LMFDB - Embedded newform 4050.2.c.t.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4050,2,Mod(649,4050)] 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("4050.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4050 = 2 \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4050.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,0,0,12,0,0,2,0,2,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$32.3394128186$
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 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4050.649
Dual form			4050.2.c.t.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +6.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{19} -6.00000i q^{22} +9.00000i q^{23} +2.00000 q^{26} -1.00000i q^{28} -3.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -8.00000i q^{37} -4.00000i q^{38} -3.00000 q^{41} +8.00000i q^{43} -6.00000 q^{44} +9.00000 q^{46} +3.00000i q^{47} +6.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{56} +3.00000i q^{58} -6.00000 q^{59} -13.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +13.0000i q^{67} -6.00000 q^{71} -4.00000i q^{73} -8.00000 q^{74} -4.00000 q^{76} +6.00000i q^{77} +10.0000 q^{79} +3.00000i q^{82} -9.00000i q^{83} +8.00000 q^{86} +6.00000i q^{88} -9.00000 q^{89} -2.00000 q^{91} -9.00000i q^{92} +3.00000 q^{94} -2.00000i q^{97} -6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 12 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{26} - 6 q^{29} - 8 q^{31} - 6 q^{41} - 12 q^{44} + 18 q^{46} + 12 q^{49} - 2 q^{56} - 12 q^{59} - 26 q^{61} - 2 q^{64} - 12 q^{71} - 16 q^{74}+ \cdots + 6 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 12 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^26 - 6 * q^29 - 8 * q^31 - 6 * q^41 - 12 * q^44 + 18 * q^46 + 12 * q^49 - 2 * q^56 - 12 * q^59 - 26 * q^61 - 2 * q^64 - 12 * q^71 - 16 * q^74 - 8 * q^76 + 20 * q^79 + 16 * q^86 - 18 * q^89 - 4 * q^91 + 6 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times$.

$n$	$2351$	$3727$
$\chi(n)$	$1$	$-1$

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
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	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
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	− 6.00000i	− 1.27920i
$23$	9.00000i	1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$	9.00000i	1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	9.00000	1.32698
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	3.00000i	0.393919i
$59$	−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
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	13.0000i	1.58820i	0.607785	+	0.794101i	$0.292058\pi$
$67$	13.0000i	1.58820i	−0.607785	+	0.794101i	$0.707942\pi$
$68$	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
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	−4.00000	−0.458831
$77$	6.00000i	0.683763i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	0	0
$82$	3.00000i	0.331295i
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	6.00000i	0.639602i
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 9.00000i	− 0.938315i
$93$	0	0
$94$	3.00000	0.309426
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 6.00000i	− 0.606092i
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	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
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	3.00000	0.278543
$117$	0	0
$118$	6.00000i	0.552345i
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	13.0000i	1.17696i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	7.00000i	0.621150i	0.950549	+	0.310575i	$0.100522\pi$
