LMFDB - Embedded newform 900.4.j.a.557.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,4,Mod(557,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("900.557");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	900.j (of order $4$, degree $2$, minimal)

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:	no
Analytic conductor:	$53.1017190052$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.695825051222016.28
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 13121x^{4} + 40960000 $ x^8 + 13121*x^4 + 40960000
Coefficient ring:	$\Z[a_1, \ldots, a_{49}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			557.1
Root			$-6.68798 - 6.68798i$ of defining polynomial
Character	$\chi$	$=$	900.557
Dual form			900.4.j.a.593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-17.9165 - 17.9165i) q^{7} +O(q^{10})$	$q+(-17.9165 - 17.9165i) q^{7} -4.24264i q^{11} +(-17.9165 + 17.9165i) q^{13} +(-76.0132 + 76.0132i) q^{17} +26.0000i q^{19} +(76.0132 + 76.0132i) q^{23} +110.309 q^{29} +52.0000 q^{31} +(-17.9165 - 17.9165i) q^{37} -199.404i q^{41} +(304.053 - 304.053i) q^{47} +299.000i q^{49} +(-380.066 - 380.066i) q^{53} +717.006 q^{59} +350.000 q^{61} +(465.828 + 465.828i) q^{67} -517.602i q^{71} +(465.828 - 465.828i) q^{73} +(-76.0132 + 76.0132i) q^{77} +1000.00i q^{79} +(456.079 + 456.079i) q^{83} +1429.77 q^{89} +642.000 q^{91} +(-788.325 - 788.325i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 416 q^{31} + 2800 q^{61} + 5136 q^{91}+O(q^{100}) $ 8 * q + 416 * q^31 + 2800 * q^61 + 5136 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−17.9165	−	17.9165i	−0.967399	−	0.967399i	0.0320865	−	0.999485i	$-0.489785\pi$
$7$	−17.9165	−	17.9165i	−0.967399	−	0.967399i	−0.999485	+	0.0320865i	$0.989785\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.24264i		−	0.116291i	−0.998308	−	0.0581456i	$-0.981481\pi$
$11$		−	4.24264i		−	0.116291i	0.998308	−	0.0581456i	$-0.0185188\pi$
$12$		0			0
$13$	−17.9165	+	17.9165i	−0.382241	+	0.382241i	−0.871909	−	0.489668i	$-0.837118\pi$
$13$	−17.9165	+	17.9165i	−0.382241	+	0.382241i	0.489668	+	0.871909i	$0.337118\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−76.0132	+	76.0132i	−1.08446	+	1.08446i	−0.0883776	+	0.996087i	$0.528168\pi$
$17$	−76.0132	+	76.0132i	−1.08446	+	1.08446i	−0.996087	+	0.0883776i	$0.971832\pi$
$18$		0			0
$19$			26.0000i			0.313937i	0.987604	+	0.156969i	$0.0501722\pi$
$19$			26.0000i			0.313937i	−0.987604	+	0.156969i	$0.949828\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	76.0132	+	76.0132i	0.689123	+	0.689123i	0.962038	−	0.272915i	$-0.0879877\pi$
$23$	76.0132	+	76.0132i	0.689123	+	0.689123i	−0.272915	+	0.962038i	$0.587988\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	110.309			0.706338			0.353169	−	0.935560i	$-0.385104\pi$
$29$	110.309			0.706338			0.353169	+	0.935560i	$0.385104\pi$
$30$		0			0
$31$	52.0000			0.301273			0.150637	−	0.988589i	$-0.451868\pi$
$31$	52.0000			0.301273			0.150637	+	0.988589i	$0.451868\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−17.9165	−	17.9165i	−0.0796068	−	0.0796068i	0.666182	−	0.745789i	$-0.267927\pi$
$37$	−17.9165	−	17.9165i	−0.0796068	−	0.0796068i	−0.745789	+	0.666182i	$0.767927\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	199.404i		−	0.759553i	−0.925078	−	0.379777i	$-0.876001\pi$
$41$		−	199.404i		−	0.759553i	0.925078	−	0.379777i	$-0.123999\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	304.053	−	304.053i	0.943631	−	0.943631i	−0.0548634	−	0.998494i	$-0.517472\pi$
$47$	304.053	−	304.053i	0.943631	−	0.943631i	0.998494	+	0.0548634i	$0.0174723\pi$
$48$		0			0
$49$			299.000i			0.871720i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−380.066	−	380.066i	−0.985020	−	0.985020i	0.0148696	−	0.999889i	$-0.495267\pi$
$53$	−380.066	−	380.066i	−0.985020	−	0.985020i	−0.999889	+	0.0148696i	$0.995267\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	717.006			1.58214			0.791070	−	0.611726i	$-0.209524\pi$
$59$	717.006			1.58214			0.791070	+	0.611726i	$0.209524\pi$
$60$		0			0
$61$	350.000			0.734638			0.367319	−	0.930095i	$-0.380276\pi$
$61$	350.000			0.734638			0.367319	+	0.930095i	$0.380276\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	465.828	+	465.828i	0.849403	+	0.849403i	0.990059	−	0.140656i	$-0.0449211\pi$
$67$	465.828	+	465.828i	0.849403	+	0.849403i	−0.140656	+	0.990059i	$0.544921\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	517.602i		−	0.865184i	−0.901590	−	0.432592i	$-0.857599\pi$
$71$		−	517.602i		−	0.865184i	0.901590	−	0.432592i	$-0.142401\pi$
$72$		0			0
$73$	465.828	−	465.828i	0.746864	−	0.746864i	−0.227025	−	0.973889i	$-0.572900\pi$
$73$	465.828	−	465.828i	0.746864	−	0.746864i	0.973889	+	0.227025i	$0.0728999\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−76.0132	+	76.0132i	−0.112500	+	0.112500i
$78$		0			0
$79$			1000.00i			1.42416i	0.702097	+	0.712081i	$0.252247\pi$
