LMFDB - Embedded newform 5148.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, 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("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.210756$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.32331 q^{5} -4.95558 q^{7} +O(q^{10})$	$q-3.32331 q^{5} -4.95558 q^{7} +1.00000 q^{11} +1.00000 q^{13} -1.21076 q^{17} +6.02372 q^{19} -1.46593 q^{23} +6.04442 q^{25} +4.27890 q^{29} +10.1663 q^{31} +16.4690 q^{35} +7.48965 q^{37} -6.95558 q^{41} +0.278896 q^{43} -7.91116 q^{47} +17.5578 q^{49} -10.3327 q^{53} -3.32331 q^{55} +6.64663 q^{59} +0.843024 q^{61} -3.32331 q^{65} -8.58785 q^{67} +5.48965 q^{71} -4.44523 q^{73} -4.95558 q^{77} -12.5040 q^{79} +6.64663 q^{83} +4.02372 q^{85} +13.1456 q^{89} -4.95558 q^{91} -20.0187 q^{95} -10.6466 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{5} - 4 q^{7}+O(q^{10}) $ 3 * q - 4 * q^5 - 4 * q^7	$ 3 q - 4 q^{5} - 4 q^{7} + 3 q^{11} + 3 q^{13} - 2 q^{17} - 8 q^{19} - 12 q^{23} + 29 q^{25} - 4 q^{29} + 18 q^{31} - 6 q^{35} + 4 q^{37} - 10 q^{41} - 16 q^{43} - 2 q^{47} + 19 q^{49} - 6 q^{53} - 4 q^{55} + 8 q^{59} - 4 q^{61} - 4 q^{65} - 10 q^{67} - 2 q^{71} + 16 q^{73} - 4 q^{77} - 12 q^{79} + 8 q^{83} - 14 q^{85} - 10 q^{89} - 4 q^{91} - 22 q^{95} - 20 q^{97}+O(q^{100}) $ 3 * q - 4 * q^5 - 4 * q^7 + 3 * q^11 + 3 * q^13 - 2 * q^17 - 8 * q^19 - 12 * q^23 + 29 * q^25 - 4 * q^29 + 18 * q^31 - 6 * q^35 + 4 * q^37 - 10 * q^41 - 16 * q^43 - 2 * q^47 + 19 * q^49 - 6 * q^53 - 4 * q^55 + 8 * q^59 - 4 * q^61 - 4 * q^65 - 10 * q^67 - 2 * q^71 + 16 * q^73 - 4 * q^77 - 12 * q^79 + 8 * q^83 - 14 * q^85 - 10 * q^89 - 4 * q^91 - 22 * q^95 - 20 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.32331	−1.48623	−0.743116	−	0.669163i	$-0.766653\pi$
$5$	−3.32331	−1.48623	−0.743116	+	0.669163i	$0.766653\pi$
$6$	0	0
$7$	−4.95558	−1.87303	−0.936517	−	0.350622i	$-0.885970\pi$
$7$	−4.95558	−1.87303	−0.936517	+	0.350622i	$0.885970\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.21076	−0.293651	−0.146826	−	0.989162i	$-0.546906\pi$
$17$	−1.21076	−0.293651	−0.146826	+	0.989162i	$0.546906\pi$
$18$	0	0
$19$	6.02372	1.38194	0.690968	−	0.722885i	$-0.257184\pi$
$19$	6.02372	1.38194	0.690968	+	0.722885i	$0.257184\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.46593	−0.305667	−0.152834	−	0.988252i	$-0.548840\pi$
$23$	−1.46593	−0.305667	−0.152834	+	0.988252i	$0.548840\pi$
$24$	0	0
$25$	6.04442	1.20888
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.27890	0.794571	0.397286	−	0.917695i	$-0.369952\pi$
$29$	4.27890	0.794571	0.397286	+	0.917695i	$0.369952\pi$
$30$	0	0
$31$	10.1663	1.82593	0.912964	−	0.408040i	$-0.133788\pi$
$31$	10.1663	1.82593	0.912964	+	0.408040i	$0.133788\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	16.4690	2.78376
$36$	0	0
$37$	7.48965	1.23129	0.615646	−	0.788023i	$-0.288895\pi$
$37$	7.48965	1.23129	0.615646	+	0.788023i	$0.288895\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.95558	−1.08628	−0.543140	−	0.839642i	$-0.682765\pi$
$41$	−6.95558	−1.08628	−0.543140	+	0.839642i	$0.682765\pi$
$42$	0	0
$43$	0.278896	0.0425313	0.0212656	−	0.999774i	$-0.493230\pi$
$43$	0.278896	0.0425313	0.0212656	+	0.999774i	$0.493230\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.91116	−1.15396	−0.576981	−	0.816758i	$-0.695769\pi$
$47$	−7.91116	−1.15396	−0.576981	+	0.816758i	$0.695769\pi$
$48$	0	0
$49$	17.5578	2.50826
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.3327	−1.41930	−0.709651	−	0.704553i	$-0.751147\pi$
$53$	−10.3327	−1.41930	−0.709651	+	0.704553i	$0.751147\pi$
$54$	0	0
$55$	−3.32331	−0.448116
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.64663	0.865317	0.432659	−	0.901558i	$-0.357576\pi$
$59$	6.64663	0.865317	0.432659	+	0.901558i	$0.357576\pi$
$60$	0	0
$61$	0.843024	0.107938	0.0539691	−	0.998543i	$-0.482813\pi$
$61$	0.843024	0.107938	0.0539691	+	0.998543i	$0.482813\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.32331	−0.412206
$66$	0	0
$67$	−8.58785	−1.04917	−0.524586	−	0.851357i	$-0.675780\pi$
$67$	−8.58785	−1.04917	−0.524586	+	0.851357i	$0.675780\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.48965	0.651502	0.325751	−	0.945456i	$-0.394383\pi$
$71$	5.48965	0.651502	0.325751	+	0.945456i	$0.394383\pi$
$72$	0	0
$73$	−4.44523	−0.520275	−0.260138	−	0.965572i	$-0.583768\pi$
$73$	−4.44523	−0.520275	−0.260138	+	0.965572i	$0.583768\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.95558	−0.564741
$78$	0	0
