LMFDB - Embedded newform 675.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(1,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61803 q^{2} +4.85410 q^{4} +7.47214 q^{8} +9.85410 q^{16} +3.76393 q^{17} -8.70820 q^{19} -1.47214 q^{23} +2.70820 q^{31} +10.8541 q^{32} +9.85410 q^{34} -22.7984 q^{38} -3.85410 q^{46} -8.94427 q^{47} -7.00000 q^{49} +14.2361 q^{53} -14.4164 q^{61} +7.09017 q^{62} +8.70820 q^{64} +18.2705 q^{68} -42.2705 q^{76} -14.7082 q^{79} +11.9443 q^{83} -7.14590 q^{92} -23.4164 q^{94} -18.3262 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} + 13 q^{16} + 12 q^{17} - 4 q^{19} + 6 q^{23} - 8 q^{31} + 15 q^{32} + 13 q^{34} - 21 q^{38} - q^{46} - 14 q^{49} + 24 q^{53} - 2 q^{61} + 3 q^{62} + 4 q^{64} + 3 q^{68}+ \cdots - 21 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 + 13 * q^16 + 12 * q^17 - 4 * q^19 + 6 * q^23 - 8 * q^31 + 15 * q^32 + 13 * q^34 - 21 * q^38 - q^46 - 14 * q^49 + 24 * q^53 - 2 * q^61 + 3 * q^62 + 4 * q^64 + 3 * q^68 - 51 * q^76 - 16 * q^79 + 6 * q^83 - 21 * q^92 - 20 * q^94 - 21 * q^98

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 $

$2$	2.61803	1.85123	0.925615	−	0.378467i	$-0.123549\pi$
$2$	2.61803	1.85123	0.925615	+	0.378467i	$0.123549\pi$
$3$	0	0
$3$	0	0
$4$	4.85410	2.42705
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	7.47214	2.64180
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	9.85410	2.46353
$17$	3.76393	0.912888	0.456444	−	0.889752i	$-0.349123\pi$
$17$	3.76393	0.912888	0.456444	+	0.889752i	$0.349123\pi$
$18$	0	0
$19$	−8.70820	−1.99780	−0.998899	−	0.0469020i	$-0.985065\pi$
$19$	−8.70820	−1.99780	−0.998899	+	0.0469020i	$0.985065\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.47214	−0.306962	−0.153481	−	0.988152i	$-0.549048\pi$
$23$	−1.47214	−0.306962	−0.153481	+	0.988152i	$0.549048\pi$
$24$	0	0
$25$	0	0
$26$	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
$31$	2.70820	0.486408	0.243204	−	0.969975i	$-0.421802\pi$
$31$	2.70820	0.486408	0.243204	+	0.969975i	$0.421802\pi$
$32$	10.8541	1.91875
$33$	0	0
$34$	9.85410	1.68996
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−22.7984	−3.69838
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	−3.85410	−0.568256
$47$	−8.94427	−1.30466	−0.652328	−	0.757937i	$-0.726208\pi$
$47$	−8.94427	−1.30466	−0.652328	+	0.757937i	$0.726208\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	14.2361	1.95547	0.977737	−	0.209833i	$-0.0672922\pi$
$53$	14.2361	1.95547	0.977737	+	0.209833i	$0.0672922\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−14.4164	−1.84583	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−14.4164	−1.84583	−0.922916	+	0.385002i	$0.874201\pi$
$62$	7.09017	0.900452
$63$	0	0
$64$	8.70820	1.08853
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	18.2705	2.21562
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	−42.2705	−4.84876
$77$	0	0
$78$	0	0
$79$	−14.7082	−1.65480	−0.827401	−	0.561611i	$-0.810182\pi$
$79$	−14.7082	−1.65480	−0.827401	+	0.561611i	$0.810182\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.9443	1.31105	0.655527	−	0.755172i	$-0.272447\pi$
$83$	11.9443	1.31105	0.655527	+	0.755172i	$0.272447\pi$
$84$	0	0
$85$	0	0
$86$	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
$92$	−7.14590	−0.745011
$93$	0	0
$94$	−23.4164	−2.41522
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−18.3262	−1.85123
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	37.2705	3.62003
$107$	17.8885	1.72935	0.864675	−	0.502331i	$-0.167524\pi$
$107$	17.8885	1.72935	0.864675	+	0.502331i	$0.167524\pi$
$108$	0	0
$109$	20.4164	1.95554	0.977769	−	0.209687i	$-0.0672444\pi$
