LMFDB - Embedded newform 6084.2.b.n.4393.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6084,2,Mod(4393,6084)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6084, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6084.4393");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$6084 = 2^{2} \cdot 3^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6084.b (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:	$48.5809845897$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 3x^{2} + 4$ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 468)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4393.4
Root			$1.32288 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	6084.4393
Dual form			6084.2.b.n.4393.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575i q^{5} +2.00000i q^{7} -5.29150i q^{11} -7.93725 q^{17} -6.00000i q^{19} +5.29150 q^{23} -2.00000 q^{25} +2.64575 q^{29} +4.00000i q^{31} -5.29150 q^{35} +3.00000i q^{37} +7.93725i q^{41} +2.00000 q^{43} -5.29150i q^{47} +3.00000 q^{49} +7.93725 q^{53} +14.0000 q^{55} -10.5830i q^{59} +13.0000 q^{61} -2.00000i q^{67} +5.29150i q^{71} +7.00000i q^{73} +10.5830 q^{77} -4.00000 q^{79} -15.8745i q^{83} -21.0000i q^{85} +15.8745 q^{95} +2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 8 q^{25} + 8 q^{43} + 12 q^{49} + 56 q^{55} + 52 q^{61} - 16 q^{79}+O(q^{100})$ 4 * q - 8 * q^25 + 8 * q^43 + 12 * q^49 + 56 * q^55 + 52 * q^61 - 16 * q^79

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6084\mathbb{Z}\right)^\times$ .

$n$	$677$	$3043$	$3889$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	2.64575i	1.18322i	0.806226	+	0.591608i	$0.201507\pi$
$5$	2.64575i	1.18322i	−0.806226	+	0.591608i	$0.798493\pi$
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.29150i	− 1.59545i	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	− 5.29150i	− 1.59545i	0.603023	−	0.797724i	$-0.293963\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.93725	−1.92507	−0.962533	−	0.271163i	$-0.912592\pi$
$17$	−7.93725	−1.92507	−0.962533	+	0.271163i	$0.912592\pi$
$18$	0	0
$19$	− 6.00000i	− 1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	− 6.00000i	− 1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.29150	1.10335	0.551677	−	0.834058i	$-0.313988\pi$
$23$	5.29150	1.10335	0.551677	+	0.834058i	$0.313988\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.64575	0.491304	0.245652	−	0.969358i	$-0.420998\pi$
$29$	2.64575	0.491304	0.245652	+	0.969358i	$0.420998\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.29150	−0.894427
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.93725i	1.23959i	0.784763	+	0.619795i	$0.212784\pi$
$41$	7.93725i	1.23959i	−0.784763	+	0.619795i	$0.787216\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.29150i	− 0.771845i	−0.922531	−	0.385922i	$-0.873883\pi$
$47$	− 5.29150i	− 0.771845i	0.922531	−	0.385922i	$-0.126117\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.93725	1.09027	0.545133	−	0.838350i	$-0.316479\pi$
$53$	7.93725	1.09027	0.545133	+	0.838350i	$0.316479\pi$
$54$	0	0
$55$	14.0000	1.88776
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.5830i	− 1.37779i	−0.724861	−	0.688895i	$-0.758096\pi$
$59$	− 10.5830i	− 1.37779i	0.724861	−	0.688895i	$-0.241904\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.29150i	0.627986i	0.949425	+	0.313993i	$0.101667\pi$
$71$	5.29150i	0.627986i	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	7.00000i	0.819288i	0.912245	+	0.409644i	$0.134347\pi$
