LMFDB - Embedded newform 8470.2.a.s.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +8.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} +8.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} +2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{45} -6.00000 q^{46} -6.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} +4.00000 q^{54} -1.00000 q^{56} -16.0000 q^{57} +8.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -4.00000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +2.00000 q^{67} +2.00000 q^{68} +12.0000 q^{69} +1.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -10.0000 q^{74} -2.00000 q^{75} +8.00000 q^{76} -4.00000 q^{78} +4.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} +4.00000 q^{83} +2.00000 q^{84} -2.00000 q^{85} +4.00000 q^{86} -16.0000 q^{87} -2.00000 q^{89} -1.00000 q^{90} -2.00000 q^{91} -6.00000 q^{92} +16.0000 q^{93} -6.00000 q^{94} -8.00000 q^{95} -2.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.00000	−0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−2.00000	−0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−2.00000	−0.408248
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	4.00000	0.769800
$28$	−1.00000	−0.188982
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	2.00000	0.365148
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	8.00000	1.29777
$39$	−4.00000	−0.640513
$40$	−1.00000	−0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	2.00000	0.308607
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	−6.00000	−0.884652
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−4.00000	−0.560112
$52$	2.00000	0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−16.0000	−2.11925
$58$	8.00000	1.05045
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	2.00000	0.258199
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−8.00000	−1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	2.00000	0.242536
$69$	12.0000	1.44463
$70$	1.00000	0.119523
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	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	$0.614200\pi$
$74$	−10.0000	−1.16248
$75$	−2.00000	−0.230940
$76$	8.00000	0.917663
$77$	0	0
$78$	−4.00000	−0.452911
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	2.00000	0.218218
$85$	−2.00000	−0.216930
$86$	4.00000	0.431331
$87$	−16.0000	−1.71538
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	−1.00000	−0.105409
$91$	−2.00000	−0.209657
$92$	−6.00000	−0.625543
$93$	16.0000	1.65912
$94$	−6.00000	−0.618853
$95$	−8.00000	−0.820783
$96$	−2.00000	−0.204124
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	−4.00000	−0.396059
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	2.00000	0.196116
$105$	−2.00000	−0.195180
$106$	−2.00000	−0.194257
$107$	20.0000	1.93347	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000	1.93347	0.966736	+	0.255774i	$0.0823304\pi$
$108$	4.00000	0.384900
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	20.0000	1.89832
$112$	−1.00000	−0.0944911
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	−16.0000	−1.49854
$115$	6.00000	0.559503
$116$	8.00000	0.742781
$117$	2.00000	0.184900
$118$	4.00000	0.368230
$119$	−2.00000	−0.183340
$120$	2.00000	0.182574
$121$	0	0
$122$	−4.00000	−0.362143
$123$	0	0
$124$	−8.00000	−0.718421
$125$	−1.00000	−0.0894427
$126$	−1.00000	−0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−8.00000	−0.704361
$130$	−2.00000	−0.175412
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	2.00000	0.172774
$135$	−4.00000	−0.344265
$136$	2.00000	0.171499
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	12.0000	1.02151
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	1.00000	0.0845154
$141$	12.0000	1.01058
$142$	8.00000	0.671345
