LMFDB - Embedded newform 1350.2.c.g.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(649,1350)]

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, 1]))

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

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.2.c.g.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000 q^{19} +6.00000i q^{23} +2.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +7.00000i q^{37} -1.00000i q^{38} +12.0000 q^{41} +11.0000i q^{43} +6.00000 q^{46} -12.0000i q^{47} +6.00000 q^{49} -2.00000i q^{52} -1.00000 q^{56} -6.00000i q^{58} +6.00000 q^{59} -7.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +4.00000i q^{67} +6.00000i q^{68} +6.00000 q^{71} -7.00000i q^{73} +7.00000 q^{74} -1.00000 q^{76} +1.00000 q^{79} -12.0000i q^{82} -6.00000i q^{83} +11.0000 q^{86} +6.00000 q^{89} -2.00000 q^{91} -6.00000i q^{92} -12.0000 q^{94} -5.00000i q^{97} -6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 2 q^{14} + 2 q^{16} + 2 q^{19} + 4 q^{26} + 12 q^{29} + 10 q^{31} - 12 q^{34} + 24 q^{41} + 12 q^{46} + 12 q^{49} - 2 q^{56} + 12 q^{59} - 14 q^{61} - 2 q^{64} + 12 q^{71} + 14 q^{74} - 2 q^{76} + 2 q^{79} + 22 q^{86} + 12 q^{89} - 4 q^{91} - 24 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^14 + 2 * q^16 + 2 * q^19 + 4 * q^26 + 12 * q^29 + 10 * q^31 - 12 * q^34 + 24 * q^41 + 12 * q^46 + 12 * q^49 - 2 * q^56 + 12 * q^59 - 14 * q^61 - 2 * q^64 + 12 * q^71 + 14 * q^74 - 2 * q^76 + 2 * q^79 + 22 * q^86 + 12 * q^89 - 4 * q^91 - 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	− 1.00000i	− 0.162221i
$39$	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
$43$	11.0000i	1.67748i	0.544529	+	0.838742i	$0.316708\pi$
$43$	11.0000i	1.67748i	−0.544529	+	0.838742i	$0.683292\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	− 5.00000i	− 0.635001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	6.00000i	0.727607i
$69$	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
$73$	− 7.00000i	− 0.819288i	−0.912245	−	0.409644i	$-0.865653\pi$
$73$	− 7.00000i	− 0.819288i	0.912245	−	0.409644i	$-0.134347\pi$
$74$	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	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	0	0
$82$	− 12.0000i	− 1.32518i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	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	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 6.00000i	− 0.625543i
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	0	0
$97$	− 5.00000i	− 0.507673i	−0.967247	−	0.253837i	$-0.918307\pi$
$97$	− 5.00000i	− 0.507673i	0.967247	−	0.253837i	$-0.0816925\pi$
$98$	− 6.00000i	− 0.606092i
$99$	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
$103$	5.00000i	0.492665i	0.969185	+	0.246332i	$0.0792255\pi$
$103$	5.00000i	0.492665i	−0.969185	+	0.246332i	$0.920775\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	13.0000	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	− 6.00000i	− 0.552345i
$119$	6.00000	0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	7.00000i	0.633750i
$123$	0	0
$124$	−5.00000	−0.449013
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	1.00000i	0.0883883i
$129$	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
$133$	1.00000i	0.0867110i
$134$	4.00000	0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	0	0
$142$	− 6.00000i	− 0.503509i
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−7.00000	−0.579324
$147$	0	0
$148$	− 7.00000i	− 0.575396i
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	1.00000i	0.0811107i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 17.0000i	− 1.35675i	−0.734717	−	0.678374i	$-0.762685\pi$
$157$	− 17.0000i	− 1.35675i	0.734717	−	0.678374i	$-0.237315\pi$
$158$	− 1.00000i	− 0.0795557i
$159$	0	0
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	−6.00000	−0.465690
