LMFDB - Embedded newform 8032.2.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45889 q^{3} +2.64078 q^{5} +1.82573 q^{7} +3.04612 q^{9} +O(q^{10})$	$q-2.45889 q^{3} +2.64078 q^{5} +1.82573 q^{7} +3.04612 q^{9} -2.75348 q^{11} +5.68557 q^{13} -6.49337 q^{15} +7.79809 q^{17} -0.235780 q^{19} -4.48926 q^{21} -9.00669 q^{23} +1.97370 q^{25} -0.113408 q^{27} -5.88052 q^{29} -1.02416 q^{31} +6.77048 q^{33} +4.82134 q^{35} -7.63564 q^{37} -13.9802 q^{39} -11.0721 q^{41} +8.87710 q^{43} +8.04413 q^{45} -7.55976 q^{47} -3.66671 q^{49} -19.1746 q^{51} -4.59960 q^{53} -7.27131 q^{55} +0.579756 q^{57} -5.43042 q^{59} +8.81502 q^{61} +5.56139 q^{63} +15.0143 q^{65} -11.2441 q^{67} +22.1464 q^{69} -13.7717 q^{71} -2.51619 q^{73} -4.85312 q^{75} -5.02710 q^{77} +1.99899 q^{79} -8.85951 q^{81} +2.15244 q^{83} +20.5930 q^{85} +14.4595 q^{87} -15.3577 q^{89} +10.3803 q^{91} +2.51830 q^{93} -0.622643 q^{95} -1.87046 q^{97} -8.38742 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9	$ 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9} - 13 q^{11} - 7 q^{13} - 28 q^{15} - 9 q^{17} + 17 q^{19} - 6 q^{21} - 43 q^{23} + 34 q^{25} - 12 q^{27} - q^{29} - 39 q^{31} - 17 q^{35} - q^{37} - 48 q^{39} - 3 q^{41} + 19 q^{43} - 66 q^{47} + 25 q^{49} + 14 q^{51} + 3 q^{53} - 50 q^{55} - 14 q^{57} - 27 q^{59} + 15 q^{61} - 75 q^{63} - 6 q^{65} - 8 q^{67} + 18 q^{69} - 64 q^{71} - 15 q^{73} - 9 q^{75} - 71 q^{79} + 6 q^{81} - 60 q^{83} + 15 q^{85} - 64 q^{87} - 32 q^{89} + 26 q^{91} - 4 q^{93} - 72 q^{95} - 4 q^{97} - 13 q^{99}+O(q^{100}) $ 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9 - 13 * q^11 - 7 * q^13 - 28 * q^15 - 9 * q^17 + 17 * q^19 - 6 * q^21 - 43 * q^23 + 34 * q^25 - 12 * q^27 - q^29 - 39 * q^31 - 17 * q^35 - q^37 - 48 * q^39 - 3 * q^41 + 19 * q^43 - 66 * q^47 + 25 * q^49 + 14 * q^51 + 3 * q^53 - 50 * q^55 - 14 * q^57 - 27 * q^59 + 15 * q^61 - 75 * q^63 - 6 * q^65 - 8 * q^67 + 18 * q^69 - 64 * q^71 - 15 * q^73 - 9 * q^75 - 71 * q^79 + 6 * q^81 - 60 * q^83 + 15 * q^85 - 64 * q^87 - 32 * q^89 + 26 * q^91 - 4 * q^93 - 72 * q^95 - 4 * q^97 - 13 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−2.45889	−1.41964	−0.709819	−	0.704384i	$-0.751223\pi$
$3$	−2.45889	−1.41964	−0.709819	+	0.704384i	$0.751223\pi$
$4$	0	0
$5$	2.64078	1.18099	0.590496	−	0.807041i	$-0.298932\pi$
$5$	2.64078	1.18099	0.590496	+	0.807041i	$0.298932\pi$
$6$	0	0
$7$	1.82573	0.690061	0.345030	−	0.938592i	$-0.387869\pi$
$7$	1.82573	0.690061	0.345030	+	0.938592i	$0.387869\pi$
$8$	0	0
$9$	3.04612	1.01537
$10$	0	0
$11$	−2.75348	−0.830204	−0.415102	−	0.909775i	$-0.636254\pi$
$11$	−2.75348	−0.830204	−0.415102	+	0.909775i	$0.636254\pi$
$12$	0	0
$13$	5.68557	1.57689	0.788447	−	0.615103i	$-0.210885\pi$
$13$	5.68557	1.57689	0.788447	+	0.615103i	$0.210885\pi$
$14$	0	0
$15$	−6.49337	−1.67658
$16$	0	0
$17$	7.79809	1.89131	0.945657	−	0.325166i	$-0.105420\pi$
$17$	7.79809	1.89131	0.945657	+	0.325166i	$0.105420\pi$
$18$	0	0
$19$	−0.235780	−0.0540917	−0.0270458	−	0.999634i	$-0.508610\pi$
$19$	−0.235780	−0.0540917	−0.0270458	+	0.999634i	$0.508610\pi$
$20$	0	0
$21$	−4.48926	−0.979637
$22$	0	0
$23$	−9.00669	−1.87802	−0.939012	−	0.343884i	$-0.888257\pi$
$23$	−9.00669	−1.87802	−0.939012	+	0.343884i	$0.888257\pi$
$24$	0	0
$25$	1.97370	0.394741
$26$	0	0
$27$	−0.113408	−0.0218253
$28$	0	0
$29$	−5.88052	−1.09198	−0.545992	−	0.837790i	$-0.683847\pi$
$29$	−5.88052	−1.09198	−0.545992	+	0.837790i	$0.683847\pi$
$30$	0	0
$31$	−1.02416	−0.183945	−0.0919724	−	0.995762i	$-0.529317\pi$
$31$	−1.02416	−0.183945	−0.0919724	+	0.995762i	$0.529317\pi$
$32$	0	0
$33$	6.77048	1.17859
$34$	0	0
$35$	4.82134	0.814956
$36$	0	0
$37$	−7.63564	−1.25529	−0.627646	−	0.778499i	$-0.715981\pi$
$37$	−7.63564	−1.25529	−0.627646	+	0.778499i	$0.715981\pi$
$38$	0	0
$39$	−13.9802	−2.23862
$40$	0	0
$41$	−11.0721	−1.72917	−0.864583	−	0.502490i	$-0.832417\pi$
$41$	−11.0721	−1.72917	−0.864583	+	0.502490i	$0.832417\pi$
$42$	0	0
$43$	8.87710	1.35374	0.676872	−	0.736100i	$-0.263335\pi$
$43$	8.87710	1.35374	0.676872	+	0.736100i	$0.263335\pi$
$44$	0	0
$45$	8.04413	1.19915
$46$	0	0
$47$	−7.55976	−1.10270	−0.551352	−	0.834272i	$-0.685888\pi$
$47$	−7.55976	−1.10270	−0.551352	+	0.834272i	$0.685888\pi$
$48$	0	0
$49$	−3.66671	−0.523816
$50$	0	0
$51$	−19.1746	−2.68498
$52$	0	0
$53$	−4.59960	−0.631804	−0.315902	−	0.948792i	$-0.602307\pi$
$53$	−4.59960	−0.631804	−0.315902	+	0.948792i	$0.602307\pi$
$54$	0	0
$55$	−7.27131	−0.980464
$56$	0	0
$57$	0.579756	0.0767906
$58$	0	0
$59$	−5.43042	−0.706980	−0.353490	−	0.935438i	$-0.615005\pi$
$59$	−5.43042	−0.706980	−0.353490	+	0.935438i	$0.615005\pi$
$60$	0	0
$61$	8.81502	1.12865	0.564324	−	0.825554i	$-0.309137\pi$
