LMFDB - Embedded newform 4050.2.c.n.649.2

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,-2,0,0,0,0,0,0,0,0,0,8,0,2,0,0,8] 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 162)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} -1.00000i q^{8} +1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +4.00000 q^{19} -1.00000 q^{26} +4.00000i q^{28} +9.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} -3.00000 q^{34} -1.00000i q^{37} +4.00000i q^{38} -6.00000 q^{41} -8.00000i q^{43} +12.0000i q^{47} -9.00000 q^{49} -1.00000i q^{52} -6.00000i q^{53} -4.00000 q^{56} +9.00000i q^{58} -1.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} -3.00000i q^{68} +12.0000 q^{71} -11.0000i q^{73} +1.00000 q^{74} -4.00000 q^{76} +16.0000 q^{79} -6.00000i q^{82} -12.0000i q^{83} +8.00000 q^{86} -3.00000 q^{89} +4.00000 q^{91} -12.0000 q^{94} +2.00000i q^{97} -9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 8 q^{14} + 2 q^{16} + 8 q^{19} - 2 q^{26} + 18 q^{29} - 8 q^{31} - 6 q^{34} - 12 q^{41} - 18 q^{49} - 8 q^{56} - 2 q^{61} - 2 q^{64} + 24 q^{71} + 2 q^{74} - 8 q^{76} + 32 q^{79} + 16 q^{86}+ \cdots - 24 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 8 * q^14 + 2 * q^16 + 8 * q^19 - 2 * q^26 + 18 * q^29 - 8 * q^31 - 6 * q^34 - 12 * q^41 - 18 * q^49 - 8 * q^56 - 2 * q^61 - 2 * q^64 + 24 * q^71 + 2 * q^74 - 8 * q^76 + 32 * q^79 + 16 * q^86 - 6 * q^89 + 8 * q^91 - 24 * 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
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	− 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
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	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
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	4.00000i	0.755929i
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	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
$34$	−3.00000	−0.514496
$35$	0	0
$36$	0	0
$37$	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	− 1.00000i	− 0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	4.00000i	0.648886i
$39$	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
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	− 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
$56$	−4.00000	−0.534522
$57$	0	0
$58$	9.00000i	1.18176i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	− 4.00000i	− 0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 3.00000i	− 0.363803i
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	− 11.0000i	− 1.28745i	−0.765256	−	0.643726i	$-0.777388\pi$
$73$	− 11.0000i	− 1.28745i	0.765256	−	0.643726i	$-0.222612\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	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
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 9.00000i	− 0.909137i
$99$	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
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	1.00000	0.0980581
$105$	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
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	0	0
$112$	− 4.00000i	− 0.377964i
$113$	− 15.0000i	− 1.41108i	−0.708669	−	0.705541i	$-0.750704\pi$
$113$	− 15.0000i	− 1.41108i	0.708669	−	0.705541i	$-0.249296\pi$
$114$	0	0
$115$	0	0
$116$	−9.00000	−0.835629
$117$	0	0
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 1.00000i	− 0.0905357i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	− 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
$133$	− 16.0000i	− 1.38738i
$134$	4.00000	0.345547
$135$	0	0
$136$	3.00000	0.257248
$137$	− 9.00000i	− 0.768922i	−0.923141	−	0.384461i	$-0.874387\pi$
$137$	− 9.00000i	− 0.768922i	0.923141	−	0.384461i	$-0.125613\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	12.0000i	1.00702i
$143$	0	0
$144$	0	0
$145$	0	0
$146$	11.0000	0.910366
$147$	0	0
$148$	1.00000i	0.0821995i