$127$	7.00000i	0.621150i	−0.950549	+	0.310575i	$0.899478\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	13.0000	1.12303
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	6.00000i	0.503509i
$143$	12.0000i	1.00349i
$144$	0	0
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	8.00000i	0.657596i
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	4.00000i	0.324443i
$153$	0	0
$154$	6.00000	0.483494
$155$	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	$-0.188685\pi$
$158$	− 10.0000i	− 0.795557i
$159$	0	0
$160$	0	0
$161$	−9.00000	−0.709299
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	−9.00000	−0.698535
$167$	3.00000i	0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$	3.00000i	0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	− 8.00000i	− 0.609994i
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	0	0
$178$	9.00000i	0.674579i
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	2.00000i	0.148250i
$183$	0	0
$184$	−9.00000	−0.663489
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 3.00000i	− 0.218797i
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	−2.00000	−0.143592
$195$	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
$202$	− 6.00000i	− 0.422159i
$203$	− 3.00000i	− 0.210559i
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	2.00000i	0.138675i
$209$	24.0000	1.66011
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	3.00000	0.205076
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	− 7.00000i	− 0.474100i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 1.00000i	− 0.0669650i	−0.999439	−	0.0334825i	$-0.989340\pi$
$223$	− 1.00000i	− 0.0669650i	0.999439	−	0.0334825i	$-0.0106598\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	0	0
$227$	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
$232$	− 3.00000i	− 0.196960i
$233$	− 12.0000i	− 0.786146i	−0.919507	−	0.393073i	$-0.871412\pi$
$233$	− 12.0000i	− 0.786146i	0.919507	−	0.393073i	$-0.128588\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	29.0000	1.86805	0.934027	−	0.357202i	$-0.116269\pi$
$241$	29.0000	1.86805	0.934027	+	0.357202i	$0.116269\pi$
$242$	− 25.0000i	− 1.60706i
$243$	0	0
$244$	13.0000	0.832240
$245$	0	0
$246$	0	0
$247$	8.00000i	0.509028i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	54.0000i	3.39495i
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	18.0000i	1.11204i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	0	0
$268$	− 13.0000i	− 0.794101i
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	− 16.0000i	− 0.959616i
$279$	0	0
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	0	0
$283$	− 13.0000i	− 0.772770i	−0.922338	−	0.386385i	$-0.873724\pi$
$283$	− 13.0000i	− 0.772770i	0.922338	−	0.386385i	$-0.126276\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	12.0000	0.709575
$287$	− 3.00000i	− 0.177084i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	4.00000i	0.234082i
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	8.00000	0.464991
$297$	0	0
$298$	− 3.00000i	− 0.173785i
$299$	−18.0000	−1.04097
$300$	0	0
$301$	−8.00000	−0.461112
$302$	− 14.0000i	− 0.805609i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	7.00000i	0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$	7.00000i	0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$	− 6.00000i	− 0.341882i
$309$	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
$313$	2.00000i	0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$	2.00000i	0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−10.0000	−0.562544
$317$	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
$319$	−18.0000	−1.00781
$320$	0	0
$321$	0	0
