LMFDB - Embedded newform 1350.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(1,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.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:	$10.7798042729$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{19} -6.00000 q^{23} +2.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +7.00000 q^{37} +1.00000 q^{38} +12.0000 q^{41} -11.0000 q^{43} +6.00000 q^{46} -12.0000 q^{47} -6.00000 q^{49} -2.00000 q^{52} -1.00000 q^{56} +6.00000 q^{58} -6.00000 q^{59} -7.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +4.00000 q^{67} -6.00000 q^{68} +6.00000 q^{71} +7.00000 q^{73} -7.00000 q^{74} -1.00000 q^{76} -1.00000 q^{79} -12.0000 q^{82} +6.00000 q^{83} +11.0000 q^{86} -6.00000 q^{89} -2.00000 q^{91} -6.00000 q^{92} +12.0000 q^{94} -5.00000 q^{97} +6.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	−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
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.277350
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	6.00000	0.787839
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	−5.00000	−0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	0	0
$82$	−12.0000	−1.32518
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	0	0
$86$	11.0000	1.18616
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−6.00000	−0.625543
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−5.00000	−0.492665	−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000	−0.492665	−0.246332	+	0.969185i	$0.579225\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−13.0000	−1.24517	−0.622587	−	0.782551i	$-0.713918\pi$
$109$	−13.0000	−1.24517	−0.622587	+	0.782551i	$0.713918\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	6.00000	0.552345
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	7.00000	0.633750
$123$	0	0
$124$	5.00000	0.449013
$125$	0	0
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−7.00000	−0.579324
$147$	0	0
$148$	7.00000	0.575396
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	1.00000	0.0795557
$159$	0	0
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	−6.00000	−0.465690
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−11.0000	−0.838742
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	6.00000	0.442326
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−12.0000	−0.875190
$189$	0	0
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	1.00000	0.0719816	0.0359908	−	0.999352i	$-0.488541\pi$
$193$	1.00000	0.0719816	0.0359908	+	0.999352i	$0.488541\pi$
$194$	5.00000	0.358979
$195$	0	0
$196$	−6.00000	−0.428571
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	12.0000	0.844317
$203$	−6.00000	−0.421117
$204$	0	0
$205$	0	0
$206$	5.00000	0.348367
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	0	0
$217$	5.00000	0.339422
$218$	13.0000	0.880471
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	12.0000	0.798228
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	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
$232$	6.00000	0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	6.00000	0.388922
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	−7.00000	−0.448129
$245$	0	0
$246$	0	0
$247$	2.00000	0.127257
$248$	−5.00000	−0.317500
$249$	0	0
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	0	0
$261$	0	0
$262$	−18.0000	−1.11204
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	0	0
$266$	1.00000	0.0613139
$267$	0	0
$268$	4.00000	0.244339
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	13.0000	0.779688
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	7.00000	0.409644
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	−7.00000	−0.406867
$297$	0	0
$298$	−18.0000	−1.04271
$299$	12.0000	0.693978
$300$	0	0
$301$	−11.0000	−0.634029
$302$	7.00000	0.402805
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	19.0000	1.08439	0.542194	−	0.840254i	$-0.317594\pi$
$307$	19.0000	1.08439	0.542194	+	0.840254i	$0.317594\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	17.0000	0.959366
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	6.00000	0.334367
$323$	6.00000	0.333849
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−12.0000	−0.662589
$329$	−12.0000	−0.661581
$330$	0	0
$331$	35.0000	1.92377	0.961887	−	0.273447i	$-0.0881639\pi$
$331$	35.0000	1.92377	0.961887	+	0.273447i	$0.0881639\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−18.0000	−0.984916
