LMFDB - Embedded newform 2496.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.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:	$147.268767374$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +6.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +6.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} -36.0000 q^{11} -13.0000 q^{13} +18.0000 q^{15} +66.0000 q^{17} -56.0000 q^{19} -12.0000 q^{21} +96.0000 q^{23} -89.0000 q^{25} +27.0000 q^{27} -222.000 q^{29} +260.000 q^{31} -108.000 q^{33} -24.0000 q^{35} +106.000 q^{37} -39.0000 q^{39} -90.0000 q^{41} -44.0000 q^{43} +54.0000 q^{45} +168.000 q^{47} -327.000 q^{49} +198.000 q^{51} -30.0000 q^{53} -216.000 q^{55} -168.000 q^{57} -348.000 q^{59} +346.000 q^{61} -36.0000 q^{63} -78.0000 q^{65} +256.000 q^{67} +288.000 q^{69} -168.000 q^{71} -814.000 q^{73} -267.000 q^{75} +144.000 q^{77} +200.000 q^{79} +81.0000 q^{81} -1236.00 q^{83} +396.000 q^{85} -666.000 q^{87} +318.000 q^{89} +52.0000 q^{91} +780.000 q^{93} -336.000 q^{95} -502.000 q^{97} -324.000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	0	0
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	18.0000	0.309839
$16$	0	0
$17$	66.0000	0.941609	0.470804	−	0.882238i	$-0.343964\pi$
$17$	66.0000	0.941609	0.470804	+	0.882238i	$0.343964\pi$
$18$	0	0
$19$	−56.0000	−0.676173	−0.338086	−	0.941115i	$-0.609780\pi$
$19$	−56.0000	−0.676173	−0.338086	+	0.941115i	$0.609780\pi$
$20$	0	0
$21$	−12.0000	−0.124696
$22$	0	0
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−222.000	−1.42153	−0.710765	−	0.703430i	$-0.751651\pi$
$29$	−222.000	−1.42153	−0.710765	+	0.703430i	$0.751651\pi$
$30$	0	0
$31$	260.000	1.50637	0.753184	−	0.657810i	$-0.228517\pi$
$31$	260.000	1.50637	0.753184	+	0.657810i	$0.228517\pi$
$32$	0	0
$33$	−108.000	−0.569709
$34$	0	0
$35$	−24.0000	−0.115907
$36$	0	0
$37$	106.000	0.470981	0.235490	−	0.971877i	$-0.424330\pi$
$37$	106.000	0.470981	0.235490	+	0.971877i	$0.424330\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−90.0000	−0.342820	−0.171410	−	0.985200i	$-0.554832\pi$
$41$	−90.0000	−0.342820	−0.171410	+	0.985200i	$0.554832\pi$
$42$	0	0
$43$	−44.0000	−0.156045	−0.0780225	−	0.996952i	$-0.524861\pi$
$43$	−44.0000	−0.156045	−0.0780225	+	0.996952i	$0.524861\pi$
$44$	0	0
$45$	54.0000	0.178885
$46$	0	0
$47$	168.000	0.521390	0.260695	−	0.965421i	$-0.416048\pi$
$47$	168.000	0.521390	0.260695	+	0.965421i	$0.416048\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	198.000	0.543638
$52$	0	0
$53$	−30.0000	−0.0777513	−0.0388756	−	0.999244i	$-0.512378\pi$
$53$	−30.0000	−0.0777513	−0.0388756	+	0.999244i	$0.512378\pi$
$54$	0	0
$55$	−216.000	−0.529553
$56$	0	0
$57$	−168.000	−0.390388
$58$	0	0
$59$	−348.000	−0.767894	−0.383947	−	0.923355i	$-0.625435\pi$
$59$	−348.000	−0.767894	−0.383947	+	0.923355i	$0.625435\pi$
$60$	0	0
$61$	346.000	0.726242	0.363121	−	0.931742i	$-0.381711\pi$
$61$	346.000	0.726242	0.363121	+	0.931742i	$0.381711\pi$
$62$	0	0
$63$	−36.0000	−0.0719932
$64$	0	0
$65$	−78.0000	−0.148842
$66$	0	0
$67$	256.000	0.466797	0.233398	−	0.972381i	$-0.425015\pi$
$67$	256.000	0.466797	0.233398	+	0.972381i	$0.425015\pi$
$68$	0	0
$69$	288.000	0.502480
$70$	0	0
$71$	−168.000	−0.280816	−0.140408	−	0.990094i	$-0.544841\pi$
$71$	−168.000	−0.280816	−0.140408	+	0.990094i	$0.544841\pi$
$72$	0	0
$73$	−814.000	−1.30509	−0.652544	−	0.757750i	$-0.726298\pi$
$73$	−814.000	−1.30509	−0.652544	+	0.757750i	$0.726298\pi$
$74$	0	0
$75$	−267.000	−0.411073
$76$	0	0
$77$	144.000	0.213121
$78$	0	0
$79$	200.000	0.284832	0.142416	−	0.989807i	$-0.454513\pi$
$79$	200.000	0.284832	0.142416	+	0.989807i	$0.454513\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1236.00	−1.63456	−0.817281	−	0.576240i	$-0.804520\pi$
$83$	−1236.00	−1.63456	−0.817281	+	0.576240i	$0.804520\pi$
$84$	0	0
$85$	396.000	0.505320
$86$	0	0
$87$	−666.000	−0.820721
$88$	0	0
$89$	318.000	0.378741	0.189370	−	0.981906i	$-0.439355\pi$
$89$	318.000	0.378741	0.189370	+	0.981906i	$0.439355\pi$
$90$	0	0
$91$	52.0000	0.0599020
$92$	0	0
$93$	780.000	0.869701
$94$	0	0
$95$	−336.000	−0.362872
$96$	0	0
$97$	−502.000	−0.525468	−0.262734	−	0.964868i	$-0.584624\pi$
$97$	−502.000	−0.525468	−0.262734	+	0.964868i	$0.584624\pi$
$98$	0	0
$99$	−324.000	−0.328921
$100$	0	0
$101$	−1062.00	−1.04627	−0.523133	−	0.852251i	$-0.675237\pi$
