LMFDB - Embedded newform 1521.4.a.y.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8112.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 3 $ x^4 - 5*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.4
Root			$-0.835000$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.42782 q^{2} +11.6056 q^{4} -20.3925 q^{5} +15.9647 q^{8} +O(q^{10})$	$q+4.42782 q^{2} +11.6056 q^{4} -20.3925 q^{5} +15.9647 q^{8} -90.2944 q^{10} +70.0332 q^{11} -22.1556 q^{16} -236.667 q^{20} +310.094 q^{22} +290.855 q^{25} -225.819 q^{32} -325.561 q^{40} -486.739 q^{41} -452.000 q^{43} +812.774 q^{44} -71.1653 q^{47} -343.000 q^{49} +1287.85 q^{50} -1428.15 q^{55} -696.914 q^{59} -944.654 q^{61} -822.638 q^{64} -123.807 q^{71} +418.244 q^{79} +451.809 q^{80} -2155.19 q^{82} +1509.37 q^{83} -2001.37 q^{86} +1118.06 q^{88} +155.245 q^{89} -315.107 q^{94} -1518.74 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{4}+O(q^{10}) $ 4 * q + 32 * q^4	$ 4 q + 32 q^{4} - 116 q^{10} - 204 q^{16} + 476 q^{22} + 500 q^{25} - 884 q^{40} - 1808 q^{43} - 1372 q^{49} - 3232 q^{55} - 1632 q^{64} - 2924 q^{82} + 2756 q^{88} - 5140 q^{94}+O(q^{100}) $ 4 * q + 32 * q^4 - 116 * q^10 - 204 * q^16 + 476 * q^22 + 500 * q^25 - 884 * q^40 - 1808 * q^43 - 1372 * q^49 - 3232 * q^55 - 1632 * q^64 - 2924 * q^82 + 2756 * q^88 - 5140 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	4.42782	1.56547	0.782735	−	0.622356i	$-0.213824\pi$
$2$	4.42782	1.56547	0.782735	+	0.622356i	$0.213824\pi$
$3$	0	0
$3$	0	0
$4$	11.6056	1.45069
$5$	−20.3925	−1.82396	−0.911982	−	0.410231i	$-0.865448\pi$
$5$	−20.3925	−1.82396	−0.911982	+	0.410231i	$0.865448\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	15.9647	0.705547
$9$	0	0
$10$	−90.2944	−2.85536
$11$	70.0332	1.91962	0.959810	−	0.280652i	$-0.0905506\pi$
$11$	70.0332	1.91962	0.959810	+	0.280652i	$0.0905506\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	−22.1556	−0.346181
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−236.667	−2.64601
$21$	0	0
$22$	310.094	3.00510
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	290.855	2.32684
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−225.819	−1.24748
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	−325.561	−1.28689
$41$	−486.739	−1.85405	−0.927024	−	0.375003i	$-0.877642\pi$
$41$	−486.739	−1.85405	−0.927024	+	0.375003i	$0.877642\pi$
$42$	0	0
$43$	−452.000	−1.60301	−0.801504	−	0.597989i	$-0.795967\pi$
$43$	−452.000	−1.60301	−0.801504	+	0.597989i	$0.795967\pi$
$44$	812.774	2.78478
$45$	0	0
$46$	0	0
$47$	−71.1653	−0.220862	−0.110431	−	0.993884i	$-0.535223\pi$
$47$	−71.1653	−0.220862	−0.110431	+	0.993884i	$0.535223\pi$
$48$	0	0
$49$	−343.000	−1.00000
$50$	1287.85	3.64260
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−1428.15	−3.50132
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−696.914	−1.53780	−0.768902	−	0.639367i	$-0.779197\pi$
$59$	−696.914	−1.53780	−0.768902	+	0.639367i	$0.779197\pi$
$60$	0	0
$61$	−944.654	−1.98280	−0.991398	−	0.130879i	$-0.958220\pi$
$61$	−944.654	−1.98280	−0.991398	+	0.130879i	$0.958220\pi$
$62$	0	0
$63$	0	0
$64$	−822.638	−1.60672
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−123.807	−0.206947	−0.103474	−	0.994632i	$-0.532996\pi$
$71$	−123.807	−0.206947	−0.103474	+	0.994632i	$0.532996\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	418.244	0.595647	0.297824	−	0.954621i	$-0.403739\pi$