$79$			1000.00i			1.42416i	−0.702097	+	0.712081i	$0.747753\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	456.079	+	456.079i	0.603147	+	0.603147i	0.941146	−	0.338000i	$-0.109750\pi$
$83$	456.079	+	456.079i	0.603147	+	0.603147i	−0.338000	+	0.941146i	$0.609750\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1429.77			1.70287			0.851434	−	0.524461i	$-0.175733\pi$
$89$	1429.77			1.70287			0.851434	+	0.524461i	$0.175733\pi$
$90$		0			0
$91$	642.000			0.739559
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−788.325	−	788.325i	−0.825178	−	0.825178i	0.161667	−	0.986845i	$-0.448313\pi$
$97$	−788.325	−	788.325i	−0.825178	−	0.825178i	−0.986845	+	0.161667i	$0.948313\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	1518.87i		−	1.49636i	−0.663494	−	0.748182i	$-0.730927\pi$
$101$		−	1518.87i		−	1.49636i	0.663494	−	0.748182i	$-0.269073\pi$
$102$		0			0
$103$	−1272.07	+	1272.07i	−1.21690	+	1.21690i	−0.248189	+	0.968712i	$0.579835\pi$
$103$	−1272.07	+	1272.07i	−1.21690	+	1.21690i	−0.968712	+	0.248189i	$0.920165\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−608.105	+	608.105i	−0.549418	+	0.549418i	−0.926273	−	0.376854i	$-0.877006\pi$
$107$	−608.105	+	608.105i	−0.549418	+	0.549418i	0.376854	+	0.926273i	$0.377006\pi$
$108$		0			0
$109$			1222.00i			1.07382i	0.843640	+	0.536910i	$0.180409\pi$
$109$			1222.00i			1.07382i	−0.843640	+	0.536910i	$0.819591\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	228.039	+	228.039i	0.189842	+	0.189842i	0.795628	−	0.605786i	$-0.207141\pi$
$113$	228.039	+	228.039i	0.189842	+	0.189842i	−0.605786	+	0.795628i	$0.707141\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2723.78			2.09822
$120$		0			0
$121$	1313.00			0.986476
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−806.241	−	806.241i	−0.563326	−	0.563326i	0.366925	−	0.930251i	$-0.380411\pi$
$127$	−806.241	−	806.241i	−0.563326	−	0.563326i	−0.930251	+	0.366925i	$0.880411\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	937.624i		−	0.625348i	−0.949861	−	0.312674i	$-0.898775\pi$
$131$		−	937.624i		−	0.625348i	0.949861	−	0.312674i	$-0.101225\pi$
$132$		0			0
$133$	465.828	−	465.828i	0.303703	−	0.303703i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−684.118	+	684.118i	−0.426629	+	0.426629i	−0.887478	−	0.460849i	$-0.847545\pi$
$137$	−684.118	+	684.118i	−0.426629	+	0.426629i	0.460849	+	0.887478i	$0.347545\pi$
$138$		0			0
$139$			2276.00i			1.38883i	0.719573	+	0.694417i	$0.244337\pi$
$139$			2276.00i			1.38883i	−0.719573	+	0.694417i	$0.755663\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	76.0132	+	76.0132i	0.0444513	+	0.0444513i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2240.11			1.23166			0.615830	−	0.787879i	$-0.288821\pi$
$149$	2240.11			1.23166			0.615830	+	0.787879i	$0.288821\pi$
$150$		0			0
$151$	280.000			0.150901			0.0754506	−	0.997150i	$-0.475960\pi$
$151$	280.000			0.150901			0.0754506	+	0.997150i	$0.475960\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2096.23	+	2096.23i	1.06559	+	1.06559i	0.997693	+	0.0678945i	$0.0216281\pi$
$157$	2096.23	+	2096.23i	1.06559	+	1.06559i	0.0678945	+	0.997693i	$0.478372\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2723.78i		−	1.33331i
$162$		0			0
$163$	2114.14	−	2114.14i	1.01591	−	1.01591i	0.0160336	−	0.999871i	$-0.494896\pi$
$163$	2114.14	−	2114.14i	1.01591	−	1.01591i	0.999871	−	0.0160336i	$-0.00510387\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1292.22	−	1292.22i	0.598774	−	0.598774i	−0.341212	−	0.939986i	$-0.610837\pi$
$167$	1292.22	−	1292.22i	0.598774	−	0.598774i	0.939986	+	0.341212i	$0.110837\pi$
$168$		0			0
$169$			1555.00i			0.707783i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−988.171	−	988.171i	−0.434273	−	0.434273i	0.455806	−	0.890079i	$-0.349351\pi$
$173$	−988.171	−	988.171i	−0.434273	−	0.434273i	−0.890079	+	0.455806i	$0.849351\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2948.64			1.23124			0.615618	−	0.788044i	$-0.288906\pi$
$179$	2948.64			1.23124			0.615618	+	0.788044i	$0.288906\pi$
$180$		0			0
$181$	2998.00			1.23116			0.615579	−	0.788075i	$-0.288922\pi$
$181$	2998.00			1.23116			0.615579	+	0.788075i	$0.288922\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	322.497	+	322.497i	0.126114	+	0.126114i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			313.955i			0.118937i	0.998230	+	0.0594686i	$0.0189406\pi$
$191$			313.955i			0.118937i	−0.998230	+	0.0594686i	$0.981059\pi$
$192$		0			0
$193$	−3045.80	+	3045.80i	−1.13597	+	1.13597i	−0.146800	+	0.989166i	$0.546897\pi$
$193$	−3045.80	+	3045.80i	−1.13597	+	1.13597i	−0.989166	+	0.146800i	$0.953103\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3496.61	−	3496.61i	1.26458	−	1.26458i	0.315736	−	0.948847i	$-0.397749\pi$
$197$	3496.61	−	3496.61i	1.26458	−	1.26458i	0.948847	−	0.315736i	$-0.102251\pi$
$198$		0			0
$199$			1612.00i			0.574229i	0.957896	+	0.287115i	$0.0926961\pi$
$199$			1612.00i			0.574229i	−0.957896	+	0.287115i	$0.907304\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1976.34	−	1976.34i	−0.683310	−	0.683310i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	110.309			0.0365082