$109$	20.4164	1.95554	0.977769	+	0.209687i	$0.0672444\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.47214	−0.420703	−0.210352	−	0.977626i	$-0.567461\pi$
$113$	−4.47214	−0.420703	−0.210352	+	0.977626i	$0.567461\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−37.7426	−3.41706
$123$	0	0
$124$	13.1459	1.18054
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.09017	0.0963583
$129$	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
$136$	28.1246	2.41167
$137$	17.1803	1.46782	0.733908	−	0.679249i	$-0.237694\pi$
$137$	17.1803	1.46782	0.733908	+	0.679249i	$0.237694\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−65.0689	−5.27778
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−38.5066	−3.06342
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	31.2705	2.42706
$167$	−16.5279	−1.27896	−0.639482	−	0.768806i	$-0.720851\pi$
$167$	−16.5279	−1.27896	−0.639482	+	0.768806i	$0.720851\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.819660	0.0623176	0.0311588	−	0.999514i	$-0.490080\pi$
$173$	0.819660	0.0623176	0.0311588	+	0.999514i	$0.490080\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−2.41641	−0.179610	−0.0898051	−	0.995959i	$-0.528624\pi$
$181$	−2.41641	−0.179610	−0.0898051	+	0.995959i	$0.528624\pi$
$182$	0	0
$183$	0	0
$184$	−11.0000	−0.810931
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−43.4164	−3.16647
$189$	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
$196$	−33.9787	−2.42705
$197$	21.7639	1.55062	0.775308	−	0.631583i	$-0.217595\pi$
$197$	21.7639	1.55062	0.775308	+	0.631583i	$0.217595\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.7082	1.42561	0.712806	−	0.701361i	$-0.247424\pi$
$211$	20.7082	1.42561	0.712806	+	0.701361i	$0.247424\pi$
$212$	69.1033	4.74604
$213$	0	0
$214$	46.8328	3.20143
$215$	0	0
$216$	0	0
$217$	0	0
$218$	53.4508	3.62015
$219$	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
$226$	−11.7082	−0.778818
$227$	−29.9443	−1.98747	−0.993736	−	0.111757i	$-0.964352\pi$
$227$	−29.9443	−1.98747	−0.993736	+	0.111757i	$0.964352\pi$
$228$	0	0
$229$	26.4164	1.74565	0.872823	−	0.488037i	$-0.162287\pi$
$229$	26.4164	1.74565	0.872823	+	0.488037i	$0.162287\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.3607	1.46490	0.732448	−	0.680823i	$-0.238378\pi$
$233$	22.3607	1.46490	0.732448	+	0.680823i	$0.238378\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	25.8328	1.66404	0.832019	−	0.554747i	$-0.187185\pi$
$241$	25.8328	1.66404	0.832019	+	0.554747i	$0.187185\pi$
$242$	−28.7984	−1.85123
$243$	0	0
$244$	−69.9787	−4.47993
$245$	0	0
$246$	0	0
$247$	0	0
$248$	20.2361	1.28499
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−14.5623	−0.910144
$257$	−9.65248	−0.602105	−0.301052	−	0.953608i	$-0.597338\pi$
$257$	−9.65248	−0.602105	−0.301052	+	0.953608i	$0.597338\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.94427	0.551527	0.275764	−	0.961225i	$-0.411069\pi$
$263$	8.94427	0.551527	0.275764	+	0.961225i	$0.411069\pi$
$264$	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
$271$	−9.29180	−0.564436	−0.282218	−	0.959350i	$-0.591070\pi$
$271$	−9.29180	−0.564436	−0.282218	+	0.959350i	$0.591070\pi$
$272$	37.0902	2.24892
$273$	0	0
$274$	44.9787	2.71726
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	10.4721	0.628077
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2.83282	−0.166636
$290$	0	0
$291$	0	0
$292$	0	0
$293$	27.6525	1.61547	0.807737	−	0.589542i	$-0.200692\pi$
$293$	27.6525	1.61547	0.807737	+	0.589542i	$0.200692\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	20.9443	1.20521
$303$	0	0
