LMFDB - Embedded newform 8037.2.a.q.1.8

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:	$0$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.26887 q^{2} -0.389958 q^{4} +0.358150 q^{5} +4.13444 q^{7} +3.03256 q^{8} +O(q^{10})$	\(q-1.26887 q^{2} -0.389958 q^{4} +0.358150 q^{5} +4.13444 q^{7} +3.03256 q^{8} -0.454448 q^{10} +6.43996 q^{11} -0.109695 q^{13} -5.24608 q^{14} -3.06802 q^{16} -5.26232 q^{17} -1.00000 q^{19} -0.139664 q^{20} -8.17150 q^{22} -2.78729 q^{23} -4.87173 q^{25} +0.139189 q^{26} -1.61226 q^{28} -6.08267 q^{29} +8.09293 q^{31} -2.17219 q^{32} +6.67722 q^{34} +1.48075 q^{35} -0.451671 q^{37} +1.26887 q^{38} +1.08611 q^{40} -5.75341 q^{41} +2.77320 q^{43} -2.51132 q^{44} +3.53672 q^{46} -1.00000 q^{47} +10.0936 q^{49} +6.18161 q^{50} +0.0427764 q^{52} +7.99478 q^{53} +2.30648 q^{55} +12.5379 q^{56} +7.71814 q^{58} +3.03588 q^{59} +6.50612 q^{61} -10.2689 q^{62} +8.89226 q^{64} -0.0392872 q^{65} -1.20052 q^{67} +2.05209 q^{68} -1.87889 q^{70} +7.47336 q^{71} +8.53409 q^{73} +0.573114 q^{74} +0.389958 q^{76} +26.6256 q^{77} -5.59419 q^{79} -1.09881 q^{80} +7.30035 q^{82} -14.6081 q^{83} -1.88470 q^{85} -3.51885 q^{86} +19.5296 q^{88} -0.682438 q^{89} -0.453526 q^{91} +1.08693 q^{92} +1.26887 q^{94} -0.358150 q^{95} +7.67347 q^{97} -12.8075 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8	$ 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8} - 5 q^{10} - 12 q^{11} + 9 q^{13} - 3 q^{14} + 60 q^{16} - 16 q^{17} - 23 q^{19} - 25 q^{20} + 3 q^{22} - 12 q^{23} + 54 q^{25} - 5 q^{26} + 8 q^{28} - 27 q^{29} + 10 q^{31} + 34 q^{32} + 6 q^{35} + 15 q^{37} + 2 q^{38} + 3 q^{40} - 10 q^{41} + 24 q^{43} - 39 q^{44} + 43 q^{46} - 23 q^{47} + 78 q^{49} + 32 q^{50} + 38 q^{52} + 2 q^{53} + 5 q^{55} - 58 q^{56} - 11 q^{58} + 51 q^{59} + 48 q^{61} + 22 q^{62} + 125 q^{64} - 15 q^{65} + 26 q^{67} - 26 q^{68} + 86 q^{70} - 24 q^{71} + 53 q^{73} - 26 q^{74} - 30 q^{76} - 18 q^{77} + 29 q^{79} - 5 q^{80} + 47 q^{82} + 22 q^{83} - 5 q^{85} + 28 q^{86} + 62 q^{88} - 38 q^{89} + 15 q^{91} - 15 q^{92} + 2 q^{94} + 9 q^{95} + 33 q^{97} + 30 q^{98}+O(q^{100}) $ 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8 - 5 * q^10 - 12 * q^11 + 9 * q^13 - 3 * q^14 + 60 * q^16 - 16 * q^17 - 23 * q^19 - 25 * q^20 + 3 * q^22 - 12 * q^23 + 54 * q^25 - 5 * q^26 + 8 * q^28 - 27 * q^29 + 10 * q^31 + 34 * q^32 + 6 * q^35 + 15 * q^37 + 2 * q^38 + 3 * q^40 - 10 * q^41 + 24 * q^43 - 39 * q^44 + 43 * q^46 - 23 * q^47 + 78 * q^49 + 32 * q^50 + 38 * q^52 + 2 * q^53 + 5 * q^55 - 58 * q^56 - 11 * q^58 + 51 * q^59 + 48 * q^61 + 22 * q^62 + 125 * q^64 - 15 * q^65 + 26 * q^67 - 26 * q^68 + 86 * q^70 - 24 * q^71 + 53 * q^73 - 26 * q^74 - 30 * q^76 - 18 * q^77 + 29 * q^79 - 5 * q^80 + 47 * q^82 + 22 * q^83 - 5 * q^85 + 28 * q^86 + 62 * q^88 - 38 * q^89 + 15 * q^91 - 15 * q^92 + 2 * q^94 + 9 * q^95 + 33 * q^97 + 30 * 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 $

$2$	−1.26887	−0.897230	−0.448615	−	0.893725i	$-0.648082\pi$
$2$	−1.26887	−0.897230	−0.448615	+	0.893725i	$0.648082\pi$
$3$	0	0
$3$	0	0
$4$	−0.389958	−0.194979
$5$	0.358150	0.160170	0.0800849	−	0.996788i	$-0.474481\pi$
$5$	0.358150	0.160170	0.0800849	+	0.996788i	$0.474481\pi$
$6$	0	0
$7$	4.13444	1.56267	0.781335	−	0.624111i	$-0.214539\pi$
$7$	4.13444	1.56267	0.781335	+	0.624111i	$0.214539\pi$
$8$	3.03256	1.07217
$9$	0	0
$10$	−0.454448	−0.143709
$11$	6.43996	1.94172	0.970861	−	0.239643i	$-0.0770305\pi$
$11$	6.43996	1.94172	0.970861	+	0.239643i	$0.0770305\pi$
$12$	0	0
$13$	−0.109695	−0.0304238	−0.0152119	−	0.999884i	$-0.504842\pi$
$13$	−0.109695	−0.0304238	−0.0152119	+	0.999884i	$0.504842\pi$
$14$	−5.24608	−1.40207
$15$	0	0
$16$	−3.06802	−0.767004
$17$	−5.26232	−1.27630	−0.638150	−	0.769912i	$-0.720300\pi$
$17$	−5.26232	−1.27630	−0.638150	+	0.769912i	$0.720300\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.139664	−0.0312298
$21$	0	0
$22$	−8.17150	−1.74217
$23$	−2.78729	−0.581190	−0.290595	−	0.956846i	$-0.593853\pi$
$23$	−2.78729	−0.581190	−0.290595	+	0.956846i	$0.593853\pi$
$24$	0	0
$25$	−4.87173	−0.974346
$26$	0.139189	0.0272972
$27$	0	0
$28$	−1.61226	−0.304688
$29$	−6.08267	−1.12952	−0.564761	−	0.825254i	$-0.691032\pi$
$29$	−6.08267	−1.12952	−0.564761	+	0.825254i	$0.691032\pi$
$30$	0	0
$31$	8.09293	1.45353	0.726766	−	0.686885i	$-0.241022\pi$
$31$	8.09293	1.45353	0.726766	+	0.686885i	$0.241022\pi$
$32$	−2.17219	−0.383992
$33$	0	0
$34$	6.67722	1.14513
$35$	1.48075	0.250293
$36$	0	0
$37$	−0.451671	−0.0742543	−0.0371272	−	0.999311i	$-0.511821\pi$
$37$	−0.451671	−0.0742543	−0.0371272	+	0.999311i	$0.511821\pi$
$38$	1.26887	0.205839
$39$	0	0
$40$	1.08611	0.171729
$41$	−5.75341	−0.898532	−0.449266	−	0.893398i	$-0.648314\pi$
$41$	−5.75341	−0.898532	−0.449266	+	0.893398i	$0.648314\pi$
$42$	0	0
$43$	2.77320	0.422910	0.211455	−	0.977388i	$-0.432180\pi$
$43$	2.77320	0.422910	0.211455	+	0.977388i	$0.432180\pi$
$44$	−2.51132	−0.378595
$45$	0	0
$46$	3.53672	0.521461
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	10.0936	1.44194
$50$	6.18161	0.874212
$51$	0	0
$52$	0.0427764	0.00593202
$53$	7.99478	1.09817	0.549084	−	0.835767i	$-0.314977\pi$
$53$	7.99478	1.09817	0.549084	+	0.835767i	$0.314977\pi$
