LMFDB - Embedded newform 3300.2.c.c.1849.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3300,2,Mod(1849,3300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3300.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3300.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:	$26.3506326670$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3300.1849
Dual form			3300.2.c.c.1849.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} +6.00000i q^{13} +4.00000i q^{17} +2.00000 q^{19} -2.00000 q^{21} -8.00000i q^{23} +1.00000i q^{27} +1.00000i q^{33} +6.00000i q^{37} +6.00000 q^{39} +10.0000i q^{43} +3.00000 q^{49} +4.00000 q^{51} +14.0000i q^{53} -2.00000i q^{57} +12.0000 q^{59} -14.0000 q^{61} +2.00000i q^{63} -4.00000i q^{67} -8.00000 q^{69} +6.00000i q^{73} +2.00000i q^{77} -2.00000 q^{79} +1.00000 q^{81} +16.0000i q^{83} +14.0000 q^{89} +12.0000 q^{91} +2.00000i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 2 q^{11} + 4 q^{19} - 4 q^{21} + 12 q^{39} + 6 q^{49} + 8 q^{51} + 24 q^{59} - 28 q^{61} - 16 q^{69} - 4 q^{79} + 2 q^{81} + 28 q^{89} + 24 q^{91} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 2 * q^11 + 4 * q^19 - 4 * q^21 + 12 * q^39 + 6 * q^49 + 8 * q^51 + 24 * q^59 - 28 * q^61 - 16 * q^69 - 4 * q^79 + 2 * q^81 + 28 * q^89 + 24 * q^91 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3300\mathbb{Z}\right)^\times$.

$n$	$1201$	$1651$	$2201$	$2377$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	− 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
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	1.00000i	0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	14.0000i	1.92305i	0.274721	+	0.961524i	$0.411414\pi$
$53$	14.0000i	1.92305i	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.00000i	− 0.264906i
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	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	$-0.0785709\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	0	0
$93$	0	0
$94$	0	0
$95$	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	$0.967625\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.0000i	− 1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$	− 16.0000i	− 1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$	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
$111$	6.00000	0.569495
$112$	0	0
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.00000i	− 0.554700i
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.00000i	0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$	6.00000i	0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.0000i	− 1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$	− 14.0000i	− 1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 6.00000i	− 0.501745i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	− 4.00000i	− 0.323381i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	14.0000	1.11027
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	8.00000i	0.608229i	0.952636	+	0.304114i	$0.0983605\pi$
$173$	8.00000i	0.608229i	−0.952636	+	0.304114i	$0.901639\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	0	0
$179$	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
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	14.0000i	1.03491i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 16.0000i	− 1.13995i	−0.821661	−	0.569976i	$-0.806952\pi$
$197$	− 16.0000i	− 1.13995i	0.821661	−	0.569976i	$-0.193048\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.00000i	0.556038i
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	12.0000i	0.803579i	0.915732	+	0.401790i	$0.131612\pi$
$223$	12.0000i	0.803579i	−0.915732	+	0.401790i	$0.868388\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	16.0000i	1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$	16.0000i	1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.00000i	0.129914i
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000i	0.763542i
$248$	0	0
$249$	16.0000	1.01396
$250$	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
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 14.0000i	− 0.856786i
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	− 12.0000i	− 0.726273i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	32.0000	1.90896	0.954480	−	0.298275i	$-0.0964112\pi$
$281$	32.0000	1.90896	0.954480	+	0.298275i	$0.0964112\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	− 20.0000i	− 1.16841i	−0.811605	−	0.584206i	$-0.801406\pi$
$293$	− 20.0000i	− 1.16841i	0.811605	−	0.584206i	$-0.198594\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.00000i	− 0.0580259i