$101$	−1062.00	−1.04627	−0.523133	+	0.852251i	$0.675237\pi$
$102$	0	0
$103$	−64.0000	−0.0612243	−0.0306122	−	0.999531i	$-0.509746\pi$
$103$	−64.0000	−0.0612243	−0.0306122	+	0.999531i	$0.509746\pi$
$104$	0	0
$105$	−72.0000	−0.0669189
$106$	0	0
$107$	444.000	0.401150	0.200575	−	0.979678i	$-0.435719\pi$
$107$	444.000	0.401150	0.200575	+	0.979678i	$0.435719\pi$
$108$	0	0
$109$	−1382.00	−1.21442	−0.607209	−	0.794542i	$-0.707711\pi$
$109$	−1382.00	−1.21442	−0.607209	+	0.794542i	$0.707711\pi$
$110$	0	0
$111$	318.000	0.271921
$112$	0	0
$113$	−870.000	−0.724272	−0.362136	−	0.932125i	$-0.617952\pi$
$113$	−870.000	−0.724272	−0.362136	+	0.932125i	$0.617952\pi$
$114$	0	0
$115$	576.000	0.467063
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−264.000	−0.203368
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	−270.000	−0.197927
$124$	0	0
$125$	−1284.00	−0.918756
$126$	0	0
$127$	464.000	0.324200	0.162100	−	0.986774i	$-0.448173\pi$
$127$	464.000	0.324200	0.162100	+	0.986774i	$0.448173\pi$
$128$	0	0
$129$	−132.000	−0.0900927
$130$	0	0
$131$	−1548.00	−1.03244	−0.516219	−	0.856457i	$-0.672661\pi$
$131$	−1548.00	−1.03244	−0.516219	+	0.856457i	$0.672661\pi$
$132$	0	0
$133$	224.000	0.146040
$134$	0	0
$135$	162.000	0.103280
$136$	0	0
$137$	294.000	0.183344	0.0916720	−	0.995789i	$-0.470779\pi$
$137$	294.000	0.183344	0.0916720	+	0.995789i	$0.470779\pi$
$138$	0	0
$139$	−2564.00	−1.56457	−0.782286	−	0.622919i	$-0.785947\pi$
$139$	−2564.00	−1.56457	−0.782286	+	0.622919i	$0.785947\pi$
$140$	0	0
$141$	504.000	0.301025
$142$	0	0
$143$	468.000	0.273679
$144$	0	0
$145$	−1332.00	−0.762873
$146$	0	0
$147$	−981.000	−0.550418
$148$	0	0
$149$	−114.000	−0.0626795	−0.0313397	−	0.999509i	$-0.509977\pi$
$149$	−114.000	−0.0626795	−0.0313397	+	0.999509i	$0.509977\pi$
$150$	0	0
$151$	2036.00	1.09727	0.548634	−	0.836063i	$-0.315148\pi$
$151$	2036.00	1.09727	0.548634	+	0.836063i	$0.315148\pi$
$152$	0	0
$153$	594.000	0.313870
$154$	0	0
$155$	1560.00	0.808401
$156$	0	0
$157$	−2870.00	−1.45892	−0.729462	−	0.684022i	$-0.760229\pi$
$157$	−2870.00	−1.45892	−0.729462	+	0.684022i	$0.760229\pi$
$158$	0	0
$159$	−90.0000	−0.0448897
$160$	0	0
$161$	−384.000	−0.187972
$162$	0	0
$163$	−1472.00	−0.707337	−0.353669	−	0.935371i	$-0.615066\pi$
$163$	−1472.00	−0.707337	−0.353669	+	0.935371i	$0.615066\pi$
$164$	0	0
$165$	−648.000	−0.305738
$166$	0	0
$167$	−240.000	−0.111208	−0.0556041	−	0.998453i	$-0.517708\pi$
$167$	−240.000	−0.111208	−0.0556041	+	0.998453i	$0.517708\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−504.000	−0.225391
$172$	0	0
$173$	306.000	0.134478	0.0672392	−	0.997737i	$-0.478581\pi$
$173$	306.000	0.134478	0.0672392	+	0.997737i	$0.478581\pi$
$174$	0	0
$175$	356.000	0.153778
$176$	0	0
$177$	−1044.00	−0.443344
$178$	0	0
$179$	2052.00	0.856836	0.428418	−	0.903581i	$-0.359071\pi$
$179$	2052.00	0.856836	0.428418	+	0.903581i	$0.359071\pi$
$180$	0	0
$181$	4498.00	1.84715	0.923574	−	0.383421i	$-0.125254\pi$
$181$	4498.00	1.84715	0.923574	+	0.383421i	$0.125254\pi$
$182$	0	0
$183$	1038.00	0.419296
$184$	0	0
$185$	636.000	0.252755
$186$	0	0
$187$	−2376.00	−0.929146
$188$	0	0
$189$	−108.000	−0.0415653
$190$	0	0
$191$	−4056.00	−1.53655	−0.768277	−	0.640117i	$-0.778886\pi$
$191$	−4056.00	−1.53655	−0.768277	+	0.640117i	$0.778886\pi$
$192$	0	0
$193$	−2062.00	−0.769047	−0.384523	−	0.923115i	$-0.625634\pi$
$193$	−2062.00	−0.769047	−0.384523	+	0.923115i	$0.625634\pi$
$194$	0	0
$195$	−234.000	−0.0859338
$196$	0	0
$197$	4374.00	1.58190	0.790951	−	0.611880i	$-0.209586\pi$
$197$	4374.00	1.58190	0.790951	+	0.611880i	$0.209586\pi$
$198$	0	0
$199$	−2536.00	−0.903378	−0.451689	−	0.892175i	$-0.649178\pi$
$199$	−2536.00	−0.903378	−0.451689	+	0.892175i	$0.649178\pi$
$200$	0	0
$201$	768.000	0.269505
$202$	0	0
$203$	888.000	0.307022
$204$	0	0
$205$	−540.000	−0.183977
$206$	0	0
$207$	864.000	0.290107
$208$	0	0
$209$	2016.00	0.667223
$210$	0	0
$211$	4444.00	1.44994	0.724971	−	0.688780i	$-0.241853\pi$
$211$	4444.00	1.44994	0.724971	+	0.688780i	$0.241853\pi$
$212$	0	0
$213$	−504.000	−0.162129
$214$	0	0
$215$	−264.000	−0.0837426
$216$	0	0
$217$	−1040.00	−0.325345
$218$	0	0
$219$	−2442.00	−0.753493
$220$	0	0
$221$	−858.000	−0.261155
$222$	0	0
$223$	−2716.00	−0.815591	−0.407796	−	0.913073i	$-0.633702\pi$
