LMFDB - Embedded newform 8470.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -7.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -2.00000 q^{18} +5.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +9.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -7.00000 q^{26} -5.00000 q^{27} +1.00000 q^{28} -1.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} -1.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +5.00000 q^{38} -7.00000 q^{39} -1.00000 q^{40} +12.0000 q^{41} +1.00000 q^{42} -4.00000 q^{43} +2.00000 q^{45} +9.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -7.00000 q^{52} -6.00000 q^{53} -5.00000 q^{54} +1.00000 q^{56} +5.00000 q^{57} +9.00000 q^{59} -1.00000 q^{60} +14.0000 q^{61} +2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +7.00000 q^{65} +2.00000 q^{67} -6.00000 q^{68} +9.00000 q^{69} -1.00000 q^{70} -2.00000 q^{72} -10.0000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +5.00000 q^{76} -7.00000 q^{78} +11.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} +9.00000 q^{83} +1.00000 q^{84} +6.00000 q^{85} -4.00000 q^{86} +12.0000 q^{89} +2.00000 q^{90} -7.00000 q^{91} +9.00000 q^{92} +2.00000 q^{93} +6.00000 q^{94} -5.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	1.00000	0.288675
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−2.00000	−0.471405
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	0	0
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−7.00000	−1.37281
$27$	−5.00000	−0.962250
$28$	1.00000	0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	−1.00000	−0.169031
$36$	−2.00000	−0.333333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	5.00000	0.811107
$39$	−7.00000	−1.12090
$40$	−1.00000	−0.158114
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	1.00000	0.154303
$43$	−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
$45$	2.00000	0.298142
$46$	9.00000	1.32698
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−6.00000	−0.840168
$52$	−7.00000	−0.970725
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	1.00000	0.133631
$57$	5.00000	0.662266
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	−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	$0.146275\pi$
$62$	2.00000	0.254000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	7.00000	0.868243
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−6.00000	−0.727607
$69$	9.00000	1.08347
$70$	−1.00000	−0.119523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−2.00000	−0.235702
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	5.00000	0.573539
$77$	0	0
$78$	−7.00000	−0.792594
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	12.0000	1.32518
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	1.00000	0.109109
$85$	6.00000	0.650791
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	2.00000	0.210819
$91$	−7.00000	−0.733799
$92$	9.00000	0.938315
$93$	2.00000	0.207390
$94$	6.00000	0.618853
$95$	−5.00000	−0.512989
$96$	1.00000	0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	−6.00000	−0.594089
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−7.00000	−0.686406
$105$	−1.00000	−0.0975900
$106$	−6.00000	−0.582772
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	−5.00000	−0.481125
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	1.00000	0.0944911
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	5.00000	0.468293
$115$	−9.00000	−0.839254
$116$	0	0
$117$	14.0000	1.29430
$118$	9.00000	0.828517
$119$	−6.00000	−0.550019
$120$	−1.00000	−0.0912871
$121$	0	0
$122$	14.0000	1.26750
$123$	12.0000	1.08200
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	7.00000	0.613941
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	2.00000	0.172774
$135$	5.00000	0.430331
$136$	−6.00000	−0.514496
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	9.00000	0.766131
