LMFDB - Embedded newform 495.3.h.f.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [495,3,Mod(109,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.109"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	495.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,16,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$13.4877730858$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{10}, \sqrt{22})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 16x^{2} + 9 $ x^4 - 16*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			109.1
Root			$-3.92635$ of defining polynomial
Character	$\chi$	$=$	495.109

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.92635 q^{2} +11.4162 q^{4} +5.00000 q^{5} +6.21855 q^{7} -29.1186 q^{8} -19.6317 q^{10} +11.0000 q^{11} +25.1922 q^{13} -24.4162 q^{14} +68.6648 q^{16} +12.7551 q^{17} +57.0810 q^{20} -43.1898 q^{22} +25.0000 q^{25} -98.9134 q^{26} +70.9922 q^{28} -59.3296 q^{31} -153.128 q^{32} -50.0810 q^{34} +31.0928 q^{35} -145.593 q^{40} -50.7024 q^{43} +125.578 q^{44} -10.3296 q^{49} -98.1587 q^{50} +287.599 q^{52} +55.0000 q^{55} -181.075 q^{56} +59.3296 q^{59} +232.949 q^{62} +326.573 q^{64} +125.961 q^{65} +145.615 q^{68} -122.081 q^{70} -118.659 q^{71} -106.351 q^{73} +68.4041 q^{77} +343.324 q^{80} +107.623 q^{83} +63.7756 q^{85} +199.075 q^{86} -320.304 q^{88} -2.00000 q^{89} +156.659 q^{91} +40.5575 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{4} + 20 q^{5} + 44 q^{11} - 68 q^{14} + 156 q^{16} + 80 q^{20} + 100 q^{25} - 188 q^{26} - 52 q^{34} + 176 q^{44} + 196 q^{49} + 220 q^{55} - 220 q^{56} + 624 q^{64} - 340 q^{70} + 780 q^{80}+ \cdots + 152 q^{91}+O(q^{100}) $ 4 * q + 16 * q^4 + 20 * q^5 + 44 * q^11 - 68 * q^14 + 156 * q^16 + 80 * q^20 + 100 * q^25 - 188 * q^26 - 52 * q^34 + 176 * q^44 + 196 * q^49 + 220 * q^55 - 220 * q^56 + 624 * q^64 - 340 * q^70 + 780 * q^80 + 292 * q^86 - 8 * q^89 + 152 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−3.92635	−1.96317	−0.981587	−	0.191017i	$-0.938821\pi$
$2$	−3.92635	−1.96317	−0.981587	+	0.191017i	$0.938821\pi$
$3$	0	0
$3$	0	0
$4$	11.4162	2.85405
$5$	5.00000	1.00000
$5$	5.00000	1.00000
$6$	0	0
$7$	6.21855	0.888365	0.444182	−	0.895936i	$-0.353494\pi$
$7$	6.21855	0.888365	0.444182	+	0.895936i	$0.353494\pi$
$8$	−29.1186	−3.63982
$9$	0	0
$10$	−19.6317	−1.96317
$11$	11.0000	1.00000
$11$	11.0000	1.00000
$12$	0	0
$13$	25.1922	1.93786	0.968932	−	0.247329i	$-0.0795529\pi$
$13$	25.1922	1.93786	0.968932	+	0.247329i	$0.0795529\pi$
$14$	−24.4162	−1.74401
$15$	0	0
$16$	68.6648	4.29155
$17$	12.7551	0.750301	0.375150	−	0.926964i	$-0.377591\pi$
$17$	12.7551	0.750301	0.375150	+	0.926964i	$0.377591\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	57.0810	2.85405
$21$	0	0
$22$	−43.1898	−1.96317
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	25.0000	1.00000
$26$	−98.9134	−3.80436
$27$	0	0
$28$	70.9922	2.53544
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−59.3296	−1.91386	−0.956929	−	0.290323i	$-0.906237\pi$
$31$	−59.3296	−1.91386	−0.956929	+	0.290323i	$0.906237\pi$
$32$	−153.128	−4.78524
$33$	0	0
$34$	−50.0810	−1.47297
$35$	31.0928	0.888365
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	−145.593	−3.63982
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−50.7024	−1.17913	−0.589563	−	0.807722i	$-0.700700\pi$
$43$	−50.7024	−1.17913	−0.589563	+	0.807722i	$0.700700\pi$
$44$	125.578	2.85405