$73$	7.00000i	0.819288i	−0.912245	+	0.409644i	$0.865653\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.5830	1.20605
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 15.8745i	− 1.74245i	−0.490881	−	0.871227i	$-0.663325\pi$
$83$	− 15.8745i	− 1.74245i	0.490881	−	0.871227i	$-0.336675\pi$
$84$	0	0
$85$	− 21.0000i	− 2.27777i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	15.8745	1.62869
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.64575	0.263262	0.131631	−	0.991299i	$-0.457979\pi$
$101$	2.64575	0.263262	0.131631	+	0.991299i	$0.457979\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.29150	−0.511549	−0.255774	−	0.966736i	$-0.582330\pi$
$107$	−5.29150	−0.511549	−0.255774	+	0.966736i	$0.582330\pi$
$108$	0	0
$109$	10.0000i	0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$	10.0000i	0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.64575	0.248891	0.124446	−	0.992226i	$-0.460285\pi$
$113$	2.64575	0.248891	0.124446	+	0.992226i	$0.460285\pi$
$114$	0	0
$115$	14.0000i	1.30551i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 15.8745i	− 1.45521i
$120$	0	0
$121$	−17.0000	−1.54545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.93725i	0.709930i
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.5830	0.924641	0.462321	−	0.886713i	$-0.347017\pi$
$131$	10.5830	0.924641	0.462321	+	0.886713i	$0.347017\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.93725i	0.678125i	0.940764	+	0.339063i	$0.110110\pi$
$137$	7.93725i	0.678125i	−0.940764	+	0.339063i	$0.889890\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	7.00000i	0.581318i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 7.93725i	− 0.650245i	−0.945672	−	0.325123i	$-0.894594\pi$
$149$	− 7.93725i	− 0.650245i	0.945672	−	0.325123i	$-0.105406\pi$
$150$	0	0
$151$	18.0000i	1.46482i	0.680864	+	0.732410i	$0.261604\pi$
$151$	18.0000i	1.46482i	−0.680864	+	0.732410i	$0.738396\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.5830	−0.850047
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	10.5830i	0.834058i
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.5830	−0.804611	−0.402305	−	0.915505i	$-0.631791\pi$
$173$	−10.5830	−0.804611	−0.402305	+	0.915505i	$0.631791\pi$
$174$	0	0
$175$	− 4.00000i	− 0.302372i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.8745	−1.18652	−0.593258	−	0.805012i	$-0.702159\pi$
$179$	−15.8745	−1.18652	−0.593258	+	0.805012i	$0.702159\pi$
$180$	0	0
$181$	9.00000	0.668965	0.334482	−	0.942402i	$-0.391439\pi$
$181$	9.00000	0.668965	0.334482	+	0.942402i	$0.391439\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.93725	−0.583559
$186$	0	0
$187$	42.0000i	3.07134i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.1660	1.53152	0.765759	−	0.643127i	$-0.222363\pi$
$191$	21.1660	1.53152	0.765759	+	0.643127i	$0.222363\pi$
$192$	0	0
$193$	25.0000i	1.79954i	0.436365	+	0.899770i	$0.356266\pi$
$193$	25.0000i	1.79954i	−0.436365	+	0.899770i	$0.643734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.29150i	0.371391i
$204$	0	0
$205$	−21.0000	−1.46670
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−31.7490	−2.19613
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.29150i	0.360877i
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 15.8745i	− 1.05363i	−0.849981	−	0.526814i	$-0.823386\pi$
$227$	− 15.8745i	− 1.05363i	0.849981	−	0.526814i	$-0.176614\pi$
$228$	0	0
$229$	− 6.00000i	− 0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$	− 6.00000i	− 0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.1660	−1.38663	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	−21.1660	−1.38663	−0.693316	+	0.720634i	$0.743851\pi$