$223$	−2716.00	−0.815591	−0.407796	+	0.913073i	$0.633702\pi$
$224$	0	0
$225$	−801.000	−0.237333
$226$	0	0
$227$	4692.00	1.37189	0.685945	−	0.727653i	$-0.259389\pi$
$227$	4692.00	1.37189	0.685945	+	0.727653i	$0.259389\pi$
$228$	0	0
$229$	−6446.00	−1.86010	−0.930052	−	0.367429i	$-0.880238\pi$
$229$	−6446.00	−1.86010	−0.930052	+	0.367429i	$0.880238\pi$
$230$	0	0
$231$	432.000	0.123046
$232$	0	0
$233$	−3102.00	−0.872184	−0.436092	−	0.899902i	$-0.643638\pi$
$233$	−3102.00	−0.872184	−0.436092	+	0.899902i	$0.643638\pi$
$234$	0	0
$235$	1008.00	0.279807
$236$	0	0
$237$	600.000	0.164448
$238$	0	0
$239$	816.000	0.220848	0.110424	−	0.993885i	$-0.464779\pi$
$239$	816.000	0.220848	0.110424	+	0.993885i	$0.464779\pi$
$240$	0	0
$241$	3818.00	1.02049	0.510247	−	0.860028i	$-0.329554\pi$
$241$	3818.00	1.02049	0.510247	+	0.860028i	$0.329554\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1962.00	−0.511623
$246$	0	0
$247$	728.000	0.187537
$248$	0	0
$249$	−3708.00	−0.943715
$250$	0	0
$251$	6612.00	1.66273	0.831366	−	0.555725i	$-0.187559\pi$
$251$	6612.00	1.66273	0.831366	+	0.555725i	$0.187559\pi$
$252$	0	0
$253$	−3456.00	−0.858802
$254$	0	0
$255$	1188.00	0.291747
$256$	0	0
$257$	−4806.00	−1.16650	−0.583249	−	0.812293i	$-0.698219\pi$
$257$	−4806.00	−1.16650	−0.583249	+	0.812293i	$0.698219\pi$
$258$	0	0
$259$	−424.000	−0.101722
$260$	0	0
$261$	−1998.00	−0.473843
$262$	0	0
$263$	−4584.00	−1.07476	−0.537379	−	0.843341i	$-0.680586\pi$
$263$	−4584.00	−1.07476	−0.537379	+	0.843341i	$0.680586\pi$
$264$	0	0
$265$	−180.000	−0.0417257
$266$	0	0
$267$	954.000	0.218666
$268$	0	0
$269$	−7134.00	−1.61698	−0.808490	−	0.588510i	$-0.799715\pi$
$269$	−7134.00	−1.61698	−0.808490	+	0.588510i	$0.799715\pi$
$270$	0	0
$271$	3140.00	0.703843	0.351921	−	0.936030i	$-0.385528\pi$
$271$	3140.00	0.703843	0.351921	+	0.936030i	$0.385528\pi$
$272$	0	0
$273$	156.000	0.0345844
$274$	0	0
$275$	3204.00	0.702576
$276$	0	0
$277$	4786.00	1.03813	0.519067	−	0.854734i	$-0.326280\pi$
$277$	4786.00	1.03813	0.519067	+	0.854734i	$0.326280\pi$
$278$	0	0
$279$	2340.00	0.502122
$280$	0	0
$281$	3798.00	0.806298	0.403149	−	0.915134i	$-0.367916\pi$
$281$	3798.00	0.806298	0.403149	+	0.915134i	$0.367916\pi$
$282$	0	0
$283$	−3572.00	−0.750295	−0.375147	−	0.926965i	$-0.622408\pi$
$283$	−3572.00	−0.750295	−0.375147	+	0.926965i	$0.622408\pi$
$284$	0	0
$285$	−1008.00	−0.209504
$286$	0	0
$287$	360.000	0.0740423
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	−1506.00	−0.303379
$292$	0	0
$293$	−7122.00	−1.42004	−0.710020	−	0.704182i	$-0.751314\pi$
$293$	−7122.00	−1.42004	−0.710020	+	0.704182i	$0.751314\pi$
$294$	0	0
$295$	−2088.00	−0.412095
$296$	0	0
$297$	−972.000	−0.189903
$298$	0	0
$299$	−1248.00	−0.241384
$300$	0	0
$301$	176.000	0.0337026
$302$	0	0
$303$	−3186.00	−0.604062
$304$	0	0
$305$	2076.00	0.389742
$306$	0	0
$307$	6856.00	1.27457	0.637284	−	0.770629i	$-0.280058\pi$
$307$	6856.00	1.27457	0.637284	+	0.770629i	$0.280058\pi$
$308$	0	0
$309$	−192.000	−0.0353479
$310$	0	0
$311$	−8832.00	−1.61034	−0.805172	−	0.593042i	$-0.797927\pi$
$311$	−8832.00	−1.61034	−0.805172	+	0.593042i	$0.797927\pi$
$312$	0	0
$313$	3626.00	0.654804	0.327402	−	0.944885i	$-0.393827\pi$
$313$	3626.00	0.654804	0.327402	+	0.944885i	$0.393827\pi$
$314$	0	0
$315$	−216.000	−0.0386356
$316$	0	0
$317$	−10146.0	−1.79765	−0.898827	−	0.438304i	$-0.855579\pi$
$317$	−10146.0	−1.79765	−0.898827	+	0.438304i	$0.855579\pi$
$318$	0	0
$319$	7992.00	1.40272
$320$	0	0
$321$	1332.00	0.231604
$322$	0	0
$323$	−3696.00	−0.636690
$324$	0	0
$325$	1157.00	0.197473
$326$	0	0
$327$	−4146.00	−0.701145
$328$	0	0
$329$	−672.000	−0.112610
$330$	0	0
$331$	−6536.00	−1.08535	−0.542675	−	0.839943i	$-0.682589\pi$
$331$	−6536.00	−1.08535	−0.542675	+	0.839943i	$0.682589\pi$
$332$	0	0
$333$	954.000	0.156994
$334$	0	0
$335$	1536.00	0.250509
$336$	0	0
$337$	−6094.00	−0.985048	−0.492524	−	0.870299i	$-0.663926\pi$
$337$	−6094.00	−0.985048	−0.492524	+	0.870299i	$0.663926\pi$
$338$	0	0
$339$	−2610.00	−0.418159
$340$	0	0
$341$	−9360.00	−1.48643
$342$	0	0
$343$	2680.00	0.421885
$344$	0	0
$345$	1728.00	0.269659
$346$	0	0
$347$	2724.00	0.421418	0.210709	−	0.977549i	$-0.432423\pi$
$347$	2724.00	0.421418	0.210709	+	0.977549i	$0.432423\pi$
$348$	0	0
$349$	1522.00	0.233441	0.116720	−	0.993165i	$-0.462762\pi$