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−10.3296	−0.210808
$50$	−98.1587	−1.96317
$51$	0	0
$52$	287.599	5.53076
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	55.0000	1.00000
$56$	−181.075	−3.23349
$57$	0	0
$58$	0	0
$59$	59.3296	1.00559	0.502793	−	0.864407i	$-0.332306\pi$
$59$	59.3296	1.00559	0.502793	+	0.864407i	$0.332306\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	232.949	3.75723
$63$	0	0
$64$	326.573	5.10270
$65$	125.961	1.93786
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	145.615	2.14140
$69$	0	0
$70$	−122.081	−1.74401
$71$	−118.659	−1.67126	−0.835628	−	0.549296i	$-0.814896\pi$
$71$	−118.659	−1.67126	−0.835628	+	0.549296i	$0.814896\pi$
$72$	0	0
$73$	−106.351	−1.45687	−0.728434	−	0.685116i	$-0.759752\pi$
$73$	−106.351	−1.45687	−0.728434	+	0.685116i	$0.759752\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	68.4041	0.888365
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	343.324	4.29155
$81$	0	0
$82$	0	0
$83$	107.623	1.29667	0.648334	−	0.761356i	$-0.275466\pi$
$83$	107.623	1.29667	0.648334	+	0.761356i	$0.275466\pi$
$84$	0	0
$85$	63.7756	0.750301
$86$	199.075	2.31483
$87$	0	0
$88$	−320.304	−3.63982
$89$	−2.00000	−0.0224719	−0.0112360	−	0.999937i	$-0.503577\pi$
$89$	−2.00000	−0.0224719	−0.0112360	+	0.999937i	$0.503577\pi$
$90$	0	0
$91$	156.659	1.72153
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	40.5575	0.413852
$99$	0	0
$100$	285.405	2.85405
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−733.561	−7.05347
$105$	0	0
$106$	0	0
$107$	−7.49057	−0.0700053	−0.0350027	−	0.999387i	$-0.511144\pi$
$107$	−7.49057	−0.0700053	−0.0350027	+	0.999387i	$0.511144\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−215.949	−1.96317
$111$	0	0
$112$	426.996	3.81246
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−232.949	−1.97414
$119$	79.3184	0.666541
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	−677.318	−5.46224
$125$	125.000	1.00000
$126$	0	0
$127$	156.100	1.22913	0.614566	−	0.788865i	$-0.289331\pi$
$127$	156.100	1.22913	0.614566	+	0.788865i	$0.289331\pi$
$128$	−669.727	−5.23224
$129$	0	0
$130$	−494.567	−3.80436
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−371.411	−2.73096
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	354.961	2.53544
$141$	0	0
$142$	465.897	3.28097
$143$	277.114	1.93786
$144$	0	0
$145$	0	0
$146$	417.573	2.86009
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	−268.578	−1.74401
$155$	−296.648	−1.91386
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	−765.638	−4.78524
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−422.567	−2.54558
$167$	−23.2842	−0.139426	−0.0697132	−	0.997567i	$-0.522208\pi$
$167$	−23.2842	−0.139426	−0.0697132	+	0.997567i	$0.522208\pi$
$168$	0	0
$169$	465.648	2.75531
$170$	−250.405	−1.47297
$171$	0	0
$172$	−578.829	−3.36529
$173$	−345.837	−1.99905	−0.999527	−	0.0307372i	$-0.990214\pi$
$173$	−345.837	−1.99905	−0.999527	+	0.0307372i	$0.990214\pi$
$174$	0	0
$175$	155.464	0.888365
$176$	755.313	4.29155
$177$	0	0
$178$	7.85269	0.0441163
$179$	−38.0000	−0.212291	−0.106145	−	0.994351i	$-0.533851\pi$
$179$	−38.0000	−0.212291	−0.106145	+	0.994351i	$0.533851\pi$
$180$	0	0
$181$	118.659	0.655576	0.327788	−	0.944751i	$-0.393697\pi$
$181$	118.659	0.655576	0.327788	+	0.944751i	$0.393697\pi$
$182$	−615.098	−3.37966
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	140.306	0.750301
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−338.000	−1.76963	−0.884817	−	0.465939i	$-0.845717\pi$
