LMFDB - Embedded newform 5760.2.k.m.2881.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5760,2,Mod(2881,5760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5760, 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("5760.2881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5760 = 2^{7} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5760.k (of order $2$, degree $1$, 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:	$45.9938315643$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2881.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	5760.2881
Dual form			5760.2.k.m.2881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} -2.00000 q^{7} -0.828427i q^{11} -0.828427i q^{13} -2.82843 q^{17} +5.65685 q^{23} -1.00000 q^{25} +3.65685i q^{29} +1.17157 q^{31} -2.00000i q^{35} -6.48528i q^{37} -3.65685 q^{41} -1.65685i q^{43} +1.65685 q^{47} -3.00000 q^{49} +11.6569i q^{53} +0.828427 q^{55} +4.82843i q^{59} -9.65685i q^{61} +0.828427 q^{65} -9.65685i q^{67} +13.6569 q^{71} -9.31371 q^{73} +1.65685i q^{77} +12.4853 q^{79} -2.82843i q^{85} -4.34315 q^{89} +1.65685i q^{91} +7.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7} - 4 q^{25} + 16 q^{31} + 8 q^{41} - 16 q^{47} - 12 q^{49} - 8 q^{55} - 8 q^{65} + 32 q^{71} + 8 q^{73} + 16 q^{79} - 40 q^{89} + 8 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 - 4 * q^25 + 16 * q^31 + 8 * q^41 - 16 * q^47 - 12 * q^49 - 8 * q^55 - 8 * q^65 + 32 * q^71 + 8 * q^73 + 16 * q^79 - 40 * q^89 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5760\mathbb{Z}\right)^\times$.

$n$	$641$	$901$	$2431$	$3457$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 0.828427i	− 0.249780i	−0.992171	−	0.124890i	$-0.960142\pi$
$11$	− 0.828427i	− 0.249780i	0.992171	−	0.124890i	$-0.0398578\pi$
$12$	0	0
$13$	− 0.828427i	− 0.229764i	−0.993379	−	0.114882i	$-0.963351\pi$
$13$	− 0.828427i	− 0.229764i	0.993379	−	0.114882i	$-0.0366490\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.65685i	0.679061i	0.940595	+	0.339530i	$0.110268\pi$
$29$	3.65685i	0.679061i	−0.940595	+	0.339530i	$0.889732\pi$
$30$	0	0
$31$	1.17157	0.210421	0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157	0.210421	0.105210	+	0.994450i	$0.466448\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	0	0
$37$	− 6.48528i	− 1.06617i	−0.846061	−	0.533087i	$-0.821032\pi$
$37$	− 6.48528i	− 1.06617i	0.846061	−	0.533087i	$-0.178968\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	− 1.65685i	− 0.252668i	−0.991988	−	0.126334i	$-0.959679\pi$
$43$	− 1.65685i	− 0.252668i	0.991988	−	0.126334i	$-0.0403211\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.65685	0.241677	0.120839	−	0.992672i	$-0.461442\pi$
$47$	1.65685	0.241677	0.120839	+	0.992672i	$0.461442\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.6569i	1.60119i	0.599204	+	0.800596i	$0.295484\pi$
$53$	11.6569i	1.60119i	−0.599204	+	0.800596i	$0.704516\pi$
$54$	0	0
$55$	0.828427	0.111705
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.82843i	0.628608i	0.949322	+	0.314304i	$0.101771\pi$
$59$	4.82843i	0.628608i	−0.949322	+	0.314304i	$0.898229\pi$
$60$	0	0
$61$	− 9.65685i	− 1.23643i	−0.786008	−	0.618217i	$-0.787855\pi$
$61$	− 9.65685i	− 1.23643i	0.786008	−	0.618217i	$-0.212145\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.828427	0.102754
$66$	0	0
$67$	− 9.65685i	− 1.17977i	−0.807486	−	0.589886i	$-0.799173\pi$
$67$	− 9.65685i	− 1.17977i	0.807486	−	0.589886i	$-0.200827\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.6569	1.62077	0.810385	−	0.585897i	$-0.199258\pi$
$71$	13.6569	1.62077	0.810385	+	0.585897i	$0.199258\pi$
$72$	0	0
$73$	−9.31371	−1.09009	−0.545044	−	0.838408i	$-0.683487\pi$
$73$	−9.31371	−1.09009	−0.545044	+	0.838408i	$0.683487\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.65685i	0.188816i
$78$	0	0
$79$	12.4853	1.40470	0.702352	−	0.711830i	$-0.252133\pi$
$79$	12.4853	1.40470	0.702352	+	0.711830i	$0.252133\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 2.82843i	− 0.306786i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.34315	−0.460373	−0.230186	−	0.973147i	$-0.573934\pi$
$89$	−4.34315	−0.460373	−0.230186	+	0.973147i	$0.573934\pi$
$90$	0	0
