LMFDB - Embedded newform 7800.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.87740$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.87740 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.87740 q^{7} +1.00000 q^{9} +4.15686 q^{11} -1.00000 q^{13} -1.00000 q^{17} -3.47536 q^{19} -1.87740 q^{21} +1.47536 q^{23} +1.00000 q^{27} -1.80410 q^{29} -8.96095 q^{31} +4.15686 q^{33} -1.72055 q^{37} -1.00000 q^{39} -8.83836 q^{41} +4.27945 q^{43} +1.07331 q^{47} -3.47536 q^{49} -1.00000 q^{51} +11.5932 q^{53} -3.47536 q^{57} -10.4363 q^{59} -0.195903 q^{61} -1.87740 q^{63} -6.15686 q^{67} +1.47536 q^{69} +4.03426 q^{71} -1.08355 q^{73} -7.80410 q^{77} -8.27945 q^{79} +1.00000 q^{81} +4.43631 q^{83} -1.80410 q^{87} -1.19590 q^{89} +1.87740 q^{91} -8.96095 q^{93} -4.24519 q^{97} +4.15686 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + q^7 + 3 * q^9	$ 3 q + 3 q^{3} + q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} - 3 q^{17} - 6 q^{19} + q^{21} + 3 q^{27} - q^{29} - 7 q^{31} - 3 q^{33} - 14 q^{37} - 3 q^{39} + 4 q^{43} + q^{47} - 6 q^{49} - 3 q^{51} - 5 q^{53} - 6 q^{57} - 7 q^{59} - 5 q^{61} + q^{63} - 3 q^{67} - 10 q^{71} + 10 q^{73} - 19 q^{77} - 16 q^{79} + 3 q^{81} - 11 q^{83} - q^{87} - 8 q^{89} - q^{91} - 7 q^{93} - 26 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 - 3 * q^17 - 6 * q^19 + q^21 + 3 * q^27 - q^29 - 7 * q^31 - 3 * q^33 - 14 * q^37 - 3 * q^39 + 4 * q^43 + q^47 - 6 * q^49 - 3 * q^51 - 5 * q^53 - 6 * q^57 - 7 * q^59 - 5 * q^61 + q^63 - 3 * q^67 - 10 * q^71 + 10 * q^73 - 19 * q^77 - 16 * q^79 + 3 * q^81 - 11 * q^83 - q^87 - 8 * q^89 - q^91 - 7 * q^93 - 26 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.87740	−0.709592	−0.354796	−	0.934944i	$-0.615450\pi$
$7$	−1.87740	−0.709592	−0.354796	+	0.934944i	$0.615450\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.15686	1.25334	0.626670	−	0.779285i	$-0.284418\pi$
$11$	4.15686	1.25334	0.626670	+	0.779285i	$0.284418\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−3.47536	−0.797301	−0.398651	−	0.917103i	$-0.630521\pi$
$19$	−3.47536	−0.797301	−0.398651	+	0.917103i	$0.630521\pi$
$20$	0	0
$21$	−1.87740	−0.409683
$22$	0	0
$23$	1.47536	0.307633	0.153816	−	0.988099i	$-0.450844\pi$
$23$	1.47536	0.307633	0.153816	+	0.988099i	$0.450844\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.80410	−0.335012	−0.167506	−	0.985871i	$-0.553571\pi$
$29$	−1.80410	−0.335012	−0.167506	+	0.985871i	$0.553571\pi$
$30$	0	0
$31$	−8.96095	−1.60943	−0.804717	−	0.593658i	$-0.797683\pi$
$31$	−8.96095	−1.60943	−0.804717	+	0.593658i	$0.797683\pi$
$32$	0	0
$33$	4.15686	0.723616
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.72055	−0.282856	−0.141428	−	0.989949i	$-0.545169\pi$
$37$	−1.72055	−0.282856	−0.141428	+	0.989949i	$0.545169\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−8.83836	−1.38032	−0.690160	−	0.723657i	$-0.742460\pi$
$41$	−8.83836	−1.38032	−0.690160	+	0.723657i	$0.742460\pi$
$42$	0	0
$43$	4.27945	0.652610	0.326305	−	0.945264i	$-0.394196\pi$
$43$	4.27945	0.652610	0.326305	+	0.945264i	$0.394196\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.07331	0.156558	0.0782789	−	0.996931i	$-0.475058\pi$
$47$	1.07331	0.156558	0.0782789	+	0.996931i	$0.475058\pi$
$48$	0	0
$49$	−3.47536	−0.496479
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	11.5932	1.59244	0.796222	−	0.605005i	$-0.206829\pi$
$53$	11.5932	1.59244	0.796222	+	0.605005i	$0.206829\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.47536	−0.460322
$58$	0	0
$59$	−10.4363	−1.35869	−0.679346	−	0.733818i	$-0.737736\pi$
$59$	−10.4363	−1.35869	−0.679346	+	0.733818i	$0.737736\pi$
$60$	0	0
$61$	−0.195903	−0.0250828	−0.0125414	−	0.999921i	$-0.503992\pi$
$61$	−0.195903	−0.0250828	−0.0125414	+	0.999921i	$0.503992\pi$
$62$	0	0
$63$	−1.87740	−0.236531
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.15686	−0.752180	−0.376090	−	0.926583i	$-0.622732\pi$
$67$	−6.15686	−0.752180	−0.376090	+	0.926583i	$0.622732\pi$
$68$	0	0
$69$	1.47536	0.177612
$70$	0	0
$71$	4.03426	0.478779	0.239389	−	0.970924i	$-0.423053\pi$
$71$	4.03426	0.478779	0.239389	+	0.970924i	$0.423053\pi$
$72$	0	0
$73$	−1.08355	−0.126820	−0.0634099	−	0.997988i	$-0.520198\pi$
$73$	−1.08355	−0.126820	−0.0634099	+	0.997988i	$0.520198\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.80410	−0.889359
$78$	0	0
$79$	−8.27945	−0.931511	−0.465756	−	0.884913i	$-0.654217\pi$
$79$	−8.27945	−0.931511	−0.465756	+	0.884913i	$0.654217\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.43631	0.486948	0.243474	−	0.969907i	$-0.421713\pi$
