LMFDB - Embedded newform 3840.2.k.b.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 1920)
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.b.1921.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} -2.00000i q^{11} -1.00000 q^{15} +4.00000i q^{21} -8.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} +2.00000i q^{29} +2.00000 q^{31} -2.00000 q^{33} +4.00000i q^{35} +8.00000i q^{37} +2.00000 q^{41} +4.00000i q^{43} +1.00000i q^{45} +9.00000 q^{49} +6.00000i q^{53} -2.00000 q^{55} -14.0000i q^{59} -14.0000i q^{61} +4.00000 q^{63} +4.00000i q^{67} +8.00000i q^{69} +8.00000 q^{71} +10.0000 q^{73} +1.00000i q^{75} +8.00000i q^{77} -6.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +2.00000 q^{87} -14.0000 q^{89} -2.00000i q^{93} +18.0000 q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} - 2 q^{9} - 2 q^{15} - 16 q^{23} - 2 q^{25} + 4 q^{31} - 4 q^{33} + 4 q^{41} + 18 q^{49} - 4 q^{55} + 8 q^{63} + 16 q^{71} + 20 q^{73} - 12 q^{79} + 2 q^{81} + 4 q^{87} - 28 q^{89} + 36 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 - 2 * q^9 - 2 * q^15 - 16 * q^23 - 2 * q^25 + 4 * q^31 - 4 * q^33 + 4 * q^41 + 18 * q^49 - 4 * q^55 + 8 * q^63 + 16 * q^71 + 20 * q^73 - 12 * q^79 + 2 * q^81 + 4 * q^87 - 28 * q^89 + 36 * 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
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	4.00000i	0.872872i
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	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
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	4.00000i	0.676123i
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	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
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	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
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 14.0000i	− 1.82264i	−0.411693	−	0.911322i	$-0.635063\pi$
$59$	− 14.0000i	− 1.82264i	0.411693	−	0.911322i	$-0.364937\pi$
$60$	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
$63$	4.00000	0.503953
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	8.00000i	0.963087i
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	8.00000i	0.911685i
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	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
$85$	0	0
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 2.00000i	− 0.207390i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	0	0
$99$	2.00000i	0.201008i
$100$	0	0
$101$	10.0000i	0.995037i	0.867453	+	0.497519i	$0.165755\pi$
$101$	10.0000i	0.995037i	−0.867453	+	0.497519i	$0.834245\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	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
$111$	8.00000	0.759326
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	0	0
$115$	8.00000i	0.746004i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	− 2.00000i	− 0.180334i
$124$	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
$131$	− 6.00000i	− 0.524222i	−0.965038	−	0.262111i	$-0.915581\pi$
$131$	− 6.00000i	− 0.524222i	0.965038	−	0.262111i	$-0.0844187\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	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
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	− 9.00000i	− 0.742307i
$148$	0	0
$149$	− 2.00000i	− 0.163846i	−0.996639	−	0.0819232i	$-0.973894\pi$
$149$	− 2.00000i	− 0.163846i	0.996639	−	0.0819232i	$-0.0261062\pi$
$150$	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
$153$	0	0
$154$	0	0
$155$	− 2.00000i	− 0.160644i
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	32.0000	2.52195
$162$	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
$165$	2.00000i	0.155700i
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.0000i	1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$	22.0000i	1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	−14.0000	−1.05230
$178$	0	0
$179$	6.00000i	0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$	6.00000i	0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$	0	0
$181$	2.00000i	0.148659i	0.997234	+	0.0743294i	$0.0236816\pi$
$181$	2.00000i	0.148659i	−0.997234	+	0.0743294i	$0.976318\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 4.00000i	− 0.290957i