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 4.00000i	− 0.324443i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 13.0000i	− 1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$	− 13.0000i	− 1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$	16.0000i	1.27289i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	− 3.00000i	− 0.228086i	−0.993476	−	0.114043i	$-0.963620\pi$
$173$	− 3.00000i	− 0.228086i	0.993476	−	0.114043i	$-0.0363801\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 3.00000i	− 0.224860i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	4.00000i	0.296500i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 12.0000i	− 0.875190i
$189$	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
$193$	13.0000i	0.935760i	0.883792	+	0.467880i	$0.154982\pi$
$193$	13.0000i	0.935760i	−0.883792	+	0.467880i	$0.845018\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	9.00000	0.642857
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	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
$203$	− 36.0000i	− 2.52670i
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	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
$217$	16.0000i	1.08615i
$218$	− 11.0000i	− 0.745014i
$219$	0	0
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	15.0000	0.997785
$227$	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
$229$	−23.0000	−1.51988	−0.759941	−	0.649992i	$-0.774772\pi$
$229$	−23.0000	−1.51988	−0.759941	+	0.649992i	$0.774772\pi$
$230$	0	0
$231$	0	0
$232$	− 9.00000i	− 0.590879i
$233$	21.0000i	1.37576i	0.725826	+	0.687878i	$0.241458\pi$
$233$	21.0000i	1.37576i	−0.725826	+	0.687878i	$0.758542\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	12.0000i	0.777844i
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	− 11.0000i	− 0.707107i
$243$	0	0
$244$	1.00000	0.0640184
$245$	0	0
$246$	0	0
$247$	4.00000i	0.254514i
$248$	4.00000i	0.254000i
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.0000i	0.935674i	0.883815	+	0.467837i	$0.154967\pi$
$257$	15.0000i	0.935674i	−0.883815	+	0.467837i	$0.845033\pi$
$258$	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
$266$	16.0000	0.981023
$267$	0	0
$268$	4.00000i	0.244339i
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	3.00000i	0.181902i
$273$	0	0
$274$	9.00000	0.543710
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 20.0000i	− 1.19952i
$279$	0	0
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	24.0000i	1.41668i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	11.0000i	0.643726i
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	0	0
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	9.00000i	0.521356i
$299$	0	0
$300$	0	0
$301$	−32.0000	−1.84445
$302$	8.00000i	0.460348i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	− 23.0000i	− 1.30004i	−0.759918	−	0.650018i	$-0.774761\pi$
$313$	− 23.0000i	− 1.30004i	0.759918	−	0.650018i	$-0.225239\pi$
$314$	13.0000	0.733632
$315$	0	0
$316$	−16.0000	−0.900070
$317$	− 21.0000i	− 1.17948i	−0.807594	−	0.589739i	$-0.799231\pi$
$317$	− 21.0000i	− 1.17948i	0.807594	−	0.589739i	$-0.200769\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	0	0
$326$	8.00000	0.443079
$327$	0	0
$328$	6.00000i	0.331295i
$329$	48.0000	2.64633
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	12.0000i	0.652714i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.00000i	0.431959i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	3.00000	0.161281
$347$	− 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
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	0	0
$355$	0	0
$356$	3.00000	0.159000
$357$	0	0
$358$	12.0000i	0.634220i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 10.0000i	− 0.525588i
$363$	0	0
$364$	−4.00000	−0.209657
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	0	0
$375$	0	0
$376$	12.0000	0.618853
$377$	9.00000i	0.463524i