$79$	−12.5040	−1.40681	−0.703406	−	0.710789i	$-0.748338\pi$
$79$	−12.5040	−1.40681	−0.703406	+	0.710789i	$0.748338\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.64663	0.729562	0.364781	−	0.931093i	$-0.381144\pi$
$83$	6.64663	0.729562	0.364781	+	0.931093i	$0.381144\pi$
$84$	0	0
$85$	4.02372	0.436434
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.1456	1.39344	0.696718	−	0.717345i	$-0.254643\pi$
$89$	13.1456	1.39344	0.696718	+	0.717345i	$0.254643\pi$
$90$	0	0
$91$	−4.95558	−0.519486
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−20.0187	−2.05388
$96$	0	0
$97$	−10.6466	−1.08100	−0.540501	−	0.841344i	$-0.681765\pi$
$97$	−10.6466	−1.08100	−0.540501	+	0.841344i	$0.681765\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.8367	−1.87432	−0.937160	−	0.348899i	$-0.886556\pi$
$101$	−18.8367	−1.87432	−0.937160	+	0.348899i	$0.886556\pi$
$102$	0	0
$103$	−16.8905	−1.66427	−0.832134	−	0.554575i	$-0.812881\pi$
$103$	−16.8905	−1.66427	−0.832134	+	0.554575i	$0.812881\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.510348	−0.0493372	−0.0246686	−	0.999696i	$-0.507853\pi$
$107$	−0.510348	−0.0493372	−0.0246686	+	0.999696i	$0.507853\pi$
$108$	0	0
$109$	3.60221	0.345029	0.172515	−	0.985007i	$-0.444811\pi$
$109$	3.60221	0.345029	0.172515	+	0.985007i	$0.444811\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.71477	−0.913889	−0.456944	−	0.889495i	$-0.651056\pi$
$113$	−9.71477	−0.913889	−0.456944	+	0.889495i	$0.651056\pi$
$114$	0	0
$115$	4.87175	0.454293
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.47093	−0.310449
$126$	0	0
$127$	−11.8574	−1.05217	−0.526086	−	0.850431i	$-0.676341\pi$
$127$	−11.8574	−1.05217	−0.526086	+	0.850431i	$0.676341\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.06814	0.792287	0.396144	−	0.918189i	$-0.370348\pi$
$131$	9.06814	0.792287	0.396144	+	0.918189i	$0.370348\pi$
$132$	0	0
$133$	−29.8510	−2.58841
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.3026	1.73457	0.867285	−	0.497812i	$-0.165863\pi$
$137$	20.3026	1.73457	0.867285	+	0.497812i	$0.165863\pi$
$138$	0	0
$139$	3.34704	0.283892	0.141946	−	0.989874i	$-0.454664\pi$
$139$	3.34704	0.283892	0.141946	+	0.989874i	$0.454664\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−14.2201	−1.18092
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.9349	1.46928	0.734642	−	0.678455i	$-0.237350\pi$
$149$	17.9349	1.46928	0.734642	+	0.678455i	$0.237350\pi$
$150$	0	0
$151$	10.8667	0.884323	0.442162	−	0.896935i	$-0.354212\pi$
$151$	10.8667	0.884323	0.442162	+	0.896935i	$0.354212\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−33.7859	−2.71375
$156$	0	0
$157$	21.0919	1.68331	0.841657	−	0.540013i	$-0.181581\pi$
$157$	21.0919	1.68331	0.841657	+	0.540013i	$0.181581\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.26454	0.572526
$162$	0	0
$163$	−11.0381	−0.864569	−0.432285	−	0.901737i	$-0.642292\pi$
$163$	−11.0381	−0.864569	−0.432285	+	0.901737i	$0.642292\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.91116	0.147890	0.0739452	−	0.997262i	$-0.476441\pi$
$167$	1.91116	0.147890	0.0739452	+	0.997262i	$0.476441\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.7892	−1.73263	−0.866317	−	0.499494i	$-0.833519\pi$
$173$	−22.7892	−1.73263	−0.866317	+	0.499494i	$0.833519\pi$
$174$	0	0
$175$	−29.9536	−2.26428
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.2488	−1.06501	−0.532504	−	0.846427i	$-0.678749\pi$
$179$	−14.2488	−1.06501	−0.532504	+	0.846427i	$0.678749\pi$
$180$	0	0
$181$	10.7305	0.797589	0.398795	−	0.917040i	$-0.369429\pi$
$181$	10.7305	0.797589	0.398795	+	0.917040i	$0.369429\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−24.8905	−1.82998
$186$	0	0
$187$	−1.21076	−0.0885392
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.6259	−1.27537	−0.637684	−	0.770298i	$-0.720107\pi$
$191$	−17.6259	−1.27537	−0.637684	+	0.770298i	$0.720107\pi$
$192$	0	0
$193$	4.62291	0.332764	0.166382	−	0.986061i	$-0.446792\pi$
$193$	4.62291	0.332764	0.166382	+	0.986061i	$0.446792\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.53407	−0.323039	−0.161520	−	0.986870i	$-0.551639\pi$
$197$	−4.53407	−0.323039	−0.161520	+	0.986870i	$0.551639\pi$
$198$	0	0
$199$	−17.9112	−1.26969	−0.634844	−	0.772640i	$-0.718936\pi$
$199$	−17.9112	−1.26969	−0.634844	+	0.772640i	$0.718936\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−21.2044	−1.48826
$204$	0	0
$205$	23.1156	1.61446
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.02372	0.416670
$210$	0	0