$167$	18.0000i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$	18.0000i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	− 11.0000i	− 0.838742i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	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	$0.804709\pi$
$182$	2.00000i	0.148250i
$183$	0	0
$184$	−6.00000	−0.442326
$185$	0	0
$186$	0	0
$187$	0	0
$188$	12.0000i	0.875190i
$189$	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
$193$	− 1.00000i	− 0.0719816i	−0.999352	−	0.0359908i	$-0.988541\pi$
$193$	− 1.00000i	− 0.0719816i	0.999352	−	0.0359908i	$-0.0114587\pi$
$194$	−5.00000	−0.358979
$195$	0	0
$196$	−6.00000	−0.428571
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	12.0000i	0.844317i
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	5.00000	0.348367
$207$	0	0
$208$	2.00000i	0.138675i
$209$	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
$217$	5.00000i	0.339422i
$218$	− 13.0000i	− 0.880471i
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	17.0000i	1.13840i	0.822198	+	0.569202i	$0.192748\pi$
$223$	17.0000i	1.13840i	−0.822198	+	0.569202i	$0.807252\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	12.0000	0.798228
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	− 6.00000i	− 0.388922i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	7.00000	0.448129
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	5.00000i	0.317500i
$249$	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
$257$	24.0000i	1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$	24.0000i	1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	0	0
$261$	0	0
$262$	− 18.0000i	− 1.11204i
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	0	0
$265$	0	0
$266$	1.00000	0.0613139
$267$	0	0
$268$	− 4.00000i	− 0.244339i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	− 6.00000i	− 0.363803i
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	0	0
$277$	1.00000i	0.0600842i	0.999549	+	0.0300421i	$0.00956413\pi$
$277$	1.00000i	0.0600842i	−0.999549	+	0.0300421i	$0.990436\pi$
$278$	− 13.0000i	− 0.779688i
$279$	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
$283$	5.00000i	0.297219i	0.988896	+	0.148610i	$0.0474798\pi$
$283$	5.00000i	0.297219i	−0.988896	+	0.148610i	$0.952520\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	7.00000i	0.409644i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	−7.00000	−0.406867
$297$	0	0
$298$	18.0000i	1.04271i
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−11.0000	−0.634029
$302$	7.00000i	0.402805i
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	19.0000i	1.08439i	0.840254	+	0.542194i	$0.182406\pi$
$307$	19.0000i	1.08439i	−0.840254	+	0.542194i	$0.817594\pi$
$308$	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
$313$	− 34.0000i	− 1.92179i	−0.276907	−	0.960897i	$-0.589309\pi$
$313$	− 34.0000i	− 1.92179i	0.276907	−	0.960897i	$-0.410691\pi$
$314$	−17.0000	−0.959366
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	6.00000i	0.334367i
$323$	− 6.00000i	− 0.333849i
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	12.0000i	0.662589i
$329$	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	$0.0881639\pi$
$332$	6.00000i	0.329293i
$333$	0	0
$334$	18.0000	0.984916
$335$	0	0
$336$	0	0
$337$	− 26.0000i	− 1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$	− 26.0000i	− 1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$	− 9.00000i	− 0.489535i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	13.0000i	0.701934i
$344$	−11.0000	−0.593080
$345$	0	0
$346$	−6.00000	−0.322562
$347$	6.00000i	0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	6.00000i	0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 24.0000i	− 1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$	− 24.0000i	− 1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$	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	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	22.0000i	1.15629i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	28.0000i	1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$	28.0000i	1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$	6.00000i	0.312772i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.0000i	1.50156i	0.660551	+	0.750782i	$0.270323\pi$