$234$	0	0
$235$	14.0000	0.913259
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.8745i	1.02684i	0.858138	+	0.513418i	$0.171621\pi$
$239$	15.8745i	1.02684i	−0.858138	+	0.513418i	$0.828379\pi$
$240$	0	0
$241$	− 21.0000i	− 1.35273i	−0.736567	−	0.676364i	$-0.763554\pi$
$241$	− 21.0000i	− 1.35273i	0.736567	−	0.676364i	$-0.236446\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7.93725i	0.507093i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	− 28.0000i	− 1.76034i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.93725	−0.495112	−0.247556	−	0.968874i	$-0.579627\pi$
$257$	−7.93725	−0.495112	−0.247556	+	0.968874i	$0.579627\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.8745	0.978864	0.489432	−	0.872041i	$-0.337204\pi$
$263$	15.8745	0.978864	0.489432	+	0.872041i	$0.337204\pi$
$264$	0	0
$265$	21.0000i	1.29002i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−31.7490	−1.93577	−0.967886	−	0.251390i	$-0.919112\pi$
$269$	−31.7490	−1.93577	−0.967886	+	0.251390i	$0.919112\pi$
$270$	0	0
$271$	8.00000i	0.485965i	0.970031	+	0.242983i	$0.0781258\pi$
$271$	8.00000i	0.485965i	−0.970031	+	0.242983i	$0.921874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.5830i	0.638179i
$276$	0	0
$277$	9.00000	0.540758	0.270379	−	0.962754i	$-0.412851\pi$
$277$	9.00000	0.540758	0.270379	+	0.962754i	$0.412851\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 13.2288i	− 0.789161i	−0.918861	−	0.394581i	$-0.870890\pi$
$281$	− 13.2288i	− 0.789161i	0.918861	−	0.394581i	$-0.129110\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.8745	−0.937043
$288$	0	0
$289$	46.0000	2.70588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.5203i	1.08197i	0.841034	+	0.540983i	$0.181948\pi$
$293$	18.5203i	1.08197i	−0.841034	+	0.540983i	$0.818052\pi$
$294$	0	0
$295$	28.0000	1.63022
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.00000i	0.230556i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	34.3948i	1.96944i
$306$	0	0
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.8745	0.900161	0.450080	−	0.892988i	$-0.351395\pi$
$311$	15.8745	0.900161	0.450080	+	0.892988i	$0.351395\pi$
$312$	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
$317$	7.93725i	0.445801i	0.974841	+	0.222900i	$0.0715524\pi$
$317$	7.93725i	0.445801i	−0.974841	+	0.222900i	$0.928448\pi$
$318$	0	0
$319$	− 14.0000i	− 0.783850i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	47.6235i	2.64984i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.5830	0.583460
$330$	0	0
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.29150	0.289106
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.1660	1.14620
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.8745	0.852188	0.426094	−	0.904679i	$-0.359889\pi$
$347$	15.8745	0.852188	0.426094	+	0.904679i	$0.359889\pi$
$348$	0	0
$349$	2.00000i	0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$	2.00000i	0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.2288i	0.704096i	0.935982	+	0.352048i	$0.114515\pi$
$353$	13.2288i	0.704096i	−0.935982	+	0.352048i	$0.885485\pi$
$354$	0	0
$355$	−14.0000	−0.743043
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.8745i	0.837824i	0.908027	+	0.418912i	$0.137589\pi$
$359$	15.8745i	0.837824i	−0.908027	+	0.418912i	$0.862411\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−18.5203	−0.969395
$366$	0	0
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.8745i	0.824163i