$349$	1522.00	0.233441	0.116720	+	0.993165i	$0.462762\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−1362.00	−0.205360	−0.102680	−	0.994714i	$-0.532742\pi$
$353$	−1362.00	−0.205360	−0.102680	+	0.994714i	$0.532742\pi$
$354$	0	0
$355$	−1008.00	−0.150702
$356$	0	0
$357$	−792.000	−0.117415
$358$	0	0
$359$	8880.00	1.30548	0.652742	−	0.757581i	$-0.273619\pi$
$359$	8880.00	1.30548	0.652742	+	0.757581i	$0.273619\pi$
$360$	0	0
$361$	−3723.00	−0.542790
$362$	0	0
$363$	−105.000	−0.0151820
$364$	0	0
$365$	−4884.00	−0.700384
$366$	0	0
$367$	−3712.00	−0.527970	−0.263985	−	0.964527i	$-0.585037\pi$
$367$	−3712.00	−0.527970	−0.263985	+	0.964527i	$0.585037\pi$
$368$	0	0
$369$	−810.000	−0.114273
$370$	0	0
$371$	120.000	0.0167927
$372$	0	0
$373$	−5726.00	−0.794855	−0.397428	−	0.917634i	$-0.630097\pi$
$373$	−5726.00	−0.794855	−0.397428	+	0.917634i	$0.630097\pi$
$374$	0	0
$375$	−3852.00	−0.530444
$376$	0	0
$377$	2886.00	0.394261
$378$	0	0
$379$	13168.0	1.78468	0.892341	−	0.451361i	$-0.149061\pi$
$379$	13168.0	1.78468	0.892341	+	0.451361i	$0.149061\pi$
$380$	0	0
$381$	1392.00	0.187177
$382$	0	0
$383$	4872.00	0.649994	0.324997	−	0.945715i	$-0.394637\pi$
$383$	4872.00	0.649994	0.324997	+	0.945715i	$0.394637\pi$
$384$	0	0
$385$	864.000	0.114373
$386$	0	0
$387$	−396.000	−0.0520150
$388$	0	0
$389$	1266.00	0.165010	0.0825048	−	0.996591i	$-0.473708\pi$
$389$	1266.00	0.165010	0.0825048	+	0.996591i	$0.473708\pi$
$390$	0	0
$391$	6336.00	0.819502
$392$	0	0
$393$	−4644.00	−0.596078
$394$	0	0
$395$	1200.00	0.152857
$396$	0	0
$397$	4882.00	0.617180	0.308590	−	0.951195i	$-0.400143\pi$
$397$	4882.00	0.617180	0.308590	+	0.951195i	$0.400143\pi$
$398$	0	0
$399$	672.000	0.0843160
$400$	0	0
$401$	−90.0000	−0.0112079	−0.00560397	−	0.999984i	$-0.501784\pi$
$401$	−90.0000	−0.0112079	−0.00560397	+	0.999984i	$0.501784\pi$
$402$	0	0
$403$	−3380.00	−0.417791
$404$	0	0
$405$	486.000	0.0596285
$406$	0	0
$407$	−3816.00	−0.464747
$408$	0	0
$409$	2354.00	0.284591	0.142296	−	0.989824i	$-0.454552\pi$
$409$	2354.00	0.284591	0.142296	+	0.989824i	$0.454552\pi$
$410$	0	0
$411$	882.000	0.105854
$412$	0	0
$413$	1392.00	0.165849
$414$	0	0
$415$	−7416.00	−0.877198
$416$	0	0
$417$	−7692.00	−0.903307
$418$	0	0
$419$	−7020.00	−0.818495	−0.409248	−	0.912423i	$-0.634209\pi$
$419$	−7020.00	−0.818495	−0.409248	+	0.912423i	$0.634209\pi$
$420$	0	0
$421$	−302.000	−0.0349610	−0.0174805	−	0.999847i	$-0.505564\pi$
$421$	−302.000	−0.0349610	−0.0174805	+	0.999847i	$0.505564\pi$
$422$	0	0
$423$	1512.00	0.173797
$424$	0	0
$425$	−5874.00	−0.670426
$426$	0	0
$427$	−1384.00	−0.156854
$428$	0	0
$429$	1404.00	0.158009
$430$	0	0
$431$	9816.00	1.09703	0.548515	−	0.836141i	$-0.315193\pi$
$431$	9816.00	1.09703	0.548515	+	0.836141i	$0.315193\pi$
$432$	0	0
$433$	−14782.0	−1.64059	−0.820297	−	0.571937i	$-0.806192\pi$
$433$	−14782.0	−1.64059	−0.820297	+	0.571937i	$0.806192\pi$
$434$	0	0
$435$	−3996.00	−0.440445
$436$	0	0
$437$	−5376.00	−0.588487
$438$	0	0
$439$	3584.00	0.389647	0.194823	−	0.980838i	$-0.437587\pi$
$439$	3584.00	0.389647	0.194823	+	0.980838i	$0.437587\pi$
$440$	0	0
$441$	−2943.00	−0.317784
$442$	0	0
$443$	−180.000	−0.0193049	−0.00965244	−	0.999953i	$-0.503073\pi$
$443$	−180.000	−0.0193049	−0.00965244	+	0.999953i	$0.503073\pi$
$444$	0	0
$445$	1908.00	0.203254
$446$	0	0
$447$	−342.000	−0.0361880
$448$	0	0
$449$	−3450.00	−0.362618	−0.181309	−	0.983426i	$-0.558033\pi$
$449$	−3450.00	−0.362618	−0.181309	+	0.983426i	$0.558033\pi$
$450$	0	0
$451$	3240.00	0.338283
$452$	0	0
$453$	6108.00	0.633507
$454$	0	0
$455$	312.000	0.0321468
$456$	0	0
$457$	−16654.0	−1.70469	−0.852343	−	0.522984i	$-0.824819\pi$
$457$	−16654.0	−1.70469	−0.852343	+	0.522984i	$0.824819\pi$
$458$	0	0
$459$	1782.00	0.181213
$460$	0	0
$461$	14046.0	1.41906	0.709531	−	0.704674i	$-0.248907\pi$
$461$	14046.0	1.41906	0.709531	+	0.704674i	$0.248907\pi$
$462$	0	0
$463$	−4588.00	−0.460524	−0.230262	−	0.973129i	$-0.573958\pi$
$463$	−4588.00	−0.460524	−0.230262	+	0.973129i	$0.573958\pi$
$464$	0	0
$465$	4680.00	0.466731
$466$	0	0
$467$	15372.0	1.52319	0.761597	−	0.648051i	$-0.224416\pi$
$467$	15372.0	1.52319	0.761597	+	0.648051i	$0.224416\pi$
$468$	0	0
$469$	−1024.00	−0.100819
$470$	0	0
$471$	−8610.00	−0.842310
$472$	0	0
$473$	1584.00	0.153980
$474$	0	0
$475$	4984.00	0.481435
$476$	0	0
$477$	−270.000	−0.0259171
