LMFDB - Embedded newform 8330.2.a.cw.1.3

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			1.3
Root			$-1.18925$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.18925 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.18925 q^{6} +1.00000 q^{8} -1.58568 q^{9} +1.00000 q^{10} +3.88703 q^{11} -1.18925 q^{12} -4.93935 q^{13} -1.18925 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.58568 q^{18} -0.599423 q^{19} +1.00000 q^{20} +3.88703 q^{22} +8.10107 q^{23} -1.18925 q^{24} +1.00000 q^{25} -4.93935 q^{26} +5.45353 q^{27} -2.81843 q^{29} -1.18925 q^{30} +5.25434 q^{31} +1.00000 q^{32} -4.62265 q^{33} +1.00000 q^{34} -1.58568 q^{36} -8.25283 q^{37} -0.599423 q^{38} +5.87412 q^{39} +1.00000 q^{40} -11.2863 q^{41} +10.7710 q^{43} +3.88703 q^{44} -1.58568 q^{45} +8.10107 q^{46} +6.09168 q^{47} -1.18925 q^{48} +1.00000 q^{50} -1.18925 q^{51} -4.93935 q^{52} -3.62503 q^{53} +5.45353 q^{54} +3.88703 q^{55} +0.712863 q^{57} -2.81843 q^{58} -5.61745 q^{59} -1.18925 q^{60} -2.50763 q^{61} +5.25434 q^{62} +1.00000 q^{64} -4.93935 q^{65} -4.62265 q^{66} -12.6500 q^{67} +1.00000 q^{68} -9.63420 q^{69} +16.1938 q^{71} -1.58568 q^{72} +11.5564 q^{73} -8.25283 q^{74} -1.18925 q^{75} -0.599423 q^{76} +5.87412 q^{78} +16.3602 q^{79} +1.00000 q^{80} -1.72855 q^{81} -11.2863 q^{82} -6.37518 q^{83} +1.00000 q^{85} +10.7710 q^{86} +3.35181 q^{87} +3.88703 q^{88} +3.22268 q^{89} -1.58568 q^{90} +8.10107 q^{92} -6.24873 q^{93} +6.09168 q^{94} -0.599423 q^{95} -1.18925 q^{96} -0.283036 q^{97} -6.16360 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.18925	−0.686614	−0.343307	−	0.939223i	$-0.611547\pi$
$3$	−1.18925	−0.686614	−0.343307	+	0.939223i	$0.611547\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.18925	−0.485509
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−1.58568	−0.528562
$10$	1.00000	0.316228
$11$	3.88703	1.17198	0.585991	−	0.810317i	$-0.300705\pi$
$11$	3.88703	1.17198	0.585991	+	0.810317i	$0.300705\pi$
$12$	−1.18925	−0.343307
$13$	−4.93935	−1.36993	−0.684965	−	0.728576i	$-0.740183\pi$
$13$	−4.93935	−1.36993	−0.684965	+	0.728576i	$0.740183\pi$
$14$	0	0
$15$	−1.18925	−0.307063
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.58568	−0.373749
$19$	−0.599423	−0.137517	−0.0687585	−	0.997633i	$-0.521904\pi$
$19$	−0.599423	−0.137517	−0.0687585	+	0.997633i	$0.521904\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	3.88703	0.828717
$23$	8.10107	1.68919	0.844595	−	0.535406i	$-0.179841\pi$
$23$	8.10107	1.68919	0.844595	+	0.535406i	$0.179841\pi$
$24$	−1.18925	−0.242755
$25$	1.00000	0.200000
$26$	−4.93935	−0.968687
$27$	5.45353	1.04953
$28$	0	0
$29$	−2.81843	−0.523369	−0.261684	−	0.965154i	$-0.584278\pi$
$29$	−2.81843	−0.523369	−0.261684	+	0.965154i	$0.584278\pi$
$30$	−1.18925	−0.217126
$31$	5.25434	0.943708	0.471854	−	0.881677i	$-0.343585\pi$
$31$	5.25434	0.943708	0.471854	+	0.881677i	$0.343585\pi$
$32$	1.00000	0.176777
$33$	−4.62265	−0.804699
$34$	1.00000	0.171499
$35$	0	0
$36$	−1.58568	−0.264281
$37$	−8.25283	−1.35676	−0.678379	−	0.734712i	$-0.737317\pi$
$37$	−8.25283	−1.35676	−0.678379	+	0.734712i	$0.737317\pi$
$38$	−0.599423	−0.0972392
$39$	5.87412	0.940613
$40$	1.00000	0.158114
$41$	−11.2863	−1.76262	−0.881311	−	0.472536i	$-0.843339\pi$
$41$	−11.2863	−1.76262	−0.881311	+	0.472536i	$0.843339\pi$
$42$	0	0
$43$	10.7710	1.64256	0.821279	−	0.570527i	$-0.193261\pi$
$43$	10.7710	1.64256	0.821279	+	0.570527i	$0.193261\pi$
$44$	3.88703	0.585991
$45$	−1.58568	−0.236380
$46$	8.10107	1.19444
$47$	6.09168	0.888562	0.444281	−	0.895887i	$-0.353459\pi$
$47$	6.09168	0.888562	0.444281	+	0.895887i	$0.353459\pi$
$48$	−1.18925	−0.171653
$49$	0	0
$50$	1.00000	0.141421
$51$	−1.18925	−0.166528
$52$	−4.93935	−0.684965
$53$	−3.62503	−0.497937	−0.248968	−	0.968512i	$-0.580092\pi$
$53$	−3.62503	−0.497937	−0.248968	+	0.968512i	$0.580092\pi$
$54$	5.45353	0.742131
$55$	3.88703	0.524127
$56$	0	0
$57$	0.712863	0.0944211
$58$	−2.81843	−0.370077
$59$	−5.61745	−0.731330	−0.365665	−	0.930747i	$-0.619158\pi$
$59$	−5.61745	−0.731330	−0.365665	+	0.930747i	$0.619158\pi$
$60$	−1.18925	−0.153531
$61$	−2.50763	−0.321069	−0.160534	−	0.987030i	$-0.551322\pi$
$61$	−2.50763	−0.321069	−0.160534	+	0.987030i	$0.551322\pi$
$62$	5.25434	0.667302
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.93935	−0.612651
$66$	−4.62265	−0.569008
$67$	−12.6500	−1.54544	−0.772722	−	0.634745i	$-0.781105\pi$
$67$	−12.6500	−1.54544	−0.772722	+	0.634745i	$0.781105\pi$
$68$	1.00000	0.121268
$69$	−9.63420	−1.15982
$70$	0	0
$71$	16.1938	1.92185	0.960927	−	0.276801i	$-0.0892744\pi$
$71$	16.1938	1.92185	0.960927	+	0.276801i	$0.0892744\pi$
$72$	−1.58568	−0.186875
$73$	11.5564	1.35258	0.676289	−	0.736637i	$-0.263587\pi$
$73$	11.5564	1.35258	0.676289	+	0.736637i	$0.263587\pi$
$74$	−8.25283	−0.959372
$75$	−1.18925	−0.137323
$76$	−0.599423	−0.0687585
$77$	0	0
$78$	5.87412	0.665114
$79$	16.3602	1.84067	0.920335	−	0.391132i	$-0.127916\pi$
$79$	16.3602	1.84067	0.920335	+	0.391132i	$0.127916\pi$
$80$	1.00000	0.111803
$81$	−1.72855	−0.192061
$82$	−11.2863	−1.24636
$83$	−6.37518	−0.699767	−0.349883	−	0.936793i	$-0.613779\pi$
$83$	−6.37518	−0.699767	−0.349883	+	0.936793i	$0.613779\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	10.7710	1.16146