$83$	4.43631	0.486948	0.243474	+	0.969907i	$0.421713\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.80410	−0.193420
$88$	0	0
$89$	−1.19590	−0.126765	−0.0633827	−	0.997989i	$-0.520189\pi$
$89$	−1.19590	−0.126765	−0.0633827	+	0.997989i	$0.520189\pi$
$90$	0	0
$91$	1.87740	0.196805
$92$	0	0
$93$	−8.96095	−0.929208
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.24519	−0.431034	−0.215517	−	0.976500i	$-0.569144\pi$
$97$	−4.24519	−0.431034	−0.215517	+	0.976500i	$0.569144\pi$
$98$	0	0
$99$	4.15686	0.417780
$100$	0	0
$101$	−5.67126	−0.564311	−0.282156	−	0.959369i	$-0.591049\pi$
$101$	−5.67126	−0.564311	−0.282156	+	0.959369i	$0.591049\pi$
$102$	0	0
$103$	−10.3480	−1.01962	−0.509808	−	0.860288i	$-0.670284\pi$
$103$	−10.3480	−1.01962	−0.509808	+	0.860288i	$0.670284\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.36300	0.518461	0.259230	−	0.965815i	$-0.416531\pi$
$107$	5.36300	0.518461	0.259230	+	0.965815i	$0.416531\pi$
$108$	0	0
$109$	−0.916451	−0.0877800	−0.0438900	−	0.999036i	$-0.513975\pi$
$109$	−0.916451	−0.0877800	−0.0438900	+	0.999036i	$0.513975\pi$
$110$	0	0
$111$	−1.72055	−0.163307
$112$	0	0
$113$	0.804097	0.0756431	0.0378215	−	0.999285i	$-0.487958\pi$
$113$	0.804097	0.0756431	0.0378215	+	0.999285i	$0.487958\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	1.87740	0.172101
$120$	0	0
$121$	6.27945	0.570859
$122$	0	0
$123$	−8.83836	−0.796928
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.03426	0.535454	0.267727	−	0.963495i	$-0.413727\pi$
$127$	6.03426	0.535454	0.267727	+	0.963495i	$0.413727\pi$
$128$	0	0
$129$	4.27945	0.376785
$130$	0	0
$131$	17.1521	1.49858	0.749292	−	0.662240i	$-0.230394\pi$
$131$	17.1521	1.49858	0.749292	+	0.662240i	$0.230394\pi$
$132$	0	0
$133$	6.52464	0.565758
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.636998	−0.0544224	−0.0272112	−	0.999630i	$-0.508663\pi$
$137$	−0.636998	−0.0544224	−0.0272112	+	0.999630i	$0.508663\pi$
$138$	0	0
$139$	−15.1178	−1.28228	−0.641138	−	0.767426i	$-0.721537\pi$
$139$	−15.1178	−1.28228	−0.641138	+	0.767426i	$0.721537\pi$
$140$	0	0
$141$	1.07331	0.0903887
$142$	0	0
$143$	−4.15686	−0.347614
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.47536	−0.286642
$148$	0	0
$149$	−6.14661	−0.503550	−0.251775	−	0.967786i	$-0.581014\pi$
$149$	−6.14661	−0.503550	−0.251775	+	0.967786i	$0.581014\pi$
$150$	0	0
$151$	−10.8281	−0.881179	−0.440590	−	0.897709i	$-0.645231\pi$
$151$	−10.8281	−0.881179	−0.440590	+	0.897709i	$0.645231\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−9.52464	−0.760149	−0.380075	−	0.924956i	$-0.624102\pi$
$157$	−9.52464	−0.760149	−0.380075	+	0.924956i	$0.624102\pi$
$158$	0	0
$159$	11.5932	0.919398
$160$	0	0
$161$	−2.76984	−0.218294
$162$	0	0
$163$	22.2014	1.73894	0.869472	−	0.493982i	$-0.164459\pi$
$163$	22.2014	1.73894	0.869472	+	0.493982i	$0.164459\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.27945	−0.485919	−0.242959	−	0.970036i	$-0.578118\pi$
$167$	−6.27945	−0.485919	−0.242959	+	0.970036i	$0.578118\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.47536	−0.265767
$172$	0	0
$173$	−5.83836	−0.443882	−0.221941	−	0.975060i	$-0.571239\pi$
$173$	−5.83836	−0.443882	−0.221941	+	0.975060i	$0.571239\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.4363	−0.784441
$178$	0	0
$179$	18.3480	1.37139	0.685696	−	0.727888i	$-0.259498\pi$
$179$	18.3480	1.37139	0.685696	+	0.727888i	$0.259498\pi$
$180$	0	0
$181$	−0.887647	−0.0659783	−0.0329891	−	0.999456i	$-0.510503\pi$
$181$	−0.887647	−0.0659783	−0.0329891	+	0.999456i	$0.510503\pi$
$182$	0	0
$183$	−0.195903	−0.0144816
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.15686	−0.303979
$188$	0	0
$189$	−1.87740	−0.136561
$190$	0	0
$191$	−15.6082	−1.12937	−0.564685	−	0.825307i	$-0.691002\pi$
$191$	−15.6082	−1.12937	−0.564685	+	0.825307i	$0.691002\pi$
$192$	0	0
$193$	5.96574	0.429423	0.214712	−	0.976677i	$-0.431119\pi$
$193$	5.96574	0.429423	0.214712	+	0.976677i	$0.431119\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.2302	−1.65508	−0.827540	−	0.561406i	$-0.810260\pi$
$197$	−23.2302	−1.65508	−0.827540	+	0.561406i	$0.810260\pi$
$198$	0	0
$199$	2.55890	0.181396	0.0906980	−	0.995878i	$-0.471090\pi$
$199$	2.55890	0.181396	0.0906980	+	0.995878i	$0.471090\pi$
$200$	0	0
$201$	−6.15686	−0.434271
$202$	0	0
$203$	3.38702	0.237722
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.47536	0.102544
$208$	0	0
$209$	−14.4466	−0.999289
$210$	0	0
$211$	−1.96574	−0.135327	−0.0676636	−	0.997708i	$-0.521554\pi$
