LMFDB - Embedded newform 6084.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6084,2,Mod(1,6084)]

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, 0]))

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

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

chi := DirichletCharacter("6084.1");

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.a (trivial)

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:	yes
Analytic conductor:	$48.5809845897$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 468)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	6084.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{5} +2.00000 q^{7} +5.29150 q^{11} -7.93725 q^{17} +6.00000 q^{19} +5.29150 q^{23} +2.00000 q^{25} -2.64575 q^{29} -4.00000 q^{31} +5.29150 q^{35} +3.00000 q^{37} +7.93725 q^{41} -2.00000 q^{43} +5.29150 q^{47} -3.00000 q^{49} -7.93725 q^{53} +14.0000 q^{55} +10.5830 q^{59} +13.0000 q^{61} +2.00000 q^{67} +5.29150 q^{71} +7.00000 q^{73} +10.5830 q^{77} -4.00000 q^{79} -15.8745 q^{83} -21.0000 q^{85} +15.8745 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7} + 12 q^{19} + 4 q^{25} - 8 q^{31} + 6 q^{37} - 4 q^{43} - 6 q^{49} + 28 q^{55} + 26 q^{61} + 4 q^{67} + 14 q^{73} - 8 q^{79} - 42 q^{85} - 4 q^{97}+O(q^{100}) $ 2 * q + 4 * q^7 + 12 * q^19 + 4 * q^25 - 8 * q^31 + 6 * q^37 - 4 * q^43 - 6 * q^49 + 28 * q^55 + 26 * q^61 + 4 * q^67 + 14 * q^73 - 8 * q^79 - 42 * q^85 - 4 * q^97

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
$5$	2.64575	1.18322	0.591608	−	0.806226i	$-0.298493\pi$
$5$	2.64575	1.18322	0.591608	+	0.806226i	$0.298493\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.29150	1.59545	0.797724	−	0.603023i	$-0.206037\pi$
$11$	5.29150	1.59545	0.797724	+	0.603023i	$0.206037\pi$
$12$	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
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	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	$-0.579002\pi$
$29$	−2.64575	−0.491304	−0.245652	+	0.969358i	$0.579002\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.29150	0.894427
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.93725	1.23959	0.619795	−	0.784763i	$-0.287216\pi$
$41$	7.93725	1.23959	0.619795	+	0.784763i	$0.287216\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.29150	0.771845	0.385922	−	0.922531i	$-0.373883\pi$
$47$	5.29150	0.771845	0.385922	+	0.922531i	$0.373883\pi$
$48$	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	$-0.683521\pi$
$53$	−7.93725	−1.09027	−0.545133	+	0.838350i	$0.683521\pi$
$54$	0	0
$55$	14.0000	1.88776
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.5830	1.37779	0.688895	−	0.724861i	$-0.258096\pi$
$59$	10.5830	1.37779	0.688895	+	0.724861i	$0.258096\pi$
$60$	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
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.29150	0.627986	0.313993	−	0.949425i	$-0.398333\pi$
$71$	5.29150	0.627986	0.313993	+	0.949425i	$0.398333\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	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
$83$	−15.8745	−1.74245	−0.871227	−	0.490881i	$-0.836675\pi$
$83$	−15.8745	−1.74245	−0.871227	+	0.490881i	$0.836675\pi$
$84$	0	0
$85$	−21.0000	−2.27777
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	15.8745	1.62869
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	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	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.29150	0.511549	0.255774	−	0.966736i	$-0.417670\pi$
$107$	5.29150	0.511549	0.255774	+	0.966736i	$0.417670\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.64575	−0.248891	−0.124446	−	0.992226i	$-0.539715\pi$
$113$	−2.64575	−0.248891	−0.124446	+	0.992226i	$0.539715\pi$
$114$	0	0
$115$	14.0000	1.30551
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−15.8745	−1.45521
$120$	0	0
$121$	17.0000	1.54545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.93725	−0.709930
$126$	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
$129$	0	0
$130$	0	0
$131$	−10.5830	−0.924641	−0.462321	−	0.886713i	$-0.652983\pi$
$131$	−10.5830	−0.924641	−0.462321	+	0.886713i	$0.652983\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.93725	−0.678125	−0.339063	−	0.940764i	$-0.610110\pi$
$137$	−7.93725	−0.678125	−0.339063	+	0.940764i	$0.610110\pi$
$138$	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
$145$	−7.00000	−0.581318
