LMFDB - Embedded newform 8330.2.a.cw.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8330,2,Mod(1,8330)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8330, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8330.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8330.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,10,2,10,10,2,0,10,12,10,-2,2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.5153848837$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 19x^{8} + 38x^{7} + 100x^{6} - 194x^{5} - 151x^{4} + 282x^{3} + 85x^{2} - 108x - 28 $ x^10 - 2x^9 - 19x^8 + 38x^7 + 100x^6 - 194x^5 - 151x^4 + 282x^3 + 85x^2 - 108*x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.664810$ of defining polynomial
Character	$\chi$	$=$	8330.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -0.664810 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.664810 q^{6} +1.00000 q^{8} -2.55803 q^{9} +1.00000 q^{10} -6.52002 q^{11} -0.664810 q^{12} -3.80618 q^{13} -0.664810 q^{15} +1.00000 q^{16} +1.00000 q^{17} -2.55803 q^{18} +6.05101 q^{19} +1.00000 q^{20} -6.52002 q^{22} -6.84921 q^{23} -0.664810 q^{24} +1.00000 q^{25} -3.80618 q^{26} +3.69503 q^{27} +9.62358 q^{29} -0.664810 q^{30} -8.49111 q^{31} +1.00000 q^{32} +4.33458 q^{33} +1.00000 q^{34} -2.55803 q^{36} -1.65916 q^{37} +6.05101 q^{38} +2.53039 q^{39} +1.00000 q^{40} +3.59766 q^{41} +11.1719 q^{43} -6.52002 q^{44} -2.55803 q^{45} -6.84921 q^{46} -2.67800 q^{47} -0.664810 q^{48} +1.00000 q^{50} -0.664810 q^{51} -3.80618 q^{52} +13.4462 q^{53} +3.69503 q^{54} -6.52002 q^{55} -4.02277 q^{57} +9.62358 q^{58} -0.344598 q^{59} -0.664810 q^{60} -11.5881 q^{61} -8.49111 q^{62} +1.00000 q^{64} -3.80618 q^{65} +4.33458 q^{66} +3.71729 q^{67} +1.00000 q^{68} +4.55343 q^{69} +0.0105838 q^{71} -2.55803 q^{72} -7.93373 q^{73} -1.65916 q^{74} -0.664810 q^{75} +6.05101 q^{76} +2.53039 q^{78} -0.721203 q^{79} +1.00000 q^{80} +5.21758 q^{81} +3.59766 q^{82} +6.22935 q^{83} +1.00000 q^{85} +11.1719 q^{86} -6.39786 q^{87} -6.52002 q^{88} +6.23763 q^{89} -2.55803 q^{90} -6.84921 q^{92} +5.64498 q^{93} -2.67800 q^{94} +6.05101 q^{95} -0.664810 q^{96} -17.8419 q^{97} +16.6784 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} + 2 q^{3} + 10 q^{4} + 10 q^{5} + 2 q^{6} + 10 q^{8} + 12 q^{9} + 10 q^{10} - 2 q^{11} + 2 q^{12} + 4 q^{13} + 2 q^{15} + 10 q^{16} + 10 q^{17} + 12 q^{18} + 14 q^{19} + 10 q^{20} - 2 q^{22}+ \cdots - 12 q^{99}+O(q^{100}) $ 10 * q + 10 * q^2 + 2 * q^3 + 10 * q^4 + 10 * q^5 + 2 * q^6 + 10 * q^8 + 12 * q^9 + 10 * q^10 - 2 * q^11 + 2 * q^12 + 4 * q^13 + 2 * q^15 + 10 * q^16 + 10 * q^17 + 12 * q^18 + 14 * q^19 + 10 * q^20 - 2 * q^22 + 10 * q^23 + 2 * q^24 + 10 * q^25 + 4 * q^26 - 4 * q^27 + 10 * q^29 + 2 * q^30 + 2 * q^31 + 10 * q^32 + 4 * q^33 + 10 * q^34 + 12 * q^36 + 14 * q^38 + 48 * q^39 + 10 * q^40 - 22 * q^41 + 14 * q^43 - 2 * q^44 + 12 * q^45 + 10 * q^46 + 18 * q^47 + 2 * q^48 + 10 * q^50 + 2 * q^51 + 4 * q^52 + 22 * q^53 - 4 * q^54 - 2 * q^55 + 32 * q^57 + 10 * q^58 - 6 * q^59 + 2 * q^60 - 16 * q^61 + 2 * q^62 + 10 * q^64 + 4 * q^65 + 4 * q^66 + 14 * q^67 + 10 * q^68 + 10 * q^69 + 20 * q^71 + 12 * q^72 + 12 * q^73 + 2 * q^75 + 14 * q^76 + 48 * q^78 + 44 * q^79 + 10 * q^80 + 50 * q^81 - 22 * q^82 + 20 * q^83 + 10 * q^85 + 14 * q^86 + 6 * q^87 - 2 * q^88 + 12 * q^90 + 10 * q^92 + 4 * q^93 + 18 * q^94 + 14 * q^95 + 2 * q^96 - 40 * q^97 - 12 * q^99

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
$3$	−0.664810	−0.383828	−0.191914	−	0.981412i	$-0.561470\pi$
$3$	−0.664810	−0.383828	−0.191914	+	0.981412i	$0.561470\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.664810	−0.271408
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.55803	−0.852676
$10$	1.00000	0.316228
$11$	−6.52002	−1.96586	−0.982931	−	0.183977i	$-0.941103\pi$
$11$	−6.52002	−1.96586	−0.982931	+	0.183977i	$0.941103\pi$
$12$	−0.664810	−0.191914
$13$	−3.80618	−1.05565	−0.527823	−	0.849355i	$-0.676991\pi$
$13$	−3.80618	−1.05565	−0.527823	+	0.849355i	$0.676991\pi$
$14$	0	0
$15$	−0.664810	−0.171653
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.55803	−0.602933
$19$	6.05101	1.38820	0.694098	−	0.719880i	$-0.255803\pi$
$19$	6.05101	1.38820	0.694098	+	0.719880i	$0.255803\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−6.52002	−1.39007
$23$	−6.84921	−1.42816	−0.714079	−	0.700065i	$-0.753154\pi$
$23$	−6.84921	−1.42816	−0.714079	+	0.700065i	$0.753154\pi$
$24$	−0.664810	−0.135704
$25$	1.00000	0.200000
$26$	−3.80618	−0.746454
$27$	3.69503	0.711110
$28$	0	0
$29$	9.62358	1.78705	0.893527	−	0.449009i	$-0.148223\pi$
$29$	9.62358	1.78705	0.893527	+	0.449009i	$0.148223\pi$
$30$	−0.664810	−0.121377
$31$	−8.49111	−1.52505	−0.762524	−	0.646960i	$-0.776040\pi$
$31$	−8.49111	−1.52505	−0.762524	+	0.646960i	$0.776040\pi$
$32$	1.00000	0.176777
$33$	4.33458	0.754553
$34$	1.00000	0.171499
$35$	0	0
$36$	−2.55803	−0.426338
$37$	−1.65916	−0.272765	−0.136382	−	0.990656i	$-0.543548\pi$
$37$	−1.65916	−0.272765	−0.136382	+	0.990656i	$0.543548\pi$
$38$	6.05101	0.981603
$39$	2.53039	0.405187
$40$	1.00000	0.158114
$41$	3.59766	0.561860	0.280930	−	0.959728i	$-0.409357\pi$
$41$	3.59766	0.561860	0.280930	+	0.959728i	$0.409357\pi$
$42$	0	0
$43$	11.1719	1.70370	0.851851	−	0.523784i	$-0.175480\pi$
$43$	11.1719	1.70370	0.851851	+	0.523784i	$0.175480\pi$
$44$	−6.52002	−0.982931
$45$	−2.55803	−0.381328
$46$	−6.84921	−1.00986
$47$	−2.67800	−0.390627	−0.195313	−	0.980741i	$-0.562572\pi$
$47$	−2.67800	−0.390627	−0.195313	+	0.980741i	$0.562572\pi$
$48$	−0.664810	−0.0959571
$49$	0	0
$50$	1.00000	0.141421