$191$	−338.000	−1.76963	−0.884817	+	0.465939i	$0.845717\pi$
$192$	0	0
$193$	−294.180	−1.52425	−0.762125	−	0.647430i	$-0.775844\pi$
$193$	−294.180	−1.52425	−0.762125	+	0.647430i	$0.775844\pi$
$194$	0	0
$195$	0	0
$196$	−117.925	−0.601656
$197$	332.127	1.68593	0.842963	−	0.537972i	$-0.180809\pi$
$197$	332.127	1.68593	0.842963	+	0.537972i	$0.180809\pi$
$198$	0	0
$199$	−178.000	−0.894472	−0.447236	−	0.894416i	$-0.647592\pi$
$199$	−178.000	−0.894472	−0.447236	+	0.894416i	$0.647592\pi$
$200$	−727.964	−3.63982
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	1729.82	8.31644
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	29.4106	0.137433
$215$	−253.512	−1.17913
$216$	0	0
$217$	−368.944	−1.70020
$218$	0	0
$219$	0	0
$220$	627.891	2.85405
$221$	321.330	1.45398
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−952.232	−4.25103
$225$	0	0
$226$	0	0
$227$	−154.828	−0.682061	−0.341030	−	0.940052i	$-0.610776\pi$
$227$	−154.828	−0.682061	−0.341030	+	0.940052i	$0.610776\pi$
$228$	0	0
$229$	422.000	1.84279	0.921397	−	0.388622i	$-0.127049\pi$
$229$	422.000	1.84279	0.921397	+	0.388622i	$0.127049\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−135.854	−0.583065	−0.291533	−	0.956561i	$-0.594165\pi$
$233$	−135.854	−0.583065	−0.291533	+	0.956561i	$0.594165\pi$
$234$	0	0
$235$	0	0
$236$	677.318	2.86999
$237$	0	0
$238$	−311.431	−1.30854
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−475.088	−1.96317
$243$	0	0
$244$	0	0
$245$	−51.6479	−0.210808
$246$	0	0
$247$	0	0
$248$	1727.59	6.96610
$249$	0	0
$250$	−490.793	−1.96317
$251$	415.307	1.65461	0.827305	−	0.561753i	$-0.189873\pi$
$251$	415.307	1.65461	0.827305	+	0.561753i	$0.189873\pi$
$252$	0	0
$253$	0	0
$254$	−612.902	−2.41300
$255$	0	0
$256$	1323.29	5.16910
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	1438.00	5.53076
$261$	0	0
$262$	0	0
$263$	−254.148	−0.966343	−0.483172	−	0.875526i	$-0.660515\pi$
$263$	−254.148	−0.966343	−0.483172	+	0.875526i	$0.660515\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	415.307	1.54389	0.771946	−	0.635688i	$-0.219283\pi$
$269$	415.307	1.54389	0.771946	+	0.635688i	$0.219283\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	875.827	3.21995
$273$	0	0
$274$	0	0
$275$	275.000	1.00000
$276$	0	0
$277$	553.275	1.99738	0.998691	−	0.0511493i	$-0.0162884\pi$
$277$	553.275	1.99738	0.998691	+	0.0511493i	$0.0162884\pi$
$278$	0	0
$279$	0	0
$280$	−905.377	−3.23349
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−273.934	−0.967966	−0.483983	−	0.875077i	$-0.660810\pi$
$283$	−273.934	−0.967966	−0.483983	+	0.875077i	$0.660810\pi$
$284$	−1354.64	−4.76985
$285$	0	0
$286$	−1088.05	−3.80436
$287$	0	0
$288$	0	0
$289$	−126.307	−0.437049
$290$	0	0
$291$	0	0
$292$	−1214.13	−4.15798
$293$	−423.639	−1.44587	−0.722934	−	0.690917i	$-0.757207\pi$
$293$	−423.639	−1.44587	−0.722934	+	0.690917i	$0.757207\pi$
$294$	0	0
$295$	296.648	1.00559
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−315.296	−1.04749
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	66.3196	0.216025	0.108012	−	0.994150i	$-0.465551\pi$
$307$	66.3196	0.216025	0.108012	+	0.994150i	$0.465551\pi$
$308$	780.915	2.53544
$309$	0	0
$310$	1164.74	3.75723
$311$	−474.637	−1.52616	−0.763082	−	0.646302i	$-0.776315\pi$
$311$	−474.637	−1.52616	−0.763082	+	0.646302i	$0.776315\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	1632.86	5.10270
