LMFDB - Embedded newform 5175.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5175,2,Mod(1,5175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5175 = 3^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5175.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:	$41.3225830460$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{7} +O(q^{10})$

$q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{7} -2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -5.00000 q^{17} +8.00000 q^{19} -1.00000 q^{23} -4.00000 q^{26} -2.00000 q^{28} +5.00000 q^{29} -5.00000 q^{31} -8.00000 q^{32} -10.0000 q^{34} -7.00000 q^{37} +16.0000 q^{38} +7.00000 q^{41} -4.00000 q^{43} -2.00000 q^{46} -2.00000 q^{47} -6.00000 q^{49} -4.00000 q^{52} -1.00000 q^{53} +10.0000 q^{58} -3.00000 q^{59} -6.00000 q^{61} -10.0000 q^{62} -8.00000 q^{64} -13.0000 q^{67} -10.0000 q^{68} -13.0000 q^{71} -8.00000 q^{73} -14.0000 q^{74} +16.0000 q^{76} -14.0000 q^{79} +14.0000 q^{82} -3.00000 q^{83} -8.00000 q^{86} +14.0000 q^{89} +2.00000 q^{91} -2.00000 q^{92} -4.00000 q^{94} -14.0000 q^{97} -12.0000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	−4.00000	−0.784465
$27$	0	0
$28$	−2.00000	−0.377964
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	−10.0000	−1.71499
$35$	0	0
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	16.0000	2.59554
$39$	0	0
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	−2.00000	−0.294884
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	10.0000	1.31306
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	−10.0000	−1.21268
$69$	0	0
$70$	0	0
$71$	−13.0000	−1.54282	−0.771408	−	0.636341i	$-0.780447\pi$
$71$	−13.0000	−1.54282	−0.771408	+	0.636341i	$0.780447\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	−14.0000	−1.62747
$75$	0	0
$76$	16.0000	1.83533
$77$	0	0
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	0	0
$82$	14.0000	1.54604
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−2.00000	−0.208514
$93$	0	0
$94$	−4.00000	−0.412568
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−12.0000	−1.21218
$99$	0	0
$100$	0	0
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	−1.00000	−0.0940721	−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000	−0.0940721	−0.0470360	+	0.998893i	$0.514978\pi$
$114$	0	0
$115$	0	0
$116$	10.0000	0.928477
$117$	0	0
$118$	−6.00000	−0.552345
$119$	5.00000	0.458349
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−12.0000	−1.08643
$123$	0	0
$124$	−10.0000	−0.898027
$125$	0	0
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	−26.0000	−2.24606
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	9.00000	0.763370	0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000	0.763370	0.381685	+	0.924292i	$0.375344\pi$
$140$	0	0
$141$	0	0
$142$	−26.0000	−2.18187
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−16.0000	−1.32417
$147$	0	0
$148$	−14.0000	−1.15079
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	−28.0000	−2.22756
$159$	0	0
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	14.0000	1.09322
$165$	0	0
$166$	−6.00000	−0.465690
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	28.0000	2.09869
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	4.00000	0.296500
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−4.00000	−0.291730
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	−28.0000	−2.01028
$195$	0	0
$196$	−12.0000	−0.857143
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	0	0
$202$	−30.0000	−2.11079
$203$	−5.00000	−0.350931
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	8.00000	0.554700
$209$	0	0
$210$	0	0
$211$	−9.00000	−0.619586	−0.309793	−	0.950804i	$-0.600260\pi$
$211$	−9.00000	−0.619586	−0.309793	+	0.950804i	$0.600260\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	18.0000	1.23045
$215$	0	0
$216$	0	0
$217$	5.00000	0.339422
$218$	36.0000	2.43823
$219$	0	0
$220$	0	0
$221$	10.0000	0.672673
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	8.00000	0.534522
$225$	0	0
$226$	−2.00000	−0.133038
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	10.0000	0.648204