$298$	0	0
$299$	48.0000	2.77591
$300$	0	0
$301$	20.0000	1.15278
$302$	0	0
$303$	− 8.00000i	− 0.459588i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.0000i	1.48390i	0.670456	+	0.741949i	$0.266098\pi$
$307$	26.0000i	1.48390i	−0.670456	+	0.741949i	$0.733902\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 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
$319$	0	0
$320$	0	0
$321$	−16.0000	−0.893033
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 18.0000i	− 0.995402i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 32.0000i	− 1.71785i	−0.512101	−	0.858925i	$-0.671133\pi$
$347$	− 32.0000i	− 1.71785i	0.512101	−	0.858925i	$-0.328867\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 8.00000i	− 0.423405i
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	28.0000	1.45369
$372$	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
$376$	0	0
$377$	0	0
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	0	0
$383$	8.00000i	0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$	8.00000i	0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.0000i	− 0.508329i
$388$	0	0
$389$	−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
$391$	32.0000	1.61831
$392$	0	0
$393$	− 4.00000i	− 0.201773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	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
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	0	0
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.0000i	0.685583i
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	28.0000i	1.35501i
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4.00000i	− 0.189194i
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 10.0000i	− 0.469841i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	0	0
$463$	− 12.0000i	− 0.557687i	−0.960337	−	0.278844i	$-0.910049\pi$
$463$	− 12.0000i	− 0.557687i	0.960337	−	0.278844i	$-0.0899511\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	− 10.0000i	− 0.459800i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 14.0000i	− 0.641016i
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	0	0
$483$	16.0000i	0.728025i
$484$	0	0
$485$	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	$0.881918\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	0	0
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	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
$501$	12.0000	0.536120
$502$	0	0
$503$	20.0000i	0.891756i	0.895094	+	0.445878i	$0.147108\pi$
$503$	20.0000i	0.891756i	−0.895094	+	0.445878i	$0.852892\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	23.0000i	1.02147i
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	0	0
$513$	2.00000i	0.0883022i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	8.00000	0.351161
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	− 22.0000i	− 0.961993i	−0.876723	−	0.480996i	$-0.840275\pi$
$523$	− 22.0000i	− 0.961993i	0.876723	−	0.480996i	$-0.159725\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.00000i	− 0.172613i
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	0	0
$543$	− 22.0000i	− 0.944110i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	0	0
$551$	0	0
$552$	0	0
$553$	4.00000i	0.170097i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	0	0
$559$	−60.0000	−2.53773
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	34.0000	1.42286	0.711428	−	0.702759i	$-0.248049\pi$
$571$	34.0000	1.42286	0.711428	+	0.702759i	$0.248049\pi$
$572$	0	0
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	0	0
$579$	10.0000	0.415586
$580$	0	0
$581$	32.0000	1.32758
$582$	0	0
$583$	− 14.0000i	− 0.579821i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−16.0000	−0.658152
$592$	0	0
$593$	24.0000i	0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$	24.0000i	0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.00000i	0.327418i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 34.0000i	− 1.38002i	−0.723801	−	0.690009i	$-0.757607\pi$
$607$	− 34.0000i	− 1.38002i	0.723801	−	0.690009i	$-0.242393\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 38.0000i	− 1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	− 38.0000i	− 1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	− 28.0000i	− 1.12180i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.00000i	0.0798723i
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	10.0000i	0.397464i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.0000i	0.713186i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	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
