LMFDB - Embedded newform 2475.2.c.f.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$19.7629745003$
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 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.f.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +1.00000 q^{11} +2.00000i q^{13} -1.00000 q^{16} +6.00000i q^{17} +4.00000 q^{19} +1.00000i q^{22} -4.00000i q^{23} -2.00000 q^{26} +6.00000 q^{29} -8.00000 q^{31} +5.00000i q^{32} -6.00000 q^{34} +2.00000i q^{37} +4.00000i q^{38} -2.00000 q^{41} +4.00000i q^{43} +1.00000 q^{44} +4.00000 q^{46} -12.0000i q^{47} +7.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} +6.00000i q^{58} +4.00000 q^{59} -10.0000 q^{61} -8.00000i q^{62} -7.00000 q^{64} +16.0000i q^{67} +6.00000i q^{68} -8.00000 q^{71} +14.0000i q^{73} -2.00000 q^{74} +4.00000 q^{76} -8.00000 q^{79} -2.00000i q^{82} +4.00000i q^{83} -4.00000 q^{86} +3.00000i q^{88} +10.0000 q^{89} -4.00000i q^{92} +12.0000 q^{94} -10.0000i q^{97} +7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{11} - 2 q^{16} + 8 q^{19} - 4 q^{26} + 12 q^{29} - 16 q^{31} - 12 q^{34} - 4 q^{41} + 2 q^{44} + 8 q^{46} + 14 q^{49} + 8 q^{59} - 20 q^{61} - 14 q^{64} - 16 q^{71} - 4 q^{74} + 8 q^{76}+ \cdots + 24 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^11 - 2 * q^16 + 8 * q^19 - 4 * q^26 + 12 * q^29 - 16 * q^31 - 12 * q^34 - 4 * q^41 + 2 * q^44 + 8 * q^46 + 14 * q^49 + 8 * q^59 - 20 * q^61 - 14 * q^64 - 16 * q^71 - 4 * q^74 + 8 * q^76 - 16 * q^79 - 8 * q^86 + 20 * q^89 + 24 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	3.00000i	1.06066i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	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
$22$	1.00000i	0.213201i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	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	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.00000i	0.883883i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	4.00000i	0.648886i
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	4.00000	0.589768
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	6.00000i	0.787839i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	− 8.00000i	− 1.01600i
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	16.0000i	1.95471i	0.211604	+	0.977356i	$0.432131\pi$
$67$	16.0000i	1.95471i	−0.211604	+	0.977356i	$0.567869\pi$
$68$	6.00000i	0.727607i
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	− 2.00000i	− 0.220863i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	3.00000i	0.319801i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	− 4.00000i	− 0.417029i
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	7.00000i	0.707107i
$99$	0	0
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	4.00000i	0.368230i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 10.0000i	− 0.905357i
$123$	0	0
$124$	−8.00000	−0.718421
$125$	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	$-0.251253\pi$
$128$	3.00000i	0.265165i
$129$	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
$133$	0	0
$134$	−16.0000	−1.38219
$135$	0	0
$136$	−18.0000	−1.54349
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	− 8.00000i	− 0.671345i
$143$	2.00000i	0.167248i
$144$	0	0
$145$	0	0
$146$	−14.0000	−1.15865
$147$	0	0
$148$	2.00000i	0.164399i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	12.0000i	0.973329i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	− 8.00000i	− 0.636446i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−4.00000	−0.310460
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	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
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	10.0000i	0.749532i
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	12.0000	0.884652
$185$	0	0
$186$	0	0
$187$	6.00000i	0.438763i
$188$	− 12.0000i	− 0.875190i
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	− 26.0000i	− 1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$	− 26.0000i	− 1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	7.00000	0.500000
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	10.0000i	0.703598i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	4.00000	0.276686
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	2.00000i	0.137361i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	18.0000i	1.21911i