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	629.805	1.93786
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−237.318	−0.716974	−0.358487	−	0.933535i	$-0.616707\pi$
$331$	−237.318	−0.716974	−0.358487	+	0.933535i	$0.616707\pi$
$332$	1228.65	3.70075
$333$	0	0
$334$	91.4218	0.273718
$335$	0	0
$336$	0	0
$337$	−103.171	−0.306147	−0.153073	−	0.988215i	$-0.548917\pi$
$337$	−103.171	−0.306147	−0.153073	+	0.988215i	$0.548917\pi$
$338$	−1828.30	−5.40916
$339$	0	0
$340$	728.074	2.14140
$341$	−652.625	−1.91386
$342$	0	0
$343$	−368.944	−1.07564
$344$	1476.38	4.29181
$345$	0	0
$346$	1357.87	3.92449
$347$	−686.090	−1.97721	−0.988603	−	0.150546i	$-0.951897\pi$
$347$	−686.090	−1.97721	−0.988603	+	0.150546i	$0.951897\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−610.405	−1.74401
$351$	0	0
$352$	−1684.40	−4.78524
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−593.296	−1.67126
$356$	−22.8324	−0.0641359
$357$	0	0
$358$	149.201	0.416763
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	−465.897	−1.28701
$363$	0	0
$364$	1788.45	4.91333
$365$	−531.757	−1.45687
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−318.242	−0.853195	−0.426598	−	0.904442i	$-0.640288\pi$
$373$	−318.242	−0.853195	−0.426598	+	0.904442i	$0.640288\pi$
$374$	−550.891	−1.47297
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−538.000	−1.41953	−0.709763	−	0.704441i	$-0.751198\pi$
$379$	−538.000	−1.41953	−0.709763	+	0.704441i	$0.751198\pi$
$380$	0	0
$381$	0	0
$382$	1327.11	3.47410
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	342.020	0.888365
$386$	1155.05	2.99237
$387$	0	0
$388$	0	0
$389$	771.285	1.98274	0.991368	−	0.131105i	$-0.0418526\pi$
$389$	771.285	1.98274	0.991368	+	0.131105i	$0.0418526\pi$
$390$	0	0
$391$	0	0
$392$	300.783	0.767303
$393$	0	0
$394$	−1304.05	−3.30976
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	698.890	1.75600
$399$	0	0
$400$	1716.62	4.29155
$401$	−782.000	−1.95012	−0.975062	−	0.221931i	$-0.928764\pi$
$401$	−782.000	−1.95012	−0.975062	+	0.221931i	$0.928764\pi$
$402$	0	0
$403$	−1494.64	−3.70879
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	368.944	0.893327
$414$	0	0
$415$	538.117	1.29667
$416$	−3857.62	−9.27313
$417$	0	0
$418$	0	0
$419$	442.000	1.05489	0.527446	−	0.849588i	$-0.323150\pi$
$419$	442.000	1.05489	0.527446	+	0.849588i	$0.323150\pi$
$420$	0	0
$421$	830.614	1.97296	0.986478	−	0.163895i	$-0.0524060\pi$
$421$	830.614	1.97296	0.986478	+	0.163895i	$0.0524060\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	318.878	0.750301
$426$	0	0
$427$	0	0
$428$	−85.5138	−0.199799
$429$	0	0
$430$	995.377	2.31483
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	1448.60	3.33779
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−1601.52	−3.63982
$441$	0	0
$442$	−1261.65	−2.85441
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−10.0000	−0.0224719
$446$	0	0
$447$	0	0
$448$	2030.81	4.53306
$449$	−722.000	−1.60802	−0.804009	−	0.594617i	$-0.797304\pi$
$449$	−722.000	−1.60802	−0.804009	+	0.594617i	$0.797304\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	607.908	1.33900
$455$	783.296	1.72153
$456$	0	0
$457$	872.647	1.90951	0.954756	−	0.297390i	$-0.0961161\pi$
$457$	872.647	1.90951	0.954756	+	0.297390i	$0.0961161\pi$
$458$	−1656.92	−3.61773
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	533.411	1.14466
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1727.59	−3.66015
$473$	−557.727	−1.17913