$304$	−85.8115	−4.92163
$305$	0	0
$306$	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
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	−71.3951	−4.01629
$317$	35.1803	1.97592	0.987962	−	0.154694i	$-0.0494393\pi$
$317$	35.1803	1.97592	0.987962	+	0.154694i	$0.0494393\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−32.7771	−1.82377
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	57.9787	3.18200
$333$	0	0
$334$	−43.2705	−2.36766
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	−34.0344	−1.85123
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	2.14590	0.115364
$347$	−35.7771	−1.92061	−0.960307	−	0.278944i	$-0.910016\pi$
$347$	−35.7771	−1.92061	−0.960307	+	0.278944i	$0.910016\pi$
$348$	0	0
$349$	−3.58359	−0.191825	−0.0959126	−	0.995390i	$-0.530577\pi$
$349$	−3.58359	−0.191825	−0.0959126	+	0.995390i	$0.530577\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−31.3050	−1.66619	−0.833097	−	0.553127i	$-0.813435\pi$
$353$	−31.3050	−1.66619	−0.833097	+	0.553127i	$0.813435\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	56.8328	2.99120
$362$	−6.32624	−0.332500
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−14.5066	−0.756208
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	−66.8328	−3.44664
$377$	0	0
$378$	0	0
$379$	31.5410	1.62015	0.810077	−	0.586324i	$-0.199425\pi$
$379$	31.5410	1.62015	0.810077	+	0.586324i	$0.199425\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−37.4721	−1.91474	−0.957368	−	0.288870i	$-0.906720\pi$
$383$	−37.4721	−1.91474	−0.957368	+	0.288870i	$0.906720\pi$
$384$	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
$391$	−5.54102	−0.280221
$392$	−52.3050	−2.64180
$393$	0	0
$394$	56.9787	2.87055
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	41.8885	2.09968
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−13.8328	−0.683989	−0.341994	−	0.939702i	$-0.611102\pi$
$409$	−13.8328	−0.683989	−0.341994	+	0.939702i	$0.611102\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	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
$421$	−32.4164	−1.57988	−0.789940	−	0.613185i	$-0.789888\pi$
$421$	−32.4164	−1.57988	−0.789940	+	0.613185i	$0.789888\pi$
$422$	54.2148	2.63913
$423$	0	0
$424$	106.374	5.16597
$425$	0	0
$426$	0	0
$427$	0	0
$428$	86.8328	4.19722
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	99.1033	4.74619
$437$	12.8197	0.613248
$438$	0	0
$439$	25.5410	1.21901	0.609503	−	0.792784i	$-0.291369\pi$
$439$	25.5410	1.21901	0.609503	+	0.792784i	$0.291369\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.0557	−1.14292	−0.571461	−	0.820629i	$-0.693623\pi$
$443$	−24.0557	−1.14292	−0.571461	+	0.820629i	$0.693623\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	−21.7082	−1.02107
$453$	0	0
$454$	−78.3951	−3.67927
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	69.1591	3.23159
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	58.5410	2.71186
$467$	−3.11146	−0.143981	−0.0719905	−	0.997405i	$-0.522935\pi$
$467$	−3.11146	−0.143981	−0.0719905	+	0.997405i	$0.522935\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	67.6312	3.08052
$483$	0	0
$484$	−53.3951	−2.42705
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−107.721	−4.87632
$489$	0	0
$490$	0	0
$491$	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
$496$	26.6869	1.19828
$497$	0	0
$498$	0	0
$499$	15.2918	0.684555	0.342277	−	0.939599i	$-0.388802\pi$
$499$	15.2918	0.684555	0.342277	+	0.939599i	$0.388802\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	25.3607	1.13078	0.565388	−	0.824825i	$-0.308726\pi$