$211$	13.6797	0.941750	0.470875	−	0.882200i	$-0.343938\pi$
$211$	13.6797	0.941750	0.470875	+	0.882200i	$0.343938\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.926860	−0.0632113
$216$	0	0
$217$	−50.3801	−3.42003
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.21076	−0.0814443
$222$	0	0
$223$	−11.0094	−0.737241	−0.368621	−	0.929580i	$-0.620170\pi$
$223$	−11.0094	−0.737241	−0.368621	+	0.929580i	$0.620170\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.0919	0.868937	0.434469	−	0.900687i	$-0.356936\pi$
$227$	13.0919	0.868937	0.434469	+	0.900687i	$0.356936\pi$
$228$	0	0
$229$	−14.3614	−0.949028	−0.474514	−	0.880248i	$-0.657376\pi$
$229$	−14.3614	−0.949028	−0.474514	+	0.880248i	$0.657376\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.7685	1.03303	0.516516	−	0.856278i	$-0.327229\pi$
$233$	15.7685	1.03303	0.516516	+	0.856278i	$0.327229\pi$
$234$	0	0
$235$	26.2913	1.71505
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.0712	−1.55703	−0.778517	−	0.627623i	$-0.784028\pi$
$239$	−24.0712	−1.55703	−0.778517	+	0.627623i	$0.784028\pi$
$240$	0	0
$241$	−24.6941	−1.59069	−0.795343	−	0.606160i	$-0.792709\pi$
$241$	−24.6941	−1.59069	−0.795343	+	0.606160i	$0.792709\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−58.3501	−3.72785
$246$	0	0
$247$	6.02372	0.383280
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.4690	0.660795	0.330397	−	0.943842i	$-0.392817\pi$
$251$	10.4690	0.660795	0.330397	+	0.943842i	$0.392817\pi$
$252$	0	0
$253$	−1.46593	−0.0921622
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−37.1156	−2.30625
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.35337	0.330103	0.165052	−	0.986285i	$-0.447221\pi$
$263$	5.35337	0.330103	0.165052	+	0.986285i	$0.447221\pi$
$264$	0	0
$265$	34.3387	2.10941
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.2645	0.808753	0.404376	−	0.914593i	$-0.367489\pi$
$269$	13.2645	0.808753	0.404376	+	0.914593i	$0.367489\pi$
$270$	0	0
$271$	12.6704	0.769669	0.384835	−	0.922986i	$-0.374259\pi$
$271$	12.6704	0.769669	0.384835	+	0.922986i	$0.374259\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.04442	0.364492
$276$	0	0
$277$	−9.38209	−0.563715	−0.281858	−	0.959456i	$-0.590951\pi$
$277$	−9.38209	−0.563715	−0.281858	+	0.959456i	$0.590951\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−27.5134	−1.64131	−0.820655	−	0.571424i	$-0.806391\pi$
$281$	−27.5134	−1.64131	−0.820655	+	0.571424i	$0.806391\pi$
$282$	0	0
$283$	−16.8080	−0.999130	−0.499565	−	0.866276i	$-0.666507\pi$
$283$	−16.8080	−0.999130	−0.499565	+	0.866276i	$0.666507\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	34.4690	2.03464
$288$	0	0
$289$	−15.5341	−0.913769
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.7385	1.15313	0.576567	−	0.817050i	$-0.304392\pi$
$293$	19.7385	1.15313	0.576567	+	0.817050i	$0.304392\pi$
$294$	0	0
$295$	−22.0888	−1.28606
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.46593	−0.0847769
$300$	0	0
$301$	−1.38209	−0.0796625
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.80163	−0.160421
$306$	0	0
$307$	1.79861	0.102652	0.0513259	−	0.998682i	$-0.483655\pi$
$307$	1.79861	0.102652	0.0513259	+	0.998682i	$0.483655\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.3978	−0.929833	−0.464917	−	0.885354i	$-0.653916\pi$
$311$	−16.3978	−0.929833	−0.464917	+	0.885354i	$0.653916\pi$
$312$	0	0
$313$	33.1994	1.87654	0.938271	−	0.345901i	$-0.112427\pi$
$313$	33.1994	1.87654	0.938271	+	0.345901i	$0.112427\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.14564	0.0643457	0.0321728	−	0.999482i	$-0.489757\pi$
$317$	1.14564	0.0643457	0.0321728	+	0.999482i	$0.489757\pi$
$318$	0	0
$319$	4.27890	0.239572
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.29326	−0.405808
$324$	0	0
$325$	6.04442	0.335284
$326$	0	0
$327$	0	0
$328$	0	0
$329$	39.2044	2.16141
$330$	0	0
$331$	13.8524	0.761396	0.380698	−	0.924699i	$-0.375684\pi$
$331$	13.8524	0.761396	0.380698	+	0.924699i	$0.375684\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	28.5401	1.55931
$336$	0	0
$337$	−1.03942	−0.0566207	−0.0283104	−	0.999599i	$-0.509013\pi$
$337$	−1.03942	−0.0566207	−0.0283104	+	0.999599i	$0.509013\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.1663	0.550538
$342$	0	0
$343$	−52.3200	−2.82501
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.0287	−0.645736	−0.322868	−	0.946444i	$-0.604647\pi$
$347$	−12.0287	−0.645736	−0.322868	+	0.946444i	$0.604647\pi$
$348$	0	0
$349$	−29.5371	−1.58108	−0.790542	−	0.612407i	$-0.790201\pi$
