LMFDB - Embedded newform 3840.2.k.o.1921.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3840,2,Mod(1921,3840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3840.1921"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,0,0,0,0,2,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$30.6625543762$
Analytic rank:	$1$
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 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1921.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3840.1921
Dual form			3840.2.k.o.1921.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} -4.00000i q^{11} -6.00000i q^{13} +1.00000 q^{15} -6.00000 q^{17} +4.00000i q^{19} -1.00000 q^{25} +1.00000i q^{27} +2.00000i q^{29} -8.00000 q^{31} -4.00000 q^{33} -2.00000i q^{37} -6.00000 q^{39} +6.00000 q^{41} +12.0000i q^{43} -1.00000i q^{45} +8.00000 q^{47} -7.00000 q^{49} +6.00000i q^{51} +6.00000i q^{53} +4.00000 q^{55} +4.00000 q^{57} +12.0000i q^{59} -14.0000i q^{61} +6.00000 q^{65} -4.00000i q^{67} -8.00000 q^{71} +6.00000 q^{73} +1.00000i q^{75} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -6.00000i q^{85} +2.00000 q^{87} -10.0000 q^{89} +8.00000i q^{93} -4.00000 q^{95} +2.00000 q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} + 2 q^{15} - 12 q^{17} - 2 q^{25} - 16 q^{31} - 8 q^{33} - 12 q^{39} + 12 q^{41} + 16 q^{47} - 14 q^{49} + 8 q^{55} + 8 q^{57} + 12 q^{65} - 16 q^{71} + 12 q^{73} - 16 q^{79} + 2 q^{81} + 4 q^{87}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^9 + 2 * q^15 - 12 * q^17 - 2 * q^25 - 16 * q^31 - 8 * q^33 - 12 * q^39 + 12 * q^41 + 16 * q^47 - 14 * q^49 + 8 * q^55 + 8 * q^57 + 12 * q^65 - 16 * q^71 + 12 * q^73 - 16 * q^79 + 2 * q^81 + 4 * q^87 - 20 * q^89 - 8 * q^95 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	6.00000i	0.840168i
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	12.0000i	1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$	12.0000i	1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$	0	0
$61$	− 14.0000i	− 1.79252i	−0.443533	−	0.896258i	$-0.646275\pi$
$61$	− 14.0000i	− 1.79252i	0.443533	−	0.896258i	$-0.353725\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.00000	0.744208
$66$	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
$69$	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
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	− 6.00000i	− 0.650791i
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	0	0
$101$	6.00000i	0.597022i	0.954406	+	0.298511i	$0.0964900\pi$
$101$	6.00000i	0.597022i	−0.954406	+	0.298511i	$0.903510\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	18.0000i	1.72409i	0.506834	+	0.862044i	$0.330816\pi$
$109$	18.0000i	1.72409i	−0.506834	+	0.862044i	$0.669184\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.00000i	0.554700i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	− 4.00000i	− 0.349482i	−0.984614	−	0.174741i	$-0.944091\pi$
$131$	− 4.00000i	− 0.349482i	0.984614	−	0.174741i	$-0.0559088\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	4.00000i	0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$	4.00000i	0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$	0	0
$141$	− 8.00000i	− 0.673722i
$142$	0	0
$143$	−24.0000	−2.00698
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	7.00000i	0.577350i
$148$	0	0
$149$	− 10.0000i	− 0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$	− 10.0000i	− 0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	− 8.00000i	− 0.642575i
$156$	0	0
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	− 4.00000i	− 0.311400i
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	12.0000i	0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$	12.0000i	0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	0	0
$181$	− 10.0000i	− 0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$	− 10.0000i	− 0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	24.0000i	1.75505i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	− 6.00000i	− 0.429669i
