LMFDB - Embedded newform 1125.2.a.g.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1125,2,Mod(1,1125)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1125.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1125 = 3^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1125.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-5,0,7,0,0,0,-15,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.98317022739$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.4
Root			$1.82709$ of defining polynomial
Character	$\chi$	$=$	1125.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.956295 q^{2} -1.08550 q^{4} -2.95065 q^{8} -0.650692 q^{16} +0.454262 q^{17} -6.13078 q^{19} -9.27222 q^{23} -1.91917 q^{31} +5.27904 q^{32} +0.434408 q^{34} -5.86284 q^{38} -8.86698 q^{46} -12.6476 q^{47} -7.00000 q^{49} -14.5602 q^{53} +15.3517 q^{61} -1.83529 q^{62} +6.34971 q^{64} -0.493101 q^{68} +6.65496 q^{76} +12.3111 q^{79} -8.82242 q^{83} +10.0650 q^{92} -12.0948 q^{94} -6.69407 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{2} + 7 q^{4} - 15 q^{8} + 21 q^{16} - 10 q^{17} - 4 q^{19} - 20 q^{23} - 8 q^{31} - 30 q^{32} + 10 q^{34} - 5 q^{38} + 20 q^{46} - 20 q^{47} - 28 q^{49} - 10 q^{53} - 2 q^{61} + 30 q^{62} + 53 q^{64}+ \cdots + 35 q^{98}+O(q^{100}) $ 4 * q - 5 * q^2 + 7 * q^4 - 15 * q^8 + 21 * q^16 - 10 * q^17 - 4 * q^19 - 20 * q^23 - 8 * q^31 - 30 * q^32 + 10 * q^34 - 5 * q^38 + 20 * q^46 - 20 * q^47 - 28 * q^49 - 10 * q^53 - 2 * q^61 + 30 * q^62 + 53 * q^64 + 15 * q^68 - 12 * q^76 - 16 * q^79 - 40 * q^83 - 45 * q^92 - 15 * q^94 + 35 * 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 $

$2$	0.956295	0.676203	0.338101	−	0.941110i	$-0.390215\pi$
$2$	0.956295	0.676203	0.338101	+	0.941110i	$0.390215\pi$
$3$	0	0
$3$	0	0
$4$	−1.08550	−0.542750
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−2.95065	−1.04321
$9$	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
$16$	−0.650692	−0.162673
$17$	0.454262	0.110175	0.0550873	−	0.998482i	$-0.482456\pi$
$17$	0.454262	0.110175	0.0550873	+	0.998482i	$0.482456\pi$
$18$	0	0
$19$	−6.13078	−1.40650	−0.703249	−	0.710943i	$-0.748268\pi$
$19$	−6.13078	−1.40650	−0.703249	+	0.710943i	$0.748268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.27222	−1.93339	−0.966695	−	0.255930i	$-0.917618\pi$
$23$	−9.27222	−1.93339	−0.966695	+	0.255930i	$0.917618\pi$
$24$	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
$31$	−1.91917	−0.344693	−0.172347	−	0.985036i	$-0.555135\pi$
$31$	−1.91917	−0.344693	−0.172347	+	0.985036i	$0.555135\pi$
$32$	5.27904	0.933212
$33$	0	0
$34$	0.434408	0.0745004
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−5.86284	−0.951078
$39$	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
$46$	−8.86698	−1.30736
$47$	−12.6476	−1.84484	−0.922421	−	0.386186i	$-0.873792\pi$
$47$	−12.6476	−1.84484	−0.922421	+	0.386186i	$0.873792\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−14.5602	−2.00000	−0.999998	−	0.00196501i	$-0.999375\pi$
$53$	−14.5602	−2.00000	−0.999998	+	0.00196501i	$0.999375\pi$
$54$	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
$61$	15.3517	1.96559	0.982794	−	0.184703i	$-0.0591324\pi$
$61$	15.3517	1.96559	0.982794	+	0.184703i	$0.0591324\pi$
$62$	−1.83529	−0.233083
$63$	0	0
$64$	6.34971	0.793713
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−0.493101	−0.0597973
$69$	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
$76$	6.65496	0.763377
$77$	0	0
$78$	0	0
$79$	12.3111	1.38511	0.692555	−	0.721365i	$-0.256485\pi$
$79$	12.3111	1.38511	0.692555	+	0.721365i	$0.256485\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.82242	−0.968386	−0.484193	−	0.874961i	$-0.660887\pi$
$83$	−8.82242	−0.968386	−0.484193	+	0.874961i	$0.660887\pi$
$84$	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
$92$	10.0650	1.04935
$93$	0	0
$94$	−12.0948	−1.24749
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−6.69407	−0.676203
$99$	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
$106$	−13.9238	−1.35240
$107$	−8.36369	−0.808549	−0.404274	−	0.914638i	$-0.632476\pi$
$107$	−8.36369	−0.808549	−0.404274	+	0.914638i	$0.632476\pi$
$108$	0	0
$109$	20.4322	1.95705	0.978523	−	0.206136i	$-0.0660889\pi$