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	−1.00000	−0.0845154
$141$	6.00000	0.505291
$142$	0	0
$143$	0	0
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−10.0000	−0.827606
$147$	1.00000	0.0824786
$148$	2.00000	0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	1.00000	0.0816497
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	5.00000	0.405554
$153$	12.0000	0.970143
$154$	0	0
$155$	−2.00000	−0.160644
$156$	−7.00000	−0.560449
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	11.0000	0.875113
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	9.00000	0.709299
$162$	1.00000	0.0785674
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	9.00000	0.698535
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	1.00000	0.0771517
$169$	36.0000	2.76923
$170$	6.00000	0.460179
$171$	−10.0000	−0.764719
$172$	−4.00000	−0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	9.00000	0.676481
$178$	12.0000	0.899438
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	2.00000	0.149071
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	−7.00000	−0.518875
$183$	14.0000	1.03491
$184$	9.00000	0.663489
$185$	−2.00000	−0.147043
$186$	2.00000	0.146647
$187$	0	0
$188$	6.00000	0.437595
$189$	−5.00000	−0.363696
$190$	−5.00000	−0.362738
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	1.00000	0.0721688
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	8.00000	0.574367
$195$	7.00000	0.501280
$196$	1.00000	0.0714286
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	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
$201$	2.00000	0.141069
$202$	9.00000	0.633238
$203$	0	0
$204$	−6.00000	−0.420084
$205$	−12.0000	−0.838116
$206$	8.00000	0.557386
$207$	−18.0000	−1.25109
$208$	−7.00000	−0.485363
$209$	0	0
$210$	−1.00000	−0.0690066
$211$	−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
$214$	−18.0000	−1.23045
$215$	4.00000	0.272798
$216$	−5.00000	−0.340207
$217$	2.00000	0.135769
$218$	2.00000	0.135457
$219$	−10.0000	−0.675737
$220$	0	0
$221$	42.0000	2.82523
$222$	2.00000	0.134231
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	1.00000	0.0668153
$225$	−2.00000	−0.133333
$226$	−15.0000	−0.997785
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	5.00000	0.331133
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	−9.00000	−0.593442
$231$	0	0
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	14.0000	0.915209
$235$	−6.00000	−0.391397
$236$	9.00000	0.585850
$237$	11.0000	0.714527
$238$	−6.00000	−0.388922
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	−1.00000	−0.0645497
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	14.0000	0.896258
$245$	−1.00000	−0.0638877
$246$	12.0000	0.765092
$247$	−35.0000	−2.22700
$248$	2.00000	0.127000
$249$	9.00000	0.570352
$250$	−1.00000	−0.0632456
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	5.00000	0.313728
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−4.00000	−0.249029
$259$	2.00000	0.124274
$260$	7.00000	0.434122
$261$	0	0
$262$	−3.00000	−0.185341
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	5.00000	0.306570
$267$	12.0000	0.734388
$268$	2.00000	0.122169
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	5.00000	0.304290
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−6.00000	−0.363803
$273$	−7.00000	−0.423659
$274$	−3.00000	−0.181237
$275$	0	0
$276$	9.00000	0.541736
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	5.00000	0.299880
$279$	−4.00000	−0.239474
$280$	−1.00000	−0.0597614
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	6.00000	0.357295
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	0	0
$285$	−5.00000	−0.296174
$286$	0	0
$287$	12.0000	0.708338
$288$	−2.00000	−0.117851
$289$	19.0000	1.11765
$290$	0	0
$291$	8.00000	0.468968
$292$	−10.0000	−0.585206
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	1.00000	0.0583212