$196$	0	0
$197$	− 10.0000i	− 0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$	− 10.0000i	− 0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	6.00000i	0.419058i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	8.00000i	0.548151i
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 6.00000i	− 0.405442i
$220$	0	0
$221$	36.0000i	2.42162i
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	28.0000i	1.85843i	0.369546	+	0.929213i	$0.379513\pi$
$227$	28.0000i	1.85843i	−0.369546	+	0.929213i	$0.620487\pi$
$228$	0	0
$229$	− 10.0000i	− 0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$	− 10.0000i	− 0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	8.00000i	0.521862i
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 7.00000i	− 0.447214i
$246$	0	0
$247$	24.0000	1.52708
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	28.0000i	1.76734i	0.468106	+	0.883672i	$0.344936\pi$
$251$	28.0000i	1.76734i	−0.468106	+	0.883672i	$0.655064\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−6.00000	−0.375735
$256$	0	0
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	10.0000i	0.611990i
$268$	0	0
$269$	18.0000i	1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$	18.0000i	1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.00000i	0.241209i
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	4.00000i	0.236940i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	0	0
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	14.0000	0.801638
$306$	0	0
$307$	28.0000i	1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$	28.0000i	1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	0	0
$309$	0	0
$310$	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
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	6.00000i	0.332820i
$326$	0	0
$327$	18.0000	0.995402
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 28.0000i	− 1.53902i	−0.638635	−	0.769510i	$-0.720501\pi$
$331$	− 28.0000i	− 1.53902i	0.638635	−	0.769510i	$-0.279499\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	6.00000i	0.325875i
$340$	0	0
$341$	32.0000i	1.73290i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	− 14.0000i	− 0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	− 14.0000i	− 0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	− 8.00000i	− 0.424596i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	5.00000i	0.262432i
$364$	0	0
$365$	6.00000i	0.314054i
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	0	0
$381$	8.00000i	0.409852i
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 12.0000i	− 0.609994i
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	− 8.00000i	− 0.402524i
$396$	0	0
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	48.0000i	2.39105i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	2.00000i	0.0986527i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	− 36.0000i	− 1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$	− 36.0000i	− 1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$	0	0
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	0	0
$428$	0	0
$429$	24.0000i	1.15873i
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	2.00000i	0.0958927i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	0	0
$445$	− 10.0000i	− 0.474045i
$446$	0	0
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	− 24.0000i	− 1.13012i
$452$	0	0
$453$	16.0000i	0.751746i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	− 6.00000i	− 0.280056i
$460$	0	0
$461$	− 14.0000i	− 0.652045i	−0.945362	−	0.326023i	$-0.894291\pi$
$461$	− 14.0000i	− 0.652045i	0.945362	−	0.326023i	$-0.105709\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	48.0000	2.20704
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.00000i	0.0908153i
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	− 36.0000i	− 1.62466i	−0.583200	−	0.812329i	$-0.698200\pi$