$478$	0	0
$479$	12864.0	1.22708	0.613540	−	0.789664i	$-0.289745\pi$
$479$	12864.0	1.22708	0.613540	+	0.789664i	$0.289745\pi$
$480$	0	0
$481$	−1378.00	−0.130627
$482$	0	0
$483$	−1152.00	−0.108525
$484$	0	0
$485$	−3012.00	−0.281996
$486$	0	0
$487$	−10276.0	−0.956160	−0.478080	−	0.878316i	$-0.658667\pi$
$487$	−10276.0	−0.956160	−0.478080	+	0.878316i	$0.658667\pi$
$488$	0	0
$489$	−4416.00	−0.408381
$490$	0	0
$491$	11220.0	1.03127	0.515633	−	0.856810i	$-0.327557\pi$
$491$	11220.0	1.03127	0.515633	+	0.856810i	$0.327557\pi$
$492$	0	0
$493$	−14652.0	−1.33853
$494$	0	0
$495$	−1944.00	−0.176518
$496$	0	0
$497$	672.000	0.0606505
$498$	0	0
$499$	−17264.0	−1.54878	−0.774392	−	0.632707i	$-0.781944\pi$
$499$	−17264.0	−1.54878	−0.774392	+	0.632707i	$0.781944\pi$
$500$	0	0
$501$	−720.000	−0.0642060
$502$	0	0
$503$	−1896.00	−0.168069	−0.0840343	−	0.996463i	$-0.526781\pi$
$503$	−1896.00	−0.168069	−0.0840343	+	0.996463i	$0.526781\pi$
$504$	0	0
$505$	−6372.00	−0.561486
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−5010.00	−0.436276	−0.218138	−	0.975918i	$-0.569998\pi$
$509$	−5010.00	−0.436276	−0.218138	+	0.975918i	$0.569998\pi$
$510$	0	0
$511$	3256.00	0.281873
$512$	0	0
$513$	−1512.00	−0.130129
$514$	0	0
$515$	−384.000	−0.0328564
$516$	0	0
$517$	−6048.00	−0.514489
$518$	0	0
$519$	918.000	0.0776411
$520$	0	0
$521$	8610.00	0.724013	0.362007	−	0.932176i	$-0.382092\pi$
$521$	8610.00	0.724013	0.362007	+	0.932176i	$0.382092\pi$
$522$	0	0
$523$	5308.00	0.443791	0.221895	−	0.975070i	$-0.428776\pi$
$523$	5308.00	0.443791	0.221895	+	0.975070i	$0.428776\pi$
$524$	0	0
$525$	1068.00	0.0887835
$526$	0	0
$527$	17160.0	1.41841
$528$	0	0
$529$	−2951.00	−0.242541
$530$	0	0
$531$	−3132.00	−0.255965
$532$	0	0
$533$	1170.00	0.0950813
$534$	0	0
$535$	2664.00	0.215280
$536$	0	0
$537$	6156.00	0.494695
$538$	0	0
$539$	11772.0	0.940735
$540$	0	0
$541$	−6182.00	−0.491285	−0.245642	−	0.969361i	$-0.578999\pi$
$541$	−6182.00	−0.491285	−0.245642	+	0.969361i	$0.578999\pi$
$542$	0	0
$543$	13494.0	1.06645
$544$	0	0
$545$	−8292.00	−0.651725
$546$	0	0
$547$	−1292.00	−0.100991	−0.0504954	−	0.998724i	$-0.516080\pi$
$547$	−1292.00	−0.100991	−0.0504954	+	0.998724i	$0.516080\pi$
$548$	0	0
$549$	3114.00	0.242081
$550$	0	0
$551$	12432.0	0.961200
$552$	0	0
$553$	−800.000	−0.0615180
$554$	0	0
$555$	1908.00	0.145928
$556$	0	0
$557$	12774.0	0.971727	0.485863	−	0.874035i	$-0.338505\pi$
$557$	12774.0	0.971727	0.485863	+	0.874035i	$0.338505\pi$
$558$	0	0
$559$	572.000	0.0432791
$560$	0	0
$561$	−7128.00	−0.536443
$562$	0	0
$563$	−16908.0	−1.26570	−0.632848	−	0.774276i	$-0.718114\pi$
$563$	−16908.0	−1.26570	−0.632848	+	0.774276i	$0.718114\pi$
$564$	0	0
$565$	−5220.00	−0.388685
$566$	0	0
$567$	−324.000	−0.0239977
$568$	0	0
$569$	−11214.0	−0.826213	−0.413107	−	0.910683i	$-0.635556\pi$
$569$	−11214.0	−0.826213	−0.413107	+	0.910683i	$0.635556\pi$
$570$	0	0
$571$	−25220.0	−1.84838	−0.924189	−	0.381935i	$-0.875258\pi$
$571$	−25220.0	−1.84838	−0.924189	+	0.381935i	$0.875258\pi$
$572$	0	0
$573$	−12168.0	−0.887130
$574$	0	0
$575$	−8544.00	−0.619669
$576$	0	0
$577$	−17710.0	−1.27778	−0.638888	−	0.769300i	$-0.720605\pi$
$577$	−17710.0	−1.27778	−0.638888	+	0.769300i	$0.720605\pi$
$578$	0	0
$579$	−6186.00	−0.444009
$580$	0	0
$581$	4944.00	0.353032
$582$	0	0
$583$	1080.00	0.0767222
$584$	0	0
$585$	−702.000	−0.0496139
$586$	0	0
$587$	−20028.0	−1.40825	−0.704126	−	0.710075i	$-0.748661\pi$
$587$	−20028.0	−1.40825	−0.704126	+	0.710075i	$0.748661\pi$
$588$	0	0
$589$	−14560.0	−1.01856
$590$	0	0
$591$	13122.0	0.913311
$592$	0	0
$593$	19926.0	1.37987	0.689935	−	0.723871i	$-0.257639\pi$
$593$	19926.0	1.37987	0.689935	+	0.723871i	$0.257639\pi$
$594$	0	0
$595$	−1584.00	−0.109139
$596$	0	0
$597$	−7608.00	−0.521566
$598$	0	0
$599$	−1704.00	−0.116233	−0.0581165	−	0.998310i	$-0.518509\pi$
$599$	−1704.00	−0.116233	−0.0581165	+	0.998310i	$0.518509\pi$
$600$	0	0
$601$	11018.0	0.747810	0.373905	−	0.927467i	$-0.378019\pi$
$601$	11018.0	0.747810	0.373905	+	0.927467i	$0.378019\pi$
$602$	0	0
$603$	2304.00	0.155599
$604$	0	0
$605$	−210.000	−0.0141119
$606$	0	0
$607$	−448.000	−0.0299568	−0.0149784	−	0.999888i	$-0.504768\pi$
$607$	−448.000	−0.0299568	−0.0149784	+	0.999888i	$0.504768\pi$
$608$	0	0
$609$	2664.00	0.177259
$610$	0	0
$611$	−2184.00	−0.144608
$612$	0	0
