LMFDB - Embedded newform 8037.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.48310 q^{2} +4.16581 q^{4} -4.02387 q^{5} +2.71666 q^{7} -5.37793 q^{8} +O(q^{10})$	\(q-2.48310 q^{2} +4.16581 q^{4} -4.02387 q^{5} +2.71666 q^{7} -5.37793 q^{8} +9.99168 q^{10} -3.51111 q^{11} -6.31210 q^{13} -6.74574 q^{14} +5.02235 q^{16} -0.705679 q^{17} +1.00000 q^{19} -16.7627 q^{20} +8.71845 q^{22} +0.522702 q^{23} +11.1915 q^{25} +15.6736 q^{26} +11.3171 q^{28} -0.738085 q^{29} +1.19726 q^{31} -1.71516 q^{32} +1.75227 q^{34} -10.9315 q^{35} -5.41516 q^{37} -2.48310 q^{38} +21.6401 q^{40} +3.75217 q^{41} -3.71053 q^{43} -14.6266 q^{44} -1.29792 q^{46} +1.00000 q^{47} +0.380227 q^{49} -27.7897 q^{50} -26.2950 q^{52} +2.32114 q^{53} +14.1282 q^{55} -14.6100 q^{56} +1.83274 q^{58} +2.47993 q^{59} +5.98702 q^{61} -2.97291 q^{62} -5.78579 q^{64} +25.3990 q^{65} +2.42311 q^{67} -2.93972 q^{68} +27.1440 q^{70} +13.2851 q^{71} +12.4740 q^{73} +13.4464 q^{74} +4.16581 q^{76} -9.53848 q^{77} +5.81653 q^{79} -20.2093 q^{80} -9.31704 q^{82} +7.81012 q^{83} +2.83956 q^{85} +9.21363 q^{86} +18.8825 q^{88} +5.89628 q^{89} -17.1478 q^{91} +2.17748 q^{92} -2.48310 q^{94} -4.02387 q^{95} +1.48505 q^{97} -0.944143 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q - 5 q^{2} + 31 q^{4} - 14 q^{5} - 15 q^{8}+O(q^{10}) $ 34 * q - 5 * q^2 + 31 * q^4 - 14 * q^5 - 15 * q^8	$ 34 q - 5 q^{2} + 31 q^{4} - 14 q^{5} - 15 q^{8} - 18 q^{11} - 6 q^{13} - 12 q^{14} + 21 q^{16} - 36 q^{17} + 34 q^{19} - 40 q^{20} + 12 q^{22} - 38 q^{23} + 32 q^{25} - 15 q^{26} + 28 q^{28} - 14 q^{29} - 6 q^{31} - 35 q^{32} + 10 q^{34} - 46 q^{35} - 2 q^{37} - 5 q^{38} + 31 q^{40} - 18 q^{41} - 6 q^{43} - 42 q^{44} - 14 q^{46} + 34 q^{47} + 44 q^{49} - 9 q^{50} + 2 q^{52} - 32 q^{53} + 8 q^{55} + 4 q^{56} + 8 q^{58} - 62 q^{59} - 10 q^{61} - 30 q^{62} - 37 q^{64} - 8 q^{65} - 92 q^{68} - 62 q^{70} - 4 q^{71} - 8 q^{73} - 34 q^{74} + 31 q^{76} - 52 q^{77} + 40 q^{79} - 48 q^{80} - 2 q^{82} - 110 q^{83} - 12 q^{85} - 16 q^{86} - 44 q^{88} - 2 q^{89} - 28 q^{91} - 60 q^{92} - 5 q^{94} - 14 q^{95} + 2 q^{97} - 45 q^{98}+O(q^{100}) $ 34 * q - 5 * q^2 + 31 * q^4 - 14 * q^5 - 15 * q^8 - 18 * q^11 - 6 * q^13 - 12 * q^14 + 21 * q^16 - 36 * q^17 + 34 * q^19 - 40 * q^20 + 12 * q^22 - 38 * q^23 + 32 * q^25 - 15 * q^26 + 28 * q^28 - 14 * q^29 - 6 * q^31 - 35 * q^32 + 10 * q^34 - 46 * q^35 - 2 * q^37 - 5 * q^38 + 31 * q^40 - 18 * q^41 - 6 * q^43 - 42 * q^44 - 14 * q^46 + 34 * q^47 + 44 * q^49 - 9 * q^50 + 2 * q^52 - 32 * q^53 + 8 * q^55 + 4 * q^56 + 8 * q^58 - 62 * q^59 - 10 * q^61 - 30 * q^62 - 37 * q^64 - 8 * q^65 - 92 * q^68 - 62 * q^70 - 4 * q^71 - 8 * q^73 - 34 * q^74 + 31 * q^76 - 52 * q^77 + 40 * q^79 - 48 * q^80 - 2 * q^82 - 110 * q^83 - 12 * q^85 - 16 * q^86 - 44 * q^88 - 2 * q^89 - 28 * q^91 - 60 * q^92 - 5 * q^94 - 14 * q^95 + 2 * q^97 - 45 * q^98

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 $

$2$	−2.48310	−1.75582	−0.877910	−	0.478825i	$-0.841063\pi$
$2$	−2.48310	−1.75582	−0.877910	+	0.478825i	$0.841063\pi$
$3$	0	0
$3$	0	0
$4$	4.16581	2.08290
$5$	−4.02387	−1.79953	−0.899764	−	0.436377i	$-0.856261\pi$
$5$	−4.02387	−1.79953	−0.899764	+	0.436377i	$0.856261\pi$
$6$	0	0
$7$	2.71666	1.02680	0.513400	−	0.858149i	$-0.328386\pi$
$7$	2.71666	1.02680	0.513400	+	0.858149i	$0.328386\pi$
$8$	−5.37793	−1.90139
$9$	0	0
$10$	9.99168	3.15965
$11$	−3.51111	−1.05864	−0.529319	−	0.848423i	$-0.677553\pi$
$11$	−3.51111	−1.05864	−0.529319	+	0.848423i	$0.677553\pi$
$12$	0	0
$13$	−6.31210	−1.75066	−0.875331	−	0.483525i	$-0.839356\pi$
$13$	−6.31210	−1.75066	−0.875331	+	0.483525i	$0.839356\pi$
$14$	−6.74574	−1.80288
$15$	0	0
$16$	5.02235	1.25559
$17$	−0.705679	−0.171152	−0.0855761	−	0.996332i	$-0.527273\pi$
$17$	−0.705679	−0.171152	−0.0855761	+	0.996332i	$0.527273\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−16.7627	−3.74825
$21$	0	0
$22$	8.71845	1.85878
$23$	0.522702	0.108991	0.0544954	−	0.998514i	$-0.482645\pi$
$23$	0.522702	0.108991	0.0544954	+	0.998514i	$0.482645\pi$
$24$	0	0
$25$	11.1915	2.23830
$26$	15.6736	3.07385
$27$	0	0
$28$	11.3171	2.13873
$29$	−0.738085	−0.137059	−0.0685295	−	0.997649i	$-0.521831\pi$
$29$	−0.738085	−0.137059	−0.0685295	+	0.997649i	$0.521831\pi$
$30$	0	0
$31$	1.19726	0.215033	0.107517	−	0.994203i	$-0.465710\pi$
$31$	1.19726	0.215033	0.107517	+	0.994203i	$0.465710\pi$
$32$	−1.71516	−0.303200
$33$	0	0
$34$	1.75227	0.300513
$35$	−10.9315	−1.84776
$36$	0	0
$37$	−5.41516	−0.890247	−0.445123	−	0.895469i	$-0.646840\pi$
$37$	−5.41516	−0.890247	−0.445123	+	0.895469i	$0.646840\pi$
$38$	−2.48310	−0.402813
$39$	0	0
$40$	21.6401	3.42160
$41$	3.75217	0.585991	0.292996	−	0.956114i	$-0.405348\pi$
$41$	3.75217	0.585991	0.292996	+	0.956114i	$0.405348\pi$
$42$	0	0
$43$	−3.71053	−0.565850	−0.282925	−	0.959142i	$-0.591305\pi$
$43$	−3.71053	−0.565850	−0.282925	+	0.959142i	$0.591305\pi$
$44$	−14.6266	−2.20504
$45$	0	0
$46$	−1.29792	−0.191368
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	0.380227	0.0543181
$50$	−27.7897	−3.93005
$51$	0	0
$52$	−26.2950	−3.64646
$53$	2.32114	0.318834	0.159417	−	0.987211i	$-0.449039\pi$
$53$	2.32114	0.318834	0.159417	+	0.987211i	$0.449039\pi$