$54$	0	0
$55$	2.30648	0.311005
$56$	12.5379	1.67545
$57$	0	0
$58$	7.71814	1.01344
$59$	3.03588	0.395238	0.197619	−	0.980279i	$-0.436679\pi$
$59$	3.03588	0.395238	0.197619	+	0.980279i	$0.436679\pi$
$60$	0	0
$61$	6.50612	0.833024	0.416512	−	0.909130i	$-0.363252\pi$
$61$	6.50612	0.833024	0.416512	+	0.909130i	$0.363252\pi$
$62$	−10.2689	−1.30415
$63$	0	0
$64$	8.89226	1.11153
$65$	−0.0392872	−0.00487298
$66$	0	0
$67$	−1.20052	−0.146667	−0.0733334	−	0.997307i	$-0.523364\pi$
$67$	−1.20052	−0.146667	−0.0733334	+	0.997307i	$0.523364\pi$
$68$	2.05209	0.248852
$69$	0	0
$70$	−1.87889	−0.224570
$71$	7.47336	0.886925	0.443463	−	0.896293i	$-0.353750\pi$
$71$	7.47336	0.886925	0.443463	+	0.896293i	$0.353750\pi$
$72$	0	0
$73$	8.53409	0.998839	0.499420	−	0.866360i	$-0.333547\pi$
$73$	8.53409	0.998839	0.499420	+	0.866360i	$0.333547\pi$
$74$	0.573114	0.0666232
$75$	0	0
$76$	0.389958	0.0447313
$77$	26.6256	3.03427
$78$	0	0
$79$	−5.59419	−0.629396	−0.314698	−	0.949192i	$-0.601903\pi$
$79$	−5.59419	−0.629396	−0.314698	+	0.949192i	$0.601903\pi$
$80$	−1.09881	−0.122851
$81$	0	0
$82$	7.30035	0.806189
$83$	−14.6081	−1.60345	−0.801725	−	0.597693i	$-0.796084\pi$
$83$	−14.6081	−1.60345	−0.801725	+	0.597693i	$0.796084\pi$
$84$	0	0
$85$	−1.88470	−0.204425
$86$	−3.51885	−0.379447
$87$	0	0
$88$	19.5296	2.08186
$89$	−0.682438	−0.0723382	−0.0361691	−	0.999346i	$-0.511515\pi$
$89$	−0.682438	−0.0723382	−0.0361691	+	0.999346i	$0.511515\pi$
$90$	0	0
$91$	−0.453526	−0.0475424
$92$	1.08693	0.113320
$93$	0	0
$94$	1.26887	0.130874
$95$	−0.358150	−0.0367455
$96$	0	0
$97$	7.67347	0.779123	0.389562	−	0.921000i	$-0.372627\pi$
$97$	7.67347	0.779123	0.389562	+	0.921000i	$0.372627\pi$
$98$	−12.8075	−1.29375
$99$	0	0
$100$	1.89977	0.189977
$101$	0.757294	0.0753536	0.0376768	−	0.999290i	$-0.488004\pi$
$101$	0.757294	0.0753536	0.0376768	+	0.999290i	$0.488004\pi$
$102$	0	0
$103$	17.9084	1.76457	0.882284	−	0.470718i	$-0.156005\pi$
$103$	17.9084	1.76457	0.882284	+	0.470718i	$0.156005\pi$
$104$	−0.332655	−0.0326195
$105$	0	0
$106$	−10.1444	−0.985308
$107$	7.47574	0.722707	0.361353	−	0.932429i	$-0.382315\pi$
$107$	7.47574	0.722707	0.361353	+	0.932429i	$0.382315\pi$
$108$	0	0
$109$	14.6858	1.40664	0.703322	−	0.710872i	$-0.251699\pi$
$109$	14.6858	1.40664	0.703322	+	0.710872i	$0.251699\pi$
$110$	−2.92663	−0.279043
$111$	0	0
$112$	−12.6845	−1.19857
$113$	−13.2608	−1.24747	−0.623736	−	0.781635i	$-0.714386\pi$
$113$	−13.2608	−1.24747	−0.623736	+	0.781635i	$0.714386\pi$
$114$	0	0
$115$	−0.998269	−0.0930890
$116$	2.37199	0.220233
$117$	0	0
$118$	−3.85215	−0.354619
$119$	−21.7567	−1.99444
$120$	0	0
$121$	30.4731	2.77028
$122$	−8.25545	−0.747413
$123$	0	0
$124$	−3.15590	−0.283409
$125$	−3.53556	−0.316230
$126$	0	0
$127$	17.0520	1.51312	0.756559	−	0.653926i	$-0.226879\pi$
$127$	17.0520	1.51312	0.756559	+	0.653926i	$0.226879\pi$
$128$	−6.93879	−0.613308
$129$	0	0
$130$	0.0498505	0.00437218
$131$	−0.515771	−0.0450631	−0.0225316	−	0.999746i	$-0.507173\pi$
$131$	−0.515771	−0.0450631	−0.0225316	+	0.999746i	$0.507173\pi$
$132$	0	0
$133$	−4.13444	−0.358501
$134$	1.52331	0.131594
$135$	0	0
$136$	−15.9583	−1.36841
$137$	3.56747	0.304789	0.152395	−	0.988320i	$-0.451302\pi$
$137$	3.56747	0.304789	0.152395	+	0.988320i	$0.451302\pi$
$138$	0	0
$139$	−5.66955	−0.480885	−0.240442	−	0.970663i	$-0.577293\pi$
$139$	−5.66955	−0.480885	−0.240442	+	0.970663i	$0.577293\pi$
$140$	−0.577431	−0.0488019
$141$	0	0
$142$	−9.48276	−0.795775
$143$	−0.706430	−0.0590746
$144$	0	0
$145$	−2.17851	−0.180915
$146$	−10.8287	−0.896188
$147$	0	0
$148$	0.176133	0.0144780
$149$	−0.473697	−0.0388068	−0.0194034	−	0.999812i	$-0.506177\pi$
$149$	−0.473697	−0.0388068	−0.0194034	+	0.999812i	$0.506177\pi$
$150$	0	0
$151$	−9.32366	−0.758749	−0.379374	−	0.925243i	$-0.623861\pi$
$151$	−9.32366	−0.758749	−0.379374	+	0.925243i	$0.623861\pi$
$152$	−3.03256	−0.245973
$153$	0	0
$154$	−33.7846	−2.72244
$155$	2.89849	0.232812
$156$	0	0
$157$	18.1068	1.44508	0.722542	−	0.691327i	$-0.242974\pi$
$157$	18.1068	1.44508	0.722542	+	0.691327i	$0.242974\pi$
$158$	7.09833	0.564713
$159$	0	0
$160$	−0.777970	−0.0615039
$161$	−11.5239	−0.908208
$162$	0	0
$163$	−2.21127	−0.173200	−0.0866002	−	0.996243i	$-0.527600\pi$
$163$	−2.21127	−0.173200	−0.0866002	+	0.996243i	$0.527600\pi$
$164$	2.24359	0.175195
$165$	0	0
$166$	18.5359	1.43866
$167$	10.8221	0.837437	0.418719	−	0.908116i	$-0.362479\pi$
$167$	10.8221	0.837437	0.418719	+	0.908116i	$0.362479\pi$
$168$	0	0
$169$	−12.9880	−0.999074
$170$	2.39145	0.183416
$171$	0	0
$172$	−1.08143	−0.0824586
$173$	−11.7142	−0.890616	−0.445308	−	0.895377i	$-0.646906\pi$
$173$	−11.7142	−0.890616	−0.445308	+	0.895377i	$0.646906\pi$
$174$	0	0
$175$	−20.1419	−1.52258
$176$	−19.7579	−1.48931
$177$	0	0
$178$	0.865927	0.0649040
$179$	0.143844	0.0107514	0.00537572	−	0.999986i	$-0.498289\pi$
$179$	0.143844	0.0107514	0.00537572	+	0.999986i	$0.498289\pi$
$180$	0	0
$181$	13.7912	1.02509	0.512545	−	0.858661i	$-0.328703\pi$
$181$	13.7912	1.02509	0.512545	+	0.858661i	$0.328703\pi$
$182$	0.575467	0.0426565
$183$	0	0
$184$	−8.45261	−0.623135
$185$	−0.161766	−0.0118933
$186$	0	0
$187$	−33.8892	−2.47822
$188$	0.389958	0.0284406
$189$	0	0
$190$	0.454448	0.0329691