$295$	−9.00000	−0.524000
$296$	2.00000	0.116248
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−63.0000	−3.64338
$300$	1.00000	0.0577350
$301$	−4.00000	−0.230556
$302$	5.00000	0.287718
$303$	9.00000	0.517036
$304$	5.00000	0.286770
$305$	−14.0000	−0.801638
$306$	12.0000	0.685994
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	−2.00000	−0.113592
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	−7.00000	−0.396297
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	17.0000	0.959366
$315$	2.00000	0.112687
$316$	11.0000	0.618798
$317$	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
$321$	−18.0000	−1.00466
$322$	9.00000	0.501550
$323$	−30.0000	−1.66924
$324$	1.00000	0.0555556
$325$	−7.00000	−0.388290
$326$	2.00000	0.110770
$327$	2.00000	0.110600
$328$	12.0000	0.662589
$329$	6.00000	0.330791
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	9.00000	0.493939
$333$	−4.00000	−0.219199
$334$	12.0000	0.656611
$335$	−2.00000	−0.109272
$336$	1.00000	0.0545545
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	36.0000	1.95814
$339$	−15.0000	−0.814688
$340$	6.00000	0.325396
$341$	0	0
$342$	−10.0000	−0.540738
$343$	1.00000	0.0539949
$344$	−4.00000	−0.215666
$345$	−9.00000	−0.484544
$346$	6.00000	0.322562
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	1.00000	0.0534522
$351$	35.0000	1.86816
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	9.00000	0.478345
$355$	0	0
$356$	12.0000	0.635999
$357$	−6.00000	−0.317554
$358$	−12.0000	−0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	2.00000	0.105409
$361$	6.00000	0.315789
$362$	−13.0000	−0.683265
$363$	0	0
$364$	−7.00000	−0.366900
$365$	10.0000	0.523424
$366$	14.0000	0.731792
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	9.00000	0.469157
$369$	−24.0000	−1.24939
$370$	−2.00000	−0.103975
$371$	−6.00000	−0.311504
$372$	2.00000	0.103695
$373$	−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
$376$	6.00000	0.309426
$377$	0	0
$378$	−5.00000	−0.257172
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	−5.00000	−0.256495
$381$	5.00000	0.256158
$382$	3.00000	0.153493
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−13.0000	−0.661683
$387$	8.00000	0.406663
$388$	8.00000	0.406138
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	7.00000	0.354459
$391$	−54.0000	−2.73090
$392$	1.00000	0.0505076
$393$	−3.00000	−0.151330
$394$	−12.0000	−0.604551
$395$	−11.0000	−0.553470
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	−16.0000	−0.802008
$399$	5.00000	0.250313
$400$	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	$0.351624\pi$
$402$	2.00000	0.0997509
$403$	−14.0000	−0.697390
$404$	9.00000	0.447767
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	−6.00000	−0.297044
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	−12.0000	−0.592638
$411$	−3.00000	−0.147979
$412$	8.00000	0.394132
$413$	9.00000	0.442861
$414$	−18.0000	−0.884652
$415$	−9.00000	−0.441793
$416$	−7.00000	−0.343203
$417$	5.00000	0.244851
$418$	0	0
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	−1.00000	−0.0487950
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−4.00000	−0.194717
$423$	−12.0000	−0.583460
$424$	−6.00000	−0.291386
$425$	−6.00000	−0.291043
$426$	0	0
$427$	14.0000	0.677507
$428$	−18.0000	−0.870063
$429$	0	0
$430$	4.00000	0.192897
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	−5.00000	−0.240563
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	2.00000	0.0957826
$437$	45.0000	2.15264
$438$	−10.0000	−0.477818
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	42.0000	1.99774
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	2.00000	0.0949158
$445$	−12.0000	−0.568855
$446$	2.00000	0.0947027
$447$	−6.00000	−0.283790
$448$	1.00000	0.0472456
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	−2.00000	−0.0942809
$451$	0	0
$452$	−15.0000	−0.705541
