LMFDB - Embedded newform 3150.2.g.l.2899.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3150,2,Mod(2899,3150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3150.2899"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,0,0,0,0,0,-2,0,2,0,0,-4,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(26)] == traces)

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

Self dual:	no
Analytic conductor:	$25.1528766367$
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:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3150.2899
Dual form			3150.2.g.l.2899.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} +2.00000 q^{26} +1.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} +4.00000i q^{37} +2.00000i q^{38} -6.00000 q^{41} +2.00000i q^{43} -6.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} +8.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -2.00000i q^{67} -6.00000 q^{71} +2.00000i q^{73} +4.00000 q^{74} +2.00000 q^{76} +16.0000 q^{79} +6.00000i q^{82} +2.00000 q^{86} +6.00000 q^{89} +2.00000 q^{91} -6.00000 q^{94} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{26} + 12 q^{29} + 16 q^{31} - 12 q^{41} - 2 q^{49} + 2 q^{56} + 24 q^{59} + 16 q^{61} - 2 q^{64} - 12 q^{71} + 8 q^{74} + 4 q^{76} + 32 q^{79} + 4 q^{86}+ \cdots - 12 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^14 + 2 * q^16 - 4 * q^19 + 4 * q^26 + 12 * q^29 + 16 * q^31 - 12 * q^41 - 2 * q^49 + 2 * q^56 + 24 * q^59 + 16 * q^61 - 2 * q^64 - 12 * q^71 + 8 * q^74 + 4 * q^76 + 32 * q^79 + 4 * q^86 + 12 * q^89 + 4 * q^91 - 12 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$1$

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
$7$	− 1.00000i	− 0.377964i
$8$	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
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	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
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	2.00000i	0.324443i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	− 8.00000i	− 1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	0	0
$82$	6.00000i	0.662589i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	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
$92$	0	0
$93$	0	0
$94$	−6.00000	−0.618853
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	1.00000i	0.101015i
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	− 1.00000i	− 0.0944911i
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	− 12.0000i	− 1.10469i
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 8.00000i	− 0.724286i
$123$	0	0
$124$	−8.00000	−0.718421
$125$	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
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	−2.00000	−0.172774
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	0	0
$142$	6.00000i	0.503509i
$143$	0	0
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	− 4.00000i	− 0.328798i
$149$	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
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	− 16.0000i	− 1.27289i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 22.0000i	− 1.72317i	−0.507611	−	0.861586i	$-0.669471\pi$
$163$	− 22.0000i	− 1.72317i	0.507611	−	0.861586i	$-0.330529\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	− 2.00000i	− 0.152499i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	−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
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	− 2.00000i	− 0.148250i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	6.00000i	0.437595i
$189$	0	0
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	6.00000i	0.422159i
$203$	− 6.00000i	− 0.421117i
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	− 8.00000i	− 0.543075i
$218$	− 10.0000i	− 0.677285i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−18.0000	−1.19734
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	−12.0000	−0.781133
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	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
$244$	−8.00000	−0.512148
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	8.00000i	0.508001i
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 12.0000i	− 0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$	− 12.0000i	− 0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	− 12.0000i	− 0.741362i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	2.00000	0.122628
$267$	0	0
