LMFDB - Embedded newform 3808.2.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3808,2,Mod(1,3808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3808, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.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:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4022000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 10x^{3} + 14x^{2} - 8x - 4 $ x^6 - 2x^5 - 7x^4 + 10x^3 + 14x^2 - 8*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.775247$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53790 q^{3} -0.138905 q^{5} -1.00000 q^{7} +3.44092 q^{9} -0.938171 q^{11} -3.02933 q^{13} -0.352526 q^{15} +1.00000 q^{17} -2.38098 q^{19} -2.53790 q^{21} -5.51799 q^{23} -4.98071 q^{25} +1.11900 q^{27} -4.99682 q^{29} -9.85122 q^{31} -2.38098 q^{33} +0.138905 q^{35} -6.85312 q^{37} -7.68812 q^{39} -4.61308 q^{41} +1.86156 q^{43} -0.477960 q^{45} +0.273295 q^{47} +1.00000 q^{49} +2.53790 q^{51} +6.18574 q^{53} +0.130316 q^{55} -6.04268 q^{57} +2.88562 q^{59} +5.38822 q^{61} -3.44092 q^{63} +0.420788 q^{65} +5.80476 q^{67} -14.0041 q^{69} +11.1978 q^{71} +10.3579 q^{73} -12.6405 q^{75} +0.938171 q^{77} +5.96214 q^{79} -7.48284 q^{81} -8.92830 q^{83} -0.138905 q^{85} -12.6814 q^{87} +2.88635 q^{89} +3.02933 q^{91} -25.0014 q^{93} +0.330729 q^{95} +11.9327 q^{97} -3.22817 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{27} + 2 q^{29} - 12 q^{31} + 2 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 - 8 * q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 - 10 * q^23 - 2 * q^27 + 2 * q^29 - 12 * q^31 + 2 * q^33 - 2 * q^35 + 2 * q^37 - 10 * q^39 - 8 * q^43 + 14 * q^45 - 22 * q^47 + 6 * q^49 - 2 * q^51 - 6 * q^53 - 24 * q^55 - 8 * q^57 - 8 * q^59 - 4 * q^61 - 4 * q^63 - 2 * q^65 + 8 * q^67 + 4 * q^69 - 18 * q^71 + 8 * q^73 - 30 * q^75 + 2 * q^77 - 14 * q^79 - 26 * q^81 + 2 * q^85 - 22 * q^87 + 2 * q^89 + 4 * q^91 - 8 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.53790	1.46526	0.732628	−	0.680630i	$-0.238294\pi$
$3$	2.53790	1.46526	0.732628	+	0.680630i	$0.238294\pi$
$4$	0	0
$5$	−0.138905	−0.0621201	−0.0310600	−	0.999518i	$-0.509888\pi$
$5$	−0.138905	−0.0621201	−0.0310600	+	0.999518i	$0.509888\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.44092	1.14697
$10$	0	0
$11$	−0.938171	−0.282869	−0.141435	−	0.989948i	$-0.545171\pi$
$11$	−0.938171	−0.282869	−0.141435	+	0.989948i	$0.545171\pi$
$12$	0	0
$13$	−3.02933	−0.840184	−0.420092	−	0.907482i	$-0.638002\pi$
$13$	−3.02933	−0.840184	−0.420092	+	0.907482i	$0.638002\pi$
$14$	0	0
$15$	−0.352526	−0.0910218
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.38098	−0.546234	−0.273117	−	0.961981i	$-0.588055\pi$
$19$	−2.38098	−0.546234	−0.273117	+	0.961981i	$0.588055\pi$
$20$	0	0
$21$	−2.53790	−0.553814
$22$	0	0
$23$	−5.51799	−1.15058	−0.575290	−	0.817949i	$-0.695111\pi$
$23$	−5.51799	−1.15058	−0.575290	+	0.817949i	$0.695111\pi$
$24$	0	0
$25$	−4.98071	−0.996141
$26$	0	0
$27$	1.11900	0.215352
$28$	0	0
$29$	−4.99682	−0.927887	−0.463943	−	0.885865i	$-0.653566\pi$
$29$	−4.99682	−0.927887	−0.463943	+	0.885865i	$0.653566\pi$
$30$	0	0
$31$	−9.85122	−1.76933	−0.884666	−	0.466225i	$-0.845614\pi$
$31$	−9.85122	−1.76933	−0.884666	+	0.466225i	$0.845614\pi$
$32$	0	0
$33$	−2.38098	−0.414475
$34$	0	0
$35$	0.138905	0.0234792
$36$	0	0
$37$	−6.85312	−1.12665	−0.563323	−	0.826237i	$-0.690477\pi$
$37$	−6.85312	−1.12665	−0.563323	+	0.826237i	$0.690477\pi$
$38$	0	0
$39$	−7.68812	−1.23108
$40$	0	0
$41$	−4.61308	−0.720442	−0.360221	−	0.932867i	$-0.617299\pi$
$41$	−4.61308	−0.720442	−0.360221	+	0.932867i	$0.617299\pi$
$42$	0	0
$43$	1.86156	0.283886	0.141943	−	0.989875i	$-0.454665\pi$
$43$	1.86156	0.283886	0.141943	+	0.989875i	$0.454665\pi$
$44$	0	0
$45$	−0.477960	−0.0712500
$46$	0	0
$47$	0.273295	0.0398641	0.0199321	−	0.999801i	$-0.493655\pi$
$47$	0.273295	0.0398641	0.0199321	+	0.999801i	$0.493655\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.53790	0.355377
$52$	0	0
$53$	6.18574	0.849676	0.424838	−	0.905269i	$-0.360331\pi$
$53$	6.18574	0.849676	0.424838	+	0.905269i	$0.360331\pi$
$54$	0	0
$55$	0.130316	0.0175719
$56$	0	0
$57$	−6.04268	−0.800373
$58$	0	0
$59$	2.88562	0.375676	0.187838	−	0.982200i	$-0.439852\pi$
$59$	2.88562	0.375676	0.187838	+	0.982200i	$0.439852\pi$
$60$	0	0
$61$	5.38822	0.689891	0.344946	−	0.938623i	$-0.387897\pi$
$61$	5.38822	0.689891	0.344946	+	0.938623i	$0.387897\pi$
$62$	0	0
$63$	−3.44092	−0.433515
$64$	0	0
$65$	0.420788	0.0521923
$66$	0	0
$67$	5.80476	0.709164	0.354582	−	0.935025i	$-0.384623\pi$
$67$	5.80476	0.709164	0.354582	+	0.935025i	$0.384623\pi$
$68$	0	0