$491$	− 36.0000i	− 1.62466i	0.583200	−	0.812329i	$-0.301800\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.0000i	0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$	20.0000i	0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$	0	0
$501$	16.0000i	0.714827i
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	23.0000i	1.02147i
$508$	0	0
$509$	2.00000i	0.0886484i	0.999017	+	0.0443242i	$0.0141135\pi$
$509$	2.00000i	0.0886484i	−0.999017	+	0.0443242i	$0.985887\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 32.0000i	− 1.40736i
$518$	0	0
$519$	−14.0000	−0.614532
$520$	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
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 12.0000i	− 0.520756i
$532$	0	0
$533$	− 36.0000i	− 1.55933i
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	28.0000i	1.20605i
$540$	0	0
$541$	− 30.0000i	− 1.28980i	−0.764267	−	0.644900i	$-0.776899\pi$
$541$	− 30.0000i	− 1.28980i	0.764267	−	0.644900i	$-0.223101\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	− 36.0000i	− 1.53925i	−0.638497	−	0.769624i	$-0.720443\pi$
$547$	− 36.0000i	− 1.53925i	0.638497	−	0.769624i	$-0.279557\pi$
$548$	0	0
$549$	14.0000i	0.597505i
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 2.00000i	− 0.0848953i
$556$	0	0
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	72.0000	3.04528
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	0	0
$565$	− 6.00000i	− 0.252422i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	− 12.0000i	− 0.502184i	−0.967963	−	0.251092i	$-0.919210\pi$
$571$	− 12.0000i	− 0.502184i	0.967963	−	0.251092i	$-0.0807897\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	14.0000i	0.581820i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	− 4.00000i	− 0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$	− 4.00000i	− 0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$	0	0
$589$	− 32.0000i	− 1.31854i
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	− 5.00000i	− 0.203279i
$606$	0	0
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 48.0000i	− 1.94187i
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 16.0000i	− 0.638978i
$628$	0	0
$629$	12.0000i	0.478471i
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	− 8.00000i	− 0.317470i
$636$	0	0
$637$	42.0000i	1.66410i
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	12.0000i	0.472500i
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	− 4.00000i	− 0.155818i	−0.996960	−	0.0779089i	$-0.975176\pi$
$659$	− 4.00000i	− 0.155818i	0.996960	−	0.0779089i	$-0.0248243\pi$
$660$	0	0
$661$	6.00000i	0.233373i	0.993169	+	0.116686i	$0.0372273\pi$
$661$	6.00000i	0.233373i	−0.993169	+	0.116686i	$0.962773\pi$
$662$	0	0
$663$	36.0000	1.39812
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 8.00000i	− 0.309298i
$670$	0	0
$671$	−56.0000	−2.16186
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	38.0000i	1.46046i	0.683202	+	0.730229i	$0.260587\pi$
$677$	38.0000i	1.46046i	−0.683202	+	0.730229i	$0.739413\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	28.0000	1.07296
$682$	0	0
$683$	28.0000i	1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$	28.0000i	1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$	0	0
$685$	− 2.00000i	− 0.0764161i
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	20.0000i	0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$	20.0000i	0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	2.00000i	0.0756469i
$700$	0	0
$701$	− 30.0000i	− 1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$	− 30.0000i	− 1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 10.0000i	− 0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$	− 10.0000i	− 0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 24.0000i	− 0.897549i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	14.0000i	0.520666i
$724$	0	0
$725$	− 2.00000i	− 0.0742781i
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 72.0000i	− 2.66302i
$732$	0	0
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	0	0
$735$	−7.00000	−0.258199