$349$	−29.5371	−1.58108	−0.790542	+	0.612407i	$0.790201\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.6166	0.884411	0.442205	−	0.896914i	$-0.354196\pi$
$353$	16.6166	0.884411	0.442205	+	0.896914i	$0.354196\pi$
$354$	0	0
$355$	−18.2438	−0.968282
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.24884	0.435357	0.217679	−	0.976021i	$-0.430152\pi$
$359$	8.24884	0.435357	0.217679	+	0.976021i	$0.430152\pi$
$360$	0	0
$361$	17.2852	0.909749
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.7729	0.773249
$366$	0	0
$367$	−0.960582	−0.0501420	−0.0250710	−	0.999686i	$-0.507981\pi$
$367$	−0.960582	−0.0501420	−0.0250710	+	0.999686i	$0.507981\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	51.2044	2.65840
$372$	0	0
$373$	7.62593	0.394856	0.197428	−	0.980317i	$-0.436741\pi$
$373$	7.62593	0.394856	0.197428	+	0.980317i	$0.436741\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.27890	0.220374
$378$	0	0
$379$	21.1744	1.08765	0.543827	−	0.839197i	$-0.316975\pi$
$379$	21.1744	1.08765	0.543827	+	0.839197i	$0.316975\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.44221	0.175889	0.0879443	−	0.996125i	$-0.471970\pi$
$383$	3.44221	0.175889	0.0879443	+	0.996125i	$0.471970\pi$
$384$	0	0
$385$	16.4690	0.839336
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.46896	0.226585	0.113293	−	0.993562i	$-0.463860\pi$
$389$	4.46896	0.226585	0.113293	+	0.993562i	$0.463860\pi$
$390$	0	0
$391$	1.77488	0.0897597
$392$	0	0
$393$	0	0
$394$	0	0
$395$	41.5548	2.09085
$396$	0	0
$397$	−19.3120	−0.969240	−0.484620	−	0.874725i	$-0.661042\pi$
$397$	−19.3120	−0.969240	−0.484620	+	0.874725i	$0.661042\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.5484	0.976202	0.488101	−	0.872787i	$-0.337690\pi$
$401$	19.5484	0.976202	0.488101	+	0.872787i	$0.337690\pi$
$402$	0	0
$403$	10.1663	0.506421
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.48965	0.371248
$408$	0	0
$409$	18.5341	0.916451	0.458225	−	0.888836i	$-0.348485\pi$
$409$	18.5341	0.916451	0.458225	+	0.888836i	$0.348485\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−32.9379	−1.62077
$414$	0	0
$415$	−22.0888	−1.08430
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−30.0712	−1.46907	−0.734536	−	0.678569i	$-0.762600\pi$
$419$	−30.0712	−1.46907	−0.734536	+	0.678569i	$0.762600\pi$
$420$	0	0
$421$	12.0888	0.589174	0.294587	−	0.955625i	$-0.404818\pi$
$421$	12.0888	0.589174	0.294587	+	0.955625i	$0.404818\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.31831	−0.354990
$426$	0	0
$427$	−4.17767	−0.202172
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.95256	0.190388	0.0951939	−	0.995459i	$-0.469653\pi$
$431$	3.95256	0.190388	0.0951939	+	0.995459i	$0.469653\pi$
$432$	0	0
$433$	−13.8223	−0.664259	−0.332129	−	0.943234i	$-0.607767\pi$
$433$	−13.8223	−0.664259	−0.332129	+	0.943234i	$0.607767\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.83035	−0.422413
$438$	0	0
$439$	6.41518	0.306180	0.153090	−	0.988212i	$-0.451078\pi$
$439$	6.41518	0.306180	0.153090	+	0.988212i	$0.451078\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.54977	−0.168654	−0.0843272	−	0.996438i	$-0.526874\pi$
$443$	−3.54977	−0.168654	−0.0843272	+	0.996438i	$0.526874\pi$
$444$	0	0
$445$	−43.6871	−2.07097
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.391455	−0.0184739	−0.00923694	−	0.999957i	$-0.502940\pi$
$449$	−0.391455	−0.0184739	−0.00923694	+	0.999957i	$0.502940\pi$
$450$	0	0
$451$	−6.95558	−0.327526
$452$	0	0
$453$	0	0
$454$	0	0
$455$	16.4690	0.772077
$456$	0	0
$457$	−34.8905	−1.63211	−0.816054	−	0.577976i	$-0.803843\pi$
$457$	−34.8905	−1.63211	−0.816054	+	0.577976i	$0.803843\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.7779	−1.06087	−0.530437	−	0.847725i	$-0.677972\pi$
$461$	−22.7779	−1.06087	−0.530437	+	0.847725i	$0.677972\pi$
$462$	0	0
$463$	−22.6353	−1.05195	−0.525976	−	0.850500i	$-0.676300\pi$
$463$	−22.6353	−1.05195	−0.525976	+	0.850500i	$0.676300\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.5341	0.487459	0.243729	−	0.969843i	$-0.421629\pi$
$467$	10.5341	0.487459	0.243729	+	0.969843i	$0.421629\pi$
$468$	0	0
$469$	42.5578	1.96514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.278896	0.0128237
$474$	0	0
$475$	36.4099	1.67060
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.7305	−1.22135	−0.610673	−	0.791883i	$-0.709101\pi$
$479$	−26.7305	−1.22135	−0.610673	+	0.791883i	$0.709101\pi$
$480$	0	0
$481$	7.48965	0.341499
$482$	0	0
$483$	0	0