$613$	12586.0	0.829272	0.414636	−	0.909987i	$-0.363909\pi$
$613$	12586.0	0.829272	0.414636	+	0.909987i	$0.363909\pi$
$614$	0	0
$615$	−1620.00	−0.106219
$616$	0	0
$617$	−29610.0	−1.93202	−0.966008	−	0.258513i	$-0.916768\pi$
$617$	−29610.0	−1.93202	−0.966008	+	0.258513i	$0.916768\pi$
$618$	0	0
$619$	7120.00	0.462321	0.231161	−	0.972916i	$-0.425748\pi$
$619$	7120.00	0.462321	0.231161	+	0.972916i	$0.425748\pi$
$620$	0	0
$621$	2592.00	0.167493
$622$	0	0
$623$	−1272.00	−0.0818003
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	6048.00	0.385221
$628$	0	0
$629$	6996.00	0.443480
$630$	0	0
$631$	−15580.0	−0.982932	−0.491466	−	0.870897i	$-0.663539\pi$
$631$	−15580.0	−0.982932	−0.491466	+	0.870897i	$0.663539\pi$
$632$	0	0
$633$	13332.0	0.837124
$634$	0	0
$635$	2784.00	0.173984
$636$	0	0
$637$	4251.00	0.264412
$638$	0	0
$639$	−1512.00	−0.0936053
$640$	0	0
$641$	−19806.0	−1.22042	−0.610211	−	0.792239i	$-0.708915\pi$
$641$	−19806.0	−1.22042	−0.610211	+	0.792239i	$0.708915\pi$
$642$	0	0
$643$	−24032.0	−1.47392	−0.736959	−	0.675937i	$-0.763739\pi$
$643$	−24032.0	−1.47392	−0.736959	+	0.675937i	$0.763739\pi$
$644$	0	0
$645$	−792.000	−0.0483488
$646$	0	0
$647$	−2808.00	−0.170624	−0.0853121	−	0.996354i	$-0.527189\pi$
$647$	−2808.00	−0.170624	−0.0853121	+	0.996354i	$0.527189\pi$
$648$	0	0
$649$	12528.0	0.757730
$650$	0	0
$651$	−3120.00	−0.187838
$652$	0	0
$653$	−23886.0	−1.43144	−0.715721	−	0.698386i	$-0.753902\pi$
$653$	−23886.0	−1.43144	−0.715721	+	0.698386i	$0.753902\pi$
$654$	0	0
$655$	−9288.00	−0.554064
$656$	0	0
$657$	−7326.00	−0.435030
$658$	0	0
$659$	3948.00	0.233372	0.116686	−	0.993169i	$-0.462773\pi$
$659$	3948.00	0.233372	0.116686	+	0.993169i	$0.462773\pi$
$660$	0	0
$661$	−5750.00	−0.338350	−0.169175	−	0.985586i	$-0.554110\pi$
$661$	−5750.00	−0.338350	−0.169175	+	0.985586i	$0.554110\pi$
$662$	0	0
$663$	−2574.00	−0.150778
$664$	0	0
$665$	1344.00	0.0783731
$666$	0	0
$667$	−21312.0	−1.23719
$668$	0	0
$669$	−8148.00	−0.470882
$670$	0	0
$671$	−12456.0	−0.716630
$672$	0	0
$673$	28082.0	1.60844	0.804221	−	0.594330i	$-0.202583\pi$
$673$	28082.0	1.60844	0.804221	+	0.594330i	$0.202583\pi$
$674$	0	0
$675$	−2403.00	−0.137024
$676$	0	0
$677$	27954.0	1.58694	0.793471	−	0.608608i	$-0.208272\pi$
$677$	27954.0	1.58694	0.793471	+	0.608608i	$0.208272\pi$
$678$	0	0
$679$	2008.00	0.113490
$680$	0	0
$681$	14076.0	0.792061
$682$	0	0
$683$	28428.0	1.59263	0.796316	−	0.604881i	$-0.206779\pi$
$683$	28428.0	1.59263	0.796316	+	0.604881i	$0.206779\pi$
$684$	0	0
$685$	1764.00	0.0983927
$686$	0	0
$687$	−19338.0	−1.07393
$688$	0	0
$689$	390.000	0.0215643
$690$	0	0
$691$	−21680.0	−1.19355	−0.596777	−	0.802407i	$-0.703552\pi$
$691$	−21680.0	−1.19355	−0.596777	+	0.802407i	$0.703552\pi$
$692$	0	0
$693$	1296.00	0.0710404
$694$	0	0
$695$	−15384.0	−0.839638
$696$	0	0
$697$	−5940.00	−0.322803
$698$	0	0
$699$	−9306.00	−0.503555
$700$	0	0
$701$	7482.00	0.403126	0.201563	−	0.979476i	$-0.435398\pi$
$701$	7482.00	0.403126	0.201563	+	0.979476i	$0.435398\pi$
$702$	0	0
$703$	−5936.00	−0.318464
$704$	0	0
$705$	3024.00	0.161547
$706$	0	0
$707$	4248.00	0.225972
$708$	0	0
$709$	−2270.00	−0.120242	−0.0601210	−	0.998191i	$-0.519149\pi$
$709$	−2270.00	−0.120242	−0.0601210	+	0.998191i	$0.519149\pi$
$710$	0	0
$711$	1800.00	0.0949441
$712$	0	0
$713$	24960.0	1.31102
$714$	0	0
$715$	2808.00	0.146872
$716$	0	0
$717$	2448.00	0.127507
$718$	0	0
$719$	36024.0	1.86852	0.934262	−	0.356588i	$-0.116060\pi$
$719$	36024.0	1.86852	0.934262	+	0.356588i	$0.116060\pi$
$720$	0	0
$721$	256.000	0.0132232
$722$	0	0
$723$	11454.0	0.589182
$724$	0	0
$725$	19758.0	1.01213
$726$	0	0
$727$	−21544.0	−1.09907	−0.549534	−	0.835471i	$-0.685195\pi$
$727$	−21544.0	−1.09907	−0.549534	+	0.835471i	$0.685195\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−2904.00	−0.146933
$732$	0	0
$733$	1018.00	0.0512970	0.0256485	−	0.999671i	$-0.491835\pi$
$733$	1018.00	0.0512970	0.0256485	+	0.999671i	$0.491835\pi$
$734$	0	0
$735$	−5886.00	−0.295386
$736$	0	0
$737$	−9216.00	−0.460618
$738$	0	0
$739$	24568.0	1.22293	0.611467	−	0.791270i	$-0.290580\pi$
$739$	24568.0	1.22293	0.611467	+	0.791270i	$0.290580\pi$
$740$	0	0
$741$	2184.00	0.108274
$742$	0	0
$743$	16968.0	0.837814	0.418907	−	0.908029i	$-0.362413\pi$
$743$	16968.0	0.837814	0.418907	+	0.908029i	$0.362413\pi$
$744$	0	0
