LMFDB - Embedded newform 2475.2.c.d.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 33)
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.d.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} +3.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{11} -2.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{22} -8.00000i q^{23} +2.00000 q^{26} -4.00000i q^{28} -6.00000 q^{29} -8.00000 q^{31} +5.00000i q^{32} +2.00000 q^{34} -6.00000i q^{37} +2.00000 q^{41} -1.00000 q^{44} +8.00000 q^{46} +8.00000i q^{47} -9.00000 q^{49} -2.00000i q^{52} -6.00000i q^{53} +12.0000 q^{56} -6.00000i q^{58} -4.00000 q^{59} +6.00000 q^{61} -8.00000i q^{62} -7.00000 q^{64} +4.00000i q^{67} -2.00000i q^{68} -14.0000i q^{73} +6.00000 q^{74} +4.00000i q^{77} +4.00000 q^{79} +2.00000i q^{82} -12.0000i q^{83} -3.00000i q^{88} -6.00000 q^{89} -8.00000 q^{91} -8.00000i q^{92} -8.00000 q^{94} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} - 2 q^{11} + 8 q^{14} - 2 q^{16} + 4 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 4 q^{41} - 2 q^{44} + 16 q^{46} - 18 q^{49} + 24 q^{56} - 8 q^{59} + 12 q^{61} - 14 q^{64} + 12 q^{74} + 8 q^{79} - 12 q^{89} - 16 q^{91} - 16 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^11 + 8 * q^14 - 2 * q^16 + 4 * q^26 - 12 * q^29 - 16 * q^31 + 4 * q^34 + 4 * q^41 - 2 * q^44 + 16 * q^46 - 18 * q^49 + 24 * q^56 - 8 * q^59 + 12 * q^61 - 14 * q^64 + 12 * q^74 + 8 * q^79 - 12 * q^89 - 16 * q^91 - 16 * q^94

Display 100 coefficients 10 coefficients

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
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	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	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	− 4.00000i	− 0.755929i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	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
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	8.00000	1.17954
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	12.0000	1.60357
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	− 8.00000i	− 1.01600i
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 2.00000i	− 0.242536i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	0	0
$77$	4.00000i	0.455842i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	2.00000i	0.220863i
$83$	− 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
$86$	0	0
$87$	0	0
$88$	− 3.00000i	− 0.319801i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	− 8.00000i	− 0.834058i
$93$	0	0
$94$	−8.00000	−0.825137
$95$	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	$-0.0323753\pi$
$98$	− 9.00000i	− 0.909137i
$99$	0	0
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	6.00000	0.588348
$105$	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
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	4.00000i	0.377964i
$113$	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
$119$	−8.00000	−0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.00000i	0.543214i
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	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
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.00000i	0.167248i
$144$	0	0
$145$	0	0
$146$	14.0000	1.15865
$147$	0	0
$148$	− 6.00000i	− 0.493197i
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	−4.00000	−0.322329
$155$	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	$-0.188685\pi$
$158$	4.00000i	0.318223i
$159$	0	0
$160$	0	0
$161$	−32.0000	−2.52195
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	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
$178$	− 6.00000i	− 0.449719i
$179$	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
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	− 8.00000i	− 0.592999i
$183$	0	0
$184$	24.0000	1.76930
$185$	0	0
$186$	0	0
$187$	2.00000i	0.146254i
$188$	8.00000i	0.583460i
$189$	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
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	− 14.0000i	− 0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$	− 14.0000i	− 0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	− 2.00000i	− 0.140720i
$203$	24.0000i	1.68447i
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	32.0000i	2.17230i
$218$	2.00000i	0.135457i
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	20.0000	1.33631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	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
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	− 18.0000i	− 1.18176i