$484$	0	0
$485$	35.3821	1.60662
$486$	0	0
$487$	5.56715	0.252272	0.126136	−	0.992013i	$-0.459742\pi$
$487$	5.56715	0.252272	0.126136	+	0.992013i	$0.459742\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−40.4690	−1.82634	−0.913169	−	0.407581i	$-0.866373\pi$
$491$	−40.4690	−1.82634	−0.913169	+	0.407581i	$0.866373\pi$
$492$	0	0
$493$	−5.18070	−0.233327
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.2044	−1.22029
$498$	0	0
$499$	−18.6767	−0.836083	−0.418042	−	0.908428i	$-0.637283\pi$
$499$	−18.6767	−0.836083	−0.418042	+	0.908428i	$0.637283\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.3534	0.773749	0.386874	−	0.922132i	$-0.373555\pi$
$503$	17.3534	0.773749	0.386874	+	0.922132i	$0.373555\pi$
$504$	0	0
$505$	62.6002	2.78567
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.0301	−0.887817	−0.443908	−	0.896072i	$-0.646408\pi$
$509$	−20.0301	−0.887817	−0.443908	+	0.896072i	$0.646408\pi$
$510$	0	0
$511$	22.0287	0.974493
$512$	0	0
$513$	0	0
$514$	0	0
$515$	56.1323	2.47349
$516$	0	0
$517$	−7.91116	−0.347933
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.8016	0.911336	0.455668	−	0.890150i	$-0.349400\pi$
$521$	20.8016	0.911336	0.455668	+	0.890150i	$0.349400\pi$
$522$	0	0
$523$	5.01436	0.219263	0.109631	−	0.993972i	$-0.465033\pi$
$523$	5.01436	0.219263	0.109631	+	0.993972i	$0.465033\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.3090	−0.536186
$528$	0	0
$529$	−20.8510	−0.906567
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.95558	−0.301280
$534$	0	0
$535$	1.69605	0.0733265
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17.5578	0.756268
$540$	0	0
$541$	7.11559	0.305923	0.152961	−	0.988232i	$-0.451119\pi$
$541$	7.11559	0.305923	0.152961	+	0.988232i	$0.451119\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.9713	−0.512793
$546$	0	0
$547$	−38.1487	−1.63112	−0.815560	−	0.578673i	$-0.803571\pi$
$547$	−38.1487	−1.63112	−0.815560	+	0.578673i	$0.803571\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	25.7749	1.09805
$552$	0	0
$553$	61.9647	2.63501
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.66232	0.239920	0.119960	−	0.992779i	$-0.461723\pi$
$557$	5.66232	0.239920	0.119960	+	0.992779i	$0.461723\pi$
$558$	0	0
$559$	0.278896	0.0117961
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.3821	0.985438	0.492719	−	0.870189i	$-0.336003\pi$
$563$	23.3821	0.985438	0.492719	+	0.870189i	$0.336003\pi$
$564$	0	0
$565$	32.2852	1.35825
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.4833	1.57138	0.785691	−	0.618619i	$-0.212307\pi$
$569$	37.4833	1.57138	0.785691	+	0.618619i	$0.212307\pi$
$570$	0	0
$571$	22.7479	0.951968	0.475984	−	0.879454i	$-0.342092\pi$
$571$	22.7479	0.951968	0.475984	+	0.879454i	$0.342092\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.86069	−0.369516
$576$	0	0
$577$	4.18373	0.174171	0.0870854	−	0.996201i	$-0.472245\pi$
$577$	4.18373	0.174171	0.0870854	+	0.996201i	$0.472245\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−32.9379	−1.36649
$582$	0	0
$583$	−10.3327	−0.427936
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.96058	0.369843	0.184921	−	0.982753i	$-0.440797\pi$
$587$	8.96058	0.369843	0.184921	+	0.982753i	$0.440797\pi$
$588$	0	0
$589$	61.2392	2.52332
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.5341	−0.678973	−0.339486	−	0.940611i	$-0.610253\pi$
$593$	−16.5341	−0.678973	−0.339486	+	0.940611i	$0.610253\pi$
$594$	0	0
$595$	−19.9399	−0.817456
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−38.0187	−1.55340	−0.776701	−	0.629869i	$-0.783109\pi$
$599$	−38.0187	−1.55340	−0.776701	+	0.629869i	$0.783109\pi$
$600$	0	0
$601$	−26.0949	−1.06443	−0.532216	−	0.846608i	$-0.678641\pi$
$601$	−26.0949	−1.06443	−0.532216	+	0.846608i	$0.678641\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.32331	−0.135112
$606$	0	0
$607$	−20.3076	−0.824261	−0.412130	−	0.911125i	$-0.635215\pi$
$607$	−20.3076	−0.824261	−0.412130	+	0.911125i	$0.635215\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.91116	−0.320051
$612$	0	0
$613$	−25.9162	−1.04674	−0.523372	−	0.852104i	$-0.675326\pi$
$613$	−25.9162	−1.04674	−0.523372	+	0.852104i	$0.675326\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.8811	1.44452	0.722259	−	0.691622i	$-0.243104\pi$
$617$	35.8811	1.44452	0.722259	+	0.691622i	$0.243104\pi$
$618$	0	0
$619$	−19.5959	−0.787625	−0.393812	−	0.919191i	$-0.628844\pi$
$619$	−19.5959	−0.787625	−0.393812	+	0.919191i	$0.628844\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−65.1443	−2.60995
