LMFDB - Embedded newform 3840.2.k.p.1921.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(1921,3840)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	no
Analytic conductor:	$30.6625543762$
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 480)
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.p.1921.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +4.00000i q^{11} +2.00000i q^{13} +1.00000 q^{15} -2.00000 q^{17} -8.00000i q^{19} +4.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -6.00000i q^{29} +4.00000 q^{33} -2.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} -1.00000i q^{45} +12.0000 q^{47} -7.00000 q^{49} +2.00000i q^{51} +6.00000i q^{53} -4.00000 q^{55} -8.00000 q^{57} +12.0000i q^{59} +14.0000i q^{61} -2.00000 q^{65} +12.0000i q^{67} -4.00000i q^{69} -2.00000 q^{73} +1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -2.00000i q^{85} -6.00000 q^{87} -2.00000 q^{89} +8.00000 q^{95} -14.0000 q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 2 q^{15} - 4 q^{17} + 8 q^{23} - 2 q^{25} + 8 q^{33} + 4 q^{39} + 12 q^{41} + 24 q^{47} - 14 q^{49} - 8 q^{55} - 16 q^{57} - 4 q^{65} - 4 q^{73} + 16 q^{79} + 2 q^{81} - 12 q^{87} - 4 q^{89} + 16 q^{95} - 28 q^{97}+O(q^{100}) $ 2 * q - 2 * q^9 + 2 * q^15 - 4 * q^17 + 8 * q^23 - 2 * q^25 + 8 * q^33 + 4 * q^39 + 12 * q^41 + 24 * q^47 - 14 * q^49 - 8 * q^55 - 16 * q^57 - 4 * q^65 - 4 * q^73 + 16 * q^79 + 2 * q^81 - 12 * q^87 - 4 * q^89 + 16 * q^95 - 28 * q^97

Display 100 coefficients 10 coefficients

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	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	− 8.00000i	− 1.83533i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	− 8.00000i	− 1.83533i	0.397360	−	0.917663i	$-0.369927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	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
$39$	2.00000	0.320256
$40$	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
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	2.00000i	0.280056i
$52$	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
$57$	−8.00000	−1.05963
$58$	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	$0.353725\pi$
$61$	14.0000i	1.79252i	−0.443533	+	0.896258i	$0.646275\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	− 4.00000i	− 0.481543i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	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	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	− 2.00000i	− 0.216930i
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	14.0000i	1.39305i	0.717532	+	0.696526i	$0.245272\pi$
$101$	14.0000i	1.39305i	−0.717532	+	0.696526i	$0.754728\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 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
$109$	14.0000i	1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$	14.0000i	1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	4.00000i	0.373002i
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	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
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	20.0000i	1.74741i	0.486458	+	0.873704i	$0.338289\pi$
$131$	20.0000i	1.74741i	−0.486458	+	0.873704i	$0.661711\pi$
$132$	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	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	16.0000i	1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$	16.0000i	1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$	0	0
$141$	− 12.0000i	− 1.01058i
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	7.00000i	0.577350i
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	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	$-0.188685\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	0	0
$165$	4.00000i	0.311400i
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	8.00000i	0.611775i
$172$	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	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	0	0
$181$	− 22.0000i	− 1.63525i	−0.575753	−	0.817624i	$-0.695291\pi$