$61$	8.81502	1.12865	0.564324	+	0.825554i	$0.309137\pi$
$62$	0	0
$63$	5.56139	0.700670
$64$	0	0
$65$	15.0143	1.86230
$66$	0	0
$67$	−11.2441	−1.37369	−0.686844	−	0.726805i	$-0.741004\pi$
$67$	−11.2441	−1.37369	−0.686844	+	0.726805i	$0.741004\pi$
$68$	0	0
$69$	22.1464	2.66612
$70$	0	0
$71$	−13.7717	−1.63440	−0.817200	−	0.576355i	$-0.804475\pi$
$71$	−13.7717	−1.63440	−0.817200	+	0.576355i	$0.804475\pi$
$72$	0	0
$73$	−2.51619	−0.294498	−0.147249	−	0.989099i	$-0.547042\pi$
$73$	−2.51619	−0.294498	−0.147249	+	0.989099i	$0.547042\pi$
$74$	0	0
$75$	−4.85312	−0.560390
$76$	0	0
$77$	−5.02710	−0.572891
$78$	0	0
$79$	1.99899	0.224903	0.112452	−	0.993657i	$-0.464130\pi$
$79$	1.99899	0.224903	0.112452	+	0.993657i	$0.464130\pi$
$80$	0	0
$81$	−8.85951	−0.984390
$82$	0	0
$83$	2.15244	0.236261	0.118131	−	0.992998i	$-0.462310\pi$
$83$	2.15244	0.236261	0.118131	+	0.992998i	$0.462310\pi$
$84$	0	0
$85$	20.5930	2.23363
$86$	0	0
$87$	14.4595	1.55022
$88$	0	0
$89$	−15.3577	−1.62791	−0.813955	−	0.580928i	$-0.802690\pi$
$89$	−15.3577	−1.62791	−0.813955	+	0.580928i	$0.802690\pi$
$90$	0	0
$91$	10.3803	1.08815
$92$	0	0
$93$	2.51830	0.261135
$94$	0	0
$95$	−0.622643	−0.0638818
$96$	0	0
$97$	−1.87046	−0.189916	−0.0949581	−	0.995481i	$-0.530272\pi$
$97$	−1.87046	−0.189916	−0.0949581	+	0.995481i	$0.530272\pi$
$98$	0	0
$99$	−8.38742	−0.842967
$100$	0	0
$101$	−14.5406	−1.44685	−0.723424	−	0.690404i	$-0.757433\pi$
$101$	−14.5406	−1.44685	−0.723424	+	0.690404i	$0.757433\pi$
$102$	0	0
$103$	16.1668	1.59296	0.796482	−	0.604662i	$-0.206692\pi$
$103$	16.1668	1.59296	0.796482	+	0.604662i	$0.206692\pi$
$104$	0	0
$105$	−11.8551	−1.15694
$106$	0	0
$107$	10.8370	1.04765	0.523826	−	0.851826i	$-0.324504\pi$
$107$	10.8370	1.04765	0.523826	+	0.851826i	$0.324504\pi$
$108$	0	0
$109$	7.34131	0.703170	0.351585	−	0.936156i	$-0.385643\pi$
$109$	7.34131	0.703170	0.351585	+	0.936156i	$0.385643\pi$
$110$	0	0
$111$	18.7752	1.78206
$112$	0	0
$113$	−8.35356	−0.785837	−0.392919	−	0.919573i	$-0.628535\pi$
$113$	−8.35356	−0.785837	−0.392919	+	0.919573i	$0.628535\pi$
$114$	0	0
$115$	−23.7847	−2.21793
$116$	0	0
$117$	17.3189	1.60114
$118$	0	0
$119$	14.2372	1.30512
$120$	0	0
$121$	−3.41837	−0.310761
$122$	0	0
$123$	27.2249	2.45479
$124$	0	0
$125$	−7.99177	−0.714806
$126$	0	0
$127$	−7.11307	−0.631183	−0.315592	−	0.948895i	$-0.602203\pi$
$127$	−7.11307	−0.631183	−0.315592	+	0.948895i	$0.602203\pi$
$128$	0	0
$129$	−21.8278	−1.92183
$130$	0	0
$131$	1.69741	0.148303	0.0741517	−	0.997247i	$-0.476375\pi$
$131$	1.69741	0.148303	0.0741517	+	0.997247i	$0.476375\pi$
$132$	0	0
$133$	−0.430471	−0.0373265
$134$	0	0
$135$	−0.299485	−0.0257755
$136$	0	0
$137$	0.930846	0.0795275	0.0397638	−	0.999209i	$-0.487339\pi$
$137$	0.930846	0.0795275	0.0397638	+	0.999209i	$0.487339\pi$
$138$	0	0
$139$	−22.5892	−1.91599	−0.957994	−	0.286789i	$-0.907412\pi$
$139$	−22.5892	−1.91599	−0.957994	+	0.286789i	$0.907412\pi$
$140$	0	0
$141$	18.5886	1.56544
$142$	0	0
$143$	−15.6551	−1.30914
$144$	0	0
$145$	−15.5291	−1.28962
$146$	0	0
$147$	9.01603	0.743630
$148$	0	0
$149$	−10.8789	−0.891235	−0.445618	−	0.895223i	$-0.647016\pi$
$149$	−10.8789	−0.891235	−0.445618	+	0.895223i	$0.647016\pi$
$150$	0	0
$151$	−2.07382	−0.168765	−0.0843824	−	0.996433i	$-0.526892\pi$
$151$	−2.07382	−0.168765	−0.0843824	+	0.996433i	$0.526892\pi$
$152$	0	0
$153$	23.7539	1.92039
$154$	0	0
$155$	−2.70458	−0.217237
$156$	0	0
$157$	4.24447	0.338745	0.169373	−	0.985552i	$-0.445826\pi$
$157$	4.24447	0.338745	0.169373	+	0.985552i	$0.445826\pi$
$158$	0	0
$159$	11.3099	0.896933
$160$	0	0
$161$	−16.4438	−1.29595
$162$	0	0
$163$	−1.75835	−0.137724	−0.0688622	−	0.997626i	$-0.521937\pi$
$163$	−1.75835	−0.137724	−0.0688622	+	0.997626i	$0.521937\pi$
$164$	0	0
$165$	17.8793	1.39190
$166$	0	0
$167$	5.78769	0.447865	0.223932	−	0.974605i	$-0.428111\pi$
$167$	5.78769	0.447865	0.223932	+	0.974605i	$0.428111\pi$
$168$	0	0
$169$	19.3257	1.48660
$170$	0	0
$171$	−0.718215	−0.0549233
$172$	0	0
$173$	5.31288	0.403930	0.201965	−	0.979393i	$-0.435267\pi$
$173$	5.31288	0.403930	0.201965	+	0.979393i	$0.435267\pi$
$174$	0	0
$175$	3.60345	0.272395
$176$	0	0
$177$	13.3528	1.00366
$178$	0	0
$179$	8.33307	0.622843	0.311422	−	0.950272i	$-0.399195\pi$
$179$	8.33307	0.622843	0.311422	+	0.950272i	$0.399195\pi$
$180$	0	0
$181$	6.79260	0.504890	0.252445	−	0.967611i	$-0.418765\pi$
$181$	6.79260	0.504890	0.252445	+	0.967611i	$0.418765\pi$
$182$	0	0
$183$	−21.6751	−1.60227
$184$	0	0
$185$	−20.1640	−1.48249
$186$	0	0
$187$	−21.4718	−1.57018
$188$	0	0
$189$	−0.207052	−0.0150608
$190$	0	0
$191$	5.79738	0.419484	0.209742	−	0.977757i	$-0.432738\pi$
$191$	5.79738	0.419484	0.209742	+	0.977757i	$0.432738\pi$
$192$	0	0
$193$	9.58997	0.690301	0.345150	−	0.938547i	$-0.387828\pi$