$268$	2.00000i	0.122169i
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	− 32.0000i	− 1.92269i	−0.275340	−	0.961347i	$-0.588791\pi$
$277$	− 32.0000i	− 1.92269i	0.275340	−	0.961347i	$-0.411209\pi$
$278$	− 10.0000i	− 0.599760i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	6.00000i	0.354169i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	− 2.00000i	− 0.117041i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	−4.00000	−0.232495
$297$	0	0
$298$	− 18.0000i	− 1.04271i
$299$	0	0
$300$	0	0
$301$	2.00000	0.115278
$302$	− 8.00000i	− 0.460348i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	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
$313$	26.0000i	1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$	26.0000i	1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−16.0000	−0.900070
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−22.0000	−1.21847
$327$	0	0
$328$	− 6.00000i	− 0.331295i
$329$	−6.00000	−0.330791
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	0	0
$334$	−18.0000	−0.984916
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	− 9.00000i	− 0.489535i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	−2.00000	−0.107833
$345$	0	0
$346$	6.00000	0.322562
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	36.0000i	1.91609i	0.286623	+	0.958043i	$0.407467\pi$
$353$	36.0000i	1.91609i	−0.286623	+	0.958043i	$0.592533\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000i	0.634220i
$359$	−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
$361$	−15.0000	−0.789474
$362$	16.0000i	0.840941i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	0	0
$382$	− 18.0000i	− 0.920960i
$383$	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
$386$	14.0000	0.712581
$387$	0	0
$388$	− 10.0000i	− 0.507673i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	− 4.00000i	− 0.200502i
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	16.0000i	0.797017i
$404$	6.00000	0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	0	0
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	0	0
$412$	− 8.00000i	− 0.394132i
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	0	0
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	− 8.00000i	− 0.389434i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	0	0
$446$	8.00000	0.378811
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	0	0
$452$	18.0000i	0.846649i
$453$	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	$-0.208074\pi$
$458$	20.0000i	0.934539i
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	24.0000i	1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$467$	24.0000i	1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	6.00000i	0.274434i
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	8.00000i	0.362143i
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	0	0
$494$	−4.00000	−0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	6.00000i	0.269137i
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	0	0
$502$	12.0000i	0.535586i
$503$	− 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
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 4.00000i	− 0.175750i
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	12.0000	0.523225
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	− 2.00000i	− 0.0867110i
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	2.00000	0.0863868
$537$	0	0
$538$	− 30.0000i	− 1.29339i
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	− 8.00000i	− 0.343629i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	18.0000i	0.768922i
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	−32.0000	−1.35955
$555$	0	0
$556$	−10.0000	−0.424094
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	0	0
$566$	20.0000	0.840663
$567$	0	0
$568$	− 6.00000i	− 0.251754i
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	6.00000	0.250435
$575$	0	0
$576$	0	0
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	− 17.0000i	− 0.707107i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−30.0000	−1.23929
$587$	36.0000i	1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$	36.0000i	1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	4.00000i	0.164399i
$593$	24.0000i	0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$	24.0000i	0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	0	0
$598$	0	0
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	− 2.00000i	− 0.0815139i