$474$	0	0
$475$	0	0
$476$	905.514	1.90234
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	1381.36	2.85405
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	202.788	0.413852
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−4073.85	−8.21342
$497$	−737.888	−1.48469
$498$	0	0
$499$	−982.000	−1.96794	−0.983968	−	0.178345i	$-0.942926\pi$
$499$	−982.000	−1.96794	−0.983968	+	0.178345i	$0.942926\pi$
$500$	1427.02	2.85405
$501$	0	0
$502$	−1630.64	−3.24829
$503$	915.859	1.82079	0.910397	−	0.413737i	$-0.135777\pi$
$503$	915.859	1.82079	0.910397	+	0.413737i	$0.135777\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	1782.07	3.50801
$509$	−1008.60	−1.98154	−0.990769	−	0.135560i	$-0.956717\pi$
$509$	−1008.60	−1.98154	−0.990769	+	0.135560i	$0.956717\pi$
$510$	0	0
$511$	−661.352	−1.29423
$512$	−2516.79	−4.91560
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−3667.81	−7.05347
$521$	−542.000	−1.04031	−0.520154	−	0.854073i	$-0.674125\pi$
$521$	−542.000	−1.04031	−0.520154	+	0.854073i	$0.674125\pi$
$522$	0	0
$523$	896.073	1.71333	0.856666	−	0.515871i	$-0.172532\pi$
$523$	896.073	1.71333	0.856666	+	0.515871i	$0.172532\pi$
$524$	0	0
$525$	0	0
$526$	997.874	1.89710
$527$	−756.756	−1.43597
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−37.4529	−0.0700053
$536$	0	0
$537$	0	0
$538$	−1630.64	−3.03093
$539$	−113.625	−0.210808
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−1953.16	−3.59037
$545$	0	0
$546$	0	0
$547$	−996.206	−1.82122	−0.910608	−	0.413270i	$-0.864387\pi$
$547$	−996.206	−1.82122	−0.910608	+	0.413270i	$0.864387\pi$
$548$	0	0
$549$	0	0
$550$	−1079.75	−1.96317
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−2172.35	−3.92121
$555$	0	0
$556$	0	0
$557$	1086.62	1.95085	0.975424	−	0.220338i	$-0.0707159\pi$
$557$	1086.62	1.95085	0.975424	+	0.220338i	$0.0707159\pi$
$558$	0	0
$559$	−1277.31	−2.28499
$560$	2134.98	3.81246
$561$	0	0
$562$	0	0
$563$	1125.84	1.99972	0.999859	−	0.0167879i	$-0.00534401\pi$
$563$	1125.84	1.99972	0.999859	+	0.0167879i	$0.00534401\pi$
$564$	0	0
$565$	0	0
$566$	1075.56	1.90029
$567$	0	0
$568$	3455.19	6.08307
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	3163.59	5.53076
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	495.926	0.858003
$579$	0	0
$580$	0	0
$581$	669.262	1.15191
$582$	0	0
$583$	0	0
$584$	3096.80	5.30274
$585$	0	0
$586$	1663.35	2.83849
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	−1164.74	−1.97414
$591$	0	0
$592$	0	0
$593$	913.951	1.54123	0.770616	−	0.637299i	$-0.219949\pi$
$593$	913.951	1.54123	0.770616	+	0.637299i	$0.219949\pi$
$594$	0	0
$595$	396.592	0.666541
$596$	0	0
$597$	0	0
$598$	0	0
$599$	802.000	1.33890	0.669449	−	0.742858i	$-0.266530\pi$
$599$	802.000	1.33890	0.669449	+	0.742858i	$0.266530\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	1237.96	2.05641
$603$	0	0
$604$	0	0
$605$	605.000	1.00000
$606$	0	0
$607$	1174.32	1.93463	0.967313	−	0.253586i	$-0.0816102\pi$
$607$	1174.32	1.93463	0.967313	+	0.253586i	$0.0816102\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1220.71	−1.99137	−0.995685	−	0.0927990i	$-0.970419\pi$
$613$	−1220.71	−1.99137	−0.995685	+	0.0927990i	$0.970419\pi$
$614$	−260.394	−0.424094
$615$	0	0
$616$	−1991.83	−3.23349
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	830.614	1.34186	0.670932	−	0.741519i	$-0.265894\pi$
$619$	830.614	1.34186	0.670932	+	0.741519i	$0.265894\pi$
$620$	−3386.59	−5.46224
$621$	0	0
$622$	1863.59	2.99612