$191$	11.9550	0.865032	0.432516	−	0.901626i	$-0.357626\pi$
$191$	11.9550	0.865032	0.432516	+	0.901626i	$0.357626\pi$
$192$	0	0
$193$	−16.9666	−1.22128	−0.610641	−	0.791908i	$-0.709088\pi$
$193$	−16.9666	−1.22128	−0.610641	+	0.791908i	$0.709088\pi$
$194$	−9.73667	−0.699052
$195$	0	0
$196$	−3.93608	−0.281148
$197$	26.8153	1.91051	0.955256	−	0.295780i	$-0.0955794\pi$
$197$	26.8153	1.91051	0.955256	+	0.295780i	$0.0955794\pi$
$198$	0	0
$199$	22.5494	1.59848	0.799241	−	0.601011i	$-0.205235\pi$
$199$	22.5494	1.59848	0.799241	+	0.601011i	$0.205235\pi$
$200$	−14.7738	−1.04466
$201$	0	0
$202$	−0.960911	−0.0676095
$203$	−25.1484	−1.76507
$204$	0	0
$205$	−2.06059	−0.143918
$206$	−22.7235	−1.58322
$207$	0	0
$208$	0.336545	0.0233352
$209$	−6.43996	−0.445462
$210$	0	0
$211$	−2.59530	−0.178668	−0.0893339	−	0.996002i	$-0.528474\pi$
$211$	−2.59530	−0.178668	−0.0893339	+	0.996002i	$0.528474\pi$
$212$	−3.11763	−0.214120
$213$	0	0
$214$	−9.48577	−0.648434
$215$	0.993224	0.0677373
$216$	0	0
$217$	33.4597	2.27139
$218$	−18.6344	−1.26208
$219$	0	0
$220$	−0.899430	−0.0606395
$221$	0.577249	0.0388299
$222$	0	0
$223$	19.0852	1.27804	0.639020	−	0.769190i	$-0.279340\pi$
$223$	19.0852	1.27804	0.639020	+	0.769190i	$0.279340\pi$
$224$	−8.98077	−0.600053
$225$	0	0
$226$	16.8263	1.11927
$227$	3.88980	0.258175	0.129088	−	0.991633i	$-0.458795\pi$
$227$	3.88980	0.258175	0.129088	+	0.991633i	$0.458795\pi$
$228$	0	0
$229$	−17.5735	−1.16129	−0.580644	−	0.814157i	$-0.697199\pi$
$229$	−17.5735	−1.16129	−0.580644	+	0.814157i	$0.697199\pi$
$230$	1.26668	0.0835222
$231$	0	0
$232$	−18.4460	−1.21104
$233$	−2.28417	−0.149641	−0.0748206	−	0.997197i	$-0.523838\pi$
$233$	−2.28417	−0.149641	−0.0748206	+	0.997197i	$0.523838\pi$
$234$	0	0
$235$	−0.358150	−0.0233632
$236$	−1.18387	−0.0770631
$237$	0	0
$238$	27.6066	1.78947
$239$	−5.23735	−0.338776	−0.169388	−	0.985549i	$-0.554179\pi$
$239$	−5.23735	−0.338776	−0.169388	+	0.985549i	$0.554179\pi$
$240$	0	0
$241$	−7.98761	−0.514527	−0.257264	−	0.966341i	$-0.582821\pi$
$241$	−7.98761	−0.514527	−0.257264	+	0.966341i	$0.582821\pi$
$242$	−38.6666	−2.48558
$243$	0	0
$244$	−2.53712	−0.162422
$245$	3.61502	0.230955
$246$	0	0
$247$	0.109695	0.00697971
$248$	24.5423	1.55843
$249$	0	0
$250$	4.48619	0.283731
$251$	−7.32460	−0.462325	−0.231162	−	0.972915i	$-0.574253\pi$
$251$	−7.32460	−0.462325	−0.231162	+	0.972915i	$0.574253\pi$
$252$	0	0
$253$	−17.9500	−1.12851
$254$	−21.6368	−1.35761
$255$	0	0
$256$	−8.98008	−0.561255
$257$	19.3505	1.20705	0.603526	−	0.797343i	$-0.293762\pi$
$257$	19.3505	1.20705	0.603526	+	0.797343i	$0.293762\pi$
$258$	0	0
$259$	−1.86741	−0.116035
$260$	0.0153204	0.000950130 0
$261$	0	0
$262$	0.654448	0.0404320
$263$	−12.2878	−0.757698	−0.378849	−	0.925459i	$-0.623680\pi$
$263$	−12.2878	−0.757698	−0.378849	+	0.925459i	$0.623680\pi$
$264$	0	0
$265$	2.86333	0.175893
$266$	5.24608	0.321658
$267$	0	0
$268$	0.468153	0.0285970
$269$	23.9726	1.46163	0.730817	−	0.682573i	$-0.239139\pi$
$269$	23.9726	1.46163	0.730817	+	0.682573i	$0.239139\pi$
$270$	0	0
$271$	−16.5455	−1.00507	−0.502534	−	0.864558i	$-0.667599\pi$
$271$	−16.5455	−1.00507	−0.502534	+	0.864558i	$0.667599\pi$
$272$	16.1449	0.978927
$273$	0	0
$274$	−4.52666	−0.273466
$275$	−31.3738	−1.89191
$276$	0	0
$277$	12.2806	0.737867	0.368934	−	0.929456i	$-0.379723\pi$
$277$	12.2806	0.737867	0.368934	+	0.929456i	$0.379723\pi$
$278$	7.19395	0.431464
$279$	0	0
$280$	4.49046	0.268356
$281$	14.0883	0.840438	0.420219	−	0.907423i	$-0.361953\pi$
$281$	14.0883	0.840438	0.420219	+	0.907423i	$0.361953\pi$
$282$	0	0
$283$	8.63351	0.513209	0.256605	−	0.966516i	$-0.417396\pi$
$283$	8.63351	0.513209	0.256605	+	0.966516i	$0.417396\pi$
$284$	−2.91430	−0.172932
$285$	0	0
$286$	0.896370	0.0530035
$287$	−23.7871	−1.40411
$288$	0	0
$289$	10.6920	0.628942
$290$	2.76425	0.162323
$291$	0	0
$292$	−3.32794	−0.194753
$293$	−30.8010	−1.79941	−0.899707	−	0.436494i	$-0.856220\pi$
$293$	−30.8010	−1.79941	−0.899707	+	0.436494i	$0.856220\pi$
$294$	0	0
$295$	1.08730	0.0633051
$296$	−1.36972	−0.0796133
$297$	0	0
$298$	0.601062	0.0348186
$299$	0.305751	0.0176820
$300$	0	0
$301$	11.4656	0.660868
$302$	11.8306	0.680772
$303$	0	0
$304$	3.06802	0.175963
$305$	2.33017	0.133425
$306$	0	0
$307$	1.19753	0.0683466	0.0341733	−	0.999416i	$-0.489120\pi$
$307$	1.19753	0.0683466	0.0341733	+	0.999416i	$0.489120\pi$
$308$	−10.3829	−0.591620
$309$	0	0
$310$	−3.67781	−0.208886
$311$	22.6086	1.28202	0.641009	−	0.767534i	$-0.278516\pi$
$311$	22.6086	1.28202	0.641009	+	0.767534i	$0.278516\pi$
$312$	0	0
$313$	16.4592	0.930327	0.465164	−	0.885225i	$-0.345995\pi$
$313$	16.4592	0.930327	0.465164	+	0.885225i	$0.345995\pi$
$314$	−22.9753	−1.29657
$315$	0	0
$316$	2.18150	0.122719
$317$	31.6441	1.77731	0.888654	−	0.458579i	$-0.151641\pi$
$317$	31.6441	1.77731	0.888654	+	0.458579i	$0.151641\pi$
$318$	0	0
$319$	−39.1721	−2.19322
$320$	3.18477	0.178034
$321$	0	0
$322$	14.6223	0.814871
$323$	5.26232	0.292803
$324$	0	0
$325$	0.534403	0.0296433
$326$	2.80583	0.155400
$327$	0	0
$328$	−17.4475	−0.963379
$329$	−4.13444	−0.227939
$330$	0	0
$331$	−33.9641	−1.86684	−0.933418	−	0.358790i	$-0.883189\pi$
$331$	−33.9641	−1.86684	−0.933418	+	0.358790i	$0.883189\pi$
$332$	5.69656	0.312640