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	18.0000i	1.18176i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	1.00000i	0.0642824i
$243$	0	0
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	8.00000i	0.509028i
$248$	− 24.0000i	− 1.52400i
$249$	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
$253$	− 4.00000i	− 0.251478i
$254$	16.0000	1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	12.0000i	0.741362i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	16.0000i	0.977356i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	− 6.00000i	− 0.363803i
$273$	0	0
$274$	−18.0000	−1.08742
$275$	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	$-0.0971271\pi$
$278$	− 12.0000i	− 0.719712i
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0	0
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	14.0000i	0.819288i
$293$	− 10.0000i	− 0.584206i	−0.956387	−	0.292103i	$-0.905645\pi$
$293$	− 10.0000i	− 0.584206i	0.956387	−	0.292103i	$-0.0943550\pi$
$294$	0	0
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	− 10.0000i	− 0.579284i
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	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	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	0	0
$326$	−16.0000	−0.886158
$327$	0	0
$328$	− 6.00000i	− 0.331295i
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	4.00000i	0.219529i
$333$	0	0
$334$	8.00000	0.437741
$335$	0	0
$336$	0	0
$337$	− 6.00000i	− 0.326841i	−0.986557	−	0.163420i	$-0.947747\pi$
$337$	− 6.00000i	− 0.326841i	0.986557	−	0.163420i	$-0.0522527\pi$
$338$	9.00000i	0.489535i
$339$	0	0
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	5.00000i	0.266501i
$353$	− 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
$356$	10.0000	0.529999
$357$	0	0
$358$	4.00000i	0.211407i
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 10.0000i	− 0.525588i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	4.00000i	0.208514i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	18.0000i	0.932005i	0.884783	+	0.466002i	$0.154306\pi$
$373$	18.0000i	0.932005i	−0.884783	+	0.466002i	$0.845694\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	36.0000	1.85656
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	− 8.00000i	− 0.409316i
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	0	0
$385$	0	0
$386$	26.0000	1.32337
$387$	0	0
$388$	− 10.0000i	− 0.507673i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	21.0000i	1.06066i
$393$	0	0
$394$	−2.00000	−0.100759
$395$	0	0
$396$	0	0
$397$	− 30.0000i	− 1.50566i	−0.658217	−	0.752828i	$-0.728689\pi$
$397$	− 30.0000i	− 1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	10.0000	0.497519
$405$	0	0
$406$	0	0
$407$	2.00000i	0.0991363i
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	0	0
$412$	− 4.00000i	− 0.197066i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−10.0000	−0.490290
$417$	0	0
$418$	4.00000i	0.195646i
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	− 22.0000i	− 1.05725i	−0.848855	−	0.528626i	$-0.822707\pi$
$433$	− 22.0000i	− 1.05725i	0.848855	−	0.528626i	$-0.177293\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	− 8.00000i	− 0.380091i	−0.981775	−	0.190046i	$-0.939136\pi$
$443$	− 8.00000i	− 0.380091i	0.981775	−	0.190046i	$-0.0608636\pi$
$444$	0	0
$445$	0	0
$446$	4.00000	0.189405
$447$	0	0
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	6.00000i	0.282216i
$453$	0	0
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	10.0000i	0.467269i
$459$	0	0
$460$	0	0
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	0	0
$463$	− 36.0000i	− 1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$	− 36.0000i	− 1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	4.00000i	0.183920i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	8.00000i	0.365911i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	10.0000i	0.455488i
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	− 28.0000i	− 1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$	− 28.0000i	− 1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$	− 30.0000i	− 1.35804i
$489$	0	0
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	36.0000i	1.62136i
$494$	−8.00000	−0.359937
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	16.0000i	0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$	16.0000i	0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	0	0
$505$	0	0
$506$	4.00000	0.177822