$745$	−684.000	−0.0336373
$746$	0	0
$747$	−11124.0	−0.544854
$748$	0	0
$749$	−1776.00	−0.0866404
$750$	0	0
$751$	3224.00	0.156652	0.0783259	−	0.996928i	$-0.475043\pi$
$751$	3224.00	0.156652	0.0783259	+	0.996928i	$0.475043\pi$
$752$	0	0
$753$	19836.0	0.959979
$754$	0	0
$755$	12216.0	0.588855
$756$	0	0
$757$	31570.0	1.51576	0.757881	−	0.652393i	$-0.226235\pi$
$757$	31570.0	1.51576	0.757881	+	0.652393i	$0.226235\pi$
$758$	0	0
$759$	−10368.0	−0.495829
$760$	0	0
$761$	−34890.0	−1.66197	−0.830987	−	0.556293i	$-0.812223\pi$
$761$	−34890.0	−1.66197	−0.830987	+	0.556293i	$0.812223\pi$
$762$	0	0
$763$	5528.00	0.262290
$764$	0	0
$765$	3564.00	0.168440
$766$	0	0
$767$	4524.00	0.212975
$768$	0	0
$769$	11522.0	0.540304	0.270152	−	0.962818i	$-0.412926\pi$
$769$	11522.0	0.540304	0.270152	+	0.962818i	$0.412926\pi$
$770$	0	0
$771$	−14418.0	−0.673478
$772$	0	0
$773$	28158.0	1.31018	0.655092	−	0.755549i	$-0.272630\pi$
$773$	28158.0	1.31018	0.655092	+	0.755549i	$0.272630\pi$
$774$	0	0
$775$	−23140.0	−1.07253
$776$	0	0
$777$	−1272.00	−0.0587294
$778$	0	0
$779$	5040.00	0.231806
$780$	0	0
$781$	6048.00	0.277099
$782$	0	0
$783$	−5994.00	−0.273574
$784$	0	0
$785$	−17220.0	−0.782940
$786$	0	0
$787$	−14504.0	−0.656940	−0.328470	−	0.944514i	$-0.606533\pi$
$787$	−14504.0	−0.656940	−0.328470	+	0.944514i	$0.606533\pi$
$788$	0	0
$789$	−13752.0	−0.620512
$790$	0	0
$791$	3480.00	0.156428
$792$	0	0
$793$	−4498.00	−0.201423
$794$	0	0
$795$	−540.000	−0.0240903
$796$	0	0
$797$	18090.0	0.803991	0.401995	−	0.915642i	$-0.368317\pi$
$797$	18090.0	0.803991	0.401995	+	0.915642i	$0.368317\pi$
$798$	0	0
$799$	11088.0	0.490945
$800$	0	0
$801$	2862.00	0.126247
$802$	0	0
$803$	29304.0	1.28782
$804$	0	0
$805$	−2304.00	−0.100876
$806$	0	0
$807$	−21402.0	−0.933564
$808$	0	0
$809$	36402.0	1.58199	0.790993	−	0.611826i	$-0.209565\pi$
$809$	36402.0	1.58199	0.790993	+	0.611826i	$0.209565\pi$
$810$	0	0
$811$	32368.0	1.40147	0.700736	−	0.713420i	$-0.252855\pi$
$811$	32368.0	1.40147	0.700736	+	0.713420i	$0.252855\pi$
$812$	0	0
$813$	9420.00	0.406364
$814$	0	0
$815$	−8832.00	−0.379597
$816$	0	0
$817$	2464.00	0.105513
$818$	0	0
$819$	468.000	0.0199673
$820$	0	0
$821$	−35778.0	−1.52090	−0.760451	−	0.649395i	$-0.775022\pi$
$821$	−35778.0	−1.52090	−0.760451	+	0.649395i	$0.775022\pi$
$822$	0	0
$823$	−10240.0	−0.433711	−0.216855	−	0.976204i	$-0.569580\pi$
$823$	−10240.0	−0.433711	−0.216855	+	0.976204i	$0.569580\pi$
$824$	0	0
$825$	9612.00	0.405633
$826$	0	0
$827$	16284.0	0.684704	0.342352	−	0.939572i	$-0.388776\pi$
$827$	16284.0	0.684704	0.342352	+	0.939572i	$0.388776\pi$
$828$	0	0
$829$	−14150.0	−0.592822	−0.296411	−	0.955060i	$-0.595790\pi$
$829$	−14150.0	−0.592822	−0.296411	+	0.955060i	$0.595790\pi$
$830$	0	0
$831$	14358.0	0.599366
$832$	0	0
$833$	−21582.0	−0.897685
$834$	0	0
$835$	−1440.00	−0.0596805
$836$	0	0
$837$	7020.00	0.289900
$838$	0	0
$839$	39576.0	1.62850	0.814252	−	0.580511i	$-0.197147\pi$
$839$	39576.0	1.62850	0.814252	+	0.580511i	$0.197147\pi$
$840$	0	0
$841$	24895.0	1.02075
$842$	0	0
$843$	11394.0	0.465516
$844$	0	0
$845$	1014.00	0.0412813
$846$	0	0
$847$	140.000	0.00567941
$848$	0	0
$849$	−10716.0	−0.433183
$850$	0	0
$851$	10176.0	0.409905
$852$	0	0
$853$	6922.00	0.277848	0.138924	−	0.990303i	$-0.455636\pi$
$853$	6922.00	0.277848	0.138924	+	0.990303i	$0.455636\pi$
$854$	0	0
$855$	−3024.00	−0.120957
$856$	0	0
$857$	48162.0	1.91970	0.959850	−	0.280514i	$-0.0905050\pi$
$857$	48162.0	1.91970	0.959850	+	0.280514i	$0.0905050\pi$
$858$	0	0
$859$	27652.0	1.09834	0.549170	−	0.835711i	$-0.314944\pi$
$859$	27652.0	1.09834	0.549170	+	0.835711i	$0.314944\pi$
$860$	0	0
$861$	1080.00	0.0427483
$862$	0	0
$863$	648.000	0.0255599	0.0127799	−	0.999918i	$-0.495932\pi$
$863$	648.000	0.0255599	0.0127799	+	0.999918i	$0.495932\pi$
$864$	0	0
$865$	1836.00	0.0721686
$866$	0	0
$867$	−1671.00	−0.0654558
$868$	0	0
$869$	−7200.00	−0.281062
$870$	0	0
$871$	−3328.00	−0.129466
$872$	0	0
$873$	−4518.00	−0.175156
$874$	0	0
$875$	5136.00	0.198433
$876$	0	0
$877$	−7166.00	−0.275916	−0.137958	−	0.990438i	$-0.544054\pi$
$877$	−7166.00	−0.275916	−0.137958	+	0.990438i	$0.544054\pi$
$878$	0	0
$879$	−21366.0	−0.819860
$880$	0	0
$881$	−37062.0	−1.41731	−0.708655	−	0.705555i	$-0.750698\pi$
$881$	−37062.0	−1.41731	−0.708655	+	0.705555i	$0.750698\pi$
$882$	0	0