$503$	25.3607	1.13078	0.565388	+	0.824825i	$0.308726\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	−40.3050	−1.78124
$513$	0	0
$514$	−25.2705	−1.11463
$515$	0	0
$516$	0	0
$517$	0	0
$518$	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
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	23.4164	1.02100
$527$	10.1935	0.444036
$528$	0	0
$529$	−20.8328	−0.905775
$530$	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
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−24.3262	−1.04490
$543$	0	0
$544$	40.8541	1.75161
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	83.3951	3.56246
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	19.4164	0.823439
$557$	−22.3607	−0.947452	−0.473726	−	0.880672i	$-0.657091\pi$
$557$	−22.3607	−0.947452	−0.473726	+	0.880672i	$0.657091\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	35.7771	1.50782	0.753912	−	0.656975i	$-0.228164\pi$
$563$	35.7771	1.50782	0.753912	+	0.656975i	$0.228164\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−19.5410	−0.817766	−0.408883	−	0.912587i	$-0.634082\pi$
$571$	−19.5410	−0.817766	−0.408883	+	0.912587i	$0.634082\pi$
$572$	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	$0.5\pi$
$578$	−7.41641	−0.308482
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	72.3951	2.99061
$587$	−47.9443	−1.97887	−0.989436	−	0.144971i	$-0.953691\pi$
$587$	−47.9443	−1.97887	−0.989436	+	0.144971i	$0.953691\pi$
$588$	0	0
$589$	−23.5836	−0.971745
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−39.7639	−1.63291	−0.816454	−	0.577410i	$-0.804064\pi$
$593$	−39.7639	−1.63291	−0.816454	+	0.577410i	$0.804064\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	7.83282	0.319507	0.159754	−	0.987157i	$-0.448930\pi$
$601$	7.83282	0.319507	0.159754	+	0.987157i	$0.448930\pi$
$602$	0	0
$603$	0	0
$604$	38.8328	1.58008
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−94.5197	−3.83328
$609$	0	0
$610$	0	0
$611$	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
$617$	30.5967	1.23178	0.615889	−	0.787833i	$-0.288797\pi$
$617$	30.5967	1.23178	0.615889	+	0.787833i	$0.288797\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−49.5410	−1.97220	−0.986098	−	0.166162i	$-0.946862\pi$
$631$	−49.5410	−1.97220	−0.986098	+	0.166162i	$0.946862\pi$
$632$	−109.902	−4.37165
$633$	0	0
$634$	92.1033	3.65789
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	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
$646$	−85.8115	−3.37621
$647$	−43.3607	−1.70468	−0.852342	−	0.522985i	$-0.824819\pi$
$647$	−43.3607	−1.70468	−0.852342	+	0.522985i	$0.824819\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.5967	−0.492949	−0.246474	−	0.969149i	$-0.579272\pi$
$653$	−12.5967	−0.492949	−0.246474	+	0.969149i	$0.579272\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	−73.3050	−2.84908
$663$	0	0
$664$	89.2492	3.46354
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−80.2279	−3.10411
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	−63.1033	−2.42705
$677$	31.3050	1.20315	0.601574	−	0.798817i	$-0.294541\pi$
$677$	31.3050	1.20315	0.601574	+	0.798817i	$0.294541\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−50.8885	−1.94720	−0.973598	−	0.228269i	$-0.926693\pi$
$683$	−50.8885	−1.94720	−0.973598	+	0.228269i	$0.926693\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	32.7082	1.24428	0.622139	−	0.782907i	$-0.286264\pi$
$691$	32.7082	1.24428	0.622139	+	0.782907i	$0.286264\pi$
$692$	3.97871	0.151248
$693$	0	0
$694$	−93.6656	−3.55550
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−9.38197	−0.355113
$699$	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
$704$	0	0
$705$	0	0