$507$	0	0
$508$	− 16.0000i	− 0.709885i
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	0	0
$512$	− 11.0000i	− 0.486136i
$513$	0	0
$514$	−18.0000	−0.793946
$515$	0	0
$516$	0	0
$517$	− 12.0000i	− 0.527759i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	− 48.0000i	− 2.09091i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.00000i	− 0.173259i
$534$	0	0
$535$	0	0
$536$	−48.0000	−2.07328
$537$	0	0
$538$	− 18.0000i	− 0.776035i
$539$	7.00000	0.301511
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	0	0
$544$	−30.0000	−1.28624
$545$	0	0
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	18.0000i	0.768922i
$549$	0	0
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	−12.0000	−0.508913
$557$	10.0000i	0.423714i	0.977301	+	0.211857i	$0.0679510\pi$
$557$	10.0000i	0.423714i	−0.977301	+	0.211857i	$0.932049\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	− 18.0000i	− 0.759284i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	− 24.0000i	− 1.00702i
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	2.00000i	0.0836242i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	− 19.0000i	− 0.790296i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.00000i	0.0828315i
$584$	−42.0000	−1.73797
$585$	0	0
$586$	10.0000	0.413096
$587$	− 24.0000i	− 0.990586i	−0.868726	−	0.495293i	$-0.835061\pi$
$587$	− 24.0000i	− 0.990586i	0.868726	−	0.495293i	$-0.164939\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	− 2.00000i	− 0.0821995i
$593$	− 22.0000i	− 0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$	− 22.0000i	− 0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	8.00000i	0.327144i
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	0	0
$604$	8.00000	0.325515
$605$	0	0
$606$	0	0
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	20.0000i	0.811107i
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	0	0
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	24.0000i	0.962312i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	22.0000	0.879297
$627$	0	0
$628$	2.00000i	0.0798087i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	− 24.0000i	− 0.954669i
$633$	0	0
$634$	18.0000	0.714871
$635$	0	0
$636$	0	0
$637$	14.0000i	0.554700i
$638$	6.00000i	0.237542i
$639$	0	0
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	20.0000i	0.786281i	0.919478	+	0.393141i	$0.128611\pi$
$647$	20.0000i	0.786281i	−0.919478	+	0.393141i	$0.871389\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	10.0000i	0.391330i	0.980671	+	0.195665i	$0.0626866\pi$
$653$	10.0000i	0.391330i	−0.980671	+	0.195665i	$0.937313\pi$
$654$	0	0
$655$	0	0
$656$	2.00000	0.0780869
$657$	0	0
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	4.00000i	0.155464i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	− 24.0000i	− 0.929284i
$668$	− 8.00000i	− 0.309529i
$669$	0	0
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	9.00000	0.346154
$677$	− 38.0000i	− 1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$	− 38.0000i	− 1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	− 8.00000i	− 0.306336i
$683$	16.0000i	0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$	16.0000i	0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	4.00000	0.151838
$695$	0	0
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	10.0000i	0.378506i
$699$	0	0
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	−7.00000	−0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	0	0
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	30.0000i	1.12430i
$713$	32.0000i	1.19841i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	− 32.0000i	− 1.19423i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	− 3.00000i	− 0.111648i
$723$	0	0
$724$	−10.0000	−0.371647
$725$	0	0
$726$	0	0
$727$	− 52.0000i	− 1.92857i	−0.264861	−	0.964287i	$-0.585326\pi$
$727$	− 52.0000i	− 1.92857i	0.264861	−	0.964287i	$-0.414674\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	42.0000i	1.55131i	0.631160	+	0.775653i	$0.282579\pi$
$733$	42.0000i	1.55131i	−0.631160	+	0.775653i	$0.717421\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	20.0000	0.737210
$737$	16.0000i	0.589368i
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 40.0000i	− 1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	− 40.0000i	− 1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$	0	0
$745$	0	0
$746$	−18.0000	−0.659027
$747$	0	0
$748$	6.00000i	0.219382i