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.93725	−0.650245	−0.325123	−	0.945672i	$-0.605406\pi$
$149$	−7.93725	−0.650245	−0.325123	+	0.945672i	$0.605406\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	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
$161$	10.5830	0.834058
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	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
$175$	4.00000	0.302372
$176$	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	$-0.608561\pi$
$181$	−9.00000	−0.668965	−0.334482	+	0.942402i	$0.608561\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.93725	0.583559
$186$	0	0
$187$	−42.0000	−3.07134
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.1660	−1.53152	−0.765759	−	0.643127i	$-0.777637\pi$
$191$	−21.1660	−1.53152	−0.765759	+	0.643127i	$0.777637\pi$
$192$	0	0
$193$	25.0000	1.79954	0.899770	−	0.436365i	$-0.143734\pi$
$193$	25.0000	1.79954	0.899770	+	0.436365i	$0.143734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.29150	−0.371391
$204$	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
$215$	−5.29150	−0.360877
$216$	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		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.8745	−1.05363	−0.526814	−	0.849981i	$-0.676614\pi$
$227$	−15.8745	−1.05363	−0.526814	+	0.849981i	$0.676614\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	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
$239$	15.8745	1.02684	0.513418	−	0.858138i	$-0.328379\pi$
$239$	15.8745	1.02684	0.513418	+	0.858138i	$0.328379\pi$
$240$	0	0
$241$	−21.0000	−1.35273	−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000	−1.35273	−0.676364	+	0.736567i	$0.736446\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−7.93725	−0.507093
$246$	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
$253$	28.0000	1.76034
$254$	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	$-0.662796\pi$
$263$	−15.8745	−0.978864	−0.489432	+	0.872041i	$0.662796\pi$
$264$	0	0
$265$	−21.0000	−1.29002
$266$	0	0
$267$	0	0
$268$	0	0
$269$	31.7490	1.93577	0.967886	−	0.251390i	$-0.0808877\pi$
$269$	31.7490	1.93577	0.967886	+	0.251390i	$0.0808877\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.5830	0.638179
$276$	0	0
$277$	−9.00000	−0.540758	−0.270379	−	0.962754i	$-0.587149\pi$
$277$	−9.00000	−0.540758	−0.270379	+	0.962754i	$0.587149\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.2288	0.789161	0.394581	−	0.918861i	$-0.370890\pi$
$281$	13.2288	0.789161	0.394581	+	0.918861i	$0.370890\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	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
$293$	−18.5203	−1.08197	−0.540983	−	0.841034i	$-0.681948\pi$
$293$	−18.5203	−1.08197	−0.540983	+	0.841034i	$0.681948\pi$
$294$	0	0
$295$	28.0000	1.63022
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	34.3948	1.96944
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	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
$317$	7.93725	0.445801	0.222900	−	0.974841i	$-0.428448\pi$
$317$	7.93725	0.445801	0.222900	+	0.974841i	$0.428448\pi$
$318$	0	0
$319$	−14.0000	−0.783850
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−47.6235	−2.64984
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.5830	0.583460
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	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	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−21.1660	−1.14620
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.8745	−0.852188	−0.426094	−	0.904679i	$-0.640111\pi$
$347$	−15.8745	−0.852188	−0.426094	+	0.904679i	$0.640111\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.2288	0.704096	0.352048	−	0.935982i	$-0.385485\pi$
$353$	13.2288	0.704096	0.352048	+	0.935982i	$0.385485\pi$
$354$	0	0
$355$	14.0000	0.743043
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.8745	−0.837824	−0.418912	−	0.908027i	$-0.637589\pi$
$359$	−15.8745	−0.837824	−0.418912	+	0.908027i	$0.637589\pi$
$360$	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
$371$	−15.8745	−0.824163
$372$	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
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	28.0000	1.42701
$386$	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
$395$	−10.5830	−0.532489
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.8118	1.18910	0.594551	−	0.804058i	$-0.297330\pi$