$54$	0	0
$55$	14.1282	1.90505
$56$	−14.6100	−1.95234
$57$	0	0
$58$	1.83274	0.240651
$59$	2.47993	0.322860	0.161430	−	0.986884i	$-0.448389\pi$
$59$	2.47993	0.322860	0.161430	+	0.986884i	$0.448389\pi$
$60$	0	0
$61$	5.98702	0.766560	0.383280	−	0.923632i	$-0.374794\pi$
$61$	5.98702	0.766560	0.383280	+	0.923632i	$0.374794\pi$
$62$	−2.97291	−0.377560
$63$	0	0
$64$	−5.78579	−0.723223
$65$	25.3990	3.15036
$66$	0	0
$67$	2.42311	0.296030	0.148015	−	0.988985i	$-0.452712\pi$
$67$	2.42311	0.296030	0.148015	+	0.988985i	$0.452712\pi$
$68$	−2.93972	−0.356494
$69$	0	0
$70$	27.1440	3.24433
$71$	13.2851	1.57665	0.788325	−	0.615259i	$-0.210949\pi$
$71$	13.2851	1.57665	0.788325	+	0.615259i	$0.210949\pi$
$72$	0	0
$73$	12.4740	1.45997	0.729985	−	0.683464i	$-0.239527\pi$
$73$	12.4740	1.45997	0.729985	+	0.683464i	$0.239527\pi$
$74$	13.4464	1.56311
$75$	0	0
$76$	4.16581	0.477851
$77$	−9.53848	−1.08701
$78$	0	0
$79$	5.81653	0.654410	0.327205	−	0.944953i	$-0.393893\pi$
$79$	5.81653	0.654410	0.327205	+	0.944953i	$0.393893\pi$
$80$	−20.2093	−2.25946
$81$	0	0
$82$	−9.31704	−1.02889
$83$	7.81012	0.857272	0.428636	−	0.903477i	$-0.358994\pi$
$83$	7.81012	0.857272	0.428636	+	0.903477i	$0.358994\pi$
$84$	0	0
$85$	2.83956	0.307993
$86$	9.21363	0.993531
$87$	0	0
$88$	18.8825	2.01288
$89$	5.89628	0.625005	0.312502	−	0.949917i	$-0.398833\pi$
$89$	5.89628	0.625005	0.312502	+	0.949917i	$0.398833\pi$
$90$	0	0
$91$	−17.1478	−1.79758
$92$	2.17748	0.227018
$93$	0	0
$94$	−2.48310	−0.256113
$95$	−4.02387	−0.412840
$96$	0	0
$97$	1.48505	0.150784	0.0753921	−	0.997154i	$-0.475979\pi$
$97$	1.48505	0.150784	0.0753921	+	0.997154i	$0.475979\pi$
$98$	−0.944143	−0.0953728
$99$	0	0
$100$	46.6217	4.66217
$101$	−6.16778	−0.613717	−0.306858	−	0.951755i	$-0.599278\pi$
$101$	−6.16778	−0.613717	−0.306858	+	0.951755i	$0.599278\pi$
$102$	0	0
$103$	1.53937	0.151678	0.0758391	−	0.997120i	$-0.475836\pi$
$103$	1.53937	0.151678	0.0758391	+	0.997120i	$0.475836\pi$
$104$	33.9460	3.32868
$105$	0	0
$106$	−5.76364	−0.559814
$107$	−14.9421	−1.44451	−0.722254	−	0.691628i	$-0.756894\pi$
$107$	−14.9421	−1.44451	−0.722254	+	0.691628i	$0.756894\pi$
$108$	0	0
$109$	−0.777250	−0.0744470	−0.0372235	−	0.999307i	$-0.511851\pi$
$109$	−0.777250	−0.0744470	−0.0372235	+	0.999307i	$0.511851\pi$
$110$	−35.0819	−3.34493
$111$	0	0
$112$	13.6440	1.28924
$113$	−1.89894	−0.178637	−0.0893185	−	0.996003i	$-0.528469\pi$
$113$	−1.89894	−0.178637	−0.0893185	+	0.996003i	$0.528469\pi$
$114$	0	0
$115$	−2.10328	−0.196132
$116$	−3.07472	−0.285481
$117$	0	0
$118$	−6.15794	−0.566884
$119$	−1.91709	−0.175739
$120$	0	0
$121$	1.32788	0.120716
$122$	−14.8664	−1.34594
$123$	0	0
$124$	4.98754	0.447894
$125$	−24.9138	−2.22836
$126$	0	0
$127$	15.6343	1.38732	0.693661	−	0.720302i	$-0.255997\pi$
$127$	15.6343	1.38732	0.693661	+	0.720302i	$0.255997\pi$
$128$	17.7970	1.57305
$129$	0	0
$130$	−63.0685	−5.53147
$131$	8.95697	0.782574	0.391287	−	0.920269i	$-0.372030\pi$
$131$	8.95697	0.782574	0.391287	+	0.920269i	$0.372030\pi$
$132$	0	0
$133$	2.71666	0.235564
$134$	−6.01683	−0.519775
$135$	0	0
$136$	3.79509	0.325426
$137$	−7.06481	−0.603587	−0.301794	−	0.953373i	$-0.597585\pi$
$137$	−7.06481	−0.603587	−0.301794	+	0.953373i	$0.597585\pi$
$138$	0	0
$139$	−16.9899	−1.44106	−0.720531	−	0.693423i	$-0.756102\pi$
$139$	−16.9899	−1.44106	−0.720531	+	0.693423i	$0.756102\pi$
$140$	−45.5384	−3.84870
$141$	0	0
$142$	−32.9883	−2.76831
$143$	22.1625	1.85332
$144$	0	0
$145$	2.96996	0.246641
$146$	−30.9742	−2.56344
$147$	0	0
$148$	−22.5585	−1.85430
$149$	−1.86219	−0.152557	−0.0762784	−	0.997087i	$-0.524304\pi$
$149$	−1.86219	−0.152557	−0.0762784	+	0.997087i	$0.524304\pi$
$150$	0	0
$151$	−9.18481	−0.747449	−0.373724	−	0.927540i	$-0.621919\pi$
$151$	−9.18481	−0.747449	−0.373724	+	0.927540i	$0.621919\pi$
$152$	−5.37793	−0.436208
$153$	0	0
$154$	23.6850	1.90859
$155$	−4.81760	−0.386959
$156$	0	0
$157$	−16.7838	−1.33950	−0.669748	−	0.742589i	$-0.733598\pi$
$157$	−16.7838	−1.33950	−0.669748	+	0.742589i	$0.733598\pi$
$158$	−14.4430	−1.14903
$159$	0	0
$160$	6.90156	0.545616
$161$	1.42000	0.111912
$162$	0	0
$163$	−7.25627	−0.568355	−0.284178	−	0.958772i	$-0.591721\pi$
$163$	−7.25627	−0.568355	−0.284178	+	0.958772i	$0.591721\pi$
$164$	15.6308	1.22056
$165$	0	0
$166$	−19.3933	−1.50521
$167$	−4.41417	−0.341579	−0.170789	−	0.985308i	$-0.554632\pi$
$167$	−4.41417	−0.341579	−0.170789	+	0.985308i	$0.554632\pi$
$168$	0	0
$169$	26.8426	2.06482
$170$	−7.05092	−0.540781
$171$	0	0
$172$	−15.4573	−1.17861
$173$	8.67610	0.659632	0.329816	−	0.944045i	$-0.393013\pi$
$173$	8.67610	0.659632	0.329816	+	0.944045i	$0.393013\pi$
$174$	0	0
$175$	30.4035	2.29829
$176$	−17.6340	−1.32921
$177$	0	0
$178$	−14.6411	−1.09740
$179$	19.2293	1.43726	0.718631	−	0.695392i	$-0.244769\pi$
$179$	19.2293	1.43726	0.718631	+	0.695392i	$0.244769\pi$
$180$	0	0
$181$	26.6835	1.98337	0.991684	−	0.128696i	$-0.0410790\pi$
$181$	26.6835	1.98337	0.991684	+	0.128696i	$0.0410790\pi$
$182$	42.5798	3.15623
$183$	0	0
$184$	−2.81105	−0.207234
$185$	21.7899	1.60202
$186$	0	0
$187$	2.47771	0.181188
$188$	4.16581	0.303823
$189$	0	0
$190$	9.99168	0.724873
$191$	3.91093	0.282985	0.141493	−	0.989939i	$-0.454810\pi$