$749$	0	0
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	12.0000i	0.437595i
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	− 6.00000i	− 0.218074i	−0.994038	−	0.109037i	$-0.965223\pi$
$757$	− 6.00000i	− 0.218074i	0.994038	−	0.109037i	$-0.0347767\pi$
$758$	− 20.0000i	− 0.726433i
$759$	0	0
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	0	0
$763$	0	0
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−12.0000	−0.433578
$767$	8.00000i	0.288863i
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	− 26.0000i	− 0.935760i
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	0	0
$775$	0	0
$776$	30.0000	1.07694
$777$	0	0
$778$	6.00000i	0.215110i
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−8.00000	−0.286263
$782$	24.0000i	0.858238i
$783$	0	0
$784$	−7.00000	−0.250000
$785$	0	0
$786$	0	0
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	2.00000i	0.0712470i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 20.0000i	− 0.710221i
$794$	30.0000	1.06466
$795$	0	0
$796$	0	0
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	0	0
$801$	0	0
$802$	− 2.00000i	− 0.0706225i
$803$	14.0000i	0.494049i
$804$	0	0
$805$	0	0
$806$	16.0000	0.563576
$807$	0	0
$808$	30.0000i	1.05540i
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	0	0
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	6.00000i	0.209785i
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	− 36.0000i	− 1.25488i	−0.778664	−	0.627441i	$-0.784103\pi$
$823$	− 36.0000i	− 1.25488i	0.778664	−	0.627441i	$-0.215897\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	0	0
$827$	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
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	− 14.0000i	− 0.485363i
$833$	42.0000i	1.45521i
$834$	0	0
$835$	0	0
$836$	4.00000	0.138343
$837$	0	0
$838$	− 28.0000i	− 0.967244i
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	6.00000i	0.206774i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 2.00000i	− 0.0686803i
$849$	0	0
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	2.00000i	0.0684787i	0.999414	+	0.0342393i	$0.0109009\pi$
$853$	2.00000i	0.0684787i	−0.999414	+	0.0342393i	$0.989099\pi$
$854$	0	0
$855$	0	0
$856$	−36.0000	−1.23045
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	0	0
$862$	24.0000i	0.817443i
$863$	− 4.00000i	− 0.136162i	−0.997680	−	0.0680808i	$-0.978312\pi$
$863$	− 4.00000i	− 0.136162i	0.997680	−	0.0680808i	$-0.0216876\pi$
$864$	0	0
$865$	0	0
$866$	22.0000	0.747590
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−32.0000	−1.08428
$872$	54.0000i	1.82867i
$873$	0	0
$874$	16.0000	0.541208
$875$	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	$0.878863\pi$
$878$	8.00000i	0.269987i
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	− 16.0000i	− 0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$	− 16.0000i	− 0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	8.00000	0.268765
$887$	56.0000i	1.88030i	0.340766	+	0.940148i	$0.389313\pi$
$887$	56.0000i	1.88030i	−0.340766	+	0.940148i	$0.610687\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 4.00000i	− 0.133930i
$893$	− 48.0000i	− 1.60626i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	2.00000i	0.0667409i
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−12.0000	−0.399778
$902$	− 2.00000i	− 0.0665927i
$903$	0	0
$904$	−18.0000	−0.598671
$905$	0	0
$906$	0	0
$907$	40.0000i	1.32818i	0.747653	+	0.664089i	$0.231180\pi$
$907$	40.0000i	1.32818i	−0.747653	+	0.664089i	$0.768820\pi$
$908$	− 20.0000i	− 0.663723i
$909$	0	0
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	4.00000i	0.132381i
$914$	−26.0000	−0.860004
$915$	0	0
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	34.0000i	1.11973i
$923$	− 16.0000i	− 0.526646i
$924$	0	0
$925$	0	0
$926$	36.0000	1.18303
$927$	0	0
$928$	30.0000i	0.984798i
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	− 6.00000i	− 0.196537i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 6.00000i	− 0.196011i	−0.995186	−	0.0980057i	$-0.968754\pi$
$937$	− 6.00000i	− 0.196011i	0.995186	−	0.0980057i	$-0.0312463\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	0	0
$943$	8.00000i	0.260516i
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−4.00000	−0.130051
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	0	0
$953$	42.0000i	1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$	42.0000i	1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$	0	0