$79$	418.244	0.595647	0.297824	+	0.954621i	$0.403739\pi$
$80$	451.809	0.631422
$81$	0	0
$82$	−2155.19	−2.90245
$83$	1509.37	1.99608	0.998040	−	0.0625815i	$-0.0199333\pi$
$83$	1509.37	1.99608	0.998040	+	0.0625815i	$0.0199333\pi$
$84$	0	0
$85$	0	0
$86$	−2001.37	−2.50946
$87$	0	0
$88$	1118.06	1.35438
$89$	155.245	0.184899	0.0924493	−	0.995717i	$-0.470530\pi$
$89$	155.245	0.184899	0.0924493	+	0.995717i	$0.470530\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−315.107	−0.345753
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−1518.74	−1.56547
$99$	0	0
$100$	3375.54	3.37554
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−848.000	−0.811223	−0.405611	−	0.914046i	$-0.632941\pi$
$103$	−848.000	−0.811223	−0.405611	+	0.914046i	$0.632941\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	−6323.61	−5.48120
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−3085.80	−2.40738
$119$	0	0
$120$	0	0
$121$	3573.65	2.68494
$122$	−4182.76	−3.10401
$123$	0	0
$124$	0	0
$125$	−3382.21	−2.42011
$126$	0	0
$127$	−2495.04	−1.74330	−0.871650	−	0.490129i	$-0.836950\pi$
$127$	−2495.04	−1.74330	−0.871650	+	0.490129i	$0.836950\pi$
$128$	−1835.94	−1.26778
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1082.73	−0.675208	−0.337604	−	0.941288i	$-0.609616\pi$
$137$	−1082.73	−0.675208	−0.337604	+	0.941288i	$0.609616\pi$
$138$	0	0
$139$	−340.000	−0.207471	−0.103735	−	0.994605i	$-0.533079\pi$
$139$	−340.000	−0.207471	−0.103735	+	0.994605i	$0.533079\pi$
$140$	0	0
$141$	0	0
$142$	−548.197	−0.323969
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2663.80	1.46461	0.732304	−	0.680978i	$-0.238445\pi$
$149$	2663.80	1.46461	0.732304	+	0.680978i	$0.238445\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−934.000	−0.474785	−0.237393	−	0.971414i	$-0.576293\pi$
$157$	−934.000	−0.474785	−0.237393	+	0.971414i	$0.576293\pi$
$158$	1851.91	0.932467
$159$	0	0
$160$	4605.01	2.27536
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	−5648.88	−2.68966
$165$	0	0
$166$	6683.20	3.12480
$167$	3555.90	1.64769	0.823845	−	0.566815i	$-0.191825\pi$
$167$	3555.90	1.64769	0.823845	+	0.566815i	$0.191825\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	−5245.71	−2.32547
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	−1551.63	−0.664536
$177$	0	0
$178$	687.398	0.289453
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−4430.00	−1.81922	−0.909611	−	0.415460i	$-0.863621\pi$
$181$	−4430.00	−1.81922	−0.909611	+	0.415460i	$0.863621\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−825.912	−0.320403
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	−3980.70	−1.45069
$197$	−3842.18	−1.38956	−0.694782	−	0.719220i	$-0.744499\pi$
$197$	−3842.18	−1.38956	−0.694782	+	0.719220i	$0.744499\pi$
$198$	0	0
$199$	−5610.24	−1.99849	−0.999244	−	0.0388706i	$-0.987624\pi$
$199$	−5610.24	−1.99849	−0.999244	+	0.0388706i	$0.987624\pi$
$200$	4643.42	1.64170
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9925.85	3.38171
$206$	−3754.79	−1.26994
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5740.04	1.87280	0.936399	−	0.350937i	$-0.114137\pi$
$211$	5740.04	1.87280	0.936399	+	0.350937i	$0.114137\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9217.42	2.92383
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−16574.5	−5.07934
$221$	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