$706$	−81.9574	−3.08451
$707$	0	0
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.98684	−0.149308
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	148.790	5.53740
$723$	0	0
$724$	−11.7295	−0.435923
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	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	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	−15.9787	−0.588983
$737$	0	0
$738$	0	0
$739$	−48.9574	−1.80093	−0.900464	−	0.434930i	$-0.856773\pi$
$739$	−48.9574	−1.80093	−0.900464	+	0.434930i	$0.856773\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.7214	−1.64067	−0.820334	−	0.571885i	$-0.806212\pi$
$743$	−44.7214	−1.64067	−0.820334	+	0.571885i	$0.806212\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	42.9574	1.56754	0.783769	−	0.621052i	$-0.213294\pi$
$751$	42.9574	1.56754	0.783769	+	0.621052i	$0.213294\pi$
$752$	−88.1378	−3.21405
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	82.5755	2.99928
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−98.1033	−3.54462
$767$	0	0
$768$	0	0
$769$	−49.8328	−1.79702	−0.898509	−	0.438956i	$-0.855348\pi$
$769$	−49.8328	−1.79702	−0.898509	+	0.438956i	$0.855348\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	50.2361	1.80687	0.903433	−	0.428730i	$-0.141039\pi$
$773$	50.2361	1.80687	0.903433	+	0.428730i	$0.141039\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−14.5066	−0.518754
$783$	0	0
$784$	−68.9787	−2.46353
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	105.644	3.76342
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	77.6656	2.75279
$797$	48.5967	1.72139	0.860693	−	0.509125i	$-0.170031\pi$
$797$	48.5967	1.72139	0.860693	+	0.509125i	$0.170031\pi$
$798$	0	0
$799$	−33.6656	−1.19100
$800$	0	0
$801$	0	0
$802$	0	0
$803$	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
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−36.2148	−1.26622
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.1115	−0.734117	−0.367059	−	0.930198i	$-0.619635\pi$
$827$	−21.1115	−0.734117	−0.367059	+	0.930198i	$0.619635\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−26.3475	−0.912888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−84.8673	−2.92472
$843$	0	0
$844$	100.520	3.46003
$845$	0	0
$846$	0	0
$847$	0	0
$848$	140.284	4.81736
$849$	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
$856$	133.666	4.56860
$857$	−23.0689	−0.788018	−0.394009	−	0.919107i	$-0.628912\pi$
$857$	−23.0689	−0.788018	−0.394009	+	0.919107i	$0.628912\pi$
$858$	0	0
$859$	55.5410	1.89504	0.947518	−	0.319704i	$-0.103583\pi$
$859$	55.5410	1.89504	0.947518	+	0.319704i	$0.103583\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10.6393	−0.362167	−0.181083	−	0.983468i	$-0.557960\pi$
$863$	−10.6393	−0.362167	−0.181083	+	0.983468i	$0.557960\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	152.554	5.16614
$873$	0	0
$874$	33.5623	1.13526
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	66.8673	2.25666
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	−62.9787	−2.11581
$887$	55.4721	1.86257	0.931286	−	0.364289i	$-0.118688\pi$
$887$	55.4721	1.86257	0.931286	+	0.364289i	$0.118688\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	77.8885	2.60644
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	53.5836	1.78513
$902$	0	0
$903$	0	0
$904$	−33.4164	−1.11141
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	−145.353	−4.82369
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	128.228	4.23677
$917$	0	0
$918$	0	0
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	60.9574	1.99780
$932$	108.541	3.55538
$933$	0	0
$934$	−8.14590	−0.266542