$87$	3.35181	0.359352
$88$	3.88703	0.414358
$89$	3.22268	0.341603	0.170801	−	0.985305i	$-0.445364\pi$
$89$	3.22268	0.341603	0.170801	+	0.985305i	$0.445364\pi$
$90$	−1.58568	−0.167146
$91$	0	0
$92$	8.10107	0.844595
$93$	−6.24873	−0.647963
$94$	6.09168	0.628308
$95$	−0.599423	−0.0614995
$96$	−1.18925	−0.121377
$97$	−0.283036	−0.0287380	−0.0143690	−	0.999897i	$-0.504574\pi$
$97$	−0.283036	−0.0287380	−0.0143690	+	0.999897i	$0.504574\pi$
$98$	0	0
$99$	−6.16360	−0.619465
$100$	1.00000	0.100000
$101$	0.287157	0.0285732	0.0142866	−	0.999898i	$-0.495452\pi$
$101$	0.287157	0.0285732	0.0142866	+	0.999898i	$0.495452\pi$
$102$	−1.18925	−0.117753
$103$	−11.0654	−1.09030	−0.545151	−	0.838338i	$-0.683528\pi$
$103$	−11.0654	−1.09030	−0.545151	+	0.838338i	$0.683528\pi$
$104$	−4.93935	−0.484343
$105$	0	0
$106$	−3.62503	−0.352094
$107$	−0.618021	−0.0597463	−0.0298732	−	0.999554i	$-0.509510\pi$
$107$	−0.618021	−0.0597463	−0.0298732	+	0.999554i	$0.509510\pi$
$108$	5.45353	0.524766
$109$	−6.43243	−0.616115	−0.308058	−	0.951368i	$-0.599679\pi$
$109$	−6.43243	−0.616115	−0.308058	+	0.951368i	$0.599679\pi$
$110$	3.88703	0.370613
$111$	9.81468	0.931568
$112$	0	0
$113$	20.1982	1.90008	0.950042	−	0.312122i	$-0.101040\pi$
$113$	20.1982	1.90008	0.950042	+	0.312122i	$0.101040\pi$
$114$	0.712863	0.0667658
$115$	8.10107	0.755429
$116$	−2.81843	−0.261684
$117$	7.83226	0.724092
$118$	−5.61745	−0.517128
$119$	0	0
$120$	−1.18925	−0.108563
$121$	4.10898	0.373543
$122$	−2.50763	−0.227030
$123$	13.4222	1.21024
$124$	5.25434	0.471854
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.98276	0.442148	0.221074	−	0.975257i	$-0.429044\pi$
$127$	4.98276	0.442148	0.221074	+	0.975257i	$0.429044\pi$
$128$	1.00000	0.0883883
$129$	−12.8094	−1.12780
$130$	−4.93935	−0.433210
$131$	18.7897	1.64166	0.820830	−	0.571173i	$-0.193511\pi$
$131$	18.7897	1.64166	0.820830	+	0.571173i	$0.193511\pi$
$132$	−4.62265	−0.402350
$133$	0	0
$134$	−12.6500	−1.09279
$135$	5.45353	0.469365
$136$	1.00000	0.0857493
$137$	12.4359	1.06247	0.531234	−	0.847225i	$-0.321728\pi$
$137$	12.4359	1.06247	0.531234	+	0.847225i	$0.321728\pi$
$138$	−9.63420	−0.820117
$139$	9.16981	0.777773	0.388886	−	0.921286i	$-0.372860\pi$
$139$	9.16981	0.777773	0.388886	+	0.921286i	$0.372860\pi$
$140$	0	0
$141$	−7.24452	−0.610099
$142$	16.1938	1.35896
$143$	−19.1994	−1.60553
$144$	−1.58568	−0.132140
$145$	−2.81843	−0.234058
$146$	11.5564	0.956416
$147$	0	0
$148$	−8.25283	−0.678379
$149$	−11.4109	−0.934814	−0.467407	−	0.884042i	$-0.654812\pi$
$149$	−11.4109	−0.934814	−0.467407	+	0.884042i	$0.654812\pi$
$150$	−1.18925	−0.0971018
$151$	8.95637	0.728859	0.364429	−	0.931231i	$-0.381264\pi$
$151$	8.95637	0.728859	0.364429	+	0.931231i	$0.381264\pi$
$152$	−0.599423	−0.0486196
$153$	−1.58568	−0.128195
$154$	0	0
$155$	5.25434	0.422039
$156$	5.87412	0.470306
$157$	−16.1983	−1.29277	−0.646383	−	0.763013i	$-0.723719\pi$
$157$	−16.1983	−1.29277	−0.646383	+	0.763013i	$0.723719\pi$
$158$	16.3602	1.30155
$159$	4.31107	0.341890
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.72855	−0.135808
$163$	12.0413	0.943146	0.471573	−	0.881827i	$-0.343686\pi$
$163$	12.0413	0.943146	0.471573	+	0.881827i	$0.343686\pi$
$164$	−11.2863	−0.881311
$165$	−4.62265	−0.359873
$166$	−6.37518	−0.494810
$167$	7.71706	0.597164	0.298582	−	0.954384i	$-0.403486\pi$
$167$	7.71706	0.597164	0.298582	+	0.954384i	$0.403486\pi$
$168$	0	0
$169$	11.3972	0.876708
$170$	1.00000	0.0766965
$171$	0.950495	0.0726862
$172$	10.7710	0.821279
$173$	5.78375	0.439731	0.219865	−	0.975530i	$-0.429438\pi$
$173$	5.78375	0.439731	0.219865	+	0.975530i	$0.429438\pi$
$174$	3.35181	0.254100
$175$	0	0
$176$	3.88703	0.292996
$177$	6.68055	0.502141
$178$	3.22268	0.241550
$179$	−12.5743	−0.939851	−0.469925	−	0.882706i	$-0.655719\pi$
$179$	−12.5743	−0.939851	−0.469925	+	0.882706i	$0.655719\pi$
$180$	−1.58568	−0.118190
$181$	22.5585	1.67676	0.838378	−	0.545089i	$-0.183504\pi$
$181$	22.5585	1.67676	0.838378	+	0.545089i	$0.183504\pi$
$182$	0	0
$183$	2.98219	0.220450
$184$	8.10107	0.597219
$185$	−8.25283	−0.606760
$186$	−6.24873	−0.458179
$187$	3.88703	0.284248
$188$	6.09168	0.444281
$189$	0	0
$190$	−0.599423	−0.0434867
$191$	5.50253	0.398149	0.199074	−	0.979984i	$-0.436206\pi$
$191$	5.50253	0.398149	0.199074	+	0.979984i	$0.436206\pi$
$192$	−1.18925	−0.0858267
$193$	−16.4940	−1.18726	−0.593631	−	0.804737i	$-0.702306\pi$
$193$	−16.4940	−1.18726	−0.593631	+	0.804737i	$0.702306\pi$
$194$	−0.283036	−0.0203208
$195$	5.87412	0.420655
$196$	0	0
$197$	21.2693	1.51537	0.757687	−	0.652618i	$-0.226329\pi$
$197$	21.2693	1.51537	0.757687	+	0.652618i	$0.226329\pi$
$198$	−6.16360	−0.438028
$199$	−16.9189	−1.19935	−0.599674	−	0.800245i	$-0.704703\pi$
$199$	−16.9189	−1.19935	−0.599674	+	0.800245i	$0.704703\pi$
$200$	1.00000	0.0707107
$201$	15.0440	1.06112
$202$	0.287157	0.0202043
$203$	0	0
$204$	−1.18925	−0.0832641
$205$	−11.2863	−0.788269
$206$	−11.0654	−0.770960
$207$	−12.8457	−0.892841
$208$	−4.93935	−0.342482
$209$	−2.32997	−0.161168
$210$	0	0
$211$	−6.90593	−0.475424	−0.237712	−	0.971336i	$-0.576397\pi$
$211$	−6.90593	−0.475424	−0.237712	+	0.971336i	$0.576397\pi$
$212$	−3.62503	−0.248968