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	−22.0000	−1.41421
$243$	0	0
$244$	−12.0000	−0.768221
$245$	0	0
$246$	0	0
$247$	−16.0000	−1.01806
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0	0
$254$	40.0000	2.50982
$255$	0	0
$256$	16.0000	1.00000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.0000	−0.801614	−0.400807	−	0.916162i	$-0.631270\pi$
$263$	−13.0000	−0.801614	−0.400807	+	0.916162i	$0.631270\pi$
$264$	0	0
$265$	0	0
$266$	−16.0000	−0.981023
$267$	0	0
$268$	−26.0000	−1.58820
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	20.0000	1.21268
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	18.0000	1.07957
$279$	0	0
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−11.0000	−0.653882	−0.326941	−	0.945045i	$-0.606018\pi$
$283$	−11.0000	−0.653882	−0.326941	+	0.945045i	$0.606018\pi$
$284$	−26.0000	−1.54282
$285$	0	0
$286$	0	0
$287$	−7.00000	−0.413197
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	−16.0000	−0.936329
$293$	29.0000	1.69420	0.847099	−	0.531435i	$-0.178347\pi$
$293$	29.0000	1.69420	0.847099	+	0.531435i	$0.178347\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	4.00000	0.230556
$302$	16.0000	0.920697
$303$	0	0
$304$	−32.0000	−1.83533
$305$	0	0
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	−21.0000	−1.18699	−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000	−1.18699	−0.593495	+	0.804838i	$0.702252\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	−28.0000	−1.57512
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	2.00000	0.111456
$323$	−40.0000	−2.22566
$324$	0	0
$325$	0	0
$326$	−48.0000	−2.65847
$327$	0	0
$328$	0	0
$329$	2.00000	0.110264
$330$	0	0
$331$	−5.00000	−0.274825	−0.137412	−	0.990514i	$-0.543879\pi$
$331$	−5.00000	−0.274825	−0.137412	+	0.990514i	$0.543879\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	32.0000	1.75096
$335$	0	0
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	−18.0000	−0.979071
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	23.0000	1.23116	0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000	1.23116	0.615581	+	0.788074i	$0.288921\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	28.0000	1.48400
$357$	0	0
$358$	8.00000	0.422813
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	−28.0000	−1.47165
$363$	0	0
$364$	4.00000	0.209657
$365$	0	0
$366$	0	0
$367$	13.0000	0.678594	0.339297	−	0.940679i	$-0.389811\pi$
$367$	13.0000	0.678594	0.339297	+	0.940679i	$0.389811\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	0	0
$371$	1.00000	0.0519174
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.0000	−0.515026
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	0	0
$381$	0	0
$382$	16.0000	0.818631
$383$	3.00000	0.153293	0.0766464	−	0.997058i	$-0.475579\pi$
$383$	3.00000	0.153293	0.0766464	+	0.997058i	$0.475579\pi$
$384$	0	0
$385$	0	0
$386$	24.0000	1.22157
$387$	0	0
$388$	−28.0000	−1.42148
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.0000	−0.803017	−0.401508	−	0.915855i	$-0.631514\pi$
$397$	−16.0000	−0.803017	−0.401508	+	0.915855i	$0.631514\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	−30.0000	−1.49256
$405$	0	0
$406$	−10.0000	−0.496292
$407$	0	0
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.00000	0.147620
$414$	0	0
$415$	0	0
$416$	16.0000	0.784465
$417$	0	0
$418$	0	0
$419$	2.00000	0.0977064	0.0488532	−	0.998806i	$-0.484443\pi$
$419$	2.00000	0.0977064	0.0488532	+	0.998806i	$0.484443\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−18.0000	−0.876226
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.00000	0.290360
$428$	18.0000	0.870063
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	36.0000	1.72409
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	0	0
$442$	20.0000	0.951303
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	0	0
$446$	28.0000	1.32584
$447$	0	0
$448$	8.00000	0.377964
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	0	0
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	8.00000	0.373815
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	−20.0000	−0.928477
