LMFDB - Embedded newform 900.4.j.a.557.3

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			557.3
Root			$5.98088 + 5.98088i$ of defining polynomial
Character	$\chi$	$=$	900.557
Dual form			900.4.j.a.593.4

$q$-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	0.999485	−	0.0320865i	$-0.0102152\pi$
$7$	17.9165	+	17.9165i	0.967399	+	0.967399i	−0.0320865	+	0.999485i	$0.510215\pi$
$8$		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	−0.489668	−	0.871909i	$-0.662882\pi$
$13$	17.9165	−	17.9165i	0.382241	−	0.382241i	0.871909	+	0.489668i	$0.162882\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	76.0132	−	76.0132i	1.08446	−	1.08446i	0.0883776	−	0.996087i	$-0.471832\pi$
$17$	76.0132	−	76.0132i	1.08446	−	1.08446i	0.996087	−	0.0883776i	$-0.0281682\pi$
$18$		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	0.272915	−	0.962038i	$-0.412012\pi$
$23$	−76.0132	−	76.0132i	−0.689123	−	0.689123i	−0.962038	+	0.272915i	$0.912012\pi$
$24$		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	0.745789	−	0.666182i	$-0.232073\pi$
$37$	17.9165	+	17.9165i	0.0796068	+	0.0796068i	−0.666182	+	0.745789i	$0.732073\pi$
$38$		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	−0.998494	−	0.0548634i	$-0.982528\pi$
$47$	−304.053	+	304.053i	−0.943631	+	0.943631i	0.0548634	+	0.998494i	$0.482528\pi$
$48$		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	0.999889	−	0.0148696i	$-0.00473332\pi$
$53$	380.066	+	380.066i	0.985020	+	0.985020i	−0.0148696	+	0.999889i	$0.504733\pi$
$54$		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	0.140656	−	0.990059i	$-0.455079\pi$
$67$	−465.828	−	465.828i	−0.849403	−	0.849403i	−0.990059	+	0.140656i	$0.955079\pi$
$68$		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	−0.973889	−	0.227025i	$-0.927100\pi$
$73$	−465.828	+	465.828i	−0.746864	+	0.746864i	0.227025	+	0.973889i	$0.427100\pi$
$74$		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	0.338000	−	0.941146i	$-0.390250\pi$
$83$	−456.079	−	456.079i	−0.603147	−	0.603147i	−0.941146	+	0.338000i	$0.890250\pi$
$84$		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	0.986845	−	0.161667i	$-0.0516872\pi$
$97$	788.325	+	788.325i	0.825178	+	0.825178i	−0.161667	+	0.986845i	$0.551687\pi$
$98$		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	$-0.420165\pi$
$103$	1272.07	−	1272.07i	1.21690	−	1.21690i	0.968712	−	0.248189i	$-0.0798353\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	608.105	−	608.105i	0.549418	−	0.549418i	−0.376854	−	0.926273i	$-0.622994\pi$
$107$	608.105	−	608.105i	0.549418	−	0.549418i	0.926273	+	0.376854i	$0.122994\pi$
$108$		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	0.605786	−	0.795628i	$-0.292859\pi$
$113$	−228.039	−	228.039i	−0.189842	−	0.189842i	−0.795628	+	0.605786i	$0.792859\pi$
$114$		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	0.930251	−	0.366925i	$-0.119589\pi$
$127$	806.241	+	806.241i	0.563326	+	0.563326i	−0.366925	+	0.930251i	$0.619589\pi$
$128$		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	−0.460849	−	0.887478i	$-0.652455\pi$
$137$	684.118	−	684.118i	0.426629	−	0.426629i	0.887478	+	0.460849i	$0.152455\pi$
$138$		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	$-0.978372\pi$
$157$	−2096.23	−	2096.23i	−1.06559	−	1.06559i	−0.0678945	−	0.997693i	$-0.521628\pi$