$322$	9.00000i	0.501550i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	− 3.00000i	− 0.165647i
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	9.00000i	0.493939i
$333$	0	0
$334$	3.00000	0.164153
$335$	0	0
$336$	0	0
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	− 9.00000i	− 0.489535i
$339$	0	0
$340$	0	0
$341$	−24.0000	−1.29967
$342$	0	0
$343$	13.0000i	0.701934i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	24.0000	1.29025
$347$	− 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
$349$	−23.0000	−1.23116	−0.615581	−	0.788074i	$-0.711079\pi$
$349$	−23.0000	−1.23116	−0.615581	+	0.788074i	$0.711079\pi$
$350$	0	0
$351$	0	0
$352$	− 6.00000i	− 0.319801i
$353$	24.0000i	1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$	24.0000i	1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	0	0
$355$	0	0
$356$	9.00000	0.476999
$357$	0	0
$358$	− 18.0000i	− 0.951330i
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 5.00000i	− 0.262794i
$363$	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	$-0.0669552\pi$
$368$	9.00000i	0.469157i
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	26.0000i	1.34623i	0.739538	+	0.673114i	$0.235044\pi$
$373$	26.0000i	1.34623i	−0.739538	+	0.673114i	$0.764956\pi$
$374$	0	0
$375$	0	0
$376$	−3.00000	−0.154713
$377$	− 6.00000i	− 0.309016i
$378$	0	0
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	0	0
$381$	0	0
$382$	12.0000i	0.613973i
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	0	0
$388$	2.00000i	0.101535i
$389$	−21.0000	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$389$	−21.0000	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$390$	0	0
$391$	0	0
$392$	6.00000i	0.303046i
$393$	0	0
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−3.00000	−0.148888
$407$	− 48.0000i	− 2.37927i
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	− 8.00000i	− 0.394132i
$413$	− 6.00000i	− 0.295241i
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	0	0
$418$	− 24.0000i	− 1.17388i
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	− 2.00000i	− 0.0973585i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	− 13.0000i	− 0.629114i
$428$	− 3.00000i	− 0.145010i
$429$	0	0
$430$	0	0
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−7.00000	−0.335239
$437$	36.0000i	1.72211i
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.00000i	0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$	9.00000i	0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$	0	0
$445$	0	0
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	0	0
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	− 13.0000i	− 0.607450i
$459$	0	0
$460$	0	0
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−12.0000	−0.555889
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	−13.0000	−0.600284
$470$	0	0
$471$	0	0
$472$	− 6.00000i	− 0.276172i
$473$	48.0000i	2.20704i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	− 29.0000i	− 1.32091i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	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	$-0.0580167\pi$
$488$	− 13.0000i	− 0.588482i
$489$	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
$494$	8.00000	0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	27.0000i	1.20387i	0.798545	+	0.601935i	$0.205603\pi$
$503$	27.0000i	1.20387i	−0.798545	+	0.601935i	$0.794397\pi$
$504$	0	0
$505$	0	0
$506$	54.0000	2.40059
$507$	0	0
$508$	− 7.00000i	− 0.310575i
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−18.0000	−0.793946
$515$	0	0
$516$	0	0
$517$	18.0000i	0.791639i
$518$	− 8.00000i	− 0.351500i
$519$	0	0