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	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
$386$	−13.0000	−0.661683
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	9.00000i	0.454569i
$393$	0	0
$394$	−3.00000	−0.151138
$395$	0	0
$396$	0	0
$397$	− 25.0000i	− 1.25471i	−0.778732	−	0.627357i	$-0.784137\pi$
$397$	− 25.0000i	− 1.25471i	0.778732	−	0.627357i	$-0.215863\pi$
$398$	4.00000i	0.200502i
$399$	0	0
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	36.0000	1.78665
$407$	0	0
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	0	0
$412$	− 4.00000i	− 0.197066i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	8.00000i	0.389434i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	− 11.0000i	− 0.528626i	−0.964437	−	0.264313i	$-0.914855\pi$
$433$	− 11.0000i	− 0.528626i	0.964437	−	0.264313i	$-0.0851452\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	11.0000	0.526804
$437$	0	0
$438$	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
$442$	− 3.00000i	− 0.142695i
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	8.00000	0.378811
$447$	0	0
$448$	4.00000i	0.188982i
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	15.0000i	0.705541i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	− 1.00000i	− 0.0467780i	−0.999726	−	0.0233890i	$-0.992554\pi$
$457$	− 1.00000i	− 0.0467780i	0.999726	−	0.0233890i	$-0.00744563\pi$
$458$	− 23.0000i	− 1.07472i
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	−21.0000	−0.972806
$467$	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
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−12.0000	−0.550019
$477$	0	0
$478$	12.0000i	0.548867i
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	1.00000	0.0455961
$482$	− 13.0000i	− 0.592134i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	0	0
$487$	− 4.00000i	− 0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$	− 4.00000i	− 0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$	1.00000i	0.0452679i
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	27.0000i	1.21602i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−4.00000	−0.179605
$497$	− 48.0000i	− 2.15309i
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	0	0
$502$	24.0000i	1.07117i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	16.0000i	0.709885i
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−44.0000	−1.94645
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−15.0000	−0.661622
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 4.00000i	− 0.175750i
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−12.0000	−0.523225
$527$	− 12.0000i	− 0.522728i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	16.0000i	0.693688i
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	21.0000i	0.905374i
$539$	0	0
$540$	0	0
$541$	−1.00000	−0.0429934	−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000	−0.0429934	−0.0214967	+	0.999769i	$0.506843\pi$
$542$	− 16.0000i	− 0.687259i
$543$	0	0
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	44.0000i	1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$	44.0000i	1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	9.00000i	0.384461i
$549$	0	0
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	− 64.0000i	− 2.72156i
$554$	10.0000	0.424859
$555$	0	0
$556$	20.0000	0.848189
$557$	3.00000i	0.127114i	0.997978	+	0.0635570i	$0.0202445\pi$
$557$	3.00000i	0.127114i	−0.997978	+	0.0635570i	$0.979756\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	27.0000i	1.13893i
$563$	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
$566$	−4.00000	−0.168133
$567$	0	0
$568$	− 12.0000i	− 0.503509i
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	0	0
$574$	−24.0000	−1.00174
$575$	0	0
$576$	0	0
$577$	− 25.0000i	− 1.04076i	−0.853934	−	0.520382i	$-0.825790\pi$
$577$	− 25.0000i	− 1.04076i	0.853934	−	0.520382i	$-0.174210\pi$
$578$	8.00000i	0.332756i
$579$	0	0
$580$	0	0