$213$	−19.2585	−1.31957
$214$	−0.618021	−0.0422470
$215$	10.7710	0.734574
$216$	5.45353	0.371065
$217$	0	0
$218$	−6.43243	−0.435659
$219$	−13.7435	−0.928698
$220$	3.88703	0.262063
$221$	−4.93935	−0.332257
$222$	9.81468	0.658718
$223$	18.1630	1.21629	0.608143	−	0.793828i	$-0.291915\pi$
$223$	18.1630	1.21629	0.608143	+	0.793828i	$0.291915\pi$
$224$	0	0
$225$	−1.58568	−0.105712
$226$	20.1982	1.34356
$227$	8.19929	0.544206	0.272103	−	0.962268i	$-0.412281\pi$
$227$	8.19929	0.544206	0.272103	+	0.962268i	$0.412281\pi$
$228$	0.712863	0.0472105
$229$	4.07526	0.269301	0.134651	−	0.990893i	$-0.457009\pi$
$229$	4.07526	0.269301	0.134651	+	0.990893i	$0.457009\pi$
$230$	8.10107	0.534169
$231$	0	0
$232$	−2.81843	−0.185039
$233$	24.3680	1.59640	0.798199	−	0.602394i	$-0.205786\pi$
$233$	24.3680	1.59640	0.798199	+	0.602394i	$0.205786\pi$
$234$	7.83226	0.512011
$235$	6.09168	0.397377
$236$	−5.61745	−0.365665
$237$	−19.4564	−1.26383
$238$	0	0
$239$	29.1110	1.88304	0.941518	−	0.336964i	$-0.109400\pi$
$239$	29.1110	1.88304	0.941518	+	0.336964i	$0.109400\pi$
$240$	−1.18925	−0.0767657
$241$	−12.2849	−0.791343	−0.395671	−	0.918392i	$-0.629488\pi$
$241$	−12.2849	−0.791343	−0.395671	+	0.918392i	$0.629488\pi$
$242$	4.10898	0.264135
$243$	−14.3049	−0.917660
$244$	−2.50763	−0.160534
$245$	0	0
$246$	13.4222	0.855770
$247$	2.96076	0.188389
$248$	5.25434	0.333651
$249$	7.58168	0.480469
$250$	1.00000	0.0632456
$251$	4.47707	0.282590	0.141295	−	0.989968i	$-0.454873\pi$
$251$	4.47707	0.282590	0.141295	+	0.989968i	$0.454873\pi$
$252$	0	0
$253$	31.4891	1.97970
$254$	4.98276	0.312646
$255$	−1.18925	−0.0744737
$256$	1.00000	0.0625000
$257$	12.2337	0.763115	0.381557	−	0.924345i	$-0.375388\pi$
$257$	12.2337	0.763115	0.381557	+	0.924345i	$0.375388\pi$
$258$	−12.8094	−0.797477
$259$	0	0
$260$	−4.93935	−0.306326
$261$	4.46914	0.276633
$262$	18.7897	1.16083
$263$	15.4713	0.954004	0.477002	−	0.878902i	$-0.341724\pi$
$263$	15.4713	0.954004	0.477002	+	0.878902i	$0.341724\pi$
$264$	−4.62265	−0.284504
$265$	−3.62503	−0.222684
$266$	0	0
$267$	−3.83257	−0.234549
$268$	−12.6500	−0.772722
$269$	−14.7861	−0.901527	−0.450763	−	0.892643i	$-0.648848\pi$
$269$	−14.7861	−0.901527	−0.450763	+	0.892643i	$0.648848\pi$
$270$	5.45353	0.331891
$271$	−27.6385	−1.67892	−0.839460	−	0.543421i	$-0.817129\pi$
$271$	−27.6385	−1.67892	−0.839460	+	0.543421i	$0.817129\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	12.4359	0.751279
$275$	3.88703	0.234397
$276$	−9.63420	−0.579910
$277$	8.27894	0.497433	0.248717	−	0.968576i	$-0.419991\pi$
$277$	8.27894	0.497433	0.248717	+	0.968576i	$0.419991\pi$
$278$	9.16981	0.549968
$279$	−8.33173	−0.498808
$280$	0	0
$281$	−18.8906	−1.12692	−0.563458	−	0.826145i	$-0.690529\pi$
$281$	−18.8906	−1.12692	−0.563458	+	0.826145i	$0.690529\pi$
$282$	−7.24452	−0.431405
$283$	−11.4025	−0.677809	−0.338904	−	0.940821i	$-0.610056\pi$
$283$	−11.4025	−0.677809	−0.338904	+	0.940821i	$0.610056\pi$
$284$	16.1938	0.960927
$285$	0.712863	0.0422264
$286$	−19.1994	−1.13528
$287$	0	0
$288$	−1.58568	−0.0934374
$289$	1.00000	0.0588235
$290$	−2.81843	−0.165504
$291$	0.336601	0.0197319
$292$	11.5564	0.676289
$293$	26.0316	1.52078	0.760390	−	0.649467i	$-0.225008\pi$
$293$	26.0316	1.52078	0.760390	+	0.649467i	$0.225008\pi$
$294$	0	0
$295$	−5.61745	−0.327061
$296$	−8.25283	−0.479686
$297$	21.1980	1.23003
$298$	−11.4109	−0.661013
$299$	−40.0140	−2.31407
$300$	−1.18925	−0.0686614
$301$	0	0
$302$	8.95637	0.515381
$303$	−0.341501	−0.0196187
$304$	−0.599423	−0.0343792
$305$	−2.50763	−0.143586
$306$	−1.58568	−0.0906476
$307$	−21.2275	−1.21152	−0.605759	−	0.795648i	$-0.707130\pi$
$307$	−21.2275	−1.21152	−0.605759	+	0.795648i	$0.707130\pi$
$308$	0	0
$309$	13.1595	0.748616
$310$	5.25434	0.298427
$311$	14.8608	0.842680	0.421340	−	0.906903i	$-0.361560\pi$
$311$	14.8608	0.842680	0.421340	+	0.906903i	$0.361560\pi$
$312$	5.87412	0.332557
$313$	18.1281	1.02466	0.512331	−	0.858788i	$-0.328782\pi$
$313$	18.1281	1.02466	0.512331	+	0.858788i	$0.328782\pi$
$314$	−16.1983	−0.914123
$315$	0	0
$316$	16.3602	0.920335
$317$	−26.0553	−1.46341	−0.731707	−	0.681619i	$-0.761276\pi$
$317$	−26.0553	−1.46341	−0.731707	+	0.681619i	$0.761276\pi$
$318$	4.31107	0.241753
$319$	−10.9553	−0.613379
$320$	1.00000	0.0559017
$321$	0.734981	0.0410226
$322$	0	0
$323$	−0.599423	−0.0333528
$324$	−1.72855	−0.0960305
$325$	−4.93935	−0.273986
$326$	12.0413	0.666905
$327$	7.64977	0.423033
$328$	−11.2863	−0.623181
$329$	0	0
$330$	−4.62265	−0.254468
$331$	10.2749	0.564760	0.282380	−	0.959303i	$-0.408876\pi$
$331$	10.2749	0.564760	0.282380	+	0.959303i	$0.408876\pi$
$332$	−6.37518	−0.349883
$333$	13.0864	0.717130
$334$	7.71706	0.422259
$335$	−12.6500	−0.691143
$336$	0	0
$337$	27.7967	1.51418	0.757092	−	0.653309i	$-0.226620\pi$
$337$	27.7967	1.51418	0.757092	+	0.653309i	$0.226620\pi$
$338$	11.3972	0.619926
$339$	−24.0207	−1.30462
$340$	1.00000	0.0542326
$341$	20.4238	1.10601
$342$	0.950495	0.0513969
$343$	0	0
$344$	10.7710	0.580732
$345$	−9.63420	−0.518688
$346$	5.78375	0.310937
$347$	31.7556	1.70473	0.852366	−	0.522946i	$-0.175167\pi$
$347$	31.7556	1.70473	0.852366	+	0.522946i	$0.175167\pi$
$348$	3.35181	0.179676
$349$	−31.6502	−1.69419	−0.847097	−	0.531438i	$-0.821652\pi$