$373$	29.0000i	1.50156i	−0.660551	+	0.750782i	$0.729677\pi$
$374$	0	0
$375$	0	0
$376$	12.0000	0.618853
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	0	0
$381$	0	0
$382$	24.0000i	1.22795i
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	0	0
$385$	0	0
$386$	−1.00000	−0.0508987
$387$	0	0
$388$	5.00000i	0.253837i
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	6.00000i	0.303046i
$393$	0	0
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	7.00000i	0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$	7.00000i	0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$	8.00000i	0.401004i
$399$	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
$403$	10.0000i	0.498135i
$404$	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	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	− 5.00000i	− 0.246332i
$413$	6.00000i	0.295241i
$414$	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	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	− 20.0000i	− 0.973585i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 7.00000i	− 0.338754i
$428$	6.00000i	0.290021i
$429$	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
$433$	− 13.0000i	− 0.624740i	−0.949960	−	0.312370i	$-0.898877\pi$
$433$	− 13.0000i	− 0.624740i	0.949960	−	0.312370i	$-0.101123\pi$
$434$	5.00000	0.240008
$435$	0	0
$436$	−13.0000	−0.622587
$437$	6.00000i	0.287019i
$438$	0	0
$439$	−5.00000	−0.238637	−0.119318	−	0.992856i	$-0.538071\pi$
$439$	−5.00000	−0.238637	−0.119318	+	0.992856i	$0.538071\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	30.0000i	1.42534i	0.701498	+	0.712672i	$0.252515\pi$
$443$	30.0000i	1.42534i	−0.701498	+	0.712672i	$0.747485\pi$
$444$	0	0
$445$	0	0
$446$	17.0000	0.804973
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	− 12.0000i	− 0.564433i
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	− 13.0000i	− 0.607450i
$459$	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
$463$	− 31.0000i	− 1.44069i	−0.693615	−	0.720346i	$-0.743983\pi$
$463$	− 31.0000i	− 1.44069i	0.693615	−	0.720346i	$-0.256017\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	6.00000i	0.276172i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−6.00000	−0.275010
$477$	0	0
$478$	24.0000i	1.09773i
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 7.00000i	− 0.316875i
$489$	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
$493$	− 36.0000i	− 1.62136i
$494$	2.00000	0.0899843
$495$	0	0
$496$	5.00000	0.224507
$497$	6.00000i	0.269137i
$498$	0	0
$499$	7.00000	0.313363	0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000	0.313363	0.156682	+	0.987649i	$0.449920\pi$
$500$	0	0
$501$	0	0
$502$	30.0000i	1.33897i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 16.0000i	− 0.709885i
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	24.0000	1.05859
$515$	0	0
$516$	0	0
$517$	0	0
$518$	7.00000i	0.307562i
$519$	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
$523$	23.0000i	1.00572i	0.864368	+	0.502860i	$0.167719\pi$
$523$	23.0000i	1.00572i	−0.864368	+	0.502860i	$0.832281\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	−18.0000	−0.784837
$527$	− 30.0000i	− 1.30682i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	− 1.00000i	− 0.0433555i
$533$	24.0000i	1.03956i
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	6.00000i	0.258678i
$539$	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	$0.680600\pi$
$542$	− 11.0000i	− 0.472490i
$543$	0	0
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	− 35.0000i	− 1.49649i	−0.663421	−	0.748246i	$-0.730896\pi$
$547$	− 35.0000i	− 1.49649i	0.663421	−	0.748246i	$-0.269104\pi$