$372$	0	0
$373$	17.0000	0.880227	0.440113	−	0.897942i	$-0.354938\pi$
$373$	17.0000	0.880227	0.440113	+	0.897942i	$0.354938\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 20.0000i	− 1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$	− 20.0000i	− 1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	28.0000i	1.42701i
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.93725	0.402435	0.201217	−	0.979547i	$-0.435510\pi$
$389$	7.93725	0.402435	0.201217	+	0.979547i	$0.435510\pi$
$390$	0	0
$391$	−42.0000	−2.12403
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 10.5830i	− 0.532489i
$396$	0	0
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 23.8118i	− 1.18910i	−0.804058	−	0.594551i	$-0.797330\pi$
$401$	− 23.8118i	− 1.18910i	0.804058	−	0.594551i	$-0.202670\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.8745	0.786870
$408$	0	0
$409$	− 35.0000i	− 1.73064i	−0.501221	−	0.865319i	$-0.667116\pi$
$409$	− 35.0000i	− 1.73064i	0.501221	−	0.865319i	$-0.332884\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	21.1660	1.04151
$414$	0	0
$415$	42.0000	2.06170
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−31.7490	−1.55104	−0.775520	−	0.631322i	$-0.782512\pi$
$419$	−31.7490	−1.55104	−0.775520	+	0.631322i	$0.782512\pi$
$420$	0	0
$421$	− 29.0000i	− 1.41337i	−0.707527	−	0.706687i	$-0.750189\pi$
$421$	− 29.0000i	− 1.41337i	0.707527	−	0.706687i	$-0.249811\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.8745	0.770027
$426$	0	0
$427$	26.0000i	1.25823i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 15.8745i	− 0.764648i	−0.924028	−	0.382324i	$-0.875124\pi$
$431$	− 15.8745i	− 0.764648i	0.924028	−	0.382324i	$-0.124876\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 31.7490i	− 1.51876i
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.7490	1.50844	0.754221	−	0.656621i	$-0.228015\pi$
$443$	31.7490	1.50844	0.754221	+	0.656621i	$0.228015\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 21.1660i	− 0.998886i	−0.866347	−	0.499443i	$-0.833538\pi$
$449$	− 21.1660i	− 0.998886i	0.866347	−	0.499443i	$-0.166462\pi$
$450$	0	0
$451$	42.0000	1.97770
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.0000i	1.16945i	0.811231	+	0.584725i	$0.198798\pi$
$457$	25.0000i	1.16945i	−0.811231	+	0.584725i	$0.801202\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 23.8118i	− 1.10902i	−0.832176	−	0.554512i	$-0.812905\pi$
$461$	− 23.8118i	− 1.10902i	0.832176	−	0.554512i	$-0.187095\pi$
$462$	0	0
$463$	26.0000i	1.20832i	0.796862	+	0.604161i	$0.206492\pi$
$463$	26.0000i	1.20832i	−0.796862	+	0.604161i	$0.793508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.29150	0.244862	0.122431	−	0.992477i	$-0.460931\pi$
$467$	5.29150	0.244862	0.122431	+	0.992477i	$0.460931\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 10.5830i	− 0.486607i
$474$	0	0
$475$	12.0000i	0.550598i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 31.7490i	− 1.45065i	−0.688407	−	0.725325i	$-0.741690\pi$
$479$	− 31.7490i	− 1.45065i	0.688407	−	0.725325i	$-0.258310\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.29150	−0.240275
$486$	0	0
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−37.0405	−1.67162	−0.835808	−	0.549022i	$-0.815000\pi$
$491$	−37.0405	−1.67162	−0.835808	+	0.549022i	$0.815000\pi$
$492$	0	0
$493$	−21.0000	−0.945792
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.5830	−0.474713
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−26.4575	−1.17968	−0.589841	−	0.807519i	$-0.700810\pi$