$349$	−31.6502	−1.69419	−0.847097	+	0.531438i	$0.821652\pi$
$350$	0	0
$351$	−26.9369	−1.43778
$352$	3.88703	0.207179
$353$	14.1446	0.752840	0.376420	−	0.926449i	$-0.377155\pi$
$353$	14.1446	0.752840	0.376420	+	0.926449i	$0.377155\pi$
$354$	6.68055	0.355067
$355$	16.1938	0.859479
$356$	3.22268	0.170801
$357$	0	0
$358$	−12.5743	−0.664575
$359$	−12.7131	−0.670970	−0.335485	−	0.942046i	$-0.608900\pi$
$359$	−12.7131	−0.670970	−0.335485	+	0.942046i	$0.608900\pi$
$360$	−1.58568	−0.0835729
$361$	−18.6407	−0.981089
$362$	22.5585	1.18565
$363$	−4.88660	−0.256480
$364$	0	0
$365$	11.5564	0.604891
$366$	2.98219	0.155882
$367$	−23.3607	−1.21942	−0.609709	−	0.792626i	$-0.708714\pi$
$367$	−23.3607	−1.21942	−0.609709	+	0.792626i	$0.708714\pi$
$368$	8.10107	0.422297
$369$	17.8965	0.931655
$370$	−8.25283	−0.429044
$371$	0	0
$372$	−6.24873	−0.323981
$373$	−18.0183	−0.932953	−0.466477	−	0.884534i	$-0.654477\pi$
$373$	−18.0183	−0.932953	−0.466477	+	0.884534i	$0.654477\pi$
$374$	3.88703	0.200993
$375$	−1.18925	−0.0614126
$376$	6.09168	0.314154
$377$	13.9212	0.716978
$378$	0	0
$379$	−22.4975	−1.15562	−0.577810	−	0.816171i	$-0.696093\pi$
$379$	−22.4975	−1.15562	−0.577810	+	0.816171i	$0.696093\pi$
$380$	−0.599423	−0.0307497
$381$	−5.92574	−0.303585
$382$	5.50253	0.281534
$383$	1.50666	0.0769867	0.0384934	−	0.999259i	$-0.487744\pi$
$383$	1.50666	0.0769867	0.0384934	+	0.999259i	$0.487744\pi$
$384$	−1.18925	−0.0606887
$385$	0	0
$386$	−16.4940	−0.839521
$387$	−17.0794	−0.868193
$388$	−0.283036	−0.0143690
$389$	0.804996	0.0408149	0.0204074	−	0.999792i	$-0.493504\pi$
$389$	0.804996	0.0408149	0.0204074	+	0.999792i	$0.493504\pi$
$390$	5.87412	0.297448
$391$	8.10107	0.409689
$392$	0	0
$393$	−22.3456	−1.12719
$394$	21.2693	1.07153
$395$	16.3602	0.823172
$396$	−6.16360	−0.309733
$397$	−21.6483	−1.08650	−0.543249	−	0.839571i	$-0.682806\pi$
$397$	−21.6483	−1.08650	−0.543249	+	0.839571i	$0.682806\pi$
$398$	−16.9189	−0.848067
$399$	0	0
$400$	1.00000	0.0500000
$401$	36.3123	1.81335	0.906675	−	0.421829i	$-0.138612\pi$
$401$	36.3123	1.81335	0.906675	+	0.421829i	$0.138612\pi$
$402$	15.0440	0.750327
$403$	−25.9531	−1.29281
$404$	0.287157	0.0142866
$405$	−1.72855	−0.0858923
$406$	0	0
$407$	−32.0790	−1.59010
$408$	−1.18925	−0.0588766
$409$	21.0159	1.03917	0.519584	−	0.854419i	$-0.326087\pi$
$409$	21.0159	1.03917	0.519584	+	0.854419i	$0.326087\pi$
$410$	−11.2863	−0.557390
$411$	−14.7894	−0.729506
$412$	−11.0654	−0.545151
$413$	0	0
$414$	−12.8457	−0.631334
$415$	−6.37518	−0.312945
$416$	−4.93935	−0.242172
$417$	−10.9052	−0.534029
$418$	−2.32997	−0.113963
$419$	−16.5926	−0.810603	−0.405302	−	0.914183i	$-0.632834\pi$
$419$	−16.5926	−0.810603	−0.405302	+	0.914183i	$0.632834\pi$
$420$	0	0
$421$	−0.242990	−0.0118426	−0.00592130	−	0.999982i	$-0.501885\pi$
$421$	−0.242990	−0.0118426	−0.00592130	+	0.999982i	$0.501885\pi$
$422$	−6.90593	−0.336175
$423$	−9.65948	−0.469660
$424$	−3.62503	−0.176047
$425$	1.00000	0.0485071
$426$	−19.2585	−0.933078
$427$	0	0
$428$	−0.618021	−0.0298732
$429$	22.8329	1.10238
$430$	10.7710	0.519422
$431$	−24.8767	−1.19827	−0.599135	−	0.800648i	$-0.704489\pi$
$431$	−24.8767	−1.19827	−0.599135	+	0.800648i	$0.704489\pi$
$432$	5.45353	0.262383
$433$	30.8365	1.48191	0.740953	−	0.671557i	$-0.234374\pi$
$433$	30.8365	1.48191	0.740953	+	0.671557i	$0.234374\pi$
$434$	0	0
$435$	3.35181	0.160707
$436$	−6.43243	−0.308058
$437$	−4.85596	−0.232292
$438$	−13.7435	−0.656689
$439$	−8.50071	−0.405717	−0.202858	−	0.979208i	$-0.565023\pi$
$439$	−8.50071	−0.405717	−0.202858	+	0.979208i	$0.565023\pi$
$440$	3.88703	0.185307
$441$	0	0
$442$	−4.93935	−0.234941
$443$	24.2993	1.15450	0.577248	−	0.816569i	$-0.304127\pi$
$443$	24.2993	1.15450	0.577248	+	0.816569i	$0.304127\pi$
$444$	9.81468	0.465784
$445$	3.22268	0.152770
$446$	18.1630	0.860044
$447$	13.5704	0.641856
$448$	0	0
$449$	−17.7605	−0.838170	−0.419085	−	0.907947i	$-0.637649\pi$
$449$	−17.7605	−0.838170	−0.419085	+	0.907947i	$0.637649\pi$
$450$	−1.58568	−0.0747499
$451$	−43.8701	−2.06576
$452$	20.1982	0.950042
$453$	−10.6514	−0.500444
$454$	8.19929	0.384812
$455$	0	0
$456$	0.712863	0.0333829
$457$	−20.2826	−0.948782	−0.474391	−	0.880314i	$-0.657332\pi$
$457$	−20.2826	−0.948782	−0.474391	+	0.880314i	$0.657332\pi$
$458$	4.07526	0.190425
$459$	5.45353	0.254549
$460$	8.10107	0.377714
$461$	18.0461	0.840491	0.420246	−	0.907410i	$-0.361944\pi$
$461$	18.0461	0.840491	0.420246	+	0.907410i	$0.361944\pi$
$462$	0	0
$463$	32.8829	1.52820	0.764099	−	0.645099i	$-0.223184\pi$
$463$	32.8829	1.52820	0.764099	+	0.645099i	$0.223184\pi$
$464$	−2.81843	−0.130842
$465$	−6.24873	−0.289778
$466$	24.3680	1.12882
$467$	22.4945	1.04092	0.520461	−	0.853885i	$-0.325760\pi$
$467$	22.4945	1.04092	0.520461	+	0.853885i	$0.325760\pi$
$468$	7.83226	0.362046
$469$	0	0
$470$	6.09168	0.280988
$471$	19.2638	0.887630
$472$	−5.61745	−0.258564
$473$	41.8671	1.92505
$474$	−19.4564	−0.893662
$475$	−0.599423	−0.0275034
$476$	0	0
$477$	5.74816	0.263190
$478$	29.1110	1.33151
$479$	42.0656	1.92203	0.961014	−	0.276499i	$-0.0891742\pi$
$479$	42.0656	1.92203	0.961014	+	0.276499i	$0.0891742\pi$
$480$	−1.18925	−0.0542816
$481$	40.7637	1.85866
$482$	−12.2849	−0.559564
$483$	0	0
$484$	4.10898	0.186772