$548$	12.0000i	0.512615i
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	1.00000i	0.0425243i
$554$	1.00000	0.0424859
$555$	0	0
$556$	−13.0000	−0.551323
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	0	0
$559$	−22.0000	−0.930501
$560$	0	0
$561$	0	0
$562$	12.0000i	0.506189i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	0	0
$566$	5.00000	0.210166
$567$	0	0
$568$	6.00000i	0.251754i
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	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
$577$	− 23.0000i	− 0.957503i	−0.877951	−	0.478751i	$-0.841090\pi$
$577$	− 23.0000i	− 0.957503i	0.877951	−	0.478751i	$-0.158910\pi$
$578$	19.0000i	0.790296i
$579$	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
$587$	− 6.00000i	− 0.247647i	−0.992304	−	0.123823i	$-0.960484\pi$
$587$	− 6.00000i	− 0.247647i	0.992304	−	0.123823i	$-0.0395156\pi$
$588$	0	0
$589$	5.00000	0.206021
$590$	0	0
$591$	0	0
$592$	7.00000i	0.287698i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	0	0
$598$	12.0000i	0.490716i
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	11.0000	0.448699	0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000	0.448699	0.224350	+	0.974509i	$0.427974\pi$
$602$	11.0000i	0.448327i
$603$	0	0
$604$	7.00000	0.284826
$605$	0	0
$606$	0	0
$607$	− 5.00000i	− 0.202944i	−0.994838	−	0.101472i	$-0.967645\pi$
$607$	− 5.00000i	− 0.202944i	0.994838	−	0.101472i	$-0.0323552\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	0	0
$613$	− 43.0000i	− 1.73675i	−0.495905	−	0.868377i	$-0.665164\pi$
$613$	− 43.0000i	− 1.73675i	0.495905	−	0.868377i	$-0.334836\pi$
$614$	19.0000	0.766778
$615$	0	0
$616$	0	0
$617$	− 36.0000i	− 1.44931i	−0.689114	−	0.724653i	$-0.742000\pi$
$617$	− 36.0000i	− 1.44931i	0.689114	−	0.724653i	$-0.258000\pi$
$618$	0	0
$619$	−11.0000	−0.442127	−0.221064	−	0.975259i	$-0.570953\pi$
$619$	−11.0000	−0.442127	−0.221064	+	0.975259i	$0.570953\pi$
$620$	0	0
$621$	0	0
$622$	− 12.0000i	− 0.481156i
$623$	6.00000i	0.240385i
$624$	0	0
$625$	0	0
$626$	−34.0000	−1.35891
$627$	0	0
$628$	17.0000i	0.678374i
$629$	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	$0.525370\pi$
$632$	1.00000i	0.0397779i
$633$	0	0
$634$	6.00000	0.238290
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	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
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	−6.00000	−0.236067
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	24.0000i	0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$	24.0000i	0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	0	0
$655$	0	0
$656$	12.0000	0.468521
$657$	0	0
$658$	− 12.0000i	− 0.467809i
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−31.0000	−1.20576	−0.602880	−	0.797832i	$-0.705980\pi$
$661$	−31.0000	−1.20576	−0.602880	+	0.797832i	$0.705980\pi$
$662$	− 35.0000i	− 1.36031i
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	0	0
$667$	36.0000i	1.39393i
$668$	− 18.0000i	− 0.696441i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 37.0000i	− 1.42625i	−0.701039	−	0.713123i	$-0.747280\pi$
$673$	− 37.0000i	− 1.42625i	0.701039	−	0.713123i	$-0.252720\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 24.0000i	− 0.922395i	−0.887298	−	0.461197i	$-0.847420\pi$
$677$	− 24.0000i	− 0.922395i	0.887298	−	0.461197i	$-0.152580\pi$
$678$	0	0
$679$	5.00000	0.191882
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.00000i	0.229584i	0.993390	+	0.114792i	$0.0366201\pi$
$683$	6.00000i	0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	11.0000i	0.419371i
$689$	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	$0.678779\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	6.00000	0.227757
$695$	0	0
$696$	0	0
$697$	− 72.0000i	− 2.72719i
$698$	− 19.0000i	− 0.719161i
$699$	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
$703$	7.00000i	0.264010i
$704$	0	0
$705$	0	0
$706$	−24.0000	−0.903252
$707$	− 12.0000i	− 0.451306i