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 8.00000i	− 0.561490i
$204$	0	0
$205$	− 2.00000i	− 0.139686i
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 16.0000i	− 1.06196i	−0.847385	−	0.530979i	$-0.821824\pi$
$227$	− 16.0000i	− 1.06196i	0.847385	−	0.530979i	$-0.178176\pi$
$228$	0	0
$229$	14.0000i	0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$	14.0000i	0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	20.0000	1.31024	0.655122	−	0.755523i	$-0.272617\pi$
$233$	20.0000	1.31024	0.655122	+	0.755523i	$0.272617\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.00000i	0.389742i
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 9.00000i	− 0.574989i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	− 22.0000i	− 1.38863i	−0.719672	−	0.694314i	$-0.755708\pi$
$251$	− 22.0000i	− 1.38863i	0.719672	−	0.694314i	$-0.244292\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.0000	0.998053	0.499026	−	0.866587i	$-0.333691\pi$
$257$	16.0000	0.998053	0.499026	+	0.866587i	$0.333691\pi$
$258$	0	0
$259$	− 32.0000i	− 1.98838i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	14.0000i	0.856786i
$268$	0	0
$269$	− 14.0000i	− 0.853595i	−0.904347	−	0.426798i	$-0.859642\pi$
$269$	− 14.0000i	− 0.853595i	0.904347	−	0.426798i	$-0.140358\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.00000i	0.120605i
$276$	0	0
$277$	28.0000i	1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$	28.0000i	1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	− 18.0000i	− 1.05518i
$292$	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
$295$	−14.0000	−0.815112
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 16.0000i	− 0.922225i
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	−14.0000	−0.801638
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	16.0000i	0.910208i
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	− 4.00000i	− 0.225374i
$316$	0	0
$317$	26.0000i	1.46031i	0.683284	+	0.730153i	$0.260551\pi$
$317$	26.0000i	1.46031i	−0.683284	+	0.730153i	$0.739449\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.0000	0.774202
$328$	0	0
$329$	0	0
$330$	0	0
$331$	28.0000i	1.53902i	0.638635	+	0.769510i	$0.279499\pi$
$331$	28.0000i	1.53902i	−0.638635	+	0.769510i	$0.720501\pi$
$332$	0	0
$333$	− 8.00000i	− 0.438397i
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	− 16.0000i	− 0.869001i
$340$	0	0
$341$	− 4.00000i	− 0.216612i
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	26.0000i	1.39175i	0.718164	+	0.695874i	$0.244983\pi$
$349$	26.0000i	1.39175i	−0.718164	+	0.695874i	$0.755017\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	0	0
$355$	− 8.00000i	− 0.424596i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	− 10.0000i	− 0.523424i
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	− 24.0000i	− 1.24602i
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	20.0000i	1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$	20.0000i	1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	0	0
$381$	− 16.0000i	− 0.819705i
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	− 6.00000i	− 0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$	− 6.00000i	− 0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	6.00000i	0.301893i
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	16.0000	0.793091
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	12.0000i	0.591916i
$412$	0	0
$413$	56.0000i	2.75558i
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	− 10.0000i	− 0.488532i	−0.969708	−	0.244266i	$-0.921453\pi$
$419$	− 10.0000i	− 0.488532i	0.969708	−	0.244266i	$-0.0785470\pi$
$420$	0	0
$421$	− 2.00000i	− 0.0974740i	−0.998812	−	0.0487370i	$-0.984480\pi$
$421$	− 2.00000i	− 0.0974740i	0.998812	−	0.0487370i	$-0.0155196\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	56.0000i	2.71003i
$428$	0	0
$429$	0	0
$430$	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
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	− 2.00000i	− 0.0958927i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	14.0000i	0.663664i
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	−38.0000	−1.79333	−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000	−1.79333	−0.896665	+	0.442709i	$0.854018\pi$
$450$	0	0
$451$	− 4.00000i	− 0.188353i