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	40.0000i	1.62355i	0.583970	+	0.811775i	$0.301498\pi$
$607$	40.0000i	1.62355i	−0.583970	+	0.811775i	$0.698502\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	− 28.0000i	− 1.13091i	−0.824779	−	0.565455i	$-0.808701\pi$
$613$	− 28.0000i	− 1.13091i	0.824779	−	0.565455i	$-0.191299\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	46.0000	1.84890	0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000	1.84890	0.924448	+	0.381308i	$0.124526\pi$
$620$	0	0
$621$	0	0
$622$	− 12.0000i	− 0.481156i
$623$	− 6.00000i	− 0.240385i
$624$	0	0
$625$	0	0
$626$	26.0000	1.03917
$627$	0	0
$628$	14.0000i	0.558661i
$629$	0	0
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	16.0000i	0.636446i
$633$	0	0
$634$	18.0000	0.714871
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.0000i	1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$	42.0000i	1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	22.0000i	0.861586i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	0	0
$658$	6.00000i	0.233904i
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	−40.0000	−1.55582	−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000	−1.55582	−0.777910	+	0.628376i	$0.783720\pi$
$662$	− 8.00000i	− 0.310929i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	18.0000i	0.696441i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	−9.00000	−0.346154
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	2.00000i	0.0762493i
$689$	12.0000	0.457164
$690$	0	0
$691$	26.0000	0.989087	0.494543	−	0.869153i	$-0.335335\pi$
$691$	26.0000	0.989087	0.494543	+	0.869153i	$0.335335\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 4.00000i	− 0.151402i
$699$	0	0
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	0	0
$706$	36.0000	1.35488
$707$	6.00000i	0.225653i
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	6.00000i	0.223918i
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	15.0000i	0.558242i
$723$	0	0
$724$	16.0000	0.594635
$725$	0	0
$726$	0	0
$727$	40.0000i	1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$	40.0000i	1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$	2.00000i	0.0741249i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	0	0
$742$	6.00000i	0.220267i
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	0	0
$746$	−4.00000	−0.146450
$747$	0	0
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	− 6.00000i	− 0.218797i
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	− 20.0000i	− 0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$	− 20.0000i	− 0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$	8.00000i	0.290573i
$759$	0	0
$760$	0	0
$761$	54.0000	1.95750	0.978749	−	0.205061i	$-0.0657392\pi$
$761$	54.0000	1.95750	0.978749	+	0.205061i	$0.0657392\pi$
$762$	0	0
$763$	− 10.0000i	− 0.362024i
$764$	−18.0000	−0.651217
$765$	0	0
$766$	6.00000	0.216789
$767$	24.0000i	0.866590i
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	− 14.0000i	− 0.503871i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	0	0
$775$	0	0
$776$	−10.0000	−0.358979
$777$	0	0
$778$	30.0000i	1.07555i
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	16.0000i	0.568177i
$794$	22.0000	0.780751
$795$	0	0
$796$	−4.00000	−0.141776
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	16.0000	0.563576
$807$	0	0
$808$	− 6.00000i	− 0.211079i
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	6.00000i	0.210559i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.00000i	− 0.139942i
$818$	2.00000i	0.0699284i
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	− 28.0000i	− 0.976019i	−0.872838	−	0.488009i	$-0.837723\pi$
$823$	− 28.0000i	− 0.976019i	0.872838	−	0.488009i	$-0.162277\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−12.0000	−0.417533
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	0	0
$832$	− 2.00000i	− 0.0693375i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	12.0000i	0.414533i
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000i	0.344623i
$843$	0	0
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	11.0000i	0.377964i
$848$	− 6.00000i	− 0.206041i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	12.0000	0.410152
$857$	− 12.0000i	− 0.409912i	−0.978771	−	0.204956i	$-0.934295\pi$
$857$	− 12.0000i	− 0.409912i	0.978771	−	0.204956i	$-0.0657052\pi$
$858$	0	0
$859$	−2.00000	−0.0682391	−0.0341196	−	0.999418i	$-0.510863\pi$