$401$	23.8118	1.18910	0.594551	+	0.804058i	$0.297330\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.8745	0.786870
$408$	0	0
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	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	$-0.217488\pi$
$419$	31.7490	1.55104	0.775520	+	0.631322i	$0.217488\pi$
$420$	0	0
$421$	29.0000	1.41337	0.706687	−	0.707527i	$-0.250189\pi$
$421$	29.0000	1.41337	0.706687	+	0.707527i	$0.250189\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.8745	−0.770027
$426$	0	0
$427$	26.0000	1.25823
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.8745	−0.764648	−0.382324	−	0.924028i	$-0.624876\pi$
$431$	−15.8745	−0.764648	−0.382324	+	0.924028i	$0.624876\pi$
$432$	0	0
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	31.7490	1.51876
$438$	0	0
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−31.7490	−1.50844	−0.754221	−	0.656621i	$-0.771985\pi$
$443$	−31.7490	−1.50844	−0.754221	+	0.656621i	$0.771985\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.1660	0.998886	0.499443	−	0.866347i	$-0.333538\pi$
$449$	21.1660	0.998886	0.499443	+	0.866347i	$0.333538\pi$
$450$	0	0
$451$	42.0000	1.97770
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.0000	−1.16945	−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000	−1.16945	−0.584725	+	0.811231i	$0.698798\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.8118	−1.10902	−0.554512	−	0.832176i	$-0.687095\pi$
$461$	−23.8118	−1.10902	−0.554512	+	0.832176i	$0.687095\pi$
$462$	0	0
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	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
$473$	−10.5830	−0.486607
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.7490	1.45065	0.725325	−	0.688407i	$-0.241690\pi$
$479$	31.7490	1.45065	0.725325	+	0.688407i	$0.241690\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.29150	−0.240275
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	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
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.4575	1.17968	0.589841	−	0.807519i	$-0.299190\pi$
$503$	26.4575	1.17968	0.589841	+	0.807519i	$0.299190\pi$
$504$	0	0
$505$	7.00000	0.311496
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.3948	−1.52452	−0.762261	−	0.647270i	$-0.775911\pi$
$509$	−34.3948	−1.52452	−0.762261	+	0.647270i	$0.775911\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−37.0405	−1.63220
$516$	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	$-0.835458\pi$
$521$	−39.6863	−1.73869	−0.869344	+	0.494208i	$0.835458\pi$
$522$	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
$527$	31.7490	1.38301
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	14.0000	0.605273
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−15.8745	−0.683763
$540$	0	0
$541$	13.0000	0.558914	0.279457	−	0.960158i	$-0.409846\pi$
$541$	13.0000	0.558914	0.279457	+	0.960158i	$0.409846\pi$
$542$	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
$551$	−15.8745	−0.676277
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.93725	0.336312	0.168156	−	0.985760i	$-0.446219\pi$
$557$	7.93725	0.336312	0.168156	+	0.985760i	$0.446219\pi$
$558$	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
$565$	−7.00000	−0.294492
$566$	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	$-0.716016\pi$
$571$	−30.0000	−1.25546	−0.627730	+	0.778431i	$0.716016\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10.5830	0.441342
$576$	0	0
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−31.7490	−1.31717
$582$	0	0
$583$	−42.0000	−1.73946
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.5830	0.436807	0.218404	−	0.975859i	$-0.429915\pi$
$587$	10.5830	0.436807	0.218404	+	0.975859i	$0.429915\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−29.1033	−1.19513	−0.597564	−	0.801821i	$-0.703865\pi$
$593$	−29.1033	−1.19513	−0.597564	+	0.801821i	$0.703865\pi$
$594$	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	$-0.642337\pi$
$599$	−21.1660	−0.864820	−0.432410	+	0.901677i	$0.642337\pi$
$600$	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
$605$	44.9778	1.82861
$606$	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
$613$	−5.00000	−0.201948	−0.100974	−	0.994889i	$-0.532196\pi$
$613$	−5.00000	−0.201948	−0.100974	+	0.994889i	$0.532196\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.64575	−0.106514	−0.0532570	−	0.998581i	$-0.516960\pi$