$624$	0	0
$625$	−18.6871	−0.747484
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.06814	−0.361570
$630$	0	0
$631$	−21.3420	−0.849613	−0.424807	−	0.905284i	$-0.639658\pi$
$631$	−21.3420	−0.849613	−0.424807	+	0.905284i	$0.639658\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	39.4058	1.56377
$636$	0	0
$637$	17.5578	0.695665
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.36140	0.172265	0.0861324	−	0.996284i	$-0.472549\pi$
$641$	4.36140	0.172265	0.0861324	+	0.996284i	$0.472549\pi$
$642$	0	0
$643$	25.8811	1.02065	0.510326	−	0.859981i	$-0.329525\pi$
$643$	25.8811	1.02065	0.510326	+	0.859981i	$0.329525\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.7048	0.931931	0.465965	−	0.884803i	$-0.345707\pi$
$647$	23.7048	0.931931	0.465965	+	0.884803i	$0.345707\pi$
$648$	0	0
$649$	6.64663	0.260903
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.8430	−0.893917	−0.446958	−	0.894555i	$-0.647493\pi$
$653$	−22.8430	−0.893917	−0.446958	+	0.894555i	$0.647493\pi$
$654$	0	0
$655$	−30.1363	−1.17752
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−19.2933	−0.751559	−0.375779	−	0.926709i	$-0.622625\pi$
$659$	−19.2933	−0.751559	−0.375779	+	0.926709i	$0.622625\pi$
$660$	0	0
$661$	40.6941	1.58282	0.791408	−	0.611288i	$-0.209348\pi$
$661$	40.6941	1.58282	0.791408	+	0.611288i	$0.209348\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	99.2044	3.84698
$666$	0	0
$667$	−6.27256	−0.242875
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.843024	0.0325446
$672$	0	0
$673$	−22.2438	−0.857437	−0.428719	−	0.903438i	$-0.641035\pi$
$673$	−22.2438	−0.857437	−0.428719	+	0.903438i	$0.641035\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.74180	0.335975	0.167987	−	0.985789i	$-0.446273\pi$
$677$	8.74180	0.335975	0.167987	+	0.985789i	$0.446273\pi$
$678$	0	0
$679$	52.7602	2.02475
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−40.7415	−1.55893	−0.779465	−	0.626445i	$-0.784509\pi$
$683$	−40.7415	−1.55893	−0.779465	+	0.626445i	$0.784509\pi$
$684$	0	0
$685$	−67.4720	−2.57797
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.3327	−0.393644
$690$	0	0
$691$	−34.6353	−1.31759	−0.658794	−	0.752323i	$-0.728933\pi$
$691$	−34.6353	−1.31759	−0.658794	+	0.752323i	$0.728933\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.1233	−0.421929
$696$	0	0
$697$	8.42151	0.318988
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.5641	−0.701157	−0.350579	−	0.936533i	$-0.614015\pi$
$701$	−18.5641	−0.701157	−0.350579	+	0.936533i	$0.614015\pi$
$702$	0	0
$703$	45.1156	1.70157
$704$	0	0
$705$	0	0
$706$	0	0
$707$	93.3468	3.51067
$708$	0	0
$709$	−46.4088	−1.74292	−0.871460	−	0.490466i	$-0.836827\pi$
$709$	−46.4088	−1.74292	−0.871460	+	0.490466i	$0.836827\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14.9031	−0.558127
$714$	0	0
$715$	−3.32331	−0.124285
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−40.0949	−1.49529	−0.747644	−	0.664100i	$-0.768815\pi$
$719$	−40.0949	−1.49529	−0.747644	+	0.664100i	$0.768815\pi$
$720$	0	0
$721$	83.7021	3.11723
$722$	0	0
$723$	0	0
$724$	0	0
$725$	25.8634	0.960544
$726$	0	0
$727$	−2.81430	−0.104377	−0.0521883	−	0.998637i	$-0.516620\pi$
$727$	−2.81430	−0.104377	−0.0521883	+	0.998637i	$0.516620\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.337675	−0.0124894
$732$	0	0
$733$	−25.1106	−0.927481	−0.463740	−	0.885971i	$-0.653493\pi$
$733$	−25.1106	−0.927481	−0.463740	+	0.885971i	$0.653493\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.58785	−0.316338
$738$	0	0
$739$	−50.5688	−1.86021	−0.930103	−	0.367300i	$-0.880282\pi$
$739$	−50.5688	−1.86021	−0.930103	+	0.367300i	$0.880282\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37.4008	−1.37210	−0.686051	−	0.727553i	$-0.740658\pi$
$743$	−37.4008	−1.37210	−0.686051	+	0.727553i	$0.740658\pi$
$744$	0	0
$745$	−59.6033	−2.18369
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.52907	0.0924102
$750$	0	0
$751$	−37.2331	−1.35866	−0.679328	−	0.733834i	$-0.737729\pi$
$751$	−37.2331	−1.35866	−0.679328	+	0.733834i	$0.737729\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−36.1136	−1.31431
$756$	0	0
$757$	43.7986	1.59189	0.795944	−	0.605371i	$-0.206975\pi$
$757$	43.7986	1.59189	0.795944	+	0.605371i	$0.206975\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35.1580	1.27448	0.637239	−	0.770666i	$-0.280076\pi$
$761$	35.1580	1.27448	0.637239	+	0.770666i	$0.280076\pi$
$762$	0	0
$763$	−17.8510	−0.646251
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.64663	0.239996