$143$	0	0
$144$	1.00000	0.0833333
$145$	−8.00000	−0.664364
$146$	−6.00000	−0.496564
$147$	−2.00000	−0.164957
$148$	−10.0000	−0.821995
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	−2.00000	−0.163299
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	8.00000	0.648886
$153$	2.00000	0.161690
$154$	0	0
$155$	8.00000	0.642575
$156$	−4.00000	−0.320256
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	4.00000	0.318223
$159$	4.00000	0.317221
$160$	−1.00000	−0.0790569
$161$	6.00000	0.472866
$162$	−11.0000	−0.864242
$163$	18.0000	1.40987	0.704934	−	0.709273i	$-0.250976\pi$
$163$	18.0000	1.40987	0.704934	+	0.709273i	$0.250976\pi$
$164$	0	0
$165$	0	0
$166$	4.00000	0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	2.00000	0.154303
$169$	−9.00000	−0.692308
$170$	−2.00000	−0.153393
$171$	8.00000	0.611775
$172$	4.00000	0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−16.0000	−1.21296
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−8.00000	−0.601317
$178$	−2.00000	−0.149906
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−1.00000	−0.0745356
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	−2.00000	−0.148250
$183$	8.00000	0.591377
$184$	−6.00000	−0.442326
$185$	10.0000	0.735215
$186$	16.0000	1.17318
$187$	0	0
$188$	−6.00000	−0.437595
$189$	−4.00000	−0.290957
$190$	−8.00000	−0.580381
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	−2.00000	−0.144338
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	−2.00000	−0.143592
$195$	4.00000	0.286446
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	1.00000	0.0707107
$201$	−4.00000	−0.282138
$202$	−8.00000	−0.562878
$203$	−8.00000	−0.561490
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−14.0000	−0.975426
$207$	−6.00000	−0.417029
$208$	2.00000	0.138675
$209$	0	0
$210$	−2.00000	−0.138013
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−2.00000	−0.137361
$213$	−16.0000	−1.09630
$214$	20.0000	1.36717
$215$	−4.00000	−0.272798
$216$	4.00000	0.272166
$217$	8.00000	0.543075
$218$	12.0000	0.812743
$219$	12.0000	0.810885
$220$	0	0
$221$	4.00000	0.269069
$222$	20.0000	1.34231
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	−18.0000	−1.19734
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−16.0000	−1.05963
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	8.00000	0.525226
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	2.00000	0.130744
$235$	6.00000	0.391397
$236$	4.00000	0.260378
$237$	−8.00000	−0.519656
$238$	−2.00000	−0.129641
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	2.00000	0.129099
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	−4.00000	−0.256074
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	16.0000	1.01806
$248$	−8.00000	−0.508001
$249$	−8.00000	−0.506979
$250$	−1.00000	−0.0632456
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	8.00000	0.501965
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	−8.00000	−0.498058
$259$	10.0000	0.621370
$260$	−2.00000	−0.124035
$261$	8.00000	0.495188
$262$	−20.0000	−1.23560
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	−8.00000	−0.490511
$267$	4.00000	0.244796
$268$	2.00000	0.122169
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−4.00000	−0.243432
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	2.00000	0.121268
$273$	4.00000	0.242091
$274$	6.00000	0.362473
$275$	0	0
$276$	12.0000	0.722315
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	4.00000	0.239904
$279$	−8.00000	−0.478947
$280$	1.00000	0.0597614
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	12.0000	0.714590
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	8.00000	0.474713
$285$	16.0000	0.947758
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−8.00000	−0.469776