$227$	1522.31	0.445107	0.222554	−	0.974920i	$-0.428561\pi$
$227$	1522.31	0.445107	0.222554	+	0.974920i	$0.428561\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	1451.24	0.402845
$236$	−8088.07	−2.23088
$237$	0	0
$238$	0	0
$239$	4191.63	1.13445	0.567226	−	0.823562i	$-0.308016\pi$
$239$	4191.63	1.13445	0.567226	+	0.823562i	$0.308016\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	15823.5	4.20319
$243$	0	0
$244$	−10963.2	−2.87643
$245$	6994.64	1.82396
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−14975.8	−3.78861
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	−11047.6	−2.72908
$255$	0	0
$256$	−1548.11	−0.377956
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	−4794.11	−1.05702
$275$	20369.5	4.46665
$276$	0	0
$277$	−7634.00	−1.65589	−0.827947	−	0.560806i	$-0.810491\pi$
$277$	−7634.00	−1.65589	−0.827947	+	0.560806i	$0.810491\pi$
$278$	−1505.46	−0.324789
$279$	0	0
$280$	0	0
$281$	4539.33	0.963678	0.481839	−	0.876260i	$-0.339969\pi$
$281$	4539.33	0.963678	0.481839	+	0.876260i	$0.339969\pi$
$282$	0	0
$283$	490.355	0.102999	0.0514993	−	0.998673i	$-0.483600\pi$
$283$	490.355	0.102999	0.0514993	+	0.998673i	$0.483600\pi$
$284$	−1436.85	−0.300217
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4913.00	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1396.60	−0.278465	−0.139232	−	0.990260i	$-0.544463\pi$
$293$	−1396.60	−0.278465	−0.139232	+	0.990260i	$0.544463\pi$
$294$	0	0
$295$	14211.8	2.80490
$296$	0	0
$297$	0	0
$298$	11794.8	2.29280
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	19263.9	3.61655
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−6396.25	−1.15507	−0.577536	−	0.816365i	$-0.695986\pi$
$313$	−6396.25	−1.15507	−0.577536	+	0.816365i	$0.695986\pi$
$314$	−4135.58	−0.743262
$315$	0	0
$316$	4853.95	0.864102
$317$	−8748.31	−1.55001	−0.775006	−	0.631954i	$-0.782253\pi$
$317$	−8748.31	−1.55001	−0.775006	+	0.631954i	$0.782253\pi$
$318$	0	0
$319$	0	0
$320$	16775.7	2.93059
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−7770.66	−1.30812
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	17517.0	2.89570
$333$	0	0
$334$	15744.9	2.57941
$335$	0	0
$336$	0	0
$337$	10420.0	1.68432	0.842160	−	0.539228i	$-0.181284\pi$
$337$	10420.0	1.68432	0.842160	+	0.539228i	$0.181284\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−7216.05	−1.13100
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	−15814.8	−2.39469
$353$	−11054.2	−1.66672	−0.833361	−	0.552728i	$-0.813587\pi$
$353$	−11054.2	−1.66672	−0.833361	+	0.552728i	$0.813587\pi$
$354$	0	0
$355$	2524.75	0.377464
$356$	1801.71	0.268231
$357$	0	0
$358$	0	0
$359$	7897.23	1.16100	0.580502	−	0.814259i	$-0.302857\pi$
$359$	7897.23	1.16100	0.580502	+	0.814259i	$0.302857\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	−19615.2	−2.84794
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8296.00	−1.17997	−0.589983	−	0.807416i	$-0.700866\pi$
$367$	−8296.00	−1.17997	−0.589983	+	0.807416i	$0.700866\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6526.05	−0.905914	−0.452957	−	0.891532i	$-0.649631\pi$
$373$	−6526.05	−0.905914	−0.452957	+	0.891532i	$0.649631\pi$
$374$	0	0
$375$	0	0
$376$	−1136.13	−0.155829
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7961.24	1.06214	0.531071	−	0.847327i	$-0.321790\pi$
$383$	7961.24	1.06214	0.531071	+	0.847327i	$0.321790\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	−5475.90	−0.705547