$623$	−12.4371	−0.0199633
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1042.00	−1.65135	−0.825674	−	0.564148i	$-0.809205\pi$
$631$	−1042.00	−1.65135	−0.825674	+	0.564148i	$0.809205\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	780.499	1.22913
$636$	0	0
$637$	−260.225	−0.408517
$638$	0	0
$639$	0	0
$640$	−3348.64	−5.23224
$641$	−302.000	−0.471139	−0.235569	−	0.971858i	$-0.575696\pi$
$641$	−302.000	−0.471139	−0.235569	+	0.971858i	$0.575696\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	652.625	1.00559
$650$	−2472.83	−3.80436
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−442.000	−0.668684	−0.334342	−	0.942452i	$-0.608514\pi$
$661$	−442.000	−0.668684	−0.334342	+	0.942452i	$0.608514\pi$
$662$	931.794	1.40754
$663$	0	0
$664$	−3133.84	−4.71964
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−265.817	−0.397930
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	13.8507	0.0205805	0.0102902	−	0.999947i	$-0.496724\pi$
$673$	13.8507	0.0205805	0.0102902	+	0.999947i	$0.496724\pi$
$674$	405.087	0.601019
$675$	0	0
$676$	5315.93	7.86380
$677$	1055.03	1.55840	0.779198	−	0.626777i	$-0.215626\pi$
$677$	1055.03	1.55840	0.779198	+	0.626777i	$0.215626\pi$
$678$	0	0
$679$	0	0
$680$	−1857.05	−2.73096
$681$	0	0
$682$	2562.43	3.75723
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	1448.60	2.11167
$687$	0	0
$688$	−3481.47	−5.06028
$689$	0	0
$690$	0	0
$691$	−598.000	−0.865412	−0.432706	−	0.901535i	$-0.642441\pi$
$691$	−598.000	−0.865412	−0.432706	+	0.901535i	$0.642441\pi$
$692$	−3948.14	−5.70540
$693$	0	0
$694$	2693.83	3.88160
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	1774.81	2.53544
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	3592.30	5.10270
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−237.318	−0.334723	−0.167361	−	0.985896i	$-0.553525\pi$
$709$	−237.318	−0.334723	−0.167361	+	0.985896i	$0.553525\pi$
$710$	2329.49	3.28097
$711$	0	0
$712$	58.2371	0.0817937
$713$	0	0
$714$	0	0
$715$	1385.57	1.93786
$716$	−433.816	−0.605888
$717$	0	0
$718$	0	0
$719$	718.000	0.998609	0.499305	−	0.866427i	$-0.333589\pi$
$719$	718.000	0.998609	0.499305	+	0.866427i	$0.333589\pi$
$720$	0	0
$721$	0	0
$722$	−1417.41	−1.96317
$723$	0	0
$724$	1354.64	1.87105
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	−4561.69	−6.26606
$729$	0	0
$730$	2087.86	2.86009
$731$	−646.715	−0.884700
$732$	0	0
$733$	−1368.05	−1.86637	−0.933183	−	0.359400i	$-0.882981\pi$
$733$	−1368.05	−1.86637	−0.933183	+	0.359400i	$0.882981\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−146.666	−0.197397	−0.0986987	−	0.995117i	$-0.531468\pi$
$743$	−146.666	−0.197397	−0.0986987	+	0.995117i	$0.531468\pi$
$744$	0	0
$745$	0	0
$746$	1249.53	1.67497
$747$	0	0
$748$	1601.76	2.14140
$749$	−46.5805	−0.0621903
$750$	0	0
$751$	−478.000	−0.636485	−0.318242	−	0.948009i	$-0.603093\pi$
$751$	−478.000	−0.636485	−0.318242	+	0.948009i	$0.603093\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	2112.37	2.78677
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−3858.68	−5.05062
$765$	0	0
$766$	0	0
$767$	1494.64	1.94869
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	−1342.89	−1.74401
$771$	0	0
$772$	−3358.42	−4.35028
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−1483.24	−1.91386
$776$	0	0
$777$	0	0
$778$	−3028.33	−3.89246
$779$	0	0
$780$	0	0
$781$	−1305.25	−1.67126
$782$	0	0
$783$	0	0
$784$	−709.279	−0.904693
$785$	0	0
$786$	0	0