$158$		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	$0.505104\pi$
$163$	−2114.14	+	2114.14i	−1.01591	+	1.01591i	−0.999871	+	0.0160336i	$0.994896\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1292.22	+	1292.22i	−0.598774	+	0.598774i	−0.939986	−	0.341212i	$-0.889163\pi$
$167$	−1292.22	+	1292.22i	−0.598774	+	0.598774i	0.341212	+	0.939986i	$0.389163\pi$
$168$		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	0.890079	−	0.455806i	$-0.150649\pi$
$173$	988.171	+	988.171i	0.434273	+	0.434273i	−0.455806	+	0.890079i	$0.650649\pi$
$174$		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	$-0.453103\pi$
$193$	3045.80	−	3045.80i	1.13597	−	1.13597i	0.989166	−	0.146800i	$-0.0468974\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3496.61	+	3496.61i	−1.26458	+	1.26458i	−0.315736	+	0.948847i	$0.602251\pi$
$197$	−3496.61	+	3496.61i	−1.26458	+	1.26458i	−0.948847	+	0.315736i	$0.897749\pi$
$198$		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	−0.947937	−	0.318458i	$-0.896835\pi$
$223$	−2096.23	+	2096.23i	−0.629479	+	0.629479i	0.318458	+	0.947937i	$0.396835\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1976.34	−	1976.34i	0.577861	−	0.577861i	−0.356453	−	0.934313i	$-0.616014\pi$
$227$	1976.34	−	1976.34i	0.577861	−	0.577861i	0.934313	+	0.356453i	$0.116014\pi$
$228$		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	$-0.954318\pi$
$233$	−4028.70	−	4028.70i	−1.13274	−	1.13274i	−0.143022	−	0.989720i	$-0.545682\pi$
$234$		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	$-0.427817\pi$
$257$	4940.86	−	4940.86i	1.19923	−	1.19923i	0.974397	−	0.224832i	$-0.0721835\pi$
$258$		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	0.813396	−	0.581711i	$-0.197616\pi$
$263$	988.171	+	988.171i	0.231685	+	0.231685i	−0.581711	+	0.813396i	$0.697616\pi$
$264$		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	0.789121	−	0.614238i	$-0.210537\pi$
$277$	806.241	+	806.241i	0.174882	+	0.174882i	−0.614238	+	0.789121i	$0.710537\pi$
$278$		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	−0.333197	−	0.942857i	$-0.608127\pi$
$283$	2902.47	−	2902.47i	0.609660	−	0.609660i	0.942857	+	0.333197i	$0.108127\pi$
$284$		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	0.482073	−	0.876131i	$-0.339884\pi$
$293$	−1976.34	−	1976.34i	−0.394058	−	0.394058i	−0.876131	+	0.482073i	$0.839884\pi$
$294$		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	0.905014	−	0.425382i	$-0.139860\pi$
$307$	2579.97	+	2579.97i	0.479631	+	0.479631i	−0.425382	+	0.905014i	$0.639860\pi$
$308$		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	−0.663794	−	0.747916i	$-0.731055\pi$
$313$	465.828	−	465.828i	0.0841220	−	0.0841220i	0.747916	+	0.663794i	$0.231055\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5929.03	+	5929.03i	−1.05050	+	1.05050i	−0.0518408	+	0.998655i	$0.516509\pi$
$317$	−5929.03	+	5929.03i	−1.05050	+	1.05050i	−0.998655	+	0.0518408i	$0.983491\pi$
$318$		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	0.743753	−	0.668455i	$-0.233044\pi$
$337$	465.828	+	465.828i	0.0752976	+	0.0752976i	−0.668455	+	0.743753i	$0.733044\pi$
$338$		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	$0.503627\pi$
$347$	−6537.13	+	6537.13i	−1.01133	+	1.01133i	−0.999935	+	0.0113952i	$0.996373\pi$