$581$	−48.0000	−1.99138
$582$	0	0
$583$	0	0
$584$	−11.0000	−0.455183
$585$	0	0
$586$	−9.00000	−0.371787
$587$	24.0000i	0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$	24.0000i	0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	− 1.00000i	− 0.0410997i
$593$	33.0000i	1.35515i	0.735455	+	0.677574i	$0.236969\pi$
$593$	33.0000i	1.35515i	−0.735455	+	0.677574i	$0.763031\pi$
$594$	0	0
$595$	0	0
$596$	−9.00000	−0.368654
$597$	0	0
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	− 32.0000i	− 1.30422i
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	20.0000i	0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$	20.0000i	0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	3.00000i	0.120775i	0.998175	+	0.0603877i	$0.0192337\pi$
$617$	3.00000i	0.120775i	−0.998175	+	0.0603877i	$0.980766\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	0	0
$622$	− 24.0000i	− 0.962312i
$623$	12.0000i	0.480770i
$624$	0	0
$625$	0	0
$626$	23.0000	0.919265
$627$	0	0
$628$	13.0000i	0.518756i
$629$	3.00000	0.119618
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	− 16.0000i	− 0.636446i
$633$	0	0
$634$	21.0000	0.834017
$635$	0	0
$636$	0	0
$637$	− 9.00000i	− 0.356593i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−45.0000	−1.77739	−0.888697	−	0.458496i	$-0.848388\pi$
$641$	−45.0000	−1.77739	−0.888697	+	0.458496i	$0.848388\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	0	0
$646$	−12.0000	−0.472134
$647$	− 36.0000i	− 1.41531i	−0.706560	−	0.707653i	$-0.749754\pi$
$647$	− 36.0000i	− 1.41531i	0.706560	−	0.707653i	$-0.250246\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	8.00000i	0.313304i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	0	0
$658$	48.0000i	1.87123i
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	23.0000	0.894596	0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000	0.894596	0.447298	+	0.894385i	$0.352386\pi$
$662$	20.0000i	0.777322i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	12.0000i	0.464294i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 11.0000i	− 0.424019i	−0.977268	−	0.212009i	$-0.931999\pi$
$673$	− 11.0000i	− 0.424019i	0.977268	−	0.212009i	$-0.0680008\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−12.0000	−0.461538
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	0	0
$688$	− 8.00000i	− 0.304997i
$689$	6.00000	0.228582
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	3.00000i	0.114043i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	− 18.0000i	− 0.681799i
$698$	− 14.0000i	− 0.529908i
$699$	0	0
$700$	0	0
$701$	51.0000	1.92624	0.963122	−	0.269066i	$-0.0867150\pi$
$701$	51.0000	1.92624	0.963122	+	0.269066i	$0.0867150\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	− 24.0000i	− 0.902613i
$708$	0	0
$709$	−47.0000	−1.76512	−0.882561	−	0.470198i	$-0.844183\pi$
$709$	−47.0000	−1.76512	−0.882561	+	0.470198i	$0.844183\pi$
$710$	0	0
$711$	0	0
$712$	3.00000i	0.112430i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	− 3.00000i	− 0.111648i
$723$	0	0
$724$	10.0000	0.371647
$725$	0	0
$726$	0	0
$727$	− 28.0000i	− 1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$	− 28.0000i	− 1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$	− 4.00000i	− 0.148250i
$729$	0	0
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	0	0
$737$	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
$742$	− 24.0000i	− 0.881068i
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	0	0
$745$	0	0
$746$	−10.0000	−0.366126
$747$	0	0
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	12.0000i	0.437595i
$753$	0	0
$754$	−9.00000	−0.327761
$755$	0	0
$756$	0	0
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	28.0000i	1.01701i
$759$	0	0
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	44.0000i	1.59291i
$764$	12.0000	0.434145
$765$	0	0
$766$	12.0000	0.433578
$767$	0	0
$768$	0	0