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	4.00000i	0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$	4.00000i	0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$	0	0
$741$	− 24.0000i	− 0.881662i
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	0	0
$755$	− 16.0000i	− 0.582300i
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	6.00000i	0.216930i
$766$	0	0
$767$	72.0000	2.59977
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	− 10.0000i	− 0.360141i
$772$	0	0
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24.0000i	0.859889i
$780$	0	0
$781$	32.0000i	1.14505i
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	6.00000	0.214149
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−84.0000	−2.98293
$794$	0	0
$795$	6.00000i	0.212798i
$796$	0	0
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	− 24.0000i	− 0.846942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	− 28.0000i	− 0.983213i	−0.870817	−	0.491606i	$-0.836410\pi$
$811$	− 28.0000i	− 0.983213i	0.870817	−	0.491606i	$-0.163590\pi$
$812$	0	0
$813$	24.0000i	0.841717i
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	−48.0000	−1.67931
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 10.0000i	− 0.349002i	−0.984657	−	0.174501i	$-0.944169\pi$
$821$	− 10.0000i	− 0.349002i	0.984657	−	0.174501i	$-0.0558313\pi$
$822$	0	0
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	0	0
$829$	2.00000i	0.0694629i	0.999397	+	0.0347314i	$0.0110576\pi$
$829$	2.00000i	0.0694629i	−0.999397	+	0.0347314i	$0.988942\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	42.0000	1.45521
$834$	0	0
$835$	− 16.0000i	− 0.553703i
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 22.0000i	− 0.757720i
$844$	0	0
$845$	− 23.0000i	− 0.791224i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	4.00000i	0.136478i	0.997669	+	0.0682391i	$0.0217381\pi$
$859$	4.00000i	0.136478i	−0.997669	+	0.0682391i	$0.978262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	− 19.0000i	− 0.645274i
$868$	0	0
$869$	32.0000i	1.08553i
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	0	0
$879$	22.0000	0.742042
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	− 52.0000i	− 1.74994i	−0.484178	−	0.874970i	$-0.660881\pi$
$883$	− 52.0000i	− 1.74994i	0.484178	−	0.874970i	$-0.339119\pi$
$884$	0	0
$885$	12.0000i	0.403376i
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 4.00000i	− 0.134005i
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	−12.0000	−0.401116
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 16.0000i	− 0.533630i
$900$	0	0
$901$	− 36.0000i	− 1.19933i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.0000	0.332411
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	0	0
$909$	− 6.00000i	− 0.199007i
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	− 14.0000i	− 0.462826i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	0	0
$923$	48.0000i	1.57994i
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	− 28.0000i	− 0.917663i
$932$	0	0
$933$	8.00000i	0.261908i
$934$	0	0
$935$	−24.0000	−0.784884
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	10.0000i	0.326338i
$940$	0	0
$941$	50.0000i	1.62995i	0.579494	+	0.814977i	$0.303250\pi$
$941$	50.0000i	1.62995i	−0.579494	+	0.814977i	$0.696750\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	60.0000i	1.94974i	0.222779	+	0.974869i	$0.428487\pi$
$947$	60.0000i	1.94974i	−0.222779	+	0.974869i	$0.571513\pi$
$948$	0	0
$949$	− 36.0000i	− 1.16861i
$950$	0	0
$951$	−14.0000	−0.453981
$952$	0	0
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8.00000i	− 0.258603i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	− 14.0000i	− 0.450676i
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	12.0000i	0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$	12.0000i	0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	6.00000	0.192154
$976$	0	0
$977$	58.0000	1.85558	0.927792	−	0.373097i	$-0.121704\pi$
$977$	58.0000	1.85558	0.927792	+	0.373097i	$0.121704\pi$
$978$	0	0