$453$	5.00000	0.234920
$454$	−24.0000	−1.12638
$455$	7.00000	0.328165
$456$	5.00000	0.234146
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	26.0000	1.21490
$459$	30.0000	1.40028
$460$	−9.00000	−0.419627
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	3.00000	0.138972
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	14.0000	0.647150
$469$	2.00000	0.0923514
$470$	−6.00000	−0.276759
$471$	17.0000	0.783319
$472$	9.00000	0.414259
$473$	0	0
$474$	11.0000	0.505247
$475$	5.00000	0.229416
$476$	−6.00000	−0.275010
$477$	12.0000	0.549442
$478$	−15.0000	−0.686084
$479$	−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
$481$	−14.0000	−0.638345
$482$	20.0000	0.910975
$483$	9.00000	0.409514
$484$	0	0
$485$	−8.00000	−0.363261
$486$	16.0000	0.725775
$487$	−25.0000	−1.13286	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	−25.0000	−1.13286	−0.566429	+	0.824110i	$0.691675\pi$
$488$	14.0000	0.633750
$489$	2.00000	0.0904431
$490$	−1.00000	−0.0451754
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	12.0000	0.541002
$493$	0	0
$494$	−35.0000	−1.57472
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	9.00000	0.403300
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	−1.00000	−0.0447214
$501$	12.0000	0.536120
$502$	24.0000	1.07117
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	−2.00000	−0.0890871
$505$	−9.00000	−0.400495
$506$	0	0
$507$	36.0000	1.59882
$508$	5.00000	0.221839
$509$	−9.00000	−0.398918	−0.199459	−	0.979906i	$-0.563918\pi$
$509$	−9.00000	−0.398918	−0.199459	+	0.979906i	$0.563918\pi$
$510$	6.00000	0.265684
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	−25.0000	−1.10378
$514$	18.0000	0.793946
$515$	−8.00000	−0.352522
$516$	−4.00000	−0.176090
$517$	0	0
$518$	2.00000	0.0878750
$519$	6.00000	0.263371
$520$	7.00000	0.306970
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	0	0
$523$	−7.00000	−0.306089	−0.153044	−	0.988219i	$-0.548908\pi$
$523$	−7.00000	−0.306089	−0.153044	+	0.988219i	$0.548908\pi$
$524$	−3.00000	−0.131056
$525$	1.00000	0.0436436
$526$	−21.0000	−0.915644
$527$	−12.0000	−0.522728
$528$	0	0
$529$	58.0000	2.52174
$530$	6.00000	0.260623
$531$	−18.0000	−0.781133
$532$	5.00000	0.216777
$533$	−84.0000	−3.63844
$534$	12.0000	0.519291
$535$	18.0000	0.778208
$536$	2.00000	0.0863868
$537$	−12.0000	−0.517838
$538$	15.0000	0.646696
$539$	0	0
$540$	5.00000	0.215166
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	20.0000	0.859074
$543$	−13.0000	−0.557883
$544$	−6.00000	−0.257248
$545$	−2.00000	−0.0856706
$546$	−7.00000	−0.299572
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	−3.00000	−0.128154
$549$	−28.0000	−1.19501
$550$	0	0
$551$	0	0
$552$	9.00000	0.383065
$553$	11.0000	0.467768
$554$	−28.0000	−1.18961
$555$	−2.00000	−0.0848953
$556$	5.00000	0.212047
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−4.00000	−0.169334
$559$	28.0000	1.18427
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−15.0000	−0.632737
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	6.00000	0.252646
$565$	15.0000	0.631055
$566$	5.00000	0.210166
$567$	1.00000	0.0419961
$568$	0	0
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	−5.00000	−0.209427
$571$	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
$573$	3.00000	0.125327
$574$	12.0000	0.500870
$575$	9.00000	0.375326
$576$	−2.00000	−0.0833333
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	19.0000	0.790296
$579$	−13.0000	−0.540262
$580$	0	0
$581$	9.00000	0.373383
$582$	8.00000	0.331611
$583$	0	0
$584$	−10.0000	−0.413803
$585$	−14.0000	−0.578829
$586$	−9.00000	−0.371787
$587$	15.0000	0.619116	0.309558	−	0.950881i	$-0.399819\pi$
$587$	15.0000	0.619116	0.309558	+	0.950881i	$0.399819\pi$
$588$	1.00000	0.0412393
$589$	10.0000	0.412043
$590$	−9.00000	−0.370524
$591$	−12.0000	−0.493614
$592$	2.00000	0.0821995