$193$	9.58997	0.690301	0.345150	+	0.938547i	$0.387828\pi$
$194$	0	0
$195$	−36.9185	−2.64379
$196$	0	0
$197$	10.3442	0.736996	0.368498	−	0.929628i	$-0.379872\pi$
$197$	10.3442	0.736996	0.368498	+	0.929628i	$0.379872\pi$
$198$	0	0
$199$	−14.0701	−0.997400	−0.498700	−	0.866775i	$-0.666189\pi$
$199$	−14.0701	−0.997400	−0.498700	+	0.866775i	$0.666189\pi$
$200$	0	0
$201$	27.6480	1.95014
$202$	0	0
$203$	−10.7362	−0.753535
$204$	0	0
$205$	−29.2389	−2.04213
$206$	0	0
$207$	−27.4355	−1.90690
$208$	0	0
$209$	0.649215	0.0449071
$210$	0	0
$211$	25.4126	1.74948	0.874739	−	0.484594i	$-0.161032\pi$
$211$	25.4126	1.74948	0.874739	+	0.484594i	$0.161032\pi$
$212$	0	0
$213$	33.8630	2.32026
$214$	0	0
$215$	23.4424	1.59876
$216$	0	0
$217$	−1.86984	−0.126933
$218$	0	0
$219$	6.18703	0.418081
$220$	0	0
$221$	44.3366	2.98240
$222$	0	0
$223$	14.8049	0.991409	0.495704	−	0.868491i	$-0.334910\pi$
$223$	14.8049	0.991409	0.495704	+	0.868491i	$0.334910\pi$
$224$	0	0
$225$	6.01214	0.400810
$226$	0	0
$227$	10.4396	0.692899	0.346450	−	0.938069i	$-0.387387\pi$
$227$	10.4396	0.692899	0.346450	+	0.938069i	$0.387387\pi$
$228$	0	0
$229$	−20.5544	−1.35828	−0.679138	−	0.734011i	$-0.737646\pi$
$229$	−20.5544	−1.35828	−0.679138	+	0.734011i	$0.737646\pi$
$230$	0	0
$231$	12.3611	0.813298
$232$	0	0
$233$	10.9167	0.715178	0.357589	−	0.933879i	$-0.383599\pi$
$233$	10.9167	0.715178	0.357589	+	0.933879i	$0.383599\pi$
$234$	0	0
$235$	−19.9637	−1.30229
$236$	0	0
$237$	−4.91528	−0.319282
$238$	0	0
$239$	−9.45602	−0.611659	−0.305830	−	0.952086i	$-0.598934\pi$
$239$	−9.45602	−0.611659	−0.305830	+	0.952086i	$0.598934\pi$
$240$	0	0
$241$	−19.1436	−1.23315	−0.616573	−	0.787297i	$-0.711480\pi$
$241$	−19.1436	−1.23315	−0.616573	+	0.787297i	$0.711480\pi$
$242$	0	0
$243$	22.1247	1.41930
$244$	0	0
$245$	−9.68298	−0.618623
$246$	0	0
$247$	−1.34054	−0.0852968
$248$	0	0
$249$	−5.29261	−0.335406
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	24.7997	1.55914
$254$	0	0
$255$	−50.6359	−3.17094
$256$	0	0
$257$	3.77535	0.235500	0.117750	−	0.993043i	$-0.462432\pi$
$257$	3.77535	0.235500	0.117750	+	0.993043i	$0.462432\pi$
$258$	0	0
$259$	−13.9406	−0.866227
$260$	0	0
$261$	−17.9128	−1.10877
$262$	0	0
$263$	3.24155	0.199883	0.0999413	−	0.994993i	$-0.468135\pi$
$263$	3.24155	0.199883	0.0999413	+	0.994993i	$0.468135\pi$
$264$	0	0
$265$	−12.1465	−0.746155
$266$	0	0
$267$	37.7628	2.31104
$268$	0	0
$269$	4.43890	0.270644	0.135322	−	0.990802i	$-0.456793\pi$
$269$	4.43890	0.270644	0.135322	+	0.990802i	$0.456793\pi$
$270$	0	0
$271$	2.94915	0.179148	0.0895740	−	0.995980i	$-0.471449\pi$
$271$	2.94915	0.179148	0.0895740	+	0.995980i	$0.471449\pi$
$272$	0	0
$273$	−25.5240	−1.54478
$274$	0	0
$275$	−5.43455	−0.327716
$276$	0	0
$277$	14.5734	0.875630	0.437815	−	0.899065i	$-0.355752\pi$
$277$	14.5734	0.875630	0.437815	+	0.899065i	$0.355752\pi$
$278$	0	0
$279$	−3.11972	−0.186773
$280$	0	0
$281$	20.0768	1.19768	0.598840	−	0.800869i	$-0.295628\pi$
$281$	20.0768	1.19768	0.598840	+	0.800869i	$0.295628\pi$
$282$	0	0
$283$	18.2000	1.08188	0.540938	−	0.841062i	$-0.318069\pi$
$283$	18.2000	1.08188	0.540938	+	0.841062i	$0.318069\pi$
$284$	0	0
$285$	1.53101	0.0906891
$286$	0	0
$287$	−20.2146	−1.19323
$288$	0	0
$289$	43.8102	2.57707
$290$	0	0
$291$	4.59924	0.269612
$292$	0	0
$293$	−19.9045	−1.16283	−0.581416	−	0.813606i	$-0.697501\pi$
$293$	−19.9045	−1.16283	−0.581416	+	0.813606i	$0.697501\pi$
$294$	0	0
$295$	−14.3405	−0.834938
$296$	0	0
$297$	0.312265	0.0181195
$298$	0	0
$299$	−51.2082	−2.96145
$300$	0	0
$301$	16.2072	0.934166
$302$	0	0
$303$	35.7538	2.05400
$304$	0	0
$305$	23.2785	1.33292
$306$	0	0
$307$	23.6501	1.34978	0.674891	−	0.737918i	$-0.264191\pi$
$307$	23.6501	1.34978	0.674891	+	0.737918i	$0.264191\pi$
$308$	0	0
$309$	−39.7524	−2.26143
$310$	0	0
$311$	−22.0073	−1.24792	−0.623959	−	0.781457i	$-0.714477\pi$
$311$	−22.0073	−1.24792	−0.623959	+	0.781457i	$0.714477\pi$
$312$	0	0
$313$	1.14641	0.0647987	0.0323993	−	0.999475i	$-0.489685\pi$
$313$	1.14641	0.0647987	0.0323993	+	0.999475i	$0.489685\pi$
$314$	0	0
$315$	14.6864	0.827485
$316$	0	0
$317$	30.7214	1.72549	0.862744	−	0.505641i	$-0.168744\pi$
$317$	30.7214	1.72549	0.862744	+	0.505641i	$0.168744\pi$
$318$	0	0
$319$	16.1919	0.906570
$320$	0	0
$321$	−26.6469	−1.48729
$322$	0	0
$323$	−1.83863	−0.102304
$324$	0	0
$325$	11.2216	0.622465
$326$	0	0
$327$	−18.0514	−0.998247
$328$	0	0
$329$	−13.8021	−0.760933
$330$	0	0
$331$	−17.2071	−0.945787	−0.472894	−	0.881120i	$-0.656790\pi$
$331$	−17.2071	−0.945787	−0.472894	+	0.881120i	$0.656790\pi$
$332$	0	0
$333$	−23.2591	−1.27459
$334$	0	0
$335$	−29.6932	−1.62231
$336$	0	0
$337$	−6.69436	−0.364665	−0.182333	−	0.983237i	$-0.558365\pi$