$769$	37.0000	1.33425	0.667127	−	0.744944i	$-0.267524\pi$
$769$	37.0000	1.33425	0.667127	+	0.744944i	$0.267524\pi$
$770$	0	0
$771$	0	0
$772$	− 13.0000i	− 0.467880i
$773$	− 27.0000i	− 0.971123i	−0.874203	−	0.485561i	$-0.838615\pi$
$773$	− 27.0000i	− 0.971123i	0.874203	−	0.485561i	$-0.161385\pi$
$774$	0	0
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	− 6.00000i	− 0.215110i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	− 3.00000i	− 0.106871i
$789$	0	0
$790$	0	0
$791$	−60.0000	−2.13335
$792$	0	0
$793$	− 1.00000i	− 0.0355110i
$794$	25.0000	0.887217
$795$	0	0
$796$	−4.00000	−0.141776
$797$	51.0000i	1.80651i	0.429101	+	0.903256i	$0.358830\pi$
$797$	51.0000i	1.80651i	−0.429101	+	0.903256i	$0.641170\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	0	0
$801$	0	0
$802$	3.00000i	0.105934i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	0	0
$808$	− 6.00000i	− 0.211079i
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	36.0000i	1.26335i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 32.0000i	− 1.11954i
$818$	25.0000i	0.874105i
$819$	0	0
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	0	0
$823$	− 44.0000i	− 1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$	− 44.0000i	− 1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	48.0000i	1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$	48.0000i	1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	− 27.0000i	− 0.935495i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	− 24.0000i	− 0.829066i
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	− 13.0000i	− 0.448010i
$843$	0	0
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	44.0000i	1.51186i
$848$	− 6.00000i	− 0.206041i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	39.0000i	1.33221i	0.745856	+	0.666107i	$0.232041\pi$
$857$	39.0000i	1.33221i	−0.745856	+	0.666107i	$0.767959\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	0	0
$862$	12.0000i	0.408722i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	0	0
$865$	0	0
$866$	11.0000	0.373795
$867$	0	0
$868$	− 16.0000i	− 0.543075i
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	11.0000i	0.372507i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 25.0000i	− 0.844190i	−0.906552	−	0.422095i	$-0.861295\pi$
$877$	− 25.0000i	− 0.844190i	0.906552	−	0.422095i	$-0.138705\pi$
$878$	28.0000i	0.944954i
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	12.0000	0.403148
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	0	0
$889$	−64.0000	−2.14649
$890$	0	0
$891$	0	0
$892$	8.00000i	0.267860i
$893$	48.0000i	1.60626i
$894$	0	0
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	− 18.0000i	− 0.600668i
$899$	−36.0000	−1.20067
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	0	0
$904$	−15.0000	−0.498893
$905$	0	0
$906$	0	0
$907$	− 16.0000i	− 0.531271i	−0.964073	−	0.265636i	$-0.914418\pi$
$907$	− 16.0000i	− 0.531271i	0.964073	−	0.265636i	$-0.0855818\pi$
$908$	− 12.0000i	− 0.398234i
$909$	0	0
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	0	0
$914$	1.00000	0.0330771
$915$	0	0
$916$	23.0000	0.759941
$917$	− 48.0000i	− 1.58510i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	− 18.0000i	− 0.592798i
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	8.00000	0.262896
$927$	0	0
$928$	9.00000i	0.295439i
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	− 21.0000i	− 0.687878i
$933$	0	0
$934$	−24.0000	−0.785304
$935$	0	0
$936$	0	0
$937$	− 37.0000i	− 1.20874i	−0.796705	−	0.604369i	$-0.793425\pi$
$937$	− 37.0000i	− 1.20874i	0.796705	−	0.604369i	$-0.206575\pi$
$938$	− 16.0000i	− 0.522419i
$939$	0	0
$940$	0	0
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	11.0000	0.357075
$950$	0	0
$951$	0	0
$952$	− 12.0000i	− 0.388922i
$953$	9.00000i	0.291539i	0.989319	+	0.145769i	$0.0465657\pi$
$953$	9.00000i	0.291539i	−0.989319	+	0.145769i	$0.953434\pi$
$954$	0	0
$955$	0	0
$956$	−12.0000	−0.388108