$335$	0	0
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−13.0000	−0.701934
$344$	11.0000	0.593080
$345$	0	0
$346$	−6.00000	−0.322562
$347$	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
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	22.0000	1.15629
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−29.0000	−1.50156	−0.750782	−	0.660551i	$-0.770323\pi$
$373$	−29.0000	−1.50156	−0.750782	+	0.660551i	$0.770323\pi$
$374$	0	0
$375$	0	0
$376$	12.0000	0.618853
$377$	12.0000	0.618031
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	0	0
$382$	24.0000	1.22795
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	0	0
$386$	−1.00000	−0.0508987
$387$	0	0
$388$	−5.00000	−0.253837
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	6.00000	0.303046
$393$	0	0
$394$	−12.0000	−0.604551
$395$	0	0
$396$	0	0
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	−12.0000	−0.597022
$405$	0	0
$406$	6.00000	0.297775
$407$	0	0
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	−5.00000	−0.246332
$413$	−6.00000	−0.295241
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	0	0
$418$	0	0
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−7.00000	−0.338754
$428$	−6.00000	−0.290021
$429$	0	0
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	13.0000	0.624740	0.312370	−	0.949960i	$-0.398877\pi$
$433$	13.0000	0.624740	0.312370	+	0.949960i	$0.398877\pi$
$434$	−5.00000	−0.240008
$435$	0	0
$436$	−13.0000	−0.622587
$437$	6.00000	0.287019
$438$	0	0
$439$	5.00000	0.238637	0.119318	−	0.992856i	$-0.461929\pi$
$439$	5.00000	0.238637	0.119318	+	0.992856i	$0.461929\pi$
$440$	0	0
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−30.0000	−1.42534	−0.712672	−	0.701498i	$-0.752515\pi$
$443$	−30.0000	−1.42534	−0.712672	+	0.701498i	$0.752515\pi$
$444$	0	0
$445$	0	0
$446$	17.0000	0.804973
$447$	0	0
$448$	1.00000	0.0472456
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	−12.0000	−0.564433
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	13.0000	0.607450
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	31.0000	1.44069	0.720346	−	0.693615i	$-0.243983\pi$
$463$	31.0000	1.44069	0.720346	+	0.693615i	$0.243983\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	6.00000	0.276172
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−6.00000	−0.275010
$477$	0	0
$478$	−24.0000	−1.09773
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	7.00000	0.316875
$489$	0	0
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	5.00000	0.224507
$497$	6.00000	0.269137
$498$	0	0
$499$	−7.00000	−0.313363	−0.156682	−	0.987649i	$-0.550080\pi$
$499$	−7.00000	−0.313363	−0.156682	+	0.987649i	$0.550080\pi$
$500$	0	0
$501$	0	0
$502$	30.0000	1.33897
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	16.0000	0.709885
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−24.0000	−1.05859
$515$	0	0
$516$	0	0
$517$	0	0
$518$	−7.00000	−0.307562
$519$	0	0
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	−23.0000	−1.00572	−0.502860	−	0.864368i	$-0.667719\pi$
$523$	−23.0000	−1.00572	−0.502860	+	0.864368i	$0.667719\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	−18.0000	−0.784837
$527$	−30.0000	−1.30682
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	−24.0000	−1.03956
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−6.00000	−0.258678
$539$	0	0
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	−11.0000	−0.472490
$543$	0	0
$544$	6.00000	0.257248
$545$	0	0
$546$	0	0
$547$	−35.0000	−1.49649	−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000	−1.49649	−0.748246	+	0.663421i	$0.769104\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−1.00000	−0.0425243
$554$	−1.00000	−0.0424859
$555$	0	0
$556$	−13.0000	−0.551323
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	22.0000	0.930501
$560$	0	0
$561$	0	0
$562$	12.0000	0.506189
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	5.00000	0.210166
$567$	0	0
$568$	−6.00000	−0.251754
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	0	0
$573$	0	0
$574$	−12.0000	−0.500870
$575$	0	0
$576$	0	0
$577$	−23.0000	−0.957503	−0.478751	−	0.877951i	$-0.658910\pi$
$577$	−23.0000	−0.957503	−0.478751	+	0.877951i	$0.658910\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	0	0
$584$	−7.00000	−0.289662
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	0	0
$589$	−5.00000	−0.206021
$590$	0	0
$591$	0	0
$592$	7.00000	0.287698