$181$	− 22.0000i	− 1.63525i	0.575753	−	0.817624i	$-0.304709\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	2.00000i	0.143223i
$196$	0	0
$197$	− 26.0000i	− 1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	− 26.0000i	− 1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	0	0
$204$	0	0
$205$	6.00000i	0.419058i
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	32.0000	2.21349
$210$	0	0
$211$	8.00000i	0.550743i	0.961338	+	0.275371i	$0.0888008\pi$
$211$	8.00000i	0.550743i	−0.961338	+	0.275371i	$0.911199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.00000i	0.135147i
$220$	0	0
$221$	− 4.00000i	− 0.269069i
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	18.0000i	1.18947i	0.803921	+	0.594737i	$0.202744\pi$
$229$	18.0000i	1.18947i	−0.803921	+	0.594737i	$0.797256\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	12.0000i	0.782794i
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 7.00000i	− 0.447214i
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	12.0000i	0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$	12.0000i	0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000i	0.371391i
$262$	0	0
$263$	20.0000	1.23325	0.616626	−	0.787256i	$-0.288499\pi$
$263$	20.0000	1.23325	0.616626	+	0.787256i	$0.288499\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	2.00000i	0.122398i
$268$	0	0
$269$	10.0000i	0.609711i	0.952399	+	0.304855i	$0.0986081\pi$
$269$	10.0000i	0.609711i	−0.952399	+	0.304855i	$0.901392\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	30.0000i	1.80253i	0.433273	+	0.901263i	$0.357359\pi$
$277$	30.0000i	1.80253i	−0.433273	+	0.901263i	$0.642641\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	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
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	14.0000i	0.820695i
$292$	0	0
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	8.00000i	0.462652i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	−14.0000	−0.801638
$306$	0	0
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	− 8.00000i	− 0.455104i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	16.0000i	0.890264i
$324$	0	0
$325$	− 2.00000i	− 0.110940i
$326$	0	0
$327$	14.0000	0.774202
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	0	0
$349$	− 2.00000i	− 0.107058i	−0.998566	−	0.0535288i	$-0.982953\pi$
$349$	− 2.00000i	− 0.107058i	0.998566	−	0.0535288i	$-0.0170469\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−45.0000	−2.36842
$362$	0	0
$363$	5.00000i	0.262432i
$364$	0	0
$365$	− 2.00000i	− 0.104685i
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	− 16.0000i	− 0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$	− 16.0000i	− 0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$	0	0
$381$	− 16.0000i	− 0.819705i
$382$	0	0
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	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
$391$	−8.00000	−0.404577
$392$	0	0
$393$	20.0000	1.00887
$394$	0	0
$395$	8.00000i	0.402524i
$396$	0	0
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	− 2.00000i	− 0.0986527i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	12.0000i	0.586238i	0.956076	+	0.293119i	$0.0946933\pi$
$419$	12.0000i	0.586238i	−0.956076	+	0.293119i	$0.905307\pi$
$420$	0	0
$421$	2.00000i	0.0974740i	0.998812	+	0.0487370i	$0.0155196\pi$
$421$	2.00000i	0.0974740i	−0.998812	+	0.0487370i	$0.984480\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8.00000i	0.386244i
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	− 6.00000i	− 0.287678i
$436$	0	0
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	− 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
$445$	− 2.00000i	− 0.0948091i
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	24.0000i	1.13012i
$452$	0	0
$453$	− 8.00000i	− 0.375873i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	0	0
$459$	− 2.00000i	− 0.0933520i
$460$	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
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	8.00000i	0.367065i
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 14.0000i	− 0.635707i
