LMFDB - Embedded newform 7245.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7245,2,Mod(1,7245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7245, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7245.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7245.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:	$57.8516162644$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2415)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +O(q^{10})$

$q-2.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -3.00000 q^{11} -4.00000 q^{13} +4.00000 q^{16} +6.00000 q^{17} -1.00000 q^{19} -2.00000 q^{20} +1.00000 q^{23} +1.00000 q^{25} -2.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{35} -10.0000 q^{37} +3.00000 q^{41} -10.0000 q^{43} +6.00000 q^{44} +3.00000 q^{47} +1.00000 q^{49} +8.00000 q^{52} +9.00000 q^{53} -3.00000 q^{55} +3.00000 q^{59} +11.0000 q^{61} -8.00000 q^{64} -4.00000 q^{65} -10.0000 q^{67} -12.0000 q^{68} +2.00000 q^{73} +2.00000 q^{76} -3.00000 q^{77} -10.0000 q^{79} +4.00000 q^{80} +6.00000 q^{85} -4.00000 q^{91} -2.00000 q^{92} -1.00000 q^{95} +2.00000 q^{97} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	8.00000	1.10940
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−12.0000	−1.45521
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	2.00000	0.229416
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	−2.00000	−0.208514
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	−2.00000	−0.200000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	12.0000	1.11417
$117$	0	0
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	−16.0000	−1.43684
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	0	0
$143$	12.0000	1.00349
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	20.0000	1.64399
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	0	0
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	20.0000	1.52499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−12.0000	−0.904534
$177$	0	0
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.0000	−0.735215
$186$	0	0
$187$	−18.0000	−1.31629
$188$	−6.00000	−0.437595
$189$	0	0
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	−25.0000	−1.79954	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−25.0000	−1.79954	−0.899770	+	0.436365i	$0.856266\pi$
$194$	0	0
$195$	0	0
$196$	−2.00000	−0.142857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	3.00000	0.209529
$206$	0	0
$207$	0	0
$208$	−16.0000	−1.10940
$209$	3.00000	0.207514
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−18.0000	−1.23625
$213$	0	0
$214$	0	0
$215$	−10.0000	−0.681994
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	0	0
$220$	6.00000	0.404520
$221$	−24.0000	−1.61441
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	−6.00000	−0.390567
$237$	0	0
$238$	0	0
$239$	30.0000	1.94054	0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000	1.94054	0.970269	+	0.242028i	$0.0778125\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	0	0
$244$	−22.0000	−1.40841
$245$	1.00000	0.0638877
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	8.00000	0.496139
$261$	0	0
$262$	0	0
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	0	0
$267$	0	0
$268$	20.0000	1.22169
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	26.0000	1.57939	0.789694	−	0.613501i	$-0.210239\pi$
$271$	26.0000	1.57939	0.789694	+	0.613501i	$0.210239\pi$
$272$	24.0000	1.45521
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	32.0000	1.90220	0.951101	−	0.308879i	$-0.0999539\pi$
$283$	32.0000	1.90220	0.951101	+	0.308879i	$0.0999539\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.00000	0.177084
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	−4.00000	−0.234082
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−10.0000	−0.576390
$302$	0	0
$303$	0	0
$304$	−4.00000	−0.229416
$305$	11.0000	0.629858
$306$	0	0
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	6.00000	0.341882
$309$	0	0
$310$	0	0
$311$	−33.0000	−1.87126	−0.935629	−	0.352985i	$-0.885167\pi$
$311$	−33.0000	−1.87126	−0.935629	+	0.352985i	$0.885167\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	0	0
$315$	0	0
$316$	20.0000	1.12509
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	−8.00000	−0.447214
$321$	0	0
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	0	0
$340$	−12.0000	−0.650791
$341$	−24.0000	−1.29967
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	0	0
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	8.00000	0.419314
$365$	2.00000	0.104685
$366$	0	0
$367$	−19.0000	−0.991792	−0.495896	−	0.868382i	$-0.665160\pi$
$367$	−19.0000	−0.991792	−0.495896	+	0.868382i	$0.665160\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	0	0
$388$	−4.00000	−0.203069