$210$		0			0
$211$	−1612.00			−0.525946			−0.262973	−	0.964803i	$-0.584703\pi$
$211$	−1612.00			−0.525946			−0.262973	+	0.964803i	$0.584703\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−931.657	−	931.657i	−0.291451	−	0.291451i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2723.78i		−	0.829054i
$222$		0			0
$223$	2096.23	−	2096.23i	0.629479	−	0.629479i	−0.318458	−	0.947937i	$-0.603165\pi$
$223$	2096.23	−	2096.23i	0.629479	−	0.629479i	0.947937	+	0.318458i	$0.103165\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1976.34	+	1976.34i	−0.577861	+	0.577861i	−0.934313	−	0.356453i	$-0.883986\pi$
$227$	−1976.34	+	1976.34i	−0.577861	+	0.577861i	0.356453	+	0.934313i	$0.383986\pi$
$228$		0			0
$229$		−	826.000i		−	0.238356i	−0.992873	−	0.119178i	$-0.961974\pi$
$229$		−	826.000i		−	0.238356i	0.992873	−	0.119178i	$-0.0380260\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4028.70	+	4028.70i	1.13274	+	1.13274i	0.989720	+	0.143022i	$0.0456819\pi$
$233$	4028.70	+	4028.70i	1.13274	+	1.13274i	0.143022	+	0.989720i	$0.454318\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1909.19			0.516716			0.258358	−	0.966049i	$-0.416819\pi$
$239$	1909.19			0.516716			0.258358	+	0.966049i	$0.416819\pi$
$240$		0			0
$241$	260.000			0.0694941			0.0347470	−	0.999396i	$-0.488937\pi$
$241$	260.000			0.0694941			0.0347470	+	0.999396i	$0.488937\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−465.828	−	465.828i	−0.120000	−	0.120000i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			4162.03i			1.04663i	0.852138	+	0.523317i	$0.175306\pi$
$251$			4162.03i			1.04663i	−0.852138	+	0.523317i	$0.824694\pi$
$252$		0			0
$253$	322.497	−	322.497i	0.0801391	−	0.0801391i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4940.86	+	4940.86i	−1.19923	+	1.19923i	−0.224832	+	0.974397i	$0.572183\pi$
$257$	−4940.86	+	4940.86i	−1.19923	+	1.19923i	−0.974397	+	0.224832i	$0.927817\pi$
$258$		0			0
$259$			642.000i			0.154023i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−988.171	−	988.171i	−0.231685	−	0.231685i	0.581711	−	0.813396i	$-0.302384\pi$
$263$	−988.171	−	988.171i	−0.231685	−	0.231685i	−0.813396	+	0.581711i	$0.802384\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5201.48			−1.17896			−0.589479	−	0.807784i	$-0.700667\pi$
$269$	−5201.48			−1.17896			−0.589479	+	0.807784i	$0.700667\pi$
$270$		0			0
$271$	−748.000			−0.167667			−0.0838335	−	0.996480i	$-0.526716\pi$
$271$	−748.000			−0.167667			−0.0838335	+	0.996480i	$0.526716\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−806.241	−	806.241i	−0.174882	−	0.174882i	0.614238	−	0.789121i	$-0.289463\pi$
$277$	−806.241	−	806.241i	−0.174882	−	0.174882i	−0.789121	+	0.614238i	$0.789463\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	2151.02i		−	0.456651i	−0.973585	−	0.228326i	$-0.926675\pi$
$281$		−	2151.02i		−	0.456651i	0.973585	−	0.228326i	$-0.0733251\pi$
$282$		0			0
$283$	−2902.47	+	2902.47i	−0.609660	+	0.609660i	−0.942857	−	0.333197i	$-0.891873\pi$
$283$	−2902.47	+	2902.47i	−0.609660	+	0.609660i	0.333197	+	0.942857i	$0.391873\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3572.62	+	3572.62i	−0.734791	+	0.734791i
$288$		0			0
$289$		−	6643.00i		−	1.35213i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1976.34	+	1976.34i	0.394058	+	0.394058i	0.876131	−	0.482073i	$-0.160116\pi$
$293$	1976.34	+	1976.34i	0.394058	+	0.394058i	−0.482073	+	0.876131i	$0.660116\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2723.78			−0.526823
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−2579.97	−	2579.97i	−0.479631	−	0.479631i	0.425382	−	0.905014i	$-0.360140\pi$
$307$	−2579.97	−	2579.97i	−0.479631	−	0.479631i	−0.905014	+	0.425382i	$0.860140\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			8052.53i			1.46822i	0.679029	+	0.734111i	$0.262401\pi$
$311$			8052.53i			1.46822i	−0.679029	+	0.734111i	$0.737599\pi$
$312$		0			0
$313$	−465.828	+	465.828i	−0.0841220	+	0.0841220i	−0.747916	−	0.663794i	$-0.768945\pi$
$313$	−465.828	+	465.828i	−0.0841220	+	0.0841220i	0.663794	+	0.747916i	$0.268945\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5929.03	−	5929.03i	1.05050	−	1.05050i	0.0518408	−	0.998655i	$-0.483491\pi$
$317$	5929.03	−	5929.03i	1.05050	−	1.05050i	0.998655	−	0.0518408i	$-0.0165088\pi$
$318$		0			0
$319$		−	468.000i		−	0.0821410i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1976.34	−	1976.34i	−0.340454	−	0.340454i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−10895.1			−1.82573
$330$		0			0
$331$	−10270.0			−1.70541			−0.852704	−	0.522394i	$-0.825039\pi$
$331$	−10270.0			−1.70541			−0.852704	+	0.522394i	$0.825039\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−465.828	−	465.828i	−0.0752976	−	0.0752976i	0.668455	−	0.743753i	$-0.266956\pi$
$337$	−465.828	−	465.828i	−0.0752976	−	0.0752976i	−0.743753	+	0.668455i	$0.766956\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	220.617i		−	0.0350355i