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	0	0
$598$	−12.0000	−0.490716
$599$	−48.0000	−1.96123	−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000	−1.96123	−0.980613	+	0.195952i	$0.937220\pi$
$600$	0	0
$601$	11.0000	0.448699	0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000	0.448699	0.224350	+	0.974509i	$0.427974\pi$
$602$	11.0000	0.448327
$603$	0	0
$604$	−7.00000	−0.284826
$605$	0	0
$606$	0	0
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	0	0
$613$	43.0000	1.73675	0.868377	−	0.495905i	$-0.165164\pi$
$613$	43.0000	1.73675	0.868377	+	0.495905i	$0.165164\pi$
$614$	−19.0000	−0.766778
$615$	0	0
$616$	0	0
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	0	0
$619$	11.0000	0.442127	0.221064	−	0.975259i	$-0.429047\pi$
$619$	11.0000	0.442127	0.221064	+	0.975259i	$0.429047\pi$
$620$	0	0
$621$	0	0
$622$	−12.0000	−0.481156
$623$	−6.00000	−0.240385
$624$	0	0
$625$	0	0
$626$	−34.0000	−1.35891
$627$	0	0
$628$	−17.0000	−0.678374
$629$	−42.0000	−1.67465
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	1.00000	0.0397779
$633$	0	0
$634$	−6.00000	−0.238290
$635$	0	0
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	−6.00000	−0.236067
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	0	0
$655$	0	0
$656$	12.0000	0.468521
$657$	0	0
$658$	12.0000	0.467809
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	−31.0000	−1.20576	−0.602880	−	0.797832i	$-0.705980\pi$
$661$	−31.0000	−1.20576	−0.602880	+	0.797832i	$0.705980\pi$
$662$	−35.0000	−1.36031
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	0	0
$667$	36.0000	1.39393
$668$	18.0000	0.696441
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	37.0000	1.42625	0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000	1.42625	0.713123	+	0.701039i	$0.247280\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−24.0000	−0.922395	−0.461197	−	0.887298i	$-0.652580\pi$
$677$	−24.0000	−0.922395	−0.461197	+	0.887298i	$0.652580\pi$
$678$	0	0
$679$	−5.00000	−0.191882
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	−11.0000	−0.419371
$689$	0	0
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−6.00000	−0.227757
$695$	0	0
$696$	0	0
$697$	−72.0000	−2.72719
$698$	19.0000	0.719161
$699$	0	0
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	−7.00000	−0.264010
$704$	0	0
$705$	0	0
$706$	−24.0000	−0.903252
$707$	−12.0000	−0.451306
$708$	0	0
$709$	−49.0000	−1.84023	−0.920117	−	0.391644i	$-0.871906\pi$
$709$	−49.0000	−1.84023	−0.920117	+	0.391644i	$0.871906\pi$
$710$	0	0
$711$	0	0
$712$	6.00000	0.224860
$713$	−30.0000	−1.12351
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	6.00000	0.223918
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−5.00000	−0.186210
$722$	18.0000	0.669891
$723$	0	0
$724$	−22.0000	−0.817624
$725$	0	0
$726$	0	0
$727$	−5.00000	−0.185440	−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	−5.00000	−0.185440	−0.0927199	+	0.995692i	$0.529556\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	0	0
$731$	66.0000	2.44110
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	0	0
$746$	29.0000	1.06177
$747$	0	0
$748$	0	0
$749$	−6.00000	−0.219235
$750$	0	0
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	−32.0000	−1.16229
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	−13.0000	−0.470632
$764$	−24.0000	−0.868290
$765$	0	0
$766$	6.00000	0.216789
$767$	12.0000	0.433295
$768$	0	0
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	0	0
$772$	1.00000	0.0359908
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	0	0
$776$	5.00000	0.179490
$777$	0	0
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	−36.0000	−1.28736
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	0	0
$787$	37.0000	1.31891	0.659454	−	0.751745i	$-0.270788\pi$
$787$	37.0000	1.31891	0.659454	+	0.751745i	$0.270788\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	14.0000	0.497155
$794$	−7.00000	−0.248421
$795$	0	0
$796$	8.00000	0.283552
$797$	−36.0000	−1.27519	−0.637593	−	0.770374i	$-0.720070\pi$
$797$	−36.0000	−1.27519	−0.637593	+	0.770374i	$0.720070\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	0	0
$801$	0	0
$802$	24.0000	0.847469
$803$	0	0
$804$	0	0
$805$	0	0
$806$	10.0000	0.352235
$807$	0	0
$808$	12.0000	0.422159
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	29.0000	1.01833	0.509164	−	0.860670i	$-0.329955\pi$
$811$	29.0000	1.01833	0.509164	+	0.860670i	$0.329955\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11.0000	0.384841