$211$	−1.96574	−0.135327	−0.0676636	+	0.997708i	$0.521554\pi$
$212$	0	0
$213$	4.03426	0.276423
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.8233	1.14204
$218$	0	0
$219$	−1.08355	−0.0732195
$220$	0	0
$221$	1.00000	0.0672673
$222$	0	0
$223$	−16.6617	−1.11575	−0.557874	−	0.829925i	$-0.688383\pi$
$223$	−16.6617	−1.11575	−0.557874	+	0.829925i	$0.688383\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.0102429	0.000679842 0	0.000339921	−	1.00000i	$-0.499892\pi$
$227$	0.0102429	0.000679842 0	0.000339921		1.00000i	$0.499892\pi$
$228$	0	0
$229$	−9.21639	−0.609036	−0.304518	−	0.952507i	$-0.598495\pi$
$229$	−9.21639	−0.609036	−0.304518	+	0.952507i	$0.598495\pi$
$230$	0	0
$231$	−7.80410	−0.513472
$232$	0	0
$233$	7.19590	0.471419	0.235710	−	0.971824i	$-0.424259\pi$
$233$	7.19590	0.471419	0.235710	+	0.971824i	$0.424259\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.27945	−0.537808
$238$	0	0
$239$	6.82811	0.441674	0.220837	−	0.975311i	$-0.429121\pi$
$239$	6.82811	0.441674	0.220837	+	0.975311i	$0.429121\pi$
$240$	0	0
$241$	16.2014	1.04362	0.521811	−	0.853061i	$-0.325257\pi$
$241$	16.2014	1.04362	0.521811	+	0.853061i	$0.325257\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.47536	0.221132
$248$	0	0
$249$	4.43631	0.281140
$250$	0	0
$251$	−23.9562	−1.51210	−0.756050	−	0.654514i	$-0.772873\pi$
$251$	−23.9562	−1.51210	−0.756050	+	0.654514i	$0.772873\pi$
$252$	0	0
$253$	6.13284	0.385568
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−29.6617	−1.85025	−0.925123	−	0.379669i	$-0.876038\pi$
$257$	−29.6617	−1.85025	−0.925123	+	0.379669i	$0.876038\pi$
$258$	0	0
$259$	3.23016	0.200713
$260$	0	0
$261$	−1.80410	−0.111671
$262$	0	0
$263$	6.21093	0.382983	0.191491	−	0.981494i	$-0.438668\pi$
$263$	6.21093	0.382983	0.191491	+	0.981494i	$0.438668\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.19590	−0.0731881
$268$	0	0
$269$	18.1178	1.10466	0.552331	−	0.833625i	$-0.313738\pi$
$269$	18.1178	1.10466	0.552331	+	0.833625i	$0.313738\pi$
$270$	0	0
$271$	−30.8966	−1.87684	−0.938418	−	0.345501i	$-0.887709\pi$
$271$	−30.8966	−1.87684	−0.938418	+	0.345501i	$0.887709\pi$
$272$	0	0
$273$	1.87740	0.113626
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−26.7740	−1.60870	−0.804348	−	0.594158i	$-0.797485\pi$
$277$	−26.7740	−1.60870	−0.804348	+	0.594158i	$0.797485\pi$
$278$	0	0
$279$	−8.96095	−0.536478
$280$	0	0
$281$	4.16710	0.248588	0.124294	−	0.992245i	$-0.460333\pi$
$281$	4.16710	0.248588	0.124294	+	0.992245i	$0.460333\pi$
$282$	0	0
$283$	20.4603	1.21624	0.608120	−	0.793845i	$-0.291924\pi$
$283$	20.4603	1.21624	0.608120	+	0.793845i	$0.291924\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.5932	0.979464
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−4.24519	−0.248858
$292$	0	0
$293$	−18.6274	−1.08823	−0.544113	−	0.839012i	$-0.683134\pi$
$293$	−18.6274	−1.08823	−0.544113	+	0.839012i	$0.683134\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.15686	0.241205
$298$	0	0
$299$	−1.47536	−0.0853220
$300$	0	0
$301$	−8.03426	−0.463087
$302$	0	0
$303$	−5.67126	−0.325805
$304$	0	0
$305$	0	0
$306$	0	0
$307$	25.7891	1.47186	0.735930	−	0.677058i	$-0.236745\pi$
$307$	25.7891	1.47186	0.735930	+	0.677058i	$0.236745\pi$
$308$	0	0
$309$	−10.3480	−0.588676
$310$	0	0
$311$	−16.4603	−0.933379	−0.466690	−	0.884421i	$-0.654554\pi$
$311$	−16.4603	−0.933379	−0.466690	+	0.884421i	$0.654554\pi$
$312$	0	0
$313$	33.1028	1.87108	0.935540	−	0.353221i	$-0.114914\pi$
$313$	33.1028	1.87108	0.935540	+	0.353221i	$0.114914\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.0835	−1.18417	−0.592085	−	0.805875i	$-0.701695\pi$
$317$	−21.0835	−1.18417	−0.592085	+	0.805875i	$0.701695\pi$
$318$	0	0
$319$	−7.49937	−0.419884
$320$	0	0
$321$	5.36300	0.299334
$322$	0	0
$323$	3.47536	0.193374
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.916451	−0.0506798
$328$	0	0
$329$	−2.01503	−0.111092
$330$	0	0
$331$	−5.08355	−0.279417	−0.139709	−	0.990193i	$-0.544617\pi$
$331$	−5.08355	−0.279417	−0.139709	+	0.990193i	$0.544617\pi$
$332$	0	0
$333$	−1.72055	−0.0942854
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.67126	0.199986	0.0999931	−	0.994988i	$-0.468118\pi$
$337$	3.67126	0.199986	0.0999931	+	0.994988i	$0.468118\pi$
$338$	0	0
$339$	0.804097	0.0436726
$340$	0	0
$341$	−37.2494	−2.01717
$342$	0	0
$343$	19.6665	1.06189
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.21639	−0.0652992	−0.0326496	−	0.999467i	$-0.510395\pi$
$347$	−1.21639	−0.0652992	−0.0326496	+	0.999467i	$0.510395\pi$
$348$	0	0
$349$	−17.2302	−0.922309	−0.461155	−	0.887320i	$-0.652565\pi$