$91$	1.65685i	0.173686i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.65685	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$97$	7.65685	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.31371i	0.528734i	0.964422	+	0.264367i	$0.0851630\pi$
$101$	5.31371i	0.528734i	−0.964422	+	0.264367i	$0.914837\pi$
$102$	0	0
$103$	−4.34315	−0.427943	−0.213971	−	0.976840i	$-0.568640\pi$
$103$	−4.34315	−0.427943	−0.213971	+	0.976840i	$0.568640\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.34315i	− 0.226520i	−0.993565	−	0.113260i	$-0.963871\pi$
$107$	− 2.34315i	− 0.226520i	0.993565	−	0.113260i	$-0.0361294\pi$
$108$	0	0
$109$	− 4.00000i	− 0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$	− 4.00000i	− 0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	20.4853	1.92709	0.963547	−	0.267541i	$-0.0862110\pi$
$113$	20.4853	1.92709	0.963547	+	0.267541i	$0.0862110\pi$
$114$	0	0
$115$	5.65685i	0.527504i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.65685	0.518563
$120$	0	0
$121$	10.3137	0.937610
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	15.6569	1.38932	0.694661	−	0.719338i	$-0.255555\pi$
$127$	15.6569	1.38932	0.694661	+	0.719338i	$0.255555\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 14.4853i	− 1.26558i	−0.774321	−	0.632792i	$-0.781909\pi$
$131$	− 14.4853i	− 1.26558i	0.774321	−	0.632792i	$-0.218091\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.17157	0.100094	0.0500471	−	0.998747i	$-0.484063\pi$
$137$	1.17157	0.100094	0.0500471	+	0.998747i	$0.484063\pi$
$138$	0	0
$139$	9.65685i	0.819084i	0.912291	+	0.409542i	$0.134311\pi$
$139$	9.65685i	0.819084i	−0.912291	+	0.409542i	$0.865689\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.686292	−0.0573906
$144$	0	0
$145$	−3.65685	−0.303685
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.6569i	0.954967i	0.878641	+	0.477483i	$0.158451\pi$
$149$	11.6569i	0.954967i	−0.878641	+	0.477483i	$0.841549\pi$
$150$	0	0
$151$	8.48528	0.690522	0.345261	−	0.938507i	$-0.387790\pi$
$151$	8.48528	0.690522	0.345261	+	0.938507i	$0.387790\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.17157i	0.0941030i
$156$	0	0
$157$	− 22.4853i	− 1.79452i	−0.441502	−	0.897260i	$-0.645554\pi$
$157$	− 22.4853i	− 1.79452i	0.441502	−	0.897260i	$-0.354446\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.6569	1.67586	0.837929	−	0.545779i	$-0.183766\pi$
$167$	21.6569	1.67586	0.837929	+	0.545779i	$0.183766\pi$
$168$	0	0
$169$	12.3137	0.947208
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4.34315i	− 0.330203i	−0.986277	−	0.165102i	$-0.947205\pi$
$173$	− 4.34315i	− 0.330203i	0.986277	−	0.165102i	$-0.0527952\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.4853i	1.08268i	0.840804	+	0.541340i	$0.182083\pi$
$179$	14.4853i	1.08268i	−0.840804	+	0.541340i	$0.817917\pi$
$180$	0	0
$181$	12.0000i	0.891953i	0.895045	+	0.445976i	$0.147144\pi$
$181$	12.0000i	0.891953i	−0.895045	+	0.445976i	$0.852856\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.48528	0.476807
$186$	0	0
$187$	2.34315i	0.171348i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.68629	−0.339088	−0.169544	−	0.985523i	$-0.554230\pi$
$191$	−4.68629	−0.339088	−0.169544	+	0.985523i	$0.554230\pi$
$192$	0	0
$193$	−10.9706	−0.789678	−0.394839	−	0.918750i	$-0.629200\pi$
$193$	−10.9706	−0.789678	−0.394839	+	0.918750i	$0.629200\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.31371i	− 0.0935979i	−0.998904	−	0.0467989i	$-0.985098\pi$
$197$	− 1.31371i	− 0.0935979i	0.998904	−	0.0467989i	$-0.0149020\pi$
$198$	0	0
$199$	−13.1716	−0.933708	−0.466854	−	0.884334i	$-0.654613\pi$
$199$	−13.1716	−0.933708	−0.466854	+	0.884334i	$0.654613\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 7.31371i	− 0.513322i
$204$	0	0
$205$	− 3.65685i	− 0.255406i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	14.3431i	0.987423i	0.869626	+	0.493711i	$0.164360\pi$
$211$	14.3431i	0.987423i	−0.869626	+	0.493711i	$0.835640\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.65685	0.112997
$216$	0	0
$217$	−2.34315	−0.159063
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.34315i	0.157617i
$222$	0	0
$223$	18.9706	1.27036	0.635181	−	0.772363i	$-0.280925\pi$
$223$	18.9706	1.27036	0.635181	+	0.772363i	$0.280925\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.6569i	1.17193i	0.810338	+	0.585963i	$0.199284\pi$