$818$	−26.0000	−0.909069
$819$	0	0
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	5.00000	0.174183
$825$	0	0
$826$	6.00000	0.208767
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	36.0000	1.24733
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−18.0000	−0.621800
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	1.00000	0.0344623
$843$	0	0
$844$	20.0000	0.688428
$845$	0	0
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−42.0000	−1.43974
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	7.00000	0.239535
$855$	0	0
$856$	6.00000	0.205076
$857$	36.0000	1.22974	0.614868	−	0.788630i	$-0.289209\pi$
$857$	36.0000	1.22974	0.614868	+	0.788630i	$0.289209\pi$
$858$	0	0
$859$	−13.0000	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	−13.0000	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$860$	0	0
$861$	0	0
$862$	−18.0000	−0.613082
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	0	0
$865$	0	0
$866$	−13.0000	−0.441758
$867$	0	0
$868$	5.00000	0.169711
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	13.0000	0.440236
$873$	0	0
$874$	−6.00000	−0.202953
$875$	0	0
$876$	0	0
$877$	31.0000	1.04680	0.523398	−	0.852088i	$-0.324664\pi$
$877$	31.0000	1.04680	0.523398	+	0.852088i	$0.324664\pi$
$878$	−5.00000	−0.168742
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	19.0000	0.639401	0.319700	−	0.947519i	$-0.396418\pi$
$883$	19.0000	0.639401	0.319700	+	0.947519i	$0.396418\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	30.0000	1.00787
$887$	−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
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	−17.0000	−0.569202
$893$	12.0000	0.401565
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−30.0000	−1.00056
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	12.0000	0.399114
$905$	0	0
$906$	0	0
$907$	−17.0000	−0.564476	−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000	−0.564476	−0.282238	+	0.959344i	$0.591077\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	0	0
$914$	26.0000	0.860004
$915$	0	0
$916$	−13.0000	−0.429532
$917$	18.0000	0.594412
$918$	0	0
$919$	−37.0000	−1.22052	−0.610259	−	0.792202i	$-0.708935\pi$
$919$	−37.0000	−1.22052	−0.610259	+	0.792202i	$0.708935\pi$
$920$	0	0
$921$	0	0
$922$	−6.00000	−0.197599
$923$	−12.0000	−0.394985
$924$	0	0
$925$	0	0
$926$	−31.0000	−1.01872
$927$	0	0
$928$	6.00000	0.196960
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	6.00000	0.196537
$933$	0	0
$934$	6.00000	0.196326
$935$	0	0
$936$	0	0
$937$	25.0000	0.816714	0.408357	−	0.912822i	$-0.366102\pi$
$937$	25.0000	0.816714	0.408357	+	0.912822i	$0.366102\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	0	0
$941$	−36.0000	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$942$	0	0
$943$	−72.0000	−2.34464
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	0	0
$949$	−14.0000	−0.454459
$950$	0	0
$951$	0	0
$952$	6.00000	0.194461
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	18.0000	0.581554
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	14.0000	0.451378
$963$	0	0
$964$	2.00000	0.0644157
$965$	0	0
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	−13.0000	−0.416761
$974$	8.00000	0.256337
$975$	0	0
$976$	−7.00000	−0.224065
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	24.0000	0.765871
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	0	0
$986$	−36.0000	−1.14647
$987$	0	0
$988$	2.00000	0.0636285
$989$	66.0000	2.09868
$990$	0	0
$991$	23.0000	0.730619	0.365310	−	0.930886i	$-0.380963\pi$
$991$	23.0000	0.730619	0.365310	+	0.930886i	$0.380963\pi$
$992$	−5.00000	−0.158750
$993$	0	0
$994$	−6.00000	−0.190308
$995$	0	0
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	7.00000	0.221581
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.a.g.1.1	yes	1
3.2	odd	2		1350.2.a.s.1.1	yes	1
5.2	odd	4		1350.2.c.g.649.1		2
5.3	odd	4		1350.2.c.g.649.2		2
5.4	even	2		1350.2.a.q.1.1	yes	1
15.2	even	4		1350.2.c.f.649.2		2
15.8	even	4		1350.2.c.f.649.1		2
15.14	odd	2		1350.2.a.f.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.2.a.f.1.1	✓	1	15.14	odd	2
1350.2.a.g.1.1	yes	1	1.1	even	1	trivial
1350.2.a.q.1.1	yes	1	5.4	even	2
1350.2.a.s.1.1	yes	1	3.2	odd	2
1350.2.c.f.649.1		2	15.8	even	4
1350.2.c.f.649.2		2	15.2	even	4
1350.2.c.g.649.1		2	5.2	odd	4
1350.2.c.g.649.2		2	5.3	odd	4

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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