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	−6.00000	−0.245770
$597$	−16.0000	−0.654836
$598$	−63.0000	−2.57626
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	1.00000	0.0408248
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	−4.00000	−0.163028
$603$	−4.00000	−0.162893
$604$	5.00000	0.203447
$605$	0	0
$606$	9.00000	0.365600
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	−14.0000	−0.566843
$611$	−42.0000	−1.69914
$612$	12.0000	0.485071
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	−28.0000	−1.12999
$615$	−12.0000	−0.483887
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	8.00000	0.321807
$619$	−25.0000	−1.00483	−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000	−1.00483	−0.502417	+	0.864625i	$0.667556\pi$
$620$	−2.00000	−0.0803219
$621$	−45.0000	−1.80579
$622$	−12.0000	−0.481156
$623$	12.0000	0.480770
$624$	−7.00000	−0.280224
$625$	1.00000	0.0400000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	17.0000	0.678374
$629$	−12.0000	−0.478471
$630$	2.00000	0.0796819
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	11.0000	0.437557
$633$	−4.00000	−0.158986
$634$	18.0000	0.714871
$635$	−5.00000	−0.198419
$636$	−6.00000	−0.237915
$637$	−7.00000	−0.277350
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	−18.0000	−0.710403
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	9.00000	0.354650
$645$	4.00000	0.157500
$646$	−30.0000	−1.18033
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−7.00000	−0.274563
$651$	2.00000	0.0783862
$652$	2.00000	0.0783260
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	2.00000	0.0782062
$655$	3.00000	0.117220
$656$	12.0000	0.468521
$657$	20.0000	0.780274
$658$	6.00000	0.233904
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	−7.00000	−0.272268	−0.136134	−	0.990690i	$-0.543468\pi$
$661$	−7.00000	−0.272268	−0.136134	+	0.990690i	$0.543468\pi$
$662$	−4.00000	−0.155464
$663$	42.0000	1.63114
$664$	9.00000	0.349268
$665$	−5.00000	−0.193892
$666$	−4.00000	−0.154997
$667$	0	0
$668$	12.0000	0.464294
$669$	2.00000	0.0773245
$670$	−2.00000	−0.0772667
$671$	0	0
$672$	1.00000	0.0385758
$673$	41.0000	1.58043	0.790217	−	0.612827i	$-0.209968\pi$
$673$	41.0000	1.58043	0.790217	+	0.612827i	$0.209968\pi$
$674$	23.0000	0.885927
$675$	−5.00000	−0.192450
$676$	36.0000	1.38462
$677$	33.0000	1.26829	0.634147	−	0.773213i	$-0.281352\pi$
$677$	33.0000	1.26829	0.634147	+	0.773213i	$0.281352\pi$
$678$	−15.0000	−0.576072
$679$	8.00000	0.307012
$680$	6.00000	0.230089
$681$	−24.0000	−0.919682
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−10.0000	−0.382360
$685$	3.00000	0.114624
$686$	1.00000	0.0381802
$687$	26.0000	0.991962
$688$	−4.00000	−0.152499
$689$	42.0000	1.60007
$690$	−9.00000	−0.342624
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	18.0000	0.683271
$695$	−5.00000	−0.189661
$696$	0	0
$697$	−72.0000	−2.72719
$698$	−25.0000	−0.946264
$699$	3.00000	0.113470
$700$	1.00000	0.0377964
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	35.0000	1.32099
$703$	10.0000	0.377157
$704$	0	0
$705$	−6.00000	−0.225973
$706$	18.0000	0.677439
$707$	9.00000	0.338480
$708$	9.00000	0.338241
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	−22.0000	−0.825064
$712$	12.0000	0.449719
$713$	18.0000	0.674105
$714$	−6.00000	−0.224544
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−15.0000	−0.560185
$718$	−24.0000	−0.895672
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	2.00000	0.0745356
$721$	8.00000	0.297936
$722$	6.00000	0.223297
$723$	20.0000	0.743808
$724$	−13.0000	−0.483141
$725$	0	0
$726$	0	0
$727$	20.0000	0.741759	0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000	0.741759	0.370879	+	0.928681i	$0.379056\pi$
$728$	−7.00000	−0.259437
$729$	13.0000	0.481481
$730$	10.0000	0.370117
$731$	24.0000	0.887672
$732$	14.0000	0.517455
$733$	−49.0000	−1.80986	−0.904928	−	0.425564i	$-0.860076\pi$