$191$	3.91093	0.282985	0.141493	+	0.989939i	$0.454810\pi$
$192$	0	0
$193$	−19.0089	−1.36829	−0.684146	−	0.729345i	$-0.739825\pi$
$193$	−19.0089	−1.36829	−0.684146	+	0.729345i	$0.739825\pi$
$194$	−3.68754	−0.264750
$195$	0	0
$196$	1.58395	0.113139
$197$	−19.8480	−1.41411	−0.707057	−	0.707157i	$-0.749978\pi$
$197$	−19.8480	−1.41411	−0.707057	+	0.707157i	$0.749978\pi$
$198$	0	0
$199$	−9.78540	−0.693669	−0.346834	−	0.937926i	$-0.612743\pi$
$199$	−9.78540	−0.693669	−0.346834	+	0.937926i	$0.612743\pi$
$200$	−60.1871	−4.25587
$201$	0	0
$202$	15.3152	1.07758
$203$	−2.00512	−0.140732
$204$	0	0
$205$	−15.0982	−1.05451
$206$	−3.82241	−0.266320
$207$	0	0
$208$	−31.7016	−2.19811
$209$	−3.51111	−0.242868
$210$	0	0
$211$	−27.4469	−1.88952	−0.944760	−	0.327763i	$-0.893705\pi$
$211$	−27.4469	−1.88952	−0.944760	+	0.327763i	$0.893705\pi$
$212$	9.66944	0.664100
$213$	0	0
$214$	37.1028	2.53630
$215$	14.9307	1.01826
$216$	0	0
$217$	3.25253	0.220796
$218$	1.92999	0.130716
$219$	0	0
$220$	58.8555	3.96804
$221$	4.45431	0.299630
$222$	0	0
$223$	−10.0639	−0.673928	−0.336964	−	0.941517i	$-0.609400\pi$
$223$	−10.0639	−0.673928	−0.336964	+	0.941517i	$0.609400\pi$
$224$	−4.65949	−0.311326
$225$	0	0
$226$	4.71526	0.313654
$227$	6.91197	0.458764	0.229382	−	0.973337i	$-0.426330\pi$
$227$	6.91197	0.458764	0.229382	+	0.973337i	$0.426330\pi$
$228$	0	0
$229$	4.91699	0.324924	0.162462	−	0.986715i	$-0.448057\pi$
$229$	4.91699	0.324924	0.162462	+	0.986715i	$0.448057\pi$
$230$	5.22267	0.344373
$231$	0	0
$232$	3.96937	0.260602
$233$	24.2705	1.59001	0.795007	−	0.606601i	$-0.207467\pi$
$233$	24.2705	1.59001	0.795007	+	0.606601i	$0.207467\pi$
$234$	0	0
$235$	−4.02387	−0.262488
$236$	10.3309	0.672487
$237$	0	0
$238$	4.76033	0.308566
$239$	−2.11822	−0.137016	−0.0685082	−	0.997651i	$-0.521824\pi$
$239$	−2.11822	−0.137016	−0.0685082	+	0.997651i	$0.521824\pi$
$240$	0	0
$241$	13.0676	0.841757	0.420879	−	0.907117i	$-0.361722\pi$
$241$	13.0676	0.841757	0.420879	+	0.907117i	$0.361722\pi$
$242$	−3.29726	−0.211956
$243$	0	0
$244$	24.9408	1.59667
$245$	−1.52998	−0.0977469
$246$	0	0
$247$	−6.31210	−0.401629
$248$	−6.43876	−0.408862
$249$	0	0
$250$	61.8635	3.91259
$251$	−3.85054	−0.243044	−0.121522	−	0.992589i	$-0.538777\pi$
$251$	−3.85054	−0.243044	−0.121522	+	0.992589i	$0.538777\pi$
$252$	0	0
$253$	−1.83526	−0.115382
$254$	−38.8217	−2.43589
$255$	0	0
$256$	−32.6203	−2.03877
$257$	27.0368	1.68651	0.843253	−	0.537516i	$-0.180637\pi$
$257$	27.0368	1.68651	0.843253	+	0.537516i	$0.180637\pi$
$258$	0	0
$259$	−14.7111	−0.914105
$260$	105.808	6.56191
$261$	0	0
$262$	−22.2411	−1.37406
$263$	21.7009	1.33813	0.669067	−	0.743202i	$-0.266694\pi$
$263$	21.7009	1.33813	0.669067	+	0.743202i	$0.266694\pi$
$264$	0	0
$265$	−9.33997	−0.573750
$266$	−6.74574	−0.413608
$267$	0	0
$268$	10.0942	0.616601
$269$	−29.4567	−1.79601	−0.898003	−	0.439989i	$-0.854982\pi$
$269$	−29.4567	−1.79601	−0.898003	+	0.439989i	$0.854982\pi$
$270$	0	0
$271$	32.4950	1.97393	0.986966	−	0.160929i	$-0.0514491\pi$
$271$	32.4950	1.97393	0.986966	+	0.160929i	$0.0514491\pi$
$272$	−3.54416	−0.214897
$273$	0	0
$274$	17.5427	1.05979
$275$	−39.2946	−2.36955
$276$	0	0
$277$	24.5896	1.47744	0.738722	−	0.674010i	$-0.235430\pi$
$277$	24.5896	1.47744	0.738722	+	0.674010i	$0.235430\pi$
$278$	42.1876	2.53024
$279$	0	0
$280$	58.7887	3.51330
$281$	−27.0410	−1.61313	−0.806564	−	0.591146i	$-0.798676\pi$
$281$	−27.0410	−1.61313	−0.806564	+	0.591146i	$0.798676\pi$
$282$	0	0
$283$	−8.44248	−0.501854	−0.250927	−	0.968006i	$-0.580735\pi$
$283$	−8.44248	−0.501854	−0.250927	+	0.968006i	$0.580735\pi$
$284$	55.3431	3.28401
$285$	0	0
$286$	−55.0317	−3.25409
$287$	10.1934	0.601696
$288$	0	0
$289$	−16.5020	−0.970707
$290$	−7.37471	−0.433058
$291$	0	0
$292$	51.9642	3.04098
$293$	4.16502	0.243323	0.121661	−	0.992572i	$-0.461178\pi$
$293$	4.16502	0.243323	0.121661	+	0.992572i	$0.461178\pi$
$294$	0	0
$295$	−9.97892	−0.580995
$296$	29.1224	1.69270
$297$	0	0
$298$	4.62402	0.267862
$299$	−3.29935	−0.190806
$300$	0	0
$301$	−10.0802	−0.581015
$302$	22.8068	1.31239
$303$	0	0
$304$	5.02235	0.288052
$305$	−24.0910	−1.37945
$306$	0	0
$307$	19.5898	1.11805	0.559024	−	0.829152i	$-0.311176\pi$
$307$	19.5898	1.11805	0.559024	+	0.829152i	$0.311176\pi$
$308$	−39.7355	−2.26414
$309$	0	0
$310$	11.9626	0.679430
$311$	−9.56872	−0.542592	−0.271296	−	0.962496i	$-0.587452\pi$
$311$	−9.56872	−0.542592	−0.271296	+	0.962496i	$0.587452\pi$
$312$	0	0
$313$	−17.8963	−1.01156	−0.505778	−	0.862663i	$-0.668795\pi$
$313$	−17.8963	−1.01156	−0.505778	+	0.862663i	$0.668795\pi$
$314$	41.6760	2.35191
$315$	0	0
$316$	24.2305	1.36307
$317$	9.87346	0.554548	0.277274	−	0.960791i	$-0.410569\pi$
$317$	9.87346	0.554548	0.277274	+	0.960791i	$0.410569\pi$
$318$	0	0
$319$	2.59150	0.145096
$320$	23.2812	1.30146
$321$	0	0
$322$	−3.52601	−0.196497
$323$	−0.705679	−0.0392650
$324$	0	0
$325$	−70.6419	−3.91851
$326$	18.0181	0.997930
$327$	0	0
$328$	−20.1789	−1.11420
$329$	2.71666	0.149774
$330$	0	0
$331$	16.4603	0.904741	0.452371	−	0.891830i	$-0.350578\pi$
$331$	16.4603	0.904741	0.452371	+	0.891830i	$0.350578\pi$
$332$	32.5355	1.78562
$333$	0	0
$334$	10.9609	0.599751
$335$	−9.75026	−0.532713
$336$	0	0