$51$	−0.664810	−0.0930921
$52$	−3.80618	−0.527823
$53$	13.4462	1.84698	0.923488	−	0.383628i	$-0.125326\pi$
$53$	13.4462	1.84698	0.923488	+	0.383628i	$0.125326\pi$
$54$	3.69503	0.502830
$55$	−6.52002	−0.879160
$56$	0	0
$57$	−4.02277	−0.532829
$58$	9.62358	1.26364
$59$	−0.344598	−0.0448629	−0.0224314	−	0.999748i	$-0.507141\pi$
$59$	−0.344598	−0.0448629	−0.0224314	+	0.999748i	$0.507141\pi$
$60$	−0.664810	−0.0858267
$61$	−11.5881	−1.48371	−0.741855	−	0.670560i	$-0.766054\pi$
$61$	−11.5881	−1.48371	−0.741855	+	0.670560i	$0.766054\pi$
$62$	−8.49111	−1.07837
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.80618	−0.472099
$66$	4.33458	0.533550
$67$	3.71729	0.454140	0.227070	−	0.973878i	$-0.427085\pi$
$67$	3.71729	0.454140	0.227070	+	0.973878i	$0.427085\pi$
$68$	1.00000	0.121268
$69$	4.55343	0.548168
$70$	0	0
$71$	0.0105838	0.00125607	0.000628034	−	1.00000i	$-0.499800\pi$
$71$	0.0105838	0.00125607	0.000628034		1.00000i	$0.499800\pi$
$72$	−2.55803	−0.301466
$73$	−7.93373	−0.928573	−0.464286	−	0.885685i	$-0.653689\pi$
$73$	−7.93373	−0.928573	−0.464286	+	0.885685i	$0.653689\pi$
$74$	−1.65916	−0.192874
$75$	−0.664810	−0.0767657
$76$	6.05101	0.694098
$77$	0	0
$78$	2.53039	0.286510
$79$	−0.721203	−0.0811417	−0.0405709	−	0.999177i	$-0.512918\pi$
$79$	−0.721203	−0.0811417	−0.0405709	+	0.999177i	$0.512918\pi$
$80$	1.00000	0.111803
$81$	5.21758	0.579732
$82$	3.59766	0.397295
$83$	6.22935	0.683760	0.341880	−	0.939744i	$-0.388936\pi$
$83$	6.22935	0.683760	0.341880	+	0.939744i	$0.388936\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	11.1719	1.20470
$87$	−6.39786	−0.685922
$88$	−6.52002	−0.695037
$89$	6.23763	0.661187	0.330594	−	0.943773i	$-0.392751\pi$
$89$	6.23763	0.661187	0.330594	+	0.943773i	$0.392751\pi$
$90$	−2.55803	−0.269640
$91$	0	0
$92$	−6.84921	−0.714079
$93$	5.64498	0.585357
$94$	−2.67800	−0.276215
$95$	6.05101	0.620820
$96$	−0.664810	−0.0678519
$97$	−17.8419	−1.81157	−0.905783	−	0.423741i	$-0.860717\pi$
$97$	−17.8419	−1.81157	−0.905783	+	0.423741i	$0.860717\pi$
$98$	0	0
$99$	16.6784	1.67624
$100$	1.00000	0.100000
$101$	16.0871	1.60073	0.800365	−	0.599513i	$-0.204639\pi$
$101$	16.0871	1.60073	0.800365	+	0.599513i	$0.204639\pi$
$102$	−0.664810	−0.0658260
$103$	4.93380	0.486142	0.243071	−	0.970009i	$-0.421845\pi$
$103$	4.93380	0.486142	0.243071	+	0.970009i	$0.421845\pi$
$104$	−3.80618	−0.373227
$105$	0	0
$106$	13.4462	1.30601
$107$	13.9312	1.34678	0.673390	−	0.739288i	$-0.264838\pi$
$107$	13.9312	1.34678	0.673390	+	0.739288i	$0.264838\pi$
$108$	3.69503	0.355555
$109$	11.2257	1.07522	0.537612	−	0.843192i	$-0.319327\pi$
$109$	11.2257	1.07522	0.537612	+	0.843192i	$0.319327\pi$
$110$	−6.52002	−0.621660
$111$	1.10303	0.104695
$112$	0	0
$113$	19.4266	1.82750	0.913750	−	0.406277i	$-0.133173\pi$
$113$	19.4266	1.82750	0.913750	+	0.406277i	$0.133173\pi$
$114$	−4.02277	−0.376767
$115$	−6.84921	−0.638692
$116$	9.62358	0.893527
$117$	9.73632	0.900123
$118$	−0.344598	−0.0317228
$119$	0	0
$120$	−0.664810	−0.0606886
$121$	31.5107	2.86461
$122$	−11.5881	−1.04914
$123$	−2.39176	−0.215658
$124$	−8.49111	−0.762524
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.65044	−0.235189	−0.117594	−	0.993062i	$-0.537518\pi$
$127$	−2.65044	−0.235189	−0.117594	+	0.993062i	$0.537518\pi$
$128$	1.00000	0.0883883
$129$	−7.42721	−0.653929
$130$	−3.80618	−0.333824
$131$	−2.25944	−0.197408	−0.0987039	−	0.995117i	$-0.531470\pi$
$131$	−2.25944	−0.197408	−0.0987039	+	0.995117i	$0.531470\pi$
$132$	4.33458	0.377277
$133$	0	0
$134$	3.71729	0.321125
$135$	3.69503	0.318018
$136$	1.00000	0.0857493
$137$	−5.27898	−0.451014	−0.225507	−	0.974242i	$-0.572404\pi$
$137$	−5.27898	−0.451014	−0.225507	+	0.974242i	$0.572404\pi$
$138$	4.55343	0.387613
$139$	13.2515	1.12398	0.561988	−	0.827145i	$-0.310037\pi$
$139$	13.2515	1.12398	0.561988	+	0.827145i	$0.310037\pi$
$140$	0	0
$141$	1.78036	0.149934
$142$	0.0105838	0.000888174 0
$143$	24.8164	2.07525
$144$	−2.55803	−0.213169
$145$	9.62358	0.799195
$146$	−7.93373	−0.656600
$147$	0	0
$148$	−1.65916	−0.136382
$149$	10.1867	0.834531	0.417265	−	0.908785i	$-0.362989\pi$
$149$	10.1867	0.834531	0.417265	+	0.908785i	$0.362989\pi$
$150$	−0.664810	−0.0542815
$151$	−10.3022	−0.838377	−0.419189	−	0.907899i	$-0.637685\pi$
$151$	−10.3022	−0.838377	−0.419189	+	0.907899i	$0.637685\pi$
$152$	6.05101	0.490802
$153$	−2.55803	−0.206804
$154$	0	0
$155$	−8.49111	−0.682022
$156$	2.53039	0.202593
$157$	7.23839	0.577687	0.288843	−	0.957376i	$-0.406729\pi$
$157$	7.23839	0.577687	0.288843	+	0.957376i	$0.406729\pi$
$158$	−0.721203	−0.0573759
$159$	−8.93916	−0.708922
$160$	1.00000	0.0790569
$161$	0	0
$162$	5.21758	0.409932
$163$	9.45112	0.740269	0.370135	−	0.928978i	$-0.379312\pi$
$163$	9.45112	0.740269	0.370135	+	0.928978i	$0.379312\pi$
$164$	3.59766	0.280930
$165$	4.33458	0.337447
$166$	6.22935	0.483491
$167$	7.30161	0.565016	0.282508	−	0.959265i	$-0.408834\pi$
$167$	7.30161	0.565016	0.282508	+	0.959265i	$0.408834\pi$
$168$	0	0
$169$	1.48704	0.114388
$170$	1.00000	0.0766965
$171$	−15.4786	−1.18368
$172$	11.1719	0.851851
$173$	−14.0587	−1.06886	−0.534430	−	0.845212i	$-0.679474\pi$
$173$	−14.0587	−1.06886	−0.534430	+	0.845212i	$0.679474\pi$
$174$	−6.39786	−0.485020
$175$	0	0
$176$	−6.52002	−0.491465
$177$	0.229093	0.0172197
$178$	6.23763	0.467530
$179$	−25.3304	−1.89329	−0.946643	−	0.322283i	$-0.895550\pi$
$179$	−25.3304	−1.89329	−0.946643	+	0.322283i	$0.895550\pi$
$180$	−2.55803	−0.190664
$181$	24.9847	1.85710	0.928549	−	0.371209i	$-0.121057\pi$
$181$	24.9847	1.85710	0.928549	+	0.371209i	$0.121057\pi$
$182$	0	0
$183$	7.70392	0.569490
$184$	−6.84921	−0.504930