$393$	0	0
$394$	−17012.5	−2.17532
$395$	−8529.05	−1.08644
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−24841.1	−3.12857
$399$	0	0
$400$	−6444.07	−0.805509
$401$	−4746.53	−0.591098	−0.295549	−	0.955328i	$-0.595503\pi$
$401$	−4746.53	−0.591098	−0.295549	+	0.955328i	$0.595503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	43949.8	5.29397
$411$	0	0
$412$	−9841.51	−1.17684
$413$	0	0
$414$	0	0
$415$	−30779.8	−3.64078
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	25415.8	2.93181
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	40813.1	4.57716
$431$	17892.7	1.99968	0.999840	−	0.0178985i	$-0.00569759\pi$
$431$	17892.7	1.99968	0.999840	+	0.0178985i	$0.00569759\pi$
$432$	0	0
$433$	36.0555	0.00400166	0.00200083	−	0.999998i	$-0.499363\pi$
$433$	36.0555	0.00400166	0.00200083	+	0.999998i	$0.499363\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	11320.0	1.23069	0.615346	−	0.788257i	$-0.289016\pi$
$439$	11320.0	1.23069	0.615346	+	0.788257i	$0.289016\pi$
$440$	−22800.1	−2.47034
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−3165.84	−0.337248
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−19025.4	−1.99969	−0.999846	−	0.0175361i	$-0.994418\pi$
$449$	−19025.4	−1.99969	−0.999846	+	0.0175361i	$0.994418\pi$
$450$	0	0
$451$	−34087.9	−3.55906
$452$	0	0
$453$	0	0
$454$	6740.52	0.696802
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9378.19	0.947475	0.473738	−	0.880666i	$-0.342905\pi$
$461$	9378.19	0.947475	0.473738	+	0.880666i	$0.342905\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	6425.82	0.630641
$471$	0	0
$472$	−11126.0	−1.08499
$473$	−31655.0	−3.07717
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	18559.8	1.77595
$479$	20311.6	1.93749	0.968747	−	0.248050i	$-0.0797898\pi$
$479$	20311.6	1.93749	0.968747	+	0.248050i	$0.0797898\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	41474.2	3.89502
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−15081.1	−1.39896
$489$	0	0
$490$	30971.0	2.85536
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	−39252.4	−3.51084
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−28956.3	−2.52900
$509$	116.887	0.0101786	0.00508931	−	0.999987i	$-0.498380\pi$
$509$	116.887	0.0101786	0.00508931	+	0.999987i	$0.498380\pi$
$510$	0	0
$511$	0	0
$512$	7832.81	0.676102
$513$	0	0
$514$	0	0
$515$	17292.9	1.47964
$516$	0	0
$517$	−4983.93	−0.423971
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	16148.0	1.35010	0.675050	−	0.737772i	$-0.264122\pi$
$523$	16148.0	1.35010	0.675050	+	0.737772i	$0.264122\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−24021.4	−1.91962
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	23996.0	1.87568	0.937838	−	0.347073i	$-0.112824\pi$
$547$	23996.0	1.87568	0.937838	+	0.347073i	$0.112824\pi$
$548$	−12565.6	−0.979520
$549$	0	0
$550$	90192.6	6.99241
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−33801.9	−2.59225
$555$	0	0
$556$	−3945.89	−0.300976
$557$	−13993.2	−1.06447	−0.532237	−	0.846595i	$-0.678648\pi$
$557$	−13993.2	−1.06447	−0.532237	+	0.846595i	$0.678648\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	20099.3	1.50861
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	2171.20	0.161241
$567$	0	0
$568$	−1976.55	−0.146011
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	25411.9	1.86244	0.931222	−	0.364451i	$-0.118743\pi$