$337$	−0.351451	−0.0191448	−0.00957239	−	0.999954i	$-0.503047\pi$
$337$	−0.351451	−0.0191448	−0.00957239	+	0.999954i	$0.503047\pi$
$338$	−66.6530	−3.62544
$339$	0	0
$340$	11.8291	0.641520
$341$	−4.20369	−0.227643
$342$	0	0
$343$	−17.9837	−0.971026
$344$	19.9550	1.07590
$345$	0	0
$346$	−21.5437	−1.15819
$347$	−14.3738	−0.771627	−0.385813	−	0.922577i	$-0.626079\pi$
$347$	−14.3738	−0.771627	−0.385813	+	0.922577i	$0.626079\pi$
$348$	0	0
$349$	24.1872	1.29471	0.647355	−	0.762189i	$-0.275875\pi$
$349$	24.1872	1.29471	0.647355	+	0.762189i	$0.275875\pi$
$350$	−75.4950	−4.03538
$351$	0	0
$352$	6.02210	0.320979
$353$	−35.6412	−1.89699	−0.948496	−	0.316789i	$-0.897395\pi$
$353$	−35.6412	−1.89699	−0.948496	+	0.316789i	$0.897395\pi$
$354$	0	0
$355$	−53.4574	−2.83723
$356$	24.5628	1.30183
$357$	0	0
$358$	−47.7483	−2.52357
$359$	12.9414	0.683019	0.341509	−	0.939878i	$-0.389062\pi$
$359$	12.9414	0.683019	0.341509	+	0.939878i	$0.389062\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−66.2579	−3.48244
$363$	0	0
$364$	−71.4345	−3.74419
$365$	−50.1936	−2.62725
$366$	0	0
$367$	21.7457	1.13511	0.567557	−	0.823334i	$-0.307888\pi$
$367$	21.7457	1.13511	0.567557	+	0.823334i	$0.307888\pi$
$368$	2.62519	0.136848
$369$	0	0
$370$	−54.1066	−2.81287
$371$	6.30575	0.327378
$372$	0	0
$373$	−3.05281	−0.158069	−0.0790343	−	0.996872i	$-0.525184\pi$
$373$	−3.05281	−0.158069	−0.0790343	+	0.996872i	$0.525184\pi$
$374$	−6.15242	−0.318134
$375$	0	0
$376$	−5.37793	−0.277346
$377$	4.65887	0.239944
$378$	0	0
$379$	10.7074	0.550002	0.275001	−	0.961444i	$-0.411322\pi$
$379$	10.7074	0.550002	0.275001	+	0.961444i	$0.411322\pi$
$380$	−16.7627	−0.859906
$381$	0	0
$382$	−9.71126	−0.496871
$383$	−11.8417	−0.605082	−0.302541	−	0.953136i	$-0.597835\pi$
$383$	−11.8417	−0.605082	−0.302541	+	0.953136i	$0.597835\pi$
$384$	0	0
$385$	38.3816	1.95611
$386$	47.2012	2.40248
$387$	0	0
$388$	6.18644	0.314069
$389$	16.8843	0.856066	0.428033	−	0.903763i	$-0.359207\pi$
$389$	16.8843	0.856066	0.428033	+	0.903763i	$0.359207\pi$
$390$	0	0
$391$	−0.368860	−0.0186540
$392$	−2.04483	−0.103280
$393$	0	0
$394$	49.2847	2.48293
$395$	−23.4049	−1.17763
$396$	0	0
$397$	1.76945	0.0888059	0.0444030	−	0.999014i	$-0.485861\pi$
$397$	1.76945	0.0888059	0.0444030	+	0.999014i	$0.485861\pi$
$398$	24.2982	1.21796
$399$	0	0
$400$	56.2076	2.81038
$401$	39.1719	1.95615	0.978074	−	0.208256i	$-0.0667786\pi$
$401$	39.1719	1.95615	0.978074	+	0.208256i	$0.0667786\pi$
$402$	0	0
$403$	−7.55719	−0.376451
$404$	−25.6938	−1.27831
$405$	0	0
$406$	4.97893	0.247100
$407$	19.0132	0.942450
$408$	0	0
$409$	−11.5901	−0.573093	−0.286546	−	0.958066i	$-0.592507\pi$
$409$	−11.5901	−0.573093	−0.286546	+	0.958066i	$0.592507\pi$
$410$	37.4905	1.85153
$411$	0	0
$412$	6.41271	0.315931
$413$	6.73713	0.331513
$414$	0	0
$415$	−31.4269	−1.54268
$416$	10.8262	0.530800
$417$	0	0
$418$	8.71845	0.426433
$419$	9.12758	0.445911	0.222956	−	0.974829i	$-0.428429\pi$
$419$	9.12758	0.445911	0.222956	+	0.974829i	$0.428429\pi$
$420$	0	0
$421$	−23.2545	−1.13336	−0.566678	−	0.823940i	$-0.691772\pi$
$421$	−23.2545	−1.13336	−0.566678	+	0.823940i	$0.691772\pi$
$422$	68.1534	3.31766
$423$	0	0
$424$	−12.4830	−0.606226
$425$	−7.89760	−0.383090
$426$	0	0
$427$	16.2647	0.787104
$428$	−62.2459	−3.00877
$429$	0	0
$430$	−37.0744	−1.78789
$431$	−0.484353	−0.0233305	−0.0116652	−	0.999932i	$-0.503713\pi$
$431$	−0.484353	−0.0233305	−0.0116652	+	0.999932i	$0.503713\pi$
$432$	0	0
$433$	−8.33864	−0.400729	−0.200365	−	0.979721i	$-0.564213\pi$
$433$	−8.33864	−0.400729	−0.200365	+	0.979721i	$0.564213\pi$
$434$	−8.07638	−0.387679
$435$	0	0
$436$	−3.23787	−0.155066
$437$	0.522702	0.0250042
$438$	0	0
$439$	22.1446	1.05690	0.528451	−	0.848964i	$-0.322773\pi$
$439$	22.1446	1.05690	0.528451	+	0.848964i	$0.322773\pi$
$440$	−75.9807	−3.62224
$441$	0	0
$442$	−11.0605	−0.526096
$443$	13.8181	0.656517	0.328259	−	0.944588i	$-0.393538\pi$
$443$	13.8181	0.656517	0.328259	+	0.944588i	$0.393538\pi$
$444$	0	0
$445$	−23.7259	−1.12471
$446$	24.9897	1.18330
$447$	0	0
$448$	−15.7180	−0.742605
$449$	−20.8401	−0.983506	−0.491753	−	0.870735i	$-0.663644\pi$
$449$	−20.8401	−0.983506	−0.491753	+	0.870735i	$0.663644\pi$
$450$	0	0
$451$	−13.1743	−0.620353
$452$	−7.91061	−0.372084
$453$	0	0
$454$	−17.1632	−0.805507
$455$	69.0005	3.23479
$456$	0	0
$457$	−3.32083	−0.155342	−0.0776709	−	0.996979i	$-0.524748\pi$
$457$	−3.32083	−0.155342	−0.0776709	+	0.996979i	$0.524748\pi$
$458$	−12.2094	−0.570507
$459$	0	0
$460$	−8.76187	−0.408524
$461$	14.1421	0.658664	0.329332	−	0.944214i	$-0.393176\pi$
$461$	14.1421	0.658664	0.329332	+	0.944214i	$0.393176\pi$
$462$	0	0
$463$	12.9015	0.599583	0.299791	−	0.954005i	$-0.403083\pi$
$463$	12.9015	0.599583	0.299791	+	0.954005i	$0.403083\pi$
$464$	−3.70692	−0.172090
$465$	0	0
$466$	−60.2662	−2.79178
$467$	−37.6518	−1.74232	−0.871159	−	0.491000i	$-0.836631\pi$
$467$	−37.6518	−1.74232	−0.871159	+	0.491000i	$0.836631\pi$
$468$	0	0
$469$	6.58275	0.303963
$470$	9.99168	0.460882
$471$	0	0
$472$	−13.3369	−0.613881
$473$	13.0281	0.599031
$474$	0	0
$475$	11.1915	0.513501
$476$	−7.98622	−0.366048
$477$	0	0
$478$	5.25977	0.240576
$479$	−3.54099	−0.161792	−0.0808960	−	0.996723i	$-0.525778\pi$