$979$	40.0000i	1.27841i
$980$	0	0
$981$	− 18.0000i	− 0.574696i
$982$	0	0
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	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	$0.280868\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 18.0000i	− 0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$	− 18.0000i	− 0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.k.o.1921.1		2
4.3	odd	2		3840.2.k.j.1921.2		2
8.3	odd	2		3840.2.k.j.1921.1		2
8.5	even	2	inner	3840.2.k.o.1921.2		2
16.3	odd	4		240.2.a.c.1.1		1
16.5	even	4		960.2.a.c.1.1		1
16.11	odd	4		960.2.a.j.1.1		1
16.13	even	4		120.2.a.b.1.1	✓	1
48.5	odd	4		2880.2.a.x.1.1		1
48.11	even	4		2880.2.a.bb.1.1		1
48.29	odd	4		360.2.a.b.1.1		1
48.35	even	4		720.2.a.d.1.1		1
80.3	even	4		1200.2.f.g.49.1		2
80.13	odd	4		600.2.f.b.49.2		2
80.19	odd	4		1200.2.a.o.1.1		1
80.27	even	4		4800.2.f.i.3649.1		2
80.29	even	4		600.2.a.c.1.1		1
80.37	odd	4		4800.2.f.bc.3649.2		2
80.43	even	4		4800.2.f.i.3649.2		2
80.53	odd	4		4800.2.f.bc.3649.1		2
80.59	odd	4		4800.2.a.r.1.1		1
80.67	even	4		1200.2.f.g.49.2		2
80.69	even	4		4800.2.a.cd.1.1		1
80.77	odd	4		600.2.f.b.49.1		2
112.13	odd	4		5880.2.a.a.1.1		1
144.13	even	12		3240.2.q.g.2161.1		2
144.29	odd	12		3240.2.q.q.1081.1		2
144.61	even	12		3240.2.q.g.1081.1		2
144.77	odd	12		3240.2.q.q.2161.1		2
240.29	odd	4		1800.2.a.n.1.1		1
240.77	even	4		1800.2.f.j.649.1		2
240.83	odd	4		3600.2.f.c.2449.2		2
240.173	even	4		1800.2.f.j.649.2		2
240.179	even	4		3600.2.a.t.1.1		1
240.227	odd	4		3600.2.f.c.2449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.a.b.1.1	✓	1	16.13	even	4
240.2.a.c.1.1		1	16.3	odd	4
360.2.a.b.1.1		1	48.29	odd	4
600.2.a.c.1.1		1	80.29	even	4
600.2.f.b.49.1		2	80.77	odd	4
600.2.f.b.49.2		2	80.13	odd	4
720.2.a.d.1.1		1	48.35	even	4
960.2.a.c.1.1		1	16.5	even	4
960.2.a.j.1.1		1	16.11	odd	4
1200.2.a.o.1.1		1	80.19	odd	4
1200.2.f.g.49.1		2	80.3	even	4
1200.2.f.g.49.2		2	80.67	even	4
1800.2.a.n.1.1		1	240.29	odd	4
1800.2.f.j.649.1		2	240.77	even	4
1800.2.f.j.649.2		2	240.173	even	4
2880.2.a.x.1.1		1	48.5	odd	4
2880.2.a.bb.1.1		1	48.11	even	4
3240.2.q.g.1081.1		2	144.61	even	12
3240.2.q.g.2161.1		2	144.13	even	12
3240.2.q.q.1081.1		2	144.29	odd	12
3240.2.q.q.2161.1		2	144.77	odd	12
3600.2.a.t.1.1		1	240.179	even	4
3600.2.f.c.2449.1		2	240.227	odd	4
3600.2.f.c.2449.2		2	240.83	odd	4
3840.2.k.j.1921.1		2	8.3	odd	2
3840.2.k.j.1921.2		2	4.3	odd	2
3840.2.k.o.1921.1		2	1.1	even	1	trivial
3840.2.k.o.1921.2		2	8.5	even	2	inner
4800.2.a.r.1.1		1	80.59	odd	4
4800.2.a.cd.1.1		1	80.69	even	4
4800.2.f.i.3649.1		2	80.27	even	4
4800.2.f.i.3649.2		2	80.43	even	4
4800.2.f.bc.3649.1		2	80.53	odd	4
4800.2.f.bc.3649.2		2	80.37	odd	4
5880.2.a.a.1.1		1	112.13	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} -4.00000i q^{11} -6.00000i q^{13} +1.00000 q^{15} -6.00000 q^{17} +4.00000i q^{19} -1.00000 q^{25} +1.00000i q^{27} +2.00000i q^{29} -8.00000 q^{31} -4.00000 q^{33} -2.00000i q^{37} -6.00000 q^{39} +6.00000 q^{41} +12.0000i q^{43} -1.00000i q^{45} +8.00000 q^{47} -7.00000 q^{49} +6.00000i q^{51} +6.00000i q^{53} +4.00000 q^{55} +4.00000 q^{57} +12.0000i q^{59} -14.0000i q^{61} +6.00000 q^{65} -4.00000i q^{67} -8.00000 q^{71} +6.00000 q^{73} +1.00000i q^{75} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -6.00000i q^{85} +2.00000 q^{87} -10.0000 q^{89} +8.00000i q^{93} -4.00000 q^{95} +2.00000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} + 2 q^{15} - 12 q^{17} - 2 q^{25} - 16 q^{31} - 8 q^{33} - 12 q^{39} + 12 q^{41} + 16 q^{47} - 14 q^{49} + 8 q^{55} + 8 q^{57} + 12 q^{65} - 16 q^{71} + 12 q^{73} - 16 q^{79} + 2 q^{81} + 4 q^{87}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^9 + 2 * q^15 - 12 * q^17 - 2 * q^25 - 16 * q^31 - 8 * q^33 - 12 * q^39 + 12 * q^41 + 16 * q^47 - 14 * q^49 + 8 * q^55 + 8 * q^57 + 12 * q^65 - 16 * q^71 + 12 * q^73 - 16 * q^79 + 2 * q^81 + 4 * q^87 - 20 * q^89 - 8 * q^95 + 4 * q^97

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