$291$	4.00000	0.234484
$292$	−6.00000	−0.351123
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−2.00000	−0.116642
$295$	−4.00000	−0.232889
$296$	−10.0000	−0.581238
$297$	0	0
$298$	−4.00000	−0.231714
$299$	−12.0000	−0.693978
$300$	−2.00000	−0.115470
$301$	−4.00000	−0.230556
$302$	16.0000	0.920697
$303$	16.0000	0.919176
$304$	8.00000	0.458831
$305$	4.00000	0.229039
$306$	2.00000	0.114332
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	28.0000	1.59286
$310$	8.00000	0.454369
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	−4.00000	−0.226455
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	10.0000	0.564333
$315$	1.00000	0.0563436
$316$	4.00000	0.225018
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	4.00000	0.224309
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−40.0000	−2.23258
$322$	6.00000	0.334367
$323$	16.0000	0.890264
$324$	−11.0000	−0.611111
$325$	2.00000	0.110940
$326$	18.0000	0.996928
$327$	−24.0000	−1.32720
$328$	0	0
$329$	6.00000	0.330791
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	4.00000	0.219529
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−2.00000	−0.109272
$336$	2.00000	0.109109
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−9.00000	−0.489535
$339$	36.0000	1.95525
$340$	−2.00000	−0.108465
$341$	0	0
$342$	8.00000	0.432590
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	−12.0000	−0.646058
$346$	14.0000	0.752645
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−16.0000	−0.857690
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	−1.00000	−0.0534522
$351$	8.00000	0.427008
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	−8.00000	−0.425195
$355$	−8.00000	−0.424596
$356$	−2.00000	−0.106000
$357$	4.00000	0.211702
$358$	20.0000	1.05703
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	−1.00000	−0.0527046
$361$	45.0000	2.36842
$362$	26.0000	1.36653
$363$	0	0
$364$	−2.00000	−0.104828
$365$	6.00000	0.314054
$366$	8.00000	0.418167
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	10.0000	0.519875
$371$	2.00000	0.103835
$372$	16.0000	0.829561
$373$	−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
$375$	2.00000	0.103280
$376$	−6.00000	−0.309426
$377$	16.0000	0.824042
$378$	−4.00000	−0.205738
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−8.00000	−0.410391
$381$	−16.0000	−0.819705
$382$	20.0000	1.02329
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	22.0000	1.11977
$387$	4.00000	0.203331
$388$	−2.00000	−0.101535
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	4.00000	0.202548
$391$	−12.0000	−0.606866
$392$	1.00000	0.0505076
$393$	40.0000	2.01773
$394$	−6.00000	−0.302276
$395$	−4.00000	−0.201262
$396$	0	0
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	−24.0000	−1.20301
$399$	16.0000	0.801002
$400$	1.00000	0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	−4.00000	−0.199502
$403$	−16.0000	−0.797017
$404$	−8.00000	−0.398015
$405$	11.0000	0.546594
$406$	−8.00000	−0.397033
$407$	0	0
$408$	−4.00000	−0.198030
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	−14.0000	−0.689730
$413$	−4.00000	−0.196827
$414$	−6.00000	−0.294884
$415$	−4.00000	−0.196352
$416$	2.00000	0.0980581
$417$	−8.00000	−0.391762
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	−2.00000	−0.0975900
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	12.0000	0.584151
$423$	−6.00000	−0.291730
$424$	−2.00000	−0.0971286
$425$	2.00000	0.0970143
$426$	−16.0000	−0.775203
$427$	4.00000	0.193574
$428$	20.0000	0.966736
$429$	0	0
$430$	−4.00000	−0.192897
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	4.00000	0.192450
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	8.00000	0.384012
$435$	16.0000	0.767141
$436$	12.0000	0.574696
$437$	−48.0000	−2.29615