$185$	−1.65916	−0.121984
$186$	5.64498	0.413910
$187$	−6.52002	−0.476791
$188$	−2.67800	−0.195313
$189$	0	0
$190$	6.05101	0.438986
$191$	−3.19120	−0.230907	−0.115454	−	0.993313i	$-0.536832\pi$
$191$	−3.19120	−0.230907	−0.115454	+	0.993313i	$0.536832\pi$
$192$	−0.664810	−0.0479786
$193$	10.6813	0.768858	0.384429	−	0.923155i	$-0.374398\pi$
$193$	10.6813	0.768858	0.384429	+	0.923155i	$0.374398\pi$
$194$	−17.8419	−1.28097
$195$	2.53039	0.181205
$196$	0	0
$197$	0.486090	0.0346325	0.0173162	−	0.999850i	$-0.494488\pi$
$197$	0.486090	0.0346325	0.0173162	+	0.999850i	$0.494488\pi$
$198$	16.6784	1.18528
$199$	−7.10223	−0.503464	−0.251732	−	0.967797i	$-0.581000\pi$
$199$	−7.10223	−0.503464	−0.251732	+	0.967797i	$0.581000\pi$
$200$	1.00000	0.0707107
$201$	−2.47130	−0.174312
$202$	16.0871	1.13189
$203$	0	0
$204$	−0.664810	−0.0465460
$205$	3.59766	0.251272
$206$	4.93380	0.343754
$207$	17.5205	1.21776
$208$	−3.80618	−0.263911
$209$	−39.4527	−2.72900
$210$	0	0
$211$	−3.98703	−0.274478	−0.137239	−	0.990538i	$-0.543823\pi$
$211$	−3.98703	−0.274478	−0.137239	+	0.990538i	$0.543823\pi$
$212$	13.4462	0.923488
$213$	−0.00703623	−0.000482115 0
$214$	13.9312	0.952317
$215$	11.1719	0.761919
$216$	3.69503	0.251415
$217$	0	0
$218$	11.2257	0.760298
$219$	5.27443	0.356413
$220$	−6.52002	−0.439580
$221$	−3.80618	−0.256032
$222$	1.10303	0.0740304
$223$	18.3655	1.22985	0.614923	−	0.788587i	$-0.289187\pi$
$223$	18.3655	1.22985	0.614923	+	0.788587i	$0.289187\pi$
$224$	0	0
$225$	−2.55803	−0.170535
$226$	19.4266	1.29224
$227$	−4.36764	−0.289890	−0.144945	−	0.989440i	$-0.546301\pi$
$227$	−4.36764	−0.289890	−0.144945	+	0.989440i	$0.546301\pi$
$228$	−4.02277	−0.266415
$229$	3.25162	0.214873	0.107437	−	0.994212i	$-0.465736\pi$
$229$	3.25162	0.214873	0.107437	+	0.994212i	$0.465736\pi$
$230$	−6.84921	−0.451623
$231$	0	0
$232$	9.62358	0.631819
$233$	−28.1326	−1.84303	−0.921513	−	0.388348i	$-0.873046\pi$
$233$	−28.1326	−1.84303	−0.921513	+	0.388348i	$0.873046\pi$
$234$	9.73632	0.636483
$235$	−2.67800	−0.174694
$236$	−0.344598	−0.0224314
$237$	0.479464	0.0311445
$238$	0	0
$239$	−7.08790	−0.458478	−0.229239	−	0.973370i	$-0.573624\pi$
$239$	−7.08790	−0.458478	−0.229239	+	0.973370i	$0.573624\pi$
$240$	−0.664810	−0.0429133
$241$	−9.35178	−0.602401	−0.301200	−	0.953561i	$-0.597387\pi$
$241$	−9.35178	−0.602401	−0.301200	+	0.953561i	$0.597387\pi$
$242$	31.5107	2.02559
$243$	−14.5538	−0.933627
$244$	−11.5881	−0.741855
$245$	0	0
$246$	−2.39176	−0.152493
$247$	−23.0313	−1.46544
$248$	−8.49111	−0.539186
$249$	−4.14134	−0.262447
$250$	1.00000	0.0632456
$251$	−10.9824	−0.693201	−0.346600	−	0.938013i	$-0.612664\pi$
$251$	−10.9824	−0.693201	−0.346600	+	0.938013i	$0.612664\pi$
$252$	0	0
$253$	44.6570	2.80756
$254$	−2.65044	−0.166303
$255$	−0.664810	−0.0416320
$256$	1.00000	0.0625000
$257$	8.15432	0.508652	0.254326	−	0.967118i	$-0.418146\pi$
$257$	8.15432	0.508652	0.254326	+	0.967118i	$0.418146\pi$
$258$	−7.42721	−0.462398
$259$	0	0
$260$	−3.80618	−0.236050
$261$	−24.6174	−1.52378
$262$	−2.25944	−0.139588
$263$	−12.4275	−0.766311	−0.383156	−	0.923684i	$-0.625163\pi$
$263$	−12.4275	−0.766311	−0.383156	+	0.923684i	$0.625163\pi$
$264$	4.33458	0.266775
$265$	13.4462	0.825993
$266$	0	0
$267$	−4.14684	−0.253782
$268$	3.71729	0.227070
$269$	13.1703	0.803006	0.401503	−	0.915858i	$-0.368488\pi$
$269$	13.1703	0.803006	0.401503	+	0.915858i	$0.368488\pi$
$270$	3.69503	0.224873
$271$	22.2437	1.35121	0.675605	−	0.737264i	$-0.263882\pi$
$271$	22.2437	1.35121	0.675605	+	0.737264i	$0.263882\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−5.27898	−0.318915
$275$	−6.52002	−0.393172
$276$	4.55343	0.274084
$277$	6.76821	0.406662	0.203331	−	0.979110i	$-0.434823\pi$
$277$	6.76821	0.406662	0.203331	+	0.979110i	$0.434823\pi$
$278$	13.2515	0.794771
$279$	21.7205	1.30037
$280$	0	0
$281$	13.3358	0.795547	0.397774	−	0.917484i	$-0.369783\pi$
$281$	13.3358	0.795547	0.397774	+	0.917484i	$0.369783\pi$
$282$	1.78036	0.106019
$283$	−24.7738	−1.47265	−0.736325	−	0.676628i	$-0.763441\pi$
$283$	−24.7738	−1.47265	−0.736325	+	0.676628i	$0.763441\pi$
$284$	0.0105838	0.000628034 0
$285$	−4.02277	−0.238289
$286$	24.8164	1.46743
$287$	0	0
$288$	−2.55803	−0.150733
$289$	1.00000	0.0588235
$290$	9.62358	0.565116
$291$	11.8615	0.695331
$292$	−7.93373	−0.464286
$293$	−1.20069	−0.0701449	−0.0350724	−	0.999385i	$-0.511166\pi$
$293$	−1.20069	−0.0701449	−0.0350724	+	0.999385i	$0.511166\pi$
$294$	0	0
$295$	−0.344598	−0.0200633
$296$	−1.65916	−0.0964369
$297$	−24.0917	−1.39794
$298$	10.1867	0.590102
$299$	26.0693	1.50763
$300$	−0.664810	−0.0383828
$301$	0	0
$302$	−10.3022	−0.592822
$303$	−10.6949	−0.614406
$304$	6.05101	0.347049
$305$	−11.5881	−0.663535
$306$	−2.55803	−0.146233
$307$	1.65968	0.0947230	0.0473615	−	0.998878i	$-0.484919\pi$
$307$	1.65968	0.0947230	0.0473615	+	0.998878i	$0.484919\pi$
$308$	0	0
$309$	−3.28004	−0.186595
$310$	−8.49111	−0.482262
$311$	−9.27450	−0.525908	−0.262954	−	0.964808i	$-0.584697\pi$
$311$	−9.27450	−0.525908	−0.262954	+	0.964808i	$0.584697\pi$
$312$	2.53039	0.143255
$313$	−1.10553	−0.0624884	−0.0312442	−	0.999512i	$-0.509947\pi$
$313$	−1.10553	−0.0624884	−0.0312442	+	0.999512i	$0.509947\pi$
$314$	7.23839	0.408486
$315$	0	0
$316$	−0.721203	−0.0405709
$317$	24.5824	1.38069	0.690343	−	0.723482i	$-0.257460\pi$
$317$	24.5824	1.38069	0.690343	+	0.723482i	$0.257460\pi$
$318$	−8.93916	−0.501283
$319$	−62.7460	−3.51310
$320$	1.00000	0.0559017
$321$	−9.26160	−0.516932
$322$	0	0
$323$	6.05101	0.336687
$324$	5.21758	0.289866
$325$	−3.80618	−0.211129
$326$	9.45112	0.523449
$327$	−7.46294	−0.412702