$333$	0	0
$334$	−13.7319	−0.751374
$335$	−0.429967	−0.0234916
$336$	0	0
$337$	−3.31820	−0.180754	−0.0903768	−	0.995908i	$-0.528807\pi$
$337$	−3.31820	−0.180754	−0.0903768	+	0.995908i	$0.528807\pi$
$338$	16.4801	0.896399
$339$	0	0
$340$	0.734956	0.0398586
$341$	52.1181	2.82236
$342$	0	0
$343$	12.7902	0.690606
$344$	8.40990	0.453431
$345$	0	0
$346$	14.8639	0.799087
$347$	16.5977	0.891011	0.445506	−	0.895279i	$-0.353024\pi$
$347$	16.5977	0.891011	0.445506	+	0.895279i	$0.353024\pi$
$348$	0	0
$349$	5.08697	0.272299	0.136150	−	0.990688i	$-0.456527\pi$
$349$	5.08697	0.272299	0.136150	+	0.990688i	$0.456527\pi$
$350$	25.5575	1.36610
$351$	0	0
$352$	−13.9888	−0.745606
$353$	1.25983	0.0670539	0.0335269	−	0.999438i	$-0.489326\pi$
$353$	1.25983	0.0670539	0.0335269	+	0.999438i	$0.489326\pi$
$354$	0	0
$355$	2.67659	0.142059
$356$	0.266122	0.0141045
$357$	0	0
$358$	−0.182520	−0.00964650
$359$	3.40749	0.179841	0.0899203	−	0.995949i	$-0.471339\pi$
$359$	3.40749	0.179841	0.0899203	+	0.995949i	$0.471339\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−17.4993	−0.919740
$363$	0	0
$364$	0.176856	0.00926979
$365$	3.05649	0.159984
$366$	0	0
$367$	−25.5353	−1.33293	−0.666466	−	0.745536i	$-0.732194\pi$
$367$	−25.5353	−1.33293	−0.666466	+	0.745536i	$0.732194\pi$
$368$	8.55144	0.445775
$369$	0	0
$370$	0.205261	0.0106710
$371$	33.0539	1.71607
$372$	0	0
$373$	32.7363	1.69502	0.847510	−	0.530779i	$-0.178101\pi$
$373$	32.7363	1.69502	0.847510	+	0.530779i	$0.178101\pi$
$374$	43.0011	2.22353
$375$	0	0
$376$	−3.03256	−0.156392
$377$	0.667236	0.0343644
$378$	0	0
$379$	−31.3146	−1.60853	−0.804263	−	0.594274i	$-0.797439\pi$
$379$	−31.3146	−1.60853	−0.804263	+	0.594274i	$0.797439\pi$
$380$	0.139664	0.00716460
$381$	0	0
$382$	−15.1694	−0.776132
$383$	−1.74701	−0.0892680	−0.0446340	−	0.999003i	$-0.514212\pi$
$383$	−1.74701	−0.0892680	−0.0446340	+	0.999003i	$0.514212\pi$
$384$	0	0
$385$	9.53598	0.485999
$386$	21.5285	1.09577
$387$	0	0
$388$	−2.99234	−0.151913
$389$	−3.15395	−0.159912	−0.0799558	−	0.996798i	$-0.525478\pi$
$389$	−3.15395	−0.159912	−0.0799558	+	0.996798i	$0.525478\pi$
$390$	0	0
$391$	14.6676	0.741773
$392$	30.6093	1.54601
$393$	0	0
$394$	−34.0253	−1.71417
$395$	−2.00356	−0.100810
$396$	0	0
$397$	30.5492	1.53322	0.766611	−	0.642111i	$-0.221941\pi$
$397$	30.5492	1.53322	0.766611	+	0.642111i	$0.221941\pi$
$398$	−28.6123	−1.43420
$399$	0	0
$400$	14.9465	0.747327
$401$	13.1999	0.659173	0.329587	−	0.944125i	$-0.393091\pi$
$401$	13.1999	0.659173	0.329587	+	0.944125i	$0.393091\pi$
$402$	0	0
$403$	−0.887751	−0.0442220
$404$	−0.295313	−0.0146924
$405$	0	0
$406$	31.9102	1.58367
$407$	−2.90875	−0.144181
$408$	0	0
$409$	−15.5269	−0.767755	−0.383878	−	0.923384i	$-0.625412\pi$
$409$	−15.5269	−0.767755	−0.383878	+	0.923384i	$0.625412\pi$
$410$	2.61463	0.129127
$411$	0	0
$412$	−6.98354	−0.344054
$413$	12.5516	0.617626
$414$	0	0
$415$	−5.23191	−0.256824
$416$	0.238277	0.0116825
$417$	0	0
$418$	8.17150	0.399681
$419$	27.1171	1.32475	0.662377	−	0.749170i	$-0.269548\pi$
$419$	27.1171	1.32475	0.662377	+	0.749170i	$0.269548\pi$
$420$	0	0
$421$	29.4976	1.43762	0.718812	−	0.695205i	$-0.244686\pi$
$421$	29.4976	1.43762	0.718812	+	0.695205i	$0.244686\pi$
$422$	3.29311	0.160306
$423$	0	0
$424$	24.2446	1.17742
$425$	25.6366	1.24356
$426$	0	0
$427$	26.8992	1.30174
$428$	−2.91523	−0.140913
$429$	0	0
$430$	−1.26028	−0.0607759
$431$	6.24922	0.301014	0.150507	−	0.988609i	$-0.451909\pi$
$431$	6.24922	0.301014	0.150507	+	0.988609i	$0.451909\pi$
$432$	0	0
$433$	−0.234010	−0.0112458	−0.00562289	−	0.999984i	$-0.501790\pi$
$433$	−0.234010	−0.0112458	−0.00562289	+	0.999984i	$0.501790\pi$
$434$	−42.4561	−2.03796
$435$	0	0
$436$	−5.72685	−0.274266
$437$	2.78729	0.133334
$438$	0	0
$439$	7.33636	0.350146	0.175073	−	0.984555i	$-0.443984\pi$
$439$	7.33636	0.350146	0.175073	+	0.984555i	$0.443984\pi$
$440$	6.99452	0.333451
$441$	0	0
$442$	−0.732456	−0.0348394
$443$	−19.3749	−0.920528	−0.460264	−	0.887782i	$-0.652245\pi$
$443$	−19.3749	−0.920528	−0.460264	+	0.887782i	$0.652245\pi$
$444$	0	0
$445$	−0.244415	−0.0115864
$446$	−24.2167	−1.14670
$447$	0	0
$448$	36.7645	1.73696
$449$	16.4048	0.774191	0.387096	−	0.922040i	$-0.373478\pi$
$449$	16.4048	0.774191	0.387096	+	0.922040i	$0.373478\pi$
$450$	0	0
$451$	−37.0518	−1.74470
$452$	5.17117	0.243231
$453$	0	0
$454$	−4.93567	−0.231642
$455$	−0.162431	−0.00761486
$456$	0	0
$457$	22.3212	1.04414	0.522071	−	0.852902i	$-0.325160\pi$
$457$	22.3212	1.04414	0.522071	+	0.852902i	$0.325160\pi$
$458$	22.2985	1.04194
$459$	0	0
$460$	0.389283	0.0181504
$461$	−40.5074	−1.88662	−0.943309	−	0.331915i	$-0.892305\pi$
$461$	−40.5074	−1.88662	−0.943309	+	0.331915i	$0.892305\pi$
$462$	0	0
$463$	−10.2894	−0.478190	−0.239095	−	0.970996i	$-0.576851\pi$
$463$	−10.2894	−0.478190	−0.239095	+	0.970996i	$0.576851\pi$
$464$	18.6617	0.866348
$465$	0	0
$466$	2.89833	0.134263
$467$	34.9394	1.61680	0.808402	−	0.588631i	$-0.200333\pi$
$467$	34.9394	1.61680	0.808402	+	0.588631i	$0.200333\pi$
$468$	0	0
$469$	−4.96347	−0.229192
$470$	0.454448	0.0209621
$471$	0	0
$472$	9.20647	0.423762
$473$	17.8593	0.821173
$474$	0	0
$475$	4.87173	0.223530
$476$	8.48422	0.388874
$477$	0	0
$478$	6.64554	0.303960