$227$	17.6569i	1.17193i	−0.810338	+	0.585963i	$0.800716\pi$
$228$	0	0
$229$	21.6569i	1.43113i	0.698549	+	0.715563i	$0.253830\pi$
$229$	21.6569i	1.43113i	−0.698549	+	0.715563i	$0.746170\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.7990	0.772978	0.386489	−	0.922294i	$-0.373688\pi$
$233$	11.7990	0.772978	0.386489	+	0.922294i	$0.373688\pi$
$234$	0	0
$235$	1.65685i	0.108081i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.6274	1.46365	0.731823	−	0.681495i	$-0.238670\pi$
$239$	22.6274	1.46365	0.731823	+	0.681495i	$0.238670\pi$
$240$	0	0
$241$	20.6274	1.32873	0.664364	−	0.747409i	$-0.268702\pi$
$241$	20.6274	1.32873	0.664364	+	0.747409i	$0.268702\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.82843i	0.557245i	0.960401	+	0.278623i	$0.0898779\pi$
$251$	8.82843i	0.557245i	−0.960401	+	0.278623i	$0.910122\pi$
$252$	0	0
$253$	− 4.68629i	− 0.294625i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.8284	−0.924972	−0.462486	−	0.886627i	$-0.653042\pi$
$257$	−14.8284	−0.924972	−0.462486	+	0.886627i	$0.653042\pi$
$258$	0	0
$259$	12.9706i	0.805952i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	29.6569	1.82872	0.914360	−	0.404902i	$-0.132694\pi$
$263$	29.6569	1.82872	0.914360	+	0.404902i	$0.132694\pi$
$264$	0	0
$265$	−11.6569	−0.716075
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 7.65685i	− 0.466847i	−0.972375	−	0.233423i	$-0.925007\pi$
$269$	− 7.65685i	− 0.466847i	0.972375	−	0.233423i	$-0.0749928\pi$
$270$	0	0
$271$	8.48528	0.515444	0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528	0.515444	0.257722	+	0.966219i	$0.417028\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.828427i	0.0499560i
$276$	0	0
$277$	23.1716i	1.39224i	0.717923	+	0.696122i	$0.245093\pi$
$277$	23.1716i	1.39224i	−0.717923	+	0.696122i	$0.754907\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.9706	−0.654449	−0.327224	−	0.944947i	$-0.606113\pi$
$281$	−10.9706	−0.654449	−0.327224	+	0.944947i	$0.606113\pi$
$282$	0	0
$283$	− 12.9706i	− 0.771020i	−0.922704	−	0.385510i	$-0.874026\pi$
$283$	− 12.9706i	− 0.771020i	0.922704	−	0.385510i	$-0.125974\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.31371	0.431715
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 17.3137i	− 1.01148i	−0.862687	−	0.505739i	$-0.831220\pi$
$293$	− 17.3137i	− 1.01148i	0.862687	−	0.505739i	$-0.168780\pi$
$294$	0	0
$295$	−4.82843	−0.281122
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.68629i	− 0.271015i
$300$	0	0
$301$	3.31371i	0.190999i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.65685	0.552950
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.3137	0.641542	0.320771	−	0.947157i	$-0.396058\pi$
$311$	11.3137	0.641542	0.320771	+	0.947157i	$0.396058\pi$
$312$	0	0
$313$	−22.9706	−1.29837	−0.649186	−	0.760629i	$-0.724891\pi$
$313$	−22.9706	−1.29837	−0.649186	+	0.760629i	$0.724891\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 13.3137i	− 0.747772i	−0.927475	−	0.373886i	$-0.878025\pi$
$317$	− 13.3137i	− 0.747772i	0.927475	−	0.373886i	$-0.121975\pi$
$318$	0	0
$319$	3.02944	0.169616
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.828427i	0.0459529i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.31371	−0.182691
$330$	0	0
$331$	− 32.0000i	− 1.75888i	−0.476011	−	0.879440i	$-0.657918\pi$
$331$	− 32.0000i	− 1.75888i	0.476011	−	0.879440i	$-0.342082\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.65685	0.527610
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 0.970563i	− 0.0525589i
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.65685i	− 0.518407i	−0.965823	−	0.259204i	$-0.916540\pi$
$347$	− 9.65685i	− 0.518407i	0.965823	−	0.259204i	$-0.0834600\pi$
$348$	0	0
$349$	− 3.02944i	− 0.162162i	−0.996708	−	0.0810810i	$-0.974163\pi$
$349$	− 3.02944i	− 0.162162i	0.996708	−	0.0810810i	$-0.0258373\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.4558	1.35488	0.677439	−	0.735579i	$-0.263090\pi$
$353$	25.4558	1.35488	0.677439	+	0.735579i	$0.263090\pi$
$354$	0	0
$355$	13.6569i	0.724831i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.9706	0.895672	0.447836	−	0.894116i	$-0.352195\pi$