$349$	−17.2302	−0.922309	−0.461155	+	0.887320i	$0.652565\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−30.2494	−1.61001	−0.805006	−	0.593266i	$-0.797838\pi$
$353$	−30.2494	−1.61001	−0.805006	+	0.593266i	$0.797838\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.87740	0.0993627
$358$	0	0
$359$	−6.33773	−0.334493	−0.167246	−	0.985915i	$-0.553488\pi$
$359$	−6.33773	−0.334493	−0.167246	+	0.985915i	$0.553488\pi$
$360$	0	0
$361$	−6.92191	−0.364311
$362$	0	0
$363$	6.27945	0.329586
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.88219	0.150449	0.0752246	−	0.997167i	$-0.476033\pi$
$367$	2.88219	0.150449	0.0752246	+	0.997167i	$0.476033\pi$
$368$	0	0
$369$	−8.83836	−0.460106
$370$	0	0
$371$	−21.7651	−1.12999
$372$	0	0
$373$	−22.0055	−1.13940	−0.569700	−	0.821853i	$-0.692940\pi$
$373$	−22.0055	−1.13940	−0.569700	+	0.821853i	$0.692940\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.80410	0.0929157
$378$	0	0
$379$	29.4418	1.51232	0.756161	−	0.654386i	$-0.227073\pi$
$379$	29.4418	1.51232	0.756161	+	0.654386i	$0.227073\pi$
$380$	0	0
$381$	6.03426	0.309144
$382$	0	0
$383$	24.1028	1.23159	0.615797	−	0.787905i	$-0.288834\pi$
$383$	24.1028	1.23159	0.615797	+	0.787905i	$0.288834\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.27945	0.217537
$388$	0	0
$389$	19.8233	1.00508	0.502541	−	0.864553i	$-0.332398\pi$
$389$	19.8233	1.00508	0.502541	+	0.864553i	$0.332398\pi$
$390$	0	0
$391$	−1.47536	−0.0746119
$392$	0	0
$393$	17.1521	0.865207
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.70552	0.135786	0.0678930	−	0.997693i	$-0.478372\pi$
$397$	2.70552	0.135786	0.0678930	+	0.997693i	$0.478372\pi$
$398$	0	0
$399$	6.52464	0.326641
$400$	0	0
$401$	−3.88765	−0.194140	−0.0970699	−	0.995278i	$-0.530947\pi$
$401$	−3.88765	−0.194140	−0.0970699	+	0.995278i	$0.530947\pi$
$402$	0	0
$403$	8.96095	0.446377
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.15207	−0.354515
$408$	0	0
$409$	−19.7891	−0.978506	−0.489253	−	0.872142i	$-0.662731\pi$
$409$	−19.7891	−0.978506	−0.489253	+	0.872142i	$0.662731\pi$
$410$	0	0
$411$	−0.636998	−0.0314208
$412$	0	0
$413$	19.5932	0.964117
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−15.1178	−0.740322
$418$	0	0
$419$	−9.78907	−0.478227	−0.239114	−	0.970992i	$-0.576857\pi$
$419$	−9.78907	−0.478227	−0.239114	+	0.970992i	$0.576857\pi$
$420$	0	0
$421$	20.6959	1.00866	0.504329	−	0.863511i	$-0.331740\pi$
$421$	20.6959	1.00866	0.504329	+	0.863511i	$0.331740\pi$
$422$	0	0
$423$	1.07331	0.0521860
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.367789	0.0177985
$428$	0	0
$429$	−4.15686	−0.200695
$430$	0	0
$431$	−27.5439	−1.32674	−0.663371	−	0.748291i	$-0.730875\pi$
$431$	−27.5439	−1.32674	−0.663371	+	0.748291i	$0.730875\pi$
$432$	0	0
$433$	−0.872617	−0.0419353	−0.0209676	−	0.999780i	$-0.506675\pi$
$433$	−0.872617	−0.0419353	−0.0209676	+	0.999780i	$0.506675\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.12738	−0.245276
$438$	0	0
$439$	−32.2014	−1.53689	−0.768443	−	0.639918i	$-0.778968\pi$
$439$	−32.2014	−1.53689	−0.768443	+	0.639918i	$0.778968\pi$
$440$	0	0
$441$	−3.47536	−0.165493
$442$	0	0
$443$	−23.2302	−1.10370	−0.551849	−	0.833944i	$-0.686078\pi$
$443$	−23.2302	−1.10370	−0.551849	+	0.833944i	$0.686078\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.14661	−0.290725
$448$	0	0
$449$	−8.06852	−0.380777	−0.190388	−	0.981709i	$-0.560975\pi$
$449$	−8.06852	−0.380777	−0.190388	+	0.981709i	$0.560975\pi$
$450$	0	0
$451$	−36.7398	−1.73001
$452$	0	0
$453$	−10.8281	−0.508749
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.8576	−1.11601	−0.558005	−	0.829837i	$-0.688433\pi$
$457$	−23.8576	−1.11601	−0.558005	+	0.829837i	$0.688433\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−19.0493	−0.887214	−0.443607	−	0.896221i	$-0.646301\pi$
$461$	−19.0493	−0.887214	−0.443607	+	0.896221i	$0.646301\pi$
$462$	0	0
$463$	29.8336	1.38648	0.693242	−	0.720705i	$-0.256182\pi$
$463$	29.8336	1.38648	0.693242	+	0.720705i	$0.256182\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.82333	−0.0843735	−0.0421868	−	0.999110i	$-0.513432\pi$
$467$	−1.82333	−0.0843735	−0.0421868	+	0.999110i	$0.513432\pi$
$468$	0	0
$469$	11.5589	0.533741
$470$	0	0
$471$	−9.52464	−0.438872
$472$	0	0
$473$	17.7891	0.817942
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.5932	0.530815
$478$	0	0
$479$	−4.30347	−0.196631	−0.0983153	−	0.995155i	$-0.531345\pi$
$479$	−4.30347	−0.196631	−0.0983153	+	0.995155i	$0.531345\pi$
$480$	0	0
$481$	1.72055	0.0784502
$482$	0	0
$483$	−2.76984	−0.126032