$452$	0	0
$453$	− 10.0000i	− 0.469841i
$454$	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
$459$	0	0
$460$	0	0
$461$	− 6.00000i	− 0.279448i	−0.990190	−	0.139724i	$-0.955378\pi$
$461$	− 6.00000i	− 0.279448i	0.990190	−	0.139724i	$-0.0446215\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	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
$469$	− 16.0000i	− 0.738811i
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 32.0000i	− 1.45605i
$484$	0	0
$485$	− 18.0000i	− 0.817338i
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	30.0000i	1.35388i	0.736038	+	0.676941i	$0.236695\pi$
$491$	30.0000i	1.35388i	−0.736038	+	0.676941i	$0.763305\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	−32.0000	−1.43540
$498$	0	0
$499$	12.0000i	0.537194i	0.963253	+	0.268597i	$0.0865599\pi$
$499$	12.0000i	0.537194i	−0.963253	+	0.268597i	$0.913440\pi$
$500$	0	0
$501$	− 16.0000i	− 0.714827i
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	− 13.0000i	− 0.577350i
$508$	0	0
$509$	− 10.0000i	− 0.443242i	−0.975133	−	0.221621i	$-0.928865\pi$
$509$	− 10.0000i	− 0.443242i	0.975133	−	0.221621i	$-0.0711348\pi$
$510$	0	0
$511$	−40.0000	−1.76950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.0000i	0.705044i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	22.0000	0.965693
$520$	0	0
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	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
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	14.0000i	0.607548i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	− 18.0000i	− 0.775315i
$540$	0	0
$541$	− 26.0000i	− 1.11783i	−0.829226	−	0.558914i	$-0.811218\pi$
$541$	− 26.0000i	− 1.11783i	0.829226	−	0.558914i	$-0.188782\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	0	0
$549$	14.0000i	0.597505i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	24.0000	1.02058
$554$	0	0
$555$	− 8.00000i	− 0.339581i
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	− 16.0000i	− 0.673125i
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	20.0000i	0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$	20.0000i	0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$	0	0
$573$	12.0000i	0.501307i
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	26.0000i	1.08052i
$580$	0	0
$581$	− 48.0000i	− 1.99138i
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.00000i	0.330195i	0.986277	+	0.165098i	$0.0527939\pi$
$587$	8.00000i	0.330195i	−0.986277	+	0.165098i	$0.947206\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.0000i	0.736691i
$598$	0	0
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	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
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	− 7.00000i	− 0.284590i
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	16.0000i	0.643094i	0.946894	+	0.321547i	$0.104203\pi$
$619$	16.0000i	0.643094i	−0.946894	+	0.321547i	$0.895797\pi$
$620$	0	0
$621$	− 8.00000i	− 0.321029i
$622$	0	0
$623$	56.0000	2.24359
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−10.0000	−0.398094	−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000	−0.398094	−0.199047	+	0.979990i	$0.563785\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 16.0000i	− 0.634941i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	− 4.00000i	− 0.157500i
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−28.0000	−1.09910
$650$	0	0
$651$	8.00000i	0.313545i
$652$	0	0
$653$	− 34.0000i	− 1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$	− 34.0000i	− 1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	2.00000i	0.0779089i	0.999241	+	0.0389545i	$0.0124027\pi$
$659$	2.00000i	0.0779089i	−0.999241	+	0.0389545i	$0.987597\pi$
$660$	0	0
$661$	− 2.00000i	− 0.0777910i	−0.999243	−	0.0388955i	$-0.987616\pi$
$661$	− 2.00000i	− 0.0777910i	0.999243	−	0.0388955i	$-0.0123839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.0000i	− 0.619522i
$668$	0	0
$669$	16.0000i	0.618596i
$670$	0	0
$671$	−28.0000	−1.08093
$672$	0	0
$673$	−42.0000	−1.61898	−0.809491	−	0.587133i	$-0.800257\pi$
$673$	−42.0000	−1.61898	−0.809491	+	0.587133i	$0.800257\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	0	0
$679$	−72.0000	−2.76311
$680$	0	0
$681$	−16.0000	−0.613121