$438$	12.0000	0.573382
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	4.00000	0.190261
$443$	26.0000	1.23530	0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000	1.23530	0.617649	+	0.786454i	$0.288085\pi$
$444$	20.0000	0.949158
$445$	2.00000	0.0948091
$446$	−2.00000	−0.0947027
$447$	8.00000	0.378387
$448$	−1.00000	−0.0472456
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−18.0000	−0.846649
$453$	−32.0000	−1.50349
$454$	12.0000	0.563188
$455$	2.00000	0.0937614
$456$	−16.0000	−0.749269
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	6.00000	0.280362
$459$	8.00000	0.373408
$460$	6.00000	0.279751
$461$	−32.0000	−1.49039	−0.745194	−	0.666847i	$-0.767643\pi$
$461$	−32.0000	−1.49039	−0.745194	+	0.666847i	$0.767643\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	8.00000	0.371391
$465$	−16.0000	−0.741982
$466$	26.0000	1.20443
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	2.00000	0.0924500
$469$	−2.00000	−0.0923514
$470$	6.00000	0.276759
$471$	−20.0000	−0.921551
$472$	4.00000	0.184115
$473$	0	0
$474$	−8.00000	−0.367452
$475$	8.00000	0.367065
$476$	−2.00000	−0.0916698
$477$	−2.00000	−0.0915737
$478$	20.0000	0.914779
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	2.00000	0.0912871
$481$	−20.0000	−0.911922
$482$	0	0
$483$	−12.0000	−0.546019
$484$	0	0
$485$	2.00000	0.0908153
$486$	10.0000	0.453609
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−4.00000	−0.181071
$489$	−36.0000	−1.62798
$490$	−1.00000	−0.0451754
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	16.0000	0.720604
$494$	16.0000	0.719874
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−8.00000	−0.358849
$498$	−8.00000	−0.358489
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	8.00000	0.357057
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−1.00000	−0.0445435
$505$	8.00000	0.355995
$506$	0	0
$507$	18.0000	0.799408
$508$	8.00000	0.354943
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	4.00000	0.177123
$511$	6.00000	0.265424
$512$	1.00000	0.0441942
$513$	32.0000	1.41283
$514$	−22.0000	−0.970378
$515$	14.0000	0.616914
$516$	−8.00000	−0.352180
$517$	0	0
$518$	10.0000	0.439375
$519$	−28.0000	−1.22906
$520$	−2.00000	−0.0877058
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	8.00000	0.350150
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−20.0000	−0.873704
$525$	2.00000	0.0872872
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	13.0000	0.565217
$530$	2.00000	0.0868744
$531$	4.00000	0.173585
$532$	−8.00000	−0.346844
$533$	0	0
$534$	4.00000	0.173097
$535$	−20.0000	−0.864675
$536$	2.00000	0.0863868
$537$	−40.0000	−1.72613
$538$	18.0000	0.776035
$539$	0	0
$540$	−4.00000	−0.172133
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	8.00000	0.343629
$543$	−52.0000	−2.23153
$544$	2.00000	0.0857493
$545$	−12.0000	−0.514024
$546$	4.00000	0.171184
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	6.00000	0.256307
$549$	−4.00000	−0.170716
$550$	0	0
$551$	64.0000	2.72649
$552$	12.0000	0.510754
$553$	−4.00000	−0.170097
$554$	−18.0000	−0.764747
$555$	−20.0000	−0.848953
$556$	4.00000	0.169638
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	−8.00000	−0.338667
$559$	8.00000	0.338364
$560$	1.00000	0.0422577
$561$	0	0
$562$	12.0000	0.506189
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	12.0000	0.505291
$565$	18.0000	0.757266
$566$	−12.0000	−0.504398
$567$	11.0000	0.461957
$568$	8.00000	0.335673
$569$	32.0000	1.34151	0.670755	−	0.741679i	$-0.265970\pi$
$569$	32.0000	1.34151	0.670755	+	0.741679i	$0.265970\pi$
$570$	16.0000	0.670166
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	−40.0000	−1.67102
$574$	0	0
$575$	−6.00000	−0.250217
$576$	1.00000	0.0416667