$233$	− 30.0000i	− 1.96537i	−0.185296	−	0.982683i	$-0.559325\pi$
$233$	− 30.0000i	− 1.96537i	0.185296	−	0.982683i	$-0.440675\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	− 8.00000i	− 0.518563i
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	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
$244$	6.00000	0.384111
$245$	0	0
$246$	0	0
$247$	0	0
$248$	− 24.0000i	− 1.52400i
$249$	0	0
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 14.0000i	− 0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$	− 14.0000i	− 0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	0	0
$262$	12.0000i	0.741362i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	2.00000i	0.121268i
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	8.00000i	0.479808i
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	−2.00000	−0.118262
$287$	− 8.00000i	− 0.472225i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	− 14.0000i	− 0.819288i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	18.0000	1.04623
$297$	0	0
$298$	− 22.0000i	− 1.27443i
$299$	−16.0000	−0.925304
$300$	0	0
$301$	0	0
$302$	20.0000i	1.15087i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 32.0000i	− 1.82634i	−0.407583	−	0.913168i	$-0.633628\pi$
$307$	− 32.0000i	− 1.82634i	0.407583	−	0.913168i	$-0.366372\pi$
$308$	4.00000i	0.227921i
$309$	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	$-0.213581\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	4.00000	0.225018
$317$	22.0000i	1.23564i	0.786318	+	0.617822i	$0.211985\pi$
$317$	22.0000i	1.23564i	−0.786318	+	0.617822i	$0.788015\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	− 32.0000i	− 1.78329i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	6.00000i	0.331295i
$329$	32.0000	1.76422
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	9.00000i	0.489535i
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	8.00000i	0.431959i
$344$	0	0
$345$	0	0
$346$	−6.00000	−0.322562
$347$	4.00000i	0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$	4.00000i	0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	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
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000i	0.634220i
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	22.0000i	1.15629i
$363$	0	0
$364$	−8.00000	−0.419314
$365$	0	0
$366$	0	0
$367$	32.0000i	1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$	32.0000i	1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$	8.00000i	0.417029i
$369$	0	0
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	−24.0000	−1.23771
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	− 8.00000i	− 0.409316i
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	0	0
$385$	0	0
$386$	14.0000	0.712581
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	− 27.0000i	− 1.36371i
$393$	0	0
$394$	14.0000	0.705310
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	0	0
$403$	16.0000i	0.797017i
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	−24.0000	−1.19110
$407$	6.00000i	0.297409i
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	8.00000i	0.394132i
$413$	16.0000i	0.787309i
$414$	0	0
$415$	0	0
$416$	10.0000	0.490290
$417$	0	0
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	0	0
$424$	18.0000	0.874157
$425$	0	0
$426$	0	0
$427$	− 24.0000i	− 1.16144i
$428$	− 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
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	0	0
$442$	− 4.00000i	− 0.190261i
$443$	− 28.0000i	− 1.33032i	−0.746701	−	0.665160i	$-0.768363\pi$
$443$	− 28.0000i	− 1.33032i	0.746701	−	0.665160i	$-0.231637\pi$
$444$	0	0
$445$	0	0
$446$	−16.0000	−0.757622
$447$	0	0
$448$	28.0000i	1.32288i
$449$	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
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	− 6.00000i	− 0.280362i
$459$	0	0
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	30.0000	1.38972
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	0	0
$472$	− 12.0000i	− 0.552345i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−8.00000	−0.366679
$477$	0	0
$478$	24.0000i	1.09773i
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	10.0000i	0.455488i
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	18.0000i	0.814822i
$489$	0	0
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	− 4.00000i	− 0.178529i
$503$	32.0000i	1.42681i	0.700752	+	0.713405i	$0.252848\pi$
$503$	32.0000i	1.42681i	−0.700752	+	0.713405i	$0.747152\pi$
$504$	0	0
$505$	0	0
$506$	−8.00000	−0.355643