$617$	−2.64575	−0.106514	−0.0532570	+	0.998581i	$0.516960\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	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
$629$	−23.8118	−0.949437
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	42.3320	1.67990
$636$	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
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	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	$-0.713362\pi$
$653$	−31.7490	−1.24243	−0.621217	+	0.783638i	$0.713362\pi$
$654$	0	0
$655$	−28.0000	−1.09405
$656$	0	0
$657$	0	0
$658$	0	0
$659$	31.7490	1.23677	0.618383	−	0.785877i	$-0.287788\pi$
$659$	31.7490	1.23677	0.618383	+	0.785877i	$0.287788\pi$
$660$	0	0
$661$	31.0000	1.20576	0.602880	−	0.797832i	$-0.294020\pi$
$661$	31.0000	1.20576	0.602880	+	0.797832i	$0.294020\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	31.7490	1.23117
$666$	0	0
$667$	−14.0000	−0.542082
$668$	0	0
$669$	0	0
$670$	0	0
$671$	68.7895	2.65559
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31.7490	−1.22021	−0.610107	−	0.792319i	$-0.708874\pi$
$677$	−31.7490	−1.22021	−0.610107	+	0.792319i	$0.708874\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−42.3320	−1.61979	−0.809895	−	0.586575i	$-0.800476\pi$
$683$	−42.3320	−1.61979	−0.809895	+	0.586575i	$0.800476\pi$
$684$	0	0
$685$	−21.0000	−0.802369
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	42.3320	1.60575
$696$	0	0
$697$	−63.0000	−2.38630
$698$	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
$707$	5.29150	0.199007
$708$	0	0
$709$	−11.0000	−0.413114	−0.206557	−	0.978435i	$-0.566226\pi$
$709$	−11.0000	−0.413114	−0.206557	+	0.978435i	$0.566226\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.1660	−0.792673
$714$	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
$721$	−28.0000	−1.04277
$722$	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	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.8745	0.587140
$732$	0	0
$733$	1.00000	0.0369358	0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000	0.0369358	0.0184679	+	0.999829i	$0.494121\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.5830	0.389830
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.1660	0.776506	0.388253	−	0.921553i	$-0.373079\pi$
$743$	21.1660	0.776506	0.388253	+	0.921553i	$0.373079\pi$
$744$	0	0
$745$	−21.0000	−0.769380
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.5830	0.386695
$750$	0	0
$751$	18.0000	0.656829	0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000	0.656829	0.328415	+	0.944534i	$0.393486\pi$
$752$	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		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.5830	0.380644	0.190322	−	0.981722i	$-0.439047\pi$
$773$	10.5830	0.380644	0.190322	+	0.981722i	$0.439047\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	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
$785$	44.9778	1.60533
$786$	0	0
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.29150	−0.188144
$792$	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
$799$	−42.0000	−1.48585
$800$	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	$-0.637475\pi$
$809$	−23.8118	−0.837177	−0.418588	+	0.908176i	$0.637475\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	10.5830	0.370707
$816$	0	0
$817$	−12.0000	−0.419827
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.5830	0.369349	0.184675	−	0.982800i	$-0.440877\pi$
$821$	10.5830	0.369349	0.184675	+	0.982800i	$0.440877\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	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
$839$	10.5830	0.365366	0.182683	−	0.983172i	$-0.441522\pi$
$839$	10.5830	0.365366	0.182683	+	0.983172i	$0.441522\pi$
$840$	0	0
$841$	−22.0000	−0.758621
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	34.0000	1.16825
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15.8745	0.544171
$852$	0	0
$853$	47.0000	1.60925	0.804625	−	0.593784i	$-0.202367\pi$
$853$	47.0000	1.60925	0.804625	+	0.593784i	$0.202367\pi$
$854$	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
$863$	5.29150	0.180125	0.0900624	−	0.995936i	$-0.471293\pi$
$863$	5.29150	0.180125	0.0900624	+	0.995936i	$0.471293\pi$
$864$	0	0
$865$	−28.0000	−0.952029
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−21.1660	−0.718008
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−15.8745	−0.536656
$876$	0	0
$877$	27.0000	0.911725	0.455863	−	0.890050i	$-0.349331\pi$
