0.782230
$792$	5.00000	0.177667
$793$	0	0
$794$	−3.00000	−0.106466
$795$	−6.00000	−0.212798
$796$	24.0000	0.850657
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	4.00000	0.141598
$799$	−18.0000	−0.636794
$800$	−1.00000	−0.0353553
$801$	2.00000	0.0706665
$802$	−36.0000	−1.27120
$803$	30.0000	1.05868
$804$	−16.0000	−0.564276
$805$	2.00000	0.0704907
$806$	0	0
$807$	−14.0000	−0.492823
$808$	−2.00000	−0.0703598
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−1.00000	−0.0351364
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−10.0000	−0.350931
$813$	13.0000	0.455930
$814$	15.0000	0.525750
$815$	−15.0000	−0.525427
$816$	2.00000	0.0700140
$817$	22.0000	0.769683
$818$	−30.0000	−1.04893
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	−51.0000	−1.77991	−0.889956	−	0.456046i	$-0.849265\pi$
$821$	−51.0000	−1.77991	−0.889956	+	0.456046i	$0.849265\pi$
$822$	−11.0000	−0.383669
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	−10.0000	−0.348367
$825$	5.00000	0.174078
$826$	−30.0000	−1.04383
$827$	−50.0000	−1.73867	−0.869335	−	0.494223i	$-0.835453\pi$
$827$	−50.0000	−1.73867	−0.869335	+	0.494223i	$0.835453\pi$
$828$	−1.00000	−0.0347524
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	−6.00000	−0.208263
$831$	11.0000	0.381586
$832$	0	0
$833$	6.00000	0.207888
$834$	2.00000	0.0692543
$835$	3.00000	0.103819
$836$	10.0000	0.345857
$837$	11.0000	0.380216
$838$	4.00000	0.138178
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	−2.00000	−0.0690066
$841$	−4.00000	−0.137931
$842$	16.0000	0.551396
$843$	10.0000	0.344418
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−9.00000	−0.309426
$847$	−28.0000	−0.962091
$848$	6.00000	0.206041
$849$	−19.0000	−0.652078
$850$	2.00000	0.0685994
$851$	−3.00000	−0.102839
$852$	0	0
$853$	−49.0000	−1.67773	−0.838864	−	0.544341i	$-0.816780\pi$
$853$	−49.0000	−1.67773	−0.838864	+	0.544341i	$0.816780\pi$
$854$	20.0000	0.684386
$855$	−2.00000	−0.0683986
$856$	−10.0000	−0.341793
$857$	−25.0000	−0.853984	−0.426992	−	0.904255i	$-0.640427\pi$
$857$	−25.0000	−0.853984	−0.426992	+	0.904255i	$0.640427\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	−11.0000	−0.375097
$861$	−4.00000	−0.136320
$862$	−40.0000	−1.36241
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	1.00000	0.0340207
$865$	20.0000	0.680020
$866$	−28.0000	−0.951479
$867$	13.0000	0.441503
$868$	22.0000	0.746729
$869$	55.0000	1.86575
$870$	5.00000	0.169516
$871$	0	0
$872$	2.00000	0.0677285
$873$	−2.00000	−0.0676897
$874$	−2.00000	−0.0676510
$875$	−2.00000	−0.0676123
$876$	6.00000	0.202721
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	4.00000	0.134993
$879$	−6.00000	−0.202375
$880$	−5.00000	−0.168550
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	3.00000	0.101015
$883$	−1.00000	−0.0336527	−0.0168263	−	0.999858i	$-0.505356\pi$
$883$	−1.00000	−0.0336527	−0.0168263	+	0.999858i	$0.505356\pi$
$884$	0	0
$885$	15.0000	0.504219
$886$	−20.0000	−0.671913
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	3.00000	0.100673
$889$	−4.00000	−0.134156
$890$	−2.00000	−0.0670402
$891$	−5.00000	−0.167506
$892$	26.0000	0.870544
$893$	−18.0000	−0.602347
$894$	17.0000	0.568565
$895$	−13.0000	−0.434542
$896$	2.00000	0.0668153
$897$	0	0
$898$	−12.0000	−0.400445
$899$	−55.0000	−1.83435
$900$	1.00000	0.0333333
$901$	−12.0000	−0.399778
$902$	−10.0000	−0.332964
$903$	−22.0000	−0.732114
$904$	−11.0000	−0.365855
$905$	−16.0000	−0.531858
$906$	8.00000	0.265782
$907$	19.0000	0.630885	0.315442	−	0.948945i	$-0.397847\pi$
$907$	19.0000	0.630885	0.315442	+	0.948945i	$0.397847\pi$
$908$	0	0
$909$	2.00000	0.0663358
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	2.00000	0.0662266
$913$	−30.0000	−0.992855