$337$	−6.69436	−0.364665	−0.182333	+	0.983237i	$0.558365\pi$
$338$	0	0
$339$	20.5405	1.11560
$340$	0	0
$341$	2.82000	0.152712
$342$	0	0
$343$	−19.4745	−1.05153
$344$	0	0
$345$	58.4838	3.14866
$346$	0	0
$347$	−13.0973	−0.703102	−0.351551	−	0.936169i	$-0.614346\pi$
$347$	−13.0973	−0.703102	−0.351551	+	0.936169i	$0.614346\pi$
$348$	0	0
$349$	−3.47888	−0.186220	−0.0931100	−	0.995656i	$-0.529681\pi$
$349$	−3.47888	−0.186220	−0.0931100	+	0.995656i	$0.529681\pi$
$350$	0	0
$351$	−0.644788	−0.0344162
$352$	0	0
$353$	−21.9909	−1.17046	−0.585228	−	0.810869i	$-0.698995\pi$
$353$	−21.9909	−1.17046	−0.585228	+	0.810869i	$0.698995\pi$
$354$	0	0
$355$	−36.3680	−1.93021
$356$	0	0
$357$	−35.0076	−1.85280
$358$	0	0
$359$	−11.3087	−0.596850	−0.298425	−	0.954433i	$-0.596461\pi$
$359$	−11.3087	−0.596850	−0.298425	+	0.954433i	$0.596461\pi$
$360$	0	0
$361$	−18.9444	−0.997074
$362$	0	0
$363$	8.40540	0.441169
$364$	0	0
$365$	−6.64470	−0.347799
$366$	0	0
$367$	−22.9658	−1.19881	−0.599403	−	0.800447i	$-0.704595\pi$
$367$	−22.9658	−1.19881	−0.599403	+	0.800447i	$0.704595\pi$
$368$	0	0
$369$	−33.7268	−1.75575
$370$	0	0
$371$	−8.39762	−0.435983
$372$	0	0
$373$	18.9884	0.983183	0.491592	−	0.870826i	$-0.336415\pi$
$373$	18.9884	0.983183	0.491592	+	0.870826i	$0.336415\pi$
$374$	0	0
$375$	19.6509	1.01477
$376$	0	0
$377$	−33.4341	−1.72194
$378$	0	0
$379$	19.6102	1.00731	0.503654	−	0.863906i	$-0.331989\pi$
$379$	19.6102	1.00731	0.503654	+	0.863906i	$0.331989\pi$
$380$	0	0
$381$	17.4902	0.896052
$382$	0	0
$383$	−16.5373	−0.845015	−0.422507	−	0.906360i	$-0.638850\pi$
$383$	−16.5373	−0.845015	−0.422507	+	0.906360i	$0.638850\pi$
$384$	0	0
$385$	−13.2754	−0.676580
$386$	0	0
$387$	27.0407	1.37456
$388$	0	0
$389$	−20.9145	−1.06041	−0.530204	−	0.847870i	$-0.677885\pi$
$389$	−20.9145	−1.06041	−0.530204	+	0.847870i	$0.677885\pi$
$390$	0	0
$391$	−70.2349	−3.55193
$392$	0	0
$393$	−4.17374	−0.210537
$394$	0	0
$395$	5.27888	0.265609
$396$	0	0
$397$	32.2638	1.61927	0.809636	−	0.586932i	$-0.199664\pi$
$397$	32.2638	1.61927	0.809636	+	0.586932i	$0.199664\pi$
$398$	0	0
$399$	1.05848	0.0529902
$400$	0	0
$401$	−21.9564	−1.09645	−0.548226	−	0.836330i	$-0.684697\pi$
$401$	−21.9564	−1.09645	−0.548226	+	0.836330i	$0.684697\pi$
$402$	0	0
$403$	−5.82295	−0.290062
$404$	0	0
$405$	−23.3960	−1.16256
$406$	0	0
$407$	21.0245	1.04215
$408$	0	0
$409$	−7.54183	−0.372919	−0.186460	−	0.982463i	$-0.559701\pi$
$409$	−7.54183	−0.372919	−0.186460	+	0.982463i	$0.559701\pi$
$410$	0	0
$411$	−2.28884	−0.112900
$412$	0	0
$413$	−9.91447	−0.487859
$414$	0	0
$415$	5.68412	0.279023
$416$	0	0
$417$	55.5442	2.72001
$418$	0	0
$419$	−16.1589	−0.789412	−0.394706	−	0.918807i	$-0.629154\pi$
$419$	−16.1589	−0.789412	−0.394706	+	0.918807i	$0.629154\pi$
$420$	0	0
$421$	16.5502	0.806609	0.403305	−	0.915066i	$-0.367861\pi$
$421$	16.5502	0.806609	0.403305	+	0.915066i	$0.367861\pi$
$422$	0	0
$423$	−23.0280	−1.11966
$424$	0	0
$425$	15.3911	0.746579
$426$	0	0
$427$	16.0938	0.778835
$428$	0	0
$429$	38.4941	1.85851
$430$	0	0
$431$	13.0428	0.628248	0.314124	−	0.949382i	$-0.398289\pi$
$431$	13.0428	0.628248	0.314124	+	0.949382i	$0.398289\pi$
$432$	0	0
$433$	4.40971	0.211917	0.105959	−	0.994371i	$-0.466209\pi$
$433$	4.40971	0.211917	0.105959	+	0.994371i	$0.466209\pi$
$434$	0	0
$435$	38.1844	1.83080
$436$	0	0
$437$	2.12360	0.101585
$438$	0	0
$439$	−23.2556	−1.10993	−0.554964	−	0.831874i	$-0.687268\pi$
$439$	−23.2556	−1.10993	−0.554964	+	0.831874i	$0.687268\pi$
$440$	0	0
$441$	−11.1693	−0.531869
$442$	0	0
$443$	−5.09598	−0.242117	−0.121059	−	0.992645i	$-0.538629\pi$
$443$	−5.09598	−0.242117	−0.121059	+	0.992645i	$0.538629\pi$
$444$	0	0
$445$	−40.5562	−1.92255
$446$	0	0
$447$	26.7500	1.26523
$448$	0	0
$449$	−5.26937	−0.248677	−0.124339	−	0.992240i	$-0.539681\pi$
$449$	−5.26937	−0.248677	−0.124339	+	0.992240i	$0.539681\pi$
$450$	0	0
$451$	30.4866	1.43556
$452$	0	0
$453$	5.09928	0.239585
$454$	0	0
$455$	27.4121	1.28510
$456$	0	0
$457$	10.2675	0.480292	0.240146	−	0.970737i	$-0.422805\pi$
$457$	10.2675	0.480292	0.240146	+	0.970737i	$0.422805\pi$
$458$	0	0
$459$	−0.884363	−0.0412785
$460$	0	0
$461$	20.3323	0.946970	0.473485	−	0.880802i	$-0.342996\pi$
$461$	20.3323	0.946970	0.473485	+	0.880802i	$0.342996\pi$
$462$	0	0
$463$	29.2200	1.35797	0.678985	−	0.734153i	$-0.262420\pi$
$463$	29.2200	1.35797	0.678985	+	0.734153i	$0.262420\pi$
$464$	0	0
$465$	6.65026	0.308398
$466$	0	0
$467$	−23.8062	−1.10162	−0.550809	−	0.834631i	$-0.685681\pi$
$467$	−23.8062	−1.10162	−0.550809	+	0.834631i	$0.685681\pi$
$468$	0	0
$469$	−20.5287	−0.947928
$470$	0	0
$471$	−10.4367	−0.480896
$472$	0	0
$473$	−24.4429	−1.12388
$474$	0	0
$475$	−0.465360	−0.0213522
$476$	0	0
$477$	−14.0109	−0.641517
$478$	0	0