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	12.0000i	0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$	12.0000i	0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 8.00000i	− 0.358129i	−0.983837	−	0.179065i	$-0.942693\pi$
$499$	− 8.00000i	− 0.358129i	0.983837	−	0.179065i	$-0.0573071\pi$
$500$	0	0
$501$	− 12.0000i	− 0.536120i
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	− 38.0000i	− 1.68432i	−0.539227	−	0.842160i	$-0.681284\pi$
$509$	− 38.0000i	− 1.68432i	0.539227	−	0.842160i	$-0.318716\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	8.00000i	0.352522i
$516$	0	0
$517$	48.0000i	2.11104i
$518$	0	0
$519$	−14.0000	−0.614532
$520$	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
$523$	− 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
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 12.0000i	− 0.520756i
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	12.0000	0.518805
$536$	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	$0.223101\pi$
$541$	30.0000i	1.28980i	−0.764267	+	0.644900i	$0.776899\pi$
$542$	0	0
$543$	−22.0000	−0.944110
$544$	0	0
$545$	−14.0000	−0.599694
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	− 14.0000i	− 0.597505i
$550$	0	0
$551$	−48.0000	−2.04487
$552$	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
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	6.00000i	0.252422i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	32.0000i	1.33916i	0.742741	+	0.669579i	$0.233526\pi$
$571$	32.0000i	1.33916i	−0.742741	+	0.669579i	$0.766474\pi$
$572$	0	0
$573$	16.0000i	0.668410i
$574$	0	0
$575$	−4.00000	−0.166812
$576$	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
$579$	22.0000i	0.914289i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	36.0000i	1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$	36.0000i	1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−26.0000	−1.06950
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 16.0000i	− 0.654836i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	0	0
$605$	− 5.00000i	− 0.203279i
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24.0000i	0.970936i
$612$	0	0
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	− 16.0000i	− 0.643094i	−0.946894	−	0.321547i	$-0.895797\pi$
$619$	− 16.0000i	− 0.643094i	0.946894	−	0.321547i	$-0.104203\pi$
$620$	0	0
$621$	4.00000i	0.160514i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 32.0000i	− 1.27796i
$628$	0	0
$629$	4.00000i	0.159490i
$630$	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
$633$	8.00000	0.317971
$634$	0	0
$635$	16.0000i	0.634941i
$636$	0	0
$637$	− 14.0000i	− 0.554700i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	4.00000i	0.157500i
$646$	0	0
$647$	20.0000	0.786281	0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000	0.786281	0.393141	+	0.919478i	$0.371389\pi$
$648$	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
$655$	−20.0000	−0.781465
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	− 20.0000i	− 0.779089i	−0.921008	−	0.389545i	$-0.872632\pi$
$659$	− 20.0000i	− 0.779089i	0.921008	−	0.389545i	$-0.127368\pi$
$660$	0	0
$661$	26.0000i	1.01128i	0.862744	+	0.505641i	$0.168744\pi$
$661$	26.0000i	1.01128i	−0.862744	+	0.505641i	$0.831256\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 24.0000i	− 0.929284i
$668$	0	0
$669$	24.0000i	0.927894i
$670$	0	0
$671$	−56.0000	−2.16186
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	2.00000i	0.0764161i
$686$	0	0
$687$	18.0000	0.686743
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	− 16.0000i	− 0.608669i	−0.952565	−	0.304334i	$-0.901566\pi$
$691$	− 16.0000i	− 0.608669i	0.952565	−	0.304334i	$-0.0984340\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	6.00000i	0.226941i
$700$	0	0
$701$	10.0000i	0.377695i	0.982006	+	0.188847i	$0.0604752\pi$
$701$	10.0000i	0.377695i	−0.982006	+	0.188847i	$0.939525\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.00000i	0.0751116i	0.999295	+	0.0375558i	$0.0119572\pi$
$709$	2.00000i	0.0751116i	−0.999295	+	0.0375558i	$0.988043\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 8.00000i	− 0.299183i