$571$	25411.9	1.86244	0.931222	+	0.364451i	$0.118743\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−21753.9	−1.56547
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−6183.88	−0.435928
$587$	20858.8	1.46667	0.733335	−	0.679868i	$-0.237963\pi$
$587$	20858.8	1.46667	0.733335	+	0.679868i	$0.237963\pi$
$588$	0	0
$589$	0	0
$590$	62927.4	4.39098
$591$	0	0
$592$	0	0
$593$	12660.4	0.876726	0.438363	−	0.898798i	$-0.355558\pi$
$593$	12660.4	0.876726	0.438363	+	0.898798i	$0.355558\pi$
$594$	0	0
$595$	0	0
$596$	30914.8	2.12470
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−12626.6	−0.856991	−0.428495	−	0.903544i	$-0.640956\pi$
$601$	−12626.6	−0.856991	−0.428495	+	0.903544i	$0.640956\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−72875.8	−4.89723
$606$	0	0
$607$	−25504.0	−1.70540	−0.852698	−	0.522404i	$-0.825035\pi$
$607$	−25504.0	−1.70540	−0.852698	+	0.522404i	$0.825035\pi$
$608$	0	0
$609$	0	0
$610$	85297.0	5.66160
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6923.04	0.451719	0.225860	−	0.974160i	$-0.427481\pi$
$617$	6923.04	0.451719	0.225860	+	0.974160i	$0.427481\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	32614.9	2.08735
$626$	−28321.4	−1.80823
$627$	0	0
$628$	−10839.6	−0.688768
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	6677.15	0.420257
$633$	0	0
$634$	−38735.9	−2.42650
$635$	50880.2	3.17972
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	37439.5	2.31239
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−48807.1	−2.95200
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	10784.0	0.641836
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	24096.6	1.40833
$665$	0	0
$666$	0	0
$667$	0	0
$668$	41268.2	2.39029
$669$	0	0
$670$	0	0
$671$	−66157.2	−3.80622
$672$	0	0
$673$	25522.0	1.46181	0.730907	−	0.682477i	$-0.239097\pi$
$673$	25522.0	1.46181	0.730907	+	0.682477i	$0.239097\pi$
$674$	46138.0	2.63675
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14207.2	0.795936	0.397968	−	0.917399i	$-0.369715\pi$
$683$	14207.2	0.795936	0.397968	+	0.917399i	$0.369715\pi$
$684$	0	0
$685$	22079.5	1.23155
$686$	0	0
$687$	0	0
$688$	10014.3	0.554931
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6933.46	0.378419
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	−57612.0	−3.08428
$705$	0	0
$706$	−48945.8	−2.60920
$707$	0	0
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	11179.1	0.590908
$711$	0	0
$712$	2478.45	0.130455
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	34967.5	1.81751
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	−30370.4	−1.56547
$723$	0	0
$724$	−51412.6	−2.63914
$725$	0	0
$726$	0	0
$727$	−28455.0	−1.45163	−0.725817	−	0.687888i	$-0.758538\pi$
$727$	−28455.0	−1.45163	−0.725817	+	0.687888i	$0.758538\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	−36733.2	−1.84720
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35259.5	1.74097	0.870487	−	0.492191i	$-0.163804\pi$
$743$	35259.5	1.74097	0.870487	+	0.492191i	$0.163804\pi$
$744$	0	0
$745$	−54321.5	−2.67139
$746$	−28896.1	−1.41818
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26120.0	−1.26915	−0.634575	−	0.772861i	$-0.718825\pi$
$751$	−26120.0	−1.26915	−0.634575	+	0.772861i	$0.718825\pi$
$752$	1576.71	0.0764583
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35009.9	1.68092	0.840460	−	0.541873i	$-0.182285\pi$