$479$	−13.6524	−0.623796	−0.311898	−	0.950116i	$-0.600965\pi$
$479$	−13.6524	−0.623796	−0.311898	+	0.950116i	$0.600965\pi$
$480$	0	0
$481$	0.0495460	0.00225910
$482$	10.1353	0.461649
$483$	0	0
$484$	−11.8833	−0.540148
$485$	2.74826	0.124792
$486$	0	0
$487$	31.9356	1.44714	0.723571	−	0.690250i	$-0.242500\pi$
$487$	31.9356	1.44714	0.723571	+	0.690250i	$0.242500\pi$
$488$	19.7302	0.893144
$489$	0	0
$490$	−4.58701	−0.207220
$491$	−18.0880	−0.816301	−0.408151	−	0.912915i	$-0.633826\pi$
$491$	−18.0880	−0.816301	−0.408151	+	0.912915i	$0.633826\pi$
$492$	0	0
$493$	32.0089	1.44161
$494$	−0.139189	−0.00626240
$495$	0	0
$496$	−24.8292	−1.11486
$497$	30.8982	1.38597
$498$	0	0
$499$	11.1090	0.497309	0.248654	−	0.968592i	$-0.420012\pi$
$499$	11.1090	0.497309	0.248654	+	0.968592i	$0.420012\pi$
$500$	1.37872	0.0616584
$501$	0	0
$502$	9.29400	0.414812
$503$	−43.7025	−1.94860	−0.974299	−	0.225260i	$-0.927677\pi$
$503$	−43.7025	−1.94860	−0.974299	+	0.225260i	$0.927677\pi$
$504$	0	0
$505$	0.271225	0.0120694
$506$	22.7763	1.01253
$507$	0	0
$508$	−6.64956	−0.295027
$509$	2.76799	0.122689	0.0613446	−	0.998117i	$-0.480461\pi$
$509$	2.76799	0.122689	0.0613446	+	0.998117i	$0.480461\pi$
$510$	0	0
$511$	35.2837	1.56086
$512$	25.2722	1.11688
$513$	0	0
$514$	−24.5534	−1.08300
$515$	6.41390	0.282630
$516$	0	0
$517$	−6.43996	−0.283229
$518$	2.36950	0.104110
$519$	0	0
$520$	−0.119141	−0.00522466
$521$	42.7335	1.87219	0.936095	−	0.351748i	$-0.114413\pi$
$521$	42.7335	1.87219	0.936095	+	0.351748i	$0.114413\pi$
$522$	0	0
$523$	−29.0922	−1.27211	−0.636056	−	0.771643i	$-0.719435\pi$
$523$	−29.0922	−1.27211	−0.636056	+	0.771643i	$0.719435\pi$
$524$	0.201129	0.00878637
$525$	0	0
$526$	15.5917	0.679829
$527$	−42.5876	−1.85514
$528$	0	0
$529$	−15.2310	−0.662218
$530$	−3.63321	−0.157817
$531$	0	0
$532$	1.61226	0.0699003
$533$	0.631119	0.0273368
$534$	0	0
$535$	2.67744	0.115756
$536$	−3.64064	−0.157252
$537$	0	0
$538$	−30.4182	−1.31142
$539$	65.0023	2.79985
$540$	0	0
$541$	−26.3097	−1.13114	−0.565571	−	0.824700i	$-0.691344\pi$
$541$	−26.3097	−1.13114	−0.565571	+	0.824700i	$0.691344\pi$
$542$	20.9942	0.901776
$543$	0	0
$544$	11.4307	0.490089
$545$	5.25972	0.225302
$546$	0	0
$547$	14.7356	0.630047	0.315023	−	0.949084i	$-0.397988\pi$
$547$	14.7356	0.630047	0.315023	+	0.949084i	$0.397988\pi$
$548$	−1.39116	−0.0594275
$549$	0	0
$550$	39.8093	1.69748
$551$	6.08267	0.259130
$552$	0	0
$553$	−23.1289	−0.983539
$554$	−15.5825	−0.662036
$555$	0	0
$556$	2.21089	0.0937626
$557$	−35.4822	−1.50343	−0.751715	−	0.659488i	$-0.770773\pi$
$557$	−35.4822	−1.50343	−0.751715	+	0.659488i	$0.770773\pi$
$558$	0	0
$559$	−0.304206	−0.0128665
$560$	−4.54297	−0.191975
$561$	0	0
$562$	−17.8763	−0.754066
$563$	15.5544	0.655540	0.327770	−	0.944758i	$-0.393703\pi$
$563$	15.5544	0.655540	0.327770	+	0.944758i	$0.393703\pi$
$564$	0	0
$565$	−4.74937	−0.199807
$566$	−10.9548	−0.460466
$567$	0	0
$568$	22.6634	0.950935
$569$	−19.0183	−0.797288	−0.398644	−	0.917106i	$-0.630519\pi$
$569$	−19.0183	−0.797288	−0.398644	+	0.917106i	$0.630519\pi$
$570$	0	0
$571$	−35.6118	−1.49031	−0.745153	−	0.666894i	$-0.767624\pi$
$571$	−35.6118	−1.49031	−0.745153	+	0.666894i	$0.767624\pi$
$572$	0.275478	0.0115183
$573$	0	0
$574$	30.1829	1.25981
$575$	13.5789	0.566280
$576$	0	0
$577$	−16.6909	−0.694851	−0.347425	−	0.937708i	$-0.612944\pi$
$577$	−16.6909	−0.694851	−0.347425	+	0.937708i	$0.612944\pi$
$578$	−13.5668	−0.564306
$579$	0	0
$580$	0.849528	0.0352747
$581$	−60.3964	−2.50567
$582$	0	0
$583$	51.4861	2.13234
$584$	25.8801	1.07093
$585$	0	0
$586$	39.0826	1.61449
$587$	−26.5249	−1.09480	−0.547400	−	0.836871i	$-0.684382\pi$
$587$	−26.5249	−1.09480	−0.547400	+	0.836871i	$0.684382\pi$
$588$	0	0
$589$	−8.09293	−0.333463
$590$	−1.37965	−0.0567992
$591$	0	0
$592$	1.38573	0.0569533
$593$	−30.8923	−1.26860	−0.634298	−	0.773089i	$-0.718711\pi$
$593$	−30.8923	−1.26860	−0.634298	+	0.773089i	$0.718711\pi$
$594$	0	0
$595$	−7.79219	−0.319448
$596$	0.184722	0.00756652
$597$	0	0
$598$	−0.387959	−0.0158648
$599$	−9.23460	−0.377316	−0.188658	−	0.982043i	$-0.560414\pi$
$599$	−9.23460	−0.377316	−0.188658	+	0.982043i	$0.560414\pi$
$600$	0	0
$601$	6.65298	0.271381	0.135690	−	0.990751i	$-0.456675\pi$
$601$	6.65298	0.271381	0.135690	+	0.990751i	$0.456675\pi$
$602$	−14.5485	−0.592951
$603$	0	0
$604$	3.63584	0.147940
$605$	10.9140	0.443716
$606$	0	0
$607$	−38.6090	−1.56709	−0.783546	−	0.621334i	$-0.786591\pi$
$607$	−38.6090	−1.56709	−0.783546	+	0.621334i	$0.786591\pi$
$608$	2.17219	0.0880938
$609$	0	0
$610$	−2.95669	−0.119713
$611$	0.109695	0.00443777
$612$	0	0
$613$	1.29627	0.0523558	0.0261779	−	0.999657i	$-0.491666\pi$
$613$	1.29627	0.0523558	0.0261779	+	0.999657i	$0.491666\pi$
$614$	−1.51951	−0.0613226
$615$	0	0
$616$	80.7437	3.25326
$617$	0.548387	0.0220772	0.0110386	−	0.999939i	$-0.496486\pi$
$617$	0.548387	0.0220772	0.0110386	+	0.999939i	$0.496486\pi$
$618$	0	0
$619$	−13.4742	−0.541573	−0.270787	−	0.962639i	$-0.587284\pi$
$619$	−13.4742	−0.541573	−0.270787	+	0.962639i	$0.587284\pi$
$620$	−1.13029	−0.0453935
$621$	0	0
$622$	−28.6875	−1.15026
$623$	−2.82150	−0.113041
$624$	0	0
$625$	23.0924	0.923695
$626$	−20.8846	−0.834717
$627$	0	0
$628$	−7.06092	−0.281761
$629$	2.37684	0.0947708