$328$	3.59766	0.198648
$329$	0	0
$330$	4.33458	0.238611
$331$	0.712529	0.0391641	0.0195821	−	0.999808i	$-0.493766\pi$
$331$	0.712529	0.0391641	0.0195821	+	0.999808i	$0.493766\pi$
$332$	6.22935	0.341880
$333$	4.24418	0.232580
$334$	7.30161	0.399527
$335$	3.71729	0.203097
$336$	0	0
$337$	22.0953	1.20361	0.601804	−	0.798644i	$-0.294449\pi$
$337$	22.0953	1.20361	0.601804	+	0.798644i	$0.294449\pi$
$338$	1.48704	0.0808842
$339$	−12.9150	−0.701447
$340$	1.00000	0.0542326
$341$	55.3622	2.99803
$342$	−15.4786	−0.836989
$343$	0	0
$344$	11.1719	0.602350
$345$	4.55343	0.245148
$346$	−14.0587	−0.755799
$347$	17.0139	0.913354	0.456677	−	0.889633i	$-0.349040\pi$
$347$	17.0139	0.913354	0.456677	+	0.889633i	$0.349040\pi$
$348$	−6.39786	−0.342961
$349$	2.84986	0.152550	0.0762748	−	0.997087i	$-0.475697\pi$
$349$	2.84986	0.152550	0.0762748	+	0.997087i	$0.475697\pi$
$350$	0	0
$351$	−14.0640	−0.750680
$352$	−6.52002	−0.347518
$353$	−25.4548	−1.35482	−0.677411	−	0.735604i	$-0.736898\pi$
$353$	−25.4548	−1.35482	−0.677411	+	0.735604i	$0.736898\pi$
$354$	0.229093	0.0121761
$355$	0.0105838	0.000561731 0
$356$	6.23763	0.330594
$357$	0	0
$358$	−25.3304	−1.33876
$359$	23.7835	1.25525	0.627624	−	0.778517i	$-0.284028\pi$
$359$	23.7835	1.25525	0.627624	+	0.778517i	$0.284028\pi$
$360$	−2.55803	−0.134820
$361$	17.6147	0.927089
$362$	24.9847	1.31317
$363$	−20.9486	−1.09952
$364$	0	0
$365$	−7.93373	−0.415270
$366$	7.70392	0.402690
$367$	−13.8863	−0.724858	−0.362429	−	0.932011i	$-0.618052\pi$
$367$	−13.8863	−0.724858	−0.362429	+	0.932011i	$0.618052\pi$
$368$	−6.84921	−0.357040
$369$	−9.20292	−0.479085
$370$	−1.65916	−0.0862558
$371$	0	0
$372$	5.64498	0.292678
$373$	0.0122696	0.000635294 0	0.000317647	−	1.00000i	$-0.499899\pi$
$373$	0.0122696	0.000635294 0	0.000317647		1.00000i	$0.499899\pi$
$374$	−6.52002	−0.337142
$375$	−0.664810	−0.0343307
$376$	−2.67800	−0.138107
$377$	−36.6291	−1.88650
$378$	0	0
$379$	18.3892	0.944589	0.472295	−	0.881441i	$-0.343426\pi$
$379$	18.3892	0.944589	0.472295	+	0.881441i	$0.343426\pi$
$380$	6.05101	0.310410
$381$	1.76204	0.0902721
$382$	−3.19120	−0.163276
$383$	−2.61159	−0.133446	−0.0667229	−	0.997772i	$-0.521254\pi$
$383$	−2.61159	−0.133446	−0.0667229	+	0.997772i	$0.521254\pi$
$384$	−0.664810	−0.0339260
$385$	0	0
$386$	10.6813	0.543665
$387$	−28.5781	−1.45271
$388$	−17.8419	−0.905783
$389$	4.11167	0.208470	0.104235	−	0.994553i	$-0.466761\pi$
$389$	4.11167	0.208470	0.104235	+	0.994553i	$0.466761\pi$
$390$	2.53039	0.128131
$391$	−6.84921	−0.346379
$392$	0	0
$393$	1.50210	0.0757707
$394$	0.486090	0.0244889
$395$	−0.721203	−0.0362877
$396$	16.6784	0.838121
$397$	14.2693	0.716157	0.358079	−	0.933691i	$-0.383432\pi$
$397$	14.2693	0.716157	0.358079	+	0.933691i	$0.383432\pi$
$398$	−7.10223	−0.356003
$399$	0	0
$400$	1.00000	0.0500000
$401$	−23.9281	−1.19491	−0.597456	−	0.801902i	$-0.703822\pi$
$401$	−23.9281	−1.19491	−0.597456	+	0.801902i	$0.703822\pi$
$402$	−2.47130	−0.123257
$403$	32.3187	1.60991
$404$	16.0871	0.800365
$405$	5.21758	0.259264
$406$	0	0
$407$	10.8178	0.536217
$408$	−0.664810	−0.0329130
$409$	17.0097	0.841073	0.420537	−	0.907276i	$-0.361842\pi$
$409$	17.0097	0.841073	0.420537	+	0.907276i	$0.361842\pi$
$410$	3.59766	0.177676
$411$	3.50952	0.173112
$412$	4.93380	0.243071
$413$	0	0
$414$	17.5205	0.861084
$415$	6.22935	0.305787
$416$	−3.80618	−0.186614
$417$	−8.80972	−0.431414
$418$	−39.4527	−1.92970
$419$	27.3533	1.33630	0.668149	−	0.744027i	$-0.267087\pi$
$419$	27.3533	1.33630	0.668149	+	0.744027i	$0.267087\pi$
$420$	0	0
$421$	−6.29686	−0.306890	−0.153445	−	0.988157i	$-0.549037\pi$
$421$	−6.29686	−0.306890	−0.153445	+	0.988157i	$0.549037\pi$
$422$	−3.98703	−0.194086
$423$	6.85040	0.333078
$424$	13.4462	0.653004
$425$	1.00000	0.0485071
$426$	−0.00703623	−0.000340906 0
$427$	0	0
$428$	13.9312	0.673390
$429$	−16.4982	−0.796541
$430$	11.1719	0.538758
$431$	−8.15799	−0.392956	−0.196478	−	0.980508i	$-0.562950\pi$
$431$	−8.15799	−0.392956	−0.196478	+	0.980508i	$0.562950\pi$
$432$	3.69503	0.177777
$433$	−38.0191	−1.82708	−0.913542	−	0.406745i	$-0.866664\pi$
$433$	−38.0191	−1.82708	−0.913542	+	0.406745i	$0.866664\pi$
$434$	0	0
$435$	−6.39786	−0.306754
$436$	11.2257	0.537612
$437$	−41.4446	−1.98256
$438$	5.27443	0.252022
$439$	22.1533	1.05732	0.528660	−	0.848833i	$-0.322695\pi$
$439$	22.1533	1.05732	0.528660	+	0.848833i	$0.322695\pi$
$440$	−6.52002	−0.310830
$441$	0	0
$442$	−3.80618	−0.181042
$443$	11.6498	0.553500	0.276750	−	0.960942i	$-0.410743\pi$
$443$	11.6498	0.553500	0.276750	+	0.960942i	$0.410743\pi$
$444$	1.10303	0.0523474
$445$	6.23763	0.295692
$446$	18.3655	0.869632
$447$	−6.77225	−0.320317
$448$	0	0
$449$	−30.5371	−1.44113	−0.720567	−	0.693386i	$-0.756118\pi$
$449$	−30.5371	−1.44113	−0.720567	+	0.693386i	$0.756118\pi$
$450$	−2.55803	−0.120587
$451$	−23.4568	−1.10454
$452$	19.4266	0.913750
$453$	6.84898	0.321793
$454$	−4.36764	−0.204983
$455$	0	0
$456$	−4.02277	−0.188384
$457$	4.92342	0.230308	0.115154	−	0.993348i	$-0.463264\pi$
$457$	4.92342	0.230308	0.115154	+	0.993348i	$0.463264\pi$
$458$	3.25162	0.151938
$459$	3.69503	0.172469
$460$	−6.84921	−0.319346
$461$	−29.7319	−1.38475	−0.692377	−	0.721536i	$-0.743436\pi$
$461$	−29.7319	−1.38475	−0.692377	+	0.721536i	$0.743436\pi$
$462$	0	0
$463$	18.1930	0.845502	0.422751	−	0.906246i	$-0.361064\pi$
$463$	18.1930	0.845502	0.422751	+	0.906246i	$0.361064\pi$
$464$	9.62358	0.446764
$465$	5.64498	0.261779
$466$	−28.1326	−1.30322
$467$	14.3738	0.665140	0.332570	−	0.943079i	$-0.392084\pi$
$467$	14.3738	0.665140	0.332570	+	0.943079i	$0.392084\pi$
$468$	9.73632	0.450062