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	24.0000i	0.932083i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	14.0000	0.540464
$672$	0	0
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.0000i	1.22986i	0.788582	+	0.614930i	$0.210816\pi$
$677$	32.0000i	1.22986i	−0.788582	+	0.614930i	$0.789184\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.00000i	0.0763048i
$688$	0	0
$689$	−84.0000	−3.20015
$690$	0	0
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	0	0
$693$	− 2.00000i	− 0.0759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	16.0000	0.605176
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	12.0000i	0.452589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 16.0000i	− 0.601742i
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 4.00000i	− 0.149383i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	2.00000i	0.0743808i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	48.0000i	1.78022i	0.455744	+	0.890111i	$0.349373\pi$
$727$	48.0000i	1.78022i	−0.455744	+	0.890111i	$0.650627\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−40.0000	−1.47945
$732$	0	0
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000i	0.147342i
$738$	0	0
$739$	−26.0000	−0.956425	−0.478213	−	0.878244i	$-0.658715\pi$
$739$	−26.0000	−0.956425	−0.478213	+	0.878244i	$0.658715\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	− 8.00000i	− 0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$	− 8.00000i	− 0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 16.0000i	− 0.585409i
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	4.00000i	0.145768i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	0	0
$763$	− 36.0000i	− 1.30329i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	72.0000i	2.59977i
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 12.0000i	− 0.430498i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.0000i	1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$	38.0000i	1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$	0	0
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−20.0000	−0.711118
$792$	0	0
$793$	− 84.0000i	− 2.98293i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.00000i	0.212531i	0.994338	+	0.106265i	$0.0338893\pi$
$797$	6.00000i	0.212531i	−0.994338	+	0.106265i	$0.966111\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	− 6.00000i	− 0.211735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.0000i	0.492823i
$808$	0	0
$809$	44.0000	1.54696	0.773479	−	0.633822i	$-0.218515\pi$
$809$	44.0000	1.54696	0.773479	+	0.633822i	$0.218515\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	10.0000i	0.350715i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20.0000i	0.699711i
$818$	0	0
$819$	−12.0000	−0.419314
$820$	0	0
$821$	40.0000	1.39601	0.698005	−	0.716093i	$-0.254071\pi$
$821$	40.0000	1.39601	0.698005	+	0.716093i	$0.254071\pi$
$822$	0	0
$823$	− 28.0000i	− 0.976019i	−0.872838	−	0.488009i	$-0.837723\pi$
$823$	− 28.0000i	− 0.976019i	0.872838	−	0.488009i	$-0.162277\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.0000i	0.556375i	0.960527	+	0.278187i	$0.0897336\pi$
$827$	16.0000i	0.556375i	−0.960527	+	0.278187i	$0.910266\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	12.0000i	0.415775i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	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
$841$	−29.0000	−1.00000
$842$	0	0
$843$	− 32.0000i	− 1.10214i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	0	0
$849$	−14.0000	−0.480479
$850$	0	0
$851$	48.0000	1.64542
$852$	0	0
$853$	18.0000i	0.616308i	0.951336	+	0.308154i	$0.0997113\pi$
$853$	18.0000i	0.616308i	−0.951336	+	0.308154i	$0.900289\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 44.0000i	− 1.50301i	−0.659727	−	0.751506i	$-0.729328\pi$
$857$	− 44.0000i	− 1.50301i	0.659727	−	0.751506i	$-0.270672\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.00000i	0.272323i	0.990687	+	0.136162i	$0.0434766\pi$
$863$	8.00000i	0.272323i	−0.990687	+	0.136162i	$0.956523\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	2.00000	0.0678454
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	0	0
$879$	−20.0000	−0.674583
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	− 44.0000i	− 1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$	− 44.0000i	− 1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 48.0000i	− 1.60267i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−56.0000	−1.86563
$902$	0	0
$903$	− 20.0000i	− 0.665558i