$757$	35009.9	1.68092	0.840460	+	0.541873i	$0.182285\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30422.5	1.44916	0.724582	−	0.689189i	$-0.242033\pi$
$761$	30422.5	1.44916	0.724582	+	0.689189i	$0.242033\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	35250.9	1.66275
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−42841.8	−1.99342	−0.996710	−	0.0810548i	$-0.974171\pi$
$773$	−42841.8	−1.99342	−0.996710	+	0.0810548i	$0.974171\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−8670.64	−0.397260
$782$	0	0
$783$	0	0
$784$	7599.37	0.346181
$785$	19046.6	0.865991
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	−44590.6	−2.01583
$789$	0	0
$790$	−37765.1	−1.70079
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−65109.9	−2.89920
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	−65680.5	−2.90270
$801$	0	0
$802$	−21016.8	−0.925346
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	115195.	4.90583
$821$	−33486.7	−1.42350	−0.711749	−	0.702433i	$-0.752097\pi$
$821$	−33486.7	−1.42350	−0.711749	+	0.702433i	$0.752097\pi$
$822$	0	0
$823$	−37352.0	−1.58203	−0.791014	−	0.611798i	$-0.790446\pi$
$823$	−37352.0	−1.58203	−0.791014	+	0.611798i	$0.790446\pi$
$824$	−13538.1	−0.572356
$825$	0	0
$826$	0	0
$827$	−35931.1	−1.51082	−0.755409	−	0.655253i	$-0.772562\pi$
$827$	−35931.1	−1.51082	−0.755409	+	0.655253i	$0.772562\pi$
$828$	0	0
$829$	−3079.14	−0.129002	−0.0645012	−	0.997918i	$-0.520546\pi$
$829$	−3079.14	−0.129002	−0.0645012	+	0.997918i	$0.520546\pi$
$830$	−136287.	−5.69952
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−72513.9	−3.00533
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12857.9	0.529086	0.264543	−	0.964374i	$-0.414779\pi$
$839$	12857.9	0.529086	0.264543	+	0.964374i	$0.414779\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	0	0
$843$	0	0
$844$	66616.3	2.71686
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−43324.3	−1.72085	−0.860423	−	0.509581i	$-0.829800\pi$
$859$	−43324.3	−1.72085	−0.860423	+	0.509581i	$0.829800\pi$
$860$	106973.	4.24158
$861$	0	0
$862$	79225.7	3.13044
$863$	−44880.2	−1.77027	−0.885133	−	0.465339i	$-0.845932\pi$
$863$	−44880.2	−1.77027	−0.885133	+	0.465339i	$0.845932\pi$
$864$	0	0
$865$	0	0
$866$	159.647	0.00626447
$867$	0	0
$868$	0	0
$869$	29291.0	1.14342
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	50122.9	1.92661
$879$	0	0
$880$	31641.6	1.21209
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	52410.3	1.99745	0.998724	−	0.0504988i	$-0.0160811\pi$
$883$	52410.3	1.99745	0.998724	+	0.0504988i	$0.0160811\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	−14017.8	−0.527952
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−84240.8	−3.13046
$899$	0	0
$900$	0	0
$901$	0	0
$902$	−150935.	−5.57161
$903$	0	0
$904$	0	0
$905$	90338.9	3.31820
$906$	0	0
$907$	35017.1	1.28195	0.640973	−	0.767564i	$-0.278531\pi$
$907$	35017.1	1.28195	0.640973	+	0.767564i	$0.278531\pi$
$908$	17667.3	0.645715
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	105706.	3.83171
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	36762.2	1.31956	0.659779	−	0.751460i	$-0.270650\pi$
$919$	36762.2	1.31956	0.659779	+	0.751460i	$0.270650\pi$
$920$	0	0
$921$	0	0
$922$	41524.9	1.48324
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	56151.6	1.98307	0.991536	−	0.129832i	$-0.0414439\pi$