$69$	−14.0041	−1.68589
$70$	0	0
$71$	11.1978	1.32894	0.664468	−	0.747317i	$-0.268658\pi$
$71$	11.1978	1.32894	0.664468	+	0.747317i	$0.268658\pi$
$72$	0	0
$73$	10.3579	1.21230	0.606150	−	0.795350i	$-0.292713\pi$
$73$	10.3579	1.21230	0.606150	+	0.795350i	$0.292713\pi$
$74$	0	0
$75$	−12.6405	−1.45960
$76$	0	0
$77$	0.938171	0.106914
$78$	0	0
$79$	5.96214	0.670793	0.335397	−	0.942077i	$-0.391130\pi$
$79$	5.96214	0.670793	0.335397	+	0.942077i	$0.391130\pi$
$80$	0	0
$81$	−7.48284	−0.831427
$82$	0	0
$83$	−8.92830	−0.980008	−0.490004	−	0.871720i	$-0.663005\pi$
$83$	−8.92830	−0.980008	−0.490004	+	0.871720i	$0.663005\pi$
$84$	0	0
$85$	−0.138905	−0.0150663
$86$	0	0
$87$	−12.6814	−1.35959
$88$	0	0
$89$	2.88635	0.305952	0.152976	−	0.988230i	$-0.451114\pi$
$89$	2.88635	0.305952	0.152976	+	0.988230i	$0.451114\pi$
$90$	0	0
$91$	3.02933	0.317560
$92$	0	0
$93$	−25.0014	−2.59252
$94$	0	0
$95$	0.330729	0.0339321
$96$	0	0
$97$	11.9327	1.21158	0.605791	−	0.795624i	$-0.292857\pi$
$97$	11.9327	1.21158	0.605791	+	0.795624i	$0.292857\pi$
$98$	0	0
$99$	−3.22817	−0.324443
$100$	0	0
$101$	−2.53513	−0.252255	−0.126128	−	0.992014i	$-0.540255\pi$
$101$	−2.53513	−0.252255	−0.126128	+	0.992014i	$0.540255\pi$
$102$	0	0
$103$	1.82731	0.180050	0.0900251	−	0.995939i	$-0.471305\pi$
$103$	1.82731	0.180050	0.0900251	+	0.995939i	$0.471305\pi$
$104$	0	0
$105$	0.352526	0.0344030
$106$	0	0
$107$	9.80598	0.947980	0.473990	−	0.880530i	$-0.342813\pi$
$107$	9.80598	0.947980	0.473990	+	0.880530i	$0.342813\pi$
$108$	0	0
$109$	−10.2351	−0.980342	−0.490171	−	0.871626i	$-0.663066\pi$
$109$	−10.2351	−0.980342	−0.490171	+	0.871626i	$0.663066\pi$
$110$	0	0
$111$	−17.3925	−1.65082
$112$	0	0
$113$	15.1561	1.42576	0.712881	−	0.701285i	$-0.247390\pi$
$113$	15.1561	1.42576	0.712881	+	0.701285i	$0.247390\pi$
$114$	0	0
$115$	0.766475	0.0714742
$116$	0	0
$117$	−10.4237	−0.963668
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−10.1198	−0.919985
$122$	0	0
$123$	−11.7075	−1.05563
$124$	0	0
$125$	1.38637	0.124000
$126$	0	0
$127$	−15.4576	−1.37164	−0.685819	−	0.727772i	$-0.740556\pi$
$127$	−15.4576	−1.37164	−0.685819	+	0.727772i	$0.740556\pi$
$128$	0	0
$129$	4.72445	0.415965
$130$	0	0
$131$	13.6280	1.19068	0.595342	−	0.803473i	$-0.297017\pi$
$131$	13.6280	1.19068	0.595342	+	0.803473i	$0.297017\pi$
$132$	0	0
$133$	2.38098	0.206457
$134$	0	0
$135$	−0.155435	−0.0133777
$136$	0	0
$137$	−4.07795	−0.348403	−0.174201	−	0.984710i	$-0.555734\pi$
$137$	−4.07795	−0.348403	−0.174201	+	0.984710i	$0.555734\pi$
$138$	0	0
$139$	−12.0637	−1.02323	−0.511613	−	0.859216i	$-0.670952\pi$
$139$	−12.0637	−1.02323	−0.511613	+	0.859216i	$0.670952\pi$
$140$	0	0
$141$	0.693594	0.0584111
$142$	0	0
$143$	2.84203	0.237662
$144$	0	0
$145$	0.694082	0.0576404
$146$	0	0
$147$	2.53790	0.209322
$148$	0	0
$149$	13.3610	1.09458	0.547288	−	0.836944i	$-0.315660\pi$
$149$	13.3610	1.09458	0.547288	+	0.836944i	$0.315660\pi$
$150$	0	0
$151$	−3.65958	−0.297813	−0.148906	−	0.988851i	$-0.547575\pi$
$151$	−3.65958	−0.297813	−0.148906	+	0.988851i	$0.547575\pi$
$152$	0	0
$153$	3.44092	0.278182
$154$	0	0
$155$	1.36838	0.109911
$156$	0	0
$157$	9.47275	0.756007	0.378004	−	0.925804i	$-0.376611\pi$
$157$	9.47275	0.756007	0.378004	+	0.925804i	$0.376611\pi$
$158$	0	0
$159$	15.6988	1.24499
$160$	0	0
$161$	5.51799	0.434879
$162$	0	0
$163$	−3.34703	−0.262159	−0.131080	−	0.991372i	$-0.541844\pi$
$163$	−3.34703	−0.262159	−0.131080	+	0.991372i	$0.541844\pi$
$164$	0	0
$165$	0.330729	0.0257472
$166$	0	0
$167$	−12.8902	−0.997470	−0.498735	−	0.866754i	$-0.666202\pi$
$167$	−12.8902	−0.997470	−0.498735	+	0.866754i	$0.666202\pi$
$168$	0	0
$169$	−3.82318	−0.294091
$170$	0	0
$171$	−8.19275	−0.626516
$172$	0	0
$173$	−11.8208	−0.898716	−0.449358	−	0.893352i	$-0.648347\pi$
$173$	−11.8208	−0.898716	−0.449358	+	0.893352i	$0.648347\pi$
$174$	0	0
$175$	4.98071	0.376506
$176$	0	0
$177$	7.32340	0.550460
$178$	0	0
$179$	−21.9257	−1.63880	−0.819401	−	0.573221i	$-0.805694\pi$
$179$	−21.9257	−1.63880	−0.819401	+	0.573221i	$0.805694\pi$
$180$	0	0
$181$	−9.65319	−0.717516	−0.358758	−	0.933431i	$-0.616800\pi$
$181$	−9.65319	−0.717516	−0.358758	+	0.933431i	$0.616800\pi$
$182$	0	0
$183$	13.6747	1.01087
$184$	0	0
$185$	0.951930	0.0699873
$186$	0	0
$187$	−0.938171	−0.0686058
$188$	0	0
$189$	−1.11900	−0.0813953
$190$	0	0
$191$	−26.1319	−1.89084	−0.945419	−	0.325857i	$-0.894347\pi$
$191$	−26.1319	−1.89084	−0.945419	+	0.325857i	$0.894347\pi$
$192$	0	0
$193$	7.23660	0.520902	0.260451	−	0.965487i	$-0.416129\pi$
$193$	7.23660	0.520902	0.260451	+	0.965487i	$0.416129\pi$
$194$	0	0
$195$	1.06792	0.0764750
$196$	0	0
$197$	−4.65989	−0.332003	−0.166002	−	0.986125i	$-0.553086\pi$