$768$	0	0
$769$	−10.2726	−0.370438	−0.185219	−	0.982697i	$-0.559299\pi$
$769$	−10.2726	−0.370438	−0.185219	+	0.982697i	$0.559299\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−0.362733	−0.0130466	−0.00652329	−	0.999979i	$-0.502076\pi$
$773$	−0.362733	−0.0130466	−0.00652329	+	0.999979i	$0.502076\pi$
$774$	0	0
$775$	61.4496	2.20733
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−41.8985	−1.50117
$780$	0	0
$781$	5.48965	0.196435
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−70.0949	−2.50179
$786$	0	0
$787$	23.7572	0.846853	0.423427	−	0.905930i	$-0.360827\pi$
$787$	23.7572	0.846853	0.423427	+	0.905930i	$0.360827\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	48.1423	1.71174
$792$	0	0
$793$	0.843024	0.0299366
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.02675	−0.0363693	−0.0181847	−	0.999835i	$-0.505789\pi$
$797$	−1.02675	−0.0363693	−0.0181847	+	0.999835i	$0.505789\pi$
$798$	0	0
$799$	9.57849	0.338863
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.44523	−0.156869
$804$	0	0
$805$	−24.1423	−0.850905
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−37.3758	−1.31406	−0.657031	−	0.753864i	$-0.728188\pi$
$809$	−37.3758	−1.31406	−0.657031	+	0.753864i	$0.728188\pi$
$810$	0	0
$811$	−46.7465	−1.64149	−0.820746	−	0.571293i	$-0.806442\pi$
$811$	−46.7465	−1.64149	−0.820746	+	0.571293i	$0.806442\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	36.6830	1.28495
$816$	0	0
$817$	1.67999	0.0587755
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.35640	−0.291640	−0.145820	−	0.989311i	$-0.546582\pi$
$821$	−8.35640	−0.291640	−0.145820	+	0.989311i	$0.546582\pi$
$822$	0	0
$823$	13.7562	0.479510	0.239755	−	0.970833i	$-0.422933\pi$
$823$	13.7562	0.479510	0.239755	+	0.970833i	$0.422933\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.0030	−1.35627	−0.678134	−	0.734938i	$-0.737211\pi$
$827$	−39.0030	−1.35627	−0.678134	+	0.734938i	$0.737211\pi$
$828$	0	0
$829$	−29.4295	−1.02213	−0.511065	−	0.859542i	$-0.670749\pi$
$829$	−29.4295	−1.02213	−0.511065	+	0.859542i	$0.670749\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−21.2582	−0.736553
$834$	0	0
$835$	−6.35140	−0.219799
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−28.2251	−0.974439	−0.487220	−	0.873279i	$-0.661989\pi$
$839$	−28.2251	−0.974439	−0.487220	+	0.873279i	$0.661989\pi$
$840$	0	0
$841$	−10.6910	−0.368657
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.32331	−0.114325
$846$	0	0
$847$	−4.95558	−0.170276
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.9793	−0.376366
$852$	0	0
$853$	−26.4402	−0.905296	−0.452648	−	0.891689i	$-0.649521\pi$
$853$	−26.4402	−0.905296	−0.452648	+	0.891689i	$0.649521\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.8861	0.679296	0.339648	−	0.940553i	$-0.389692\pi$
$857$	19.8861	0.679296	0.339648	+	0.940553i	$0.389692\pi$
$858$	0	0
$859$	−13.8510	−0.472592	−0.236296	−	0.971681i	$-0.575933\pi$
$859$	−13.8510	−0.472592	−0.236296	+	0.971681i	$0.575933\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	57.3180	1.95113	0.975564	−	0.219714i	$-0.0705125\pi$
$863$	57.3180	1.95113	0.975564	+	0.219714i	$0.0705125\pi$
$864$	0	0
$865$	75.7358	2.57510
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.5040	−0.424170
$870$	0	0
$871$	−8.58785	−0.290988
$872$	0	0
$873$	0	0
$874$	0	0
$875$	17.2005	0.581482
$876$	0	0
$877$	11.8510	0.400182	0.200091	−	0.979777i	$-0.435876\pi$
$877$	11.8510	0.400182	0.200091	+	0.979777i	$0.435876\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.7228	1.30460	0.652302	−	0.757959i	$-0.273803\pi$
$881$	38.7228	1.30460	0.652302	+	0.757959i	$0.273803\pi$
$882$	0	0
$883$	1.51837	0.0510974	0.0255487	−	0.999674i	$-0.491867\pi$
$883$	1.51837	0.0510974	0.0255487	+	0.999674i	$0.491867\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.29326	−0.244884	−0.122442	−	0.992476i	$-0.539072\pi$
$887$	−7.29326	−0.244884	−0.122442	+	0.992476i	$0.539072\pi$
$888$	0	0
$889$	58.7602	1.97076
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−47.6547	−1.59470
$894$	0	0
$895$	47.3534	1.58285
$896$	0	0
$897$	0	0
$898$	0	0
$899$	43.5007	1.45083
$900$	0	0
$901$	12.5103	0.416780
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−35.6607	−1.18540
$906$	0	0
$907$	14.6466	0.486333	0.243167	−	0.969985i	$-0.421814\pi$
$907$	14.6466	0.486333	0.243167	+	0.969985i	$0.421814\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.41082	−0.179268	−0.0896342	−	0.995975i	$-0.528570\pi$