$348$		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	0.807554	−	0.589793i	$-0.200791\pi$
$353$	1444.25	+	1444.25i	0.217761	+	0.217761i	−0.589793	+	0.807554i	$0.700791\pi$
$354$		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	$-0.965099\pi$
$367$	−7757.83	−	7757.83i	−1.10342	−	1.10342i	−0.109426	−	0.993995i	$-0.534901\pi$
$368$		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	$0.575180\pi$
$373$	−8689.49	+	8689.49i	−1.20623	+	1.20623i	−0.972238	+	0.233995i	$0.924820\pi$
$374$		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	0.525979	−	0.850498i	$-0.323699\pi$
$383$	−2432.42	−	2432.42i	−0.324519	−	0.324519i	−0.850498	+	0.525979i	$0.823699\pi$
$384$		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	0.00113769	−	0.999999i	$-0.499638\pi$
$397$	−7901.16	−	7901.16i	−0.998862	−	0.998862i	−0.999999	+	0.00113769i	$0.999638\pi$
$398$		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	−0.334073	−	0.942547i	$-0.608423\pi$
$433$	5482.44	−	5482.44i	0.608474	−	0.608474i	0.942547	+	0.334073i	$0.108423\pi$
$434$		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	$-0.924684\pi$
$443$	−11249.9	−	11249.9i	−1.20655	−	1.20655i	−0.234411	−	0.972138i	$-0.575316\pi$
$444$		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	$0.133417\pi$
$457$	12899.9	+	12899.9i	1.32042	+	1.32042i	0.406978	+	0.913438i	$0.366583\pi$
$458$		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	−0.884867	−	0.465844i	$-0.845751\pi$
$463$	−4174.54	+	4174.54i	−0.419022	+	0.419022i	0.465844	+	0.884867i	$0.345751\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1064.18	+	1064.18i	−0.105449	+	0.105449i	−0.757863	−	0.652414i	$-0.773756\pi$
$467$	−1064.18	+	1064.18i	−0.105449	+	0.105449i	0.652414	+	0.757863i	$0.273756\pi$
$468$		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	0.765333	−	0.643635i	$-0.222575\pi$
$487$	1307.90	+	1307.90i	0.121698	+	0.121698i	−0.643635	+	0.765333i	$0.722575\pi$
$488$		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	0.263813	−	0.964574i	$-0.415020\pi$
$503$	−7905.37	−	7905.37i	−0.700761	−	0.700761i	−0.964574	+	0.263813i	$0.915020\pi$
$504$		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	−0.162784	−	0.986662i	$-0.552047\pi$
$523$	9854.06	−	9854.06i	0.823878	−	0.823878i	0.986662	+	0.162784i	$0.0520474\pi$
$524$		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	0.0519376	−	0.998650i	$-0.483460\pi$
$547$	−12111.5	−	12111.5i	−0.946713	−	0.946713i	−0.998650	+	0.0519376i	$0.983460\pi$
$548$		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	−0.585303	−	0.810815i	$-0.699024\pi$
$557$	2964.51	−	2964.51i	0.225512	−	0.225512i	0.810815	+	0.585303i	$0.199024\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8969.55	+	8969.55i	0.671442	+	0.671442i	0.958048	−	0.286607i	$-0.0925273\pi$
$563$	8969.55	+	8969.55i	0.671442	+	0.671442i	−0.286607	+	0.958048i	$0.592527\pi$
$564$		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	0.377796	−	0.925889i	$-0.376682\pi$
$577$	−7596.58	−	7596.58i	−0.548094	−	0.548094i	−0.925889	+	0.377796i	$0.876682\pi$
$578$		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	−0.895010	−	0.446046i	$-0.852832\pi$
$587$	−6385.11	+	6385.11i	−0.448963	+	0.448963i	0.446046	+	0.895010i	$0.352832\pi$