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	−13.0000	−0.540729
$579$	−44.0000	−1.82858
$580$	−8.00000	−0.332182
$581$	−4.00000	−0.165948
$582$	4.00000	0.165805
$583$	0	0
$584$	−6.00000	−0.248282
$585$	−2.00000	−0.0826898
$586$	−6.00000	−0.247858
$587$	−26.0000	−1.07313	−0.536567	−	0.843857i	$-0.680279\pi$
$587$	−26.0000	−1.07313	−0.536567	+	0.843857i	$0.680279\pi$
$588$	−2.00000	−0.0824786
$589$	−64.0000	−2.63707
$590$	−4.00000	−0.164677
$591$	12.0000	0.493614
$592$	−10.0000	−0.410997
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	−4.00000	−0.163846
$597$	48.0000	1.96451
$598$	−12.0000	−0.490716
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	−2.00000	−0.0816497
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	−4.00000	−0.163028
$603$	2.00000	0.0814463
$604$	16.0000	0.651031
$605$	0	0
$606$	16.0000	0.649956
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	8.00000	0.324443
$609$	16.0000	0.648353
$610$	4.00000	0.161955
$611$	−12.0000	−0.485468
$612$	2.00000	0.0808452
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	28.0000	1.12633
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	8.00000	0.321288
$621$	−24.0000	−0.963087
$622$	−12.0000	−0.481156
$623$	2.00000	0.0801283
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	10.0000	0.399043
$629$	−20.0000	−0.797452
$630$	1.00000	0.0398410
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	4.00000	0.159111
$633$	−24.0000	−0.953914
$634$	−22.0000	−0.873732
$635$	−8.00000	−0.317470
$636$	4.00000	0.158610
$637$	2.00000	0.0792429
$638$	0	0
$639$	8.00000	0.316475
$640$	−1.00000	−0.0395285
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	−40.0000	−1.57867
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	6.00000	0.236433
$645$	8.00000	0.315000
$646$	16.0000	0.629512
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	2.00000	0.0784465
$651$	−16.0000	−0.627089
$652$	18.0000	0.704934
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−24.0000	−0.938474
$655$	20.0000	0.781465
$656$	0	0
$657$	−6.00000	−0.234082
$658$	6.00000	0.233904
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	−8.00000	−0.310929
$663$	−8.00000	−0.310694
$664$	4.00000	0.155230
$665$	8.00000	0.310227
$666$	−10.0000	−0.387492
$667$	−48.0000	−1.85857
$668$	0	0
$669$	4.00000	0.154649
$670$	−2.00000	−0.0772667
$671$	0	0
$672$	2.00000	0.0771517
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	22.0000	0.847408
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	36.0000	1.38257
$679$	2.00000	0.0767530
$680$	−2.00000	−0.0766965
$681$	−24.0000	−0.919682
$682$	0	0
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	8.00000	0.305888
$685$	−6.00000	−0.229248
$686$	−1.00000	−0.0381802
$687$	−12.0000	−0.457829
$688$	4.00000	0.152499
$689$	−4.00000	−0.152388
$690$	−12.0000	−0.456832
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	20.0000	0.759190
$695$	−4.00000	−0.151729
$696$	−16.0000	−0.606478
$697$	0	0
$698$	4.00000	0.151402
$699$	−52.0000	−1.96682
$700$	−1.00000	−0.0377964
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	8.00000	0.301941
$703$	−80.0000	−3.01726
$704$	0	0
$705$	−12.0000	−0.451946
$706$	26.0000	0.978523
$707$	8.00000	0.300871
$708$	−8.00000	−0.300658
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	−8.00000	−0.300235
$711$	4.00000	0.150012
$712$	−2.00000	−0.0749532
$713$	48.0000	1.79761
$714$	4.00000	0.149696
$715$	0	0
$716$	20.0000	0.747435
$717$	−40.0000	−1.49383
$718$	−4.00000	−0.149279
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	−1.00000	−0.0372678
$721$	14.0000	0.521387
$722$	45.0000	1.67473
$723$	0	0
$724$	26.0000	0.966282
$725$	8.00000	0.297113
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	−2.00000	−0.0741249
$729$	13.0000	0.481481
$730$	6.00000	0.222070