$465$	0	0
$466$	12.0000	0.555889
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	0	0
$469$	13.0000	0.600284
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	10.0000	0.458349
$477$	0	0
$478$	2.00000	0.0914779
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	14.0000	0.638345
$482$	12.0000	0.546585
$483$	0	0
$484$	−22.0000	−1.00000
$485$	0	0
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.0000	1.39901	0.699505	−	0.714628i	$-0.253404\pi$
$491$	31.0000	1.39901	0.699505	+	0.714628i	$0.253404\pi$
$492$	0	0
$493$	−25.0000	−1.12594
$494$	−32.0000	−1.43975
$495$	0	0
$496$	20.0000	0.898027
$497$	13.0000	0.583130
$498$	0	0
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	0	0
$501$	0	0
$502$	−36.0000	−1.60676
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	40.0000	1.77471
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	32.0000	1.41421
$513$	0	0
$514$	44.0000	1.94076
$515$	0	0
$516$	0	0
$517$	0	0
$518$	14.0000	0.615125
$519$	0	0
$520$	0	0
$521$	20.0000	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	0	0
$525$	0	0
$526$	−26.0000	−1.13365
$527$	25.0000	1.08902
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	−16.0000	−0.693688
$533$	−14.0000	−0.606407
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	30.0000	1.29339
$539$	0	0
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	−30.0000	−1.28861
$543$	0	0
$544$	40.0000	1.71499
$545$	0	0
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	0	0
$551$	40.0000	1.70406
$552$	0	0
$553$	14.0000	0.595341
$554$	52.0000	2.20927
$555$	0	0
$556$	18.0000	0.763370
$557$	−33.0000	−1.39825	−0.699127	−	0.714997i	$-0.746428\pi$
$557$	−33.0000	−1.39825	−0.699127	+	0.714997i	$0.746428\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	24.0000	1.01238
$563$	−25.0000	−1.05362	−0.526812	−	0.849982i	$-0.676613\pi$
$563$	−25.0000	−1.05362	−0.526812	+	0.849982i	$0.676613\pi$
$564$	0	0
$565$	0	0
$566$	−22.0000	−0.924729
$567$	0	0
$568$	0	0
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	0	0
$573$	0	0
$574$	−14.0000	−0.584349
$575$	0	0
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	0	0
$581$	3.00000	0.124461
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	58.0000	2.39596
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−40.0000	−1.64817
$590$	0	0
$591$	0	0
$592$	28.0000	1.15079
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	4.00000	0.163572
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	−64.0000	−2.59554
$609$	0	0
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	0	0
$617$	−47.0000	−1.89215	−0.946074	−	0.323949i	$-0.894989\pi$
$617$	−47.0000	−1.89215	−0.946074	+	0.323949i	$0.894989\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	0	0
$622$	−8.00000	−0.320771
$623$	−14.0000	−0.560898
$624$	0	0
$625$	0	0
$626$	−42.0000	−1.67866
$627$	0	0
$628$	6.00000	0.239426
$629$	35.0000	1.39554
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	0	0
$634$	−48.0000	−1.90632
$635$	0	0
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	−37.0000	−1.45914	−0.729569	−	0.683907i	$-0.760279\pi$
$643$	−37.0000	−1.45914	−0.729569	+	0.683907i	$0.760279\pi$
$644$	2.00000	0.0788110
$645$	0	0
$646$	−80.0000	−3.14756
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−48.0000	−1.87983
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	0	0
$655$	0	0
$656$	−28.0000	−1.09322
$657$	0	0
$658$	4.00000	0.155936
$659$	−22.0000	−0.856998	−0.428499	−	0.903542i	$-0.640958\pi$
$659$	−22.0000	−0.856998	−0.428499	+	0.903542i	$0.640958\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.00000	−0.193601
$668$	32.0000	1.23812
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−24.0000	−0.925132	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000	−0.925132	−0.462566	+	0.886585i	$0.653071\pi$
$674$	−52.0000	−2.00297
$675$	0	0
$676$	−18.0000	−0.692308
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	0	0
$682$	0	0
$683$	46.0000	1.76014	0.880071	−	0.474843i	$-0.157495\pi$
$683$	46.0000	1.76014	0.880071	+	0.474843i	$0.157495\pi$
$684$	0	0