$588$		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	0.515023	−	0.857176i	$-0.327783\pi$
$593$	−4940.86	−	4940.86i	−0.342153	−	0.342153i	−0.857176	+	0.515023i	$0.827783\pi$
$594$		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	0.957647	−	0.287945i	$-0.0929721\pi$
$607$	10015.3	+	10015.3i	0.669702	+	0.669702i	−0.287945	+	0.957647i	$0.592972\pi$
$608$		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	−0.768510	−	0.639837i	$-0.779002\pi$
$613$	−1952.90	+	1952.90i	−0.128673	+	0.128673i	0.639837	+	0.768510i	$0.279002\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18775.2	+	18775.2i	−1.22506	+	1.22506i	−0.259252	+	0.965810i	$0.583476\pi$
$617$	−18775.2	+	18775.2i	−1.22506	+	1.22506i	−0.965810	+	0.259252i	$0.916524\pi$
$618$		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	−0.909714	−	0.415235i	$-0.863700\pi$
$643$	−8062.41	+	8062.41i	−0.494480	+	0.494480i	0.415235	+	0.909714i	$0.363700\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−12770.2	+	12770.2i	−0.775964	+	0.775964i	−0.979142	−	0.203178i	$-0.934873\pi$
$647$	−12770.2	+	12770.2i	−0.775964	+	0.775964i	0.203178	+	0.979142i	$0.434873\pi$
$648$		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	0.578679	−	0.815556i	$-0.303568\pi$
$653$	−3952.68	−	3952.68i	−0.236877	−	0.236877i	−0.815556	+	0.578679i	$0.803568\pi$
$654$		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	−0.392351	−	0.919815i	$-0.628338\pi$
$673$	9209.07	−	9209.07i	0.527464	−	0.527464i	0.919815	+	0.392351i	$0.128338\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−608.105	+	608.105i	−0.0345220	+	0.0345220i	−0.724157	−	0.689635i	$-0.757771\pi$
$677$	−608.105	+	608.105i	−0.0345220	+	0.0345220i	0.689635	+	0.724157i	$0.257771\pi$
$678$		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	0.577743	−	0.816219i	$-0.303934\pi$
$683$	−4256.74	−	4256.74i	−0.238477	−	0.238477i	−0.816219	+	0.577743i	$0.803934\pi$
$684$		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	0.758552	−	0.651613i	$-0.225907\pi$
$727$	2096.23	+	2096.23i	0.106939	+	0.106939i	−0.651613	+	0.758552i	$0.725907\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	14996.1	−	14996.1i	0.755652	−	0.755652i	−0.219876	−	0.975528i	$-0.570565\pi$
$733$	14996.1	−	14996.1i	0.755652	−	0.755652i	0.975528	+	0.219876i	$0.0705652\pi$
$734$		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	0.350906	−	0.936411i	$-0.385874\pi$
$743$	−11858.1	−	11858.1i	−0.585504	−	0.585504i	−0.936411	+	0.350906i	$0.885874\pi$
$744$		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	0.933179	−	0.359413i	$-0.117023\pi$
$757$	11950.3	+	11950.3i	0.573766	+	0.573766i	−0.359413	+	0.933179i	$0.617023\pi$
$758$		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	0.0969257	−	0.995292i	$-0.469099\pi$
$773$	−19307.3	−	19307.3i	−0.898366	−	0.898366i	−0.995292	+	0.0969257i	$0.969099\pi$
$774$		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	0.993992	−	0.109453i	$-0.0349099\pi$
$787$	19529.0	+	19529.0i	0.884539	+	0.884539i	−0.109453	+	0.993992i	$0.534910\pi$
$788$		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	−0.454793	−	0.890597i	$-0.650287\pi$
$797$	9805.70	−	9805.70i	0.435804	−	0.435804i	0.890597	+	0.454793i	$0.150287\pi$
$798$		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	−0.212914	−	0.977071i	$-0.568296\pi$