$479$	31.8416	1.45488	0.727440	−	0.686171i	$-0.240710\pi$
$479$	31.8416	1.45488	0.727440	+	0.686171i	$0.240710\pi$
$480$	0	0
$481$	−43.4130	−1.97946
$482$	0	0
$483$	40.4334	1.83978
$484$	0	0
$485$	−4.93946	−0.224289
$486$	0	0
$487$	−40.0470	−1.81470	−0.907352	−	0.420372i	$-0.861900\pi$
$487$	−40.0470	−1.81470	−0.907352	+	0.420372i	$0.861900\pi$
$488$	0	0
$489$	4.32357	0.195519
$490$	0	0
$491$	−9.07936	−0.409746	−0.204873	−	0.978789i	$-0.565678\pi$
$491$	−9.07936	−0.409746	−0.204873	+	0.978789i	$0.565678\pi$
$492$	0	0
$493$	−45.8568	−2.06529
$494$	0	0
$495$	−22.1493	−0.995537
$496$	0	0
$497$	−25.1434	−1.12783
$498$	0	0
$499$	−5.97557	−0.267503	−0.133752	−	0.991015i	$-0.542702\pi$
$499$	−5.97557	−0.267503	−0.133752	+	0.991015i	$0.542702\pi$
$500$	0	0
$501$	−14.2313	−0.635806
$502$	0	0
$503$	−1.34215	−0.0598437	−0.0299218	−	0.999552i	$-0.509526\pi$
$503$	−1.34215	−0.0598437	−0.0299218	+	0.999552i	$0.509526\pi$
$504$	0	0
$505$	−38.3986	−1.70871
$506$	0	0
$507$	−47.5198	−2.11043
$508$	0	0
$509$	15.3783	0.681632	0.340816	−	0.940130i	$-0.389297\pi$
$509$	15.3783	0.681632	0.340816	+	0.940130i	$0.389297\pi$
$510$	0	0
$511$	−4.59388	−0.203221
$512$	0	0
$513$	0.0267393	0.00118057
$514$	0	0
$515$	42.6930	1.88128
$516$	0	0
$517$	20.8156	0.915470
$518$	0	0
$519$	−13.0638	−0.573435
$520$	0	0
$521$	29.0171	1.27126	0.635632	−	0.771992i	$-0.280740\pi$
$521$	29.0171	1.27126	0.635632	+	0.771992i	$0.280740\pi$
$522$	0	0
$523$	−20.1525	−0.881206	−0.440603	−	0.897702i	$-0.645235\pi$
$523$	−20.1525	−0.881206	−0.440603	+	0.897702i	$0.645235\pi$
$524$	0	0
$525$	−8.86047	−0.386703
$526$	0	0
$527$	−7.98650	−0.347897
$528$	0	0
$529$	58.1204	2.52698
$530$	0	0
$531$	−16.5417	−0.717849
$532$	0	0
$533$	−62.9510	−2.72671
$534$	0	0
$535$	28.6181	1.23727
$536$	0	0
$537$	−20.4901	−0.884212
$538$	0	0
$539$	10.0962	0.434874
$540$	0	0
$541$	−18.6165	−0.800388	−0.400194	−	0.916431i	$-0.631057\pi$
$541$	−18.6165	−0.800388	−0.400194	+	0.916431i	$0.631057\pi$
$542$	0	0
$543$	−16.7022	−0.716762
$544$	0	0
$545$	19.3868	0.830438
$546$	0	0
$547$	−35.3903	−1.51318	−0.756591	−	0.653889i	$-0.773136\pi$
$547$	−35.3903	−1.51318	−0.756591	+	0.653889i	$0.773136\pi$
$548$	0	0
$549$	26.8516	1.14600
$550$	0	0
$551$	1.38651	0.0590673
$552$	0	0
$553$	3.64961	0.155197
$554$	0	0
$555$	49.5810	2.10460
$556$	0	0
$557$	−20.9542	−0.887859	−0.443929	−	0.896062i	$-0.646416\pi$
$557$	−20.9542	−0.887859	−0.443929	+	0.896062i	$0.646416\pi$
$558$	0	0
$559$	50.4714	2.13471
$560$	0	0
$561$	52.7968	2.22908
$562$	0	0
$563$	42.7762	1.80280	0.901400	−	0.432987i	$-0.142541\pi$
$563$	42.7762	1.80280	0.901400	+	0.432987i	$0.142541\pi$
$564$	0	0
$565$	−22.0599	−0.928067
$566$	0	0
$567$	−16.1751	−0.679289
$568$	0	0
$569$	−36.3588	−1.52424	−0.762120	−	0.647436i	$-0.775841\pi$
$569$	−36.3588	−1.52424	−0.762120	+	0.647436i	$0.775841\pi$
$570$	0	0
$571$	−8.53168	−0.357040	−0.178520	−	0.983936i	$-0.557131\pi$
$571$	−8.53168	−0.357040	−0.178520	+	0.983936i	$0.557131\pi$
$572$	0	0
$573$	−14.2551	−0.595515
$574$	0	0
$575$	−17.7765	−0.741333
$576$	0	0
$577$	−30.8784	−1.28549	−0.642743	−	0.766082i	$-0.722204\pi$
$577$	−30.8784	−1.28549	−0.642743	+	0.766082i	$0.722204\pi$
$578$	0	0
$579$	−23.5806	−0.979978
$580$	0	0
$581$	3.92978	0.163035
$582$	0	0
$583$	12.6649	0.524526
$584$	0	0
$585$	45.7355	1.89093
$586$	0	0
$587$	1.05137	0.0433947	0.0216974	−	0.999765i	$-0.493093\pi$
$587$	1.05137	0.0433947	0.0216974	+	0.999765i	$0.493093\pi$
$588$	0	0
$589$	0.241477	0.00994988
$590$	0	0
$591$	−25.4353	−1.04627
$592$	0	0
$593$	−21.1372	−0.868002	−0.434001	−	0.900912i	$-0.642899\pi$
$593$	−21.1372	−0.868002	−0.434001	+	0.900912i	$0.642899\pi$
$594$	0	0
$595$	37.5973	1.54134
$596$	0	0
$597$	34.5967	1.41595
$598$	0	0
$599$	−0.677286	−0.0276732	−0.0138366	−	0.999904i	$-0.504404\pi$
$599$	−0.677286	−0.0276732	−0.0138366	+	0.999904i	$0.504404\pi$
$600$	0	0
$601$	45.0681	1.83837	0.919183	−	0.393831i	$-0.128851\pi$
$601$	45.0681	1.83837	0.919183	+	0.393831i	$0.128851\pi$
$602$	0	0
$603$	−34.2510	−1.39481
$604$	0	0
$605$	−9.02717	−0.367007
$606$	0	0
$607$	5.06464	0.205568	0.102784	−	0.994704i	$-0.467225\pi$
$607$	5.06464	0.205568	0.102784	+	0.994704i	$0.467225\pi$
$608$	0	0
$609$	26.3992	1.06975
$610$	0	0
$611$	−42.9816	−1.73885
$612$	0	0
$613$	−8.61778	−0.348069	−0.174034	−	0.984740i	$-0.555680\pi$
$613$	−8.61778	−0.348069	−0.174034	+	0.984740i	$0.555680\pi$
$614$	0	0
$615$	71.8950	2.89909
$616$	0	0
$617$	−14.0242	−0.564594	−0.282297	−	0.959327i	$-0.591096\pi$
$617$	−14.0242	−0.564594	−0.282297	+	0.959327i	$0.591096\pi$
$618$	0	0
$619$	−41.6982	−1.67599	−0.837997	−	0.545675i	$-0.816273\pi$
$619$	−41.6982	−1.67599	−0.837997	+	0.545675i	$0.816273\pi$
$620$	0	0
$621$	1.02143	0.0409885