$485$	−0.283036	−0.0128520
$486$	−14.3049	−0.648883
$487$	9.82480	0.445204	0.222602	−	0.974909i	$-0.428545\pi$
$487$	9.82480	0.445204	0.222602	+	0.974909i	$0.428545\pi$
$488$	−2.50763	−0.113515
$489$	−14.3201	−0.647577
$490$	0	0
$491$	21.3078	0.961609	0.480804	−	0.876828i	$-0.340345\pi$
$491$	21.3078	0.961609	0.480804	+	0.876828i	$0.340345\pi$
$492$	13.4222	0.605120
$493$	−2.81843	−0.126936
$494$	2.96076	0.133211
$495$	−6.16360	−0.277033
$496$	5.25434	0.235927
$497$	0	0
$498$	7.58168	0.339743
$499$	−15.1163	−0.676699	−0.338350	−	0.941020i	$-0.609869\pi$
$499$	−15.1163	−0.676699	−0.338350	+	0.941020i	$0.609869\pi$
$500$	1.00000	0.0447214
$501$	−9.17752	−0.410021
$502$	4.47707	0.199822
$503$	18.3323	0.817398	0.408699	−	0.912669i	$-0.365983\pi$
$503$	18.3323	0.817398	0.408699	+	0.912669i	$0.365983\pi$
$504$	0	0
$505$	0.287157	0.0127783
$506$	31.4891	1.39986
$507$	−13.5541	−0.601960
$508$	4.98276	0.221074
$509$	33.6416	1.49114	0.745569	−	0.666428i	$-0.232178\pi$
$509$	33.6416	1.49114	0.745569	+	0.666428i	$0.232178\pi$
$510$	−1.18925	−0.0526609
$511$	0	0
$512$	1.00000	0.0441942
$513$	−3.26897	−0.144328
$514$	12.2337	0.539604
$515$	−11.0654	−0.487598
$516$	−12.8094	−0.563901
$517$	23.6785	1.04138
$518$	0	0
$519$	−6.87833	−0.301925
$520$	−4.93935	−0.216605
$521$	−4.62222	−0.202503	−0.101252	−	0.994861i	$-0.532285\pi$
$521$	−4.62222	−0.202503	−0.101252	+	0.994861i	$0.532285\pi$
$522$	4.46914	0.195609
$523$	22.5126	0.984406	0.492203	−	0.870480i	$-0.336192\pi$
$523$	22.5126	0.984406	0.492203	+	0.870480i	$0.336192\pi$
$524$	18.7897	0.820830
$525$	0	0
$526$	15.4713	0.674582
$527$	5.25434	0.228883
$528$	−4.62265	−0.201175
$529$	42.6273	1.85336
$530$	−3.62503	−0.157461
$531$	8.90751	0.386553
$532$	0	0
$533$	55.7470	2.41467
$534$	−3.83257	−0.165851
$535$	−0.618021	−0.0267194
$536$	−12.6500	−0.546397
$537$	14.9540	0.645314
$538$	−14.7861	−0.637476
$539$	0	0
$540$	5.45353	0.234682
$541$	−13.8007	−0.593336	−0.296668	−	0.954981i	$-0.595876\pi$
$541$	−13.8007	−0.593336	−0.296668	+	0.954981i	$0.595876\pi$
$542$	−27.6385	−1.18718
$543$	−26.8276	−1.15128
$544$	1.00000	0.0428746
$545$	−6.43243	−0.275535
$546$	0	0
$547$	5.73930	0.245395	0.122697	−	0.992444i	$-0.460846\pi$
$547$	5.73930	0.245395	0.122697	+	0.992444i	$0.460846\pi$
$548$	12.4359	0.531234
$549$	3.97630	0.169705
$550$	3.88703	0.165743
$551$	1.68943	0.0719721
$552$	−9.63420	−0.410059
$553$	0	0
$554$	8.27894	0.351738
$555$	9.81468	0.416610
$556$	9.16981	0.388886
$557$	15.7899	0.669041	0.334520	−	0.942389i	$-0.391426\pi$
$557$	15.7899	0.669041	0.334520	+	0.942389i	$0.391426\pi$
$558$	−8.33173	−0.352710
$559$	−53.2016	−2.25019
$560$	0	0
$561$	−4.62265	−0.195168
$562$	−18.8906	−0.796850
$563$	0.731332	0.0308220	0.0154110	−	0.999881i	$-0.495094\pi$
$563$	0.731332	0.0308220	0.0154110	+	0.999881i	$0.495094\pi$
$564$	−7.24452	−0.305049
$565$	20.1982	0.849743
$566$	−11.4025	−0.479283
$567$	0	0
$568$	16.1938	0.679478
$569$	−3.58359	−0.150232	−0.0751160	−	0.997175i	$-0.523933\pi$
$569$	−3.58359	−0.150232	−0.0751160	+	0.997175i	$0.523933\pi$
$570$	0.712863	0.0298586
$571$	29.6676	1.24155	0.620776	−	0.783988i	$-0.286818\pi$
$571$	29.6676	1.24155	0.620776	+	0.783988i	$0.286818\pi$
$572$	−19.1994	−0.802767
$573$	−6.54388	−0.273374
$574$	0	0
$575$	8.10107	0.337838
$576$	−1.58568	−0.0660702
$577$	−12.4198	−0.517042	−0.258521	−	0.966006i	$-0.583235\pi$
$577$	−12.4198	−0.517042	−0.258521	+	0.966006i	$0.583235\pi$
$578$	1.00000	0.0415945
$579$	19.6155	0.815191
$580$	−2.81843	−0.117029
$581$	0	0
$582$	0.336601	0.0139526
$583$	−14.0906	−0.583573
$584$	11.5564	0.478208
$585$	7.83226	0.323824
$586$	26.0316	1.07535
$587$	−14.4642	−0.597000	−0.298500	−	0.954410i	$-0.596486\pi$
$587$	−14.4642	−0.597000	−0.298500	+	0.954410i	$0.596486\pi$
$588$	0	0
$589$	−3.14957	−0.129776
$590$	−5.61745	−0.231267
$591$	−25.2945	−1.04048
$592$	−8.25283	−0.339189
$593$	47.4570	1.94882	0.974412	−	0.224767i	$-0.0721622\pi$
$593$	47.4570	1.94882	0.974412	+	0.224767i	$0.0721622\pi$
$594$	21.1980	0.869764
$595$	0	0
$596$	−11.4109	−0.467407
$597$	20.1208	0.823488
$598$	−40.0140	−1.63630
$599$	−30.1094	−1.23023	−0.615117	−	0.788435i	$-0.710891\pi$
$599$	−30.1094	−1.23023	−0.615117	+	0.788435i	$0.710891\pi$
$600$	−1.18925	−0.0485509
$601$	−34.1137	−1.39153	−0.695764	−	0.718270i	$-0.744934\pi$
$601$	−34.1137	−1.39153	−0.695764	+	0.718270i	$0.744934\pi$
$602$	0	0
$603$	20.0589	0.816862
$604$	8.95637	0.364429
$605$	4.10898	0.167054
$606$	−0.341501	−0.0138725
$607$	22.1081	0.897339	0.448670	−	0.893698i	$-0.351898\pi$
$607$	22.1081	0.897339	0.448670	+	0.893698i	$0.351898\pi$
$608$	−0.599423	−0.0243098
$609$	0	0
$610$	−2.50763	−0.101531
$611$	−30.0889	−1.21727
$612$	−1.58568	−0.0640975
$613$	−23.4663	−0.947795	−0.473897	−	0.880580i	$-0.657153\pi$
$613$	−23.4663	−0.947795	−0.473897	+	0.880580i	$0.657153\pi$
$614$	−21.2275	−0.856672
$615$	13.4222	0.541236
$616$	0	0
$617$	−29.7155	−1.19630	−0.598151	−	0.801383i	$-0.704098\pi$
$617$	−29.7155	−1.19630	−0.598151	+	0.801383i	$0.704098\pi$
$618$	13.1595	0.529352
$619$	4.18746	0.168308	0.0841541	−	0.996453i	$-0.473181\pi$
$619$	4.18746	0.168308	0.0841541	+	0.996453i	$0.473181\pi$
$620$	5.25434	0.211020
$621$	44.1794	1.77286
$622$	14.8608	0.595865