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.0557	1.36663	0.683314	−	0.730125i	$-0.260538\pi$
$947$	42.0557	1.36663	0.683314	+	0.730125i	$0.260538\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−58.1378	−1.88327	−0.941634	−	0.336640i	$-0.890710\pi$
$953$	−58.1378	−1.88327	−0.941634	+	0.336640i	$0.890710\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.6656	−0.763407
$962$	0	0
$963$	0	0
$964$	125.395	4.03870
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−82.1935	−2.64180
$969$	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
$976$	−142.061	−4.54725
$977$	4.47214	0.143076	0.0715382	−	0.997438i	$-0.477209\pi$
$977$	4.47214	0.143076	0.0715382	+	0.997438i	$0.477209\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.3050	−0.902788	−0.451394	−	0.892325i	$-0.649073\pi$
$983$	−28.3050	−0.902788	−0.451394	+	0.892325i	$0.649073\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	30.9574	0.983395	0.491698	−	0.870766i	$-0.336377\pi$
$991$	30.9574	0.983395	0.491698	+	0.870766i	$0.336377\pi$
$992$	29.3951	0.933296
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	40.0344	1.26727
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.a.q.1.2		2
3.2	odd	2		675.2.a.j.1.1		2
5.2	odd	4		135.2.b.a.109.4	yes	4
5.3	odd	4		135.2.b.a.109.1	✓	4
5.4	even	2		675.2.a.j.1.1		2
15.2	even	4		135.2.b.a.109.1	✓	4
15.8	even	4		135.2.b.a.109.4	yes	4
15.14	odd	2	CM	675.2.a.q.1.2		2
20.3	even	4		2160.2.f.j.1729.1		4
20.7	even	4		2160.2.f.j.1729.4		4
45.2	even	12		405.2.j.h.109.1		8
45.7	odd	12		405.2.j.h.109.4		8
45.13	odd	12		405.2.j.h.379.4		8
45.22	odd	12		405.2.j.h.379.1		8
45.23	even	12		405.2.j.h.379.1		8
45.32	even	12		405.2.j.h.379.4		8
45.38	even	12		405.2.j.h.109.4		8
45.43	odd	12		405.2.j.h.109.1		8
60.23	odd	4		2160.2.f.j.1729.4		4
60.47	odd	4		2160.2.f.j.1729.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.b.a.109.1	✓	4	5.3	odd	4
135.2.b.a.109.1	✓	4	15.2	even	4
135.2.b.a.109.4	yes	4	5.2	odd	4
135.2.b.a.109.4	yes	4	15.8	even	4
405.2.j.h.109.1		8	45.2	even	12
405.2.j.h.109.1		8	45.43	odd	12
405.2.j.h.109.4		8	45.7	odd	12
405.2.j.h.109.4		8	45.38	even	12
405.2.j.h.379.1		8	45.22	odd	12
405.2.j.h.379.1		8	45.23	even	12
405.2.j.h.379.4		8	45.13	odd	12
405.2.j.h.379.4		8	45.32	even	12
675.2.a.j.1.1		2	3.2	odd	2
675.2.a.j.1.1		2	5.4	even	2
675.2.a.q.1.2		2	1.1	even	1	trivial
675.2.a.q.1.2		2	15.14	odd	2	CM
2160.2.f.j.1729.1		4	20.3	even	4
2160.2.f.j.1729.1		4	60.47	odd	4
2160.2.f.j.1729.4		4	20.7	even	4
2160.2.f.j.1729.4		4	60.23	odd	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61803 q^{2} +4.85410 q^{4} +7.47214 q^{8} +9.85410 q^{16} +3.76393 q^{17} -8.70820 q^{19} -1.47214 q^{23} +2.70820 q^{31} +10.8541 q^{32} +9.85410 q^{34} -22.7984 q^{38} -3.85410 q^{46} -8.94427 q^{47} -7.00000 q^{49} +14.2361 q^{53} -14.4164 q^{61} +7.09017 q^{62} +8.70820 q^{64} +18.2705 q^{68} -42.2705 q^{76} -14.7082 q^{79} +11.9443 q^{83} -7.14590 q^{92} -23.4164 q^{94} -18.3262 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} + 13 q^{16} + 12 q^{17} - 4 q^{19} + 6 q^{23} - 8 q^{31} + 15 q^{32} + 13 q^{34} - 21 q^{38} - q^{46} - 14 q^{49} + 24 q^{53} - 2 q^{61} + 3 q^{62} + 4 q^{64} + 3 q^{68}+ \cdots - 21 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 + 13 * q^16 + 12 * q^17 - 4 * q^19 + 6 * q^23 - 8 * q^31 + 15 * q^32 + 13 * q^34 - 21 * q^38 - q^46 - 14 * q^49 + 24 * q^53 - 2 * q^61 + 3 * q^62 + 4 * q^64 + 3 * q^68 - 51 * q^76 - 16 * q^79 + 6 * q^83 - 21 * q^92 - 20 * q^94 - 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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