$716$	0	0
$717$	− 8.00000i	− 0.298765i
$718$	0	0
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 8.00000i	− 0.295891i
$732$	0	0
$733$	10.0000i	0.369358i	0.982799	+	0.184679i	$0.0591246\pi$
$733$	10.0000i	0.369358i	−0.982799	+	0.184679i	$0.940875\pi$
$734$	0	0
$735$	−7.00000	−0.258199
$736$	0	0
$737$	−48.0000	−1.76810
$738$	0	0
$739$	8.00000i	0.294285i	0.989115	+	0.147142i	$0.0470076\pi$
$739$	8.00000i	0.294285i	−0.989115	+	0.147142i	$0.952992\pi$
$740$	0	0
$741$	− 16.0000i	− 0.587775i
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	8.00000i	0.291150i
$756$	0	0
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	0	0
$759$	16.0000	0.580763
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.00000i	0.0723102i
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	− 6.00000i	− 0.216085i
$772$	0	0
$773$	− 10.0000i	− 0.359675i	−0.983696	−	0.179838i	$-0.942443\pi$
$773$	− 10.0000i	− 0.359675i	0.983696	−	0.179838i	$-0.0575572\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 48.0000i	− 1.71978i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	− 20.0000i	− 0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$	− 20.0000i	− 0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$	0	0
$789$	− 20.0000i	− 0.712019i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−28.0000	−0.994309
$794$	0	0
$795$	6.00000i	0.212798i
$796$	0	0
$797$	34.0000i	1.20434i	0.798367	+	0.602171i	$0.205697\pi$
$797$	34.0000i	1.20434i	−0.798367	+	0.602171i	$0.794303\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	− 8.00000i	− 0.282314i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	24.0000i	0.842754i	0.906886	+	0.421377i	$0.138453\pi$
$811$	24.0000i	0.842754i	−0.906886	+	0.421377i	$0.861547\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.0000i	1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$	30.0000i	1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	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
$829$	30.0000i	1.04194i	0.853574	+	0.520972i	$0.174430\pi$
$829$	30.0000i	1.04194i	−0.853574	+	0.520972i	$0.825570\pi$
$830$	0	0
$831$	30.0000	1.04069
$832$	0	0
$833$	14.0000	0.485071
$834$	0	0
$835$	12.0000i	0.415277i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	2.00000i	0.0688837i
$844$	0	0
$845$	9.00000i	0.309609i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	− 8.00000i	− 0.274236i
$852$	0	0
$853$	30.0000i	1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$	30.0000i	1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	− 32.0000i	− 1.09183i	−0.837842	−	0.545913i	$-0.816183\pi$
$859$	− 32.0000i	− 1.09183i	0.837842	−	0.545913i	$-0.183817\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	32.0000i	1.08553i
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	0	0
$879$	−26.0000	−0.876958
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	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
$885$	12.0000i	0.403376i
$886$	0	0
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000i	0.134005i
$892$	0	0
$893$	− 96.0000i	− 3.21252i
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	− 36.0000i	− 1.19536i	−0.801735	−	0.597680i	$-0.796089\pi$
$907$	− 36.0000i	− 1.19536i	0.801735	−	0.597680i	$-0.203911\pi$
$908$	0	0
$909$	− 14.0000i	− 0.464351i
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	14.0000i	0.462826i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	−8.00000	−0.262754
$928$	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
$931$	56.0000i	1.83533i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.00000	0.261628
$936$	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
$939$	− 14.0000i	− 0.456873i
$940$	0	0
$941$	34.0000i	1.10837i	0.832394	+	0.554184i	$0.186970\pi$
$941$	34.0000i	1.10837i	−0.832394	+	0.554184i	$0.813030\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	− 4.00000i	− 0.129845i
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	− 16.0000i	− 0.517748i
$956$	0	0