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.33353	−0.105742	−0.0528711	−	0.998601i	$-0.516837\pi$
$487$	−2.33353	−0.105742	−0.0528711	+	0.998601i	$0.516837\pi$
$488$	0	0
$489$	22.2014	1.00398
$490$	0	0
$491$	18.6274	0.840644	0.420322	−	0.907375i	$-0.361917\pi$
$491$	18.6274	0.840644	0.420322	+	0.907375i	$0.361917\pi$
$492$	0	0
$493$	1.80410	0.0812524
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.57393	−0.339737
$498$	0	0
$499$	15.9254	0.712921	0.356460	−	0.934310i	$-0.383984\pi$
$499$	15.9254	0.712921	0.356460	+	0.934310i	$0.383984\pi$
$500$	0	0
$501$	−6.27945	−0.280545
$502$	0	0
$503$	−18.4946	−0.824633	−0.412316	−	0.911041i	$-0.635280\pi$
$503$	−18.4946	−0.824633	−0.412316	+	0.911041i	$0.635280\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−23.6425	−1.04793	−0.523967	−	0.851739i	$-0.675548\pi$
$509$	−23.6425	−1.04793	−0.523967	+	0.851739i	$0.675548\pi$
$510$	0	0
$511$	2.03426	0.0899904
$512$	0	0
$513$	−3.47536	−0.153441
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.46158	0.196220
$518$	0	0
$519$	−5.83836	−0.256275
$520$	0	0
$521$	0.391806	0.0171653	0.00858266	−	0.999963i	$-0.497268\pi$
$521$	0.391806	0.0171653	0.00858266	+	0.999963i	$0.497268\pi$
$522$	0	0
$523$	18.6370	0.814939	0.407470	−	0.913219i	$-0.366411\pi$
$523$	18.6370	0.814939	0.407470	+	0.913219i	$0.366411\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.96095	0.390345
$528$	0	0
$529$	−20.8233	−0.905362
$530$	0	0
$531$	−10.4363	−0.452897
$532$	0	0
$533$	8.83836	0.382832
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18.3480	0.791773
$538$	0	0
$539$	−14.4466	−0.622257
$540$	0	0
$541$	32.8864	1.41390	0.706948	−	0.707265i	$-0.250071\pi$
$541$	32.8864	1.41390	0.706948	+	0.707265i	$0.250071\pi$
$542$	0	0
$543$	−0.887647	−0.0380926
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.23016	−0.0525980	−0.0262990	−	0.999654i	$-0.508372\pi$
$547$	−1.23016	−0.0525980	−0.0262990	+	0.999654i	$0.508372\pi$
$548$	0	0
$549$	−0.195903	−0.00836093
$550$	0	0
$551$	6.26988	0.267106
$552$	0	0
$553$	15.5439	0.660993
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.1028	0.428069	0.214034	−	0.976826i	$-0.431340\pi$
$557$	10.1028	0.428069	0.214034	+	0.976826i	$0.431340\pi$
$558$	0	0
$559$	−4.27945	−0.181002
$560$	0	0
$561$	−4.15686	−0.175503
$562$	0	0
$563$	−13.7410	−0.579116	−0.289558	−	0.957161i	$-0.593508\pi$
$563$	−13.7410	−0.579116	−0.289558	+	0.957161i	$0.593508\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.87740	−0.0788435
$568$	0	0
$569$	35.4850	1.48761	0.743805	−	0.668397i	$-0.233019\pi$
$569$	35.4850	1.48761	0.743805	+	0.668397i	$0.233019\pi$
$570$	0	0
$571$	15.8672	0.664020	0.332010	−	0.943276i	$-0.392273\pi$
$571$	15.8672	0.664020	0.332010	+	0.943276i	$0.392273\pi$
$572$	0	0
$573$	−15.6082	−0.652042
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.5000	0.728536	0.364268	−	0.931294i	$-0.381319\pi$
$577$	17.5000	0.728536	0.364268	+	0.931294i	$0.381319\pi$
$578$	0	0
$579$	5.96574	0.247928
$580$	0	0
$581$	−8.32874	−0.345534
$582$	0	0
$583$	48.1911	1.99587
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.3227	1.45792	0.728962	−	0.684554i	$-0.240003\pi$
$587$	35.3227	1.45792	0.728962	+	0.684554i	$0.240003\pi$
$588$	0	0
$589$	31.1425	1.28320
$590$	0	0
$591$	−23.2302	−0.955561
$592$	0	0
$593$	7.04929	0.289480	0.144740	−	0.989470i	$-0.453765\pi$
$593$	7.04929	0.289480	0.144740	+	0.989470i	$0.453765\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.55890	0.104729
$598$	0	0
$599$	1.57393	0.0643092	0.0321546	−	0.999483i	$-0.489763\pi$
$599$	1.57393	0.0643092	0.0321546	+	0.999483i	$0.489763\pi$
$600$	0	0
$601$	−15.7603	−0.642875	−0.321437	−	0.946931i	$-0.604166\pi$
$601$	−15.7603	−0.642875	−0.321437	+	0.946931i	$0.604166\pi$
$602$	0	0
$603$	−6.15686	−0.250727
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.0205	0.487896	0.243948	−	0.969788i	$-0.421557\pi$
$607$	12.0205	0.487896	0.243948	+	0.969788i	$0.421557\pi$
$608$	0	0
$609$	3.38702	0.137249
$610$	0	0
$611$	−1.07331	−0.0434213
$612$	0	0
$613$	14.6274	0.590796	0.295398	−	0.955374i	$-0.404548\pi$
$613$	14.6274	0.590796	0.295398	+	0.955374i	$0.404548\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.9219	−0.882543	−0.441271	−	0.897374i	$-0.645472\pi$
$617$	−21.9219	−0.882543	−0.441271	+	0.897374i	$0.645472\pi$
$618$	0	0
$619$	−22.9069	−0.920705	−0.460353	−	0.887736i	$-0.652277\pi$
$619$	−22.9069	−0.920705	−0.460353	+	0.887736i	$0.652277\pi$
$620$	0	0
$621$	1.47536	0.0592040
$622$	0	0
$623$	2.24519	0.0899517