$787$	582.954	0.740729	0.370365	−	0.928886i	$-0.379233\pi$
$787$	582.954	0.740729	0.370365	+	0.928886i	$0.379233\pi$
$788$	3791.63	4.81172
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−2032.08	−2.55287
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−3828.19	−4.78524
$801$	0	0
$802$	3070.40	3.82843
$803$	−1169.87	−1.45687
$804$	0	0
$805$	0	0
$806$	5868.49	7.28101
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	−1448.60	−1.75376
$827$	1653.92	1.99991	0.999954	−	0.00959135i	$-0.00305307\pi$
$827$	1653.92	1.99991	0.999954	+	0.00959135i	$0.00305307\pi$
$828$	0	0
$829$	1186.59	1.43135	0.715677	−	0.698432i	$-0.246118\pi$
$829$	1186.59	1.43135	0.715677	+	0.698432i	$0.246118\pi$
$830$	−2112.83	−2.54558
$831$	0	0
$832$	8227.09	9.88833
$833$	−131.755	−0.158169
$834$	0	0
$835$	−116.421	−0.139426
$836$	0	0
$837$	0	0
$838$	−1735.45	−2.07094
$839$	−830.614	−0.990005	−0.495003	−	0.868892i	$-0.664833\pi$
$839$	−830.614	−0.990005	−0.495003	+	0.868892i	$0.664833\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	−3261.28	−3.87325
$843$	0	0
$844$	0	0
$845$	2328.24	2.75531
$846$	0	0
$847$	752.445	0.888365
$848$	0	0
$849$	0	0
$850$	−1252.02	−1.47297
$851$	0	0
$852$	0	0
$853$	−210.760	−0.247081	−0.123540	−	0.992340i	$-0.539425\pi$
$853$	−210.760	−0.247081	−0.123540	+	0.992340i	$0.539425\pi$
$854$	0	0
$855$	0	0
$856$	218.115	0.254807
$857$	191.680	0.223664	0.111832	−	0.993727i	$-0.464328\pi$
$857$	191.680	0.223664	0.111832	+	0.993727i	$0.464328\pi$
$858$	0	0
$859$	−1661.23	−1.93391	−0.966955	−	0.254948i	$-0.917942\pi$
$859$	−1661.23	−1.93391	−0.966955	+	0.254948i	$0.917942\pi$
$860$	−2894.15	−3.36529
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−1729.18	−1.99905
$866$	0	0
$867$	0	0
$868$	−4211.94	−4.85247
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	777.319	0.888365
$876$	0	0
$877$	−1061.29	−1.21013	−0.605067	−	0.796174i	$-0.706854\pi$
$877$	−1061.29	−1.21013	−0.605067	+	0.796174i	$0.706854\pi$
$878$	0	0
$879$	0	0
$880$	3776.56	4.29155
$881$	−118.659	−0.134687	−0.0673435	−	0.997730i	$-0.521452\pi$
$881$	−118.659	−0.134687	−0.0673435	+	0.997730i	$0.521452\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	3668.36	4.14973
$885$	0	0
$886$	0	0
$887$	−1688.51	−1.90362	−0.951812	−	0.306682i	$-0.900781\pi$
$887$	−1688.51	−1.90362	−0.951812	+	0.306682i	$0.900781\pi$
$888$	0	0
$889$	970.715	1.09192
$890$	39.2635	0.0441163
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−190.000	−0.212291
$896$	−4164.73	−4.64814
$897$	0	0
$898$	2834.82	3.15682
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	593.296	0.655576
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−1767.55	−1.94664
$909$	0	0
$910$	−3075.49	−3.37966
$911$	−1742.00	−1.91218	−0.956092	−	0.293066i	$-0.905324\pi$
$911$	−1742.00	−1.91218	−0.956092	+	0.293066i	$0.905324\pi$
$912$	0	0
$913$	1183.86	1.29667
$914$	−3426.32	−3.74870
$915$	0	0
$916$	4817.64	5.25943
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2989.29	−3.23867
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−830.614	−0.894095	−0.447047	−	0.894510i	$-0.647525\pi$
$929$	−830.614	−0.894095	−0.447047	+	0.894510i	$0.647525\pi$
$930$	0	0
$931$	0	0
$932$	−1550.94	−1.66410
$933$	0	0
$934$	0	0
$935$	701.531	0.750301
$936$	0	0
$937$	883.459	0.942859	0.471430	−	0.881904i	$-0.343738\pi$
$937$	883.459	0.942859	0.471430	+	0.881904i	$0.343738\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	4073.85	4.31552