$342$		0			0
$343$	−788.325	+	788.325i	−0.124098	+	0.124098i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6537.13	−	6537.13i	1.01133	−	1.01133i	0.0113952	−	0.999935i	$-0.496373\pi$
$347$	6537.13	−	6537.13i	1.01133	−	1.01133i	0.999935	−	0.0113952i	$-0.00362728\pi$
$348$		0			0
$349$		−	5518.00i		−	0.846337i	−0.906051	−	0.423169i	$-0.860918\pi$
$349$		−	5518.00i		−	0.846337i	0.906051	−	0.423169i	$-0.139082\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1444.25	−	1444.25i	−0.217761	−	0.217761i	0.589793	−	0.807554i	$-0.299209\pi$
$353$	−1444.25	−	1444.25i	−0.217761	−	0.217761i	−0.807554	+	0.589793i	$0.799209\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2868.03			−0.421639			−0.210820	−	0.977525i	$-0.567613\pi$
$359$	−2868.03			−0.421639			−0.210820	+	0.977525i	$0.567613\pi$
$360$		0			0
$361$	6183.00			0.901443
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7757.83	+	7757.83i	1.10342	+	1.10342i	0.993995	+	0.109426i	$0.0349014\pi$
$367$	7757.83	+	7757.83i	1.10342	+	1.10342i	0.109426	+	0.993995i	$0.465099\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			13618.9i			1.90581i
$372$		0			0
$373$	8689.49	−	8689.49i	1.20623	−	1.20623i	0.233995	−	0.972238i	$-0.424820\pi$
$373$	8689.49	−	8689.49i	1.20623	−	1.20623i	0.972238	−	0.233995i	$-0.0751798\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1976.34	+	1976.34i	−0.269991	+	0.269991i
$378$		0			0
$379$		−	10036.0i		−	1.36020i	−0.733121	−	0.680099i	$-0.761937\pi$
$379$		−	10036.0i		−	1.36020i	0.733121	−	0.680099i	$-0.238063\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2432.42	+	2432.42i	0.324519	+	0.324519i	0.850498	−	0.525979i	$-0.176301\pi$
$383$	2432.42	+	2432.42i	0.324519	+	0.324519i	−0.525979	+	0.850498i	$0.676301\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13237.0			−1.72531			−0.862654	−	0.505795i	$-0.831199\pi$
$389$	−13237.0			−1.72531			−0.862654	+	0.505795i	$0.831199\pi$
$390$		0			0
$391$	−11556.0			−1.49466
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7901.16	+	7901.16i	0.998862	+	0.998862i	0.999999	−	0.00113769i	$-0.000362138\pi$
$397$	7901.16	+	7901.16i	0.998862	+	0.998862i	−0.00113769	+	0.999999i	$0.500362\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	3372.90i		−	0.420036i	−0.977697	−	0.210018i	$-0.932648\pi$
$401$		−	3372.90i		−	0.420036i	0.977697	−	0.210018i	$-0.0673523\pi$
$402$		0			0
$403$	−931.657	+	931.657i	−0.115159	+	0.115159i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−76.0132	+	76.0132i	−0.00925757	+	0.00925757i
$408$		0			0
$409$		−	8866.00i		−	1.07187i	−0.844259	−	0.535936i	$-0.819959\pi$
$409$		−	8866.00i		−	1.07187i	0.844259	−	0.535936i	$-0.180041\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12846.2	−	12846.2i	−1.53056	−	1.53056i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8107.69			−0.945314			−0.472657	−	0.881247i	$-0.656705\pi$
$419$	−8107.69			−0.945314			−0.472657	+	0.881247i	$0.656705\pi$
$420$		0			0
$421$	−3662.00			−0.423931			−0.211966	−	0.977277i	$-0.567986\pi$
$421$	−3662.00			−0.423931			−0.211966	+	0.977277i	$0.567986\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−6270.77	−	6270.77i	−0.710688	−	0.710688i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			11624.8i			1.29918i	0.760283	+	0.649592i	$0.225060\pi$
$431$			11624.8i			1.29918i	−0.760283	+	0.649592i	$0.774940\pi$
$432$		0			0
$433$	−5482.44	+	5482.44i	−0.608474	+	0.608474i	−0.942547	−	0.334073i	$-0.891577\pi$
$433$	−5482.44	+	5482.44i	−0.608474	+	0.608474i	0.334073	+	0.942547i	$0.391577\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1976.34	+	1976.34i	−0.216342	+	0.216342i
$438$		0			0
$439$			3580.00i			0.389212i	0.980881	+	0.194606i	$0.0623428\pi$
$439$			3580.00i			0.389212i	−0.980881	+	0.194606i	$0.937657\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11249.9	+	11249.9i	1.20655	+	1.20655i	0.972138	+	0.234411i	$0.0753162\pi$
$443$	11249.9	+	11249.9i	1.20655	+	1.20655i	0.234411	+	0.972138i	$0.424684\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12800.0			−1.34537			−0.672685	−	0.739929i	$-0.734859\pi$
$449$	−12800.0			−1.34537			−0.672685	+	0.739929i	$0.734859\pi$
$450$		0			0
$451$	−846.000			−0.0883295
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−12899.9	−	12899.9i	−1.32042	−	1.32042i	−0.913438	−	0.406978i	$-0.866583\pi$
$457$	−12899.9	−	12899.9i	−1.32042	−	1.32042i	−0.406978	−	0.913438i	$-0.633417\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	2834.08i		−	0.286326i	−0.989699	−	0.143163i	$-0.954273\pi$
$461$		−	2834.08i		−	0.286326i	0.989699	−	0.143163i	$-0.0457274\pi$
$462$		0			0
$463$	4174.54	−	4174.54i	0.419022	−	0.419022i	−0.465844	−	0.884867i	$-0.654249\pi$
$463$	4174.54	−	4174.54i	0.419022	−	0.419022i	0.884867	+	0.465844i	$0.154249\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1064.18	−	1064.18i	0.105449	−	0.105449i	−0.652414	−	0.757863i	$-0.726244\pi$
$467$	1064.18	−	1064.18i	0.105449	−	0.105449i	0.757863	+	0.652414i	$0.226244\pi$