$359$	16.9706	0.895672	0.447836	+	0.894116i	$0.352195\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 9.31371i	− 0.487502i
$366$	0	0
$367$	−2.68629	−0.140223	−0.0701116	−	0.997539i	$-0.522336\pi$
$367$	−2.68629	−0.140223	−0.0701116	+	0.997539i	$0.522336\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 23.3137i	− 1.21039i
$372$	0	0
$373$	30.4853i	1.57847i	0.614093	+	0.789234i	$0.289522\pi$
$373$	30.4853i	1.57847i	−0.614093	+	0.789234i	$0.710478\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.02944	0.156024
$378$	0	0
$379$	24.2843i	1.24740i	0.781664	+	0.623700i	$0.214371\pi$
$379$	24.2843i	1.24740i	−0.781664	+	0.623700i	$0.785629\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.34315	−0.324120	−0.162060	−	0.986781i	$-0.551814\pi$
$383$	−6.34315	−0.324120	−0.162060	+	0.986781i	$0.551814\pi$
$384$	0	0
$385$	−1.65685	−0.0844411
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 13.3137i	− 0.675032i	−0.941320	−	0.337516i	$-0.890413\pi$
$389$	− 13.3137i	− 0.675032i	0.941320	−	0.337516i	$-0.109587\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.4853i	0.628203i
$396$	0	0
$397$	6.48528i	0.325487i	0.986668	+	0.162743i	$0.0520343\pi$
$397$	6.48528i	0.325487i	−0.986668	+	0.162743i	$0.947966\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−36.6274	−1.82909	−0.914543	−	0.404489i	$-0.867449\pi$
$401$	−36.6274	−1.82909	−0.914543	+	0.404489i	$0.867449\pi$
$402$	0	0
$403$	− 0.970563i	− 0.0483472i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.37258	−0.266309
$408$	0	0
$409$	2.68629	0.132829	0.0664143	−	0.997792i	$-0.478844\pi$
$409$	2.68629	0.132829	0.0664143	+	0.997792i	$0.478844\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 9.65685i	− 0.475183i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.48528i	0.121414i	0.998156	+	0.0607070i	$0.0193355\pi$
$419$	2.48528i	0.121414i	−0.998156	+	0.0607070i	$0.980664\pi$
$420$	0	0
$421$	16.9706i	0.827095i	0.910483	+	0.413547i	$0.135710\pi$
$421$	16.9706i	0.827095i	−0.910483	+	0.413547i	$0.864290\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.82843	0.137199
$426$	0	0
$427$	19.3137i	0.934656i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.6569	1.04317	0.521587	−	0.853198i	$-0.325340\pi$
$431$	21.6569	1.04317	0.521587	+	0.853198i	$0.325340\pi$
$432$	0	0
$433$	−32.6274	−1.56797	−0.783987	−	0.620777i	$-0.786817\pi$
$433$	−32.6274	−1.56797	−0.783987	+	0.620777i	$0.786817\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	22.1421	1.05679	0.528393	−	0.849000i	$-0.322795\pi$
$439$	22.1421	1.05679	0.528393	+	0.849000i	$0.322795\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 17.6569i	− 0.838902i	−0.907778	−	0.419451i	$-0.862223\pi$
$443$	− 17.6569i	− 0.838902i	0.907778	−	0.419451i	$-0.137777\pi$
$444$	0	0
$445$	− 4.34315i	− 0.205885i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.31371	−0.0619977	−0.0309989	−	0.999519i	$-0.509869\pi$
$449$	−1.31371	−0.0619977	−0.0309989	+	0.999519i	$0.509869\pi$
$450$	0	0
$451$	3.02944i	0.142651i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.65685	−0.0776745
$456$	0	0
$457$	14.9706	0.700293	0.350147	−	0.936695i	$-0.386132\pi$
$457$	14.9706	0.700293	0.350147	+	0.936695i	$0.386132\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 19.6569i	− 0.915511i	−0.889078	−	0.457755i	$-0.848654\pi$
$461$	− 19.6569i	− 0.915511i	0.889078	−	0.457755i	$-0.151346\pi$
$462$	0	0
$463$	−24.6274	−1.14453	−0.572267	−	0.820068i	$-0.693936\pi$
$463$	−24.6274	−1.14453	−0.572267	+	0.820068i	$0.693936\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.6863i	0.587052i	0.955951	+	0.293526i	$0.0948287\pi$
$467$	12.6863i	0.587052i	−0.955951	+	0.293526i	$0.905171\pi$
$468$	0	0
$469$	19.3137i	0.891824i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.37258	−0.0631114
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.3431	0.472590	0.236295	−	0.971681i	$-0.424067\pi$
$479$	10.3431	0.472590	0.236295	+	0.971681i	$0.424067\pi$
$480$	0	0
$481$	−5.37258	−0.244969
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.65685i	0.347680i
$486$	0	0
$487$	2.97056	0.134609	0.0673045	−	0.997732i	$-0.478560\pi$
$487$	2.97056	0.134609	0.0673045	+	0.997732i	$0.478560\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.1716i	1.22624i	0.789991	+	0.613118i	$0.210085\pi$