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−14.4466	−0.576940
$628$	0	0
$629$	1.72055	0.0686027
$630$	0	0
$631$	−15.3287	−0.610228	−0.305114	−	0.952316i	$-0.598694\pi$
$631$	−15.3287	−0.610228	−0.305114	+	0.952316i	$0.598694\pi$
$632$	0	0
$633$	−1.96574	−0.0781312
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.47536	0.137699
$638$	0	0
$639$	4.03426	0.159593
$640$	0	0
$641$	17.1370	0.676872	0.338436	−	0.940989i	$-0.390102\pi$
$641$	17.1370	0.676872	0.338436	+	0.940989i	$0.390102\pi$
$642$	0	0
$643$	15.4068	0.607586	0.303793	−	0.952738i	$-0.401747\pi$
$643$	15.4068	0.607586	0.303793	+	0.952738i	$0.401747\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.8041	0.424753	0.212376	−	0.977188i	$-0.431880\pi$
$647$	10.8041	0.424753	0.212376	+	0.977188i	$0.431880\pi$
$648$	0	0
$649$	−43.3822	−1.70290
$650$	0	0
$651$	16.8233	0.659358
$652$	0	0
$653$	−28.8438	−1.12875	−0.564373	−	0.825520i	$-0.690882\pi$
$653$	−28.8438	−1.12875	−0.564373	+	0.825520i	$0.690882\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.08355	−0.0422733
$658$	0	0
$659$	39.7891	1.54996	0.774981	−	0.631985i	$-0.217759\pi$
$659$	39.7891	1.54996	0.774981	+	0.631985i	$0.217759\pi$
$660$	0	0
$661$	40.6822	1.58235	0.791177	−	0.611588i	$-0.209469\pi$
$661$	40.6822	1.58235	0.791177	+	0.611588i	$0.209469\pi$
$662$	0	0
$663$	1.00000	0.0388368
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.66168	−0.103061
$668$	0	0
$669$	−16.6617	−0.644178
$670$	0	0
$671$	−0.814340	−0.0314372
$672$	0	0
$673$	21.7698	0.839166	0.419583	−	0.907717i	$-0.362176\pi$
$673$	21.7698	0.839166	0.419583	+	0.907717i	$0.362176\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.9507	−0.497736	−0.248868	−	0.968537i	$-0.580059\pi$
$677$	−12.9507	−0.497736	−0.248868	+	0.968537i	$0.580059\pi$
$678$	0	0
$679$	7.96994	0.305858
$680$	0	0
$681$	0.0102429	0.000392507 0
$682$	0	0
$683$	−6.89243	−0.263732	−0.131866	−	0.991268i	$-0.542097\pi$
$683$	−6.89243	−0.263732	−0.131866	+	0.991268i	$0.542097\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−9.21639	−0.351627
$688$	0	0
$689$	−11.5932	−0.441664
$690$	0	0
$691$	34.0830	1.29658	0.648289	−	0.761395i	$-0.275485\pi$
$691$	34.0830	1.29658	0.648289	+	0.761395i	$0.275485\pi$
$692$	0	0
$693$	−7.80410	−0.296453
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.83836	0.334777
$698$	0	0
$699$	7.19590	0.272174
$700$	0	0
$701$	−4.58771	−0.173275	−0.0866377	−	0.996240i	$-0.527612\pi$
$701$	−4.58771	−0.173275	−0.0866377	+	0.996240i	$0.527612\pi$
$702$	0	0
$703$	5.97951	0.225522
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.6472	0.400431
$708$	0	0
$709$	15.7891	0.592971	0.296485	−	0.955037i	$-0.404185\pi$
$709$	15.7891	0.592971	0.296485	+	0.955037i	$0.404185\pi$
$710$	0	0
$711$	−8.27945	−0.310504
$712$	0	0
$713$	−13.2206	−0.495115
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.82811	0.255001
$718$	0	0
$719$	41.7891	1.55847	0.779235	−	0.626732i	$-0.215608\pi$
$719$	41.7891	1.55847	0.779235	+	0.626732i	$0.215608\pi$
$720$	0	0
$721$	19.4273	0.723511
$722$	0	0
$723$	16.2014	0.602535
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0.378032	0.0140204	0.00701021	−	0.999975i	$-0.497769\pi$
$727$	0.378032	0.0140204	0.00701021	+	0.999975i	$0.497769\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.27945	−0.158281
$732$	0	0
$733$	41.8138	1.54443	0.772213	−	0.635364i	$-0.219150\pi$
$733$	41.8138	1.54443	0.772213	+	0.635364i	$0.219150\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.5932	−0.942736
$738$	0	0
$739$	−15.9117	−0.585320	−0.292660	−	0.956217i	$-0.594540\pi$
$739$	−15.9117	−0.585320	−0.292660	+	0.956217i	$0.594540\pi$
$740$	0	0
$741$	3.47536	0.127670
$742$	0	0
$743$	−25.4213	−0.932616	−0.466308	−	0.884622i	$-0.654416\pi$
$743$	−25.4213	−0.932616	−0.466308	+	0.884622i	$0.654416\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.43631	0.162316
$748$	0	0
$749$	−10.0685	−0.367896
$750$	0	0
$751$	31.7795	1.15965	0.579825	−	0.814741i	$-0.303121\pi$
$751$	31.7795	1.15965	0.579825	+	0.814741i	$0.303121\pi$
$752$	0	0
$753$	−23.9562	−0.873011
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−23.2356	−0.844513	−0.422256	−	0.906476i	$-0.638762\pi$
$757$	−23.2356	−0.844513	−0.422256	+	0.906476i	$0.638762\pi$
$758$	0	0
$759$	6.13284	0.222608
$760$	0	0
$761$	−5.08355	−0.184279	−0.0921393	−	0.995746i	$-0.529370\pi$
$761$	−5.08355	−0.184279	−0.0921393	+	0.995746i	$0.529370\pi$
$762$	0	0
$763$	1.72055	0.0622880
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.4363	0.376833
$768$	0	0