$469$	0	0
$470$	−2.67800	−0.123527
$471$	−4.81216	−0.221733
$472$	−0.344598	−0.0158614
$473$	−72.8412	−3.34924
$474$	0.479464	0.0220225
$475$	6.05101	0.277639
$476$	0	0
$477$	−34.3957	−1.57487
$478$	−7.08790	−0.324193
$479$	−32.1649	−1.46965	−0.734825	−	0.678257i	$-0.762736\pi$
$479$	−32.1649	−1.46965	−0.734825	+	0.678257i	$0.762736\pi$
$480$	−0.664810	−0.0303443
$481$	6.31508	0.287943
$482$	−9.35178	−0.425962
$483$	0	0
$484$	31.5107	1.43230
$485$	−17.8419	−0.810157
$486$	−14.5538	−0.660174
$487$	8.05695	0.365095	0.182548	−	0.983197i	$-0.441566\pi$
$487$	8.05695	0.365095	0.182548	+	0.983197i	$0.441566\pi$
$488$	−11.5881	−0.524571
$489$	−6.28321	−0.284136
$490$	0	0
$491$	−38.8366	−1.75267	−0.876335	−	0.481703i	$-0.840018\pi$
$491$	−38.8366	−1.75267	−0.876335	+	0.481703i	$0.840018\pi$
$492$	−2.39176	−0.107829
$493$	9.62358	0.433424
$494$	−23.0313	−1.03622
$495$	16.6784	0.749638
$496$	−8.49111	−0.381262
$497$	0	0
$498$	−4.14134	−0.185578
$499$	25.5511	1.14382	0.571912	−	0.820315i	$-0.306202\pi$
$499$	25.5511	1.14382	0.571912	+	0.820315i	$0.306202\pi$
$500$	1.00000	0.0447214
$501$	−4.85419	−0.216869
$502$	−10.9824	−0.490167
$503$	8.21895	0.366465	0.183232	−	0.983070i	$-0.441344\pi$
$503$	8.21895	0.366465	0.183232	+	0.983070i	$0.441344\pi$
$504$	0	0
$505$	16.0871	0.715868
$506$	44.6570	1.98525
$507$	−0.988599	−0.0439052
$508$	−2.65044	−0.117594
$509$	−11.3042	−0.501052	−0.250526	−	0.968110i	$-0.580604\pi$
$509$	−11.3042	−0.501052	−0.250526	+	0.968110i	$0.580604\pi$
$510$	−0.664810	−0.0294383
$511$	0	0
$512$	1.00000	0.0441942
$513$	22.3587	0.987160
$514$	8.15432	0.359672
$515$	4.93380	0.217409
$516$	−7.42721	−0.326965
$517$	17.4606	0.767918
$518$	0	0
$519$	9.34635	0.410259
$520$	−3.80618	−0.166912
$521$	−5.13750	−0.225078	−0.112539	−	0.993647i	$-0.535898\pi$
$521$	−5.13750	−0.225078	−0.112539	+	0.993647i	$0.535898\pi$
$522$	−24.6174	−1.07747
$523$	−17.1699	−0.750787	−0.375394	−	0.926865i	$-0.622492\pi$
$523$	−17.1699	−0.750787	−0.375394	+	0.926865i	$0.622492\pi$
$524$	−2.25944	−0.0987039
$525$	0	0
$526$	−12.4275	−0.541864
$527$	−8.49111	−0.369878
$528$	4.33458	0.188638
$529$	23.9117	1.03964
$530$	13.4462	0.584065
$531$	0.881492	0.0382535
$532$	0	0
$533$	−13.6934	−0.593125
$534$	−4.14684	−0.179451
$535$	13.9312	0.602298
$536$	3.71729	0.160563
$537$	16.8399	0.726697
$538$	13.1703	0.567811
$539$	0	0
$540$	3.69503	0.159009
$541$	35.4584	1.52448	0.762238	−	0.647296i	$-0.224100\pi$
$541$	35.4584	1.52448	0.762238	+	0.647296i	$0.224100\pi$
$542$	22.2437	0.955450
$543$	−16.6101	−0.712807
$544$	1.00000	0.0428746
$545$	11.2257	0.480855
$546$	0	0
$547$	10.3432	0.442241	0.221121	−	0.975246i	$-0.429029\pi$
$547$	10.3432	0.442241	0.221121	+	0.975246i	$0.429029\pi$
$548$	−5.27898	−0.225507
$549$	29.6428	1.26512
$550$	−6.52002	−0.278015
$551$	58.2324	2.48078
$552$	4.55343	0.193807
$553$	0	0
$554$	6.76821	0.287554
$555$	1.10303	0.0468210
$556$	13.2515	0.561988
$557$	−13.5080	−0.572352	−0.286176	−	0.958177i	$-0.592384\pi$
$557$	−13.5080	−0.572352	−0.286176	+	0.958177i	$0.592384\pi$
$558$	21.7205	0.919501
$559$	−42.5224	−1.79851
$560$	0	0
$561$	4.33458	0.183006
$562$	13.3358	0.562537
$563$	−24.7467	−1.04295	−0.521474	−	0.853267i	$-0.674618\pi$
$563$	−24.7467	−1.04295	−0.521474	+	0.853267i	$0.674618\pi$
$564$	1.78036	0.0749669
$565$	19.4266	0.817283
$566$	−24.7738	−1.04132
$567$	0	0
$568$	0.0105838	0.000444087 0
$569$	−1.24375	−0.0521409	−0.0260705	−	0.999660i	$-0.508299\pi$
$569$	−1.24375	−0.0521409	−0.0260705	+	0.999660i	$0.508299\pi$
$570$	−4.02277	−0.168495
$571$	11.1564	0.466880	0.233440	−	0.972371i	$-0.425002\pi$
$571$	11.1564	0.466880	0.233440	+	0.972371i	$0.425002\pi$
$572$	24.8164	1.03763
$573$	2.12154	0.0886288
$574$	0	0
$575$	−6.84921	−0.285632
$576$	−2.55803	−0.106584
$577$	22.1393	0.921671	0.460836	−	0.887485i	$-0.347550\pi$
$577$	22.1393	0.921671	0.460836	+	0.887485i	$0.347550\pi$
$578$	1.00000	0.0415945
$579$	−7.10105	−0.295110
$580$	9.62358	0.399598
$581$	0	0
$582$	11.8615	0.491673
$583$	−87.6694	−3.63090
$584$	−7.93373	−0.328300
$585$	9.73632	0.402547
$586$	−1.20069	−0.0495999
$587$	−32.3171	−1.33387	−0.666935	−	0.745116i	$-0.732394\pi$
$587$	−32.3171	−1.33387	−0.666935	+	0.745116i	$0.732394\pi$
$588$	0	0
$589$	−51.3797	−2.11707
$590$	−0.344598	−0.0141869
$591$	−0.323158	−0.0132929
$592$	−1.65916	−0.0681912
$593$	43.3915	1.78188	0.890938	−	0.454124i	$-0.150048\pi$
$593$	43.3915	1.78188	0.890938	+	0.454124i	$0.150048\pi$
$594$	−24.0917	−0.988495
$595$	0	0
$596$	10.1867	0.417265
$597$	4.72164	0.193244
$598$	26.0693	1.06605
$599$	8.53846	0.348872	0.174436	−	0.984669i	$-0.444190\pi$
$599$	8.53846	0.348872	0.174436	+	0.984669i	$0.444190\pi$
$600$	−0.664810	−0.0271408
$601$	15.8289	0.645674	0.322837	−	0.946455i	$-0.395363\pi$
$601$	15.8289	0.645674	0.322837	+	0.946455i	$0.395363\pi$
$602$	0	0
$603$	−9.50894	−0.387234
$604$	−10.3022	−0.419189
$605$	31.5107	1.28109
$606$	−10.6949	−0.434451
$607$	24.6895	1.00212	0.501059	−	0.865413i	$-0.332944\pi$
$607$	24.6895	1.00212	0.501059	+	0.865413i	$0.332944\pi$
$608$	6.05101	0.245401
$609$	0	0
$610$	−11.5881	−0.469190
$611$	10.1930	0.412364
$612$	−2.55803	−0.103402
$613$	−3.79420	−0.153246	−0.0766231	−	0.997060i	$-0.524414\pi$
$613$	−3.79420	−0.153246	−0.0766231	+	0.997060i	$0.524414\pi$
$614$	1.65968	0.0669793
$615$	−2.39176	−0.0964452
$616$	0	0
$617$	26.4180	1.06355	0.531775	−	0.846886i	$-0.321525\pi$
$617$	26.4180	1.06355	0.531775	+	0.846886i	$0.321525\pi$
$618$	−3.28004	−0.131943
$619$	2.16086	0.0868523	0.0434262	−	0.999057i	$-0.486173\pi$
$619$	2.16086	0.0868523	0.0434262	+	0.999057i	$0.486173\pi$