$491$	27.1716i	1.22624i	−0.789991	+	0.613118i	$0.789915\pi$
$492$	0	0
$493$	− 10.3431i	− 0.465832i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.3137	−1.22519
$498$	0	0
$499$	6.62742i	0.296684i	0.988936	+	0.148342i	$0.0473936\pi$
$499$	6.62742i	0.296684i	−0.988936	+	0.148342i	$0.952606\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.65685	−0.430578	−0.215289	−	0.976550i	$-0.569069\pi$
$503$	−9.65685	−0.430578	−0.215289	+	0.976550i	$0.569069\pi$
$504$	0	0
$505$	−5.31371	−0.236457
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	18.6274	0.824028
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 4.34315i	− 0.191382i
$516$	0	0
$517$	− 1.37258i	− 0.0603661i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.31371	−0.232798	−0.116399	−	0.993203i	$-0.537135\pi$
$521$	−5.31371	−0.232798	−0.116399	+	0.993203i	$0.537135\pi$
$522$	0	0
$523$	− 37.9411i	− 1.65905i	−0.558470	−	0.829525i	$-0.688611\pi$
$523$	− 37.9411i	− 1.65905i	0.558470	−	0.829525i	$-0.311389\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.31371	−0.144347
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.02944i	0.131219i
$534$	0	0
$535$	2.34315	0.101303
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.48528i	0.107049i
$540$	0	0
$541$	− 12.0000i	− 0.515920i	−0.966156	−	0.257960i	$-0.916950\pi$
$541$	− 12.0000i	− 0.515920i	0.966156	−	0.257960i	$-0.0830503\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	− 25.6569i	− 1.09701i	−0.836148	−	0.548504i	$-0.815198\pi$
$547$	− 25.6569i	− 1.09701i	0.836148	−	0.548504i	$-0.184802\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−24.9706	−1.06186
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23.6569i	− 1.00237i	−0.865339	−	0.501187i	$-0.832897\pi$
$557$	− 23.6569i	− 1.00237i	0.865339	−	0.501187i	$-0.167103\pi$
$558$	0	0
$559$	−1.37258	−0.0580541
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 28.9706i	− 1.22096i	−0.792030	−	0.610482i	$-0.790976\pi$
$563$	− 28.9706i	− 1.22096i	0.792030	−	0.610482i	$-0.209024\pi$
$564$	0	0
$565$	20.4853i	0.861822i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.31371	−0.222762	−0.111381	−	0.993778i	$-0.535527\pi$
$569$	−5.31371	−0.222762	−0.111381	+	0.993778i	$0.535527\pi$
$570$	0	0
$571$	− 38.6274i	− 1.61651i	−0.588835	−	0.808254i	$-0.700413\pi$
$571$	− 38.6274i	− 1.61651i	0.588835	−	0.808254i	$-0.299587\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.65685	−0.235907
$576$	0	0
$577$	−5.02944	−0.209378	−0.104689	−	0.994505i	$-0.533385\pi$
$577$	−5.02944	−0.209378	−0.104689	+	0.994505i	$0.533385\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.65685	0.399946
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 29.6569i	− 1.22407i	−0.790831	−	0.612035i	$-0.790351\pi$
$587$	− 29.6569i	− 1.22407i	0.790831	−	0.612035i	$-0.209649\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	39.7990	1.63435	0.817174	−	0.576391i	$-0.195539\pi$
$593$	39.7990	1.63435	0.817174	+	0.576391i	$0.195539\pi$
$594$	0	0
$595$	5.65685i	0.231908i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.6569	−1.86549	−0.932744	−	0.360539i	$-0.882593\pi$
$599$	−45.6569	−1.86549	−0.932744	+	0.360539i	$0.882593\pi$
$600$	0	0
$601$	27.9411	1.13974	0.569871	−	0.821734i	$-0.306993\pi$
$601$	27.9411	1.13974	0.569871	+	0.821734i	$0.306993\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.3137i	0.419312i
$606$	0	0
$607$	−24.6274	−0.999596	−0.499798	−	0.866142i	$-0.666592\pi$
$607$	−24.6274	−0.999596	−0.499798	+	0.866142i	$0.666592\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.37258i	− 0.0555288i
$612$	0	0
$613$	9.51472i	0.384296i	0.981366	+	0.192148i	$0.0615453\pi$
$613$	9.51472i	0.384296i	−0.981366	+	0.192148i	$0.938455\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.82843	−0.113868	−0.0569341	−	0.998378i	$-0.518132\pi$
$617$	−2.82843	−0.113868	−0.0569341	+	0.998378i	$0.518132\pi$
$618$	0	0
$619$	44.9706i	1.80752i	0.428040	+	0.903760i	$0.359204\pi$
$619$	44.9706i	1.80752i	−0.428040	+	0.903760i	$0.640796\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.68629	0.348009
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.3431i	0.731389i
$630$	0	0
$631$	19.5147	0.776869	0.388434	−	0.921476i	$-0.373016\pi$