$197$	−4.65989	−0.332003	−0.166002	+	0.986125i	$0.553086\pi$
$198$	0	0
$199$	−3.39673	−0.240788	−0.120394	−	0.992726i	$-0.538416\pi$
$199$	−3.39673	−0.240788	−0.120394	+	0.992726i	$0.538416\pi$
$200$	0	0
$201$	14.7319	1.03911
$202$	0	0
$203$	4.99682	0.350708
$204$	0	0
$205$	0.640778	0.0447539
$206$	0	0
$207$	−18.9870	−1.31968
$208$	0	0
$209$	2.23377	0.154513
$210$	0	0
$211$	2.15000	0.148012	0.0740060	−	0.997258i	$-0.476422\pi$
$211$	2.15000	0.148012	0.0740060	+	0.997258i	$0.476422\pi$
$212$	0	0
$213$	28.4189	1.94723
$214$	0	0
$215$	−0.258580	−0.0176350
$216$	0	0
$217$	9.85122	0.668745
$218$	0	0
$219$	26.2873	1.77633
$220$	0	0
$221$	−3.02933	−0.203775
$222$	0	0
$223$	−18.1006	−1.21211	−0.606053	−	0.795424i	$-0.707248\pi$
$223$	−18.1006	−1.21211	−0.606053	+	0.795424i	$0.707248\pi$
$224$	0	0
$225$	−17.1382	−1.14255
$226$	0	0
$227$	−15.5802	−1.03410	−0.517049	−	0.855956i	$-0.672969\pi$
$227$	−15.5802	−1.03410	−0.517049	+	0.855956i	$0.672969\pi$
$228$	0	0
$229$	22.0701	1.45844	0.729218	−	0.684282i	$-0.239884\pi$
$229$	22.0701	1.45844	0.729218	+	0.684282i	$0.239884\pi$
$230$	0	0
$231$	2.38098	0.156657
$232$	0	0
$233$	−3.15371	−0.206607	−0.103303	−	0.994650i	$-0.532941\pi$
$233$	−3.15371	−0.206607	−0.103303	+	0.994650i	$0.532941\pi$
$234$	0	0
$235$	−0.0379619	−0.00247636
$236$	0	0
$237$	15.1313	0.982883
$238$	0	0
$239$	−1.33493	−0.0863492	−0.0431746	−	0.999068i	$-0.513747\pi$
$239$	−1.33493	−0.0863492	−0.0431746	+	0.999068i	$0.513747\pi$
$240$	0	0
$241$	14.7400	0.949487	0.474743	−	0.880124i	$-0.342541\pi$
$241$	14.7400	0.949487	0.474743	+	0.880124i	$0.342541\pi$
$242$	0	0
$243$	−22.3477	−1.43360
$244$	0	0
$245$	−0.138905	−0.00887430
$246$	0	0
$247$	7.21277	0.458937
$248$	0	0
$249$	−22.6591	−1.43596
$250$	0	0
$251$	−25.0232	−1.57945	−0.789725	−	0.613462i	$-0.789777\pi$
$251$	−25.0232	−1.57945	−0.789725	+	0.613462i	$0.789777\pi$
$252$	0	0
$253$	5.17682	0.325464
$254$	0	0
$255$	−0.352526	−0.0220760
$256$	0	0
$257$	19.9222	1.24271	0.621356	−	0.783528i	$-0.286582\pi$
$257$	19.9222	1.24271	0.621356	+	0.783528i	$0.286582\pi$
$258$	0	0
$259$	6.85312	0.425832
$260$	0	0
$261$	−17.1937	−1.06426
$262$	0	0
$263$	−22.2979	−1.37495	−0.687473	−	0.726210i	$-0.741280\pi$
$263$	−22.2979	−1.37495	−0.687473	+	0.726210i	$0.741280\pi$
$264$	0	0
$265$	−0.859228	−0.0527820
$266$	0	0
$267$	7.32525	0.448298
$268$	0	0
$269$	−10.9673	−0.668686	−0.334343	−	0.942451i	$-0.608514\pi$
$269$	−10.9673	−0.668686	−0.334343	+	0.942451i	$0.608514\pi$
$270$	0	0
$271$	7.55873	0.459160	0.229580	−	0.973290i	$-0.426265\pi$
$271$	7.55873	0.459160	0.229580	+	0.973290i	$0.426265\pi$
$272$	0	0
$273$	7.68812	0.465306
$274$	0	0
$275$	4.67275	0.281778
$276$	0	0
$277$	25.7682	1.54826	0.774130	−	0.633027i	$-0.218188\pi$
$277$	25.7682	1.54826	0.774130	+	0.633027i	$0.218188\pi$
$278$	0	0
$279$	−33.8972	−2.02937
$280$	0	0
$281$	−3.20981	−0.191481	−0.0957405	−	0.995406i	$-0.530522\pi$
$281$	−3.20981	−0.191481	−0.0957405	+	0.995406i	$0.530522\pi$
$282$	0	0
$283$	5.57421	0.331353	0.165676	−	0.986180i	$-0.447019\pi$
$283$	5.57421	0.331353	0.165676	+	0.986180i	$0.447019\pi$
$284$	0	0
$285$	0.839357	0.0497192
$286$	0	0
$287$	4.61308	0.272301
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	30.2839	1.77528
$292$	0	0
$293$	23.0232	1.34503	0.672514	−	0.740084i	$-0.265214\pi$
$293$	23.0232	1.34503	0.672514	+	0.740084i	$0.265214\pi$
$294$	0	0
$295$	−0.400826	−0.0233370
$296$	0	0
$297$	−1.04981	−0.0609164
$298$	0	0
$299$	16.7158	0.966700
$300$	0	0
$301$	−1.86156	−0.107299
$302$	0	0
$303$	−6.43390	−0.369618
$304$	0	0
$305$	−0.748449	−0.0428561
$306$	0	0
$307$	−11.7365	−0.669839	−0.334920	−	0.942247i	$-0.608709\pi$
$307$	−11.7365	−0.669839	−0.334920	+	0.942247i	$0.608709\pi$
$308$	0	0
$309$	4.63752	0.263819
$310$	0	0
$311$	−12.8083	−0.726293	−0.363146	−	0.931732i	$-0.618297\pi$
$311$	−12.8083	−0.726293	−0.363146	+	0.931732i	$0.618297\pi$
$312$	0	0
$313$	−31.3719	−1.77325	−0.886624	−	0.462492i	$-0.846955\pi$
$313$	−31.3719	−1.77325	−0.886624	+	0.462492i	$0.846955\pi$
$314$	0	0
$315$	0.477960	0.0269300
$316$	0	0
$317$	29.2659	1.64374	0.821869	−	0.569677i	$-0.192932\pi$
$317$	29.2659	1.64374	0.821869	+	0.569677i	$0.192932\pi$
$318$	0	0
$319$	4.68787	0.262471
$320$	0	0
$321$	24.8866	1.38903
$322$	0	0
$323$	−2.38098	−0.132481
$324$	0	0
$325$	15.0882	0.836942
$326$	0	0
$327$	−25.9755	−1.43645
$328$	0	0
$329$	−0.273295	−0.0150672
$330$	0	0
$331$	11.0653	0.608202	0.304101	−	0.952640i	$-0.401644\pi$
$331$	11.0653	0.608202	0.304101	+	0.952640i	$0.401644\pi$
$332$	0	0
$333$	−23.5810	−1.29223
$334$	0	0
$335$	−0.806308	−0.0440533
$336$	0	0