$708$	0	0
$709$	49.0000	1.84023	0.920117	−	0.391644i	$-0.128094\pi$
$709$	49.0000	1.84023	0.920117	+	0.391644i	$0.128094\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	30.0000i	1.12351i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	− 6.00000i	− 0.223918i
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	−5.00000	−0.186210
$722$	18.0000i	0.669891i
$723$	0	0
$724$	22.0000	0.817624
$725$	0	0
$726$	0	0
$727$	− 5.00000i	− 0.185440i	−0.995692	−	0.0927199i	$-0.970444\pi$
$727$	− 5.00000i	− 0.185440i	0.995692	−	0.0927199i	$-0.0295561\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	0	0
$730$	0	0
$731$	66.0000	2.44110
$732$	0	0
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	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	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	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	$0.362156\pi$
$752$	− 12.0000i	− 0.437595i
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	− 23.0000i	− 0.835949i	−0.908459	−	0.417975i	$-0.862740\pi$
$757$	− 23.0000i	− 0.835949i	0.908459	−	0.417975i	$-0.137260\pi$
$758$	32.0000i	1.16229i
$759$	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
$763$	13.0000i	0.470632i
$764$	24.0000	0.868290
$765$	0	0
$766$	6.00000	0.216789
$767$	12.0000i	0.433295i
$768$	0	0
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	0	0
$772$	1.00000i	0.0359908i
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	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
$782$	− 36.0000i	− 1.28736i
$783$	0	0
$784$	6.00000	0.214286
$785$	0	0
$786$	0	0
$787$	37.0000i	1.31891i	0.751745	+	0.659454i	$0.229212\pi$
$787$	37.0000i	1.31891i	−0.751745	+	0.659454i	$0.770788\pi$
$788$	− 12.0000i	− 0.427482i
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	− 14.0000i	− 0.497155i
$794$	7.00000	0.248421
$795$	0	0
$796$	8.00000	0.283552
$797$	− 36.0000i	− 1.27519i	−0.770374	−	0.637593i	$-0.779930\pi$
$797$	− 36.0000i	− 1.27519i	0.770374	−	0.637593i	$-0.220070\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	0	0
$801$	0	0
$802$	24.0000i	0.847469i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	10.0000	0.352235
$807$	0	0
$808$	− 12.0000i	− 0.422159i
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	29.0000	1.01833	0.509164	−	0.860670i	$-0.329955\pi$
$811$	29.0000	1.01833	0.509164	+	0.860670i	$0.329955\pi$
$812$	− 6.00000i	− 0.210559i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11.0000i	0.384841i
$818$	26.0000i	0.909069i
$819$	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
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	−5.00000	−0.174183
$825$	0	0
$826$	6.00000	0.208767
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	0	0
$832$	− 2.00000i	− 0.0693375i
$833$	− 36.0000i	− 1.24733i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	18.0000i	0.621800i
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	1.00000i	0.0344623i
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	− 11.0000i	− 0.377964i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−42.0000	−1.43974
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	−7.00000	−0.239535
$855$	0	0
$856$	6.00000	0.205076
$857$	36.0000i	1.22974i	0.788630	+	0.614868i	$0.210791\pi$
$857$	36.0000i	1.22974i	−0.788630	+	0.614868i	$0.789209\pi$
$858$	0	0
$859$	13.0000	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$860$	0	0
$861$	0	0
$862$	− 18.0000i	− 0.613082i
$863$	54.0000i	1.83818i	0.394046	+	0.919091i	$0.371075\pi$
$863$	54.0000i	1.83818i	−0.394046	+	0.919091i	$0.628925\pi$
$864$	0	0
$865$	0	0
$866$	−13.0000	−0.441758
$867$	0	0
$868$	− 5.00000i	− 0.169711i
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	13.0000i	0.440236i
$873$	0	0
$874$	6.00000	0.202953
$875$	0	0
$876$	0	0
$877$	31.0000i	1.04680i	0.852088	+	0.523398i	$0.175336\pi$
$877$	31.0000i	1.04680i	−0.852088	+	0.523398i	$0.824664\pi$
$878$	5.00000i	0.168742i