$479$	−3.54099	−0.161792	−0.0808960	+	0.996723i	$0.525778\pi$
$480$	0	0
$481$	34.1810	1.55852
$482$	−32.4482	−1.47797
$483$	0	0
$484$	5.53168	0.251440
$485$	−5.97565	−0.271340
$486$	0	0
$487$	33.3111	1.50947	0.754734	−	0.656031i	$-0.227766\pi$
$487$	33.3111	1.50947	0.754734	+	0.656031i	$0.227766\pi$
$488$	−32.1978	−1.45753
$489$	0	0
$490$	3.79910	0.171626
$491$	6.40800	0.289189	0.144595	−	0.989491i	$-0.453812\pi$
$491$	6.40800	0.289189	0.144595	+	0.989491i	$0.453812\pi$
$492$	0	0
$493$	0.520851	0.0234579
$494$	15.6736	0.705189
$495$	0	0
$496$	6.01303	0.269993
$497$	36.0910	1.61890
$498$	0	0
$499$	32.6169	1.46013	0.730067	−	0.683376i	$-0.239489\pi$
$499$	32.6169	1.46013	0.730067	+	0.683376i	$0.239489\pi$
$500$	−103.786	−4.64145
$501$	0	0
$502$	9.56129	0.426741
$503$	−38.0548	−1.69678	−0.848390	−	0.529372i	$-0.822428\pi$
$503$	−38.0548	−1.69678	−0.848390	+	0.529372i	$0.822428\pi$
$504$	0	0
$505$	24.8183	1.10440
$506$	4.55715	0.202590
$507$	0	0
$508$	65.1296	2.88966
$509$	−8.86920	−0.393121	−0.196560	−	0.980492i	$-0.562977\pi$
$509$	−8.86920	−0.393121	−0.196560	+	0.980492i	$0.562977\pi$
$510$	0	0
$511$	33.8875	1.49910
$512$	45.4056	2.00666
$513$	0	0
$514$	−67.1351	−2.96120
$515$	−6.19421	−0.272949
$516$	0	0
$517$	−3.51111	−0.154418
$518$	36.5293	1.60500
$519$	0	0
$520$	−136.594	−5.99006
$521$	−25.0665	−1.09819	−0.549093	−	0.835762i	$-0.685027\pi$
$521$	−25.0665	−1.09819	−0.549093	+	0.835762i	$0.685027\pi$
$522$	0	0
$523$	8.06425	0.352625	0.176312	−	0.984334i	$-0.443583\pi$
$523$	8.06425	0.352625	0.176312	+	0.984334i	$0.443583\pi$
$524$	37.3130	1.63003
$525$	0	0
$526$	−53.8856	−2.34952
$527$	−0.844877	−0.0368034
$528$	0	0
$529$	−22.7268	−0.988121
$530$	23.1921	1.00740
$531$	0	0
$532$	11.3171	0.490657
$533$	−23.6841	−1.02587
$534$	0	0
$535$	60.1250	2.59943
$536$	−13.0313	−0.562866
$537$	0	0
$538$	73.1440	3.15346
$539$	−1.33502	−0.0575032
$540$	0	0
$541$	−17.2656	−0.742306	−0.371153	−	0.928572i	$-0.621037\pi$
$541$	−17.2656	−0.742306	−0.371153	+	0.928572i	$0.621037\pi$
$542$	−80.6885	−3.46587
$543$	0	0
$544$	1.21035	0.0518933
$545$	3.12755	0.133969
$546$	0	0
$547$	29.6990	1.26984	0.634918	−	0.772579i	$-0.281034\pi$
$547$	29.6990	1.26984	0.634918	+	0.772579i	$0.281034\pi$
$548$	−29.4306	−1.25722
$549$	0	0
$550$	97.5725	4.16051
$551$	−0.738085	−0.0314435
$552$	0	0
$553$	15.8015	0.671949
$554$	−61.0585	−2.59413
$555$	0	0
$556$	−70.7765	−3.00159
$557$	−37.8834	−1.60517	−0.802586	−	0.596537i	$-0.796543\pi$
$557$	−37.8834	−1.60517	−0.802586	+	0.596537i	$0.796543\pi$
$558$	0	0
$559$	23.4212	0.990612
$560$	−54.9016	−2.32002
$561$	0	0
$562$	67.1455	2.83236
$563$	−28.4212	−1.19781	−0.598906	−	0.800819i	$-0.704398\pi$
$563$	−28.4212	−1.19781	−0.598906	+	0.800819i	$0.704398\pi$
$564$	0	0
$565$	7.64107	0.321462
$566$	20.9636	0.881165
$567$	0	0
$568$	−71.4463	−2.99782
$569$	2.23553	0.0937184	0.0468592	−	0.998902i	$-0.485079\pi$
$569$	2.23553	0.0937184	0.0468592	+	0.998902i	$0.485079\pi$
$570$	0	0
$571$	−40.8552	−1.70974	−0.854869	−	0.518844i	$-0.826363\pi$
$571$	−40.8552	−1.70974	−0.854869	+	0.518844i	$0.826363\pi$
$572$	92.3246	3.86028
$573$	0	0
$574$	−25.3112	−1.05647
$575$	5.84982	0.243954
$576$	0	0
$577$	−34.0664	−1.41820	−0.709102	−	0.705106i	$-0.750900\pi$
$577$	−34.0664	−1.41820	−0.709102	+	0.705106i	$0.750900\pi$
$578$	40.9762	1.70439
$579$	0	0
$580$	12.3723	0.513731
$581$	21.2174	0.880246
$582$	0	0
$583$	−8.14978	−0.337530
$584$	−67.0842	−2.77596
$585$	0	0
$586$	−10.3422	−0.427231
$587$	1.25295	0.0517149	0.0258575	−	0.999666i	$-0.491768\pi$
$587$	1.25295	0.0517149	0.0258575	+	0.999666i	$0.491768\pi$
$588$	0	0
$589$	1.19726	0.0493320
$590$	24.7787	1.02012
$591$	0	0
$592$	−27.1968	−1.11778
$593$	−22.2595	−0.914089	−0.457045	−	0.889444i	$-0.651092\pi$
$593$	−22.2595	−0.914089	−0.457045	+	0.889444i	$0.651092\pi$
$594$	0	0
$595$	7.71410	0.316247
$596$	−7.75754	−0.317761
$597$	0	0
$598$	8.19262	0.335021
$599$	18.8669	0.770881	0.385440	−	0.922733i	$-0.374050\pi$
$599$	18.8669	0.770881	0.385440	+	0.922733i	$0.374050\pi$
$600$	0	0
$601$	−34.9178	−1.42433	−0.712163	−	0.702014i	$-0.752284\pi$
$601$	−34.9178	−1.42433	−0.712163	+	0.702014i	$0.752284\pi$
$602$	25.0303	1.02016
$603$	0	0
$604$	−38.2622	−1.55686
$605$	−5.34320	−0.217232
$606$	0	0
$607$	−25.4651	−1.03359	−0.516797	−	0.856108i	$-0.672876\pi$
$607$	−25.4651	−1.03359	−0.516797	+	0.856108i	$0.672876\pi$
$608$	−1.71516	−0.0695588
$609$	0	0
$610$	59.8204	2.42206
$611$	−6.31210	−0.255360
$612$	0	0
$613$	5.32500	0.215075	0.107537	−	0.994201i	$-0.465703\pi$
$613$	5.32500	0.215075	0.107537	+	0.994201i	$0.465703\pi$
$614$	−48.6434	−1.96309
$615$	0	0
$616$	51.2973	2.06683
$617$	−0.330525	−0.0133064	−0.00665321	−	0.999978i	$-0.502118\pi$
$617$	−0.330525	−0.0133064	−0.00665321	+	0.999978i	$0.502118\pi$
$618$	0	0
$619$	36.3272	1.46011	0.730056	−	0.683387i	$-0.239494\pi$
$619$	36.3272	1.46011	0.730056	+	0.683387i	$0.239494\pi$
$620$	−20.0692	−0.805998
$621$	0	0
$622$	23.7601	0.952695
$623$	16.0182	0.641755
$624$	0	0
$625$	44.2922	1.77169
$626$	44.4383	1.77611
$627$	0	0
$628$	−69.9182	−2.79004
$629$	3.82136	0.152368
$630$	0	0
$631$	3.85083	0.153299	0.0766496	−	0.997058i	$-0.475578\pi$