$503$	−26.4575	−1.17968	−0.589841	+	0.807519i	$0.700810\pi$
$504$	0	0
$505$	7.00000i	0.311496i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 34.3948i	− 1.52452i	−0.647270	−	0.762261i	$-0.724089\pi$
$509$	− 34.3948i	− 1.52452i	0.647270	−	0.762261i	$-0.275911\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	0	0
$514$	0	0
$515$	37.0405i	1.63220i
$516$	0	0
$517$	−28.0000	−1.23144
$518$	0	0
$519$	0	0
$520$	0	0
$521$	39.6863	1.73869	0.869344	−	0.494208i	$-0.164542\pi$
$521$	39.6863	1.73869	0.869344	+	0.494208i	$0.164542\pi$
$522$	0	0
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 31.7490i	− 1.38301i
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 14.0000i	− 0.605273i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 15.8745i	− 0.683763i
$540$	0	0
$541$	13.0000i	0.558914i	0.960158	+	0.279457i	$0.0901544\pi$
$541$	13.0000i	0.558914i	−0.960158	+	0.279457i	$0.909846\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−26.4575	−1.13332
$546$	0	0
$547$	34.0000	1.45374	0.726868	−	0.686778i	$-0.240975\pi$
$547$	34.0000	1.45374	0.726868	+	0.686778i	$0.240975\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 15.8745i	− 0.676277i
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.93725i	− 0.336312i	−0.985760	−	0.168156i	$-0.946219\pi$
$557$	− 7.93725i	− 0.336312i	0.985760	−	0.168156i	$-0.0537813\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−42.3320	−1.78408	−0.892041	−	0.451955i	$-0.850727\pi$
$563$	−42.3320	−1.78408	−0.892041	+	0.451955i	$0.850727\pi$
$564$	0	0
$565$	7.00000i	0.294492i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.5830	0.443663	0.221831	−	0.975085i	$-0.428797\pi$
$569$	10.5830	0.443663	0.221831	+	0.975085i	$0.428797\pi$
$570$	0	0
$571$	30.0000	1.25546	0.627730	−	0.778431i	$-0.283984\pi$
$571$	30.0000	1.25546	0.627730	+	0.778431i	$0.283984\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−10.5830	−0.441342
$576$	0	0
$577$	3.00000i	0.124892i	0.998048	+	0.0624458i	$0.0198901\pi$
$577$	3.00000i	0.124892i	−0.998048	+	0.0624458i	$0.980110\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	31.7490	1.31717
$582$	0	0
$583$	− 42.0000i	− 1.73946i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.5830i	0.436807i	0.975859	+	0.218404i	$0.0700850\pi$
$587$	10.5830i	0.436807i	−0.975859	+	0.218404i	$0.929915\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	0	0
$593$	29.1033i	1.19513i	0.801821	+	0.597564i	$0.203865\pi$
$593$	29.1033i	1.19513i	−0.801821	+	0.597564i	$0.796135\pi$
$594$	0	0
$595$	42.0000	1.72183
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21.1660	0.864820	0.432410	−	0.901677i	$-0.357663\pi$
$599$	21.1660	0.864820	0.432410	+	0.901677i	$0.357663\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 44.9778i	− 1.82861i
$606$	0	0
$607$	−36.0000	−1.46119	−0.730597	−	0.682808i	$-0.760758\pi$
$607$	−36.0000	−1.46119	−0.730597	+	0.682808i	$0.760758\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	5.00000i	0.201948i	0.994889	+	0.100974i	$0.0321959\pi$
$613$	5.00000i	0.201948i	−0.994889	+	0.100974i	$0.967804\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.64575i	− 0.106514i	−0.998581	−	0.0532570i	$-0.983040\pi$
$617$	− 2.64575i	− 0.106514i	0.998581	−	0.0532570i	$-0.0169602\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 23.8118i	− 0.949437i
$630$	0	0
$631$	8.00000i	0.318475i	0.987240	+	0.159237i	$0.0509036\pi$
$631$	8.00000i	0.318475i	−0.987240	+	0.159237i	$0.949096\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 42.3320i	− 1.67990i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.5203	0.731506	0.365753	−	0.930712i	$-0.380811\pi$