$620$	−8.49111	−0.341011
$621$	−25.3081	−1.01558
$622$	−9.27450	−0.371873
$623$	0	0
$624$	2.53039	0.101297
$625$	1.00000	0.0400000
$626$	−1.10553	−0.0441860
$627$	26.2286	1.04747
$628$	7.23839	0.288843
$629$	−1.65916	−0.0661551
$630$	0	0
$631$	−5.38707	−0.214456	−0.107228	−	0.994234i	$-0.534197\pi$
$631$	−5.38707	−0.214456	−0.107228	+	0.994234i	$0.534197\pi$
$632$	−0.721203	−0.0286879
$633$	2.65062	0.105353
$634$	24.5824	0.976292
$635$	−2.65044	−0.105180
$636$	−8.93916	−0.354461
$637$	0	0
$638$	−62.7460	−2.48414
$639$	−0.0270737	−0.00107102
$640$	1.00000	0.0395285
$641$	−36.6389	−1.44715	−0.723574	−	0.690247i	$-0.757502\pi$
$641$	−36.6389	−1.44715	−0.723574	+	0.690247i	$0.757502\pi$
$642$	−9.26160	−0.365526
$643$	6.88063	0.271346	0.135673	−	0.990754i	$-0.456680\pi$
$643$	6.88063	0.271346	0.135673	+	0.990754i	$0.456680\pi$
$644$	0	0
$645$	−7.42721	−0.292446
$646$	6.05101	0.238074
$647$	23.9917	0.943210	0.471605	−	0.881810i	$-0.343675\pi$
$647$	23.9917	0.943210	0.471605	+	0.881810i	$0.343675\pi$
$648$	5.21758	0.204966
$649$	2.24679	0.0881942
$650$	−3.80618	−0.149291
$651$	0	0
$652$	9.45112	0.370135
$653$	3.68368	0.144153	0.0720767	−	0.997399i	$-0.477037\pi$
$653$	3.68368	0.144153	0.0720767	+	0.997399i	$0.477037\pi$
$654$	−7.46294	−0.291824
$655$	−2.25944	−0.0882834
$656$	3.59766	0.140465
$657$	20.2947	0.791772
$658$	0	0
$659$	15.3093	0.596366	0.298183	−	0.954509i	$-0.403619\pi$
$659$	15.3093	0.596366	0.298183	+	0.954509i	$0.403619\pi$
$660$	4.33458	0.168723
$661$	33.2293	1.29247	0.646234	−	0.763139i	$-0.276343\pi$
$661$	33.2293	1.29247	0.646234	+	0.763139i	$0.276343\pi$
$662$	0.712529	0.0276932
$663$	2.53039	0.0982722
$664$	6.22935	0.241746
$665$	0	0
$666$	4.24418	0.164459
$667$	−65.9139	−2.55220
$668$	7.30161	0.282508
$669$	−12.2096	−0.472050
$670$	3.71729	0.143612
$671$	75.5550	2.91677
$672$	0	0
$673$	39.2272	1.51210	0.756049	−	0.654514i	$-0.227127\pi$
$673$	39.2272	1.51210	0.756049	+	0.654514i	$0.227127\pi$
$674$	22.0953	0.851079
$675$	3.69503	0.142222
$676$	1.48704	0.0571938
$677$	−17.8082	−0.684425	−0.342213	−	0.939623i	$-0.611176\pi$
$677$	−17.8082	−0.684425	−0.342213	+	0.939623i	$0.611176\pi$
$678$	−12.9150	−0.495998
$679$	0	0
$680$	1.00000	0.0383482
$681$	2.90365	0.111268
$682$	55.3622	2.11993
$683$	17.1689	0.656949	0.328474	−	0.944513i	$-0.393466\pi$
$683$	17.1689	0.656949	0.328474	+	0.944513i	$0.393466\pi$
$684$	−15.4786	−0.591841
$685$	−5.27898	−0.201700
$686$	0	0
$687$	−2.16171	−0.0824744
$688$	11.1719	0.425925
$689$	−51.1787	−1.94975
$690$	4.55343	0.173346
$691$	−39.1575	−1.48962	−0.744812	−	0.667275i	$-0.767461\pi$
$691$	−39.1575	−1.48962	−0.744812	+	0.667275i	$0.767461\pi$
$692$	−14.0587	−0.534430
$693$	0	0
$694$	17.0139	0.645839
$695$	13.2515	0.502657
$696$	−6.39786	−0.242510
$697$	3.59766	0.136271
$698$	2.84986	0.107869
$699$	18.7028	0.707406
$700$	0	0
$701$	26.0042	0.982163	0.491082	−	0.871113i	$-0.336602\pi$
$701$	26.0042	0.982163	0.491082	+	0.871113i	$0.336602\pi$
$702$	−14.0640	−0.530811
$703$	−10.0396	−0.378651
$704$	−6.52002	−0.245733
$705$	1.78036	0.0670524
$706$	−25.4548	−0.958004
$707$	0	0
$708$	0.229093	0.00860983
$709$	12.7581	0.479141	0.239570	−	0.970879i	$-0.422993\pi$
$709$	12.7581	0.479141	0.239570	+	0.970879i	$0.422993\pi$
$710$	0.0105838	0.000397204 0
$711$	1.84486	0.0691876
$712$	6.23763	0.233765
$713$	58.1574	2.17801
$714$	0	0
$715$	24.8164	0.928081
$716$	−25.3304	−0.946643
$717$	4.71211	0.175977
$718$	23.7835	0.887594
$719$	−39.6642	−1.47923	−0.739613	−	0.673033i	$-0.764991\pi$
$719$	−39.6642	−1.47923	−0.739613	+	0.673033i	$0.764991\pi$
$720$	−2.55803	−0.0953320
$721$	0	0
$722$	17.6147	0.655551
$723$	6.21716	0.231219
$724$	24.9847	0.928549
$725$	9.62358	0.357411
$726$	−20.9486	−0.777477
$727$	−12.3643	−0.458565	−0.229283	−	0.973360i	$-0.573638\pi$
$727$	−12.3643	−0.458565	−0.229283	+	0.973360i	$0.573638\pi$
$728$	0	0
$729$	−5.97723	−0.221379
$730$	−7.93373	−0.293641
$731$	11.1719	0.413208
$732$	7.70392	0.284745
$733$	−43.4894	−1.60632	−0.803158	−	0.595766i	$-0.796849\pi$
$733$	−43.4894	−1.60632	−0.803158	+	0.595766i	$0.796849\pi$
$734$	−13.8863	−0.512552
$735$	0	0
$736$	−6.84921	−0.252465
$737$	−24.2368	−0.892776
$738$	−9.20292	−0.338764
$739$	19.0251	0.699849	0.349925	−	0.936778i	$-0.386207\pi$
$739$	19.0251	0.699849	0.349925	+	0.936778i	$0.386207\pi$
$740$	−1.65916	−0.0609920
$741$	15.3114	0.562479
$742$	0	0
$743$	28.2320	1.03573	0.517865	−	0.855462i	$-0.326727\pi$
$743$	28.2320	1.03573	0.517865	+	0.855462i	$0.326727\pi$
$744$	5.64498	0.206955
$745$	10.1867	0.373213
$746$	0.0122696	0.000449220 0
$747$	−15.9348	−0.583026
$748$	−6.52002	−0.238396
$749$	0	0
$750$	−0.664810	−0.0242754
$751$	49.1270	1.79267	0.896334	−	0.443379i	$-0.146220\pi$
$751$	49.1270	1.79267	0.896334	+	0.443379i	$0.146220\pi$
$752$	−2.67800	−0.0976567
$753$	7.30119	0.266070
$754$	−36.6291	−1.33395
$755$	−10.3022	−0.374934
$756$	0	0
$757$	31.3820	1.14060	0.570299	−	0.821437i	$-0.306827\pi$
$757$	31.3820	1.14060	0.570299	+	0.821437i	$0.306827\pi$
$758$	18.3892	0.667925
$759$	−29.6884	−1.07762
$760$	6.05101	0.219493
$761$	38.1464	1.38281	0.691403	−	0.722469i	$-0.256993\pi$
$761$	38.1464	1.38281	0.691403	+	0.722469i	$0.256993\pi$
$762$	1.76204	0.0638320
$763$	0	0
$764$	−3.19120	−0.115454
$765$	−2.55803	−0.0924857
$766$	−2.61159	−0.0943605
$767$	1.31160	0.0473593
$768$	−0.664810	−0.0239893
$769$	−29.9068	−1.07847	−0.539233	−	0.842157i	$-0.681286\pi$
$769$	−29.9068	−1.07847	−0.539233	+	0.842157i	$0.681286\pi$
$770$	0	0
$771$	−5.42108	−0.195235
$772$	10.6813	0.384429
$773$	29.8061	1.07205	0.536026	−	0.844202i	$-0.319925\pi$