$877$	27.0000	0.911725	0.455863	+	0.890050i	$0.349331\pi$
$878$	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	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−52.9150	−1.77671	−0.888356	−	0.459155i	$-0.848152\pi$
$887$	−52.9150	−1.77671	−0.888356	+	0.459155i	$0.848152\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	0	0
$891$	0	0
$892$	0	0
$893$	31.7490	1.06244
$894$	0	0
$895$	−42.0000	−1.40391
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.5830	0.352963
$900$	0	0
$901$	63.0000	2.09883
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.8118	−0.791530
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.7490	1.05189	0.525946	−	0.850518i	$-0.323711\pi$
$911$	31.7490	1.05189	0.525946	+	0.850518i	$0.323711\pi$
$912$	0	0
$913$	−84.0000	−2.77999
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21.1660	−0.698963
$918$	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
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.93725	−0.260413	−0.130206	−	0.991487i	$-0.541564\pi$
$929$	−7.93725	−0.260413	−0.130206	+	0.991487i	$0.541564\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	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
$941$	−10.5830	−0.344996	−0.172498	−	0.985010i	$-0.555184\pi$
$941$	−10.5830	−0.344996	−0.172498	+	0.985010i	$0.555184\pi$
$942$	0	0
$943$	42.0000	1.36771
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	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
$955$	−56.0000	−1.81212
$956$	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
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−31.7490	−1.01887	−0.509437	−	0.860508i	$-0.670146\pi$
$971$	−31.7490	−1.01887	−0.509437	+	0.860508i	$0.670146\pi$
$972$	0	0
$973$	32.0000	1.02587
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.6863	−1.26968	−0.634838	−	0.772645i	$-0.718933\pi$
$977$	−39.6863	−1.26968	−0.634838	+	0.772645i	$0.718933\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	10.5830	0.337545	0.168773	−	0.985655i	$-0.446020\pi$
$983$	10.5830	0.337545	0.168773	+	0.985655i	$0.446020\pi$
$984$	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
$995$	−26.4575	−0.838760
$996$	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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6084.2.a.w.1.2		2
3.2	odd	2	inner	6084.2.a.w.1.1		2
13.4	even	6		468.2.l.e.289.1	yes	4
13.5	odd	4		6084.2.b.n.4393.2		4
13.8	odd	4		6084.2.b.n.4393.3		4
13.10	even	6		468.2.l.e.217.1	✓	4
13.12	even	2		6084.2.a.q.1.1		2
39.5	even	4		6084.2.b.n.4393.4		4
39.8	even	4		6084.2.b.n.4393.1		4
39.17	odd	6		468.2.l.e.289.2	yes	4
39.23	odd	6		468.2.l.e.217.2	yes	4
39.38	odd	2		6084.2.a.q.1.2		2
52.23	odd	6		1872.2.t.o.1153.1		4
52.43	odd	6		1872.2.t.o.289.1		4
156.23	even	6		1872.2.t.o.1153.2		4
156.95	even	6		1872.2.t.o.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.l.e.217.1	✓	4	13.10	even	6
468.2.l.e.217.2	yes	4	39.23	odd	6
468.2.l.e.289.1	yes	4	13.4	even	6
468.2.l.e.289.2	yes	4	39.17	odd	6
1872.2.t.o.289.1		4	52.43	odd	6
1872.2.t.o.289.2		4	156.95	even	6
1872.2.t.o.1153.1		4	52.23	odd	6
1872.2.t.o.1153.2		4	156.23	even	6
6084.2.a.q.1.1		2	13.12	even	2
6084.2.a.q.1.2		2	39.38	odd	2
6084.2.a.w.1.1		2	3.2	odd	2	inner
6084.2.a.w.1.2		2	1.1	even	1	trivial
6084.2.b.n.4393.1		4	39.8	even	4
6084.2.b.n.4393.2		4	13.5	odd	4
6084.2.b.n.4393.3		4	13.8	odd	4
6084.2.b.n.4393.4		4	39.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6084.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{5} +2.00000 q^{7} +5.29150 q^{11} -7.93725 q^{17} +6.00000 q^{19} +5.29150 q^{23} +2.00000 q^{25} -2.64575 q^{29} -4.00000 q^{31} +5.29150 q^{35} +3.00000 q^{37} +7.93725 q^{41} -2.00000 q^{43} +5.29150 q^{47} -3.00000 q^{49} -7.93725 q^{53} +14.0000 q^{55} +10.5830 q^{59} +13.0000 q^{61} +2.00000 q^{67} +5.29150 q^{71} +7.00000 q^{73} +10.5830 q^{77} -4.00000 q^{79} -15.8745 q^{83} -21.0000 q^{85} +15.8745 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7} + 12 q^{19} + 4 q^{25} - 8 q^{31} + 6 q^{37} - 4 q^{43} - 6 q^{49} + 28 q^{55} + 26 q^{61} + 4 q^{67} + 14 q^{73} - 8 q^{79} - 42 q^{85} - 4 q^{97}+O(q^{100}) \) 2 * q + 4 * q^7 + 12 * q^19 + 4 * q^25 - 8 * q^31 + 6 * q^37 - 4 * q^43 - 6 * q^49 + 28 * q^55 + 26 * q^61 + 4 * q^67 + 14 * q^73 - 8 * q^79 - 42 * q^85 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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