$109$	20.4322	1.95705	0.978523	+	0.206136i	$0.0660889\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.3854	1.72955	0.864775	−	0.502159i	$-0.167461\pi$
$113$	18.3854	1.72955	0.864775	+	0.502159i	$0.167461\pi$
$114$	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
$122$	14.6808	1.32914
$123$	0	0
$124$	2.08326	0.187082
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−4.48589	−0.396501
$129$	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
$136$	−1.34037	−0.114935
$137$	−22.3607	−1.91040	−0.955201	−	0.295958i	$-0.904361\pi$
$137$	−22.3607	−1.91040	−0.955201	+	0.295958i	$0.904361\pi$
$138$	0	0
$139$	10.4228	0.884053	0.442026	−	0.897002i	$-0.354260\pi$
$139$	10.4228	0.884053	0.442026	+	0.897002i	$0.354260\pi$
$140$	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
$151$	24.5727	1.99970	0.999849	−	0.0173966i	$-0.00553779\pi$
$151$	24.5727	1.99970	0.999849	+	0.0173966i	$0.00553779\pi$
$152$	18.0898	1.46728
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	11.7731	0.936615
$159$	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
$166$	−8.43684	−0.654826
$167$	20.2980	1.57070	0.785352	−	0.619050i	$-0.212482\pi$
$167$	20.2980	1.57070	0.785352	+	0.619050i	$0.212482\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.3607	1.70005	0.850026	−	0.526742i	$-0.176586\pi$
$173$	22.3607	1.70005	0.850026	+	0.526742i	$0.176586\pi$
$174$	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
$181$	−21.5321	−1.60047	−0.800233	−	0.599689i	$-0.795291\pi$
$181$	−21.5321	−1.60047	−0.800233	+	0.599689i	$0.795291\pi$
$182$	0	0
$183$	0	0
$184$	27.3590	2.01694
$185$	0	0
$186$	0	0
$187$	0	0
$188$	13.7290	1.00129
$189$	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
$196$	7.59850	0.542750
$197$	−19.9072	−1.41833	−0.709165	−	0.705042i	$-0.750928\pi$
$197$	−19.9072	−1.41833	−0.709165	+	0.705042i	$0.750928\pi$
$198$	0	0
$199$	−26.6032	−1.88585	−0.942924	−	0.333007i	$-0.891937\pi$
$199$	−26.6032	−1.88585	−0.942924	+	0.333007i	$0.891937\pi$
$200$	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
$211$	18.0995	1.24602	0.623011	−	0.782213i	$-0.285909\pi$
$211$	18.0995	1.24602	0.623011	+	0.782213i	$0.285909\pi$
$212$	15.8051	1.08550
$213$	0	0
$214$	−7.99816	−0.546743
$215$	0	0
$216$	0	0
$217$	0	0
$218$	19.5392	1.32336
$219$	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
$226$	17.5818	1.16953
$227$	−28.7252	−1.90656	−0.953278	−	0.302094i	$-0.902314\pi$
$227$	−28.7252	−1.90656	−0.953278	+	0.302094i	$0.902314\pi$
$228$	0	0
$229$	11.9285	0.788258	0.394129	−	0.919055i	$-0.371046\pi$
$229$	11.9285	0.788258	0.394129	+	0.919055i	$0.371046\pi$
$230$	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
$241$	16.5938	1.06890	0.534451	−	0.845199i	$-0.320518\pi$
$241$	16.5938	1.06890	0.534451	+	0.845199i	$0.320518\pi$
$242$	−10.5192	−0.676203
$243$	0	0
$244$	−16.6643	−1.06682
$245$	0	0
$246$	0	0
$247$	0	0
$248$	5.66280	0.359588
$249$	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
$256$	−16.9893	−1.06183
$257$	3.08465	0.192415	0.0962076	−	0.995361i	$-0.469329\pi$
$257$	3.08465	0.192415	0.0962076	+	0.995361i	$0.469329\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.0893	0.683793	0.341897	−	0.939738i	$-0.388931\pi$
$263$	11.0893	0.683793	0.341897	+	0.939738i	$0.388931\pi$
$264$	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
$271$	−30.4415	−1.84919	−0.924595	−	0.380952i	$-0.875596\pi$
$271$	−30.4415	−1.84919	−0.924595	+	0.380952i	$0.875596\pi$
$272$	−0.295584	−0.0179224
$273$	0	0
$274$	−21.3834	−1.29182
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	9.96730	0.597799
$279$	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
$289$	−16.7936	−0.987862
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.1816	1.00376	0.501881	−	0.864937i	$-0.332641\pi$
$293$	17.1816	1.00376	0.501881	+	0.864937i	$0.332641\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	23.4987	1.35220
$303$	0	0
$304$	3.98925	0.228799
$305$	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
$316$	−13.3637	−0.751768
$317$	−22.3607	−1.25590	−0.627950	−	0.778253i	$-0.716106\pi$
$317$	−22.3607	−1.25590	−0.627950	+	0.778253i	$0.716106\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.78498	−0.154960
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−30.7530	−1.69034	−0.845170	−	0.534498i	$-0.820501\pi$