$337$	29.8288	1.62488	0.812440	−	0.583045i	$-0.198139\pi$
$337$	29.8288	1.62488	0.812440	+	0.583045i	$0.198139\pi$
$338$	0	0
$339$	38.4645	2.08911
$340$	0	0
$341$	9.24213	0.500489
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.94523	0.104728
$346$	0	0
$347$	−7.22486	−0.387851	−0.193925	−	0.981016i	$-0.562122\pi$
$347$	−7.22486	−0.387851	−0.193925	+	0.981016i	$0.562122\pi$
$348$	0	0
$349$	18.8155	1.00717	0.503584	−	0.863946i	$-0.332014\pi$
$349$	18.8155	1.00717	0.503584	+	0.863946i	$0.332014\pi$
$350$	0	0
$351$	−3.38982	−0.180935
$352$	0	0
$353$	−3.99112	−0.212426	−0.106213	−	0.994343i	$-0.533873\pi$
$353$	−3.99112	−0.212426	−0.106213	+	0.994343i	$0.533873\pi$
$354$	0	0
$355$	−1.55543	−0.0825536
$356$	0	0
$357$	−2.53790	−0.134320
$358$	0	0
$359$	−9.36828	−0.494439	−0.247219	−	0.968960i	$-0.579517\pi$
$359$	−9.36828	−0.494439	−0.247219	+	0.968960i	$0.579517\pi$
$360$	0	0
$361$	−13.3309	−0.701628
$362$	0	0
$363$	−25.6831	−1.34801
$364$	0	0
$365$	−1.43876	−0.0753082
$366$	0	0
$367$	−15.9267	−0.831368	−0.415684	−	0.909509i	$-0.636458\pi$
$367$	−15.9267	−0.831368	−0.415684	+	0.909509i	$0.636458\pi$
$368$	0	0
$369$	−15.8732	−0.826327
$370$	0	0
$371$	−6.18574	−0.321148
$372$	0	0
$373$	−17.4079	−0.901347	−0.450674	−	0.892689i	$-0.648816\pi$
$373$	−17.4079	−0.901347	−0.450674	+	0.892689i	$0.648816\pi$
$374$	0	0
$375$	3.51846	0.181692
$376$	0	0
$377$	15.1370	0.779596
$378$	0	0
$379$	24.9504	1.28161	0.640807	−	0.767702i	$-0.278600\pi$
$379$	24.9504	1.28161	0.640807	+	0.767702i	$0.278600\pi$
$380$	0	0
$381$	−39.2297	−2.00980
$382$	0	0
$383$	23.5500	1.20335	0.601675	−	0.798741i	$-0.294500\pi$
$383$	23.5500	1.20335	0.601675	+	0.798741i	$0.294500\pi$
$384$	0	0
$385$	−0.130316	−0.00664154
$386$	0	0
$387$	6.40548	0.325609
$388$	0	0
$389$	17.3562	0.879995	0.439997	−	0.897999i	$-0.354979\pi$
$389$	17.3562	0.879995	0.439997	+	0.897999i	$0.354979\pi$
$390$	0	0
$391$	−5.51799	−0.279057
$392$	0	0
$393$	34.5864	1.74465
$394$	0	0
$395$	−0.828169	−0.0416697
$396$	0	0
$397$	−27.6590	−1.38817	−0.694084	−	0.719894i	$-0.744190\pi$
$397$	−27.6590	−1.38817	−0.694084	+	0.719894i	$0.744190\pi$
$398$	0	0
$399$	6.04268	0.302512
$400$	0	0
$401$	2.40006	0.119853	0.0599266	−	0.998203i	$-0.480913\pi$
$401$	2.40006	0.119853	0.0599266	+	0.998203i	$0.480913\pi$
$402$	0	0
$403$	29.8426	1.48656
$404$	0	0
$405$	1.03940	0.0516483
$406$	0	0
$407$	6.42939	0.318693
$408$	0	0
$409$	−20.2419	−1.00090	−0.500450	−	0.865765i	$-0.666832\pi$
$409$	−20.2419	−1.00090	−0.500450	+	0.865765i	$0.666832\pi$
$410$	0	0
$411$	−10.3494	−0.510499
$412$	0	0
$413$	−2.88562	−0.141992
$414$	0	0
$415$	1.24018	0.0608782
$416$	0	0
$417$	−30.6163	−1.49929
$418$	0	0
$419$	31.9192	1.55935	0.779677	−	0.626181i	$-0.215383\pi$
$419$	31.9192	1.55935	0.779677	+	0.626181i	$0.215383\pi$
$420$	0	0
$421$	19.5192	0.951309	0.475655	−	0.879632i	$-0.342211\pi$
$421$	19.5192	0.951309	0.475655	+	0.879632i	$0.342211\pi$
$422$	0	0
$423$	0.940384	0.0457230
$424$	0	0
$425$	−4.98071	−0.241600
$426$	0	0
$427$	−5.38822	−0.260754
$428$	0	0
$429$	7.21277	0.348236
$430$	0	0
$431$	−17.7756	−0.856222	−0.428111	−	0.903726i	$-0.640821\pi$
$431$	−17.7756	−0.856222	−0.428111	+	0.903726i	$0.640821\pi$
$432$	0	0
$433$	−16.1099	−0.774194	−0.387097	−	0.922039i	$-0.626522\pi$
$433$	−16.1099	−0.774194	−0.387097	+	0.922039i	$0.626522\pi$
$434$	0	0
$435$	1.76151	0.0844579
$436$	0	0
$437$	13.1382	0.628487
$438$	0	0
$439$	10.6115	0.506461	0.253230	−	0.967406i	$-0.418507\pi$
$439$	10.6115	0.506461	0.253230	+	0.967406i	$0.418507\pi$
$440$	0	0
$441$	3.44092	0.163853
$442$	0	0
$443$	5.75390	0.273376	0.136688	−	0.990614i	$-0.456354\pi$
$443$	5.75390	0.273376	0.136688	+	0.990614i	$0.456354\pi$
$444$	0	0
$445$	−0.400927	−0.0190058
$446$	0	0
$447$	33.9089	1.60383
$448$	0	0
$449$	−24.4469	−1.15372	−0.576860	−	0.816843i	$-0.695722\pi$
$449$	−24.4469	−1.15372	−0.576860	+	0.816843i	$0.695722\pi$
$450$	0	0
$451$	4.32786	0.203791
$452$	0	0
$453$	−9.28764	−0.436371
$454$	0	0
$455$	−0.420788	−0.0197268
$456$	0	0
$457$	−6.71086	−0.313921	−0.156960	−	0.987605i	$-0.550170\pi$
$457$	−6.71086	−0.313921	−0.156960	+	0.987605i	$0.550170\pi$
$458$	0	0
$459$	1.11900	0.0522305
$460$	0	0
$461$	36.0532	1.67917	0.839583	−	0.543231i	$-0.182799\pi$
$461$	36.0532	1.67917	0.839583	+	0.543231i	$0.182799\pi$
$462$	0	0
$463$	26.0397	1.21017	0.605083	−	0.796162i	$-0.293140\pi$
$463$	26.0397	1.21017	0.605083	+	0.796162i	$0.293140\pi$
$464$	0	0
$465$	3.47281	0.161048
$466$	0	0
$467$	−14.7790	−0.683889	−0.341945	−	0.939720i	$-0.611086\pi$
$467$	−14.7790	−0.683889	−0.341945	+	0.939720i	$0.611086\pi$
$468$	0	0
$469$	−5.80476	−0.268039
$470$	0	0