$823$	18041.9	−	18041.9i	0.764156	−	0.764156i	0.977071	+	0.212914i	$0.0682956\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4256.74	−	4256.74i	0.178986	−	0.178986i	−0.611928	−	0.790914i	$-0.709606\pi$
$827$	4256.74	−	4256.74i	0.178986	−	0.178986i	0.790914	+	0.611928i	$0.209606\pi$
$828$		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	−0.541954	−	0.840408i	$-0.682315\pi$
$853$	7435.34	−	7435.34i	0.298454	−	0.298454i	0.840408	+	0.541954i	$0.182315\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15886.7	+	15886.7i	−0.633233	+	0.633233i	−0.948878	−	0.315644i	$-0.897779\pi$
$857$	−15886.7	+	15886.7i	−0.633233	+	0.633233i	0.315644	+	0.948878i	$0.397779\pi$
$858$		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	0.913515	−	0.406805i	$-0.133357\pi$
$863$	12846.2	+	12846.2i	0.506710	+	0.506710i	−0.406805	+	0.913515i	$0.633357\pi$
$864$		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	0.352784	−	0.935705i	$-0.385235\pi$
$877$	−15139.4	−	15139.4i	−0.582921	−	0.582921i	−0.935705	+	0.352784i	$0.885235\pi$
$878$		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	−0.942349	−	0.334632i	$-0.891388\pi$
$883$	−15945.7	+	15945.7i	−0.607717	+	0.607717i	0.334632	+	0.942349i	$0.391388\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18167.1	+	18167.1i	−0.687704	+	0.687704i	−0.961724	−	0.274020i	$-0.911646\pi$
$887$	−18167.1	+	18167.1i	−0.687704	+	0.687704i	0.274020	+	0.961724i	$0.411646\pi$
$888$		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	$-0.985599\pi$
$907$	−28523.0	−	28523.0i	−1.04420	−	1.04420i	−0.0452255	−	0.998977i	$-0.514401\pi$
$908$		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	0.170758	−	0.985313i	$-0.445379\pi$
$937$	−23363.1	−	23363.1i	−0.814556	−	0.814556i	−0.985313	+	0.170758i	$0.945379\pi$
$938$		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	−0.957191	−	0.289457i	$-0.906525\pi$
$947$	−19459.4	+	19459.4i	−0.667734	+	0.667734i	0.289457	+	0.957191i	$0.406525\pi$
$948$		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	0.972873	−	0.231339i	$-0.0743106\pi$
$953$	21815.8	+	21815.8i	0.741534	+	0.741534i	−0.231339	+	0.972873i	$0.574311\pi$
$954$		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	$-0.979887\pi$
$967$	−31909.2	−	31909.2i	−1.06115	−	1.06115i	−0.0631455	−	0.998004i	$-0.520113\pi$
$968$		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	−0.997273	−	0.0738067i	$-0.976485\pi$
$977$	−28200.9	+	28200.9i	−0.923466	+	0.923466i	0.0738067	+	0.997273i	$0.476485\pi$
$978$		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	0.309608	−	0.950864i	$-0.399802\pi$
$983$	−19763.4	−	19763.4i	−0.641257	−	0.641257i	−0.950864	+	0.309608i	$0.899802\pi$
$984$		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	$0.0697485\pi$
$997$	37570.8	+	37570.8i	1.19346	+	1.19346i	0.217372	+	0.976089i	$0.430251\pi$
$998$		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.557.3	yes	8
3.2	odd	2	inner	900.4.j.a.557.4	yes	8
5.2	odd	4	inner	900.4.j.a.593.1	yes	8
5.3	odd	4	inner	900.4.j.a.593.3	yes	8
5.4	even	2	inner	900.4.j.a.557.1	✓	8
15.2	even	4	inner	900.4.j.a.593.2	yes	8
15.8	even	4	inner	900.4.j.a.593.4	yes	8
15.14	odd	2	inner	900.4.j.a.557.2	yes	8

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

Properties

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