$769$	−5.75481	−0.207524	−0.103762	−	0.994602i	$-0.533088\pi$
$769$	−5.75481	−0.207524	−0.103762	+	0.994602i	$0.533088\pi$
$770$	0	0
$771$	−29.6617	−1.06824
$772$	0	0
$773$	−8.24519	−0.296559	−0.148279	−	0.988945i	$-0.547374\pi$
$773$	−8.24519	−0.296559	−0.148279	+	0.988945i	$0.547374\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.23016	0.115881
$778$	0	0
$779$	30.7164	1.10053
$780$	0	0
$781$	16.7698	0.600072
$782$	0	0
$783$	−1.80410	−0.0644732
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.8624	0.672371	0.336186	−	0.941796i	$-0.390863\pi$
$787$	18.8624	0.672371	0.336186	+	0.941796i	$0.390863\pi$
$788$	0	0
$789$	6.21093	0.221115
$790$	0	0
$791$	−1.50961	−0.0536757
$792$	0	0
$793$	0.195903	0.00695671
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.0097	0.850466	0.425233	−	0.905084i	$-0.360192\pi$
$797$	24.0097	0.850466	0.425233	+	0.905084i	$0.360192\pi$
$798$	0	0
$799$	−1.07331	−0.0379709
$800$	0	0
$801$	−1.19590	−0.0422551
$802$	0	0
$803$	−4.50416	−0.158948
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.1178	0.637777
$808$	0	0
$809$	−34.1466	−1.20053	−0.600265	−	0.799801i	$-0.704938\pi$
$809$	−34.1466	−1.20053	−0.600265	+	0.799801i	$0.704938\pi$
$810$	0	0
$811$	−5.77882	−0.202922	−0.101461	−	0.994840i	$-0.532352\pi$
$811$	−5.77882	−0.202922	−0.101461	+	0.994840i	$0.532352\pi$
$812$	0	0
$813$	−30.8966	−1.08359
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−14.8726	−0.520327
$818$	0	0
$819$	1.87740	0.0656018
$820$	0	0
$821$	−34.9069	−1.21826	−0.609129	−	0.793071i	$-0.708481\pi$
$821$	−34.9069	−1.21826	−0.609129	+	0.793071i	$0.708481\pi$
$822$	0	0
$823$	27.1383	0.945981	0.472991	−	0.881067i	$-0.343174\pi$
$823$	27.1383	0.945981	0.472991	+	0.881067i	$0.343174\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.13763	0.109106	0.0545530	−	0.998511i	$-0.482627\pi$
$827$	3.13763	0.109106	0.0545530	+	0.998511i	$0.482627\pi$
$828$	0	0
$829$	−9.34377	−0.324523	−0.162261	−	0.986748i	$-0.551879\pi$
$829$	−9.34377	−0.324523	−0.162261	+	0.986748i	$0.551879\pi$
$830$	0	0
$831$	−26.7740	−0.928781
$832$	0	0
$833$	3.47536	0.120414
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.96095	−0.309736
$838$	0	0
$839$	−0.661684	−0.0228439	−0.0114219	−	0.999935i	$-0.503636\pi$
$839$	−0.661684	−0.0228439	−0.0114219	+	0.999935i	$0.503636\pi$
$840$	0	0
$841$	−25.7452	−0.887767
$842$	0	0
$843$	4.16710	0.143523
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−11.7891	−0.405077
$848$	0	0
$849$	20.4603	0.702197
$850$	0	0
$851$	−2.53842	−0.0870159
$852$	0	0
$853$	−37.4110	−1.28093	−0.640465	−	0.767988i	$-0.721258\pi$
$853$	−37.4110	−1.28093	−0.640465	+	0.767988i	$0.721258\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.2164	0.861375	0.430688	−	0.902501i	$-0.358271\pi$
$857$	25.2164	0.861375	0.430688	+	0.902501i	$0.358271\pi$
$858$	0	0
$859$	38.0247	1.29739	0.648693	−	0.761050i	$-0.275316\pi$
$859$	38.0247	1.29739	0.648693	+	0.761050i	$0.275316\pi$
$860$	0	0
$861$	16.5932	0.565494
$862$	0	0
$863$	15.9802	0.543972	0.271986	−	0.962301i	$-0.412320\pi$
$863$	15.9802	0.543972	0.271986	+	0.962301i	$0.412320\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.0000	−0.543388
$868$	0	0
$869$	−34.4165	−1.16750
$870$	0	0
$871$	6.15686	0.208617
$872$	0	0
$873$	−4.24519	−0.143678
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.6562	0.528673	0.264337	−	0.964430i	$-0.414847\pi$
$877$	15.6562	0.528673	0.264337	+	0.964430i	$0.414847\pi$
$878$	0	0
$879$	−18.6274	−0.628287
$880$	0	0
$881$	2.08355	0.0701966	0.0350983	−	0.999384i	$-0.488826\pi$
$881$	2.08355	0.0701966	0.0350983	+	0.999384i	$0.488826\pi$
$882$	0	0
$883$	−9.89185	−0.332887	−0.166444	−	0.986051i	$-0.553228\pi$
$883$	−9.89185	−0.332887	−0.166444	+	0.986051i	$0.553228\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.44655	0.216454	0.108227	−	0.994126i	$-0.465483\pi$
$887$	6.44655	0.216454	0.108227	+	0.994126i	$0.465483\pi$
$888$	0	0
$889$	−11.3287	−0.379954
$890$	0	0
$891$	4.15686	0.139260
$892$	0	0
$893$	−3.73012	−0.124824
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.47536	−0.0492607
$898$	0	0
$899$	16.1664	0.539181
$900$	0	0
$901$	−11.5932	−0.386224
$902$	0	0
$903$	−8.03426	−0.267363
$904$	0	0
$905$	0	0
$906$	0	0
$907$	29.4110	0.976577	0.488289	−	0.872682i	$-0.337621\pi$
$907$	29.4110	0.976577	0.488289	+	0.872682i	$0.337621\pi$
$908$	0	0
$909$	−5.67126	−0.188104
$910$	0	0
$911$	−16.7603	−0.555292	−0.277646	−	0.960683i	$-0.589554\pi$
$911$	−16.7603	−0.555292	−0.277646	+	0.960683i	$0.589554\pi$
$912$	0	0
$913$	18.4411	0.610311