$631$	19.5147	0.776869	0.388434	+	0.921476i	$0.373016\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15.6569i	0.621323i
$636$	0	0
$637$	2.48528i	0.0984704i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23.6569	−0.934390	−0.467195	−	0.884154i	$-0.654735\pi$
$641$	−23.6569	−0.934390	−0.467195	+	0.884154i	$0.654735\pi$
$642$	0	0
$643$	− 42.6274i	− 1.68106i	−0.541764	−	0.840531i	$-0.682243\pi$
$643$	− 42.6274i	− 1.68106i	0.541764	−	0.840531i	$-0.317757\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.34315	0.249375	0.124687	−	0.992196i	$-0.460207\pi$
$647$	6.34315	0.249375	0.124687	+	0.992196i	$0.460207\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.68629i	0.261655i	0.991405	+	0.130827i	$0.0417634\pi$
$653$	6.68629i	0.261655i	−0.991405	+	0.130827i	$0.958237\pi$
$654$	0	0
$655$	14.4853	0.565987
$656$	0	0
$657$	0	0
$658$	0	0
$659$	44.8284i	1.74627i	0.487481	+	0.873134i	$0.337916\pi$
$659$	44.8284i	1.74627i	−0.487481	+	0.873134i	$0.662084\pi$
$660$	0	0
$661$	3.02944i	0.117831i	0.998263	+	0.0589157i	$0.0187643\pi$
$661$	3.02944i	0.117831i	−0.998263	+	0.0589157i	$0.981236\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.6863i	0.800976i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	10.6863	0.411926	0.205963	−	0.978560i	$-0.433967\pi$
$673$	10.6863	0.411926	0.205963	+	0.978560i	$0.433967\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.3137i	0.665420i	0.943029	+	0.332710i	$0.107963\pi$
$677$	17.3137i	0.665420i	−0.943029	+	0.332710i	$0.892037\pi$
$678$	0	0
$679$	−15.3137	−0.587686
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.31371i	− 0.126796i	−0.997988	−	0.0633978i	$-0.979806\pi$
$683$	− 3.31371i	− 0.126796i	0.997988	−	0.0633978i	$-0.0201937\pi$
$684$	0	0
$685$	1.17157i	0.0447635i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.65685	0.367897
$690$	0	0
$691$	− 6.62742i	− 0.252119i	−0.992023	−	0.126059i	$-0.959767\pi$
$691$	− 6.62742i	− 0.252119i	0.992023	−	0.126059i	$-0.0402330\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.65685	−0.366305
$696$	0	0
$697$	10.3431	0.391775
$698$	0	0
$699$	0	0
$700$	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
$707$	− 10.6274i	− 0.399685i
$708$	0	0
$709$	− 36.2843i	− 1.36268i	−0.731965	−	0.681342i	$-0.761397\pi$
$709$	− 36.2843i	− 1.36268i	0.731965	−	0.681342i	$-0.238603\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.62742	0.248199
$714$	0	0
$715$	− 0.686292i	− 0.0256658i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−12.2843	−0.458126	−0.229063	−	0.973412i	$-0.573566\pi$
$719$	−12.2843	−0.458126	−0.229063	+	0.973412i	$0.573566\pi$
$720$	0	0
$721$	8.68629	0.323494
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.65685i	− 0.135812i
$726$	0	0
$727$	50.9706	1.89039	0.945197	−	0.326501i	$-0.105870\pi$
$727$	50.9706	1.89039	0.945197	+	0.326501i	$0.105870\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.68629i	0.173329i
$732$	0	0
$733$	− 21.7990i	− 0.805164i	−0.915384	−	0.402582i	$-0.868113\pi$
$733$	− 21.7990i	− 0.805164i	0.915384	−	0.402582i	$-0.131887\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	41.9411i	1.54283i	0.636333	+	0.771415i	$0.280451\pi$
$739$	41.9411i	1.54283i	−0.636333	+	0.771415i	$0.719549\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.97056	0.182352	0.0911761	−	0.995835i	$-0.470937\pi$
$743$	4.97056	0.182352	0.0911761	+	0.995835i	$0.470937\pi$
$744$	0	0
$745$	−11.6569	−0.427074
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.68629i	0.171233i
$750$	0	0
$751$	−45.4558	−1.65871	−0.829354	−	0.558724i	$-0.811291\pi$
$751$	−45.4558	−1.65871	−0.829354	+	0.558724i	$0.811291\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.48528i	0.308811i
$756$	0	0
$757$	34.4853i	1.25339i	0.779265	+	0.626694i	$0.215593\pi$
$757$	34.4853i	1.25339i	−0.779265	+	0.626694i	$0.784407\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.3137	1.20762	0.603810	−	0.797128i	$-0.293648\pi$
$761$	33.3137	1.20762	0.603810	+	0.797128i	$0.293648\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	13.3137	0.480105	0.240052	−	0.970760i	$-0.422835\pi$
$769$	13.3137	0.480105	0.240052	+	0.970760i	$0.422835\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39.2548i	1.41190i	0.708263	+	0.705949i	$0.249479\pi$