$883$	−24716.0	−0.941970	−0.470985	−	0.882141i	$-0.656101\pi$
$883$	−24716.0	−0.941970	−0.470985	+	0.882141i	$0.656101\pi$
$884$	0	0
$885$	−6264.00	−0.237923
$886$	0	0
$887$	48672.0	1.84244	0.921221	−	0.389040i	$-0.127193\pi$
$887$	48672.0	1.84244	0.921221	+	0.389040i	$0.127193\pi$
$888$	0	0
$889$	−1856.00	−0.0700205
$890$	0	0
$891$	−2916.00	−0.109640
$892$	0	0
$893$	−9408.00	−0.352550
$894$	0	0
$895$	12312.0	0.459827
$896$	0	0
$897$	−3744.00	−0.139363
$898$	0	0
$899$	−57720.0	−2.14135
$900$	0	0
$901$	−1980.00	−0.0732113
$902$	0	0
$903$	528.000	0.0194582
$904$	0	0
$905$	26988.0	0.991283
$906$	0	0
$907$	9484.00	0.347201	0.173600	−	0.984816i	$-0.444460\pi$
$907$	9484.00	0.347201	0.173600	+	0.984816i	$0.444460\pi$
$908$	0	0
$909$	−9558.00	−0.348756
$910$	0	0
$911$	12792.0	0.465223	0.232611	−	0.972570i	$-0.425273\pi$
$911$	12792.0	0.465223	0.232611	+	0.972570i	$0.425273\pi$
$912$	0	0
$913$	44496.0	1.61293
$914$	0	0
$915$	6228.00	0.225018
$916$	0	0
$917$	6192.00	0.222986
$918$	0	0
$919$	−18592.0	−0.667349	−0.333674	−	0.942688i	$-0.608289\pi$
$919$	−18592.0	−0.667349	−0.333674	+	0.942688i	$0.608289\pi$
$920$	0	0
$921$	20568.0	0.735873
$922$	0	0
$923$	2184.00	0.0778843
$924$	0	0
$925$	−9434.00	−0.335338
$926$	0	0
$927$	−576.000	−0.0204081
$928$	0	0
$929$	−15378.0	−0.543096	−0.271548	−	0.962425i	$-0.587536\pi$
$929$	−15378.0	−0.543096	−0.271548	+	0.962425i	$0.587536\pi$
$930$	0	0
$931$	18312.0	0.644631
$932$	0	0
$933$	−26496.0	−0.929732
$934$	0	0
$935$	−14256.0	−0.498632
$936$	0	0
$937$	−37078.0	−1.29273	−0.646364	−	0.763030i	$-0.723711\pi$
$937$	−37078.0	−1.29273	−0.646364	+	0.763030i	$0.723711\pi$
$938$	0	0
$939$	10878.0	0.378051
$940$	0	0
$941$	−10842.0	−0.375599	−0.187800	−	0.982207i	$-0.560136\pi$
$941$	−10842.0	−0.375599	−0.187800	+	0.982207i	$0.560136\pi$
$942$	0	0
$943$	−8640.00	−0.298364
$944$	0	0
$945$	−648.000	−0.0223063
$946$	0	0
$947$	−41508.0	−1.42432	−0.712159	−	0.702018i	$-0.752282\pi$
$947$	−41508.0	−1.42432	−0.712159	+	0.702018i	$0.752282\pi$
$948$	0	0
$949$	10582.0	0.361967
$950$	0	0
$951$	−30438.0	−1.03788
$952$	0	0
$953$	38706.0	1.31565	0.657823	−	0.753173i	$-0.271478\pi$
$953$	38706.0	1.31565	0.657823	+	0.753173i	$0.271478\pi$
$954$	0	0
$955$	−24336.0	−0.824602
$956$	0	0
$957$	23976.0	0.809858
$958$	0	0
$959$	−1176.00	−0.0395986
$960$	0	0
$961$	37809.0	1.26914
$962$	0	0
$963$	3996.00	0.133717
$964$	0	0
$965$	−12372.0	−0.412714
$966$	0	0
$967$	−24388.0	−0.811029	−0.405515	−	0.914089i	$-0.632908\pi$
$967$	−24388.0	−0.811029	−0.405515	+	0.914089i	$0.632908\pi$
$968$	0	0
$969$	−11088.0	−0.367593
$970$	0	0
$971$	14100.0	0.466005	0.233002	−	0.972476i	$-0.425145\pi$
$971$	14100.0	0.466005	0.233002	+	0.972476i	$0.425145\pi$
$972$	0	0
$973$	10256.0	0.337916
$974$	0	0
$975$	3471.00	0.114011
$976$	0	0
$977$	44838.0	1.46826	0.734132	−	0.679006i	$-0.237589\pi$
$977$	44838.0	1.46826	0.734132	+	0.679006i	$0.237589\pi$
$978$	0	0
$979$	−11448.0	−0.373728
$980$	0	0
$981$	−12438.0	−0.404806
$982$	0	0
$983$	13176.0	0.427517	0.213758	−	0.976887i	$-0.431429\pi$
$983$	13176.0	0.427517	0.213758	+	0.976887i	$0.431429\pi$
$984$	0	0
$985$	26244.0	0.848937
$986$	0	0
$987$	−2016.00	−0.0650152
$988$	0	0
$989$	−4224.00	−0.135809
$990$	0	0
$991$	−43648.0	−1.39912	−0.699558	−	0.714576i	$-0.746620\pi$
$991$	−43648.0	−1.39912	−0.699558	+	0.714576i	$0.746620\pi$
$992$	0	0
$993$	−19608.0	−0.626627
$994$	0	0
$995$	−15216.0	−0.484804
$996$	0	0
$997$	−62750.0	−1.99329	−0.996646	−	0.0818317i	$-0.973923\pi$
$997$	−62750.0	−1.99329	−0.996646	+	0.0818317i	$0.973923\pi$
$998$	0	0
$999$	2862.00	0.0906403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.n.1.1		1
4.3	odd	2		2496.4.a.e.1.1		1
8.3	odd	2		624.4.a.h.1.1		1
8.5	even	2		156.4.a.a.1.1	✓	1
24.5	odd	2		468.4.a.b.1.1		1
24.11	even	2		1872.4.a.j.1.1		1
104.5	odd	4		2028.4.b.a.337.2		2
104.21	odd	4		2028.4.b.a.337.1		2
104.77	even	2		2028.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.4.a.a.1.1	✓	1	8.5	even	2
468.4.a.b.1.1		1	24.5	odd	2
624.4.a.h.1.1		1	8.3	odd	2
1872.4.a.j.1.1		1	24.11	even	2
2028.4.a.a.1.1		1	104.77	even	2
2028.4.b.a.337.1		2	104.21	odd	4
2028.4.b.a.337.2		2	104.5	odd	4
2496.4.a.e.1.1		1	4.3	odd	2
2496.4.a.n.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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