$471$	24.0408	1.10774
$472$	0	0
$473$	−1.74646	−0.0803025
$474$	0	0
$475$	11.8590	0.544126
$476$	0	0
$477$	21.2846	0.974555
$478$	0	0
$479$	10.0317	0.458360	0.229180	−	0.973384i	$-0.426396\pi$
$479$	10.0317	0.458360	0.229180	+	0.973384i	$0.426396\pi$
$480$	0	0
$481$	20.7603	0.946589
$482$	0	0
$483$	14.0041	0.637208
$484$	0	0
$485$	−1.65751	−0.0752636
$486$	0	0
$487$	−20.9436	−0.949047	−0.474524	−	0.880243i	$-0.657380\pi$
$487$	−20.9436	−0.949047	−0.474524	+	0.880243i	$0.657380\pi$
$488$	0	0
$489$	−8.49440	−0.384130
$490$	0	0
$491$	−8.90982	−0.402095	−0.201047	−	0.979582i	$-0.564435\pi$
$491$	−8.90982	−0.402095	−0.201047	+	0.979582i	$0.564435\pi$
$492$	0	0
$493$	−4.99682	−0.225046
$494$	0	0
$495$	0.448408	0.0201544
$496$	0	0
$497$	−11.1978	−0.502290
$498$	0	0
$499$	25.3335	1.13408	0.567041	−	0.823690i	$-0.308088\pi$
$499$	25.3335	1.13408	0.567041	+	0.823690i	$0.308088\pi$
$500$	0	0
$501$	−32.7139	−1.46155
$502$	0	0
$503$	−6.23099	−0.277826	−0.138913	−	0.990305i	$-0.544361\pi$
$503$	−6.23099	−0.277826	−0.138913	+	0.990305i	$0.544361\pi$
$504$	0	0
$505$	0.352142	0.0156701
$506$	0	0
$507$	−9.70283	−0.430918
$508$	0	0
$509$	−15.7013	−0.695949	−0.347974	−	0.937504i	$-0.613130\pi$
$509$	−15.7013	−0.695949	−0.347974	+	0.937504i	$0.613130\pi$
$510$	0	0
$511$	−10.3579	−0.458207
$512$	0	0
$513$	−2.66432	−0.117633
$514$	0	0
$515$	−0.253822	−0.0111847
$516$	0	0
$517$	−0.256397	−0.0112763
$518$	0	0
$519$	−29.9999	−1.31685
$520$	0	0
$521$	−6.95507	−0.304707	−0.152354	−	0.988326i	$-0.548685\pi$
$521$	−6.95507	−0.304707	−0.152354	+	0.988326i	$0.548685\pi$
$522$	0	0
$523$	26.8463	1.17391	0.586954	−	0.809620i	$-0.300327\pi$
$523$	26.8463	1.17391	0.586954	+	0.809620i	$0.300327\pi$
$524$	0	0
$525$	12.6405	0.551677
$526$	0	0
$527$	−9.85122	−0.429126
$528$	0	0
$529$	7.44824	0.323837
$530$	0	0
$531$	9.92917	0.430889
$532$	0	0
$533$	13.9745	0.605304
$534$	0	0
$535$	−1.36210	−0.0588886
$536$	0	0
$537$	−55.6451	−2.40126
$538$	0	0
$539$	−0.938171	−0.0404099
$540$	0	0
$541$	23.3480	1.00381	0.501904	−	0.864923i	$-0.332633\pi$
$541$	23.3480	1.00381	0.501904	+	0.864923i	$0.332633\pi$
$542$	0	0
$543$	−24.4988	−1.05134
$544$	0	0
$545$	1.42170	0.0608989
$546$	0	0
$547$	−0.980079	−0.0419052	−0.0209526	−	0.999780i	$-0.506670\pi$
$547$	−0.980079	−0.0419052	−0.0209526	+	0.999780i	$0.506670\pi$
$548$	0	0
$549$	18.5404	0.791286
$550$	0	0
$551$	11.8973	0.506844
$552$	0	0
$553$	−5.96214	−0.253536
$554$	0	0
$555$	2.41590	0.102549
$556$	0	0
$557$	2.24073	0.0949428	0.0474714	−	0.998873i	$-0.484884\pi$
$557$	2.24073	0.0949428	0.0474714	+	0.998873i	$0.484884\pi$
$558$	0	0
$559$	−5.63928	−0.238516
$560$	0	0
$561$	−2.38098	−0.100525
$562$	0	0
$563$	−10.5876	−0.446213	−0.223107	−	0.974794i	$-0.571620\pi$
$563$	−10.5876	−0.446213	−0.223107	+	0.974794i	$0.571620\pi$
$564$	0	0
$565$	−2.10525	−0.0885685
$566$	0	0
$567$	7.48284	0.314250
$568$	0	0
$569$	−30.0567	−1.26004	−0.630021	−	0.776578i	$-0.716954\pi$
$569$	−30.0567	−1.26004	−0.630021	+	0.776578i	$0.716954\pi$
$570$	0	0
$571$	17.8476	0.746900	0.373450	−	0.927650i	$-0.378175\pi$
$571$	17.8476	0.746900	0.373450	+	0.927650i	$0.378175\pi$
$572$	0	0
$573$	−66.3201	−2.77056
$574$	0	0
$575$	27.4835	1.14614
$576$	0	0
$577$	−0.188036	−0.00782805	−0.00391402	−	0.999992i	$-0.501246\pi$
$577$	−0.188036	−0.00782805	−0.00391402	+	0.999992i	$0.501246\pi$
$578$	0	0
$579$	18.3657	0.763254
$580$	0	0
$581$	8.92830	0.370408
$582$	0	0
$583$	−5.80328	−0.240347
$584$	0	0
$585$	1.44790	0.0598631
$586$	0	0
$587$	−36.4145	−1.50299	−0.751493	−	0.659742i	$-0.770666\pi$
$587$	−36.4145	−1.50299	−0.751493	+	0.659742i	$0.770666\pi$
$588$	0	0
$589$	23.4556	0.966470
$590$	0	0
$591$	−11.8263	−0.486469
$592$	0	0
$593$	3.58451	0.147198	0.0735990	−	0.997288i	$-0.476551\pi$
$593$	3.58451	0.147198	0.0735990	+	0.997288i	$0.476551\pi$
$594$	0	0
$595$	0.138905	0.00569454
$596$	0	0
$597$	−8.62055	−0.352816
$598$	0	0
$599$	−32.3830	−1.32313	−0.661566	−	0.749887i	$-0.730108\pi$
$599$	−32.3830	−1.32313	−0.661566	+	0.749887i	$0.730108\pi$
$600$	0	0
$601$	−10.9279	−0.445757	−0.222879	−	0.974846i	$-0.571545\pi$
$601$	−10.9279	−0.445757	−0.222879	+	0.974846i	$0.571545\pi$
$602$	0	0
$603$	19.9737	0.813391
$604$	0	0
$605$	1.40569	0.0571495
$606$	0	0
$607$	11.2301	0.455818	0.227909	−	0.973682i	$-0.426811\pi$
$607$	11.2301	0.455818	0.227909	+	0.973682i	$0.426811\pi$
$608$	0	0
$609$	12.6814	0.513877
$610$	0	0
$611$	−0.827899	−0.0334932
$612$	0	0
$613$	−14.0099	−0.565854	−0.282927	−	0.959141i	$-0.591305\pi$
$613$	−14.0099	−0.565854	−0.282927	+	0.959141i	$0.591305\pi$
$614$	0	0
$615$	1.62623	0.0655759
$616$	0	0