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	− 19.0000i	− 0.830812i	−0.909636	−	0.415406i	$-0.863640\pi$
$523$	− 19.0000i	− 0.830812i	0.909636	−	0.415406i	$-0.136360\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−58.0000	−2.52174
$530$	0	0
$531$	0	0
$532$	− 4.00000i	− 0.173422i
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	−13.0000	−0.561514
$537$	0	0
$538$	21.0000i	0.905374i
$539$	36.0000	1.55063
$540$	0	0
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	4.00000i	0.171815i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	43.0000i	1.83855i	0.393619	+	0.919274i	$0.371223\pi$
$547$	43.0000i	1.83855i	−0.393619	+	0.919274i	$0.628777\pi$
$548$	− 12.0000i	− 0.512615i
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	10.0000i	0.425243i
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−16.0000	−0.678551
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	15.0000i	0.632737i
$563$	− 3.00000i	− 0.126435i	−0.998000	−	0.0632175i	$-0.979864\pi$
$563$	− 3.00000i	− 0.126435i	0.998000	−	0.0632175i	$-0.0201362\pi$
$564$	0	0
$565$	0	0
$566$	−13.0000	−0.546431
$567$	0	0
$568$	− 6.00000i	− 0.251754i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	− 12.0000i	− 0.501745i
$573$	0	0
$574$	−3.00000	−0.125218
$575$	0	0
$576$	0	0
$577$	10.0000i	0.416305i	0.978096	+	0.208153i	$0.0667451\pi$
$577$	10.0000i	0.416305i	−0.978096	+	0.208153i	$0.933255\pi$
$578$	− 17.0000i	− 0.707107i
$579$	0	0
$580$	0	0
$581$	9.00000	0.373383
$582$	0	0
$583$	36.0000i	1.49097i
$584$	4.00000	0.165521
$585$	0	0
$586$	0	0
$587$	15.0000i	0.619116i	0.950881	+	0.309558i	$0.100181\pi$
$587$	15.0000i	0.619116i	−0.950881	+	0.309558i	$0.899819\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	− 8.00000i	− 0.328798i
$593$	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
$596$	−3.00000	−0.122885
$597$	0	0
$598$	18.0000i	0.736075i
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	8.00000i	0.326056i
$603$	0	0
$604$	−14.0000	−0.569652
$605$	0	0
$606$	0	0
$607$	− 29.0000i	− 1.17707i	−0.808470	−	0.588537i	$-0.799704\pi$
$607$	− 29.0000i	− 1.17707i	0.808470	−	0.588537i	$-0.200296\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	− 40.0000i	− 1.61558i	−0.589467	−	0.807792i	$-0.700662\pi$
$613$	− 40.0000i	− 1.61558i	0.589467	−	0.807792i	$-0.299338\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	−6.00000	−0.241747
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	0	0
$622$	− 12.0000i	− 0.481156i
$623$	− 9.00000i	− 0.360577i
$624$	0	0
$625$	0	0
$626$	2.00000	0.0799361
$627$	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	$0.280196\pi$
$632$	10.0000i	0.397779i
$633$	0	0
$634$	6.00000	0.238290
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	18.0000i	0.712627i
$639$	0	0
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	− 31.0000i	− 1.22252i	−0.791430	−	0.611260i	$-0.790663\pi$
$643$	− 31.0000i	− 1.22252i	0.791430	−	0.611260i	$-0.209337\pi$
$644$	9.00000	0.354650
$645$	0	0
$646$	0	0
$647$	3.00000i	0.117942i	0.998260	+	0.0589711i	$0.0187820\pi$
$647$	3.00000i	0.117942i	−0.998260	+	0.0589711i	$0.981218\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−3.00000	−0.117130
$657$	0	0
$658$	3.00000i	0.116952i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	10.0000i	0.388661i
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	0	0
$667$	− 27.0000i	− 1.04544i
$668$	− 3.00000i	− 0.116073i
$669$	0	0
$670$	0	0
$671$	−78.0000	−3.01116
$672$	0	0
$673$	− 46.0000i	− 1.77317i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	− 46.0000i	− 1.77317i	0.462566	−	0.886585i	$-0.346929\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	24.0000i	0.919007i