$859$	−2.00000	−0.0682391	−0.0341196	+	0.999418i	$0.510863\pi$
$860$	0	0
$861$	0	0
$862$	6.00000i	0.204361i
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	0	0
$865$	0	0
$866$	14.0000	0.475739
$867$	0	0
$868$	8.00000i	0.271538i
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	10.0000i	0.338643i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	8.00000i	0.269987i
$879$	0	0
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	2.00000i	0.0673054i	0.999434	+	0.0336527i	$0.0107140\pi$
$883$	2.00000i	0.0673054i	−0.999434	+	0.0336527i	$0.989286\pi$
$884$	0	0
$885$	0	0
$886$	36.0000	1.20944
$887$	− 18.0000i	− 0.604381i	−0.953248	−	0.302190i	$-0.902282\pi$
$887$	− 18.0000i	− 0.604381i	0.953248	−	0.302190i	$-0.0977178\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	− 8.00000i	− 0.267860i
$893$	12.0000i	0.401565i
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	36.0000i	1.20134i
$899$	48.0000	1.60089
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	18.0000	0.598671
$905$	0	0
$906$	0	0
$907$	− 2.00000i	− 0.0664089i	−0.999449	−	0.0332045i	$-0.989429\pi$
$907$	− 2.00000i	− 0.0664089i	0.999449	−	0.0332045i	$-0.0105712\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	0	0
$914$	−26.0000	−0.860004
$915$	0	0
$916$	20.0000	0.660819
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	30.0000i	0.987997i
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	− 6.00000i	− 0.196537i
$933$	0	0
$934$	24.0000	0.785304
$935$	0	0
$936$	0	0
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	2.00000i	0.0653023i
$939$	0	0
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	0	0
$955$	0	0
$956$	6.00000	0.194054
$957$	0	0
$958$	− 36.0000i	− 1.16311i
$959$	−18.0000	−0.581250
$960$	0	0
$961$	33.0000	1.06452
$962$	8.00000i	0.257930i
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	− 10.0000i	− 0.320585i
$974$	16.0000	0.512673
$975$	0	0
$976$	8.00000	0.256074
$977$	− 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
$982$	− 12.0000i	− 0.382935i
$983$	54.0000i	1.72233i	0.508323	+	0.861166i	$0.330265\pi$
$983$	54.0000i	1.72233i	−0.508323	+	0.861166i	$0.669735\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	4.00000i	0.127257i
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	− 8.00000i	− 0.254000i
$993$	0	0
$994$	6.00000	0.190308
$995$	0	0
$996$	0	0
$997$	46.0000i	1.45683i	0.685134	+	0.728417i	$0.259744\pi$
$997$	46.0000i	1.45683i	−0.685134	+	0.728417i	$0.740256\pi$
$998$	− 16.0000i	− 0.506471i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.l.2899.1		2
3.2	odd	2		3150.2.g.m.2899.2		2
5.2	odd	4		630.2.a.j.1.1	yes	1
5.3	odd	4		3150.2.a.g.1.1		1
5.4	even	2	inner	3150.2.g.l.2899.2		2
15.2	even	4		630.2.a.c.1.1	✓	1
15.8	even	4		3150.2.a.bb.1.1		1
15.14	odd	2		3150.2.g.m.2899.1		2
20.7	even	4		5040.2.a.z.1.1		1
35.27	even	4		4410.2.a.y.1.1		1
60.47	odd	4		5040.2.a.f.1.1		1
105.62	odd	4		4410.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.a.c.1.1	✓	1	15.2	even	4
630.2.a.j.1.1	yes	1	5.2	odd	4
3150.2.a.g.1.1		1	5.3	odd	4
3150.2.a.bb.1.1		1	15.8	even	4
3150.2.g.l.2899.1		2	1.1	even	1	trivial
3150.2.g.l.2899.2		2	5.4	even	2	inner
3150.2.g.m.2899.1		2	15.14	odd	2
3150.2.g.m.2899.2		2	3.2	odd	2
4410.2.a.p.1.1		1	105.62	odd	4
4410.2.a.y.1.1		1	35.27	even	4
5040.2.a.f.1.1		1	60.47	odd	4
5040.2.a.z.1.1		1	20.7	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} +2.00000 q^{26} +1.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} +4.00000i q^{37} +2.00000i q^{38} -6.00000 q^{41} +2.00000i q^{43} -6.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} +8.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -2.00000i q^{67} -6.00000 q^{71} +2.00000i q^{73} +4.00000 q^{74} +2.00000 q^{76} +16.0000 q^{79} +6.00000i q^{82} +2.00000 q^{86} +6.00000 q^{89} +2.00000 q^{91} -6.00000 q^{94} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{26} + 12 q^{29} + 16 q^{31} - 12 q^{41} - 2 q^{49} + 2 q^{56} + 24 q^{59} + 16 q^{61} - 2 q^{64} - 12 q^{71} + 8 q^{74} + 4 q^{76} + 32 q^{79} + 4 q^{86}+ \cdots - 12 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^14 + 2 * q^16 - 4 * q^19 + 4 * q^26 + 12 * q^29 + 16 * q^31 - 12 * q^41 - 2 * q^49 + 2 * q^56 + 24 * q^59 + 16 * q^61 - 2 * q^64 - 12 * q^71 + 8 * q^74 + 4 * q^76 + 32 * q^79 + 4 * q^86 + 12 * q^89 + 4 * q^91 - 12 * q^94

Introduction

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