$685$	0	0
$686$	26.0000	0.992685
$687$	0	0
$688$	16.0000	0.609994
$689$	2.00000	0.0761939
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	16.0000	0.607352
$695$	0	0
$696$	0	0
$697$	−35.0000	−1.32572
$698$	46.0000	1.74113
$699$	0	0
$700$	0	0
$701$	28.0000	1.05755	0.528773	−	0.848763i	$-0.322652\pi$
$701$	28.0000	1.05755	0.528773	+	0.848763i	$0.322652\pi$
$702$	0	0
$703$	−56.0000	−2.11208
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.0000	0.564133
$708$	0	0
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.00000	0.187251
$714$	0	0
$715$	0	0
$716$	8.00000	0.298974
$717$	0	0
$718$	12.0000	0.447836
$719$	45.0000	1.67822	0.839108	−	0.543964i	$-0.183077\pi$
$719$	45.0000	1.67822	0.839108	+	0.543964i	$0.183077\pi$
$720$	0	0
$721$	0	0
$722$	90.0000	3.34945
$723$	0	0
$724$	−28.0000	−1.04061
$725$	0	0
$726$	0	0
$727$	−27.0000	−1.00137	−0.500687	−	0.865628i	$-0.666919\pi$
$727$	−27.0000	−1.00137	−0.500687	+	0.865628i	$0.666919\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	20.0000	0.739727
$732$	0	0
$733$	−7.00000	−0.258551	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000	−0.258551	−0.129275	+	0.991609i	$0.541265\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	8.00000	0.294884
$737$	0	0
$738$	0	0
$739$	7.00000	0.257499	0.128750	−	0.991677i	$-0.458904\pi$
$739$	7.00000	0.257499	0.128750	+	0.991677i	$0.458904\pi$
$740$	0	0
$741$	0	0
$742$	2.00000	0.0734223
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	0	0
$746$	12.0000	0.439351
$747$	0	0
$748$	0	0
$749$	−9.00000	−0.328853
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	−20.0000	−0.728357
$755$	0	0
$756$	0	0
$757$	35.0000	1.27210	0.636048	−	0.771649i	$-0.280568\pi$
$757$	35.0000	1.27210	0.636048	+	0.771649i	$0.280568\pi$
$758$	12.0000	0.435860
$759$	0	0
$760$	0	0
$761$	−25.0000	−0.906249	−0.453125	−	0.891447i	$-0.649691\pi$
$761$	−25.0000	−0.906249	−0.453125	+	0.891447i	$0.649691\pi$
$762$	0	0
$763$	−18.0000	−0.651644
$764$	16.0000	0.578860
$765$	0	0
$766$	6.00000	0.216789
$767$	6.00000	0.216647
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	0	0
$772$	24.0000	0.863779
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−44.0000	−1.57748
$779$	56.0000	2.00641
$780$	0	0
$781$	0	0
$782$	10.0000	0.357599
$783$	0	0
$784$	24.0000	0.857143
$785$	0	0
$786$	0	0
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.00000	0.0355559
$792$	0	0
$793$	12.0000	0.426132
$794$	−32.0000	−1.13564
$795$	0	0
$796$	4.00000	0.141776
$797$	−15.0000	−0.531327	−0.265664	−	0.964066i	$-0.585591\pi$
$797$	−15.0000	−0.531327	−0.265664	+	0.964066i	$0.585591\pi$
$798$	0	0
$799$	10.0000	0.353775
$800$	0	0
$801$	0	0
$802$	−4.00000	−0.141245
$803$	0	0
$804$	0	0
$805$	0	0
$806$	20.0000	0.704470
$807$	0	0
$808$	0	0
$809$	25.0000	0.878953	0.439477	−	0.898254i	$-0.355164\pi$
$809$	25.0000	0.878953	0.439477	+	0.898254i	$0.355164\pi$
$810$	0	0
$811$	31.0000	1.08856	0.544279	−	0.838905i	$-0.316803\pi$
$811$	31.0000	1.08856	0.544279	+	0.838905i	$0.316803\pi$
$812$	−10.0000	−0.350931
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−32.0000	−1.11954
$818$	−38.0000	−1.32864
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	34.0000	1.18517	0.592583	−	0.805510i	$-0.298108\pi$
$823$	34.0000	1.18517	0.592583	+	0.805510i	$0.298108\pi$
$824$	0	0
$825$	0	0
$826$	6.00000	0.208767
$827$	−9.00000	−0.312961	−0.156480	−	0.987681i	$-0.550015\pi$
$827$	−9.00000	−0.312961	−0.156480	+	0.987681i	$0.550015\pi$
$828$	0	0
$829$	37.0000	1.28506	0.642532	−	0.766259i	$-0.277884\pi$
$829$	37.0000	1.28506	0.642532	+	0.766259i	$0.277884\pi$
$830$	0	0
$831$	0	0
$832$	16.0000	0.554700
$833$	30.0000	1.03944
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	4.00000	0.138178
$839$	−14.0000	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$839$	−14.0000	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	20.0000	0.689246
$843$	0	0
$844$	−18.0000	−0.619586
$845$	0	0
$846$	0	0
$847$	11.0000	0.377964
$848$	4.00000	0.137361
$849$	0	0
$850$	0	0
$851$	7.00000	0.239957
$852$	0	0
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	0	0