$623$	0	0
$624$	5.87412	0.235153
$625$	1.00000	0.0400000
$626$	18.1281	0.724546
$627$	2.77092	0.110660
$628$	−16.1983	−0.646383
$629$	−8.25283	−0.329062
$630$	0	0
$631$	11.7004	0.465787	0.232894	−	0.972502i	$-0.425181\pi$
$631$	11.7004	0.465787	0.232894	+	0.972502i	$0.425181\pi$
$632$	16.3602	0.650775
$633$	8.21288	0.326432
$634$	−26.0553	−1.03479
$635$	4.98276	0.197735
$636$	4.31107	0.170945
$637$	0	0
$638$	−10.9553	−0.433724
$639$	−25.6783	−1.01582
$640$	1.00000	0.0395285
$641$	−5.26640	−0.208010	−0.104005	−	0.994577i	$-0.533166\pi$
$641$	−5.26640	−0.208010	−0.104005	+	0.994577i	$0.533166\pi$
$642$	0.734981	0.0290074
$643$	30.5288	1.20394	0.601968	−	0.798520i	$-0.294383\pi$
$643$	30.5288	1.20394	0.601968	+	0.798520i	$0.294383\pi$
$644$	0	0
$645$	−12.8094	−0.504369
$646$	−0.599423	−0.0235840
$647$	−18.7768	−0.738193	−0.369097	−	0.929391i	$-0.620333\pi$
$647$	−18.7768	−0.738193	−0.369097	+	0.929391i	$0.620333\pi$
$648$	−1.72855	−0.0679039
$649$	−21.8352	−0.857106
$650$	−4.93935	−0.193737
$651$	0	0
$652$	12.0413	0.471573
$653$	−50.9459	−1.99367	−0.996833	−	0.0795252i	$-0.974660\pi$
$653$	−50.9459	−1.99367	−0.996833	+	0.0795252i	$0.974660\pi$
$654$	7.64977	0.299130
$655$	18.7897	0.734172
$656$	−11.2863	−0.440656
$657$	−18.3248	−0.714920
$658$	0	0
$659$	−12.4244	−0.483987	−0.241994	−	0.970278i	$-0.577801\pi$
$659$	−12.4244	−0.483987	−0.241994	+	0.970278i	$0.577801\pi$
$660$	−4.62265	−0.179936
$661$	8.30350	0.322969	0.161484	−	0.986875i	$-0.448372\pi$
$661$	8.30350	0.322969	0.161484	+	0.986875i	$0.448372\pi$
$662$	10.2749	0.399345
$663$	5.87412	0.228132
$664$	−6.37518	−0.247405
$665$	0	0
$666$	13.0864	0.507087
$667$	−22.8323	−0.884069
$668$	7.71706	0.298582
$669$	−21.6004	−0.835119
$670$	−12.6500	−0.488712
$671$	−9.74721	−0.376287
$672$	0	0
$673$	−37.8645	−1.45957	−0.729785	−	0.683677i	$-0.760380\pi$
$673$	−37.8645	−1.45957	−0.729785	+	0.683677i	$0.760380\pi$
$674$	27.7967	1.07069
$675$	5.45353	0.209906
$676$	11.3972	0.438354
$677$	16.0861	0.618239	0.309119	−	0.951023i	$-0.399966\pi$
$677$	16.0861	0.618239	0.309119	+	0.951023i	$0.399966\pi$
$678$	−24.0207	−0.922508
$679$	0	0
$680$	1.00000	0.0383482
$681$	−9.75100	−0.373659
$682$	20.4238	0.782067
$683$	−22.8684	−0.875035	−0.437517	−	0.899210i	$-0.644142\pi$
$683$	−22.8684	−0.875035	−0.437517	+	0.899210i	$0.644142\pi$
$684$	0.950495	0.0363431
$685$	12.4359	0.475150
$686$	0	0
$687$	−4.84651	−0.184906
$688$	10.7710	0.410640
$689$	17.9053	0.682138
$690$	−9.63420	−0.366768
$691$	−0.474226	−0.0180404	−0.00902021	−	0.999959i	$-0.502871\pi$
$691$	−0.474226	−0.0180404	−0.00902021	+	0.999959i	$0.502871\pi$
$692$	5.78375	0.219865
$693$	0	0
$694$	31.7556	1.20543
$695$	9.16981	0.347831
$696$	3.35181	0.127050
$697$	−11.2863	−0.427499
$698$	−31.6502	−1.19798
$699$	−28.9796	−1.09611
$700$	0	0
$701$	−5.08326	−0.191992	−0.0959961	−	0.995382i	$-0.530604\pi$
$701$	−5.08326	−0.191992	−0.0959961	+	0.995382i	$0.530604\pi$
$702$	−26.9369	−1.01667
$703$	4.94694	0.186577
$704$	3.88703	0.146498
$705$	−7.24452	−0.272845
$706$	14.1446	0.532338
$707$	0	0
$708$	6.68055	0.251071
$709$	30.0471	1.12844	0.564222	−	0.825623i	$-0.309176\pi$
$709$	30.0471	1.12844	0.564222	+	0.825623i	$0.309176\pi$
$710$	16.1938	0.607744
$711$	−25.9422	−0.972907
$712$	3.22268	0.120775
$713$	42.5658	1.59410
$714$	0	0
$715$	−19.1994	−0.718017
$716$	−12.5743	−0.469925
$717$	−34.6203	−1.29292
$718$	−12.7131	−0.474448
$719$	46.9309	1.75023	0.875113	−	0.483918i	$-0.160787\pi$
$719$	46.9309	1.75023	0.875113	+	0.483918i	$0.160787\pi$
$720$	−1.58568	−0.0590950
$721$	0	0
$722$	−18.6407	−0.693735
$723$	14.6099	0.543347
$724$	22.5585	0.838378
$725$	−2.81843	−0.104674
$726$	−4.88660	−0.181359
$727$	−27.1740	−1.00783	−0.503914	−	0.863754i	$-0.668107\pi$
$727$	−27.1740	−1.00783	−0.503914	+	0.863754i	$0.668107\pi$
$728$	0	0
$729$	22.1977	0.822139
$730$	11.5564	0.427722
$731$	10.7710	0.398379
$732$	2.98219	0.110225
$733$	28.1822	1.04093	0.520467	−	0.853882i	$-0.325758\pi$
$733$	28.1822	1.04093	0.520467	+	0.853882i	$0.325758\pi$
$734$	−23.3607	−0.862258
$735$	0	0
$736$	8.10107	0.298609
$737$	−49.1709	−1.81123
$738$	17.8965	0.658779
$739$	−25.2536	−0.928967	−0.464484	−	0.885582i	$-0.653760\pi$
$739$	−25.2536	−0.928967	−0.464484	+	0.885582i	$0.653760\pi$
$740$	−8.25283	−0.303380
$741$	−3.52108	−0.129350
$742$	0	0
$743$	−7.42054	−0.272233	−0.136117	−	0.990693i	$-0.543462\pi$
$743$	−7.42054	−0.272233	−0.136117	+	0.990693i	$0.543462\pi$
$744$	−6.24873	−0.229089
$745$	−11.4109	−0.418061
$746$	−18.0183	−0.659698
$747$	10.1090	0.369870
$748$	3.88703	0.142124
$749$	0	0
$750$	−1.18925	−0.0434253
$751$	8.68030	0.316749	0.158374	−	0.987379i	$-0.449375\pi$
$751$	8.68030	0.316749	0.158374	+	0.987379i	$0.449375\pi$
$752$	6.09168	0.222141
$753$	−5.32436	−0.194030
$754$	13.9212	0.506980
$755$	8.95637	0.325956
$756$	0	0
$757$	−31.7815	−1.15512	−0.577559	−	0.816349i	$-0.695995\pi$
$757$	−31.7815	−1.15512	−0.577559	+	0.816349i	$0.695995\pi$
$758$	−22.4975	−0.817147
$759$	−37.4484	−1.35929
$760$	−0.599423	−0.0217433
$761$	−16.8455	−0.610650	−0.305325	−	0.952248i	$-0.598765\pi$
$761$	−16.8455	−0.610650	−0.305325	+	0.952248i	$0.598765\pi$
$762$	−5.92574	−0.214667
$763$	0	0
$764$	5.50253	0.199074