$879$	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
$883$	− 19.0000i	− 0.639401i	−0.947519	−	0.319700i	$-0.896418\pi$
$883$	− 19.0000i	− 0.639401i	0.947519	−	0.319700i	$-0.103582\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	30.0000	1.00787
$887$	− 36.0000i	− 1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$	− 36.0000i	− 1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	− 17.0000i	− 0.569202i
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	30.0000i	1.00111i
$899$	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
$907$	− 17.0000i	− 0.564476i	−0.959344	−	0.282238i	$-0.908923\pi$
$907$	− 17.0000i	− 0.564476i	0.959344	−	0.282238i	$-0.0910767\pi$
$908$	− 12.0000i	− 0.398234i
$909$	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
$917$	18.0000i	0.594412i
$918$	0	0
$919$	37.0000	1.22052	0.610259	−	0.792202i	$-0.291065\pi$
$919$	37.0000	1.22052	0.610259	+	0.792202i	$0.291065\pi$
$920$	0	0
$921$	0	0
$922$	− 6.00000i	− 0.197599i
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	−31.0000	−1.01872
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	6.00000i	0.196537i
$933$	0	0
$934$	−6.00000	−0.196326
$935$	0	0
$936$	0	0
$937$	25.0000i	0.816714i	0.912822	+	0.408357i	$0.133898\pi$
$937$	25.0000i	0.816714i	−0.912822	+	0.408357i	$0.866102\pi$
$938$	4.00000i	0.130605i
$939$	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
$943$	72.0000i	2.34464i
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	14.0000	0.454459
$950$	0	0
$951$	0	0
$952$	6.00000i	0.194461i
$953$	24.0000i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$	24.0000i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	− 18.0000i	− 0.581554i
$959$	12.0000	0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	14.0000i	0.451378i
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	13.0000i	0.418052i	0.977910	+	0.209026i	$0.0670293\pi$
$967$	13.0000i	0.418052i	−0.977910	+	0.209026i	$0.932971\pi$
$968$	− 11.0000i	− 0.353553i
$969$	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
$973$	13.0000i	0.416761i
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−7.00000	−0.224065
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	24.0000i	0.765871i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	−36.0000	−1.14647
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	−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	$0.380963\pi$
$992$	− 5.00000i	− 0.158750i
$993$	0	0
$994$	6.00000	0.190308
$995$	0	0
$996$	0	0
$997$	10.0000i	0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$	10.0000i	0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$	− 7.00000i	− 0.221581i
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000 q^{19} +6.00000i q^{23} +2.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +7.00000i q^{37} -1.00000i q^{38} +12.0000 q^{41} +11.0000i q^{43} +6.00000 q^{46} -12.0000i q^{47} +6.00000 q^{49} -2.00000i q^{52} -1.00000 q^{56} -6.00000i q^{58} +6.00000 q^{59} -7.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +4.00000i q^{67} +6.00000i q^{68} +6.00000 q^{71} -7.00000i q^{73} +7.00000 q^{74} -1.00000 q^{76} +1.00000 q^{79} -12.0000i q^{82} -6.00000i q^{83} +11.0000 q^{86} +6.00000 q^{89} -2.00000 q^{91} -6.00000i q^{92} -12.0000 q^{94} -5.00000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 2 q^{14} + 2 q^{16} + 2 q^{19} + 4 q^{26} + 12 q^{29} + 10 q^{31} - 12 q^{34} + 24 q^{41} + 12 q^{46} + 12 q^{49} - 2 q^{56} + 12 q^{59} - 14 q^{61} - 2 q^{64} + 12 q^{71} + 14 q^{74} - 2 q^{76} + 2 q^{79} + 22 q^{86} + 12 q^{89} - 4 q^{91} - 24 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^14 + 2 * q^16 + 2 * q^19 + 4 * q^26 + 12 * q^29 + 10 * q^31 - 12 * q^34 + 24 * q^41 + 12 * q^46 + 12 * q^49 - 2 * q^56 + 12 * q^59 - 14 * q^61 - 2 * q^64 + 12 * q^71 + 14 * q^74 - 2 * q^76 + 2 * q^79 + 22 * q^86 + 12 * q^89 - 4 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more