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−32.2014	−1.06338
$918$	0	0
$919$	−13.0973	−0.432041	−0.216020	−	0.976389i	$-0.569308\pi$
$919$	−13.0973	−0.432041	−0.216020	+	0.976389i	$0.569308\pi$
$920$	0	0
$921$	25.7891	0.849779
$922$	0	0
$923$	−4.03426	−0.132789
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.3480	−0.339872
$928$	0	0
$929$	12.7945	0.419775	0.209887	−	0.977726i	$-0.432690\pi$
$929$	12.7945	0.419775	0.209887	+	0.977726i	$0.432690\pi$
$930$	0	0
$931$	12.0781	0.395844
$932$	0	0
$933$	−16.4603	−0.538887
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0397	0.720006	0.360003	−	0.932951i	$-0.382776\pi$
$937$	22.0397	0.720006	0.360003	+	0.932951i	$0.382776\pi$
$938$	0	0
$939$	33.1028	1.08027
$940$	0	0
$941$	−58.7603	−1.91553	−0.957765	−	0.287552i	$-0.907158\pi$
$941$	−58.7603	−1.91553	−0.957765	+	0.287552i	$0.907158\pi$
$942$	0	0
$943$	−13.0397	−0.424632
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.2747	1.53622	0.768110	−	0.640318i	$-0.221197\pi$
$947$	47.2747	1.53622	0.768110	+	0.640318i	$0.221197\pi$
$948$	0	0
$949$	1.08355	0.0351735
$950$	0	0
$951$	−21.0835	−0.683681
$952$	0	0
$953$	15.8083	0.512081	0.256040	−	0.966666i	$-0.417582\pi$
$953$	15.8083	0.512081	0.256040	+	0.966666i	$0.417582\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−7.49937	−0.242420
$958$	0	0
$959$	1.19590	0.0386177
$960$	0	0
$961$	49.2987	1.59028
$962$	0	0
$963$	5.36300	0.172820
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−24.6377	−0.792294	−0.396147	−	0.918187i	$-0.629653\pi$
$967$	−24.6377	−0.792294	−0.396147	+	0.918187i	$0.629653\pi$
$968$	0	0
$969$	3.47536	0.111644
$970$	0	0
$971$	−8.60694	−0.276210	−0.138105	−	0.990418i	$-0.544101\pi$
$971$	−8.60694	−0.276210	−0.138105	+	0.990418i	$0.544101\pi$
$972$	0	0
$973$	28.3822	0.909893
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19.4411	−0.621976	−0.310988	−	0.950414i	$-0.600660\pi$
$977$	−19.4411	−0.621976	−0.310988	+	0.950414i	$0.600660\pi$
$978$	0	0
$979$	−4.97120	−0.158880
$980$	0	0
$981$	−0.916451	−0.0292600
$982$	0	0
$983$	−30.5691	−0.975004	−0.487502	−	0.873122i	$-0.662092\pi$
$983$	−30.5691	−0.975004	−0.487502	+	0.873122i	$0.662092\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.01503	−0.0641391
$988$	0	0
$989$	6.31371	0.200764
$990$	0	0
$991$	18.1713	0.577230	0.288615	−	0.957445i	$-0.406805\pi$
$991$	18.1713	0.577230	0.288615	+	0.957445i	$0.406805\pi$
$992$	0	0
$993$	−5.08355	−0.161322
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−45.1275	−1.42920	−0.714601	−	0.699533i	$-0.753392\pi$
$997$	−45.1275	−1.42920	−0.714601	+	0.699533i	$0.753392\pi$
$998$	0	0
$999$	−1.72055	−0.0544357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.br.1.1	yes	3
5.4	even	2		7800.2.a.bg.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bg.1.3	✓	3	5.4	even	2
7800.2.a.br.1.1	yes	3	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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.87740 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.87740 q^{7} +1.00000 q^{9} +4.15686 q^{11} -1.00000 q^{13} -1.00000 q^{17} -3.47536 q^{19} -1.87740 q^{21} +1.47536 q^{23} +1.00000 q^{27} -1.80410 q^{29} -8.96095 q^{31} +4.15686 q^{33} -1.72055 q^{37} -1.00000 q^{39} -8.83836 q^{41} +4.27945 q^{43} +1.07331 q^{47} -3.47536 q^{49} -1.00000 q^{51} +11.5932 q^{53} -3.47536 q^{57} -10.4363 q^{59} -0.195903 q^{61} -1.87740 q^{63} -6.15686 q^{67} +1.47536 q^{69} +4.03426 q^{71} -1.08355 q^{73} -7.80410 q^{77} -8.27945 q^{79} +1.00000 q^{81} +4.43631 q^{83} -1.80410 q^{87} -1.19590 q^{89} +1.87740 q^{91} -8.96095 q^{93} -4.24519 q^{97} +4.15686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + q^7 + 3 * q^9	\( 3 q + 3 q^{3} + q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} - 3 q^{17} - 6 q^{19} + q^{21} + 3 q^{27} - q^{29} - 7 q^{31} - 3 q^{33} - 14 q^{37} - 3 q^{39} + 4 q^{43} + q^{47} - 6 q^{49} - 3 q^{51} - 5 q^{53} - 6 q^{57} - 7 q^{59} - 5 q^{61} + q^{63} - 3 q^{67} - 10 q^{71} + 10 q^{73} - 19 q^{77} - 16 q^{79} + 3 q^{81} - 11 q^{83} - q^{87} - 8 q^{89} - q^{91} - 7 q^{93} - 26 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 - 3 * q^17 - 6 * q^19 + q^21 + 3 * q^27 - q^29 - 7 * q^31 - 3 * q^33 - 14 * q^37 - 3 * q^39 + 4 * q^43 + q^47 - 6 * q^49 - 3 * q^51 - 5 * q^53 - 6 * q^57 - 7 * q^59 - 5 * q^61 + q^63 - 3 * q^67 - 10 * q^71 + 10 * q^73 - 19 * q^77 - 16 * q^79 + 3 * q^81 - 11 * q^83 - q^87 - 8 * q^89 - q^91 - 7 * q^93 - 26 * q^97 - 3 * q^99

Introduction

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