$731$	8.00000	0.295891
$732$	8.00000	0.295689
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	−22.0000	−0.812035
$735$	2.00000	0.0737711
$736$	−6.00000	−0.221163
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	10.0000	0.367607
$741$	−32.0000	−1.17555
$742$	2.00000	0.0734223
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	16.0000	0.586588
$745$	4.00000	0.146549
$746$	−22.0000	−0.805477
$747$	4.00000	0.146352
$748$	0	0
$749$	−20.0000	−0.730784
$750$	2.00000	0.0730297
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	−6.00000	−0.218797
$753$	−16.0000	−0.583072
$754$	16.0000	0.582686
$755$	−16.0000	−0.582300
$756$	−4.00000	−0.145479
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−8.00000	−0.290573
$759$	0	0
$760$	−8.00000	−0.290191
$761$	28.0000	1.01500	0.507500	−	0.861652i	$-0.330570\pi$
$761$	28.0000	1.01500	0.507500	+	0.861652i	$0.330570\pi$
$762$	−16.0000	−0.579619
$763$	−12.0000	−0.434429
$764$	20.0000	0.723575
$765$	−2.00000	−0.0723102
$766$	18.0000	0.650366
$767$	8.00000	0.288863
$768$	−2.00000	−0.0721688
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	44.0000	1.58462
$772$	22.0000	0.791797
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	4.00000	0.143777
$775$	−8.00000	−0.287368
$776$	−2.00000	−0.0717958
$777$	−20.0000	−0.717496
$778$	−2.00000	−0.0717035
$779$	0	0
$780$	4.00000	0.143223
$781$	0	0
$782$	−12.0000	−0.429119
$783$	32.0000	1.14359
$784$	1.00000	0.0357143
$785$	−10.0000	−0.356915
$786$	40.0000	1.42675
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−4.00000	−0.142314
$791$	18.0000	0.640006
$792$	0	0
$793$	−8.00000	−0.284088
$794$	10.0000	0.354887
$795$	−4.00000	−0.141865
$796$	−24.0000	−0.850657
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	16.0000	0.566394
$799$	−12.0000	−0.424529
$800$	1.00000	0.0353553
$801$	−2.00000	−0.0706665
$802$	10.0000	0.353112
$803$	0	0
$804$	−4.00000	−0.141069
$805$	−6.00000	−0.211472
$806$	−16.0000	−0.563576
$807$	−36.0000	−1.26726
$808$	−8.00000	−0.281439
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	11.0000	0.386501
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	−8.00000	−0.280745
$813$	−16.0000	−0.561144
$814$	0	0
$815$	−18.0000	−0.630512
$816$	−4.00000	−0.140028
$817$	32.0000	1.11954
$818$	20.0000	0.699284
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	−12.0000	−0.418548
$823$	54.0000	1.88232	0.941161	−	0.337959i	$-0.109737\pi$
$823$	54.0000	1.88232	0.941161	+	0.337959i	$0.109737\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−6.00000	−0.208514
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−4.00000	−0.138842
$831$	36.0000	1.24883
$832$	2.00000	0.0693375
$833$	2.00000	0.0692959
$834$	−8.00000	−0.277017
$835$	0	0
$836$	0	0
$837$	−32.0000	−1.10608
$838$	24.0000	0.829066
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	−2.00000	−0.0690066
$841$	35.0000	1.20690
$842$	−6.00000	−0.206774
$843$	−24.0000	−0.826604
$844$	12.0000	0.413057
$845$	9.00000	0.309609
$846$	−6.00000	−0.206284
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	24.0000	0.823678
$850$	2.00000	0.0685994
$851$	60.0000	2.05677
$852$	−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	$0.422959\pi$
$854$	4.00000	0.136877
$855$	−8.00000	−0.273594
$856$	20.0000	0.683586
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	12.0000	0.408722
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	4.00000	0.136083
$865$	−14.0000	−0.476014
$866$	−10.0000	−0.339814
$867$	26.0000	0.883006
$868$	8.00000	0.271538
$869$	0	0
$870$	16.0000	0.542451
$871$	4.00000	0.135535
$872$	12.0000	0.406371
$873$	−2.00000	−0.0676897
$874$	−48.0000	−1.62362
$875$	1.00000	0.0338062
$876$	12.0000	0.405442
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−40.0000	−1.34993