$630$	0	0
$631$	33.3241	1.32661	0.663305	−	0.748349i	$-0.269153\pi$
$631$	33.3241	1.32661	0.663305	+	0.748349i	$0.269153\pi$
$632$	−16.9647	−0.674820
$633$	0	0
$634$	−40.1523	−1.59465
$635$	6.10717	0.242356
$636$	0	0
$637$	−1.10721	−0.0438693
$638$	49.7045	1.96782
$639$	0	0
$640$	−2.48513	−0.0982334
$641$	34.3759	1.35777	0.678883	−	0.734246i	$-0.262464\pi$
$641$	34.3759	1.35777	0.678883	+	0.734246i	$0.262464\pi$
$642$	0	0
$643$	32.1822	1.26914	0.634571	−	0.772865i	$-0.281177\pi$
$643$	32.1822	1.26914	0.634571	+	0.772865i	$0.281177\pi$
$644$	4.49383	0.177082
$645$	0	0
$646$	−6.67722	−0.262712
$647$	−22.1807	−0.872013	−0.436007	−	0.899943i	$-0.643608\pi$
$647$	−22.1807	−0.872013	−0.436007	+	0.899943i	$0.643608\pi$
$648$	0	0
$649$	19.5509	0.767441
$650$	−0.678090	−0.0265969
$651$	0	0
$652$	0.862305	0.0337705
$653$	30.3898	1.18924	0.594622	−	0.804005i	$-0.297302\pi$
$653$	30.3898	1.18924	0.594622	+	0.804005i	$0.297302\pi$
$654$	0	0
$655$	−0.184724	−0.00721775
$656$	17.6516	0.689177
$657$	0	0
$658$	5.24608	0.204514
$659$	0.523232	0.0203822	0.0101911	−	0.999948i	$-0.496756\pi$
$659$	0.523232	0.0203822	0.0101911	+	0.999948i	$0.496756\pi$
$660$	0	0
$661$	12.8045	0.498039	0.249020	−	0.968498i	$-0.419892\pi$
$661$	12.8045	0.498039	0.249020	+	0.968498i	$0.419892\pi$
$662$	43.0962	1.67498
$663$	0	0
$664$	−44.3000	−1.71917
$665$	−1.48075	−0.0574211
$666$	0	0
$667$	16.9541	0.656467
$668$	−4.22016	−0.163283
$669$	0	0
$670$	0.545573	0.0210773
$671$	41.8992	1.61750
$672$	0	0
$673$	20.0956	0.774630	0.387315	−	0.921947i	$-0.373403\pi$
$673$	20.0956	0.774630	0.387315	+	0.921947i	$0.373403\pi$
$674$	4.21037	0.162177
$675$	0	0
$676$	5.06477	0.194799
$677$	−41.2342	−1.58476	−0.792380	−	0.610028i	$-0.791158\pi$
$677$	−41.2342	−1.58476	−0.792380	+	0.610028i	$0.791158\pi$
$678$	0	0
$679$	31.7255	1.21751
$680$	−5.71547	−0.219178
$681$	0	0
$682$	−66.1314	−2.53230
$683$	−32.3774	−1.23889	−0.619444	−	0.785041i	$-0.712642\pi$
$683$	−32.3774	−1.23889	−0.619444	+	0.785041i	$0.712642\pi$
$684$	0	0
$685$	1.27769	0.0488180
$686$	−16.2292	−0.619632
$687$	0	0
$688$	−8.50823	−0.324373
$689$	−0.876985	−0.0334105
$690$	0	0
$691$	−34.6817	−1.31935	−0.659677	−	0.751549i	$-0.729307\pi$
$691$	−34.6817	−1.31935	−0.659677	+	0.751549i	$0.729307\pi$
$692$	4.56806	0.173652
$693$	0	0
$694$	−21.0604	−0.799442
$695$	−2.03055	−0.0770232
$696$	0	0
$697$	30.2763	1.14680
$698$	−6.45472	−0.244315
$699$	0	0
$700$	7.85449	0.296872
$701$	−45.9314	−1.73481	−0.867403	−	0.497606i	$-0.834212\pi$
$701$	−45.9314	−1.73481	−0.867403	+	0.497606i	$0.834212\pi$
$702$	0	0
$703$	0.451671	0.0170351
$704$	57.2658	2.15829
$705$	0	0
$706$	−1.59856	−0.0601627
$707$	3.13099	0.117753
$708$	0	0
$709$	26.6271	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$709$	26.6271	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$710$	−3.39625	−0.127459
$711$	0	0
$712$	−2.06953	−0.0775589
$713$	−22.5573	−0.844778
$714$	0	0
$715$	−0.253008	−0.00946197
$716$	−0.0560933	−0.00209631
$717$	0	0
$718$	−4.32368	−0.161358
$719$	24.1269	0.899781	0.449890	−	0.893084i	$-0.351463\pi$
$719$	24.1269	0.899781	0.449890	+	0.893084i	$0.351463\pi$
$720$	0	0
$721$	74.0412	2.75744
$722$	−1.26887	−0.0472226
$723$	0	0
$724$	−5.37798	−0.199871
$725$	29.6331	1.10055
$726$	0	0
$727$	25.8003	0.956879	0.478439	−	0.878121i	$-0.341203\pi$
$727$	25.8003	0.956879	0.478439	+	0.878121i	$0.341203\pi$
$728$	−1.37534	−0.0509736
$729$	0	0
$730$	−3.87830	−0.143542
$731$	−14.5935	−0.539760
$732$	0	0
$733$	−31.4784	−1.16268	−0.581340	−	0.813661i	$-0.697471\pi$
$733$	−31.4784	−1.16268	−0.581340	+	0.813661i	$0.697471\pi$
$734$	32.4011	1.19595
$735$	0	0
$736$	6.05451	0.223172
$737$	−7.73130	−0.284786
$738$	0	0
$739$	−45.5126	−1.67421	−0.837105	−	0.547042i	$-0.815754\pi$
$739$	−45.5126	−1.67421	−0.837105	+	0.547042i	$0.815754\pi$
$740$	0.0630821	0.00231895
$741$	0	0
$742$	−41.9413	−1.53971
$743$	9.50434	0.348680	0.174340	−	0.984685i	$-0.444221\pi$
$743$	9.50434	0.348680	0.174340	+	0.984685i	$0.444221\pi$
$744$	0	0
$745$	−0.169655	−0.00621567
$746$	−41.5382	−1.52082
$747$	0	0
$748$	13.2154	0.483201
$749$	30.9080	1.12935
$750$	0	0
$751$	42.3212	1.54432	0.772162	−	0.635426i	$-0.219175\pi$
$751$	42.3212	1.54432	0.772162	+	0.635426i	$0.219175\pi$
$752$	3.06802	0.111879
$753$	0	0
$754$	−0.846639	−0.0308328
$755$	−3.33927	−0.121529
$756$	0	0
$757$	−33.7738	−1.22753	−0.613765	−	0.789489i	$-0.710346\pi$
$757$	−33.7738	−1.22753	−0.613765	+	0.789489i	$0.710346\pi$
$758$	39.7343	1.44322
$759$	0	0
$760$	−1.08611	−0.0393974
$761$	25.8925	0.938604	0.469302	−	0.883038i	$-0.344506\pi$
$761$	25.8925	0.938604	0.469302	+	0.883038i	$0.344506\pi$
$762$	0	0
$763$	60.7175	2.19812
$764$	−4.66194	−0.168663
$765$	0	0
$766$	2.21673	0.0800938
$767$	−0.333020	−0.0120246
$768$	0	0
$769$	−14.4159	−0.519851	−0.259926	−	0.965629i	$-0.583698\pi$
$769$	−14.4159	−0.519851	−0.259926	+	0.965629i	$0.583698\pi$
$770$	−12.1000	−0.436052
$771$	0	0
$772$	6.61627	0.238125
$773$	5.59277	0.201158	0.100579	−	0.994929i	$-0.467931\pi$
$773$	5.59277	0.201158	0.100579	+	0.994929i	$0.467931\pi$
$774$	0	0
$775$	−39.4265	−1.41624
$776$	23.2702	0.835353
$777$	0	0
$778$	4.00197	0.143477
$779$	5.75341	0.206137
$780$	0	0