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	0	0
$685$	12.0000i	0.458496i
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	0	0
$690$	0	0
$691$	40.0000i	1.52167i	0.648944	+	0.760836i	$0.275211\pi$
$691$	40.0000i	1.52167i	−0.648944	+	0.760836i	$0.724789\pi$
$692$	0	0
$693$	− 8.00000i	− 0.303895i
$694$	0	0
$695$	16.0000	0.606915
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 20.0000i	− 0.756469i
$700$	0	0
$701$	− 42.0000i	− 1.58632i	−0.609015	−	0.793159i	$-0.708435\pi$
$701$	− 42.0000i	− 1.58632i	0.609015	−	0.793159i	$-0.291565\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 40.0000i	− 1.50435i
$708$	0	0
$709$	− 18.0000i	− 0.676004i	−0.941145	−	0.338002i	$-0.890249\pi$
$709$	− 18.0000i	− 0.676004i	0.941145	−	0.338002i	$-0.109751\pi$
$710$	0	0
$711$	6.00000	0.225018
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.00000i	0.149383i
$718$	0	0
$719$	−44.0000	−1.64092	−0.820462	−	0.571702i	$-0.806283\pi$
$719$	−44.0000	−1.64092	−0.820462	+	0.571702i	$0.806283\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	0	0
$723$	− 14.0000i	− 0.520666i
$724$	0	0
$725$	− 2.00000i	− 0.0742781i
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 44.0000i	− 1.62518i	−0.582838	−	0.812589i	$-0.698058\pi$
$733$	− 44.0000i	− 1.62518i	0.582838	−	0.812589i	$-0.301942\pi$
$734$	0	0
$735$	−9.00000	−0.331970
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	34.0000	1.24068	0.620339	−	0.784334i	$-0.286995\pi$
$751$	34.0000	1.24068	0.620339	+	0.784334i	$0.286995\pi$
$752$	0	0
$753$	−22.0000	−0.801725
$754$	0	0
$755$	− 10.0000i	− 0.363937i
$756$	0	0
$757$	− 28.0000i	− 1.01768i	−0.860862	−	0.508839i	$-0.830075\pi$
$757$	− 28.0000i	− 1.01768i	0.860862	−	0.508839i	$-0.169925\pi$
$758$	0	0
$759$	16.0000	0.580763
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	− 56.0000i	− 2.02734i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	− 16.0000i	− 0.576226i
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	−32.0000	−1.14799
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 16.0000i	− 0.572525i
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−4.00000	−0.142766
$786$	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
$789$	− 16.0000i	− 0.569615i
$790$	0	0
$791$	−64.0000	−2.27558
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 6.00000i	− 0.212798i
$796$	0	0
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	14.0000	0.494666
$802$	0	0
$803$	− 20.0000i	− 0.705785i
$804$	0	0
$805$	− 32.0000i	− 1.12785i
$806$	0	0
$807$	−14.0000	−0.492823
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	28.0000i	0.983213i	0.870817	+	0.491606i	$0.163590\pi$
$811$	28.0000i	0.983213i	−0.870817	+	0.491606i	$0.836410\pi$
$812$	0	0
$813$	22.0000i	0.771574i
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	0	0
$818$	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
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	0	0
$829$	− 2.00000i	− 0.0694629i	−0.999397	−	0.0347314i	$-0.988942\pi$
$829$	− 2.00000i	− 0.0694629i	0.999397	−	0.0347314i	$-0.0110576\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 16.0000i	− 0.553703i
$836$	0	0
$837$	2.00000i	0.0691301i
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 2.00000i	− 0.0688837i
$844$	0	0
$845$	− 13.0000i	− 0.447214i
$846$	0	0
$847$	−28.0000	−0.962091
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	− 64.0000i	− 2.19389i
$852$	0	0
$853$	8.00000i	0.273915i	0.990577	+	0.136957i	$0.0437323\pi$
$853$	8.00000i	0.273915i	−0.990577	+	0.136957i	$0.956268\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	0	0
$859$	− 4.00000i	− 0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$	− 4.00000i	− 0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	0	0
$861$	8.00000i	0.272639i
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	17.0000i	0.577350i
$868$	0	0
$869$	12.0000i	0.407072i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−18.0000	−0.609208
$874$	0	0
$875$	− 4.00000i	− 0.135225i
$876$	0	0
$877$	− 4.00000i	− 0.135070i	−0.997717	−	0.0675352i	$-0.978487\pi$
$877$	− 4.00000i	− 0.135070i	0.997717	−	0.0675352i	$-0.0215135\pi$
$878$	0	0
$879$	−26.0000	−0.876958
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	14.0000i	0.470605i
$886$	0	0