$641$	18.5203	0.731506	0.365753	+	0.930712i	$0.380811\pi$
$642$	0	0
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.3320	1.66424	0.832122	−	0.554593i	$-0.187126\pi$
$647$	42.3320	1.66424	0.832122	+	0.554593i	$0.187126\pi$
$648$	0	0
$649$	−56.0000	−2.19819
$650$	0	0
$651$	0	0
$652$	0	0
$653$	31.7490	1.24243	0.621217	−	0.783638i	$-0.286638\pi$
$653$	31.7490	1.24243	0.621217	+	0.783638i	$0.286638\pi$
$654$	0	0
$655$	28.0000i	1.09405i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−31.7490	−1.23677	−0.618383	−	0.785877i	$-0.712212\pi$
$659$	−31.7490	−1.23677	−0.618383	+	0.785877i	$0.712212\pi$
$660$	0	0
$661$	31.0000i	1.20576i	0.797832	+	0.602880i	$0.205980\pi$
$661$	31.0000i	1.20576i	−0.797832	+	0.602880i	$0.794020\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	31.7490i	1.23117i
$666$	0	0
$667$	14.0000	0.542082
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 68.7895i	− 2.65559i
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.7490	1.22021	0.610107	−	0.792319i	$-0.291126\pi$
$677$	31.7490	1.22021	0.610107	+	0.792319i	$0.291126\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	42.3320i	1.61979i	0.586575	+	0.809895i	$0.300476\pi$
$683$	42.3320i	1.61979i	−0.586575	+	0.809895i	$0.699524\pi$
$684$	0	0
$685$	−21.0000	−0.802369
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 14.0000i	− 0.532585i	−0.963892	−	0.266293i	$-0.914201\pi$
$691$	− 14.0000i	− 0.532585i	0.963892	−	0.266293i	$-0.0857987\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	42.3320i	1.60575i
$696$	0	0
$697$	− 63.0000i	− 2.38630i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31.7490	1.19914	0.599572	−	0.800321i	$-0.295338\pi$
$701$	31.7490	1.19914	0.599572	+	0.800321i	$0.295338\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.29150i	0.199007i
$708$	0	0
$709$	− 11.0000i	− 0.413114i	−0.978435	−	0.206557i	$-0.933774\pi$
$709$	− 11.0000i	− 0.413114i	0.978435	−	0.206557i	$-0.0662258\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	21.1660i	0.792673i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.5830	−0.394679	−0.197340	−	0.980335i	$-0.563230\pi$
$719$	−10.5830	−0.394679	−0.197340	+	0.980335i	$0.563230\pi$
$720$	0	0
$721$	28.0000i	1.04277i
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.29150	−0.196521
$726$	0	0
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−15.8745	−0.587140
$732$	0	0
$733$	− 1.00000i	− 0.0369358i	−0.999829	−	0.0184679i	$-0.994121\pi$
$733$	− 1.00000i	− 0.0369358i	0.999829	−	0.0184679i	$-0.00587886\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.5830	−0.389830
$738$	0	0
$739$	− 12.0000i	− 0.441427i	−0.975339	−	0.220714i	$-0.929161\pi$
$739$	− 12.0000i	− 0.441427i	0.975339	−	0.220714i	$-0.0708386\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.1660i	0.776506i	0.921553	+	0.388253i	$0.126921\pi$
$743$	21.1660i	0.776506i	−0.921553	+	0.388253i	$0.873079\pi$
$744$	0	0
$745$	21.0000	0.769380
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 10.5830i	− 0.386695i
$750$	0	0
$751$	−18.0000	−0.656829	−0.328415	−	0.944534i	$-0.606514\pi$
$751$	−18.0000	−0.656829	−0.328415	+	0.944534i	$0.606514\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−47.6235	−1.73320
$756$	0	0
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	30.0000i	1.08183i	0.841078	+	0.540914i	$0.181921\pi$
$769$	30.0000i	1.08183i	−0.841078	+	0.540914i	$0.818079\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.5830i	0.380644i	0.981722	+	0.190322i	$0.0609532\pi$