$617$	2.36529	0.0952231	0.0476115	−	0.998866i	$-0.484839\pi$
$617$	2.36529	0.0952231	0.0476115	+	0.998866i	$0.484839\pi$
$618$	0	0
$619$	−30.4323	−1.22318	−0.611588	−	0.791177i	$-0.709469\pi$
$619$	−30.4323	−1.22318	−0.611588	+	0.791177i	$0.709469\pi$
$620$	0	0
$621$	−6.17464	−0.247780
$622$	0	0
$623$	−2.88635	−0.115639
$624$	0	0
$625$	24.7110	0.988438
$626$	0	0
$627$	5.66907	0.226401
$628$	0	0
$629$	−6.85312	−0.273252
$630$	0	0
$631$	−2.85815	−0.113781	−0.0568906	−	0.998380i	$-0.518119\pi$
$631$	−2.85815	−0.113781	−0.0568906	+	0.998380i	$0.518119\pi$
$632$	0	0
$633$	5.45647	0.216875
$634$	0	0
$635$	2.14713	0.0852063
$636$	0	0
$637$	−3.02933	−0.120026
$638$	0	0
$639$	38.5307	1.52425
$640$	0	0
$641$	−38.3550	−1.51493	−0.757465	−	0.652875i	$-0.773562\pi$
$641$	−38.3550	−1.51493	−0.757465	+	0.652875i	$0.773562\pi$
$642$	0	0
$643$	1.82210	0.0718564	0.0359282	−	0.999354i	$-0.488561\pi$
$643$	1.82210	0.0718564	0.0359282	+	0.999354i	$0.488561\pi$
$644$	0	0
$645$	−0.656249	−0.0258398
$646$	0	0
$647$	−30.1683	−1.18604	−0.593019	−	0.805188i	$-0.702064\pi$
$647$	−30.1683	−1.18604	−0.593019	+	0.805188i	$0.702064\pi$
$648$	0	0
$649$	−2.70720	−0.106267
$650$	0	0
$651$	25.0014	0.979881
$652$	0	0
$653$	−15.9106	−0.622632	−0.311316	−	0.950306i	$-0.600770\pi$
$653$	−15.9106	−0.622632	−0.311316	+	0.950306i	$0.600770\pi$
$654$	0	0
$655$	−1.89299	−0.0739653
$656$	0	0
$657$	35.6407	1.39048
$658$	0	0
$659$	−47.3825	−1.84576	−0.922880	−	0.385088i	$-0.874171\pi$
$659$	−47.3825	−1.84576	−0.922880	+	0.385088i	$0.874171\pi$
$660$	0	0
$661$	15.9297	0.619594	0.309797	−	0.950803i	$-0.399739\pi$
$661$	15.9297	0.619594	0.309797	+	0.950803i	$0.399739\pi$
$662$	0	0
$663$	−7.68812	−0.298582
$664$	0	0
$665$	−0.330729	−0.0128251
$666$	0	0
$667$	27.5724	1.06761
$668$	0	0
$669$	−45.9375	−1.77604
$670$	0	0
$671$	−5.05507	−0.195149
$672$	0	0
$673$	−50.8478	−1.96004	−0.980020	−	0.198898i	$-0.936264\pi$
$673$	−50.8478	−1.96004	−0.980020	+	0.198898i	$0.936264\pi$
$674$	0	0
$675$	−5.57341	−0.214521
$676$	0	0
$677$	9.74262	0.374439	0.187220	−	0.982318i	$-0.440052\pi$
$677$	9.74262	0.374439	0.187220	+	0.982318i	$0.440052\pi$
$678$	0	0
$679$	−11.9327	−0.457935
$680$	0	0
$681$	−39.5410	−1.51522
$682$	0	0
$683$	−37.5069	−1.43516	−0.717581	−	0.696475i	$-0.754751\pi$
$683$	−37.5069	−1.43516	−0.717581	+	0.696475i	$0.754751\pi$
$684$	0	0
$685$	0.566446	0.0216428
$686$	0	0
$687$	56.0117	2.13698
$688$	0	0
$689$	−18.7386	−0.713885
$690$	0	0
$691$	−8.46101	−0.321872	−0.160936	−	0.986965i	$-0.551451\pi$
$691$	−8.46101	−0.321872	−0.160936	+	0.986965i	$0.551451\pi$
$692$	0	0
$693$	3.22817	0.122628
$694$	0	0
$695$	1.67570	0.0635629
$696$	0	0
$697$	−4.61308	−0.174733
$698$	0	0
$699$	−8.00380	−0.302732
$700$	0	0
$701$	7.07563	0.267243	0.133621	−	0.991032i	$-0.457339\pi$
$701$	7.07563	0.267243	0.133621	+	0.991032i	$0.457339\pi$
$702$	0	0
$703$	16.3171	0.615412
$704$	0	0
$705$	−0.0963434	−0.00362850
$706$	0	0
$707$	2.53513	0.0953434
$708$	0	0
$709$	17.7504	0.666632	0.333316	−	0.942815i	$-0.391832\pi$
$709$	17.7504	0.666632	0.333316	+	0.942815i	$0.391832\pi$
$710$	0	0
$711$	20.5152	0.769381
$712$	0	0
$713$	54.3590	2.03576
$714$	0	0
$715$	−0.394771	−0.0147636
$716$	0	0
$717$	−3.38790	−0.126524
$718$	0	0
$719$	27.2756	1.01721	0.508604	−	0.861000i	$-0.330162\pi$
$719$	27.2756	1.01721	0.508604	+	0.861000i	$0.330162\pi$
$720$	0	0
$721$	−1.82731	−0.0680526
$722$	0	0
$723$	37.4086	1.39124
$724$	0	0
$725$	24.8877	0.924306
$726$	0	0
$727$	−20.7713	−0.770365	−0.385182	−	0.922840i	$-0.625861\pi$
$727$	−20.7713	−0.770365	−0.385182	+	0.922840i	$0.625861\pi$
$728$	0	0
$729$	−34.2676	−1.26917
$730$	0	0
$731$	1.86156	0.0688524
$732$	0	0
$733$	−46.9928	−1.73572	−0.867859	−	0.496810i	$-0.834505\pi$
$733$	−46.9928	−1.73572	−0.867859	+	0.496810i	$0.834505\pi$
$734$	0	0
$735$	−0.352526	−0.0130031
$736$	0	0
$737$	−5.44585	−0.200601
$738$	0	0
$739$	28.8966	1.06298	0.531490	−	0.847065i	$-0.321632\pi$
$739$	28.8966	1.06298	0.531490	+	0.847065i	$0.321632\pi$
$740$	0	0
$741$	18.3053	0.672460
$742$	0	0
$743$	−11.3956	−0.418064	−0.209032	−	0.977909i	$-0.567031\pi$
$743$	−11.3956	−0.418064	−0.209032	+	0.977909i	$0.567031\pi$
$744$	0	0
$745$	−1.85591	−0.0679952
$746$	0	0
$747$	−30.7215	−1.12404
$748$	0	0
$749$	−9.80598	−0.358303
$750$	0	0
$751$	17.1482	0.625746	0.312873	−	0.949795i	$-0.398709\pi$
$751$	17.1482	0.625746	0.312873	+	0.949795i	$0.398709\pi$
$752$	0	0
$753$	−63.5062	−2.31430
$754$	0	0
$755$	0.508333	0.0185001
$756$	0	0
$757$	37.2242	1.35293	0.676467	−	0.736473i	$-0.263510\pi$
$757$	37.2242	1.35293	0.676467	+	0.736473i	$0.263510\pi$
$758$	0	0
$759$	13.1382	0.476888
$760$	0	0