$733$	−49.0000	−1.80986	−0.904928	+	0.425564i	$0.860076\pi$
$734$	−28.0000	−1.03350
$735$	−1.00000	−0.0368856
$736$	9.00000	0.331744
$737$	0	0
$738$	−24.0000	−0.883452
$739$	38.0000	1.39785	0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000	1.39785	0.698926	+	0.715194i	$0.253662\pi$
$740$	−2.00000	−0.0735215
$741$	−35.0000	−1.28576
$742$	−6.00000	−0.220267
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	2.00000	0.0733236
$745$	6.00000	0.219823
$746$	−22.0000	−0.805477
$747$	−18.0000	−0.658586
$748$	0	0
$749$	−18.0000	−0.657706
$750$	−1.00000	−0.0365148
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	6.00000	0.218797
$753$	24.0000	0.874609
$754$	0	0
$755$	−5.00000	−0.181969
$756$	−5.00000	−0.181848
$757$	−16.0000	−0.581530	−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000	−0.581530	−0.290765	+	0.956795i	$0.593910\pi$
$758$	−10.0000	−0.363216
$759$	0	0
$760$	−5.00000	−0.181369
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	5.00000	0.181131
$763$	2.00000	0.0724049
$764$	3.00000	0.108536
$765$	−12.0000	−0.433861
$766$	24.0000	0.867155
$767$	−63.0000	−2.27480
$768$	1.00000	0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	−13.0000	−0.467880
$773$	27.0000	0.971123	0.485561	−	0.874203i	$-0.338615\pi$
$773$	27.0000	0.971123	0.485561	+	0.874203i	$0.338615\pi$
$774$	8.00000	0.287554
$775$	2.00000	0.0718421
$776$	8.00000	0.287183
$777$	2.00000	0.0717496
$778$	18.0000	0.645331
$779$	60.0000	2.14972
$780$	7.00000	0.250640
$781$	0	0
$782$	−54.0000	−1.93104
$783$	0	0
$784$	1.00000	0.0357143
$785$	−17.0000	−0.606756
$786$	−3.00000	−0.107006
$787$	−52.0000	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$788$	−12.0000	−0.427482
$789$	−21.0000	−0.747620
$790$	−11.0000	−0.391362
$791$	−15.0000	−0.533339
$792$	0	0
$793$	−98.0000	−3.48008
$794$	−34.0000	−1.20661
$795$	6.00000	0.212798
$796$	−16.0000	−0.567105
$797$	−27.0000	−0.956389	−0.478195	−	0.878254i	$-0.658709\pi$
$797$	−27.0000	−0.956389	−0.478195	+	0.878254i	$0.658709\pi$
$798$	5.00000	0.176998
$799$	−36.0000	−1.27359
$800$	1.00000	0.0353553
$801$	−24.0000	−0.847998
$802$	18.0000	0.635602
$803$	0	0
$804$	2.00000	0.0705346
$805$	−9.00000	−0.317208
$806$	−14.0000	−0.493129
$807$	15.0000	0.528025
$808$	9.00000	0.316619
$809$	−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
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	−6.00000	−0.210042
$817$	−20.0000	−0.699711
$818$	−4.00000	−0.139857
$819$	14.0000	0.489200
$820$	−12.0000	−0.419058
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−3.00000	−0.104637
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	9.00000	0.313150
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	−18.0000	−0.625543
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	−9.00000	−0.312395
$831$	−28.0000	−0.971309
$832$	−7.00000	−0.242681
$833$	−6.00000	−0.207888
$834$	5.00000	0.173136
$835$	−12.0000	−0.415277
$836$	0	0
$837$	−10.0000	−0.345651
$838$	−3.00000	−0.103633
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	−1.00000	−0.0345033
$841$	−29.0000	−1.00000
$842$	8.00000	0.275698
$843$	−15.0000	−0.516627
$844$	−4.00000	−0.137686
$845$	−36.0000	−1.23844
$846$	−12.0000	−0.412568
$847$	0	0
$848$	−6.00000	−0.206041
$849$	5.00000	0.171600
$850$	−6.00000	−0.205798
$851$	18.0000	0.617032
$852$	0	0
$853$	−7.00000	−0.239675	−0.119838	−	0.992793i	$-0.538237\pi$
$853$	−7.00000	−0.239675	−0.119838	+	0.992793i	$0.538237\pi$
$854$	14.0000	0.479070
$855$	10.0000	0.341993
$856$	−18.0000	−0.615227
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	4.00000	0.136399
$861$	12.0000	0.408959
$862$	33.0000	1.12398
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−5.00000	−0.170103
$865$	−6.00000	−0.204006
$866$	8.00000	0.271851
$867$	19.0000	0.645274
$868$	2.00000	0.0678844
$869$	0	0
$870$	0	0
$871$	−14.0000	−0.474372
$872$	2.00000	0.0677285