$468$		0			0
$469$		−	16692.0i		−	1.64342i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1926.16			0.183734			0.0918669	−	0.995771i	$-0.470717\pi$
$479$	1926.16			0.183734			0.0918669	+	0.995771i	$0.470717\pi$
$480$		0			0
$481$	642.000			0.0608580
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−1307.90	−	1307.90i	−0.121698	−	0.121698i	0.643635	−	0.765333i	$-0.277425\pi$
$487$	−1307.90	−	1307.90i	−0.121698	−	0.121698i	−0.765333	+	0.643635i	$0.777425\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			17619.7i			1.61948i	0.586788	+	0.809741i	$0.300392\pi$
$491$			17619.7i			1.61948i	−0.586788	+	0.809741i	$0.699608\pi$
$492$		0			0
$493$	−8384.91	+	8384.91i	−0.765999	+	0.765999i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9273.61	+	9273.61i	−0.836978	+	0.836978i
$498$		0			0
$499$			3454.00i			0.309864i	0.987925	+	0.154932i	$0.0495159\pi$
$499$			3454.00i			0.309864i	−0.987925	+	0.154932i	$0.950484\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7905.37	+	7905.37i	0.700761	+	0.700761i	0.964574	−	0.263813i	$-0.0849801\pi$
$503$	7905.37	+	7905.37i	0.700761	+	0.700761i	−0.263813	+	0.964574i	$0.584980\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9851.41			−0.857871			−0.428935	−	0.903335i	$-0.641111\pi$
$509$	−9851.41			−0.857871			−0.428935	+	0.903335i	$0.641111\pi$
$510$		0			0
$511$	−16692.0			−1.44503
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−1289.99	−	1289.99i	−0.109736	−	0.109736i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	7649.48i		−	0.643243i	−0.946868	−	0.321622i	$-0.895772\pi$
$521$		−	7649.48i		−	0.643243i	0.946868	−	0.321622i	$-0.104228\pi$
$522$		0			0
$523$	−9854.06	+	9854.06i	−0.823878	+	0.823878i	−0.986662	−	0.162784i	$-0.947953\pi$
$523$	−9854.06	+	9854.06i	−0.823878	+	0.823878i	0.162784	+	0.986662i	$0.447953\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3952.68	+	3952.68i	−0.326720	+	0.326720i
$528$		0			0
$529$		−	611.000i		−	0.0502178i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3572.62	+	3572.62i	0.290333	+	0.290333i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1268.55			0.101373
$540$		0			0
$541$	22490.0			1.78728			0.893642	−	0.448781i	$-0.148142\pi$
$541$	22490.0			1.78728			0.893642	+	0.448781i	$0.148142\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	12111.5	+	12111.5i	0.946713	+	0.946713i	0.998650	−	0.0519376i	$-0.0165397\pi$
$547$	12111.5	+	12111.5i	0.946713	+	0.946713i	−0.0519376	+	0.998650i	$0.516540\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			2868.03i			0.221746i
$552$		0			0
$553$	17916.5	−	17916.5i	1.37773	−	1.37773i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2964.51	+	2964.51i	−0.225512	+	0.225512i	−0.810815	−	0.585303i	$-0.800976\pi$
$557$	−2964.51	+	2964.51i	−0.225512	+	0.225512i	0.585303	+	0.810815i	$0.300976\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8969.55	−	8969.55i	−0.671442	−	0.671442i	0.286607	−	0.958048i	$-0.407473\pi$
$563$	−8969.55	−	8969.55i	−0.671442	−	0.671442i	−0.958048	+	0.286607i	$0.907473\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6563.37			−0.483569			−0.241784	−	0.970330i	$-0.577733\pi$
$569$	−6563.37			−0.483569			−0.241784	+	0.970330i	$0.577733\pi$
$570$		0			0
$571$	−14734.0			−1.07986			−0.539929	−	0.841711i	$-0.681549\pi$
$571$	−14734.0			−1.07986			−0.539929	+	0.841711i	$0.681549\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	7596.58	+	7596.58i	0.548094	+	0.548094i	0.925889	−	0.377796i	$-0.123318\pi$
$577$	7596.58	+	7596.58i	0.548094	+	0.548094i	−0.377796	+	0.925889i	$0.623318\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	16342.7i		−	1.16697i
$582$		0			0
$583$	−1612.48	+	1612.48i	−0.114549	+	0.114549i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6385.11	−	6385.11i	0.448963	−	0.448963i	−0.446046	−	0.895010i	$-0.647168\pi$
$587$	6385.11	−	6385.11i	0.448963	−	0.448963i	0.895010	+	0.446046i	$0.147168\pi$
$588$		0			0
$589$			1352.00i			0.0945810i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4940.86	+	4940.86i	0.342153	+	0.342153i	0.857176	−	0.515023i	$-0.172217\pi$
$593$	4940.86	+	4940.86i	0.342153	+	0.342153i	−0.515023	+	0.857176i	$0.672217\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	26049.8			1.77691			0.888453	−	0.458968i	$-0.151781\pi$
$599$	26049.8			1.77691			0.888453	+	0.458968i	$0.151781\pi$
$600$		0			0
$601$	−740.000			−0.0502250			−0.0251125	−	0.999685i	$-0.507994\pi$
$601$	−740.000			−0.0502250			−0.0251125	+	0.999685i	$0.507994\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−10015.3	−	10015.3i	−0.669702	−	0.669702i	0.287945	−	0.957647i	$-0.407028\pi$
$607$	−10015.3	−	10015.3i	−0.669702	−	0.669702i	−0.957647	+	0.287945i	$0.907028\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			10895.1i			0.721389i
$612$		0			0
$613$	1952.90	−	1952.90i	0.128673	−	0.128673i	−0.639837	−	0.768510i	$-0.720998\pi$