$914$	−38.0000	−1.25693
$915$	−10.0000	−0.330590
$916$	10.0000	0.330409
$917$	2.00000	0.0660458
$918$	−2.00000	−0.0660098
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	1.00000	0.0329690
$921$	0	0
$922$	39.0000	1.28440
$923$	0	0
$924$	−10.0000	−0.328976
$925$	3.00000	0.0986394
$926$	−14.0000	−0.460069
$927$	10.0000	0.328443
$928$	−5.00000	−0.164133
$929$	−16.0000	−0.524943	−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000	−0.524943	−0.262471	+	0.964940i	$0.584538\pi$
$930$	−11.0000	−0.360704
$931$	6.00000	0.196642
$932$	1.00000	0.0327561
$933$	−20.0000	−0.654771
$934$	−6.00000	−0.196326
$935$	10.0000	0.327035
$936$	0	0
$937$	−50.0000	−1.63343	−0.816714	−	0.577042i	$-0.804207\pi$
$937$	−50.0000	−1.63343	−0.816714	+	0.577042i	$0.804207\pi$
$938$	32.0000	1.04484
$939$	−20.0000	−0.652675
$940$	9.00000	0.293548
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	−7.00000	−0.228072
$943$	2.00000	0.0651290
$944$	−15.0000	−0.488208
$945$	2.00000	0.0650600
$946$	−55.0000	−1.78820
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	11.0000	0.357263
$949$	0	0
$950$	2.00000	0.0648886
$951$	−16.0000	−0.518836
$952$	−4.00000	−0.129641
$953$	−51.0000	−1.65205	−0.826026	−	0.563632i	$-0.809404\pi$
$953$	−51.0000	−1.65205	−0.826026	+	0.563632i	$0.809404\pi$
$954$	−6.00000	−0.194257
$955$	4.00000	0.129437
$956$	−20.0000	−0.646846
$957$	25.0000	0.808135
$958$	24.0000	0.775405
$959$	22.0000	0.710417
$960$	−1.00000	−0.0322749
$961$	90.0000	2.90323
$962$	0	0
$963$	10.0000	0.322245
$964$	−7.00000	−0.225455
$965$	−24.0000	−0.772587
$966$	2.00000	0.0643489
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	−14.0000	−0.449977
$969$	−4.00000	−0.128499
$970$	2.00000	0.0642161
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	−1.00000	−0.0320750
$973$	−4.00000	−0.128234
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	10.0000	0.320092
$977$	−21.0000	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$978$	−15.0000	−0.479647
$979$	−10.0000	−0.319601
$980$	−3.00000	−0.0958315
$981$	−2.00000	−0.0638551
$982$	16.0000	0.510581
$983$	−3.00000	−0.0956851	−0.0478426	−	0.998855i	$-0.515235\pi$
$983$	−3.00000	−0.0956851	−0.0478426	+	0.998855i	$0.515235\pi$
$984$	−2.00000	−0.0637577
$985$	−8.00000	−0.254901
$986$	10.0000	0.318465
$987$	18.0000	0.572946
$988$	0	0
$989$	11.0000	0.349780
$990$	5.00000	0.158910
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	11.0000	0.349250
$993$	28.0000	0.888553
$994$	0	0
$995$	24.0000	0.760851
$996$	−6.00000	−0.190117
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	−28.0000	−0.886325
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.d.1.1		1
13.3	even	3		390.2.i.f.61.1	✓	2
13.5	odd	4		5070.2.b.h.1351.2		2
13.8	odd	4		5070.2.b.h.1351.1		2
13.9	even	3		390.2.i.f.211.1	yes	2
13.12	even	2		5070.2.a.p.1.1		1
39.29	odd	6		1170.2.i.a.451.1		2
39.35	odd	6		1170.2.i.a.991.1		2
65.3	odd	12		1950.2.z.e.1699.1		4
65.9	even	6		1950.2.i.d.601.1		2
65.22	odd	12		1950.2.z.e.1849.1		4
65.29	even	6		1950.2.i.d.451.1		2
65.42	odd	12		1950.2.z.e.1699.2		4
65.48	odd	12		1950.2.z.e.1849.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.f.61.1	✓	2	13.3	even	3
390.2.i.f.211.1	yes	2	13.9	even	3
1170.2.i.a.451.1		2	39.29	odd	6
1170.2.i.a.991.1		2	39.35	odd	6
1950.2.i.d.451.1		2	65.29	even	6
1950.2.i.d.601.1		2	65.9	even	6
1950.2.z.e.1699.1		4	65.3	odd	12
1950.2.z.e.1699.2		4	65.42	odd	12
1950.2.z.e.1849.1		4	65.22	odd	12
1950.2.z.e.1849.2		4	65.48	odd	12
5070.2.a.d.1.1		1	1.1	even	1	trivial
5070.2.a.p.1.1		1	13.12	even	2
5070.2.b.h.1351.1		2	13.8	odd	4
5070.2.b.h.1351.2		2	13.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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