$507$	0	0
$508$	4.00000i	0.177471i
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	− 11.0000i	− 0.486136i
$513$	0	0
$514$	14.0000	0.617514
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	− 24.0000i	− 1.05450i
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	16.0000i	0.696971i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.00000i	− 0.173259i
$534$	0	0
$535$	0	0
$536$	−12.0000	−0.518321
$537$	0	0
$538$	− 2.00000i	− 0.0862261i
$539$	9.00000	0.387657
$540$	0	0
$541$	46.0000	1.97769	0.988847	−	0.148933i	$-0.0475840\pi$
$541$	46.0000	1.97769	0.988847	+	0.148933i	$0.0475840\pi$
$542$	20.0000i	0.859074i
$543$	0	0
$544$	10.0000	0.428746
$545$	0	0
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	2.00000i	0.0854358i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	−26.0000	−1.10463
$555$	0	0
$556$	8.00000	0.339276
$557$	− 14.0000i	− 0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$	− 14.0000i	− 0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	44.0000i	1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$	44.0000i	1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$	0	0
$565$	0	0
$566$	−16.0000	−0.672530
$567$	0	0
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	2.00000i	0.0836242i
$573$	0	0
$574$	8.00000	0.333914
$575$	0	0
$576$	0	0
$577$	30.0000i	1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$577$	30.0000i	1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$578$	13.0000i	0.540729i
$579$	0	0
$580$	0	0
$581$	−48.0000	−1.99138
$582$	0	0
$583$	6.00000i	0.248495i
$584$	42.0000	1.73797
$585$	0	0
$586$	−6.00000	−0.247858
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	6.00000i	0.246598i
$593$	− 38.0000i	− 1.56047i	−0.625485	−	0.780236i	$-0.715099\pi$
$593$	− 38.0000i	− 1.56047i	0.625485	−	0.780236i	$-0.284901\pi$
$594$	0	0
$595$	0	0
$596$	−22.0000	−0.901155
$597$	0	0
$598$	− 16.0000i	− 0.654289i
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	0	0
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	4.00000i	0.162355i	0.996700	+	0.0811775i	$0.0258681\pi$
$607$	4.00000i	0.162355i	−0.996700	+	0.0811775i	$0.974132\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	−12.0000	−0.483494
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	24.0000i	0.962312i
$623$	24.0000i	0.961540i
$624$	0	0
$625$	0	0
$626$	22.0000	0.879297
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	12.0000i	0.477334i
$633$	0	0
$634$	−22.0000	−0.873732
$635$	0	0
$636$	0	0
$637$	18.0000i	0.713186i
$638$	6.00000i	0.237542i
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	−32.0000	−1.26098
$645$	0	0
$646$	0	0
$647$	8.00000i	0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$	8.00000i	0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	2.00000i	0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$	2.00000i	0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$	0	0
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	32.0000i	1.24749i
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	− 20.0000i	− 0.777322i
$663$	0	0
$664$	36.0000	1.39707
$665$	0	0
$666$	0	0
$667$	48.0000i	1.85857i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	− 46.0000i	− 1.77317i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	− 46.0000i	− 1.77317i	0.462566	−	0.886585i	$-0.346929\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	9.00000	0.346154
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	0	0
$682$	8.00000i	0.306336i
$683$	− 20.0000i	− 0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$	− 20.0000i	− 0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	0	0
$688$	0	0
$689$	−12.0000	−0.457164
$690$	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
$697$	− 4.00000i	− 0.151511i
$698$	− 6.00000i	− 0.227103i
$699$	0	0
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	0	0
$704$	7.00000	0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	8.00000i	0.300871i
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	0	0
$712$	− 18.0000i	− 0.674579i
$713$	64.0000i	2.39682i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	− 8.00000i	− 0.298557i
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	− 19.0000i	− 0.707107i
$723$	0	0
$724$	22.0000	0.817624
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	− 24.0000i	− 0.889499i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	30.0000i	1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$	30.0000i	1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	40.0000	1.47442