$929$	56151.6	1.98307	0.991536	+	0.129832i	$0.0414439\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13275.6	−0.462856	−0.231428	−	0.972852i	$-0.574340\pi$
$937$	−13275.6	−0.462856	−0.231428	+	0.972852i	$0.574340\pi$
$938$	0	0
$939$	0	0
$940$	16842.4	0.584404
$941$	55622.6	1.92693	0.963467	−	0.267826i	$-0.0863052\pi$
$941$	55622.6	1.92693	0.963467	+	0.267826i	$0.0863052\pi$
$942$	0	0
$943$	0	0
$944$	15440.5	0.532359
$945$	0	0
$946$	−140163.	−4.81721
$947$	−14319.3	−0.491356	−0.245678	−	0.969352i	$-0.579011\pi$
$947$	−14319.3	−0.491356	−0.245678	+	0.969352i	$0.579011\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	48646.2	1.64574
$957$	0	0
$958$	89935.9	3.03309
$959$	0	0
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	57052.4	1.89435
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	20929.4	0.686407
$977$	31890.7	1.04429	0.522146	−	0.852856i	$-0.325132\pi$
$977$	31890.7	1.04429	0.522146	+	0.852856i	$0.325132\pi$
$978$	0	0
$979$	10872.3	0.354935
$980$	81176.6	2.64601
$981$	0	0
$982$	0	0
$983$	−11061.5	−0.358908	−0.179454	−	0.983766i	$-0.557433\pi$
$983$	−11061.5	−0.358908	−0.179454	+	0.983766i	$0.557433\pi$
$984$	0	0
$985$	78351.8	2.53451
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−23272.0	−0.745973	−0.372987	−	0.927837i	$-0.621666\pi$
$991$	−23272.0	−0.745973	−0.372987	+	0.927837i	$0.621666\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	114407.	3.64517
$996$	0	0
$997$	−21634.0	−0.687217	−0.343609	−	0.939113i	$-0.611649\pi$
$997$	−21634.0	−0.687217	−0.343609	+	0.939113i	$0.611649\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.y.1.4		4
3.2	odd	2	inner	1521.4.a.y.1.1		4
13.5	odd	4		117.4.b.c.64.1	✓	4
13.8	odd	4		117.4.b.c.64.4	yes	4
13.12	even	2	inner	1521.4.a.y.1.1		4
39.5	even	4		117.4.b.c.64.4	yes	4
39.8	even	4		117.4.b.c.64.1	✓	4
39.38	odd	2	CM	1521.4.a.y.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.b.c.64.1	✓	4	13.5	odd	4
117.4.b.c.64.1	✓	4	39.8	even	4
117.4.b.c.64.4	yes	4	13.8	odd	4
117.4.b.c.64.4	yes	4	39.5	even	4
1521.4.a.y.1.1		4	3.2	odd	2	inner
1521.4.a.y.1.1		4	13.12	even	2	inner
1521.4.a.y.1.4		4	1.1	even	1	trivial
1521.4.a.y.1.4		4	39.38	odd	2	CM

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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+4.42782 q^{2} +11.6056 q^{4} -20.3925 q^{5} +15.9647 q^{8} +O(q^{10})\)	\(q+4.42782 q^{2} +11.6056 q^{4} -20.3925 q^{5} +15.9647 q^{8} -90.2944 q^{10} +70.0332 q^{11} -22.1556 q^{16} -236.667 q^{20} +310.094 q^{22} +290.855 q^{25} -225.819 q^{32} -325.561 q^{40} -486.739 q^{41} -452.000 q^{43} +812.774 q^{44} -71.1653 q^{47} -343.000 q^{49} +1287.85 q^{50} -1428.15 q^{55} -696.914 q^{59} -944.654 q^{61} -822.638 q^{64} -123.807 q^{71} +418.244 q^{79} +451.809 q^{80} -2155.19 q^{82} +1509.37 q^{83} -2001.37 q^{86} +1118.06 q^{88} +155.245 q^{89} -315.107 q^{94} -1518.74 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{4}+O(q^{10}) \) 4 * q + 32 * q^4	\( 4 q + 32 q^{4} - 116 q^{10} - 204 q^{16} + 476 q^{22} + 500 q^{25} - 884 q^{40} - 1808 q^{43} - 1372 q^{49} - 3232 q^{55} - 1632 q^{64} - 2924 q^{82} + 2756 q^{88} - 5140 q^{94}+O(q^{100}) \) 4 * q + 32 * q^4 - 116 * q^10 - 204 * q^16 + 476 * q^22 + 500 * q^25 - 884 * q^40 - 1808 * q^43 - 1372 * q^49 - 3232 * q^55 - 1632 * q^64 - 2924 * q^82 + 2756 * q^88 - 5140 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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