$613$	1952.90	−	1952.90i	0.128673	−	0.128673i	0.768510	+	0.639837i	$0.220998\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18775.2	−	18775.2i	1.22506	−	1.22506i	0.259252	−	0.965810i	$-0.416524\pi$
$617$	18775.2	−	18775.2i	1.22506	−	1.22506i	0.965810	−	0.259252i	$-0.0834760\pi$
$618$		0			0
$619$			11044.0i			0.717118i	0.933507	+	0.358559i	$0.116732\pi$
$619$			11044.0i			0.717118i	−0.933507	+	0.358559i	$0.883268\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−25616.4	−	25616.4i	−1.64735	−	1.64735i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2723.78			0.172661
$630$		0			0
$631$	24596.0			1.55175			0.775873	−	0.630890i	$-0.217310\pi$
$631$	24596.0			1.55175			0.775873	+	0.630890i	$0.217310\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5357.03	−	5357.03i	−0.333207	−	0.333207i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	13843.7i		−	0.853034i	−0.904480	−	0.426517i	$-0.859740\pi$
$641$		−	13843.7i		−	0.853034i	0.904480	−	0.426517i	$-0.140260\pi$
$642$		0			0
$643$	8062.41	−	8062.41i	0.494480	−	0.494480i	−0.415235	−	0.909714i	$-0.636300\pi$
$643$	8062.41	−	8062.41i	0.494480	−	0.494480i	0.909714	+	0.415235i	$0.136300\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12770.2	−	12770.2i	0.775964	−	0.775964i	−0.203178	−	0.979142i	$-0.565127\pi$
$647$	12770.2	−	12770.2i	0.775964	−	0.775964i	0.979142	+	0.203178i	$0.0651269\pi$
$648$		0			0
$649$		−	3042.00i		−	0.183989i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3952.68	+	3952.68i	0.236877	+	0.236877i	0.815556	−	0.578679i	$-0.196432\pi$
$653$	3952.68	+	3952.68i	0.236877	+	0.236877i	−0.578679	+	0.815556i	$0.696432\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13071.6			0.772680			0.386340	−	0.922356i	$-0.373739\pi$
$659$	13071.6			0.772680			0.386340	+	0.922356i	$0.373739\pi$
$660$		0			0
$661$	1774.00			0.104388			0.0521941	−	0.998637i	$-0.483379\pi$
$661$	1774.00			0.104388			0.0521941	+	0.998637i	$0.483379\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8384.91	+	8384.91i	0.486754	+	0.486754i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1484.92i		−	0.0854320i
$672$		0			0
$673$	−9209.07	+	9209.07i	−0.527464	+	0.527464i	−0.919815	−	0.392351i	$-0.871662\pi$
$673$	−9209.07	+	9209.07i	−0.527464	+	0.527464i	0.392351	+	0.919815i	$0.371662\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	608.105	−	608.105i	0.0345220	−	0.0345220i	−0.689635	−	0.724157i	$-0.742229\pi$
$677$	608.105	−	608.105i	0.0345220	−	0.0345220i	0.724157	+	0.689635i	$0.242229\pi$
$678$		0			0
$679$			28248.0i			1.59655i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4256.74	+	4256.74i	0.238477	+	0.238477i	0.816219	−	0.577743i	$-0.196066\pi$
$683$	4256.74	+	4256.74i	0.238477	+	0.238477i	−0.577743	+	0.816219i	$0.696066\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	13618.9			0.753030
$690$		0			0
$691$	−24158.0			−1.32998			−0.664988	−	0.746854i	$-0.731563\pi$
$691$	−24158.0			−1.32998			−0.664988	+	0.746854i	$0.731563\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	15157.3	+	15157.3i	0.823709	+	0.823709i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	29673.0i		−	1.59877i	−0.600822	−	0.799383i	$-0.705160\pi$
$701$		−	29673.0i		−	1.59877i	0.600822	−	0.799383i	$-0.294840\pi$
$702$		0			0
$703$	465.828	−	465.828i	0.0249915	−	0.0249915i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−27212.7	+	27212.7i	−1.44758	+	1.44758i
$708$		0			0
$709$			22490.0i			1.19130i	0.803245	+	0.595649i	$0.203105\pi$
$709$			22490.0i			1.19130i	−0.803245	+	0.595649i	$0.796895\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3952.68	+	3952.68i	0.207615	+	0.207615i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	30309.4			1.57212			0.786058	−	0.618153i	$-0.212119\pi$
$719$	30309.4			1.57212			0.786058	+	0.618153i	$0.212119\pi$
$720$		0			0
$721$	45582.0			2.35446
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−2096.23	−	2096.23i	−0.106939	−	0.106939i	0.651613	−	0.758552i	$-0.274093\pi$
$727$	−2096.23	−	2096.23i	−0.106939	−	0.106939i	−0.758552	+	0.651613i	$0.774093\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−14996.1	+	14996.1i	−0.755652	+	0.755652i	−0.975528	−	0.219876i	$-0.929435\pi$
$733$	−14996.1	+	14996.1i	−0.755652	+	0.755652i	0.219876	+	0.975528i	$0.429435\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1976.34	−	1976.34i	0.0987781	−	0.0987781i
$738$		0			0
$739$		−	8242.00i		−	0.410266i	−0.978734	−	0.205133i	$-0.934237\pi$
$739$		−	8242.00i		−	0.410266i	0.978734	−	0.205133i	$-0.0657628\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11858.1	+	11858.1i	0.585504	+	0.585504i	0.936411	−	0.350906i	$-0.114126\pi$
$743$	11858.1	+	11858.1i	0.585504	+	0.585504i	−0.350906	+	0.936411i	$0.614126\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	21790.2			1.06301
$750$		0			0
$751$	6976.00			0.338959			0.169479	−	0.985534i	$-0.445791\pi$
$751$	6976.00			0.338959			0.169479	+	0.985534i	$0.445791\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−11950.3	−	11950.3i	−0.573766	−	0.573766i	0.359413	−	0.933179i	$-0.382977\pi$