$957$	0	0
$958$	− 12.0000i	− 0.387702i
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	1.00000i	0.0322413i
$963$	0	0
$964$	13.0000	0.418702
$965$	0	0
$966$	0	0
$967$	− 4.00000i	− 0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$	− 4.00000i	− 0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$	11.0000i	0.353553i
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	80.0000i	2.56468i
$974$	4.00000	0.128168
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	0	0
$985$	0	0
$986$	−27.0000	−0.859855
$987$	0	0
$988$	− 4.00000i	− 0.127257i
$989$	0	0
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	− 4.00000i	− 0.127000i
$993$	0	0
$994$	48.0000	1.52247
$995$	0	0
$996$	0	0
$997$	− 37.0000i	− 1.17180i	−0.810383	−	0.585901i	$-0.800741\pi$
$997$	− 37.0000i	− 1.17180i	0.810383	−	0.585901i	$-0.199259\pi$
$998$	40.0000i	1.26618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.c.n.649.2		2
3.2	odd	2		4050.2.c.g.649.1		2
5.2	odd	4		4050.2.a.r.1.1		1
5.3	odd	4		162.2.a.d.1.1	yes	1
5.4	even	2	inner	4050.2.c.n.649.1		2
15.2	even	4		4050.2.a.bh.1.1		1
15.8	even	4		162.2.a.a.1.1	✓	1
15.14	odd	2		4050.2.c.g.649.2		2
20.3	even	4		1296.2.a.l.1.1		1
35.13	even	4		7938.2.a.s.1.1		1
40.3	even	4		5184.2.a.h.1.1		1
40.13	odd	4		5184.2.a.c.1.1		1
45.13	odd	12		162.2.c.a.55.1		2
45.23	even	12		162.2.c.d.55.1		2
45.38	even	12		162.2.c.d.109.1		2
45.43	odd	12		162.2.c.a.109.1		2
60.23	odd	4		1296.2.a.c.1.1		1
105.83	odd	4		7938.2.a.n.1.1		1
120.53	even	4		5184.2.a.y.1.1		1
120.83	odd	4		5184.2.a.bd.1.1		1
180.23	odd	12		1296.2.i.n.865.1		2
180.43	even	12		1296.2.i.b.433.1		2
180.83	odd	12		1296.2.i.n.433.1		2
180.103	even	12		1296.2.i.b.865.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.2.a.a.1.1	✓	1	15.8	even	4
162.2.a.d.1.1	yes	1	5.3	odd	4
162.2.c.a.55.1		2	45.13	odd	12
162.2.c.a.109.1		2	45.43	odd	12
162.2.c.d.55.1		2	45.23	even	12
162.2.c.d.109.1		2	45.38	even	12
1296.2.a.c.1.1		1	60.23	odd	4
1296.2.a.l.1.1		1	20.3	even	4
1296.2.i.b.433.1		2	180.43	even	12
1296.2.i.b.865.1		2	180.103	even	12
1296.2.i.n.433.1		2	180.83	odd	12
1296.2.i.n.865.1		2	180.23	odd	12
4050.2.a.r.1.1		1	5.2	odd	4
4050.2.a.bh.1.1		1	15.2	even	4
4050.2.c.g.649.1		2	3.2	odd	2
4050.2.c.g.649.2		2	15.14	odd	2
4050.2.c.n.649.1		2	5.4	even	2	inner
4050.2.c.n.649.2		2	1.1	even	1	trivial
5184.2.a.c.1.1		1	40.13	odd	4
5184.2.a.h.1.1		1	40.3	even	4
5184.2.a.y.1.1		1	120.53	even	4
5184.2.a.bd.1.1		1	120.83	odd	4
7938.2.a.n.1.1		1	105.83	odd	4
7938.2.a.s.1.1		1	35.13	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:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} -1.00000i q^{8} +1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +4.00000 q^{19} -1.00000 q^{26} +4.00000i q^{28} +9.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} -3.00000 q^{34} -1.00000i q^{37} +4.00000i q^{38} -6.00000 q^{41} -8.00000i q^{43} +12.0000i q^{47} -9.00000 q^{49} -1.00000i q^{52} -6.00000i q^{53} -4.00000 q^{56} +9.00000i q^{58} -1.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} -3.00000i q^{68} +12.0000 q^{71} -11.0000i q^{73} +1.00000 q^{74} -4.00000 q^{76} +16.0000 q^{79} -6.00000i q^{82} -12.0000i q^{83} +8.00000 q^{86} -3.00000 q^{89} +4.00000 q^{91} -12.0000 q^{94} +2.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 8 q^{14} + 2 q^{16} + 8 q^{19} - 2 q^{26} + 18 q^{29} - 8 q^{31} - 6 q^{34} - 12 q^{41} - 18 q^{49} - 8 q^{56} - 2 q^{61} - 2 q^{64} + 24 q^{71} + 2 q^{74} - 8 q^{76} + 32 q^{79} + 16 q^{86}+ \cdots - 24 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 8 * q^14 + 2 * q^16 + 8 * q^19 - 2 * q^26 + 18 * q^29 - 8 * q^31 - 6 * q^34 - 12 * q^41 - 18 * q^49 - 8 * q^56 - 2 * q^61 - 2 * q^64 + 24 * q^71 + 2 * q^74 - 8 * q^76 + 32 * q^79 + 16 * q^86 - 6 * q^89 + 8 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

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