$622$	0	0
$623$	−28.0389	−1.12336
$624$	0	0
$625$	−30.9730	−1.23892
$626$	0	0
$627$	−1.59634	−0.0637519
$628$	0	0
$629$	−59.5434	−2.37415
$630$	0	0
$631$	−9.71454	−0.386730	−0.193365	−	0.981127i	$-0.561940\pi$
$631$	−9.71454	−0.386730	−0.193365	+	0.981127i	$0.561940\pi$
$632$	0	0
$633$	−62.4868	−2.48363
$634$	0	0
$635$	−18.7840	−0.745422
$636$	0	0
$637$	−20.8474	−0.826003
$638$	0	0
$639$	−41.9502	−1.65953
$640$	0	0
$641$	49.9219	1.97180	0.985898	−	0.167345i	$-0.0535194\pi$
$641$	49.9219	1.97180	0.985898	+	0.167345i	$0.0535194\pi$
$642$	0	0
$643$	21.8689	0.862424	0.431212	−	0.902251i	$-0.358086\pi$
$643$	21.8689	0.862424	0.431212	+	0.902251i	$0.358086\pi$
$644$	0	0
$645$	−57.6423	−2.26966
$646$	0	0
$647$	37.2231	1.46339	0.731696	−	0.681631i	$-0.238729\pi$
$647$	37.2231	1.46339	0.731696	+	0.681631i	$0.238729\pi$
$648$	0	0
$649$	14.9525	0.586938
$650$	0	0
$651$	4.59773	0.180199
$652$	0	0
$653$	−26.8031	−1.04889	−0.524443	−	0.851446i	$-0.675726\pi$
$653$	−26.8031	−1.04889	−0.524443	+	0.851446i	$0.675726\pi$
$654$	0	0
$655$	4.48248	0.175145
$656$	0	0
$657$	−7.66462	−0.299025
$658$	0	0
$659$	15.0950	0.588019	0.294010	−	0.955802i	$-0.405010\pi$
$659$	15.0950	0.588019	0.294010	+	0.955802i	$0.405010\pi$
$660$	0	0
$661$	−5.57864	−0.216984	−0.108492	−	0.994097i	$-0.534602\pi$
$661$	−5.57864	−0.216984	−0.108492	+	0.994097i	$0.534602\pi$
$662$	0	0
$663$	−109.019	−4.23393
$664$	0	0
$665$	−1.13678	−0.0440823
$666$	0	0
$667$	52.9640	2.05077
$668$	0	0
$669$	−36.4035	−1.40744
$670$	0	0
$671$	−24.2719	−0.937007
$672$	0	0
$673$	−46.9525	−1.80988	−0.904942	−	0.425535i	$-0.860086\pi$
$673$	−46.9525	−1.80988	−0.904942	+	0.425535i	$0.860086\pi$
$674$	0	0
$675$	−0.223833	−0.00861535
$676$	0	0
$677$	8.75915	0.336642	0.168321	−	0.985732i	$-0.446166\pi$
$677$	8.75915	0.336642	0.168321	+	0.985732i	$0.446166\pi$
$678$	0	0
$679$	−3.41495	−0.131054
$680$	0	0
$681$	−25.6697	−0.983666
$682$	0	0
$683$	47.4270	1.81474	0.907372	−	0.420328i	$-0.138085\pi$
$683$	47.4270	1.81474	0.907372	+	0.420328i	$0.138085\pi$
$684$	0	0
$685$	2.45816	0.0939213
$686$	0	0
$687$	50.5410	1.92826
$688$	0	0
$689$	−26.1514	−0.996287
$690$	0	0
$691$	−43.6980	−1.66235	−0.831175	−	0.556011i	$-0.812331\pi$
$691$	−43.6980	−1.66235	−0.831175	+	0.556011i	$0.812331\pi$
$692$	0	0
$693$	−15.3132	−0.581699
$694$	0	0
$695$	−59.6529	−2.26276
$696$	0	0
$697$	−86.3409	−3.27040
$698$	0	0
$699$	−26.8430	−1.01529
$700$	0	0
$701$	10.8868	0.411190	0.205595	−	0.978637i	$-0.434087\pi$
$701$	10.8868	0.411190	0.205595	+	0.978637i	$0.434087\pi$
$702$	0	0
$703$	1.80033	0.0679008
$704$	0	0
$705$	49.0883	1.84877
$706$	0	0
$707$	−26.5473	−0.998412
$708$	0	0
$709$	40.5893	1.52436	0.762181	−	0.647364i	$-0.224129\pi$
$709$	40.5893	1.52436	0.762181	+	0.647364i	$0.224129\pi$
$710$	0	0
$711$	6.08915	0.228361
$712$	0	0
$713$	9.22431	0.345453
$714$	0	0
$715$	−41.3416	−1.54609
$716$	0	0
$717$	23.2513	0.868335
$718$	0	0
$719$	−44.6286	−1.66437	−0.832183	−	0.554501i	$-0.812909\pi$
$719$	−44.6286	−1.66437	−0.832183	+	0.554501i	$0.812909\pi$
$720$	0	0
$721$	29.5162	1.09924
$722$	0	0
$723$	47.0719	1.75062
$724$	0	0
$725$	−11.6064	−0.431051
$726$	0	0
$727$	44.4577	1.64884	0.824422	−	0.565975i	$-0.191500\pi$
$727$	44.4577	1.64884	0.824422	+	0.565975i	$0.191500\pi$
$728$	0	0
$729$	−27.8237	−1.03051
$730$	0	0
$731$	69.2244	2.56036
$732$	0	0
$733$	9.49609	0.350746	0.175373	−	0.984502i	$-0.443887\pi$
$733$	9.49609	0.350746	0.175373	+	0.984502i	$0.443887\pi$
$734$	0	0
$735$	23.8093	0.878221
$736$	0	0
$737$	30.9604	1.14044
$738$	0	0
$739$	−21.2446	−0.781494	−0.390747	−	0.920498i	$-0.627783\pi$
$739$	−21.2446	−0.781494	−0.390747	+	0.920498i	$0.627783\pi$
$740$	0	0
$741$	3.29625	0.121091
$742$	0	0
$743$	43.1652	1.58358	0.791789	−	0.610794i	$-0.209150\pi$
$743$	43.1652	1.58358	0.791789	+	0.610794i	$0.209150\pi$
$744$	0	0
$745$	−28.7288	−1.05254
$746$	0	0
$747$	6.55660	0.239894
$748$	0	0
$749$	19.7854	0.722943
$750$	0	0
$751$	−47.5441	−1.73491	−0.867454	−	0.497517i	$-0.834245\pi$
$751$	−47.5441	−1.73491	−0.867454	+	0.497517i	$0.834245\pi$
$752$	0	0
$753$	2.45889	0.0896068
$754$	0	0
$755$	−5.47649	−0.199310
$756$	0	0
$757$	23.3441	0.848457	0.424229	−	0.905555i	$-0.360545\pi$
$757$	23.3441	0.848457	0.424229	+	0.905555i	$0.360545\pi$
$758$	0	0
$759$	−60.9796	−2.21342
$760$	0	0
$761$	−26.5364	−0.961945	−0.480972	−	0.876736i	$-0.659716\pi$
$761$	−26.5364	−0.961945	−0.480972	+	0.876736i	$0.659716\pi$
$762$	0	0
$763$	13.4032	0.485230
$764$	0	0
$765$	62.7288	2.26797
$766$	0	0
$767$	−30.8750	−1.11483
$768$	0	0
$769$	−9.67209	−0.348784	−0.174392	−	0.984676i	$-0.555796\pi$
$769$	−9.67209	−0.348784	−0.174392	+	0.984676i	$0.555796\pi$
$770$	0	0
$771$	−9.28316	−0.334325
$772$	0	0