$757$	−11950.3	−	11950.3i	−0.573766	−	0.573766i	−0.933179	+	0.359413i	$0.882977\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	18256.1i		−	0.869622i	−0.900522	−	0.434811i	$-0.856815\pi$
$761$		−	18256.1i		−	0.869622i	0.900522	−	0.434811i	$-0.143185\pi$
$762$		0			0
$763$	21893.9	−	21893.9i	1.03881	−	1.03881i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12846.2	+	12846.2i	−0.604759	+	0.604759i
$768$		0			0
$769$			12952.0i			0.607362i	0.952774	+	0.303681i	$0.0982156\pi$
$769$			12952.0i			0.607362i	−0.952774	+	0.303681i	$0.901784\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19307.3	+	19307.3i	0.898366	+	0.898366i	0.995292	−	0.0969257i	$-0.0309009\pi$
$773$	19307.3	+	19307.3i	0.898366	+	0.898366i	−0.0969257	+	0.995292i	$0.530901\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5184.51			0.238452
$780$		0			0
$781$	−2196.00			−0.100613
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−19529.0	−	19529.0i	−0.884539	−	0.884539i	0.109453	−	0.993992i	$-0.465090\pi$
$787$	−19529.0	−	19529.0i	−0.884539	−	0.884539i	−0.993992	+	0.109453i	$0.965090\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	8171.33i		−	0.367306i
$792$		0			0
$793$	−6270.77	+	6270.77i	−0.280809	+	0.280809i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9805.70	+	9805.70i	−0.435804	+	0.435804i	−0.890597	−	0.454793i	$-0.849713\pi$
$797$	−9805.70	+	9805.70i	−0.435804	+	0.435804i	0.454793	+	0.890597i	$0.349713\pi$
$798$		0			0
$799$			46224.0i			2.04667i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1976.34	−	1976.34i	−0.0868538	−	0.0868538i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	21455.0			0.932409			0.466204	−	0.884677i	$-0.345621\pi$
$809$	21455.0			0.932409			0.466204	+	0.884677i	$0.345621\pi$
$810$		0			0
$811$	24388.0			1.05595			0.527977	−	0.849259i	$-0.322951\pi$
$811$	24388.0			1.05595			0.527977	+	0.849259i	$0.322951\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	18234.9i		−	0.775154i	−0.921837	−	0.387577i	$-0.873312\pi$
$821$		−	18234.9i		−	0.775154i	0.921837	−	0.387577i	$-0.126688\pi$
$822$		0			0
$823$	−18041.9	+	18041.9i	−0.764156	+	0.764156i	−0.977071	−	0.212914i	$-0.931704\pi$
$823$	−18041.9	+	18041.9i	−0.764156	+	0.764156i	0.212914	+	0.977071i	$0.431704\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4256.74	+	4256.74i	−0.178986	+	0.178986i	−0.790914	−	0.611928i	$-0.790394\pi$
$827$	−4256.74	+	4256.74i	−0.178986	+	0.178986i	0.611928	+	0.790914i	$0.290394\pi$
$828$		0			0
$829$		−	15418.0i		−	0.645946i	−0.946408	−	0.322973i	$-0.895318\pi$
$829$		−	15418.0i		−	0.645946i	0.946408	−	0.322973i	$-0.104682\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−22727.9	−	22727.9i	−0.945350	−	0.945350i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	45837.5			1.88616			0.943079	−	0.332570i	$-0.107916\pi$
$839$	45837.5			1.88616			0.943079	+	0.332570i	$0.107916\pi$
$840$		0			0
$841$	−12221.0			−0.501087
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−23524.3	−	23524.3i	−0.954316	−	0.954316i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	2723.78i		−	0.109718i
$852$		0			0
$853$	−7435.34	+	7435.34i	−0.298454	+	0.298454i	−0.840408	−	0.541954i	$-0.817685\pi$
$853$	−7435.34	+	7435.34i	−0.298454	+	0.298454i	0.541954	+	0.840408i	$0.317685\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15886.7	−	15886.7i	0.633233	−	0.633233i	−0.315644	−	0.948878i	$-0.602221\pi$
$857$	15886.7	−	15886.7i	0.633233	−	0.633233i	0.948878	+	0.315644i	$0.102221\pi$
$858$		0			0
$859$			14572.0i			0.578801i	0.957208	+	0.289401i	$0.0934560\pi$
$859$			14572.0i			0.578801i	−0.957208	+	0.289401i	$0.906544\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12846.2	−	12846.2i	−0.506710	−	0.506710i	0.406805	−	0.913515i	$-0.366643\pi$
$863$	−12846.2	−	12846.2i	−0.506710	−	0.506710i	−0.913515	+	0.406805i	$0.866643\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4242.64			0.165618
$870$		0			0
$871$	−16692.0			−0.649353
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	15139.4	+	15139.4i	0.582921	+	0.582921i	0.935705	−	0.352784i	$-0.114765\pi$
$877$	15139.4	+	15139.4i	0.582921	+	0.582921i	−0.352784	+	0.935705i	$0.614765\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	14921.4i		−	0.570617i	−0.958436	−	0.285309i	$-0.907904\pi$
$881$		−	14921.4i		−	0.570617i	0.958436	−	0.285309i	$-0.0920961\pi$
$882$		0			0
$883$	15945.7	−	15945.7i	0.607717	−	0.607717i	−0.334632	−	0.942349i	$-0.608612\pi$
$883$	15945.7	−	15945.7i	0.607717	−	0.607717i	0.942349	+	0.334632i	$0.108612\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18167.1	−	18167.1i	0.687704	−	0.687704i	−0.274020	−	0.961724i	$-0.588354\pi$
$887$	18167.1	−	18167.1i	0.687704	−	0.687704i	0.961724	+	0.274020i	$0.0883536\pi$
$888$		0			0
$889$			28890.0i			1.08992i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7905.37	+	7905.37i	0.296241	+	0.296241i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5736.05			0.212801