$761$	−9.38531	−0.340217	−0.170109	−	0.985425i	$-0.554412\pi$
$761$	−9.38531	−0.340217	−0.170109	+	0.985425i	$0.554412\pi$
$762$	0	0
$763$	10.2351	0.370534
$764$	0	0
$765$	−0.477960	−0.0172807
$766$	0	0
$767$	−8.74148	−0.315637
$768$	0	0
$769$	7.29898	0.263208	0.131604	−	0.991302i	$-0.457987\pi$
$769$	7.29898	0.263208	0.131604	+	0.991302i	$0.457987\pi$
$770$	0	0
$771$	50.5604	1.82089
$772$	0	0
$773$	−12.3839	−0.445418	−0.222709	−	0.974885i	$-0.571490\pi$
$773$	−12.3839	−0.445418	−0.222709	+	0.974885i	$0.571490\pi$
$774$	0	0
$775$	49.0660	1.76250
$776$	0	0
$777$	17.3925	0.623952
$778$	0	0
$779$	10.9837	0.393530
$780$	0	0
$781$	−10.5055	−0.375915
$782$	0	0
$783$	−5.59145	−0.199822
$784$	0	0
$785$	−1.31581	−0.0469632
$786$	0	0
$787$	38.5306	1.37347	0.686733	−	0.726910i	$-0.259044\pi$
$787$	38.5306	1.37347	0.686733	+	0.726910i	$0.259044\pi$
$788$	0	0
$789$	−56.5897	−2.01465
$790$	0	0
$791$	−15.1561	−0.538888
$792$	0	0
$793$	−16.3227	−0.579636
$794$	0	0
$795$	−2.18063	−0.0773390
$796$	0	0
$797$	20.7736	0.735839	0.367920	−	0.929858i	$-0.380070\pi$
$797$	20.7736	0.735839	0.367920	+	0.929858i	$0.380070\pi$
$798$	0	0
$799$	0.273295	0.00966847
$800$	0	0
$801$	9.93168	0.350919
$802$	0	0
$803$	−9.71748	−0.342922
$804$	0	0
$805$	−0.766475	−0.0270147
$806$	0	0
$807$	−27.8338	−0.979796
$808$	0	0
$809$	4.39047	0.154361	0.0771803	−	0.997017i	$-0.475408\pi$
$809$	4.39047	0.154361	0.0771803	+	0.997017i	$0.475408\pi$
$810$	0	0
$811$	−21.0547	−0.739330	−0.369665	−	0.929165i	$-0.620528\pi$
$811$	−21.0547	−0.739330	−0.369665	+	0.929165i	$0.620528\pi$
$812$	0	0
$813$	19.1833	0.672787
$814$	0	0
$815$	0.464918	0.0162854
$816$	0	0
$817$	−4.43234	−0.155068
$818$	0	0
$819$	10.4237	0.364232
$820$	0	0
$821$	36.9979	1.29124	0.645618	−	0.763661i	$-0.276600\pi$
$821$	36.9979	1.29124	0.645618	+	0.763661i	$0.276600\pi$
$822$	0	0
$823$	−40.4809	−1.41107	−0.705537	−	0.708673i	$-0.749294\pi$
$823$	−40.4809	−1.41107	−0.705537	+	0.708673i	$0.749294\pi$
$824$	0	0
$825$	11.8590	0.412876
$826$	0	0
$827$	7.83822	0.272562	0.136281	−	0.990670i	$-0.456485\pi$
$827$	7.83822	0.272562	0.136281	+	0.990670i	$0.456485\pi$
$828$	0	0
$829$	14.3043	0.496810	0.248405	−	0.968656i	$-0.420094\pi$
$829$	14.3043	0.496810	0.248405	+	0.968656i	$0.420094\pi$
$830$	0	0
$831$	65.3969	2.26859
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	1.79050	0.0619629
$836$	0	0
$837$	−11.0235	−0.381029
$838$	0	0
$839$	28.6526	0.989198	0.494599	−	0.869121i	$-0.335315\pi$
$839$	28.6526	0.989198	0.494599	+	0.869121i	$0.335315\pi$
$840$	0	0
$841$	−4.03175	−0.139026
$842$	0	0
$843$	−8.14615	−0.280568
$844$	0	0
$845$	0.531058	0.0182689
$846$	0	0
$847$	10.1198	0.347722
$848$	0	0
$849$	14.1468	0.485516
$850$	0	0
$851$	37.8154	1.29630
$852$	0	0
$853$	−36.5582	−1.25173	−0.625865	−	0.779931i	$-0.715254\pi$
$853$	−36.5582	−1.25173	−0.625865	+	0.779931i	$0.715254\pi$
$854$	0	0
$855$	1.13801	0.0389192
$856$	0	0
$857$	−9.14903	−0.312525	−0.156262	−	0.987716i	$-0.549945\pi$
$857$	−9.14903	−0.312525	−0.156262	+	0.987716i	$0.549945\pi$
$858$	0	0
$859$	−12.9610	−0.442223	−0.221112	−	0.975248i	$-0.570969\pi$
$859$	−12.9610	−0.442223	−0.221112	+	0.975248i	$0.570969\pi$
$860$	0	0
$861$	11.7075	0.398991
$862$	0	0
$863$	7.92253	0.269686	0.134843	−	0.990867i	$-0.456947\pi$
$863$	7.92253	0.269686	0.134843	+	0.990867i	$0.456947\pi$
$864$	0	0
$865$	1.64196	0.0558283
$866$	0	0
$867$	2.53790	0.0861915
$868$	0	0
$869$	−5.59351	−0.189747
$870$	0	0
$871$	−17.5845	−0.595828
$872$	0	0
$873$	41.0594	1.38965
$874$	0	0
$875$	−1.38637	−0.0468678
$876$	0	0
$877$	−2.34688	−0.0792484	−0.0396242	−	0.999215i	$-0.512616\pi$
$877$	−2.34688	−0.0792484	−0.0396242	+	0.999215i	$0.512616\pi$
$878$	0	0
$879$	58.4304	1.97081
$880$	0	0
$881$	33.2775	1.12115	0.560573	−	0.828105i	$-0.310581\pi$
$881$	33.2775	1.12115	0.560573	+	0.828105i	$0.310581\pi$
$882$	0	0
$883$	15.6831	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$883$	15.6831	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$884$	0	0
$885$	−1.01725	−0.0341946
$886$	0	0
$887$	1.95317	0.0655811	0.0327906	−	0.999462i	$-0.489561\pi$
$887$	1.95317	0.0655811	0.0327906	+	0.999462i	$0.489561\pi$
$888$	0	0
$889$	15.4576	0.518431
$890$	0	0
$891$	7.02018	0.235185
$892$	0	0
$893$	−0.650709	−0.0217752
$894$	0	0
$895$	3.04558	0.101802
$896$	0	0
$897$	42.4230	1.41646
$898$	0	0
$899$	49.2248	1.64174
$900$	0	0
$901$	6.18574	0.206077
$902$	0	0
$903$	−4.72445	−0.157220
$904$	0	0
$905$	1.34087	0.0445722
$906$	0	0
$907$	25.5462	0.848247	0.424123	−	0.905604i	$-0.360582\pi$
$907$	25.5462	0.848247	0.424123	+	0.905604i	$0.360582\pi$
$908$	0	0
$909$	−8.72318	−0.289329
$910$	0	0