$737$	− 4.00000i	− 0.147342i
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	0	0
$742$	− 24.0000i	− 0.881068i
$743$	− 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
$746$	2.00000	0.0732252
$747$	0	0
$748$	2.00000i	0.0731272i
$749$	−48.0000	−1.75388
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	− 8.00000i	− 0.291730i
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	− 28.0000i	− 1.01701i
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	− 8.00000i	− 0.289619i
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−16.0000	−0.578103
$767$	8.00000i	0.288863i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	− 14.0000i	− 0.503871i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	− 18.0000i	− 0.645331i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 16.0000i	− 0.572159i
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	0	0
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	− 14.0000i	− 0.498729i
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	− 12.0000i	− 0.426132i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	0	0
$797$	− 10.0000i	− 0.354218i	−0.984191	−	0.177109i	$-0.943325\pi$
$797$	− 10.0000i	− 0.354218i	0.984191	−	0.177109i	$-0.0566745\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	0	0
$802$	− 26.0000i	− 0.918092i
$803$	14.0000i	0.494049i
$804$	0	0
$805$	0	0
$806$	−16.0000	−0.563576
$807$	0	0
$808$	− 6.00000i	− 0.211079i
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	−56.0000	−1.96643	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	−56.0000	−1.96643	−0.983213	+	0.182462i	$0.941593\pi$
$812$	24.0000i	0.842235i
$813$	0	0
$814$	−6.00000	−0.210300
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 18.0000i	− 0.629355i
$819$	0	0
$820$	0	0
$821$	14.0000	0.488603	0.244302	−	0.969699i	$-0.421441\pi$
$821$	14.0000	0.488603	0.244302	+	0.969699i	$0.421441\pi$
$822$	0	0
$823$	24.0000i	0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$	24.0000i	0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	−16.0000	−0.556711
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	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
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	− 4.00000i	− 0.138178i
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 26.0000i	− 0.896019i
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 4.00000i	− 0.137442i
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	− 34.0000i	− 1.16414i	−0.813139	−	0.582069i	$-0.802243\pi$
$853$	− 34.0000i	− 1.16414i	0.813139	−	0.582069i	$-0.197757\pi$
$854$	24.0000	0.821263
$855$	0	0
$856$	36.0000	1.23045
$857$	− 10.0000i	− 0.341593i	−0.985306	−	0.170797i	$-0.945366\pi$
$857$	− 10.0000i	− 0.341593i	0.985306	−	0.170797i	$-0.0546341\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	0	0
$862$	24.0000i	0.817443i
$863$	− 48.0000i	− 1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$	− 48.0000i	− 1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$	0	0
$865$	0	0
$866$	−34.0000	−1.15537
$867$	0	0
$868$	32.0000i	1.08615i
$869$	−4.00000	−0.135691
$870$	0	0
$871$	8.00000	0.271070
$872$	6.00000i	0.203186i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 6.00000i	− 0.202606i	−0.994856	−	0.101303i	$-0.967699\pi$
$877$	− 6.00000i	− 0.202606i	0.994856	−	0.101303i	$-0.0323011\pi$
$878$	20.0000i	0.674967i
$879$	0	0
$880$	0	0
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	28.0000	0.940678
$887$	8.00000i	0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$	8.00000i	0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	16.0000i	0.535720i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	12.0000	0.400892
$897$	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
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	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
$913$	12.0000i	0.397142i
$914$	18.0000	0.595387
$915$	0	0
$916$	−6.00000	−0.198246
$917$	− 48.0000i	− 1.58510i
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	0	0
$922$	30.0000i	0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−16.0000	−0.525793
$927$	0	0
$928$	− 30.0000i	− 0.984798i
$929$	−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
$931$	0	0
$932$	− 30.0000i	− 0.982683i
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	16.0000i	0.522419i
$939$	0	0
$940$	0	0
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	0	0
$943$	− 16.0000i	− 0.521032i
$944$	4.00000	0.130189
$945$	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