$773$	10.5830i	0.380644i	−0.981722	+	0.190322i	$0.939047\pi$
$774$	0	0
$775$	− 8.00000i	− 0.287368i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	47.6235	1.70629
$780$	0	0
$781$	28.0000	1.00192
$782$	0	0
$783$	0	0
$784$	0	0
$785$	44.9778i	1.60533i
$786$	0	0
$787$	52.0000i	1.85360i	0.375555	+	0.926800i	$0.377452\pi$
$787$	52.0000i	1.85360i	−0.375555	+	0.926800i	$0.622548\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.29150i	0.188144i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	42.0000i	1.48585i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	37.0405	1.30713
$804$	0	0
$805$	−28.0000	−0.986870
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.8118	0.837177	0.418588	−	0.908176i	$-0.362525\pi$
$809$	23.8118	0.837177	0.418588	+	0.908176i	$0.362525\pi$
$810$	0	0
$811$	− 16.0000i	− 0.561836i	−0.959732	−	0.280918i	$-0.909361\pi$
$811$	− 16.0000i	− 0.561836i	0.959732	−	0.280918i	$-0.0906389\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.5830	−0.370707
$816$	0	0
$817$	− 12.0000i	− 0.419827i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.5830i	0.369349i	0.982800	+	0.184675i	$0.0591232\pi$
$821$	10.5830i	0.369349i	−0.982800	+	0.184675i	$0.940877\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−7.00000	−0.243120	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000	−0.243120	−0.121560	+	0.992584i	$0.538790\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−23.8118	−0.825029
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 10.5830i	− 0.365366i	−0.983172	−	0.182683i	$-0.941522\pi$
$839$	− 10.5830i	− 0.365366i	0.983172	−	0.182683i	$-0.0584782\pi$
$840$	0	0
$841$	−22.0000	−0.758621
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 34.0000i	− 1.16825i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15.8745i	0.544171i
$852$	0	0
$853$	47.0000i	1.60925i	0.593784	+	0.804625i	$0.297633\pi$
$853$	47.0000i	1.60925i	−0.593784	+	0.804625i	$0.702367\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.3948	1.17490	0.587451	−	0.809259i	$-0.300131\pi$
$857$	34.3948	1.17490	0.587451	+	0.809259i	$0.300131\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.29150i	0.180125i	0.995936	+	0.0900624i	$0.0287067\pi$
$863$	5.29150i	0.180125i	−0.995936	+	0.0900624i	$0.971293\pi$
$864$	0	0
$865$	− 28.0000i	− 0.952029i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	21.1660i	0.718008i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−15.8745	−0.536656
$876$	0	0
$877$	− 27.0000i	− 0.911725i	−0.890050	−	0.455863i	$-0.849331\pi$
$877$	− 27.0000i	− 0.911725i	0.890050	−	0.455863i	$-0.150669\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.93725	0.267413	0.133706	−	0.991021i	$-0.457312\pi$
$881$	7.93725	0.267413	0.133706	+	0.991021i	$0.457312\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.9150	1.77671	0.888356	−	0.459155i	$-0.151848\pi$
$887$	52.9150	1.77671	0.888356	+	0.459155i	$0.151848\pi$
$888$	0	0
$889$	− 32.0000i	− 1.07325i
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−31.7490	−1.06244
$894$	0	0
$895$	− 42.0000i	− 1.40391i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.5830i	0.352963i
$900$	0	0
$901$	−63.0000	−2.09883
$902$	0	0
$903$	0	0
$904$	0	0
$905$	23.8118i	0.791530i
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.7490	−1.05189	−0.525946	−	0.850518i	$-0.676289\pi$
$911$	−31.7490	−1.05189	−0.525946	+	0.850518i	$0.676289\pi$
$912$	0	0
$913$	−84.0000	−2.77999