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	43.0000	1.46714	0.733571	−	0.679613i	$-0.237852\pi$
$859$	43.0000	1.46714	0.733571	+	0.679613i	$0.237852\pi$
$860$	0	0
$861$	0	0
$862$	24.0000	0.817443
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	0	0
$865$	0	0
$866$	−38.0000	−1.29129
$867$	0	0
$868$	10.0000	0.339422
$869$	0	0
$870$	0	0
$871$	26.0000	0.880976
$872$	0	0
$873$	0	0
$874$	−16.0000	−0.541208
$875$	0	0
$876$	0	0
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	−56.0000	−1.88991
$879$	0	0
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	20.0000	0.672673
$885$	0	0
$886$	−48.0000	−1.61259
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	0	0
$892$	28.0000	0.937509
$893$	−16.0000	−0.535420
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−25.0000	−0.833797
$900$	0	0
$901$	5.00000	0.166574
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−9.00000	−0.298840	−0.149420	−	0.988774i	$-0.547741\pi$
$907$	−9.00000	−0.298840	−0.149420	+	0.988774i	$0.547741\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	8.00000	0.264327
$917$	0	0
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	0	0
$922$	36.0000	1.18560
$923$	26.0000	0.855800
$924$	0	0
$925$	0	0
$926$	8.00000	0.262896
$927$	0	0
$928$	−40.0000	−1.31306
$929$	−35.0000	−1.14831	−0.574156	−	0.818746i	$-0.694670\pi$
$929$	−35.0000	−1.14831	−0.574156	+	0.818746i	$0.694670\pi$
$930$	0	0
$931$	−48.0000	−1.57314
$932$	12.0000	0.393073
$933$	0	0
$934$	−54.0000	−1.76693
$935$	0	0
$936$	0	0
$937$	−54.0000	−1.76410	−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000	−1.76410	−0.882052	+	0.471153i	$0.843838\pi$
$938$	26.0000	0.848930
$939$	0	0
$940$	0	0
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	0	0
$943$	−7.00000	−0.227951
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	14.0000	0.454939	0.227469	−	0.973785i	$-0.426955\pi$
$947$	14.0000	0.454939	0.227469	+	0.973785i	$0.426955\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	0	0
$955$	0	0
$956$	2.00000	0.0646846
$957$	0	0
$958$	56.0000	1.80928
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	28.0000	0.902756
$963$	0	0
$964$	12.0000	0.386494
$965$	0	0
$966$	0	0
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	0	0
$973$	−9.00000	−0.288527
$974$	64.0000	2.05069
$975$	0	0
$976$	24.0000	0.768221
$977$	−39.0000	−1.24772	−0.623860	−	0.781536i	$-0.714437\pi$
$977$	−39.0000	−1.24772	−0.623860	+	0.781536i	$0.714437\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	62.0000	1.97850
$983$	21.0000	0.669796	0.334898	−	0.942254i	$-0.391298\pi$
$983$	21.0000	0.669796	0.334898	+	0.942254i	$0.391298\pi$
$984$	0	0
$985$	0	0
$986$	−50.0000	−1.59232
$987$	0	0
$988$	−32.0000	−1.01806
$989$	4.00000	0.127193
$990$	0	0
$991$	19.0000	0.603555	0.301777	−	0.953378i	$-0.402420\pi$
$991$	19.0000	0.603555	0.301777	+	0.953378i	$0.402420\pi$
$992$	40.0000	1.27000
$993$	0	0
$994$	26.0000	0.824670
$995$	0	0
$996$	0	0
$997$	−20.0000	−0.633406	−0.316703	−	0.948525i	$-0.602576\pi$
$997$	−20.0000	−0.633406	−0.316703	+	0.948525i	$0.602576\pi$
$998$	−22.0000	−0.696398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.z.1.1		1
3.2	odd	2		575.2.a.a.1.1		1
5.2	odd	4		1035.2.b.a.829.2		2
5.3	odd	4		1035.2.b.a.829.1		2
5.4	even	2		5175.2.a.a.1.1		1
12.11	even	2		9200.2.a.bg.1.1		1
15.2	even	4		115.2.b.a.24.1	✓	2
15.8	even	4		115.2.b.a.24.2	yes	2
15.14	odd	2		575.2.a.e.1.1		1
60.23	odd	4		1840.2.e.b.369.2		2
60.47	odd	4		1840.2.e.b.369.1		2
60.59	even	2		9200.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.a.24.1	✓	2	15.2	even	4
115.2.b.a.24.2	yes	2	15.8	even	4
575.2.a.a.1.1		1	3.2	odd	2
575.2.a.e.1.1		1	15.14	odd	2
1035.2.b.a.829.1		2	5.3	odd	4
1035.2.b.a.829.2		2	5.2	odd	4
1840.2.e.b.369.1		2	60.47	odd	4
1840.2.e.b.369.2		2	60.23	odd	4
5175.2.a.a.1.1		1	5.4	even	2
5175.2.a.z.1.1		1	1.1	even	1	trivial
9200.2.a.g.1.1		1	60.59	even	2
9200.2.a.bg.1.1		1	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:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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