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	8.00000i	0.304997i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	− 24.0000i	− 0.912343i
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	0	0
$698$	23.0000i	0.870563i
$699$	0	0
$700$	0	0
$701$	−45.0000	−1.69963	−0.849813	−	0.527084i	$-0.823285\pi$
$701$	−45.0000	−1.69963	−0.849813	+	0.527084i	$0.823285\pi$
$702$	0	0
$703$	− 32.0000i	− 1.20690i
$704$	−6.00000	−0.226134
$705$	0	0
$706$	24.0000	0.903252
$707$	6.00000i	0.225653i
$708$	0	0
$709$	−11.0000	−0.413114	−0.206557	−	0.978435i	$-0.566226\pi$
$709$	−11.0000	−0.413114	−0.206557	+	0.978435i	$0.566226\pi$
$710$	0	0
$711$	0	0
$712$	− 9.00000i	− 0.337289i
$713$	− 36.0000i	− 1.34821i
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	0	0
$718$	− 12.0000i	− 0.447836i
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	3.00000i	0.111648i
$723$	0	0
$724$	−5.00000	−0.185824
$725$	0	0
$726$	0	0
$727$	− 53.0000i	− 1.96566i	−0.184510	−	0.982831i	$-0.559070\pi$
$727$	− 53.0000i	− 1.96566i	0.184510	−	0.982831i	$-0.440930\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	9.00000	0.331744
$737$	78.0000i	2.87317i
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	0	0
$742$	6.00000i	0.220267i
$743$	− 15.0000i	− 0.550297i	−0.961402	−	0.275148i	$-0.911273\pi$
$743$	− 15.0000i	− 0.550297i	0.961402	−	0.275148i	$-0.0887270\pi$
$744$	0	0
$745$	0	0
$746$	26.0000	0.951928
$747$	0	0
$748$	0	0
$749$	−3.00000	−0.109618
$750$	0	0
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	3.00000i	0.109399i
$753$	0	0
$754$	−6.00000	−0.218507
$755$	0	0
$756$	0	0
$757$	46.0000i	1.67190i	0.548807	+	0.835949i	$0.315082\pi$
$757$	46.0000i	1.67190i	−0.548807	+	0.835949i	$0.684918\pi$
$758$	− 22.0000i	− 0.799076i
$759$	0	0
$760$	0	0
$761$	−33.0000	−1.19625	−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000	−1.19625	−0.598125	+	0.801403i	$0.704087\pi$
$762$	0	0
$763$	7.00000i	0.253417i
$764$	12.0000	0.434145
$765$	0	0
$766$	−12.0000	−0.433578
$767$	− 12.0000i	− 0.433295i
$768$	0	0
$769$	−29.0000	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$769$	−29.0000	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$770$	0	0
$771$	0	0
$772$	− 2.00000i	− 0.0719816i
$773$	− 48.0000i	− 1.72644i	−0.504828	−	0.863220i	$-0.668444\pi$
$773$	− 48.0000i	− 1.72644i	0.504828	−	0.863220i	$-0.331556\pi$
$774$	0	0
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	21.0000i	0.752886i
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	6.00000	0.214286
$785$	0	0
$786$	0	0
$787$	− 20.0000i	− 0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$	− 20.0000i	− 0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$	− 12.0000i	− 0.427482i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 26.0000i	− 0.923287i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	8.00000	0.283552
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 6.00000i	− 0.211867i
$803$	− 24.0000i	− 0.846942i
$804$	0	0
$805$	0	0
$806$	−8.00000	−0.281788
$807$	0	0
$808$	6.00000i	0.211079i
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	3.00000i	0.105279i
$813$	0	0
$814$	−48.0000	−1.68240
$815$	0	0
$816$	0	0
$817$	32.0000i	1.11954i
$818$	− 10.0000i	− 0.349642i
$819$	0	0
$820$	0	0
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	0	0
$823$	− 25.0000i	− 0.871445i	−0.900081	−	0.435723i	$-0.856493\pi$
$823$	− 25.0000i	− 0.871445i	0.900081	−	0.435723i	$-0.143507\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−6.00000	−0.208767
$827$	− 3.00000i	− 0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$	− 3.00000i	− 0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	0	0
$831$	0	0