$765$	−1.58568	−0.0573306
$766$	1.50666	0.0544378
$767$	27.7466	1.00187
$768$	−1.18925	−0.0429134
$769$	7.79513	0.281100	0.140550	−	0.990074i	$-0.455113\pi$
$769$	7.79513	0.281100	0.140550	+	0.990074i	$0.455113\pi$
$770$	0	0
$771$	−14.5489	−0.523965
$772$	−16.4940	−0.593631
$773$	−35.3846	−1.27269	−0.636347	−	0.771403i	$-0.719556\pi$
$773$	−35.3846	−1.27269	−0.636347	+	0.771403i	$0.719556\pi$
$774$	−17.0794	−0.613905
$775$	5.25434	0.188742
$776$	−0.283036	−0.0101604
$777$	0	0
$778$	0.804996	0.0288605
$779$	6.76526	0.242391
$780$	5.87412	0.210327
$781$	62.9459	2.25238
$782$	8.10107	0.289694
$783$	−15.3704	−0.549292
$784$	0	0
$785$	−16.1983	−0.578142
$786$	−22.3456	−0.797041
$787$	52.7134	1.87903	0.939514	−	0.342510i	$-0.111277\pi$
$787$	52.7134	1.87903	0.939514	+	0.342510i	$0.111277\pi$
$788$	21.2693	0.757687
$789$	−18.3993	−0.655032
$790$	16.3602	0.582071
$791$	0	0
$792$	−6.16360	−0.219014
$793$	12.3861	0.439842
$794$	−21.6483	−0.768270
$795$	4.31107	0.152898
$796$	−16.9189	−0.599674
$797$	30.5020	1.08044	0.540219	−	0.841525i	$-0.318341\pi$
$797$	30.5020	1.08044	0.540219	+	0.841525i	$0.318341\pi$
$798$	0	0
$799$	6.09168	0.215508
$800$	1.00000	0.0353553
$801$	−5.11015	−0.180558
$802$	36.3123	1.28223
$803$	44.9201	1.58520
$804$	15.0440	0.530561
$805$	0	0
$806$	−25.9531	−0.914157
$807$	17.5844	0.619001
$808$	0.287157	0.0101021
$809$	−53.5184	−1.88161	−0.940803	−	0.338954i	$-0.889927\pi$
$809$	−53.5184	−1.88161	−0.940803	+	0.338954i	$0.889927\pi$
$810$	−1.72855	−0.0607351
$811$	32.3827	1.13711	0.568555	−	0.822645i	$-0.307503\pi$
$811$	32.3827	1.13711	0.568555	+	0.822645i	$0.307503\pi$
$812$	0	0
$813$	32.8691	1.15277
$814$	−32.0790	−1.12437
$815$	12.0413	0.421788
$816$	−1.18925	−0.0416321
$817$	−6.45637	−0.225880
$818$	21.0159	0.734803
$819$	0	0
$820$	−11.2863	−0.394134
$821$	16.6610	0.581472	0.290736	−	0.956803i	$-0.406100\pi$
$821$	16.6610	0.581472	0.290736	+	0.956803i	$0.406100\pi$
$822$	−14.7894	−0.515838
$823$	−4.62168	−0.161102	−0.0805508	−	0.996751i	$-0.525668\pi$
$823$	−4.62168	−0.161102	−0.0805508	+	0.996751i	$0.525668\pi$
$824$	−11.0654	−0.385480
$825$	−4.62265	−0.160940
$826$	0	0
$827$	−39.6791	−1.37978	−0.689889	−	0.723915i	$-0.742341\pi$
$827$	−39.6791	−1.37978	−0.689889	+	0.723915i	$0.742341\pi$
$828$	−12.8457	−0.446420
$829$	17.8306	0.619282	0.309641	−	0.950854i	$-0.399791\pi$
$829$	17.8306	0.619282	0.309641	+	0.950854i	$0.399791\pi$
$830$	−6.37518	−0.221286
$831$	−9.84573	−0.341544
$832$	−4.93935	−0.171241
$833$	0	0
$834$	−10.9052	−0.377616
$835$	7.71706	0.267060
$836$	−2.32997	−0.0805838
$837$	28.6547	0.990451
$838$	−16.5926	−0.573183
$839$	20.5528	0.709560	0.354780	−	0.934950i	$-0.384556\pi$
$839$	20.5528	0.709560	0.354780	+	0.934950i	$0.384556\pi$
$840$	0	0
$841$	−21.0565	−0.726085
$842$	−0.242990	−0.00837399
$843$	22.4656	0.773756
$844$	−6.90593	−0.237712
$845$	11.3972	0.392076
$846$	−9.65948	−0.332100
$847$	0	0
$848$	−3.62503	−0.124484
$849$	13.5604	0.465393
$850$	1.00000	0.0342997
$851$	−66.8568	−2.29182
$852$	−19.2585	−0.659786
$853$	−43.0042	−1.47243	−0.736217	−	0.676745i	$-0.763390\pi$
$853$	−43.0042	−1.47243	−0.736217	+	0.676745i	$0.763390\pi$
$854$	0	0
$855$	0.950495	0.0325063
$856$	−0.618021	−0.0211235
$857$	−28.3183	−0.967334	−0.483667	−	0.875252i	$-0.660695\pi$
$857$	−28.3183	−0.967334	−0.483667	+	0.875252i	$0.660695\pi$
$858$	22.8329	0.779502
$859$	−37.3408	−1.27405	−0.637026	−	0.770842i	$-0.719836\pi$
$859$	−37.3408	−1.27405	−0.637026	+	0.770842i	$0.719836\pi$
$860$	10.7710	0.367287
$861$	0	0
$862$	−24.8767	−0.847305
$863$	−52.9397	−1.80209	−0.901045	−	0.433727i	$-0.857198\pi$
$863$	−52.9397	−1.80209	−0.901045	+	0.433727i	$0.857198\pi$
$864$	5.45353	0.185533
$865$	5.78375	0.196654
$866$	30.8365	1.04787
$867$	−1.18925	−0.0403890
$868$	0	0
$869$	63.5926	2.15723
$870$	3.35181	0.113637
$871$	62.4828	2.11715
$872$	−6.43243	−0.217830
$873$	0.448807	0.0151898
$874$	−4.85596	−0.164255
$875$	0	0
$876$	−13.7435	−0.464349
$877$	−43.0042	−1.45215	−0.726074	−	0.687617i	$-0.758657\pi$
$877$	−43.0042	−1.45215	−0.726074	+	0.687617i	$0.758657\pi$
$878$	−8.50071	−0.286885
$879$	−30.9580	−1.04419
$880$	3.88703	0.131032
$881$	−6.02631	−0.203032	−0.101516	−	0.994834i	$-0.532369\pi$
$881$	−6.02631	−0.203032	−0.101516	+	0.994834i	$0.532369\pi$
$882$	0	0
$883$	−13.6728	−0.460125	−0.230062	−	0.973176i	$-0.573893\pi$
$883$	−13.6728	−0.460125	−0.230062	+	0.973176i	$0.573893\pi$
$884$	−4.93935	−0.166128
$885$	6.68055	0.224564
$886$	24.2993	0.816352
$887$	0.521217	0.0175008	0.00875038	−	0.999962i	$-0.497215\pi$
$887$	0.521217	0.0175008	0.00875038	+	0.999962i	$0.497215\pi$
$888$	9.81468	0.329359
$889$	0	0
$890$	3.22268	0.108024
$891$	−6.71892	−0.225092
$892$	18.1630	0.608143
$893$	−3.65149	−0.122192
$894$	13.5704	0.453861
$895$	−12.5743	−0.420314
$896$	0	0
$897$	47.5867	1.58887
$898$	−17.7605	−0.592675
$899$	−14.8090	−0.493907
$900$	−1.58568	−0.0528562
$901$	−3.62503	−0.120767
$902$	−43.8701	−1.46072
$903$	0	0
$904$	20.1982	0.671781
$905$	22.5585	0.749868
$906$	−10.6514	−0.353868
$907$	53.2011	1.76651	0.883257	−	0.468889i	$-0.155346\pi$
$907$	53.2011	1.76651	0.883257	+	0.468889i	$0.155346\pi$
$908$	8.19929	0.272103
$909$	−0.455340	−0.0151027