$773$	39.2548i	1.41190i	−0.708263	+	0.705949i	$0.750521\pi$
$774$	0	0
$775$	−1.17157	−0.0420841
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 11.3137i	− 0.404836i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.4853	0.802534
$786$	0	0
$787$	− 16.6863i	− 0.594802i	−0.954753	−	0.297401i	$-0.903880\pi$
$787$	− 16.6863i	− 0.594802i	0.954753	−	0.297401i	$-0.0961198\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−40.9706	−1.45675
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.9706i	0.813659i	0.913504	+	0.406830i	$0.133366\pi$
$797$	22.9706i	0.813659i	−0.913504	+	0.406830i	$0.866634\pi$
$798$	0	0
$799$	−4.68629	−0.165789
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.71573i	0.272282i
$804$	0	0
$805$	− 11.3137i	− 0.398756i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.6569	0.831731	0.415865	−	0.909426i	$-0.363479\pi$
$809$	23.6569	0.831731	0.415865	+	0.909426i	$0.363479\pi$
$810$	0	0
$811$	− 35.5980i	− 1.25001i	−0.780619	−	0.625007i	$-0.785096\pi$
$811$	− 35.5980i	− 1.25001i	0.780619	−	0.625007i	$-0.214904\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34.9706i	1.22048i	0.792216	+	0.610241i	$0.208927\pi$
$821$	34.9706i	1.22048i	−0.792216	+	0.610241i	$0.791073\pi$
$822$	0	0
$823$	−18.2843	−0.637350	−0.318675	−	0.947864i	$-0.603238\pi$
$823$	−18.2843	−0.637350	−0.318675	+	0.947864i	$0.603238\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 39.3137i	− 1.36707i	−0.729917	−	0.683536i	$-0.760441\pi$
$827$	− 39.3137i	− 1.36707i	0.729917	−	0.683536i	$-0.239559\pi$
$828$	0	0
$829$	− 37.9411i	− 1.31775i	−0.752253	−	0.658875i	$-0.771033\pi$
$829$	− 37.9411i	− 1.31775i	0.752253	−	0.658875i	$-0.228967\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.48528	0.293998
$834$	0	0
$835$	21.6569i	0.749466i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.2843	−0.424100	−0.212050	−	0.977259i	$-0.568014\pi$
$839$	−12.2843	−0.424100	−0.212050	+	0.977259i	$0.568014\pi$
$840$	0	0
$841$	15.6274	0.538876
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.3137i	0.423604i
$846$	0	0
$847$	−20.6274	−0.708766
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 36.6863i	− 1.25759i
$852$	0	0
$853$	1.51472i	0.0518630i	0.999664	+	0.0259315i	$0.00825517\pi$
$853$	1.51472i	0.0518630i	−0.999664	+	0.0259315i	$0.991745\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.1421	0.756361	0.378180	−	0.925732i	$-0.376550\pi$
$857$	22.1421	0.756361	0.378180	+	0.925732i	$0.376550\pi$
$858$	0	0
$859$	46.9117i	1.60061i	0.599596	+	0.800303i	$0.295328\pi$
$859$	46.9117i	1.60061i	−0.599596	+	0.800303i	$0.704672\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.6569	0.737208	0.368604	−	0.929587i	$-0.379836\pi$
$863$	21.6569	0.737208	0.368604	+	0.929587i	$0.379836\pi$
$864$	0	0
$865$	4.34315	0.147671
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 10.3431i	− 0.350867i
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.00000i	0.0676123i
$876$	0	0
$877$	− 14.4853i	− 0.489133i	−0.969632	−	0.244567i	$-0.921354\pi$
$877$	− 14.4853i	− 0.489133i	0.969632	−	0.244567i	$-0.0786457\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−24.3431	−0.820141	−0.410071	−	0.912054i	$-0.634496\pi$
$881$	−24.3431	−0.820141	−0.410071	+	0.912054i	$0.634496\pi$
$882$	0	0
$883$	− 29.9411i	− 1.00760i	−0.863821	−	0.503800i	$-0.831935\pi$
$883$	− 29.9411i	− 1.00760i	0.863821	−	0.503800i	$-0.168065\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39.5980	−1.32957	−0.664785	−	0.747035i	$-0.731477\pi$
$887$	−39.5980	−1.32957	−0.664785	+	0.747035i	$0.731477\pi$
$888$	0	0
$889$	−31.3137	−1.05023
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−14.4853	−0.484190
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.28427i	0.142888i
$900$	0	0
$901$	− 32.9706i	− 1.09841i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	20.9706i	0.696316i	0.937436	+	0.348158i	$0.113193\pi$
$907$	20.9706i	0.696316i	−0.937436	+	0.348158i	$0.886807\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.6863	0.420316	0.210158	−	0.977667i	$-0.432602\pi$
$911$	12.6863	0.420316	0.210158	+	0.977667i	$0.432602\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	28.9706i	0.956692i
$918$	0	0
$919$	27.5147	0.907627	0.453813	−	0.891097i	$-0.350063\pi$