$832$	− 2.00000i	− 0.0693375i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−24.0000	−0.830057
$837$	0	0
$838$	− 30.0000i	− 1.03633i
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	22.0000i	0.758170i
$843$	0	0
$844$	−2.00000	−0.0688428
$845$	0	0
$846$	0	0
$847$	25.0000i	0.859010i
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	72.0000	2.46813
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	−13.0000	−0.444851
$855$	0	0
$856$	−3.00000	−0.102538
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	0	0
$859$	34.0000	1.16007	0.580033	−	0.814593i	$-0.303040\pi$
$859$	34.0000	1.16007	0.580033	+	0.814593i	$0.303040\pi$
$860$	0	0
$861$	0	0
$862$	− 6.00000i	− 0.204361i
$863$	3.00000i	0.102121i	0.998696	+	0.0510606i	$0.0162602\pi$
$863$	3.00000i	0.102121i	−0.998696	+	0.0510606i	$0.983740\pi$
$864$	0	0
$865$	0	0
$866$	−16.0000	−0.543702
$867$	0	0
$868$	4.00000i	0.135769i
$869$	60.0000	2.03536
$870$	0	0
$871$	−26.0000	−0.880976
$872$	7.00000i	0.237050i
$873$	0	0
$874$	36.0000	1.21772
$875$	0	0
$876$	0	0
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	− 28.0000i	− 0.944954i
$879$	0	0
$880$	0	0
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	0	0
$883$	− 31.0000i	− 1.04323i	−0.853180	−	0.521617i	$-0.825329\pi$
$883$	− 31.0000i	− 1.04323i	0.853180	−	0.521617i	$-0.174671\pi$
$884$	0	0
$885$	0	0
$886$	9.00000	0.302361
$887$	− 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
$889$	−7.00000	−0.234772
$890$	0	0
$891$	0	0
$892$	1.00000i	0.0334825i
$893$	12.0000i	0.401565i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	6.00000i	0.200223i
$899$	12.0000	0.400222
$900$	0	0
$901$	0	0
$902$	18.0000i	0.599334i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	37.0000i	1.22856i	0.789086	+	0.614282i	$0.210554\pi$
$907$	37.0000i	1.22856i	−0.789086	+	0.614282i	$0.789446\pi$
$908$	− 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
$913$	− 54.0000i	− 1.78714i
$914$	−8.00000	−0.264616
$915$	0	0
$916$	−13.0000	−0.429532
$917$	− 18.0000i	− 0.594412i
$918$	0	0
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	0	0
$922$	27.0000i	0.889198i
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	0	0
$928$	3.00000i	0.0984798i
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	12.0000i	0.393073i
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	0	0
$937$	− 56.0000i	− 1.82944i	−0.404088	−	0.914720i	$-0.632411\pi$
$937$	− 56.0000i	− 1.82944i	0.404088	−	0.914720i	$-0.367589\pi$
$938$	13.0000i	0.424465i
$939$	0	0
$940$	0	0
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	0	0
$943$	− 27.0000i	− 0.879241i
$944$	−6.00000	−0.195283
$945$	0	0
$946$	48.0000	1.56061
$947$	− 39.0000i	− 1.26733i	−0.773608	−	0.633665i	$-0.781550\pi$
$947$	− 39.0000i	− 1.26733i	0.773608	−	0.633665i	$-0.218450\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	− 30.0000i	− 0.969256i
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 16.0000i	− 0.515861i
$963$	0	0
$964$	−29.0000	−0.934027
$965$	0	0
$966$	0	0
$967$	37.0000i	1.18984i	0.803785	+	0.594920i	$0.202816\pi$
$967$	37.0000i	1.18984i	−0.803785	+	0.594920i	$0.797184\pi$
$968$	25.0000i	0.803530i
$969$	0	0
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	16.0000i	0.512936i
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−13.0000	−0.416120
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	0	0
$979$	−54.0000	−1.72585
$980$	0	0
$981$	0	0
$982$	− 12.0000i	− 0.382935i
$983$	9.00000i	0.287055i	0.989646	+	0.143528i	$0.0458446\pi$
$983$	9.00000i	0.287055i	−0.989646	+	0.143528i	$0.954155\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	−72.0000	−2.28947