$910$	0	0
$911$	8.68545	0.287762	0.143881	−	0.989595i	$-0.454042\pi$
$911$	8.68545	0.287762	0.143881	+	0.989595i	$0.454042\pi$
$912$	0.712863	0.0236053
$913$	−24.7805	−0.820114
$914$	−20.2826	−0.670890
$915$	2.98219	0.0985883
$916$	4.07526	0.134651
$917$	0	0
$918$	5.45353	0.179993
$919$	1.15896	0.0382307	0.0191154	−	0.999817i	$-0.493915\pi$
$919$	1.15896	0.0382307	0.0191154	+	0.999817i	$0.493915\pi$
$920$	8.10107	0.267084
$921$	25.2448	0.831844
$922$	18.0461	0.594317
$923$	−79.9871	−2.63281
$924$	0	0
$925$	−8.25283	−0.271352
$926$	32.8829	1.08060
$927$	17.5462	0.576292
$928$	−2.81843	−0.0925194
$929$	−53.4986	−1.75523	−0.877615	−	0.479365i	$-0.840867\pi$
$929$	−53.4986	−1.75523	−0.877615	+	0.479365i	$0.840867\pi$
$930$	−6.24873	−0.204904
$931$	0	0
$932$	24.3680	0.798199
$933$	−17.6732	−0.578596
$934$	22.4945	0.736044
$935$	3.88703	0.127119
$936$	7.83226	0.256005
$937$	−34.8486	−1.13845	−0.569227	−	0.822180i	$-0.692758\pi$
$937$	−34.8486	−1.13845	−0.569227	+	0.822180i	$0.692758\pi$
$938$	0	0
$939$	−21.5589	−0.703547
$940$	6.09168	0.198689
$941$	−3.88775	−0.126737	−0.0633685	−	0.997990i	$-0.520184\pi$
$941$	−3.88775	−0.126737	−0.0633685	+	0.997990i	$0.520184\pi$
$942$	19.2638	0.627649
$943$	−91.4310	−2.97740
$944$	−5.61745	−0.182832
$945$	0	0
$946$	41.8671	1.36122
$947$	1.40947	0.0458017	0.0229008	−	0.999738i	$-0.492710\pi$
$947$	1.40947	0.0458017	0.0229008	+	0.999738i	$0.492710\pi$
$948$	−19.4564	−0.631914
$949$	−57.0813	−1.85294
$950$	−0.599423	−0.0194478
$951$	30.9863	1.00480
$952$	0	0
$953$	11.5558	0.374330	0.187165	−	0.982329i	$-0.440070\pi$
$953$	11.5558	0.374330	0.187165	+	0.982329i	$0.440070\pi$
$954$	5.74816	0.186104
$955$	5.50253	0.178058
$956$	29.1110	0.941518
$957$	13.0286	0.421154
$958$	42.0656	1.35908
$959$	0	0
$960$	−1.18925	−0.0383829
$961$	−3.39187	−0.109415
$962$	40.7637	1.31427
$963$	0.979986	0.0315796
$964$	−12.2849	−0.395671
$965$	−16.4940	−0.530960
$966$	0	0
$967$	26.7891	0.861481	0.430740	−	0.902476i	$-0.358252\pi$
$967$	26.7891	0.861481	0.430740	+	0.902476i	$0.358252\pi$
$968$	4.10898	0.132068
$969$	0.712863	0.0229005
$970$	−0.283036	−0.00908775
$971$	−1.27869	−0.0410352	−0.0205176	−	0.999789i	$-0.506531\pi$
$971$	−1.27869	−0.0410352	−0.0205176	+	0.999789i	$0.506531\pi$
$972$	−14.3049	−0.458830
$973$	0	0
$974$	9.82480	0.314807
$975$	5.87412	0.188123
$976$	−2.50763	−0.0802672
$977$	−43.0651	−1.37777	−0.688887	−	0.724869i	$-0.741900\pi$
$977$	−43.0651	−1.37777	−0.688887	+	0.724869i	$0.741900\pi$
$978$	−14.3201	−0.457906
$979$	12.5266	0.400353
$980$	0	0
$981$	10.1998	0.325655
$982$	21.3078	0.679960
$983$	−1.40219	−0.0447229	−0.0223614	−	0.999750i	$-0.507118\pi$
$983$	−1.40219	−0.0447229	−0.0223614	+	0.999750i	$0.507118\pi$
$984$	13.4222	0.427885
$985$	21.2693	0.677696
$986$	−2.81843	−0.0897570
$987$	0	0
$988$	2.96076	0.0941943
$989$	87.2564	2.77459
$990$	−6.16360	−0.195892
$991$	21.2478	0.674960	0.337480	−	0.941333i	$-0.390426\pi$
$991$	21.2478	0.674960	0.337480	+	0.941333i	$0.390426\pi$
$992$	5.25434	0.166826
$993$	−12.2194	−0.387772
$994$	0	0
$995$	−16.9189	−0.536364
$996$	7.58168	0.240235
$997$	26.5267	0.840109	0.420054	−	0.907499i	$-0.362011\pi$
$997$	26.5267	0.840109	0.420054	+	0.907499i	$0.362011\pi$
$998$	−15.1163	−0.478499
$999$	−45.0070	−1.42396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8330.2.a.cw.1.3	yes	10
7.6	odd	2		8330.2.a.cv.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8330.2.a.cv.1.8	✓	10	7.6	odd	2
8330.2.a.cw.1.3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.18925 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.18925 q^{6} +1.00000 q^{8} -1.58568 q^{9} +1.00000 q^{10} +3.88703 q^{11} -1.18925 q^{12} -4.93935 q^{13} -1.18925 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.58568 q^{18} -0.599423 q^{19} +1.00000 q^{20} +3.88703 q^{22} +8.10107 q^{23} -1.18925 q^{24} +1.00000 q^{25} -4.93935 q^{26} +5.45353 q^{27} -2.81843 q^{29} -1.18925 q^{30} +5.25434 q^{31} +1.00000 q^{32} -4.62265 q^{33} +1.00000 q^{34} -1.58568 q^{36} -8.25283 q^{37} -0.599423 q^{38} +5.87412 q^{39} +1.00000 q^{40} -11.2863 q^{41} +10.7710 q^{43} +3.88703 q^{44} -1.58568 q^{45} +8.10107 q^{46} +6.09168 q^{47} -1.18925 q^{48} +1.00000 q^{50} -1.18925 q^{51} -4.93935 q^{52} -3.62503 q^{53} +5.45353 q^{54} +3.88703 q^{55} +0.712863 q^{57} -2.81843 q^{58} -5.61745 q^{59} -1.18925 q^{60} -2.50763 q^{61} +5.25434 q^{62} +1.00000 q^{64} -4.93935 q^{65} -4.62265 q^{66} -12.6500 q^{67} +1.00000 q^{68} -9.63420 q^{69} +16.1938 q^{71} -1.58568 q^{72} +11.5564 q^{73} -8.25283 q^{74} -1.18925 q^{75} -0.599423 q^{76} +5.87412 q^{78} +16.3602 q^{79} +1.00000 q^{80} -1.72855 q^{81} -11.2863 q^{82} -6.37518 q^{83} +1.00000 q^{85} +10.7710 q^{86} +3.35181 q^{87} +3.88703 q^{88} +3.22268 q^{89} -1.58568 q^{90} +8.10107 q^{92} -6.24873 q^{93} +6.09168 q^{94} -0.599423 q^{95} -1.18925 q^{96} -0.283036 q^{97} -6.16360 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} + 2 q^{3} + 10 q^{4} + 10 q^{5} + 2 q^{6} + 10 q^{8} + 12 q^{9} + 10 q^{10} - 2 q^{11} + 2 q^{12} + 4 q^{13} + 2 q^{15} + 10 q^{16} + 10 q^{17} + 12 q^{18} + 14 q^{19} + 10 q^{20} - 2 q^{22}+ \cdots - 12 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more