$945$	0	0
$946$	2189.83	2.31483
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	−2679.23	−2.82321
$950$	0	0
$951$	0	0
$952$	−2309.64	−2.42609
$953$	1782.29	1.87019	0.935093	−	0.354402i	$-0.115316\pi$
$953$	1782.29	1.87019	0.935093	+	0.354402i	$0.115316\pi$
$954$	0	0
$955$	−1690.00	−1.76963
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2559.00	2.66285
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1470.90	−1.52425
$966$	0	0
$967$	−1696.96	−1.75487	−0.877435	−	0.479696i	$-0.840747\pi$
$967$	−1696.96	−1.75487	−0.877435	+	0.479696i	$0.840747\pi$
$968$	−3523.35	−3.63982
$969$	0	0
$970$	0	0
$971$	−1622.00	−1.67044	−0.835221	−	0.549914i	$-0.814661\pi$
$971$	−1622.00	−1.67044	−0.835221	+	0.549914i	$0.814661\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	−22.0000	−0.0224719
$980$	−589.623	−0.601656
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	1660.64	1.68593
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−415.307	−0.419079	−0.209539	−	0.977800i	$-0.567196\pi$
$991$	−415.307	−0.419079	−0.209539	+	0.977800i	$0.567196\pi$
$992$	9084.99	9.15826
$993$	0	0
$994$	2897.21	2.91469
$995$	−890.000	−0.894472
$996$	0	0
$997$	51.6914	0.0518469	0.0259235	−	0.999664i	$-0.491747\pi$
$997$	51.6914	0.0518469	0.0259235	+	0.999664i	$0.491747\pi$
$998$	3855.67	3.86340
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.3.h.f.109.1	yes	4
3.2	odd	2		495.3.h.e.109.4	yes	4
5.4	even	2	inner	495.3.h.f.109.4	yes	4
11.10	odd	2	inner	495.3.h.f.109.4	yes	4
15.14	odd	2		495.3.h.e.109.1	✓	4
33.32	even	2		495.3.h.e.109.1	✓	4
55.54	odd	2	CM	495.3.h.f.109.1	yes	4
165.164	even	2		495.3.h.e.109.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.3.h.e.109.1	✓	4	15.14	odd	2
495.3.h.e.109.1	✓	4	33.32	even	2
495.3.h.e.109.4	yes	4	3.2	odd	2
495.3.h.e.109.4	yes	4	165.164	even	2
495.3.h.f.109.1	yes	4	1.1	even	1	trivial
495.3.h.f.109.1	yes	4	55.54	odd	2	CM
495.3.h.f.109.4	yes	4	5.4	even	2	inner
495.3.h.f.109.4	yes	4	11.10	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	495.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.92635 q^{2} +11.4162 q^{4} +5.00000 q^{5} +6.21855 q^{7} -29.1186 q^{8} -19.6317 q^{10} +11.0000 q^{11} +25.1922 q^{13} -24.4162 q^{14} +68.6648 q^{16} +12.7551 q^{17} +57.0810 q^{20} -43.1898 q^{22} +25.0000 q^{25} -98.9134 q^{26} +70.9922 q^{28} -59.3296 q^{31} -153.128 q^{32} -50.0810 q^{34} +31.0928 q^{35} -145.593 q^{40} -50.7024 q^{43} +125.578 q^{44} -10.3296 q^{49} -98.1587 q^{50} +287.599 q^{52} +55.0000 q^{55} -181.075 q^{56} +59.3296 q^{59} +232.949 q^{62} +326.573 q^{64} +125.961 q^{65} +145.615 q^{68} -122.081 q^{70} -118.659 q^{71} -106.351 q^{73} +68.4041 q^{77} +343.324 q^{80} +107.623 q^{83} +63.7756 q^{85} +199.075 q^{86} -320.304 q^{88} -2.00000 q^{89} +156.659 q^{91} +40.5575 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4} + 20 q^{5} + 44 q^{11} - 68 q^{14} + 156 q^{16} + 80 q^{20} + 100 q^{25} - 188 q^{26} - 52 q^{34} + 176 q^{44} + 196 q^{49} + 220 q^{55} - 220 q^{56} + 624 q^{64} - 340 q^{70} + 780 q^{80}+ \cdots + 152 q^{91}+O(q^{100}) \) 4 * q + 16 * q^4 + 20 * q^5 + 44 * q^11 - 68 * q^14 + 156 * q^16 + 80 * q^20 + 100 * q^25 - 188 * q^26 - 52 * q^34 + 176 * q^44 + 196 * q^49 + 220 * q^55 - 220 * q^56 + 624 * q^64 - 340 * q^70 + 780 * q^80 + 292 * q^86 - 8 * q^89 + 152 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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