$879$	12.0000	0.404750
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	1.00000	0.0336718
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	4.00000	0.134535
$885$	8.00000	0.268917
$886$	26.0000	0.873487
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	20.0000	0.671156
$889$	−8.00000	−0.268311
$890$	2.00000	0.0670402
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	−48.0000	−1.60626
$894$	8.00000	0.267560
$895$	−20.0000	−0.668526
$896$	−1.00000	−0.0334077
$897$	24.0000	0.801337
$898$	2.00000	0.0667409
$899$	−64.0000	−2.13452
$900$	1.00000	0.0333333
$901$	−4.00000	−0.133259
$902$	0	0
$903$	8.00000	0.266223
$904$	−18.0000	−0.598671
$905$	−26.0000	−0.864269
$906$	−32.0000	−1.06313
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	12.0000	0.398234
$909$	−8.00000	−0.265343
$910$	2.00000	0.0662994
$911$	4.00000	0.132526	0.0662630	−	0.997802i	$-0.478892\pi$
$911$	4.00000	0.132526	0.0662630	+	0.997802i	$0.478892\pi$
$912$	−16.0000	−0.529813
$913$	0	0
$914$	−22.0000	−0.727695
$915$	−8.00000	−0.264472
$916$	6.00000	0.198246
$917$	20.0000	0.660458
$918$	8.00000	0.264039
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	6.00000	0.197814
$921$	−40.0000	−1.31804
$922$	−32.0000	−1.05386
$923$	16.0000	0.526646
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−22.0000	−0.722965
$927$	−14.0000	−0.459820
$928$	8.00000	0.262613
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−16.0000	−0.524661
$931$	8.00000	0.262189
$932$	26.0000	0.851658
$933$	24.0000	0.785725
$934$	−2.00000	−0.0654420
$935$	0	0
$936$	2.00000	0.0653720
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	−2.00000	−0.0653023
$939$	44.0000	1.43589
$940$	6.00000	0.195698
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	−20.0000	−0.651635
$943$	0	0
$944$	4.00000	0.130189
$945$	4.00000	0.130120
$946$	0	0
$947$	−14.0000	−0.454939	−0.227469	−	0.973785i	$-0.573045\pi$
$947$	−14.0000	−0.454939	−0.227469	+	0.973785i	$0.573045\pi$
$948$	−8.00000	−0.259828
$949$	−12.0000	−0.389536
$950$	8.00000	0.259554
$951$	44.0000	1.42680
$952$	−2.00000	−0.0648204
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−2.00000	−0.0647524
$955$	−20.0000	−0.647185
$956$	20.0000	0.646846
$957$	0	0
$958$	−20.0000	−0.646171
$959$	−6.00000	−0.193750
$960$	2.00000	0.0645497
$961$	33.0000	1.06452
$962$	−20.0000	−0.644826
$963$	20.0000	0.644491
$964$	0	0
$965$	−22.0000	−0.708205
$966$	−12.0000	−0.386094
$967$	−56.0000	−1.80084	−0.900419	−	0.435023i	$-0.856740\pi$
$967$	−56.0000	−1.80084	−0.900419	+	0.435023i	$0.856740\pi$
$968$	0	0
$969$	−32.0000	−1.02799
$970$	2.00000	0.0642161
$971$	44.0000	1.41203	0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000	1.41203	0.706014	+	0.708198i	$0.250492\pi$
$972$	10.0000	0.320750
$973$	−4.00000	−0.128234
$974$	−2.00000	−0.0640841
$975$	−4.00000	−0.128103
$976$	−4.00000	−0.128037
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−36.0000	−1.15115
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	12.0000	0.383131
$982$	−8.00000	−0.255290
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	16.0000	0.509544
$987$	−12.0000	−0.381964
$988$	16.0000	0.509028
$989$	−24.0000	−0.763156
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	−8.00000	−0.254000
$993$	16.0000	0.507745
$994$	−8.00000	−0.253745
$995$	24.0000	0.760851
$996$	−8.00000	−0.253490
$997$	−34.0000	−1.07679	−0.538395	−	0.842692i	$-0.680969\pi$
$997$	−34.0000	−1.07679	−0.538395	+	0.842692i	$0.680969\pi$
$998$	−24.0000	−0.759707
$999$	−40.0000	−1.26554

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.d.1.1	✓	1	11.10	odd	2
8470.2.a.s.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

Related objects

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