$919$	27.5147	0.907627	0.453813	+	0.891097i	$0.350063\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 11.3137i	− 0.372395i
$924$	0	0
$925$	6.48528i	0.213235i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.34315	−0.0766291
$936$	0	0
$937$	9.31371	0.304266	0.152133	−	0.988360i	$-0.451386\pi$
$937$	9.31371	0.304266	0.152133	+	0.988360i	$0.451386\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	55.2548i	1.80126i	0.434591	+	0.900628i	$0.356893\pi$
$941$	55.2548i	1.80126i	−0.434591	+	0.900628i	$0.643107\pi$
$942$	0	0
$943$	−20.6863	−0.673638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.9411i	0.712991i	0.934297	+	0.356495i	$0.116028\pi$
$947$	21.9411i	0.712991i	−0.934297	+	0.356495i	$0.883972\pi$
$948$	0	0
$949$	7.71573i	0.250463i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38.8284	−1.25778	−0.628888	−	0.777496i	$-0.716490\pi$
$953$	−38.8284	−1.25778	−0.628888	+	0.777496i	$0.716490\pi$
$954$	0	0
$955$	− 4.68629i	− 0.151645i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.34315	−0.0756641
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 10.9706i	− 0.353155i
$966$	0	0
$967$	−44.9117	−1.44426	−0.722131	−	0.691756i	$-0.756837\pi$
$967$	−44.9117	−1.44426	−0.722131	+	0.691756i	$0.756837\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.1421i	1.41659i	0.705917	+	0.708294i	$0.250535\pi$
$971$	44.1421i	1.41659i	−0.705917	+	0.708294i	$0.749465\pi$
$972$	0	0
$973$	− 19.3137i	− 0.619169i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.1716	0.677339	0.338669	−	0.940905i	$-0.390023\pi$
$977$	21.1716	0.677339	0.338669	+	0.940905i	$0.390023\pi$
$978$	0	0
$979$	3.59798i	0.114992i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−52.9706	−1.68950	−0.844749	−	0.535162i	$-0.820250\pi$
$983$	−52.9706	−1.68950	−0.844749	+	0.535162i	$0.820250\pi$
$984$	0	0
$985$	1.31371	0.0418582
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 9.37258i	− 0.298031i
$990$	0	0
$991$	−32.4853	−1.03193	−0.515964	−	0.856610i	$-0.672566\pi$
$991$	−32.4853	−1.03193	−0.515964	+	0.856610i	$0.672566\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 13.1716i	− 0.417567i
$996$	0	0
$997$	− 16.1421i	− 0.511227i	−0.966779	−	0.255613i	$-0.917723\pi$
$997$	− 16.1421i	− 0.511227i	0.966779	−	0.255613i	$-0.0822774\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5760.2.k.m.2881.3		4
3.2	odd	2		1920.2.k.j.961.4	yes	4
4.3	odd	2		5760.2.k.x.2881.4		4
8.3	odd	2		5760.2.k.x.2881.1		4
8.5	even	2	inner	5760.2.k.m.2881.2		4
12.11	even	2		1920.2.k.k.961.1	yes	4
24.5	odd	2		1920.2.k.j.961.1	✓	4
24.11	even	2		1920.2.k.k.961.4	yes	4
48.5	odd	4		3840.2.a.bm.1.1		2
48.11	even	4		3840.2.a.bg.1.2		2
48.29	odd	4		3840.2.a.bd.1.2		2
48.35	even	4		3840.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.k.j.961.1	✓	4	24.5	odd	2
1920.2.k.j.961.4	yes	4	3.2	odd	2
1920.2.k.k.961.1	yes	4	12.11	even	2
1920.2.k.k.961.4	yes	4	24.11	even	2
3840.2.a.bd.1.2		2	48.29	odd	4
3840.2.a.bg.1.2		2	48.11	even	4
3840.2.a.bj.1.1		2	48.35	even	4
3840.2.a.bm.1.1		2	48.5	odd	4
5760.2.k.m.2881.2		4	8.5	even	2	inner
5760.2.k.m.2881.3		4	1.1	even	1	trivial
5760.2.k.x.2881.1		4	8.3	odd	2
5760.2.k.x.2881.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5760 = 2^{7} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5760.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} -2.00000 q^{7} -0.828427i q^{11} -0.828427i q^{13} -2.82843 q^{17} +5.65685 q^{23} -1.00000 q^{25} +3.65685i q^{29} +1.17157 q^{31} -2.00000i q^{35} -6.48528i q^{37} -3.65685 q^{41} -1.65685i q^{43} +1.65685 q^{47} -3.00000 q^{49} +11.6569i q^{53} +0.828427 q^{55} +4.82843i q^{59} -9.65685i q^{61} +0.828427 q^{65} -9.65685i q^{67} +13.6569 q^{71} -9.31371 q^{73} +1.65685i q^{77} +12.4853 q^{79} -2.82843i q^{85} -4.34315 q^{89} +1.65685i q^{91} +7.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7} - 4 q^{25} + 16 q^{31} + 8 q^{41} - 16 q^{47} - 12 q^{49} - 8 q^{55} - 8 q^{65} + 32 q^{71} + 8 q^{73} + 16 q^{79} - 40 q^{89} + 8 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 - 4 * q^25 + 16 * q^31 + 8 * q^41 - 16 * q^47 - 12 * q^49 - 8 * q^55 - 8 * q^65 + 32 * q^71 + 8 * q^73 + 16 * q^79 - 40 * q^89 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more