$781$	48.1282	1.72216
$782$	−18.6113	−0.665540
$783$	0	0
$784$	−30.9673	−1.10597
$785$	6.48498	0.231459
$786$	0	0
$787$	−33.5709	−1.19667	−0.598337	−	0.801244i	$-0.704172\pi$
$787$	−33.5709	−1.19667	−0.598337	+	0.801244i	$0.704172\pi$
$788$	−10.4569	−0.372510
$789$	0	0
$790$	2.54227	0.0904499
$791$	−54.8260	−1.94939
$792$	0	0
$793$	−0.713687	−0.0253438
$794$	−38.7631	−1.37565
$795$	0	0
$796$	−8.79331	−0.311671
$797$	21.2549	0.752887	0.376443	−	0.926440i	$-0.377147\pi$
$797$	21.2549	0.752887	0.376443	+	0.926440i	$0.377147\pi$
$798$	0	0
$799$	5.26232	0.186168
$800$	10.5823	0.374141
$801$	0	0
$802$	−16.7491	−0.591430
$803$	54.9592	1.93947
$804$	0	0
$805$	−4.12728	−0.145467
$806$	1.12644	0.0396773
$807$	0	0
$808$	2.29654	0.0807919
$809$	−54.8007	−1.92669	−0.963345	−	0.268266i	$-0.913549\pi$
$809$	−54.8007	−1.92669	−0.963345	+	0.268266i	$0.913549\pi$
$810$	0	0
$811$	36.5839	1.28464	0.642318	−	0.766439i	$-0.277973\pi$
$811$	36.5839	1.28464	0.642318	+	0.766439i	$0.277973\pi$
$812$	9.80683	0.344152
$813$	0	0
$814$	3.69083	0.129364
$815$	−0.791969	−0.0277415
$816$	0	0
$817$	−2.77320	−0.0970221
$818$	19.7017	0.688853
$819$	0	0
$820$	0.803543	0.0280609
$821$	−1.43285	−0.0500069	−0.0250034	−	0.999687i	$-0.507960\pi$
$821$	−1.43285	−0.0500069	−0.0250034	+	0.999687i	$0.507960\pi$
$822$	0	0
$823$	−22.1157	−0.770904	−0.385452	−	0.922728i	$-0.625954\pi$
$823$	−22.1157	−0.770904	−0.385452	+	0.922728i	$0.625954\pi$
$824$	54.3083	1.89192
$825$	0	0
$826$	−15.9265	−0.554152
$827$	27.1407	0.943774	0.471887	−	0.881659i	$-0.343573\pi$
$827$	27.1407	0.943774	0.471887	+	0.881659i	$0.343573\pi$
$828$	0	0
$829$	−3.68181	−0.127875	−0.0639373	−	0.997954i	$-0.520366\pi$
$829$	−3.68181	−0.127875	−0.0639373	+	0.997954i	$0.520366\pi$
$830$	6.63863	0.230430
$831$	0	0
$832$	−0.975434	−0.0338171
$833$	−53.1156	−1.84035
$834$	0	0
$835$	3.87593	0.134132
$836$	2.51132	0.0868557
$837$	0	0
$838$	−34.4081	−1.18861
$839$	−12.2084	−0.421479	−0.210740	−	0.977542i	$-0.567587\pi$
$839$	−12.2084	−0.421479	−0.210740	+	0.977542i	$0.567587\pi$
$840$	0	0
$841$	7.99882	0.275821
$842$	−37.4287	−1.28988
$843$	0	0
$844$	1.01206	0.0348365
$845$	−4.65165	−0.160022
$846$	0	0
$847$	125.989	4.32904
$848$	−24.5281	−0.842299
$849$	0	0
$850$	−32.5296	−1.11576
$851$	1.25894	0.0431558
$852$	0	0
$853$	−0.104751	−0.00358661	−0.00179331	−	0.999998i	$-0.500571\pi$
$853$	−0.104751	−0.00358661	−0.00179331	+	0.999998i	$0.500571\pi$
$854$	−34.1317	−1.16796
$855$	0	0
$856$	22.6706	0.774865
$857$	9.79527	0.334600	0.167300	−	0.985906i	$-0.446495\pi$
$857$	9.79527	0.334600	0.167300	+	0.985906i	$0.446495\pi$
$858$	0	0
$859$	10.4005	0.354859	0.177430	−	0.984133i	$-0.443222\pi$
$859$	10.4005	0.354859	0.177430	+	0.984133i	$0.443222\pi$
$860$	−0.387316	−0.0132074
$861$	0	0
$862$	−7.92947	−0.270079
$863$	42.6205	1.45082	0.725409	−	0.688318i	$-0.241651\pi$
$863$	42.6205	1.45082	0.725409	+	0.688318i	$0.241651\pi$
$864$	0	0
$865$	−4.19545	−0.142650
$866$	0.296929	0.0100901
$867$	0	0
$868$	−13.0479	−0.442874
$869$	−36.0264	−1.22211
$870$	0	0
$871$	0.131691	0.00446217
$872$	44.5355	1.50816
$873$	0	0
$874$	−3.53672	−0.119631
$875$	−14.6176	−0.494164
$876$	0	0
$877$	−30.0245	−1.01386	−0.506928	−	0.861988i	$-0.669219\pi$
$877$	−30.0245	−1.01386	−0.506928	+	0.861988i	$0.669219\pi$
$878$	−9.30892	−0.314161
$879$	0	0
$880$	−7.07630	−0.238542
$881$	−12.6811	−0.427238	−0.213619	−	0.976917i	$-0.568525\pi$
$881$	−12.6811	−0.427238	−0.213619	+	0.976917i	$0.568525\pi$
$882$	0	0
$883$	34.5598	1.16303	0.581515	−	0.813536i	$-0.302460\pi$
$883$	34.5598	1.16303	0.581515	+	0.813536i	$0.302460\pi$
$884$	−0.225103	−0.00757103
$885$	0	0
$886$	24.5843	0.825925
$887$	−38.9442	−1.30762	−0.653808	−	0.756660i	$-0.726830\pi$
$887$	−38.9442	−1.30762	−0.653808	+	0.756660i	$0.726830\pi$
$888$	0	0
$889$	70.5003	2.36451
$890$	0.310132	0.0103957
$891$	0	0
$892$	−7.44244	−0.249191
$893$	1.00000	0.0334637
$894$	0	0
$895$	0.0515179	0.00172205
$896$	−28.6880	−0.958399
$897$	0	0
$898$	−20.8156	−0.694627
$899$	−49.2266	−1.64180
$900$	0	0
$901$	−42.0711	−1.40159
$902$	47.0140	1.56540
$903$	0	0
$904$	−40.2142	−1.33750
$905$	4.93931	0.164188
$906$	0	0
$907$	−24.9947	−0.829935	−0.414968	−	0.909836i	$-0.636207\pi$
$907$	−24.9947	−0.829935	−0.414968	+	0.909836i	$0.636207\pi$
$908$	−1.51686	−0.0503388
$909$	0	0
$910$	0.206104	0.00683228
$911$	12.9994	0.430690	0.215345	−	0.976538i	$-0.430912\pi$
$911$	12.9994	0.430690	0.215345	+	0.976538i	$0.430912\pi$
$912$	0	0
$913$	−94.0758	−3.11346
$914$	−28.3228	−0.936834
$915$	0	0
$916$	6.85293	0.226427
$917$	−2.13242	−0.0704188
$918$	0	0
$919$	19.1281	0.630977	0.315488	−	0.948929i	$-0.397832\pi$
$919$	19.1281	0.630977	0.315488	+	0.948929i	$0.397832\pi$
$920$	−3.02731	−0.0998073
$921$	0	0
$922$	51.3988	1.69273
$923$	−0.819788	−0.0269837
$924$	0	0
$925$	2.20042	0.0723494
$926$	13.0560	0.429047
$927$	0	0
$928$	13.2127	0.433728
$929$	−37.9576	−1.24535	−0.622675	−	0.782481i	$-0.713954\pi$
$929$	−37.9576	−1.24535	−0.622675	+	0.782481i	$0.713954\pi$
$930$	0	0
$931$	−10.0936	−0.330804
$932$	0.890733	0.0291769
$933$	0	0
$934$	−44.3337	−1.45064
$935$	−12.1374	−0.396936
$936$	0	0
$937$	−24.1950	−0.790417	−0.395209	−	0.918591i	$-0.629328\pi$