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	− 24.0000i	− 0.777844i
$953$	− 22.0000i	− 0.712650i	−0.934362	−	0.356325i	$-0.884030\pi$
$953$	− 22.0000i	− 0.712650i	0.934362	−	0.356325i	$-0.115970\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	8.00000i	0.258468i
$959$	8.00000	0.258333
$960$	0	0
$961$	33.0000	1.06452
$962$	− 12.0000i	− 0.386896i
$963$	0	0
$964$	10.0000	0.322078
$965$	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	$-0.0204865\pi$
$968$	3.00000i	0.0964237i
$969$	0	0
$970$	0	0
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	0	0
$973$	− 32.0000i	− 1.02587i
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−6.00000	−0.192055
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	0	0
$982$	− 4.00000i	− 0.127645i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	− 40.0000i	− 1.27000i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	4.00000i	0.126618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.d.199.2		2
3.2	odd	2		825.2.c.a.199.1		2
5.2	odd	4		99.2.a.b.1.1		1
5.3	odd	4		2475.2.a.g.1.1		1
5.4	even	2	inner	2475.2.c.d.199.1		2
15.2	even	4		33.2.a.a.1.1	✓	1
15.8	even	4		825.2.a.a.1.1		1
15.14	odd	2		825.2.c.a.199.2		2
20.7	even	4		1584.2.a.o.1.1		1
35.27	even	4		4851.2.a.b.1.1		1
40.27	even	4		6336.2.a.n.1.1		1
40.37	odd	4		6336.2.a.x.1.1		1
45.2	even	12		891.2.e.e.595.1		2
45.7	odd	12		891.2.e.g.595.1		2
45.22	odd	12		891.2.e.g.298.1		2
45.32	even	12		891.2.e.e.298.1		2
55.32	even	4		1089.2.a.j.1.1		1
60.47	odd	4		528.2.a.g.1.1		1
105.62	odd	4		1617.2.a.j.1.1		1
120.77	even	4		2112.2.a.bb.1.1		1
120.107	odd	4		2112.2.a.j.1.1		1
165.2	odd	20		363.2.e.g.202.1		4
165.17	odd	20		363.2.e.g.124.1		4
165.32	odd	4		363.2.a.b.1.1		1
165.47	even	20		363.2.e.e.130.1		4
165.62	odd	20		363.2.e.g.148.1		4
165.92	even	20		363.2.e.e.148.1		4
165.98	odd	4		9075.2.a.q.1.1		1
165.107	odd	20		363.2.e.g.130.1		4
165.137	even	20		363.2.e.e.124.1		4
165.152	even	20		363.2.e.e.202.1		4
195.77	even	4		5577.2.a.a.1.1		1
255.152	even	4		9537.2.a.m.1.1		1
660.527	even	4		5808.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.a.a.1.1	✓	1	15.2	even	4
99.2.a.b.1.1		1	5.2	odd	4
363.2.a.b.1.1		1	165.32	odd	4
363.2.e.e.124.1		4	165.137	even	20
363.2.e.e.130.1		4	165.47	even	20
363.2.e.e.148.1		4	165.92	even	20
363.2.e.e.202.1		4	165.152	even	20
363.2.e.g.124.1		4	165.17	odd	20
363.2.e.g.130.1		4	165.107	odd	20
363.2.e.g.148.1		4	165.62	odd	20
363.2.e.g.202.1		4	165.2	odd	20
528.2.a.g.1.1		1	60.47	odd	4
825.2.a.a.1.1		1	15.8	even	4
825.2.c.a.199.1		2	3.2	odd	2
825.2.c.a.199.2		2	15.14	odd	2
891.2.e.e.298.1		2	45.32	even	12
891.2.e.e.595.1		2	45.2	even	12
891.2.e.g.298.1		2	45.22	odd	12
891.2.e.g.595.1		2	45.7	odd	12
1089.2.a.j.1.1		1	55.32	even	4
1584.2.a.o.1.1		1	20.7	even	4
1617.2.a.j.1.1		1	105.62	odd	4
2112.2.a.j.1.1		1	120.107	odd	4
2112.2.a.bb.1.1		1	120.77	even	4
2475.2.a.g.1.1		1	5.3	odd	4
2475.2.c.d.199.1		2	5.4	even	2	inner
2475.2.c.d.199.2		2	1.1	even	1	trivial
4851.2.a.b.1.1		1	35.27	even	4
5577.2.a.a.1.1		1	195.77	even	4
5808.2.a.t.1.1		1	660.527	even	4
6336.2.a.n.1.1		1	40.27	even	4
6336.2.a.x.1.1		1	40.37	odd	4
9075.2.a.q.1.1		1	165.98	odd	4
9537.2.a.m.1.1		1	255.152	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{11} -2.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{22} -8.00000i q^{23} +2.00000 q^{26} -4.00000i q^{28} -6.00000 q^{29} -8.00000 q^{31} +5.00000i q^{32} +2.00000 q^{34} -6.00000i q^{37} +2.00000 q^{41} -1.00000 q^{44} +8.00000 q^{46} +8.00000i q^{47} -9.00000 q^{49} -2.00000i q^{52} -6.00000i q^{53} +12.0000 q^{56} -6.00000i q^{58} -4.00000 q^{59} +6.00000 q^{61} -8.00000i q^{62} -7.00000 q^{64} +4.00000i q^{67} -2.00000i q^{68} -14.0000i q^{73} +6.00000 q^{74} +4.00000i q^{77} +4.00000 q^{79} +2.00000i q^{82} -12.0000i q^{83} -3.00000i q^{88} -6.00000 q^{89} -8.00000 q^{91} -8.00000i q^{92} -8.00000 q^{94} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} - 2 q^{11} + 8 q^{14} - 2 q^{16} + 4 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 4 q^{41} - 2 q^{44} + 16 q^{46} - 18 q^{49} + 24 q^{56} - 8 q^{59} + 12 q^{61} - 14 q^{64} + 12 q^{74} + 8 q^{79} - 12 q^{89} - 16 q^{91} - 16 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^11 + 8 * q^14 - 2 * q^16 + 4 * q^26 - 12 * q^29 - 16 * q^31 + 4 * q^34 + 4 * q^41 - 2 * q^44 + 16 * q^46 - 18 * q^49 + 24 * q^56 - 8 * q^59 + 12 * q^61 - 14 * q^64 + 12 * q^74 + 8 * q^79 - 12 * q^89 - 16 * q^91 - 16 * q^94

Introduction

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