$773$	−48.0700	−1.72896	−0.864480	−	0.502668i	$-0.832352\pi$
$773$	−48.0700	−1.72896	−0.864480	+	0.502668i	$0.832352\pi$
$774$	0	0
$775$	−2.02139	−0.0726106
$776$	0	0
$777$	34.2784	1.22973
$778$	0	0
$779$	2.61057	0.0935334
$780$	0	0
$781$	37.9200	1.35688
$782$	0	0
$783$	0.666896	0.0238329
$784$	0	0
$785$	11.2087	0.400055
$786$	0	0
$787$	15.2702	0.544325	0.272163	−	0.962251i	$-0.412261\pi$
$787$	15.2702	0.544325	0.272163	+	0.962251i	$0.412261\pi$
$788$	0	0
$789$	−7.97060	−0.283761
$790$	0	0
$791$	−15.2513	−0.542275
$792$	0	0
$793$	50.1184	1.77976
$794$	0	0
$795$	29.8669	1.05927
$796$	0	0
$797$	27.1953	0.963307	0.481653	−	0.876362i	$-0.340036\pi$
$797$	27.1953	0.963307	0.481653	+	0.876362i	$0.340036\pi$
$798$	0	0
$799$	−58.9517	−2.08556
$800$	0	0
$801$	−46.7813	−1.65294
$802$	0	0
$803$	6.92827	0.244493
$804$	0	0
$805$	−43.4243	−1.53051
$806$	0	0
$807$	−10.9147	−0.384217
$808$	0	0
$809$	−15.0073	−0.527628	−0.263814	−	0.964574i	$-0.584981\pi$
$809$	−15.0073	−0.527628	−0.263814	+	0.964574i	$0.584981\pi$
$810$	0	0
$811$	−1.94228	−0.0682026	−0.0341013	−	0.999418i	$-0.510857\pi$
$811$	−1.94228	−0.0682026	−0.0341013	+	0.999418i	$0.510857\pi$
$812$	0	0
$813$	−7.25162	−0.254325
$814$	0	0
$815$	−4.64340	−0.162651
$816$	0	0
$817$	−2.09304	−0.0732263
$818$	0	0
$819$	31.6197	1.10488
$820$	0	0
$821$	−3.96292	−0.138307	−0.0691534	−	0.997606i	$-0.522030\pi$
$821$	−3.96292	−0.138307	−0.0691534	+	0.997606i	$0.522030\pi$
$822$	0	0
$823$	−1.35971	−0.0473964	−0.0236982	−	0.999719i	$-0.507544\pi$
$823$	−1.35971	−0.0473964	−0.0236982	+	0.999719i	$0.507544\pi$
$824$	0	0
$825$	13.3629	0.465238
$826$	0	0
$827$	5.31877	0.184952	0.0924759	−	0.995715i	$-0.470522\pi$
$827$	5.31877	0.184952	0.0924759	+	0.995715i	$0.470522\pi$
$828$	0	0
$829$	13.9689	0.485159	0.242579	−	0.970132i	$-0.422006\pi$
$829$	13.9689	0.485159	0.242579	+	0.970132i	$0.422006\pi$
$830$	0	0
$831$	−35.8343	−1.24308
$832$	0	0
$833$	−28.5934	−0.990701
$834$	0	0
$835$	15.2840	0.528924
$836$	0	0
$837$	0.116148	0.00401466
$838$	0	0
$839$	26.7180	0.922407	0.461204	−	0.887294i	$-0.347418\pi$
$839$	26.7180	0.922407	0.461204	+	0.887294i	$0.347418\pi$
$840$	0	0
$841$	5.58047	0.192430
$842$	0	0
$843$	−49.3665	−1.70027
$844$	0	0
$845$	51.0350	1.75566
$846$	0	0
$847$	−6.24103	−0.214444
$848$	0	0
$849$	−44.7517	−1.53587
$850$	0	0
$851$	68.7718	2.35747
$852$	0	0
$853$	−58.0923	−1.98904	−0.994521	−	0.104540i	$-0.966663\pi$
$853$	−58.0923	−1.98904	−0.994521	+	0.104540i	$0.966663\pi$
$854$	0	0
$855$	−1.89665	−0.0648639
$856$	0	0
$857$	9.16837	0.313186	0.156593	−	0.987663i	$-0.449949\pi$
$857$	9.16837	0.313186	0.156593	+	0.987663i	$0.449949\pi$
$858$	0	0
$859$	30.5219	1.04139	0.520697	−	0.853742i	$-0.325672\pi$
$859$	30.5219	1.04139	0.520697	+	0.853742i	$0.325672\pi$
$860$	0	0
$861$	49.7054	1.69395
$862$	0	0
$863$	−48.7995	−1.66116	−0.830578	−	0.556902i	$-0.811990\pi$
$863$	−48.7995	−1.66116	−0.830578	+	0.556902i	$0.811990\pi$
$864$	0	0
$865$	14.0301	0.477038
$866$	0	0
$867$	−107.724	−3.65851
$868$	0	0
$869$	−5.50416	−0.186716
$870$	0	0
$871$	−63.9292	−2.16616
$872$	0	0
$873$	−5.69764	−0.192836
$874$	0	0
$875$	−14.5908	−0.493259
$876$	0	0
$877$	21.9900	0.742548	0.371274	−	0.928523i	$-0.378921\pi$
$877$	21.9900	0.742548	0.371274	+	0.928523i	$0.378921\pi$
$878$	0	0
$879$	48.9429	1.65080
$880$	0	0
$881$	23.4835	0.791180	0.395590	−	0.918427i	$-0.370540\pi$
$881$	23.4835	0.791180	0.395590	+	0.918427i	$0.370540\pi$
$882$	0	0
$883$	19.2558	0.648011	0.324005	−	0.946055i	$-0.394970\pi$
$883$	19.2558	0.648011	0.324005	+	0.946055i	$0.394970\pi$
$884$	0	0
$885$	35.2617	1.18531
$886$	0	0
$887$	25.6892	0.862560	0.431280	−	0.902218i	$-0.358062\pi$
$887$	25.6892	0.862560	0.431280	+	0.902218i	$0.358062\pi$
$888$	0	0
$889$	−12.9865	−0.435555
$890$	0	0
$891$	24.3944	0.817244
$892$	0	0
$893$	1.78244	0.0596471
$894$	0	0
$895$	22.0058	0.735573
$896$	0	0
$897$	125.915	4.20418
$898$	0	0
$899$	6.02260	0.200865
$900$	0	0
$901$	−35.8681	−1.19494
$902$	0	0
$903$	−39.8516	−1.32618
$904$	0	0
$905$	17.9378	0.596271
$906$	0	0
$907$	4.84468	0.160865	0.0804325	−	0.996760i	$-0.474370\pi$
$907$	4.84468	0.160865	0.0804325	+	0.996760i	$0.474370\pi$
$908$	0	0
$909$	−44.2925	−1.46909
$910$	0	0
$911$	31.8741	1.05604	0.528019	−	0.849233i	$-0.322935\pi$
$911$	31.8741	1.05604	0.528019	+	0.849233i	$0.322935\pi$
$912$	0	0
$913$	−5.92670	−0.196145
$914$	0	0
$915$	−57.2392	−1.89227
$916$	0	0
$917$	3.09901	0.102338
$918$	0	0
$919$	−41.1376	−1.35700	−0.678502	−	0.734599i	$-0.737371\pi$
$919$	−41.1376	−1.35700	−0.678502	+	0.734599i	$0.737371\pi$
$920$	0	0
$921$	−58.1528	−1.91620
$922$	0	0
$923$	−78.2999	−2.57727
$924$	0	0
$925$	−15.0705	−0.495515
$926$	0	0
$927$	49.2461	1.61745
$928$	0	0