$631$	3.85083	0.153299	0.0766496	+	0.997058i	$0.475578\pi$
$632$	−31.2809	−1.24429
$633$	0	0
$634$	−24.5168	−0.973687
$635$	−62.9104	−2.49652
$636$	0	0
$637$	−2.40003	−0.0950926
$638$	−6.43496	−0.254762
$639$	0	0
$640$	−71.6129	−2.83075
$641$	−45.4303	−1.79439	−0.897194	−	0.441636i	$-0.854398\pi$
$641$	−45.4303	−1.79439	−0.897194	+	0.441636i	$0.854398\pi$
$642$	0	0
$643$	29.6911	1.17090	0.585451	−	0.810708i	$-0.300917\pi$
$643$	29.6911	1.17090	0.585451	+	0.810708i	$0.300917\pi$
$644$	5.91546	0.233102
$645$	0	0
$646$	1.75227	0.0689423
$647$	−20.9434	−0.823370	−0.411685	−	0.911326i	$-0.635060\pi$
$647$	−20.9434	−0.823370	−0.411685	+	0.911326i	$0.635060\pi$
$648$	0	0
$649$	−8.70732	−0.341792
$650$	175.411	6.88019
$651$	0	0
$652$	−30.2283	−1.18383
$653$	−37.9052	−1.48335	−0.741673	−	0.670762i	$-0.765967\pi$
$653$	−37.9052	−1.48335	−0.741673	+	0.670762i	$0.765967\pi$
$654$	0	0
$655$	−36.0417	−1.40826
$656$	18.8447	0.735763
$657$	0	0
$658$	−6.74574	−0.262977
$659$	31.2734	1.21824	0.609120	−	0.793078i	$-0.291523\pi$
$659$	31.2734	1.21824	0.609120	+	0.793078i	$0.291523\pi$
$660$	0	0
$661$	−42.1412	−1.63910	−0.819551	−	0.573006i	$-0.805777\pi$
$661$	−42.1412	−1.63910	−0.819551	+	0.573006i	$0.805777\pi$
$662$	−40.8727	−1.58856
$663$	0	0
$664$	−42.0023	−1.63000
$665$	−10.9315	−0.423904
$666$	0	0
$667$	−0.385799	−0.0149382
$668$	−18.3886	−0.711476
$669$	0	0
$670$	24.2109	0.935349
$671$	−21.0211	−0.811510
$672$	0	0
$673$	−27.8189	−1.07234	−0.536171	−	0.844109i	$-0.680130\pi$
$673$	−27.8189	−1.07234	−0.536171	+	0.844109i	$0.680130\pi$
$674$	0.872691	0.0336148
$675$	0	0
$676$	111.821	4.30081
$677$	31.0310	1.19262	0.596309	−	0.802755i	$-0.296633\pi$
$677$	31.0310	1.19262	0.596309	+	0.802755i	$0.296633\pi$
$678$	0	0
$679$	4.03438	0.154825
$680$	−15.2709	−0.585614
$681$	0	0
$682$	10.4382	0.399700
$683$	−18.3316	−0.701439	−0.350720	−	0.936481i	$-0.614063\pi$
$683$	−18.3316	−0.701439	−0.350720	+	0.936481i	$0.614063\pi$
$684$	0	0
$685$	28.4278	1.08617
$686$	44.6553	1.70495
$687$	0	0
$688$	−18.6356	−0.710474
$689$	−14.6513	−0.558170
$690$	0	0
$691$	−8.78243	−0.334099	−0.167050	−	0.985948i	$-0.553424\pi$
$691$	−8.78243	−0.334099	−0.167050	+	0.985948i	$0.553424\pi$
$692$	36.1430	1.37395
$693$	0	0
$694$	35.6917	1.35484
$695$	68.3649	2.59323
$696$	0	0
$697$	−2.64783	−0.100294
$698$	−60.0593	−2.27328
$699$	0	0
$700$	126.655	4.78711
$701$	2.16668	0.0818345	0.0409172	−	0.999163i	$-0.486972\pi$
$701$	2.16668	0.0818345	0.0409172	+	0.999163i	$0.486972\pi$
$702$	0	0
$703$	−5.41516	−0.204237
$704$	20.3145	0.765632
$705$	0	0
$706$	88.5009	3.33078
$707$	−16.7557	−0.630165
$708$	0	0
$709$	−21.9588	−0.824678	−0.412339	−	0.911030i	$-0.635288\pi$
$709$	−21.9588	−0.824678	−0.412339	+	0.911030i	$0.635288\pi$
$710$	132.740	4.98166
$711$	0	0
$712$	−31.7098	−1.18838
$713$	0.625808	0.0234367
$714$	0	0
$715$	−89.1788	−3.33510
$716$	80.1054	2.99368
$717$	0	0
$718$	−32.1347	−1.19926
$719$	33.9274	1.26528	0.632640	−	0.774446i	$-0.281971\pi$
$719$	33.9274	1.26528	0.632640	+	0.774446i	$0.281971\pi$
$720$	0	0
$721$	4.18193	0.155743
$722$	−2.48310	−0.0924116
$723$	0	0
$724$	111.158	4.13117
$725$	−8.26028	−0.306779
$726$	0	0
$727$	−20.7454	−0.769404	−0.384702	−	0.923041i	$-0.625696\pi$
$727$	−20.7454	−0.769404	−0.384702	+	0.923041i	$0.625696\pi$
$728$	92.2198	3.41789
$729$	0	0
$730$	124.636	4.61299
$731$	2.61844	0.0968465
$732$	0	0
$733$	−2.33208	−0.0861375	−0.0430688	−	0.999072i	$-0.513713\pi$
$733$	−2.33208	−0.0861375	−0.0430688	+	0.999072i	$0.513713\pi$
$734$	−53.9968	−1.99306
$735$	0	0
$736$	−0.896516	−0.0330460
$737$	−8.50779	−0.313388
$738$	0	0
$739$	39.9089	1.46807	0.734036	−	0.679110i	$-0.237634\pi$
$739$	39.9089	1.46807	0.734036	+	0.679110i	$0.237634\pi$
$740$	90.7725	3.33686
$741$	0	0
$742$	−15.6578	−0.574817
$743$	−20.6443	−0.757365	−0.378682	−	0.925527i	$-0.623623\pi$
$743$	−20.6443	−0.757365	−0.378682	+	0.925527i	$0.623623\pi$
$744$	0	0
$745$	7.49321	0.274530
$746$	7.58045	0.277540
$747$	0	0
$748$	10.3217	0.377398
$749$	−40.5926	−1.48322
$750$	0	0
$751$	22.3288	0.814790	0.407395	−	0.913252i	$-0.366437\pi$
$751$	22.3288	0.814790	0.407395	+	0.913252i	$0.366437\pi$
$752$	5.02235	0.183146
$753$	0	0
$754$	−11.5685	−0.421298
$755$	36.9584	1.34506
$756$	0	0
$757$	−2.93386	−0.106633	−0.0533165	−	0.998578i	$-0.516979\pi$
$757$	−2.93386	−0.106633	−0.0533165	+	0.998578i	$0.516979\pi$
$758$	−26.5876	−0.965704
$759$	0	0
$760$	21.6401	0.784968
$761$	−31.8557	−1.15477	−0.577384	−	0.816472i	$-0.695927\pi$
$761$	−31.8557	−1.15477	−0.577384	+	0.816472i	$0.695927\pi$
$762$	0	0
$763$	−2.11152	−0.0764422
$764$	16.2922	0.589431
$765$	0	0
$766$	29.4042	1.06242
$767$	−15.6536	−0.565218
$768$	0	0
$769$	41.8882	1.51053	0.755264	−	0.655421i	$-0.227509\pi$
$769$	41.8882	1.51053	0.755264	+	0.655421i	$0.227509\pi$
$770$	−95.3054	−3.43457
$771$	0	0
$772$	−79.1876	−2.85002
$773$	−16.1073	−0.579338	−0.289669	−	0.957127i	$-0.593545\pi$
$773$	−16.1073	−0.579338	−0.289669	+	0.957127i	$0.593545\pi$
$774$	0	0
$775$	13.3991	0.481309
$776$	−7.98651	−0.286699
$777$	0	0
$778$	−41.9254	−1.50310
$779$	3.75217	0.134436
$780$	0	0
$781$	−46.6454	−1.66910
$782$	0.915917	0.0327531
$783$	0	0
$784$	1.90963	0.0682011