$773$	29.8061	1.07205	0.536026	+	0.844202i	$0.319925\pi$
$774$	−28.5781	−1.02722
$775$	−8.49111	−0.305010
$776$	−17.8419	−0.640486
$777$	0	0
$778$	4.11167	0.147410
$779$	21.7695	0.779973
$780$	2.53039	0.0906025
$781$	−0.0690067	−0.00246925
$782$	−6.84921	−0.244927
$783$	35.5595	1.27079
$784$	0	0
$785$	7.23839	0.258349
$786$	1.50210	0.0535780
$787$	51.2500	1.82687	0.913433	−	0.406989i	$-0.133421\pi$
$787$	51.2500	1.82687	0.913433	+	0.406989i	$0.133421\pi$
$788$	0.486090	0.0173162
$789$	8.26192	0.294132
$790$	−0.721203	−0.0256593
$791$	0	0
$792$	16.6784	0.592641
$793$	44.1066	1.56627
$794$	14.2693	0.506399
$795$	−8.93916	−0.317039
$796$	−7.10223	−0.251732
$797$	23.1273	0.819211	0.409605	−	0.912263i	$-0.365666\pi$
$797$	23.1273	0.819211	0.409605	+	0.912263i	$0.365666\pi$
$798$	0	0
$799$	−2.67800	−0.0947409
$800$	1.00000	0.0353553
$801$	−15.9560	−0.563778
$802$	−23.9281	−0.844930
$803$	51.7281	1.82545
$804$	−2.47130	−0.0871559
$805$	0	0
$806$	32.3187	1.13838
$807$	−8.75574	−0.308217
$808$	16.0871	0.565944
$809$	8.53616	0.300115	0.150058	−	0.988677i	$-0.452054\pi$
$809$	8.53616	0.300115	0.150058	+	0.988677i	$0.452054\pi$
$810$	5.21758	0.183327
$811$	−32.3685	−1.13661	−0.568305	−	0.822818i	$-0.692401\pi$
$811$	−32.3685	−1.13661	−0.568305	+	0.822818i	$0.692401\pi$
$812$	0	0
$813$	−14.7879	−0.518633
$814$	10.8178	0.379163
$815$	9.45112	0.331058
$816$	−0.664810	−0.0232730
$817$	67.6014	2.36507
$818$	17.0097	0.594729
$819$	0	0
$820$	3.59766	0.125636
$821$	−37.4067	−1.30550	−0.652751	−	0.757573i	$-0.726385\pi$
$821$	−37.4067	−1.30550	−0.652751	+	0.757573i	$0.726385\pi$
$822$	3.50952	0.122409
$823$	29.3865	1.02435	0.512174	−	0.858882i	$-0.328840\pi$
$823$	29.3865	1.02435	0.512174	+	0.858882i	$0.328840\pi$
$824$	4.93380	0.171877
$825$	4.33458	0.150911
$826$	0	0
$827$	−20.1831	−0.701835	−0.350918	−	0.936406i	$-0.614130\pi$
$827$	−20.1831	−0.701835	−0.350918	+	0.936406i	$0.614130\pi$
$828$	17.5205	0.608878
$829$	−20.5850	−0.714948	−0.357474	−	0.933923i	$-0.616362\pi$
$829$	−20.5850	−0.714948	−0.357474	+	0.933923i	$0.616362\pi$
$830$	6.22935	0.216224
$831$	−4.49958	−0.156089
$832$	−3.80618	−0.131956
$833$	0	0
$834$	−8.80972	−0.305056
$835$	7.30161	0.252683
$836$	−39.4527	−1.36450
$837$	−31.3749	−1.08448
$838$	27.3533	0.944906
$839$	−48.6129	−1.67830	−0.839152	−	0.543897i	$-0.816948\pi$
$839$	−48.6129	−1.67830	−0.839152	+	0.543897i	$0.816948\pi$
$840$	0	0
$841$	63.6133	2.19356
$842$	−6.29686	−0.217004
$843$	−8.86578	−0.305354
$844$	−3.98703	−0.137239
$845$	1.48704	0.0511557
$846$	6.85040	0.235522
$847$	0	0
$848$	13.4462	0.461744
$849$	16.4699	0.565245
$850$	1.00000	0.0342997
$851$	11.3639	0.389551
$852$	−0.00703623	−0.000241057 0
$853$	−4.82501	−0.165205	−0.0826026	−	0.996583i	$-0.526323\pi$
$853$	−4.82501	−0.165205	−0.0826026	+	0.996583i	$0.526323\pi$
$854$	0	0
$855$	−15.4786	−0.529358
$856$	13.9312	0.476158
$857$	−22.2382	−0.759644	−0.379822	−	0.925060i	$-0.624015\pi$
$857$	−22.2382	−0.759644	−0.379822	+	0.925060i	$0.624015\pi$
$858$	−16.4982	−0.563240
$859$	−36.0342	−1.22947	−0.614736	−	0.788733i	$-0.710737\pi$
$859$	−36.0342	−1.22947	−0.614736	+	0.788733i	$0.710737\pi$
$860$	11.1719	0.380959
$861$	0	0
$862$	−8.15799	−0.277862
$863$	29.0548	0.989038	0.494519	−	0.869167i	$-0.335344\pi$
$863$	29.0548	0.989038	0.494519	+	0.869167i	$0.335344\pi$
$864$	3.69503	0.125708
$865$	−14.0587	−0.478009
$866$	−38.0191	−1.29194
$867$	−0.664810	−0.0225781
$868$	0	0
$869$	4.70226	0.159513
$870$	−6.39786	−0.216908
$871$	−14.1487	−0.479411
$872$	11.2257	0.380149
$873$	45.6400	1.54468
$874$	−41.4446	−1.40188
$875$	0	0
$876$	5.27443	0.178206
$877$	−51.2995	−1.73226	−0.866130	−	0.499819i	$-0.833400\pi$
$877$	−51.2995	−1.73226	−0.866130	+	0.499819i	$0.833400\pi$
$878$	22.1533	0.747639
$879$	0.798229	0.0269236
$880$	−6.52002	−0.219790
$881$	1.96766	0.0662922	0.0331461	−	0.999451i	$-0.489447\pi$
$881$	1.96766	0.0662922	0.0331461	+	0.999451i	$0.489447\pi$
$882$	0	0
$883$	−45.1553	−1.51960	−0.759798	−	0.650159i	$-0.774702\pi$
$883$	−45.1553	−1.51960	−0.759798	+	0.650159i	$0.774702\pi$
$884$	−3.80618	−0.128016
$885$	0.229093	0.00770086
$886$	11.6498	0.391384
$887$	−11.7817	−0.395591	−0.197795	−	0.980243i	$-0.563378\pi$
$887$	−11.7817	−0.395591	−0.197795	+	0.980243i	$0.563378\pi$
$888$	1.10303	0.0370152
$889$	0	0
$890$	6.23763	0.209086
$891$	−34.0188	−1.13967
$892$	18.3655	0.614923
$893$	−16.2046	−0.542267
$894$	−6.77225	−0.226498
$895$	−25.3304	−0.846703
$896$	0	0
$897$	−17.3312	−0.578671
$898$	−30.5371	−1.01903
$899$	−81.7149	−2.72534
$900$	−2.55803	−0.0852676
$901$	13.4462	0.447957
$902$	−23.4568	−0.781027
$903$	0	0
$904$	19.4266	0.646119
$905$	24.9847	0.830520
$906$	6.84898	0.227542
$907$	6.94531	0.230615	0.115308	−	0.993330i	$-0.463215\pi$
$907$	6.94531	0.230615	0.115308	+	0.993330i	$0.463215\pi$
$908$	−4.36764	−0.144945
$909$	−41.1513	−1.36490
$910$	0	0
$911$	19.3452	0.640935	0.320468	−	0.947259i	$-0.396160\pi$
$911$	19.3452	0.640935	0.320468	+	0.947259i	$0.396160\pi$
$912$	−4.02277	−0.133207
$913$	−40.6155	−1.34418
$914$	4.92342	0.162852
$915$	7.70392	0.254684
$916$	3.25162	0.107437
$917$	0	0
$918$	3.69503	0.121954
$919$	57.3011	1.89019	0.945095	−	0.326796i	$-0.105969\pi$
$919$	57.3011	1.89019	0.945095	+	0.326796i	$0.105969\pi$
$920$	−6.84921	−0.225812
$921$	−1.10337	−0.0363574
$922$	−29.7319	−0.979169
$923$	−0.0402840	−0.00132596
$924$	0	0
$925$	−1.65916	−0.0545529
$926$	18.1930	0.597860
$927$	−12.6208	−0.414521
$928$	9.62358	0.315910
$929$	−42.8599	−1.40619	−0.703093	−	0.711098i	$-0.748198\pi$