$873$	−16.0000	−0.541518
$874$	45.0000	1.52215
$875$	−1.00000	−0.0338062
$876$	−10.0000	−0.337869
$877$	−52.0000	−1.75592	−0.877958	−	0.478738i	$-0.841094\pi$
$877$	−52.0000	−1.75592	−0.877958	+	0.478738i	$0.841094\pi$
$878$	−28.0000	−0.944954
$879$	−9.00000	−0.303562
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−2.00000	−0.0673435
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	42.0000	1.41261
$885$	−9.00000	−0.302532
$886$	−24.0000	−0.806296
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	2.00000	0.0671156
$889$	5.00000	0.167695
$890$	−12.0000	−0.402241
$891$	0	0
$892$	2.00000	0.0669650
$893$	30.0000	1.00391
$894$	−6.00000	−0.200670
$895$	12.0000	0.401116
$896$	1.00000	0.0334077
$897$	−63.0000	−2.10351
$898$	−27.0000	−0.901002
$899$	0	0
$900$	−2.00000	−0.0666667
$901$	36.0000	1.19933
$902$	0	0
$903$	−4.00000	−0.133112
$904$	−15.0000	−0.498893
$905$	13.0000	0.432135
$906$	5.00000	0.166114
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−24.0000	−0.796468
$909$	−18.0000	−0.597022
$910$	7.00000	0.232048
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	5.00000	0.165567
$913$	0	0
$914$	−1.00000	−0.0330771
$915$	−14.0000	−0.462826
$916$	26.0000	0.859064
$917$	−3.00000	−0.0990687
$918$	30.0000	0.990148
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−9.00000	−0.296721
$921$	−28.0000	−0.922631
$922$	−18.0000	−0.592798
$923$	0	0
$924$	0	0
$925$	2.00000	0.0657596
$926$	5.00000	0.164310
$927$	−16.0000	−0.525509
$928$	0	0
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	−2.00000	−0.0655826
$931$	5.00000	0.163868
$932$	3.00000	0.0982683
$933$	−12.0000	−0.392862
$934$	21.0000	0.687141
$935$	0	0
$936$	14.0000	0.457604
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	2.00000	0.0653023
$939$	−10.0000	−0.326338
$940$	−6.00000	−0.195698
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	17.0000	0.553890
$943$	108.000	3.51696
$944$	9.00000	0.292925
$945$	5.00000	0.162650
$946$	0	0
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	11.0000	0.357263
$949$	70.0000	2.27230
$950$	5.00000	0.162221
$951$	18.0000	0.583690
$952$	−6.00000	−0.194461
$953$	−33.0000	−1.06897	−0.534487	−	0.845176i	$-0.679495\pi$
$953$	−33.0000	−1.06897	−0.534487	+	0.845176i	$0.679495\pi$
$954$	12.0000	0.388514
$955$	−3.00000	−0.0970777
$956$	−15.0000	−0.485135
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−3.00000	−0.0968751
$960$	−1.00000	−0.0322749
$961$	−27.0000	−0.870968
$962$	−14.0000	−0.451378
$963$	36.0000	1.16008
$964$	20.0000	0.644157
$965$	13.0000	0.418485
$966$	9.00000	0.289570
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	−30.0000	−0.963739
$970$	−8.00000	−0.256865
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	16.0000	0.513200
$973$	5.00000	0.160293
$974$	−25.0000	−0.801052
$975$	−7.00000	−0.224179
$976$	14.0000	0.448129
$977$	45.0000	1.43968	0.719839	−	0.694141i	$-0.244216\pi$
$977$	45.0000	1.43968	0.719839	+	0.694141i	$0.244216\pi$
$978$	2.00000	0.0639529
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−4.00000	−0.127710
$982$	6.00000	0.191468
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	12.0000	0.382546
$985$	12.0000	0.382352
$986$	0	0
$987$	6.00000	0.190982
$988$	−35.0000	−1.11350
$989$	−36.0000	−1.14473
$990$	0	0
$991$	23.0000	0.730619	0.365310	−	0.930886i	$-0.380963\pi$
$991$	23.0000	0.730619	0.365310	+	0.930886i	$0.380963\pi$
$992$	2.00000	0.0635001
$993$	−4.00000	−0.126936
$994$	0	0
$995$	16.0000	0.507234
$996$	9.00000	0.285176
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	14.0000	0.443162
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bc.1.1	yes	1
11.10	odd	2		8470.2.a.l.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.l.1.1	✓	1	11.10	odd	2
8470.2.a.bc.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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