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	24.0000i	0.775405i
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	− 4.00000i	− 0.128965i
$963$	0	0
$964$	10.0000	0.322078
$965$	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	$-0.0410584\pi$
$968$	3.00000i	0.0964237i
$969$	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
$973$	0	0
$974$	28.0000	0.897178
$975$	0	0
$976$	10.0000	0.320092
$977$	10.0000i	0.319928i	0.987123	+	0.159964i	$0.0511379\pi$
$977$	10.0000i	0.319928i	−0.987123	+	0.159964i	$0.948862\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	0	0
$982$	28.0000i	0.893516i
$983$	4.00000i	0.127580i	0.997963	+	0.0637901i	$0.0203188\pi$
$983$	4.00000i	0.127580i	−0.997963	+	0.0637901i	$0.979681\pi$
$984$	0	0
$985$	0	0
$986$	−36.0000	−1.14647
$987$	0	0
$988$	8.00000i	0.254514i
$989$	16.0000	0.508770
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	− 40.0000i	− 1.27000i
$993$	0	0
$994$	0	0
$995$	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	$0.740256\pi$
$998$	36.0000i	1.13956i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.f.199.2		2
3.2	odd	2		275.2.b.b.199.1		2
5.2	odd	4		495.2.a.a.1.1		1
5.3	odd	4		2475.2.a.i.1.1		1
5.4	even	2	inner	2475.2.c.f.199.1		2
12.11	even	2		4400.2.b.n.4049.2		2
15.2	even	4		55.2.a.a.1.1	✓	1
15.8	even	4		275.2.a.a.1.1		1
15.14	odd	2		275.2.b.b.199.2		2
20.7	even	4		7920.2.a.i.1.1		1
55.32	even	4		5445.2.a.i.1.1		1
60.23	odd	4		4400.2.a.p.1.1		1
60.47	odd	4		880.2.a.h.1.1		1
60.59	even	2		4400.2.b.n.4049.1		2
105.62	odd	4		2695.2.a.c.1.1		1
120.77	even	4		3520.2.a.p.1.1		1
120.107	odd	4		3520.2.a.n.1.1		1
165.2	odd	20		605.2.g.c.81.1		4
165.17	odd	20		605.2.g.c.366.1		4
165.32	odd	4		605.2.a.b.1.1		1
165.47	even	20		605.2.g.a.251.1		4
165.62	odd	20		605.2.g.c.511.1		4
165.92	even	20		605.2.g.a.511.1		4
165.98	odd	4		3025.2.a.f.1.1		1
165.107	odd	20		605.2.g.c.251.1		4
165.137	even	20		605.2.g.a.366.1		4
165.152	even	20		605.2.g.a.81.1		4
195.77	even	4		9295.2.a.b.1.1		1
660.527	even	4		9680.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.a.1.1	✓	1	15.2	even	4
275.2.a.a.1.1		1	15.8	even	4
275.2.b.b.199.1		2	3.2	odd	2
275.2.b.b.199.2		2	15.14	odd	2
495.2.a.a.1.1		1	5.2	odd	4
605.2.a.b.1.1		1	165.32	odd	4
605.2.g.a.81.1		4	165.152	even	20
605.2.g.a.251.1		4	165.47	even	20
605.2.g.a.366.1		4	165.137	even	20
605.2.g.a.511.1		4	165.92	even	20
605.2.g.c.81.1		4	165.2	odd	20
605.2.g.c.251.1		4	165.107	odd	20
605.2.g.c.366.1		4	165.17	odd	20
605.2.g.c.511.1		4	165.62	odd	20
880.2.a.h.1.1		1	60.47	odd	4
2475.2.a.i.1.1		1	5.3	odd	4
2475.2.c.f.199.1		2	5.4	even	2	inner
2475.2.c.f.199.2		2	1.1	even	1	trivial
2695.2.a.c.1.1		1	105.62	odd	4
3025.2.a.f.1.1		1	165.98	odd	4
3520.2.a.n.1.1		1	120.107	odd	4
3520.2.a.p.1.1		1	120.77	even	4
4400.2.a.p.1.1		1	60.23	odd	4
4400.2.b.n.4049.1		2	60.59	even	2
4400.2.b.n.4049.2		2	12.11	even	2
5445.2.a.i.1.1		1	55.32	even	4
7920.2.a.i.1.1		1	20.7	even	4
9295.2.a.b.1.1		1	195.77	even	4
9680.2.a.r.1.1		1	660.527	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +1.00000 q^{11} +2.00000i q^{13} -1.00000 q^{16} +6.00000i q^{17} +4.00000 q^{19} +1.00000i q^{22} -4.00000i q^{23} -2.00000 q^{26} +6.00000 q^{29} -8.00000 q^{31} +5.00000i q^{32} -6.00000 q^{34} +2.00000i q^{37} +4.00000i q^{38} -2.00000 q^{41} +4.00000i q^{43} +1.00000 q^{44} +4.00000 q^{46} -12.0000i q^{47} +7.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} +6.00000i q^{58} +4.00000 q^{59} -10.0000 q^{61} -8.00000i q^{62} -7.00000 q^{64} +16.0000i q^{67} +6.00000i q^{68} -8.00000 q^{71} +14.0000i q^{73} -2.00000 q^{74} +4.00000 q^{76} -8.00000 q^{79} -2.00000i q^{82} +4.00000i q^{83} -4.00000 q^{86} +3.00000i q^{88} +10.0000 q^{89} -4.00000i q^{92} +12.0000 q^{94} -10.0000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{11} - 2 q^{16} + 8 q^{19} - 4 q^{26} + 12 q^{29} - 16 q^{31} - 12 q^{34} - 4 q^{41} + 2 q^{44} + 8 q^{46} + 14 q^{49} + 8 q^{59} - 20 q^{61} - 14 q^{64} - 16 q^{71} - 4 q^{74} + 8 q^{76}+ \cdots + 24 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^11 - 2 * q^16 + 8 * q^19 - 4 * q^26 + 12 * q^29 - 16 * q^31 - 12 * q^34 - 4 * q^41 + 2 * q^44 + 8 * q^46 + 14 * q^49 + 8 * q^59 - 20 * q^61 - 14 * q^64 - 16 * q^71 - 4 * q^74 + 8 * q^76 - 16 * q^79 - 8 * q^86 + 20 * q^89 + 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more