$929$	−42.8599	−1.40619	−0.703093	+	0.711098i	$0.748198\pi$
$930$	5.64498	0.185106
$931$	0	0
$932$	−28.1326	−0.921513
$933$	6.16578	0.201859
$934$	14.3738	0.470325
$935$	−6.52002	−0.213228
$936$	9.73632	0.318242
$937$	24.4116	0.797491	0.398745	−	0.917062i	$-0.369446\pi$
$937$	24.4116	0.797491	0.398745	+	0.917062i	$0.369446\pi$
$938$	0	0
$939$	0.734969	0.0239848
$940$	−2.67800	−0.0873468
$941$	50.9708	1.66160	0.830801	−	0.556570i	$-0.187883\pi$
$941$	50.9708	1.66160	0.830801	+	0.556570i	$0.187883\pi$
$942$	−4.81216	−0.156789
$943$	−24.6411	−0.802426
$944$	−0.344598	−0.0112157
$945$	0	0
$946$	−72.8412	−2.36827
$947$	−8.40434	−0.273104	−0.136552	−	0.990633i	$-0.543602\pi$
$947$	−8.40434	−0.273104	−0.136552	+	0.990633i	$0.543602\pi$
$948$	0.479464	0.0155723
$949$	30.1972	0.980244
$950$	6.05101	0.196321
$951$	−16.3426	−0.529946
$952$	0	0
$953$	16.5830	0.537176	0.268588	−	0.963255i	$-0.413443\pi$
$953$	16.5830	0.537176	0.268588	+	0.963255i	$0.413443\pi$
$954$	−34.3957	−1.11360
$955$	−3.19120	−0.103265
$956$	−7.08790	−0.229239
$957$	41.7142	1.34843
$958$	−32.1649	−1.03920
$959$	0	0
$960$	−0.664810	−0.0214567
$961$	41.0989	1.32577
$962$	6.31508	0.203606
$963$	−35.6364	−1.14837
$964$	−9.35178	−0.301200
$965$	10.6813	0.343844
$966$	0	0
$967$	10.0435	0.322976	0.161488	−	0.986875i	$-0.448371\pi$
$967$	10.0435	0.322976	0.161488	+	0.986875i	$0.448371\pi$
$968$	31.5107	1.01279
$969$	−4.02277	−0.129230
$970$	−17.8419	−0.572868
$971$	−8.61844	−0.276579	−0.138289	−	0.990392i	$-0.544160\pi$
$971$	−8.61844	−0.276579	−0.138289	+	0.990392i	$0.544160\pi$
$972$	−14.5538	−0.466814
$973$	0	0
$974$	8.05695	0.258161
$975$	2.53039	0.0810374
$976$	−11.5881	−0.370927
$977$	45.5811	1.45827	0.729135	−	0.684370i	$-0.239923\pi$
$977$	45.5811	1.45827	0.729135	+	0.684370i	$0.239923\pi$
$978$	−6.28321	−0.200915
$979$	−40.6695	−1.29980
$980$	0	0
$981$	−28.7156	−0.916817
$982$	−38.8366	−1.23932
$983$	−36.3190	−1.15840	−0.579198	−	0.815187i	$-0.696634\pi$
$983$	−36.3190	−1.15840	−0.579198	+	0.815187i	$0.696634\pi$
$984$	−2.39176	−0.0762466
$985$	0.486090	0.0154881
$986$	9.62358	0.306477
$987$	0	0
$988$	−23.0313	−0.732722
$989$	−76.5188	−2.43316
$990$	16.6784	0.530074
$991$	−33.3208	−1.05847	−0.529235	−	0.848476i	$-0.677521\pi$
$991$	−33.3208	−1.05847	−0.529235	+	0.848476i	$0.677521\pi$
$992$	−8.49111	−0.269593
$993$	−0.473697	−0.0150323
$994$	0	0
$995$	−7.10223	−0.225156
$996$	−4.14134	−0.131223
$997$	−4.30519	−0.136347	−0.0681733	−	0.997673i	$-0.521717\pi$
$997$	−4.30519	−0.136347	−0.0681733	+	0.997673i	$0.521717\pi$
$998$	25.5511	0.808805
$999$	−6.13066	−0.193966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8330.2.a.cw.1.4	yes	10
7.6	odd	2		8330.2.a.cv.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8330.2.a.cv.1.7	✓	10	7.6	odd	2
8330.2.a.cw.1.4	yes	10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8330.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.664810 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.664810 q^{6} +1.00000 q^{8} -2.55803 q^{9} +1.00000 q^{10} -6.52002 q^{11} -0.664810 q^{12} -3.80618 q^{13} -0.664810 q^{15} +1.00000 q^{16} +1.00000 q^{17} -2.55803 q^{18} +6.05101 q^{19} +1.00000 q^{20} -6.52002 q^{22} -6.84921 q^{23} -0.664810 q^{24} +1.00000 q^{25} -3.80618 q^{26} +3.69503 q^{27} +9.62358 q^{29} -0.664810 q^{30} -8.49111 q^{31} +1.00000 q^{32} +4.33458 q^{33} +1.00000 q^{34} -2.55803 q^{36} -1.65916 q^{37} +6.05101 q^{38} +2.53039 q^{39} +1.00000 q^{40} +3.59766 q^{41} +11.1719 q^{43} -6.52002 q^{44} -2.55803 q^{45} -6.84921 q^{46} -2.67800 q^{47} -0.664810 q^{48} +1.00000 q^{50} -0.664810 q^{51} -3.80618 q^{52} +13.4462 q^{53} +3.69503 q^{54} -6.52002 q^{55} -4.02277 q^{57} +9.62358 q^{58} -0.344598 q^{59} -0.664810 q^{60} -11.5881 q^{61} -8.49111 q^{62} +1.00000 q^{64} -3.80618 q^{65} +4.33458 q^{66} +3.71729 q^{67} +1.00000 q^{68} +4.55343 q^{69} +0.0105838 q^{71} -2.55803 q^{72} -7.93373 q^{73} -1.65916 q^{74} -0.664810 q^{75} +6.05101 q^{76} +2.53039 q^{78} -0.721203 q^{79} +1.00000 q^{80} +5.21758 q^{81} +3.59766 q^{82} +6.22935 q^{83} +1.00000 q^{85} +11.1719 q^{86} -6.39786 q^{87} -6.52002 q^{88} +6.23763 q^{89} -2.55803 q^{90} -6.84921 q^{92} +5.64498 q^{93} -2.67800 q^{94} +6.05101 q^{95} -0.664810 q^{96} -17.8419 q^{97} +16.6784 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} + 2 q^{3} + 10 q^{4} + 10 q^{5} + 2 q^{6} + 10 q^{8} + 12 q^{9} + 10 q^{10} - 2 q^{11} + 2 q^{12} + 4 q^{13} + 2 q^{15} + 10 q^{16} + 10 q^{17} + 12 q^{18} + 14 q^{19} + 10 q^{20} - 2 q^{22}+ \cdots - 12 q^{99}+O(q^{100}) \) 10 * q + 10 * q^2 + 2 * q^3 + 10 * q^4 + 10 * q^5 + 2 * q^6 + 10 * q^8 + 12 * q^9 + 10 * q^10 - 2 * q^11 + 2 * q^12 + 4 * q^13 + 2 * q^15 + 10 * q^16 + 10 * q^17 + 12 * q^18 + 14 * q^19 + 10 * q^20 - 2 * q^22 + 10 * q^23 + 2 * q^24 + 10 * q^25 + 4 * q^26 - 4 * q^27 + 10 * q^29 + 2 * q^30 + 2 * q^31 + 10 * q^32 + 4 * q^33 + 10 * q^34 + 12 * q^36 + 14 * q^38 + 48 * q^39 + 10 * q^40 - 22 * q^41 + 14 * q^43 - 2 * q^44 + 12 * q^45 + 10 * q^46 + 18 * q^47 + 2 * q^48 + 10 * q^50 + 2 * q^51 + 4 * q^52 + 22 * q^53 - 4 * q^54 - 2 * q^55 + 32 * q^57 + 10 * q^58 - 6 * q^59 + 2 * q^60 - 16 * q^61 + 2 * q^62 + 10 * q^64 + 4 * q^65 + 4 * q^66 + 14 * q^67 + 10 * q^68 + 10 * q^69 + 20 * q^71 + 12 * q^72 + 12 * q^73 + 2 * q^75 + 14 * q^76 + 48 * q^78 + 44 * q^79 + 10 * q^80 + 50 * q^81 - 22 * q^82 + 20 * q^83 + 10 * q^85 + 14 * q^86 + 6 * q^87 - 2 * q^88 + 12 * q^90 + 10 * q^92 + 4 * q^93 + 18 * q^94 + 14 * q^95 + 2 * q^96 - 40 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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