$911$	18.9725	0.628586	0.314293	−	0.949326i	$-0.398233\pi$
$911$	18.9725	0.628586	0.314293	+	0.949326i	$0.398233\pi$
$912$	0	0
$913$	8.37627	0.277214
$914$	0	0
$915$	−1.89949	−0.0627951
$916$	0	0
$917$	−13.6280	−0.450036
$918$	0	0
$919$	−16.0055	−0.527973	−0.263987	−	0.964526i	$-0.585037\pi$
$919$	−16.0055	−0.527973	−0.263987	+	0.964526i	$0.585037\pi$
$920$	0	0
$921$	−29.7861	−0.981485
$922$	0	0
$923$	−33.9218	−1.11655
$924$	0	0
$925$	34.1334	1.12230
$926$	0	0
$927$	6.28762	0.206513
$928$	0	0
$929$	−34.2945	−1.12516	−0.562582	−	0.826741i	$-0.690192\pi$
$929$	−34.2945	−1.12516	−0.562582	+	0.826741i	$0.690192\pi$
$930$	0	0
$931$	−2.38098	−0.0780335
$932$	0	0
$933$	−32.5062	−1.06420
$934$	0	0
$935$	0.130316	0.00426180
$936$	0	0
$937$	46.4116	1.51620	0.758101	−	0.652137i	$-0.226127\pi$
$937$	46.4116	1.51620	0.758101	+	0.652137i	$0.226127\pi$
$938$	0	0
$939$	−79.6187	−2.59826
$940$	0	0
$941$	33.8074	1.10209	0.551044	−	0.834476i	$-0.314230\pi$
$941$	33.8074	1.10209	0.551044	+	0.834476i	$0.314230\pi$
$942$	0	0
$943$	25.4549	0.828927
$944$	0	0
$945$	0.155435	0.00505629
$946$	0	0
$947$	−19.4012	−0.630454	−0.315227	−	0.949016i	$-0.602081\pi$
$947$	−19.4012	−0.630454	−0.315227	+	0.949016i	$0.602081\pi$
$948$	0	0
$949$	−31.3775	−1.01856
$950$	0	0
$951$	74.2738	2.40849
$952$	0	0
$953$	−32.4748	−1.05196	−0.525981	−	0.850497i	$-0.676302\pi$
$953$	−32.4748	−1.05196	−0.525981	+	0.850497i	$0.676302\pi$
$954$	0	0
$955$	3.62985	0.117459
$956$	0	0
$957$	11.8973	0.384586
$958$	0	0
$959$	4.07795	0.131684
$960$	0	0
$961$	66.0466	2.13054
$962$	0	0
$963$	33.7416	1.08731
$964$	0	0
$965$	−1.00520	−0.0323585
$966$	0	0
$967$	53.5070	1.72067	0.860334	−	0.509730i	$-0.170255\pi$
$967$	53.5070	1.72067	0.860334	+	0.509730i	$0.170255\pi$
$968$	0	0
$969$	−6.04268	−0.194119
$970$	0	0
$971$	−9.00403	−0.288953	−0.144476	−	0.989508i	$-0.546150\pi$
$971$	−9.00403	−0.288953	−0.144476	+	0.989508i	$0.546150\pi$
$972$	0	0
$973$	12.0637	0.386743
$974$	0	0
$975$	38.2922	1.22633
$976$	0	0
$977$	−38.0369	−1.21691	−0.608454	−	0.793589i	$-0.708210\pi$
$977$	−38.0369	−1.21691	−0.608454	+	0.793589i	$0.708210\pi$
$978$	0	0
$979$	−2.70789	−0.0865445
$980$	0	0
$981$	−35.2180	−1.12442
$982$	0	0
$983$	11.3788	0.362927	0.181464	−	0.983398i	$-0.441917\pi$
$983$	11.3788	0.362927	0.181464	+	0.983398i	$0.441917\pi$
$984$	0	0
$985$	0.647280	0.0206241
$986$	0	0
$987$	−0.693594	−0.0220773
$988$	0	0
$989$	−10.2721	−0.326633
$990$	0	0
$991$	−46.7450	−1.48490	−0.742452	−	0.669900i	$-0.766337\pi$
$991$	−46.7450	−1.48490	−0.742452	+	0.669900i	$0.766337\pi$
$992$	0	0
$993$	28.0825	0.891171
$994$	0	0
$995$	0.471822	0.0149578
$996$	0	0
$997$	43.5528	1.37933	0.689665	−	0.724128i	$-0.257758\pi$
$997$	43.5528	1.37933	0.689665	+	0.724128i	$0.257758\pi$
$998$	0	0
$999$	−7.66864	−0.242625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.j.1.6	✓	6
4.3	odd	2		3808.2.a.n.1.1	yes	6
8.3	odd	2		7616.2.a.bw.1.6		6
8.5	even	2		7616.2.a.ca.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.j.1.6	✓	6	1.1	even	1	trivial
3808.2.a.n.1.1	yes	6	4.3	odd	2
7616.2.a.bw.1.6		6	8.3	odd	2
7616.2.a.ca.1.1		6	8.5	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53790 q^{3} -0.138905 q^{5} -1.00000 q^{7} +3.44092 q^{9} -0.938171 q^{11} -3.02933 q^{13} -0.352526 q^{15} +1.00000 q^{17} -2.38098 q^{19} -2.53790 q^{21} -5.51799 q^{23} -4.98071 q^{25} +1.11900 q^{27} -4.99682 q^{29} -9.85122 q^{31} -2.38098 q^{33} +0.138905 q^{35} -6.85312 q^{37} -7.68812 q^{39} -4.61308 q^{41} +1.86156 q^{43} -0.477960 q^{45} +0.273295 q^{47} +1.00000 q^{49} +2.53790 q^{51} +6.18574 q^{53} +0.130316 q^{55} -6.04268 q^{57} +2.88562 q^{59} +5.38822 q^{61} -3.44092 q^{63} +0.420788 q^{65} +5.80476 q^{67} -14.0041 q^{69} +11.1978 q^{71} +10.3579 q^{73} -12.6405 q^{75} +0.938171 q^{77} +5.96214 q^{79} -7.48284 q^{81} -8.92830 q^{83} -0.138905 q^{85} -12.6814 q^{87} +2.88635 q^{89} +3.02933 q^{91} -25.0014 q^{93} +0.330729 q^{95} +11.9327 q^{97} -3.22817 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{27} + 2 q^{29} - 12 q^{31} + 2 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 - 8 * q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 - 10 * q^23 - 2 * q^27 + 2 * q^29 - 12 * q^31 + 2 * q^33 - 2 * q^35 + 2 * q^37 - 10 * q^39 - 8 * q^43 + 14 * q^45 - 22 * q^47 + 6 * q^49 - 2 * q^51 - 6 * q^53 - 24 * q^55 - 8 * q^57 - 8 * q^59 - 4 * q^61 - 4 * q^63 - 2 * q^65 + 8 * q^67 + 4 * q^69 - 18 * q^71 + 8 * q^73 - 30 * q^75 + 2 * q^77 - 14 * q^79 - 26 * q^81 + 2 * q^85 - 22 * q^87 + 2 * q^89 + 4 * q^91 - 8 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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