$957$	− 24.0000i	− 0.775810i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	− 22.0000i	− 0.708205i
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	28.0000i	0.898563i	0.893390	+	0.449281i	$0.148320\pi$
$971$	28.0000i	0.898563i	−0.893390	+	0.449281i	$0.851680\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	− 8.00000i	− 0.255681i
$980$	0	0
$981$	− 14.0000i	− 0.446986i
$982$	0	0
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	26.0000	0.828429
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.0000i	0.508770i
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.0000i	0.507234i
$996$	0	0
$997$	6.00000i	0.190022i	0.995476	+	0.0950110i	$0.0302886\pi$
$997$	6.00000i	0.190022i	−0.995476	+	0.0950110i	$0.969711\pi$
$998$	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.p.1921.1		2
4.3	odd	2		3840.2.k.k.1921.2		2
8.3	odd	2		3840.2.k.k.1921.1		2
8.5	even	2	inner	3840.2.k.p.1921.2		2
16.3	odd	4		960.2.a.f.1.1		1
16.5	even	4		480.2.a.b.1.1	✓	1
16.11	odd	4		480.2.a.e.1.1	yes	1
16.13	even	4		960.2.a.o.1.1		1
48.5	odd	4		1440.2.a.k.1.1		1
48.11	even	4		1440.2.a.j.1.1		1
48.29	odd	4		2880.2.a.i.1.1		1
48.35	even	4		2880.2.a.j.1.1		1
80.3	even	4		4800.2.f.j.3649.1		2
80.13	odd	4		4800.2.f.ba.3649.2		2
80.19	odd	4		4800.2.a.ca.1.1		1
80.27	even	4		2400.2.f.n.1249.1		2
80.29	even	4		4800.2.a.u.1.1		1
80.37	odd	4		2400.2.f.e.1249.2		2
80.43	even	4		2400.2.f.n.1249.2		2
80.53	odd	4		2400.2.f.e.1249.1		2
80.59	odd	4		2400.2.a.j.1.1		1
80.67	even	4		4800.2.f.j.3649.2		2
80.69	even	4		2400.2.a.y.1.1		1
80.77	odd	4		4800.2.f.ba.3649.1		2
240.53	even	4		7200.2.f.bb.6049.2		2
240.59	even	4		7200.2.a.u.1.1		1
240.107	odd	4		7200.2.f.b.6049.1		2
240.149	odd	4		7200.2.a.bg.1.1		1
240.197	even	4		7200.2.f.bb.6049.1		2
240.203	odd	4		7200.2.f.b.6049.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.a.b.1.1	✓	1	16.5	even	4
480.2.a.e.1.1	yes	1	16.11	odd	4
960.2.a.f.1.1		1	16.3	odd	4
960.2.a.o.1.1		1	16.13	even	4
1440.2.a.j.1.1		1	48.11	even	4
1440.2.a.k.1.1		1	48.5	odd	4
2400.2.a.j.1.1		1	80.59	odd	4
2400.2.a.y.1.1		1	80.69	even	4
2400.2.f.e.1249.1		2	80.53	odd	4
2400.2.f.e.1249.2		2	80.37	odd	4
2400.2.f.n.1249.1		2	80.27	even	4
2400.2.f.n.1249.2		2	80.43	even	4
2880.2.a.i.1.1		1	48.29	odd	4
2880.2.a.j.1.1		1	48.35	even	4
3840.2.k.k.1921.1		2	8.3	odd	2
3840.2.k.k.1921.2		2	4.3	odd	2
3840.2.k.p.1921.1		2	1.1	even	1	trivial
3840.2.k.p.1921.2		2	8.5	even	2	inner
4800.2.a.u.1.1		1	80.29	even	4
4800.2.a.ca.1.1		1	80.19	odd	4
4800.2.f.j.3649.1		2	80.3	even	4
4800.2.f.j.3649.2		2	80.67	even	4
4800.2.f.ba.3649.1		2	80.77	odd	4
4800.2.f.ba.3649.2		2	80.13	odd	4
7200.2.a.u.1.1		1	240.59	even	4
7200.2.a.bg.1.1		1	240.149	odd	4
7200.2.f.b.6049.1		2	240.107	odd	4
7200.2.f.b.6049.2		2	240.203	odd	4
7200.2.f.bb.6049.1		2	240.197	even	4
7200.2.f.bb.6049.2		2	240.53	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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +4.00000i q^{11} +2.00000i q^{13} +1.00000 q^{15} -2.00000 q^{17} -8.00000i q^{19} +4.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -6.00000i q^{29} +4.00000 q^{33} -2.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} -1.00000i q^{45} +12.0000 q^{47} -7.00000 q^{49} +2.00000i q^{51} +6.00000i q^{53} -4.00000 q^{55} -8.00000 q^{57} +12.0000i q^{59} +14.0000i q^{61} -2.00000 q^{65} +12.0000i q^{67} -4.00000i q^{69} -2.00000 q^{73} +1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -2.00000i q^{85} -6.00000 q^{87} -2.00000 q^{89} +8.00000 q^{95} -14.0000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 2 q^{15} - 4 q^{17} + 8 q^{23} - 2 q^{25} + 8 q^{33} + 4 q^{39} + 12 q^{41} + 24 q^{47} - 14 q^{49} - 8 q^{55} - 16 q^{57} - 4 q^{65} - 4 q^{73} + 16 q^{79} + 2 q^{81} - 12 q^{87} - 4 q^{89} + 16 q^{95} - 28 q^{97}+O(q^{100}) \) 2 * q - 2 * q^9 + 2 * q^15 - 4 * q^17 + 8 * q^23 - 2 * q^25 + 8 * q^33 + 4 * q^39 + 12 * q^41 + 24 * q^47 - 14 * q^49 - 8 * q^55 - 16 * q^57 - 4 * q^65 - 4 * q^73 + 16 * q^79 + 2 * q^81 - 12 * q^87 - 4 * q^89 + 16 * q^95 - 28 * q^97

Introduction

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