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.0000	−0.503155
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	0	0
$403$	−32.0000	−1.59403
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	30.0000	1.48704
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	0	0
$412$	2.00000	0.0985329
$413$	3.00000	0.147620
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	11.0000	0.532327
$428$	−24.0000	−1.16008
$429$	0	0
$430$	0	0
$431$	−27.0000	−1.30054	−0.650272	−	0.759701i	$-0.725345\pi$
$431$	−27.0000	−1.30054	−0.650272	+	0.759701i	$0.725345\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	0	0
$436$	20.0000	0.957826
$437$	−1.00000	−0.0478365
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−8.00000	−0.377964
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	−12.0000	−0.564433
$453$	0	0
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	0	0
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	−25.0000	−1.16185	−0.580924	−	0.813958i	$-0.697309\pi$
$463$	−25.0000	−1.16185	−0.580924	+	0.813958i	$0.697309\pi$
$464$	−24.0000	−1.11417
$465$	0	0
$466$	0	0
$467$	−30.0000	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$468$	0	0
$469$	−10.0000	−0.461757
$470$	0	0
$471$	0	0
$472$	0	0
$473$	30.0000	1.37940
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−12.0000	−0.550019
$477$	0	0
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	0	0
$483$	0	0
$484$	4.00000	0.181818
$485$	2.00000	0.0908153
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	0	0
$495$	0	0
$496$	32.0000	1.43684
$497$	0	0
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	−2.00000	−0.0894427
$501$	0	0
$502$	0	0
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	0	0
$505$	3.00000	0.133498
$506$	0	0
$507$	0	0
$508$	26.0000	1.15356
$509$	27.0000	1.19675	0.598377	−	0.801215i	$-0.295813\pi$
$509$	27.0000	1.19675	0.598377	+	0.801215i	$0.295813\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.00000	−0.0440653
$516$	0	0
$517$	−9.00000	−0.395820
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	−31.0000	−1.35554	−0.677768	−	0.735276i	$-0.737052\pi$
$523$	−31.0000	−1.35554	−0.677768	+	0.735276i	$0.737052\pi$
$524$	30.0000	1.31056
$525$	0	0
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	2.00000	0.0867110
$533$	−12.0000	−0.519778
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	0	0
$556$	32.0000	1.35710
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	4.00000	0.169031
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	−24.0000	−1.00349
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	0	0
$579$	0	0
$580$	12.0000	0.498273
$581$	0	0
$582$	0	0
$583$	−27.0000	−1.11823
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.0000	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$587$	27.0000	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	0	0
$592$	−40.0000	−1.64399
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	38.0000	1.54620
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−16.0000	−0.642575
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	−34.0000	−1.35675
$629$	−60.0000	−2.39236
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.0000	−0.515889
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	0	0
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	0	0
$652$	−22.0000	−0.861586
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	0	0
$655$	−15.0000	−0.586098
$656$	12.0000	0.468521
$657$	0	0
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.00000	−0.0387783
$666$	0	0
$667$	−6.00000	−0.232321
$668$	−30.0000	−1.16073
$669$	0	0
$670$	0	0
$671$	−33.0000	−1.27395
$672$	0	0
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	0	0
$675$	0	0
$676$	−6.00000	−0.230769
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	−3.00000	−0.114624
$686$	0	0
$687$	0	0
$688$	−40.0000	−1.52499
$689$	−36.0000	−1.37149
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	18.0000	0.681799
$698$	0	0
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	51.0000	1.92624	0.963122	−	0.269066i	$-0.0867150\pi$
$701$	51.0000	1.92624	0.963122	+	0.269066i	$0.0867150\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	24.0000	0.904534
$705$	0	0
$706$	0	0
$707$	3.00000	0.112827
$708$	0	0
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	12.0000	0.448775
$716$	24.0000	0.896922
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−1.00000	−0.0372419
$722$	0	0
$723$	0	0
$724$	20.0000	0.743294
$725$	−6.00000	−0.222834
$726$	0	0
$727$	47.0000	1.74313	0.871567	−	0.490277i	$-0.163104\pi$
$727$	47.0000	1.74313	0.871567	+	0.490277i	$0.163104\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−60.0000	−2.21918
$732$	0	0
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30.0000	1.10506