$900$		0			0
$901$	57780.0			2.13644
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	28523.0	+	28523.0i	1.04420	+	1.04420i	0.998977	+	0.0452255i	$0.0144006\pi$
$907$	28523.0	+	28523.0i	1.04420	+	1.04420i	0.0452255	+	0.998977i	$0.485599\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	15867.5i		−	0.577072i	−0.957469	−	0.288536i	$-0.906831\pi$
$911$		−	15867.5i		−	0.577072i	0.957469	−	0.288536i	$-0.0931686\pi$
$912$		0			0
$913$	1934.98	−	1934.98i	0.0701407	−	0.0701407i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−16798.9	+	16798.9i	−0.604961	+	0.604961i
$918$		0			0
$919$		−	17692.0i		−	0.635044i	−0.948251	−	0.317522i	$-0.897149\pi$
$919$		−	17692.0i		−	0.635044i	0.948251	−	0.317522i	$-0.102851\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9273.61	+	9273.61i	0.330709	+	0.330709i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7742.82			−0.273449			−0.136724	−	0.990609i	$-0.543657\pi$
$929$	−7742.82			−0.273449			−0.136724	+	0.990609i	$0.543657\pi$
$930$		0			0
$931$	−7774.00			−0.273665
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	23363.1	+	23363.1i	0.814556	+	0.814556i	0.985313	−	0.170758i	$-0.0546215\pi$
$937$	23363.1	+	23363.1i	0.814556	+	0.814556i	−0.170758	+	0.985313i	$0.554621\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			10844.2i			0.375675i	0.982200	+	0.187838i	$0.0601479\pi$
$941$			10844.2i			0.375675i	−0.982200	+	0.187838i	$0.939852\pi$
$942$		0			0
$943$	15157.3	−	15157.3i	0.523426	−	0.523426i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19459.4	−	19459.4i	0.667734	−	0.667734i	−0.289457	−	0.957191i	$-0.593475\pi$
$947$	19459.4	−	19459.4i	0.667734	−	0.667734i	0.957191	+	0.289457i	$0.0934746\pi$
$948$		0			0
$949$			16692.0i			0.570964i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21815.8	−	21815.8i	−0.741534	−	0.741534i	0.231339	−	0.972873i	$-0.425689\pi$
$953$	−21815.8	−	21815.8i	−0.741534	−	0.741534i	−0.972873	+	0.231339i	$0.925689\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24514.0			0.825441
$960$		0			0
$961$	−27087.0			−0.909234
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	31909.2	+	31909.2i	1.06115	+	1.06115i	0.998004	+	0.0631455i	$0.0201132\pi$
$967$	31909.2	+	31909.2i	1.06115	+	1.06115i	0.0631455	+	0.998004i	$0.479887\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			11069.0i			0.365832i	0.983129	+	0.182916i	$0.0585536\pi$
$971$			11069.0i			0.365832i	−0.983129	+	0.182916i	$0.941446\pi$
$972$		0			0
$973$	40777.9	−	40777.9i	1.34356	−	1.34356i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28200.9	−	28200.9i	0.923466	−	0.923466i	−0.0738067	−	0.997273i	$-0.523515\pi$
$977$	28200.9	−	28200.9i	0.923466	−	0.923466i	0.997273	+	0.0738067i	$0.0235148\pi$
$978$		0			0
$979$		−	6066.00i		−	0.198029i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	19763.4	+	19763.4i	0.641257	+	0.641257i	0.950864	−	0.309608i	$-0.100198\pi$
$983$	19763.4	+	19763.4i	0.641257	+	0.641257i	−0.309608	+	0.950864i	$0.600198\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	40364.0			1.29385			0.646925	−	0.762554i	$-0.276055\pi$
$991$	40364.0			1.29385			0.646925	+	0.762554i	$0.276055\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−37570.8	−	37570.8i	−1.19346	−	1.19346i	−0.976089	−	0.217372i	$-0.930251\pi$
$997$	−37570.8	−	37570.8i	−1.19346	−	1.19346i	−0.217372	−	0.976089i	$-0.569749\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.4.j.a.557.1	✓	8
3.2	odd	2	inner	900.4.j.a.557.2	yes	8
5.2	odd	4	inner	900.4.j.a.593.3	yes	8
5.3	odd	4	inner	900.4.j.a.593.1	yes	8
5.4	even	2	inner	900.4.j.a.557.3	yes	8
15.2	even	4	inner	900.4.j.a.593.4	yes	8
15.8	even	4	inner	900.4.j.a.593.2	yes	8
15.14	odd	2	inner	900.4.j.a.557.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.4.j.a.557.1	✓	8	1.1	even	1	trivial
900.4.j.a.557.2	yes	8	3.2	odd	2	inner
900.4.j.a.557.3	yes	8	5.4	even	2	inner
900.4.j.a.557.4	yes	8	15.14	odd	2	inner
900.4.j.a.593.1	yes	8	5.3	odd	4	inner
900.4.j.a.593.2	yes	8	15.8	even	4	inner
900.4.j.a.593.3	yes	8	5.2	odd	4	inner
900.4.j.a.593.4	yes	8	15.2	even	4	inner

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-17.9165 - 17.9165i) q^{7} +O(q^{10})\)	\(q+(-17.9165 - 17.9165i) q^{7} -4.24264i q^{11} +(-17.9165 + 17.9165i) q^{13} +(-76.0132 + 76.0132i) q^{17} +26.0000i q^{19} +(76.0132 + 76.0132i) q^{23} +110.309 q^{29} +52.0000 q^{31} +(-17.9165 - 17.9165i) q^{37} -199.404i q^{41} +(304.053 - 304.053i) q^{47} +299.000i q^{49} +(-380.066 - 380.066i) q^{53} +717.006 q^{59} +350.000 q^{61} +(465.828 + 465.828i) q^{67} -517.602i q^{71} +(465.828 - 465.828i) q^{73} +(-76.0132 + 76.0132i) q^{77} +1000.00i q^{79} +(456.079 + 456.079i) q^{83} +1429.77 q^{89} +642.000 q^{91} +(-788.325 - 788.325i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 416 q^{31} + 2800 q^{61} + 5136 q^{91}+O(q^{100}) \) 8 * q + 416 * q^31 + 2800 * q^61 + 5136 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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