$911$	−5.41082	−0.179268	−0.0896342	+	0.995975i	$0.528570\pi$
$912$	0	0
$913$	6.64663	0.219971
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−44.9379	−1.48398
$918$	0	0
$919$	27.9649	0.922478	0.461239	−	0.887276i	$-0.347405\pi$
$919$	27.9649	0.922478	0.461239	+	0.887276i	$0.347405\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.48965	0.180694
$924$	0	0
$925$	45.2706	1.48849
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.92053	0.292673	0.146336	−	0.989235i	$-0.453252\pi$
$929$	8.92053	0.292673	0.146336	+	0.989235i	$0.453252\pi$
$930$	0	0
$931$	105.763	3.46625
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.02372	0.131590
$936$	0	0
$937$	51.4069	1.67939	0.839695	−	0.543059i	$-0.182734\pi$
$937$	51.4069	1.67939	0.839695	+	0.543059i	$0.182734\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	39.0030	1.27146	0.635731	−	0.771911i	$-0.280699\pi$
$941$	39.0030	1.27146	0.635731	+	0.771911i	$0.280699\pi$
$942$	0	0
$943$	10.1964	0.332040
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−55.7809	−1.81264	−0.906318	−	0.422595i	$-0.861119\pi$
$947$	−55.7809	−1.81264	−0.906318	+	0.422595i	$0.861119\pi$
$948$	0	0
$949$	−4.44523	−0.144298
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.51668	−0.146310	−0.0731548	−	0.997321i	$-0.523307\pi$
$953$	−4.51668	−0.146310	−0.0731548	+	0.997321i	$0.523307\pi$
$954$	0	0
$955$	58.5765	1.89549
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−100.611	−3.24891
$960$	0	0
$961$	72.3544	2.33401
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.3634	−0.494564
$966$	0	0
$967$	16.4927	0.530369	0.265184	−	0.964198i	$-0.414567\pi$
$967$	16.4927	0.530369	0.265184	+	0.964198i	$0.414567\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.24081	−0.296552	−0.148276	−	0.988946i	$-0.547372\pi$
$971$	−9.24081	−0.296552	−0.148276	+	0.988946i	$0.547372\pi$
$972$	0	0
$973$	−16.5865	−0.531739
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.3420	1.13069	0.565346	−	0.824854i	$-0.308743\pi$
$977$	35.3420	1.13069	0.565346	+	0.824854i	$0.308743\pi$
$978$	0	0
$979$	13.1456	0.420137
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−24.6179	−0.785189	−0.392595	−	0.919712i	$-0.628422\pi$
$983$	−24.6179	−0.785189	−0.392595	+	0.919712i	$0.628422\pi$
$984$	0	0
$985$	15.0681	0.480111
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.408842	−0.0130004
$990$	0	0
$991$	0.0601141	0.00190959	0.000954794	−	1.00000i	$-0.499696\pi$
$991$	0.0601141	0.00190959	0.000954794		1.00000i	$0.499696\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	59.5244	1.88705
$996$	0	0
$997$	−12.8243	−0.406150	−0.203075	−	0.979163i	$-0.565093\pi$
$997$	−12.8243	−0.406150	−0.203075	+	0.979163i	$0.565093\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.l.1.2		3
3.2	odd	2		1716.2.a.h.1.2	✓	3
12.11	even	2		6864.2.a.bt.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.h.1.2	✓	3	3.2	odd	2
5148.2.a.l.1.2		3	1.1	even	1	trivial
6864.2.a.bt.1.2		3	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q-3.32331 q^{5} -4.95558 q^{7} +O(q^{10})\)	\(q-3.32331 q^{5} -4.95558 q^{7} +1.00000 q^{11} +1.00000 q^{13} -1.21076 q^{17} +6.02372 q^{19} -1.46593 q^{23} +6.04442 q^{25} +4.27890 q^{29} +10.1663 q^{31} +16.4690 q^{35} +7.48965 q^{37} -6.95558 q^{41} +0.278896 q^{43} -7.91116 q^{47} +17.5578 q^{49} -10.3327 q^{53} -3.32331 q^{55} +6.64663 q^{59} +0.843024 q^{61} -3.32331 q^{65} -8.58785 q^{67} +5.48965 q^{71} -4.44523 q^{73} -4.95558 q^{77} -12.5040 q^{79} +6.64663 q^{83} +4.02372 q^{85} +13.1456 q^{89} -4.95558 q^{91} -20.0187 q^{95} -10.6466 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) 3 * q - 4 * q^5 - 4 * q^7	\( 3 q - 4 q^{5} - 4 q^{7} + 3 q^{11} + 3 q^{13} - 2 q^{17} - 8 q^{19} - 12 q^{23} + 29 q^{25} - 4 q^{29} + 18 q^{31} - 6 q^{35} + 4 q^{37} - 10 q^{41} - 16 q^{43} - 2 q^{47} + 19 q^{49} - 6 q^{53} - 4 q^{55} + 8 q^{59} - 4 q^{61} - 4 q^{65} - 10 q^{67} - 2 q^{71} + 16 q^{73} - 4 q^{77} - 12 q^{79} + 8 q^{83} - 14 q^{85} - 10 q^{89} - 4 q^{91} - 22 q^{95} - 20 q^{97}+O(q^{100}) \) 3 * q - 4 * q^5 - 4 * q^7 + 3 * q^11 + 3 * q^13 - 2 * q^17 - 8 * q^19 - 12 * q^23 + 29 * q^25 - 4 * q^29 + 18 * q^31 - 6 * q^35 + 4 * q^37 - 10 * q^41 - 16 * q^43 - 2 * q^47 + 19 * q^49 - 6 * q^53 - 4 * q^55 + 8 * q^59 - 4 * q^61 - 4 * q^65 - 10 * q^67 - 2 * q^71 + 16 * q^73 - 4 * q^77 - 12 * q^79 + 8 * q^83 - 14 * q^85 - 10 * q^89 - 4 * q^91 - 22 * q^95 - 20 * q^97

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