$929$	29.0828	0.954176	0.477088	−	0.878855i	$-0.341692\pi$
$929$	29.0828	0.954176	0.477088	+	0.878855i	$0.341692\pi$
$930$	0	0
$931$	0.864538	0.0283341
$932$	0	0
$933$	54.1133	1.77159
$934$	0	0
$935$	−56.7023	−1.85437
$936$	0	0
$937$	−52.2058	−1.70549	−0.852744	−	0.522328i	$-0.825064\pi$
$937$	−52.2058	−1.70549	−0.852744	+	0.522328i	$0.825064\pi$
$938$	0	0
$939$	−2.81888	−0.0919907
$940$	0	0
$941$	40.9415	1.33466	0.667328	−	0.744764i	$-0.267438\pi$
$941$	40.9415	1.33466	0.667328	+	0.744764i	$0.267438\pi$
$942$	0	0
$943$	99.7226	3.24742
$944$	0	0
$945$	−0.546778	−0.0177867
$946$	0	0
$947$	−2.35562	−0.0765475	−0.0382737	−	0.999267i	$-0.512186\pi$
$947$	−2.35562	−0.0765475	−0.0382737	+	0.999267i	$0.512186\pi$
$948$	0	0
$949$	−14.3060	−0.464392
$950$	0	0
$951$	−75.5405	−2.44957
$952$	0	0
$953$	−18.0465	−0.584584	−0.292292	−	0.956329i	$-0.594418\pi$
$953$	−18.0465	−0.584584	−0.292292	+	0.956329i	$0.594418\pi$
$954$	0	0
$955$	15.3096	0.495407
$956$	0	0
$957$	−39.8139	−1.28700
$958$	0	0
$959$	1.69947	0.0548788
$960$	0	0
$961$	−29.9511	−0.966164
$962$	0	0
$963$	33.0108	1.06376
$964$	0	0
$965$	25.3250	0.815240
$966$	0	0
$967$	33.6472	1.08202	0.541011	−	0.841016i	$-0.318042\pi$
$967$	33.6472	1.08202	0.541011	+	0.841016i	$0.318042\pi$
$968$	0	0
$969$	4.52099	0.145235
$970$	0	0
$971$	53.2221	1.70798	0.853989	−	0.520292i	$-0.174177\pi$
$971$	53.2221	1.70798	0.853989	+	0.520292i	$0.174177\pi$
$972$	0	0
$973$	−41.2417	−1.32215
$974$	0	0
$975$	−27.5927	−0.883675
$976$	0	0
$977$	5.63782	0.180370	0.0901849	−	0.995925i	$-0.471254\pi$
$977$	5.63782	0.180370	0.0901849	+	0.995925i	$0.471254\pi$
$978$	0	0
$979$	42.2869	1.35150
$980$	0	0
$981$	22.3625	0.713980
$982$	0	0
$983$	−11.6074	−0.370218	−0.185109	−	0.982718i	$-0.559264\pi$
$983$	−11.6074	−0.370218	−0.185109	+	0.982718i	$0.559264\pi$
$984$	0	0
$985$	27.3168	0.870386
$986$	0	0
$987$	33.9377	1.08025
$988$	0	0
$989$	−79.9532	−2.54237
$990$	0	0
$991$	22.6740	0.720265	0.360132	−	0.932901i	$-0.382732\pi$
$991$	22.6740	0.720265	0.360132	+	0.932901i	$0.382732\pi$
$992$	0	0
$993$	42.3103	1.34268
$994$	0	0
$995$	−37.1559	−1.17792
$996$	0	0
$997$	28.4606	0.901357	0.450679	−	0.892686i	$-0.351182\pi$
$997$	28.4606	0.901357	0.450679	+	0.892686i	$0.351182\pi$
$998$	0	0
$999$	0.865940	0.0273971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.h.1.6	✓	30
4.3	odd	2		8032.2.a.i.1.25	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.6	✓	30	1.1	even	1	trivial
8032.2.a.i.1.25	yes	30	4.3	odd	2

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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-2.45889 q^{3} +2.64078 q^{5} +1.82573 q^{7} +3.04612 q^{9} +O(q^{10})\)	\(q-2.45889 q^{3} +2.64078 q^{5} +1.82573 q^{7} +3.04612 q^{9} -2.75348 q^{11} +5.68557 q^{13} -6.49337 q^{15} +7.79809 q^{17} -0.235780 q^{19} -4.48926 q^{21} -9.00669 q^{23} +1.97370 q^{25} -0.113408 q^{27} -5.88052 q^{29} -1.02416 q^{31} +6.77048 q^{33} +4.82134 q^{35} -7.63564 q^{37} -13.9802 q^{39} -11.0721 q^{41} +8.87710 q^{43} +8.04413 q^{45} -7.55976 q^{47} -3.66671 q^{49} -19.1746 q^{51} -4.59960 q^{53} -7.27131 q^{55} +0.579756 q^{57} -5.43042 q^{59} +8.81502 q^{61} +5.56139 q^{63} +15.0143 q^{65} -11.2441 q^{67} +22.1464 q^{69} -13.7717 q^{71} -2.51619 q^{73} -4.85312 q^{75} -5.02710 q^{77} +1.99899 q^{79} -8.85951 q^{81} +2.15244 q^{83} +20.5930 q^{85} +14.4595 q^{87} -15.3577 q^{89} +10.3803 q^{91} +2.51830 q^{93} -0.622643 q^{95} -1.87046 q^{97} -8.38742 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9	\( 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9} - 13 q^{11} - 7 q^{13} - 28 q^{15} - 9 q^{17} + 17 q^{19} - 6 q^{21} - 43 q^{23} + 34 q^{25} - 12 q^{27} - q^{29} - 39 q^{31} - 17 q^{35} - q^{37} - 48 q^{39} - 3 q^{41} + 19 q^{43} - 66 q^{47} + 25 q^{49} + 14 q^{51} + 3 q^{53} - 50 q^{55} - 14 q^{57} - 27 q^{59} + 15 q^{61} - 75 q^{63} - 6 q^{65} - 8 q^{67} + 18 q^{69} - 64 q^{71} - 15 q^{73} - 9 q^{75} - 71 q^{79} + 6 q^{81} - 60 q^{83} + 15 q^{85} - 64 q^{87} - 32 q^{89} + 26 q^{91} - 4 q^{93} - 72 q^{95} - 4 q^{97} - 13 q^{99}+O(q^{100}) \) 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9 - 13 * q^11 - 7 * q^13 - 28 * q^15 - 9 * q^17 + 17 * q^19 - 6 * q^21 - 43 * q^23 + 34 * q^25 - 12 * q^27 - q^29 - 39 * q^31 - 17 * q^35 - q^37 - 48 * q^39 - 3 * q^41 + 19 * q^43 - 66 * q^47 + 25 * q^49 + 14 * q^51 + 3 * q^53 - 50 * q^55 - 14 * q^57 - 27 * q^59 + 15 * q^61 - 75 * q^63 - 6 * q^65 - 8 * q^67 + 18 * q^69 - 64 * q^71 - 15 * q^73 - 9 * q^75 - 71 * q^79 + 6 * q^81 - 60 * q^83 + 15 * q^85 - 64 * q^87 - 32 * q^89 + 26 * q^91 - 4 * q^93 - 72 * q^95 - 4 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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