$937$	−24.1950	−0.790417	−0.395209	+	0.918591i	$0.629328\pi$
$938$	6.29802	0.205638
$939$	0	0
$940$	0.139664	0.00455533
$941$	−12.8340	−0.418376	−0.209188	−	0.977875i	$-0.567082\pi$
$941$	−12.8340	−0.418376	−0.209188	+	0.977875i	$0.567082\pi$
$942$	0	0
$943$	16.0364	0.522218
$944$	−9.31412	−0.303149
$945$	0	0
$946$	−22.6612	−0.736780
$947$	−36.1664	−1.17525	−0.587625	−	0.809134i	$-0.699937\pi$
$947$	−36.1664	−1.17525	−0.587625	+	0.809134i	$0.699937\pi$
$948$	0	0
$949$	−0.936144	−0.0303885
$950$	−6.18161	−0.200558
$951$	0	0
$952$	−65.9785	−2.13838
$953$	27.3318	0.885363	0.442681	−	0.896679i	$-0.354027\pi$
$953$	27.3318	0.885363	0.442681	+	0.896679i	$0.354027\pi$
$954$	0	0
$955$	4.28168	0.138552
$956$	2.04235	0.0660543
$957$	0	0
$958$	17.3232	0.559688
$959$	14.7495	0.476285
$960$	0	0
$961$	34.4954	1.11276
$962$	−0.0628676	−0.00202693
$963$	0	0
$964$	3.11484	0.100322
$965$	−6.07659	−0.195612
$966$	0	0
$967$	−29.4635	−0.947482	−0.473741	−	0.880664i	$-0.657097\pi$
$967$	−29.4635	−0.947482	−0.473741	+	0.880664i	$0.657097\pi$
$968$	92.4115	2.97022
$969$	0	0
$970$	−3.48719	−0.111967
$971$	−32.0182	−1.02751	−0.513756	−	0.857936i	$-0.671746\pi$
$971$	−32.0182	−1.02751	−0.513756	+	0.857936i	$0.671746\pi$
$972$	0	0
$973$	−23.4404	−0.751465
$974$	−40.5223	−1.29842
$975$	0	0
$976$	−19.9609	−0.638932
$977$	−34.9099	−1.11687	−0.558434	−	0.829549i	$-0.688598\pi$
$977$	−34.9099	−1.11687	−0.558434	+	0.829549i	$0.688598\pi$
$978$	0	0
$979$	−4.39487	−0.140461
$980$	−1.40971	−0.0450315
$981$	0	0
$982$	22.9514	0.732410
$983$	−31.3028	−0.998404	−0.499202	−	0.866486i	$-0.666373\pi$
$983$	−31.3028	−0.998404	−0.499202	+	0.866486i	$0.666373\pi$
$984$	0	0
$985$	9.60392	0.306006
$986$	−40.6153	−1.29345
$987$	0	0
$988$	−0.0427764	−0.00136090
$989$	−7.72972	−0.245791
$990$	0	0
$991$	−28.3725	−0.901284	−0.450642	−	0.892705i	$-0.648805\pi$
$991$	−28.3725	−0.901284	−0.450642	+	0.892705i	$0.648805\pi$
$992$	−17.5794	−0.558145
$993$	0	0
$994$	−39.2059	−1.24353
$995$	8.07606	0.256028
$996$	0	0
$997$	57.8011	1.83058	0.915290	−	0.402795i	$-0.131961\pi$
$997$	57.8011	1.83058	0.915290	+	0.402795i	$0.131961\pi$
$998$	−14.0960	−0.446200
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.q.1.8		23
3.2	odd	2		2679.2.a.m.1.16	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.m.1.16	✓	23	3.2	odd	2
8037.2.a.q.1.8		23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-1.26887 q^{2} -0.389958 q^{4} +0.358150 q^{5} +4.13444 q^{7} +3.03256 q^{8} +O(q^{10})\)	\(q-1.26887 q^{2} -0.389958 q^{4} +0.358150 q^{5} +4.13444 q^{7} +3.03256 q^{8} -0.454448 q^{10} +6.43996 q^{11} -0.109695 q^{13} -5.24608 q^{14} -3.06802 q^{16} -5.26232 q^{17} -1.00000 q^{19} -0.139664 q^{20} -8.17150 q^{22} -2.78729 q^{23} -4.87173 q^{25} +0.139189 q^{26} -1.61226 q^{28} -6.08267 q^{29} +8.09293 q^{31} -2.17219 q^{32} +6.67722 q^{34} +1.48075 q^{35} -0.451671 q^{37} +1.26887 q^{38} +1.08611 q^{40} -5.75341 q^{41} +2.77320 q^{43} -2.51132 q^{44} +3.53672 q^{46} -1.00000 q^{47} +10.0936 q^{49} +6.18161 q^{50} +0.0427764 q^{52} +7.99478 q^{53} +2.30648 q^{55} +12.5379 q^{56} +7.71814 q^{58} +3.03588 q^{59} +6.50612 q^{61} -10.2689 q^{62} +8.89226 q^{64} -0.0392872 q^{65} -1.20052 q^{67} +2.05209 q^{68} -1.87889 q^{70} +7.47336 q^{71} +8.53409 q^{73} +0.573114 q^{74} +0.389958 q^{76} +26.6256 q^{77} -5.59419 q^{79} -1.09881 q^{80} +7.30035 q^{82} -14.6081 q^{83} -1.88470 q^{85} -3.51885 q^{86} +19.5296 q^{88} -0.682438 q^{89} -0.453526 q^{91} +1.08693 q^{92} +1.26887 q^{94} -0.358150 q^{95} +7.67347 q^{97} -12.8075 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8	\( 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8} - 5 q^{10} - 12 q^{11} + 9 q^{13} - 3 q^{14} + 60 q^{16} - 16 q^{17} - 23 q^{19} - 25 q^{20} + 3 q^{22} - 12 q^{23} + 54 q^{25} - 5 q^{26} + 8 q^{28} - 27 q^{29} + 10 q^{31} + 34 q^{32} + 6 q^{35} + 15 q^{37} + 2 q^{38} + 3 q^{40} - 10 q^{41} + 24 q^{43} - 39 q^{44} + 43 q^{46} - 23 q^{47} + 78 q^{49} + 32 q^{50} + 38 q^{52} + 2 q^{53} + 5 q^{55} - 58 q^{56} - 11 q^{58} + 51 q^{59} + 48 q^{61} + 22 q^{62} + 125 q^{64} - 15 q^{65} + 26 q^{67} - 26 q^{68} + 86 q^{70} - 24 q^{71} + 53 q^{73} - 26 q^{74} - 30 q^{76} - 18 q^{77} + 29 q^{79} - 5 q^{80} + 47 q^{82} + 22 q^{83} - 5 q^{85} + 28 q^{86} + 62 q^{88} - 38 q^{89} + 15 q^{91} - 15 q^{92} + 2 q^{94} + 9 q^{95} + 33 q^{97} + 30 q^{98}+O(q^{100}) \) 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8 - 5 * q^10 - 12 * q^11 + 9 * q^13 - 3 * q^14 + 60 * q^16 - 16 * q^17 - 23 * q^19 - 25 * q^20 + 3 * q^22 - 12 * q^23 + 54 * q^25 - 5 * q^26 + 8 * q^28 - 27 * q^29 + 10 * q^31 + 34 * q^32 + 6 * q^35 + 15 * q^37 + 2 * q^38 + 3 * q^40 - 10 * q^41 + 24 * q^43 - 39 * q^44 + 43 * q^46 - 23 * q^47 + 78 * q^49 + 32 * q^50 + 38 * q^52 + 2 * q^53 + 5 * q^55 - 58 * q^56 - 11 * q^58 + 51 * q^59 + 48 * q^61 + 22 * q^62 + 125 * q^64 - 15 * q^65 + 26 * q^67 - 26 * q^68 + 86 * q^70 - 24 * q^71 + 53 * q^73 - 26 * q^74 - 30 * q^76 - 18 * q^77 + 29 * q^79 - 5 * q^80 + 47 * q^82 + 22 * q^83 - 5 * q^85 + 28 * q^86 + 62 * q^88 - 38 * q^89 + 15 * q^91 - 15 * q^92 + 2 * q^94 + 9 * q^95 + 33 * q^97 + 30 * q^98

Introduction

L-functions

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