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	4.00000i	0.127000i
$993$	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	$0.949382\pi$
$998$	32.0000i	1.01294i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.c.t.649.1		2
3.2	odd	2		4050.2.c.a.649.2		2
5.2	odd	4		810.2.a.g.1.1		1
5.3	odd	4		4050.2.a.n.1.1		1
5.4	even	2	inner	4050.2.c.t.649.2		2
9.2	odd	6		1350.2.j.e.199.2		4
9.4	even	3		450.2.j.c.349.2		4
9.5	odd	6		1350.2.j.e.1099.1		4
9.7	even	3		450.2.j.c.49.1		4
15.2	even	4		810.2.a.b.1.1		1
15.8	even	4		4050.2.a.ba.1.1		1
15.14	odd	2		4050.2.c.a.649.1		2
20.7	even	4		6480.2.a.g.1.1		1
45.2	even	12		270.2.e.b.91.1		2
45.4	even	6		450.2.j.c.349.1		4
45.7	odd	12		90.2.e.a.31.1	✓	2
45.13	odd	12		450.2.e.e.151.1		2
45.14	odd	6		1350.2.j.e.1099.2		4
45.22	odd	12		90.2.e.a.61.1	yes	2
45.23	even	12		1350.2.e.b.451.1		2
45.29	odd	6		1350.2.j.e.199.1		4
45.32	even	12		270.2.e.b.181.1		2
45.34	even	6		450.2.j.c.49.2		4
45.38	even	12		1350.2.e.b.901.1		2
45.43	odd	12		450.2.e.e.301.1		2
60.47	odd	4		6480.2.a.v.1.1		1
180.7	even	12		720.2.q.b.481.1		2
180.47	odd	12		2160.2.q.b.1441.1		2
180.67	even	12		720.2.q.b.241.1		2
180.167	odd	12		2160.2.q.b.721.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.e.a.31.1	✓	2	45.7	odd	12
90.2.e.a.61.1	yes	2	45.22	odd	12
270.2.e.b.91.1		2	45.2	even	12
270.2.e.b.181.1		2	45.32	even	12
450.2.e.e.151.1		2	45.13	odd	12
450.2.e.e.301.1		2	45.43	odd	12
450.2.j.c.49.1		4	9.7	even	3
450.2.j.c.49.2		4	45.34	even	6
450.2.j.c.349.1		4	45.4	even	6
450.2.j.c.349.2		4	9.4	even	3
720.2.q.b.241.1		2	180.67	even	12
720.2.q.b.481.1		2	180.7	even	12
810.2.a.b.1.1		1	15.2	even	4
810.2.a.g.1.1		1	5.2	odd	4
1350.2.e.b.451.1		2	45.23	even	12
1350.2.e.b.901.1		2	45.38	even	12
1350.2.j.e.199.1		4	45.29	odd	6
1350.2.j.e.199.2		4	9.2	odd	6
1350.2.j.e.1099.1		4	9.5	odd	6
1350.2.j.e.1099.2		4	45.14	odd	6
2160.2.q.b.721.1		2	180.167	odd	12
2160.2.q.b.1441.1		2	180.47	odd	12
4050.2.a.n.1.1		1	5.3	odd	4
4050.2.a.ba.1.1		1	15.8	even	4
4050.2.c.a.649.1		2	15.14	odd	2
4050.2.c.a.649.2		2	3.2	odd	2
4050.2.c.t.649.1		2	1.1	even	1	trivial
4050.2.c.t.649.2		2	5.4	even	2	inner
6480.2.a.g.1.1		1	20.7	even	4
6480.2.a.v.1.1		1	60.47	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +6.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{19} -6.00000i q^{22} +9.00000i q^{23} +2.00000 q^{26} -1.00000i q^{28} -3.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -8.00000i q^{37} -4.00000i q^{38} -3.00000 q^{41} +8.00000i q^{43} -6.00000 q^{44} +9.00000 q^{46} +3.00000i q^{47} +6.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{56} +3.00000i q^{58} -6.00000 q^{59} -13.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +13.0000i q^{67} -6.00000 q^{71} -4.00000i q^{73} -8.00000 q^{74} -4.00000 q^{76} +6.00000i q^{77} +10.0000 q^{79} +3.00000i q^{82} -9.00000i q^{83} +8.00000 q^{86} +6.00000i q^{88} -9.00000 q^{89} -2.00000 q^{91} -9.00000i q^{92} +3.00000 q^{94} -2.00000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 12 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{26} - 6 q^{29} - 8 q^{31} - 6 q^{41} - 12 q^{44} + 18 q^{46} + 12 q^{49} - 2 q^{56} - 12 q^{59} - 26 q^{61} - 2 q^{64} - 12 q^{71} - 16 q^{74}+ \cdots + 6 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 12 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^26 - 6 * q^29 - 8 * q^31 - 6 * q^41 - 12 * q^44 + 18 * q^46 + 12 * q^49 - 2 * q^56 - 12 * q^59 - 26 * q^61 - 2 * q^64 - 12 * q^71 - 16 * q^74 - 8 * q^76 + 20 * q^79 + 16 * q^86 - 18 * q^89 - 4 * q^91 + 6 * q^94

Introduction

L-functions

Modular forms

Varieties

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