$738$	0	0
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	20.0000	0.735215
$741$	0	0
$742$	0	0
$743$	−33.0000	−1.21065	−0.605326	−	0.795977i	$-0.706957\pi$
$743$	−33.0000	−1.21065	−0.605326	+	0.795977i	$0.706957\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	0	0
$747$	0	0
$748$	36.0000	1.31629
$749$	12.0000	0.438470
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	0	0
$755$	−19.0000	−0.691481
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	−30.0000	−1.08536
$765$	0	0
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	0	0
$772$	50.0000	1.79954
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.00000	−0.107486
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	4.00000	0.142857
$785$	17.0000	0.606756
$786$	0	0
$787$	−25.0000	−0.891154	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000	−0.891154	−0.445577	+	0.895244i	$0.647001\pi$
$788$	24.0000	0.854965
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	−44.0000	−1.56249
$794$	0	0
$795$	0	0
$796$	26.0000	0.921546
$797$	24.0000	0.850124	0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000	0.850124	0.425062	+	0.905164i	$0.360252\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	1.00000	0.0352454
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	12.0000	0.421117
$813$	0	0
$814$	0	0
$815$	11.0000	0.385313
$816$	0	0
$817$	10.0000	0.349856
$818$	0	0
$819$	0	0
$820$	−6.00000	−0.209529
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−7.00000	−0.244005	−0.122002	−	0.992530i	$-0.538932\pi$
$823$	−7.00000	−0.244005	−0.122002	+	0.992530i	$0.538932\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	0	0
$832$	32.0000	1.10940
$833$	6.00000	0.207888
$834$	0	0
$835$	15.0000	0.519096
$836$	−6.00000	−0.207514
$837$	0	0
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	−10.0000	−0.344214
$845$	3.00000	0.103203
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	36.0000	1.23625
$849$	0	0
$850$	0	0
$851$	−10.0000	−0.342796
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	20.0000	0.681994
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	0	0
$868$	−16.0000	−0.543075
$869$	30.0000	1.01768
$870$	0	0
$871$	40.0000	1.35535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	5.00000	0.168838	0.0844190	−	0.996430i	$-0.473097\pi$
$877$	5.00000	0.168838	0.0844190	+	0.996430i	$0.473097\pi$
$878$	0	0
$879$	0	0
$880$	−12.0000	−0.404520
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	0	0
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	48.0000	1.61441
$885$	0	0
$886$	0	0
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	0	0
$891$	0	0
$892$	32.0000	1.07144
$893$	−3.00000	−0.100391
$894$	0	0
$895$	−12.0000	−0.401116
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−48.0000	−1.60089
$900$	0	0
$901$	54.0000	1.79900
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	48.0000	1.59294
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	26.0000	0.859064
$917$	−15.0000	−0.495344
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−10.0000	−0.328798
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	12.0000	0.393073
$933$	0	0
$934$	0	0
$935$	−18.0000	−0.588663
$936$	0	0
$937$	23.0000	0.751377	0.375689	−	0.926746i	$-0.377406\pi$
$937$	23.0000	0.751377	0.375689	+	0.926746i	$0.377406\pi$
$938$	0	0
$939$	0	0
$940$	−6.00000	−0.195698
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	0	0
$943$	3.00000	0.0976934
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.0000	0.680257	0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000	0.680257	0.340128	+	0.940379i	$0.389529\pi$
$954$	0	0
$955$	15.0000	0.485389
$956$	−60.0000	−1.94054
$957$	0	0
$958$	0	0
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	−34.0000	−1.09507
$965$	−25.0000	−0.804778
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	0	0
$976$	44.0000	1.40841
$977$	21.0000	0.671850	0.335925	−	0.941889i	$-0.390951\pi$
$977$	21.0000	0.671850	0.335925	+	0.941889i	$0.390951\pi$
$978$	0	0
$979$	0	0
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	0	0
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−10.0000	−0.317982
$990$	0	0
$991$	41.0000	1.30241	0.651204	−	0.758903i	$-0.274264\pi$
$991$	41.0000	1.30241	0.651204	+	0.758903i	$0.274264\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−13.0000	−0.412128
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7245.2.a.j.1.1		1
3.2	odd	2		2415.2.a.g.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.g.1.1	✓	1	3.2	odd	2
7245.2.a.j.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:	\( N \)	\(=\)	\( 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7245.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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