$904$	0	0
$905$	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	$-0.0855818\pi$
$908$	0	0
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	− 16.0000i	− 0.529523i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.00000i	− 0.264183i
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	26.0000	0.856729
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.0000i	0.394132i
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	8.00000i	0.261908i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	−36.0000	−1.16861
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	12.0000i	0.388718i	0.980930	+	0.194359i	$0.0622627\pi$
$953$	12.0000i	0.388718i	−0.980930	+	0.194359i	$0.937737\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−28.0000	−0.904167
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	16.0000i	0.515593i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 34.0000i	− 1.09337i	−0.837340	−	0.546683i	$-0.815890\pi$
$967$	− 34.0000i	− 1.09337i	0.837340	−	0.546683i	$-0.184110\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	0	0
$973$	28.0000i	0.897639i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	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
$979$	−14.0000	−0.447442
$980$	0	0
$981$	−18.0000	−0.574696
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	80.0000	2.54385
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	− 8.00000i	− 0.253872i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3300.2.c.c.1849.1		2
3.2	odd	2		9900.2.c.k.5149.1		2
5.2	odd	4		132.2.a.a.1.1	✓	1
5.3	odd	4		3300.2.a.k.1.1		1
5.4	even	2	inner	3300.2.c.c.1849.2		2
15.2	even	4		396.2.a.b.1.1		1
15.8	even	4		9900.2.a.g.1.1		1
15.14	odd	2		9900.2.c.k.5149.2		2
20.7	even	4		528.2.a.h.1.1		1
35.27	even	4		6468.2.a.l.1.1		1
40.27	even	4		2112.2.a.b.1.1		1
40.37	odd	4		2112.2.a.t.1.1		1
45.2	even	12		3564.2.i.g.2377.1		2
45.7	odd	12		3564.2.i.b.2377.1		2
45.22	odd	12		3564.2.i.b.1189.1		2
45.32	even	12		3564.2.i.g.1189.1		2
55.2	even	20		1452.2.i.l.565.1		4
55.7	even	20		1452.2.i.l.1237.1		4
55.17	even	20		1452.2.i.l.1213.1		4
55.27	odd	20		1452.2.i.k.1213.1		4
55.32	even	4		1452.2.a.c.1.1		1
55.37	odd	20		1452.2.i.k.1237.1		4
55.42	odd	20		1452.2.i.k.565.1		4
55.47	odd	20		1452.2.i.k.493.1		4
55.52	even	20		1452.2.i.l.493.1		4
60.47	odd	4		1584.2.a.c.1.1		1
120.77	even	4		6336.2.a.cf.1.1		1
120.107	odd	4		6336.2.a.bz.1.1		1
165.32	odd	4		4356.2.a.c.1.1		1
220.87	odd	4		5808.2.a.bf.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.2.a.a.1.1	✓	1	5.2	odd	4
396.2.a.b.1.1		1	15.2	even	4
528.2.a.h.1.1		1	20.7	even	4
1452.2.a.c.1.1		1	55.32	even	4
1452.2.i.k.493.1		4	55.47	odd	20
1452.2.i.k.565.1		4	55.42	odd	20
1452.2.i.k.1213.1		4	55.27	odd	20
1452.2.i.k.1237.1		4	55.37	odd	20
1452.2.i.l.493.1		4	55.52	even	20
1452.2.i.l.565.1		4	55.2	even	20
1452.2.i.l.1213.1		4	55.17	even	20
1452.2.i.l.1237.1		4	55.7	even	20
1584.2.a.c.1.1		1	60.47	odd	4
2112.2.a.b.1.1		1	40.27	even	4
2112.2.a.t.1.1		1	40.37	odd	4
3300.2.a.k.1.1		1	5.3	odd	4
3300.2.c.c.1849.1		2	1.1	even	1	trivial
3300.2.c.c.1849.2		2	5.4	even	2	inner
3564.2.i.b.1189.1		2	45.22	odd	12
3564.2.i.b.2377.1		2	45.7	odd	12
3564.2.i.g.1189.1		2	45.32	even	12
3564.2.i.g.2377.1		2	45.2	even	12
4356.2.a.c.1.1		1	165.32	odd	4
5808.2.a.bf.1.1		1	220.87	odd	4
6336.2.a.bz.1.1		1	120.107	odd	4
6336.2.a.cf.1.1		1	120.77	even	4
6468.2.a.l.1.1		1	35.27	even	4
9900.2.a.g.1.1		1	15.8	even	4
9900.2.c.k.5149.1		2	3.2	odd	2
9900.2.c.k.5149.2		2	15.14	odd	2

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:	\( N \)	\(=\)	\( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} +6.00000i q^{13} +4.00000i q^{17} +2.00000 q^{19} -2.00000 q^{21} -8.00000i q^{23} +1.00000i q^{27} +1.00000i q^{33} +6.00000i q^{37} +6.00000 q^{39} +10.0000i q^{43} +3.00000 q^{49} +4.00000 q^{51} +14.0000i q^{53} -2.00000i q^{57} +12.0000 q^{59} -14.0000 q^{61} +2.00000i q^{63} -4.00000i q^{67} -8.00000 q^{69} +6.00000i q^{73} +2.00000i q^{77} -2.00000 q^{79} +1.00000 q^{81} +16.0000i q^{83} +14.0000 q^{89} +12.0000 q^{91} +2.00000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 2 q^{11} + 4 q^{19} - 4 q^{21} + 12 q^{39} + 6 q^{49} + 8 q^{51} + 24 q^{59} - 28 q^{61} - 16 q^{69} - 4 q^{79} + 2 q^{81} + 28 q^{89} + 24 q^{91} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 2 * q^11 + 4 * q^19 - 4 * q^21 + 12 * q^39 + 6 * q^49 + 8 * q^51 + 24 * q^59 - 28 * q^61 - 16 * q^69 - 4 * q^79 + 2 * q^81 + 28 * q^89 + 24 * q^91 + 2 * q^99

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