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	−64.0000	−2.14649
$890$	0	0
$891$	− 2.00000i	− 0.0670025i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	6.00000	0.200558
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.00000i	0.133407i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	2.00000	0.0664822
$906$	0	0
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	0	0
$909$	− 10.0000i	− 0.331679i
$910$	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
$913$	24.0000	0.794284
$914$	0	0
$915$	14.0000i	0.462826i
$916$	0	0
$917$	24.0000i	0.792550i
$918$	0	0
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 8.00000i	− 0.263038i
$926$	0	0
$927$	16.0000	0.525509
$928$	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
$931$	0	0
$932$	0	0
$933$	− 12.0000i	− 0.392862i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	− 2.00000i	− 0.0652675i
$940$	0	0
$941$	− 18.0000i	− 0.586783i	−0.955992	−	0.293392i	$-0.905216\pi$
$941$	− 18.0000i	− 0.586783i	0.955992	−	0.293392i	$-0.0947840\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	20.0000	0.647864	0.323932	−	0.946080i	$-0.394995\pi$
$953$	20.0000	0.647864	0.323932	+	0.946080i	$0.394995\pi$
$954$	0	0
$955$	12.0000i	0.388311i
$956$	0	0
$957$	− 4.00000i	− 0.129302i
$958$	0	0
$959$	48.0000	1.55000
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	26.0000i	0.836970i
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26.0000i	0.834380i	0.908819	+	0.417190i	$0.136985\pi$
$971$	26.0000i	0.834380i	−0.908819	+	0.417190i	$0.863015\pi$
$972$	0	0
$973$	− 64.0000i	− 2.05175i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	0	0
$979$	28.0000i	0.894884i
$980$	0	0
$981$	− 14.0000i	− 0.446986i
$982$	0	0
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 32.0000i	− 1.01754i
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	18.0000i	0.570638i
$996$	0	0
$997$	12.0000i	0.380044i	0.981780	+	0.190022i	$0.0608559\pi$
$997$	12.0000i	0.380044i	−0.981780	+	0.190022i	$0.939144\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.k.b.1921.1		2
4.3	odd	2		3840.2.k.ba.1921.2		2
8.3	odd	2		3840.2.k.ba.1921.1		2
8.5	even	2	inner	3840.2.k.b.1921.2		2
16.3	odd	4		1920.2.a.a.1.1	✓	1
16.5	even	4		1920.2.a.l.1.1	yes	1
16.11	odd	4		1920.2.a.s.1.1	yes	1
16.13	even	4		1920.2.a.r.1.1	yes	1
48.5	odd	4		5760.2.a.w.1.1		1
48.11	even	4		5760.2.a.b.1.1		1
48.29	odd	4		5760.2.a.bv.1.1		1
48.35	even	4		5760.2.a.y.1.1		1
80.19	odd	4		9600.2.a.cc.1.1		1
80.29	even	4		9600.2.a.b.1.1		1
80.59	odd	4		9600.2.a.z.1.1		1
80.69	even	4		9600.2.a.be.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.a.a.1.1	✓	1	16.3	odd	4
1920.2.a.l.1.1	yes	1	16.5	even	4
1920.2.a.r.1.1	yes	1	16.13	even	4
1920.2.a.s.1.1	yes	1	16.11	odd	4
3840.2.k.b.1921.1		2	1.1	even	1	trivial
3840.2.k.b.1921.2		2	8.5	even	2	inner
3840.2.k.ba.1921.1		2	8.3	odd	2
3840.2.k.ba.1921.2		2	4.3	odd	2
5760.2.a.b.1.1		1	48.11	even	4
5760.2.a.w.1.1		1	48.5	odd	4
5760.2.a.y.1.1		1	48.35	even	4
5760.2.a.bv.1.1		1	48.29	odd	4
9600.2.a.b.1.1		1	80.29	even	4
9600.2.a.z.1.1		1	80.59	odd	4
9600.2.a.be.1.1		1	80.69	even	4
9600.2.a.cc.1.1		1	80.19	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} -2.00000i q^{11} -1.00000 q^{15} +4.00000i q^{21} -8.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} +2.00000i q^{29} +2.00000 q^{31} -2.00000 q^{33} +4.00000i q^{35} +8.00000i q^{37} +2.00000 q^{41} +4.00000i q^{43} +1.00000i q^{45} +9.00000 q^{49} +6.00000i q^{53} -2.00000 q^{55} -14.0000i q^{59} -14.0000i q^{61} +4.00000 q^{63} +4.00000i q^{67} +8.00000i q^{69} +8.00000 q^{71} +10.0000 q^{73} +1.00000i q^{75} +8.00000i q^{77} -6.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +2.00000 q^{87} -14.0000 q^{89} -2.00000i q^{93} +18.0000 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} - 2 q^{9} - 2 q^{15} - 16 q^{23} - 2 q^{25} + 4 q^{31} - 4 q^{33} + 4 q^{41} + 18 q^{49} - 4 q^{55} + 8 q^{63} + 16 q^{71} + 20 q^{73} - 12 q^{79} + 2 q^{81} + 4 q^{87} - 28 q^{89} + 36 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 - 2 * q^9 - 2 * q^15 - 16 * q^23 - 2 * q^25 + 4 * q^31 - 4 * q^33 + 4 * q^41 + 18 * q^49 - 4 * q^55 + 8 * q^63 + 16 * q^71 + 20 * q^73 - 12 * q^79 + 2 * q^81 + 4 * q^87 - 28 * q^89 + 36 * q^97

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