$785$	67.5359	2.41046
$786$	0	0
$787$	−3.25989	−0.116203	−0.0581013	−	0.998311i	$-0.518505\pi$
$787$	−3.25989	−0.116203	−0.0581013	+	0.998311i	$0.518505\pi$
$788$	−82.6831	−2.94546
$789$	0	0
$790$	58.1169	2.06771
$791$	−5.15876	−0.183424
$792$	0	0
$793$	−37.7907	−1.34199
$794$	−4.39372	−0.155927
$795$	0	0
$796$	−40.7641	−1.44485
$797$	49.6445	1.75850	0.879249	−	0.476362i	$-0.158045\pi$
$797$	49.6445	1.75850	0.879249	+	0.476362i	$0.158045\pi$
$798$	0	0
$799$	−0.705679	−0.0249651
$800$	−19.1952	−0.678652
$801$	0	0
$802$	−97.2678	−3.43465
$803$	−43.7975	−1.54558
$804$	0	0
$805$	−5.71390	−0.201388
$806$	18.7653	0.660980
$807$	0	0
$808$	33.1699	1.16691
$809$	9.78574	0.344048	0.172024	−	0.985093i	$-0.444969\pi$
$809$	9.78574	0.344048	0.172024	+	0.985093i	$0.444969\pi$
$810$	0	0
$811$	−6.11784	−0.214826	−0.107413	−	0.994214i	$-0.534257\pi$
$811$	−6.11784	−0.214826	−0.107413	+	0.994214i	$0.534257\pi$
$812$	−8.35297	−0.293132
$813$	0	0
$814$	−47.2118	−1.65477
$815$	29.1983	1.02277
$816$	0	0
$817$	−3.71053	−0.129815
$818$	28.7794	1.00625
$819$	0	0
$820$	−62.8964	−2.19644
$821$	32.6120	1.13817	0.569084	−	0.822279i	$-0.307298\pi$
$821$	32.6120	1.13817	0.569084	+	0.822279i	$0.307298\pi$
$822$	0	0
$823$	−35.1621	−1.22567	−0.612837	−	0.790209i	$-0.709972\pi$
$823$	−35.1621	−1.22567	−0.612837	+	0.790209i	$0.709972\pi$
$824$	−8.27861	−0.288399
$825$	0	0
$826$	−16.7290	−0.582076
$827$	−14.9603	−0.520220	−0.260110	−	0.965579i	$-0.583759\pi$
$827$	−14.9603	−0.520220	−0.260110	+	0.965579i	$0.583759\pi$
$828$	0	0
$829$	−48.1491	−1.67229	−0.836143	−	0.548511i	$-0.815195\pi$
$829$	−48.1491	−1.67229	−0.836143	+	0.548511i	$0.815195\pi$
$830$	78.0362	2.70868
$831$	0	0
$832$	36.5204	1.26612
$833$	−0.268318	−0.00929666
$834$	0	0
$835$	17.7620	0.614681
$836$	−14.6266	−0.505872
$837$	0	0
$838$	−22.6647	−0.782940
$839$	48.7780	1.68400	0.842002	−	0.539474i	$-0.181377\pi$
$839$	48.7780	1.68400	0.842002	+	0.539474i	$0.181377\pi$
$840$	0	0
$841$	−28.4552	−0.981215
$842$	57.7434	1.98997
$843$	0	0
$844$	−114.338	−3.93569
$845$	−108.011	−3.71569
$846$	0	0
$847$	3.60739	0.123951
$848$	11.6576	0.400323
$849$	0	0
$850$	19.6106	0.672637
$851$	−2.83051	−0.0970288
$852$	0	0
$853$	−14.7844	−0.506207	−0.253103	−	0.967439i	$-0.581451\pi$
$853$	−14.7844	−0.506207	−0.253103	+	0.967439i	$0.581451\pi$
$854$	−40.3869	−1.38201
$855$	0	0
$856$	80.3576	2.74657
$857$	15.6506	0.534613	0.267306	−	0.963612i	$-0.413866\pi$
$857$	15.6506	0.534613	0.267306	+	0.963612i	$0.413866\pi$
$858$	0	0
$859$	−21.9538	−0.749054	−0.374527	−	0.927216i	$-0.622195\pi$
$859$	−21.9538	−0.749054	−0.374527	+	0.927216i	$0.622195\pi$
$860$	62.1983	2.12094
$861$	0	0
$862$	1.20270	0.0409641
$863$	−40.2100	−1.36876	−0.684382	−	0.729124i	$-0.739928\pi$
$863$	−40.2100	−1.36876	−0.684382	+	0.729124i	$0.739928\pi$
$864$	0	0
$865$	−34.9115	−1.18703
$866$	20.7057	0.703609
$867$	0	0
$868$	13.5494	0.459898
$869$	−20.4225	−0.692784
$870$	0	0
$871$	−15.2949	−0.518247
$872$	4.18000	0.141553
$873$	0	0
$874$	−1.29792	−0.0439029
$875$	−67.6822	−2.28808
$876$	0	0
$877$	24.8198	0.838107	0.419053	−	0.907962i	$-0.362362\pi$
$877$	24.8198	0.838107	0.419053	+	0.907962i	$0.362362\pi$
$878$	−54.9872	−1.85573
$879$	0	0
$880$	70.9569	2.39196
$881$	−35.4747	−1.19517	−0.597587	−	0.801804i	$-0.703874\pi$
$881$	−35.4747	−1.19517	−0.597587	+	0.801804i	$0.703874\pi$
$882$	0	0
$883$	−15.8821	−0.534476	−0.267238	−	0.963631i	$-0.586111\pi$
$883$	−15.8821	−0.534476	−0.267238	+	0.963631i	$0.586111\pi$
$884$	18.5558	0.624100
$885$	0	0
$886$	−34.3118	−1.15273
$887$	−40.8498	−1.37160	−0.685801	−	0.727789i	$-0.740548\pi$
$887$	−40.8498	−1.37160	−0.685801	+	0.727789i	$0.740548\pi$
$888$	0	0
$889$	42.4731	1.42450
$890$	58.9138	1.97479
$891$	0	0
$892$	−41.9243	−1.40373
$893$	1.00000	0.0334637
$894$	0	0
$895$	−77.3760	−2.58639
$896$	48.3484	1.61521
$897$	0	0
$898$	51.7482	1.72686
$899$	−0.883676	−0.0294723
$900$	0	0
$901$	−1.63798	−0.0545691
$902$	32.7131	1.08923
$903$	0	0
$904$	10.2124	0.339658
$905$	−107.371	−3.56913
$906$	0	0
$907$	−17.5869	−0.583962	−0.291981	−	0.956424i	$-0.594314\pi$
$907$	−17.5869	−0.583962	−0.291981	+	0.956424i	$0.594314\pi$
$908$	28.7940	0.955561
$909$	0	0
$910$	−171.335	−5.67972
$911$	10.3080	0.341520	0.170760	−	0.985313i	$-0.445378\pi$
$911$	10.3080	0.341520	0.170760	+	0.985313i	$0.445378\pi$
$912$	0	0
$913$	−27.4222	−0.907541
$914$	8.24596	0.272752
$915$	0	0
$916$	20.4832	0.676785
$917$	24.3330	0.803547
$918$	0	0
$919$	−29.4560	−0.971663	−0.485832	−	0.874052i	$-0.661483\pi$
$919$	−29.4560	−0.971663	−0.485832	+	0.874052i	$0.661483\pi$
$920$	11.3113	0.372923
$921$	0	0
$922$	−35.1164	−1.15650
$923$	−83.8568	−2.76018
$924$	0	0
$925$	−60.6038	−1.99264
$926$	−32.0357	−1.05276
$927$	0	0
$928$	1.26593	0.0415563
$929$	−36.7989	−1.20733	−0.603666	−	0.797237i	$-0.706294\pi$
$929$	−36.7989	−1.20733	−0.603666	+	0.797237i	$0.706294\pi$
$930$	0	0
$931$	0.380227	0.0124614
$932$	101.106	3.31185
$933$	0	0
$934$	93.4934	3.05920
$935$	−9.96999	−0.326054
$936$	0	0
$937$	−51.6675	−1.68790	−0.843952	−	0.536419i	$-0.819777\pi$
$937$	−51.6675	−1.68790	−0.843952	+	0.536419i	$0.819777\pi$
$938$	−16.3457	−0.533705
$939$	0	0