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.1660i	0.698963i
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 6.00000i	− 0.197279i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 7.93725i	− 0.260413i	−0.991487	−	0.130206i	$-0.958436\pi$
$929$	− 7.93725i	− 0.260413i	0.991487	−	0.130206i	$-0.0415640\pi$
$930$	0	0
$931$	− 18.0000i	− 0.589926i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−111.122	−3.63406
$936$	0	0
$937$	−9.00000	−0.294017	−0.147009	−	0.989135i	$-0.546964\pi$
$937$	−9.00000	−0.294017	−0.147009	+	0.989135i	$0.546964\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 10.5830i	− 0.344996i	−0.985010	−	0.172498i	$-0.944816\pi$
$941$	− 10.5830i	− 0.344996i	0.985010	−	0.172498i	$-0.0551839\pi$
$942$	0	0
$943$	42.0000i	1.36771i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−10.5830	−0.342817	−0.171409	−	0.985200i	$-0.554832\pi$
$953$	−10.5830	−0.342817	−0.171409	+	0.985200i	$0.554832\pi$
$954$	0	0
$955$	56.0000i	1.81212i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15.8745	−0.512615
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−66.1438	−2.12924
$966$	0	0
$967$	− 22.0000i	− 0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$	− 22.0000i	− 0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.7490	1.01887	0.509437	−	0.860508i	$-0.329854\pi$
$971$	31.7490	1.01887	0.509437	+	0.860508i	$0.329854\pi$
$972$	0	0
$973$	32.0000i	1.02587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 39.6863i	− 1.26968i	−0.772645	−	0.634838i	$-0.781067\pi$
$977$	− 39.6863i	− 1.26968i	0.772645	−	0.634838i	$-0.218933\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 10.5830i	− 0.337545i	−0.985655	−	0.168773i	$-0.946020\pi$
$983$	− 10.5830i	− 0.337545i	0.985655	−	0.168773i	$-0.0539804\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.5830	0.336520
$990$	0	0
$991$	−54.0000	−1.71537	−0.857683	−	0.514178i	$-0.828097\pi$
$991$	−54.0000	−1.71537	−0.857683	+	0.514178i	$0.828097\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	26.4575i	0.838760i
$996$	0	0
$997$	−7.00000	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$997$	−7.00000	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6084.2.b.n.4393.4		4
3.2	odd	2	inner	6084.2.b.n.4393.2		4
13.2	odd	12		468.2.l.e.217.2	yes	4
13.5	odd	4		6084.2.a.q.1.2		2
13.6	odd	12		468.2.l.e.289.2	yes	4
13.8	odd	4		6084.2.a.w.1.1		2
13.12	even	2	inner	6084.2.b.n.4393.1		4
39.2	even	12		468.2.l.e.217.1	✓	4
39.5	even	4		6084.2.a.q.1.1		2
39.8	even	4		6084.2.a.w.1.2		2
39.32	even	12		468.2.l.e.289.1	yes	4
39.38	odd	2	inner	6084.2.b.n.4393.3		4
52.15	even	12		1872.2.t.o.1153.2		4
52.19	even	12		1872.2.t.o.289.2		4
156.71	odd	12		1872.2.t.o.289.1		4
156.119	odd	12		1872.2.t.o.1153.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.l.e.217.1	✓	4	39.2	even	12
468.2.l.e.217.2	yes	4	13.2	odd	12
468.2.l.e.289.1	yes	4	39.32	even	12
468.2.l.e.289.2	yes	4	13.6	odd	12
1872.2.t.o.289.1		4	156.71	odd	12
1872.2.t.o.289.2		4	52.19	even	12
1872.2.t.o.1153.1		4	156.119	odd	12
1872.2.t.o.1153.2		4	52.15	even	12
6084.2.a.q.1.1		2	39.5	even	4
6084.2.a.q.1.2		2	13.5	odd	4
6084.2.a.w.1.1		2	13.8	odd	4
6084.2.a.w.1.2		2	39.8	even	4
6084.2.b.n.4393.1		4	13.12	even	2	inner
6084.2.b.n.4393.2		4	3.2	odd	2	inner
6084.2.b.n.4393.3		4	39.38	odd	2	inner
6084.2.b.n.4393.4		4	1.1	even	1	trivial

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	6084.2.b.n.4393.4

Level	$6084$
Weight	$2$
Character	6084.4393
Analytic conductor	$48.581$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$