$940$	−16.7627	−0.546738
$941$	42.7326	1.39304	0.696521	−	0.717536i	$-0.254730\pi$
$941$	42.7326	1.39304	0.696521	+	0.717536i	$0.254730\pi$
$942$	0	0
$943$	1.96127	0.0638677
$944$	12.4551	0.405379
$945$	0	0
$946$	−32.3500	−1.05179
$947$	−26.9514	−0.875804	−0.437902	−	0.899023i	$-0.644278\pi$
$947$	−26.9514	−0.875804	−0.437902	+	0.899023i	$0.644278\pi$
$948$	0	0
$949$	−78.7370	−2.55591
$950$	−27.7897	−0.901616
$951$	0	0
$952$	10.3100	0.334148
$953$	16.6470	0.539248	0.269624	−	0.962966i	$-0.413101\pi$
$953$	16.6470	0.539248	0.269624	+	0.962966i	$0.413101\pi$
$954$	0	0
$955$	−15.7371	−0.509240
$956$	−8.82412	−0.285392
$957$	0	0
$958$	8.79266	0.284078
$959$	−19.1927	−0.619763
$960$	0	0
$961$	−29.5666	−0.953761
$962$	−84.8751	−2.73648
$963$	0	0
$964$	54.4371	1.75330
$965$	76.4894	2.46228
$966$	0	0
$967$	−50.1680	−1.61329	−0.806647	−	0.591033i	$-0.798720\pi$
$967$	−50.1680	−1.61329	−0.806647	+	0.591033i	$0.798720\pi$
$968$	−7.14123	−0.229528
$969$	0	0
$970$	14.8382	0.476425
$971$	−40.0861	−1.28643	−0.643213	−	0.765687i	$-0.722399\pi$
$971$	−40.0861	−1.28643	−0.643213	+	0.765687i	$0.722399\pi$
$972$	0	0
$973$	−46.1556	−1.47968
$974$	−82.7149	−2.65036
$975$	0	0
$976$	30.0689	0.962483
$977$	19.1271	0.611931	0.305966	−	0.952043i	$-0.401021\pi$
$977$	19.1271	0.611931	0.305966	+	0.952043i	$0.401021\pi$
$978$	0	0
$979$	−20.7025	−0.661654
$980$	−6.37361	−0.203598
$981$	0	0
$982$	−15.9117	−0.507764
$983$	25.0065	0.797584	0.398792	−	0.917041i	$-0.369430\pi$
$983$	25.0065	0.797584	0.398792	+	0.917041i	$0.369430\pi$
$984$	0	0
$985$	79.8658	2.54474
$986$	−1.29333	−0.0411879
$987$	0	0
$988$	−26.2950	−0.836555
$989$	−1.93950	−0.0616725
$990$	0	0
$991$	22.1003	0.702040	0.351020	−	0.936368i	$-0.385835\pi$
$991$	22.1003	0.702040	0.351020	+	0.936368i	$0.385835\pi$
$992$	−2.05348	−0.0651981
$993$	0	0
$994$	−89.6178	−2.84250
$995$	39.3752	1.24828
$996$	0	0
$997$	−54.5238	−1.72679	−0.863393	−	0.504533i	$-0.831665\pi$
$997$	−54.5238	−1.72679	−0.863393	+	0.504533i	$0.831665\pi$
$998$	−80.9912	−2.56373
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.u.1.4	✓	34
3.2	odd	2		8037.2.a.x.1.31	yes	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.u.1.4	✓	34	1.1	even	1	trivial
8037.2.a.x.1.31	yes	34	3.2	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-2.48310 q^{2} +4.16581 q^{4} -4.02387 q^{5} +2.71666 q^{7} -5.37793 q^{8} +O(q^{10})\)	\(q-2.48310 q^{2} +4.16581 q^{4} -4.02387 q^{5} +2.71666 q^{7} -5.37793 q^{8} +9.99168 q^{10} -3.51111 q^{11} -6.31210 q^{13} -6.74574 q^{14} +5.02235 q^{16} -0.705679 q^{17} +1.00000 q^{19} -16.7627 q^{20} +8.71845 q^{22} +0.522702 q^{23} +11.1915 q^{25} +15.6736 q^{26} +11.3171 q^{28} -0.738085 q^{29} +1.19726 q^{31} -1.71516 q^{32} +1.75227 q^{34} -10.9315 q^{35} -5.41516 q^{37} -2.48310 q^{38} +21.6401 q^{40} +3.75217 q^{41} -3.71053 q^{43} -14.6266 q^{44} -1.29792 q^{46} +1.00000 q^{47} +0.380227 q^{49} -27.7897 q^{50} -26.2950 q^{52} +2.32114 q^{53} +14.1282 q^{55} -14.6100 q^{56} +1.83274 q^{58} +2.47993 q^{59} +5.98702 q^{61} -2.97291 q^{62} -5.78579 q^{64} +25.3990 q^{65} +2.42311 q^{67} -2.93972 q^{68} +27.1440 q^{70} +13.2851 q^{71} +12.4740 q^{73} +13.4464 q^{74} +4.16581 q^{76} -9.53848 q^{77} +5.81653 q^{79} -20.2093 q^{80} -9.31704 q^{82} +7.81012 q^{83} +2.83956 q^{85} +9.21363 q^{86} +18.8825 q^{88} +5.89628 q^{89} -17.1478 q^{91} +2.17748 q^{92} -2.48310 q^{94} -4.02387 q^{95} +1.48505 q^{97} -0.944143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 5 q^{2} + 31 q^{4} - 14 q^{5} - 15 q^{8}+O(q^{10}) \) 34 * q - 5 * q^2 + 31 * q^4 - 14 * q^5 - 15 * q^8	\( 34 q - 5 q^{2} + 31 q^{4} - 14 q^{5} - 15 q^{8} - 18 q^{11} - 6 q^{13} - 12 q^{14} + 21 q^{16} - 36 q^{17} + 34 q^{19} - 40 q^{20} + 12 q^{22} - 38 q^{23} + 32 q^{25} - 15 q^{26} + 28 q^{28} - 14 q^{29} - 6 q^{31} - 35 q^{32} + 10 q^{34} - 46 q^{35} - 2 q^{37} - 5 q^{38} + 31 q^{40} - 18 q^{41} - 6 q^{43} - 42 q^{44} - 14 q^{46} + 34 q^{47} + 44 q^{49} - 9 q^{50} + 2 q^{52} - 32 q^{53} + 8 q^{55} + 4 q^{56} + 8 q^{58} - 62 q^{59} - 10 q^{61} - 30 q^{62} - 37 q^{64} - 8 q^{65} - 92 q^{68} - 62 q^{70} - 4 q^{71} - 8 q^{73} - 34 q^{74} + 31 q^{76} - 52 q^{77} + 40 q^{79} - 48 q^{80} - 2 q^{82} - 110 q^{83} - 12 q^{85} - 16 q^{86} - 44 q^{88} - 2 q^{89} - 28 q^{91} - 60 q^{92} - 5 q^{94} - 14 q^{95} + 2 q^{97} - 45 q^{98}+O(q^{100}) \) 34 * q - 5 * q^2 + 31 * q^4 - 14 * q^5 - 15 * q^8 - 18 * q^11 - 6 * q^13 - 12 * q^14 + 21 * q^16 - 36 * q^17 + 34 * q^19 - 40 * q^20 + 12 * q^22 - 38 * q^23 + 32 * q^25 - 15 * q^26 + 28 * q^28 - 14 * q^29 - 6 * q^31 - 35 * q^32 + 10 * q^34 - 46 * q^35 - 2 * q^37 - 5 * q^38 + 31 * q^40 - 18 * q^41 - 6 * q^43 - 42 * q^44 - 14 * q^46 + 34 * q^47 + 44 * q^49 - 9 * q^50 + 2 * q^52 - 32 * q^53 + 8 * q^55 + 4 * q^56 + 8 * q^58 - 62 * q^59 - 10 * q^61 - 30 * q^62 - 37 * q^64 - 8 * q^65 - 92 * q^68 - 62 * q^70 - 4 * q^71 - 8 * q^73 - 34 * q^74 + 31 * q^76 - 52 * q^77 + 40 * q^79 - 48 * q^80 - 2 * q^82 - 110 * q^83 - 12 * q^85 - 16 * q^86 - 44 * q^88 - 2 * q^89 - 28 * q^91 - 60 * q^92 - 5 * q^94 - 14 * q^95 + 2 * q^97 - 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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