$331$	−30.7530	−1.69034	−0.845170	+	0.534498i	$0.820501\pi$
$332$	9.57673	0.525591
$333$	0	0
$334$	19.4108	1.06211
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	−12.4318	−0.676203
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	21.3834	1.14958
$347$	−1.17206	−0.0629194	−0.0314597	−	0.999505i	$-0.510016\pi$
$347$	−1.17206	−0.0629194	−0.0314597	+	0.999505i	$0.510016\pi$
$348$	0	0
$349$	−36.6125	−1.95982	−0.979911	−	0.199434i	$-0.936090\pi$
$349$	−36.6125	−1.95982	−0.979911	+	0.199434i	$0.936090\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	37.5431	1.99822	0.999109	−	0.0421935i	$-0.0134346\pi$
$353$	37.5431	1.99822	0.999109	+	0.0421935i	$0.0134346\pi$
$354$	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
$361$	18.5865	0.978238
$362$	−20.5910	−1.08224
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	6.03336	0.314510
$369$	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
$376$	37.3186	1.92456
$377$	0	0
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−29.6337	−1.51421	−0.757106	−	0.653293i	$-0.773387\pi$
$383$	−29.6337	−1.51421	−0.757106	+	0.653293i	$0.773387\pi$
$384$	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
$391$	−4.21201	−0.213011
$392$	20.6545	1.04321
$393$	0	0
$394$	−19.0372	−0.959079
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−25.4405	−1.27522
$399$	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
$409$	21.4330	1.05979	0.529896	−	0.848063i	$-0.322231\pi$
$409$	21.4330	1.05979	0.529896	+	0.848063i	$0.322231\pi$
$410$	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
$421$	−2.99106	−0.145775	−0.0728876	−	0.997340i	$-0.523221\pi$
$421$	−2.99106	−0.145775	−0.0728876	+	0.997340i	$0.523221\pi$
$422$	17.3085	0.842563
$423$	0	0
$424$	42.9620	2.08642
$425$	0	0
$426$	0	0
$427$	0	0
$428$	9.07878	0.438840
$429$	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
$436$	−22.1791	−1.06219
$437$	56.8460	2.71931
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−41.7680	−1.98446	−0.992228	−	0.124430i	$-0.960290\pi$
$443$	−41.7680	−1.98446	−0.992228	+	0.124430i	$0.960290\pi$
$444$	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
$452$	−19.9573	−0.938713
$453$	0	0
$454$	−27.4697	−1.28922
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	11.4072	0.533022
$459$	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
$466$	21.3834	0.990567
$467$	−34.1176	−1.57878	−0.789388	−	0.613895i	$-0.789602\pi$
$467$	−34.1176	−1.57878	−0.789388	+	0.613895i	$0.789602\pi$
$468$	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
$482$	15.8686	0.722795
$483$	0	0
$484$	11.9405	0.542750
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−45.2976	−2.05053
$489$	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
$496$	1.24879	0.0560723
$497$	0	0
$498$	0	0
$499$	−6.22990	−0.278888	−0.139444	−	0.990230i	$-0.544532\pi$
$499$	−6.22990	−0.278888	−0.139444	+	0.990230i	$0.544532\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−44.7214	−1.99403	−0.997013	−	0.0772283i	$-0.975393\pi$
$503$	−44.7214	−1.99403	−0.997013	+	0.0772283i	$0.975393\pi$
$504$	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
$512$	−7.27496	−0.321511
$513$	0	0
$514$	2.94984	0.130112
$515$	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
$526$	10.6046	0.462383
$527$	−0.871806	−0.0379765
$528$	0	0
$529$	62.9740	2.73800
$530$	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
$541$	−28.9358	−1.24405	−0.622024	−	0.782998i	$-0.713689\pi$
$541$	−28.9358	−1.24405	−0.622024	+	0.782998i	$0.713689\pi$
$542$	−29.1111	−1.25043
$543$	0	0
$544$	2.39807	0.102816
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	24.2725	1.03687
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−11.3140	−0.479819
$557$	−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
$563$	−47.2696	−1.99218	−0.996088	−	0.0883656i	$-0.971836\pi$
$563$	−47.2696	−1.99218	−0.996088	+	0.0883656i	$0.971836\pi$
$564$	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
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	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	$0.5\pi$
$578$	−16.0597	−0.667995
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	16.4307	0.678747
$587$	11.9978	0.495202	0.247601	−	0.968862i	$-0.420358\pi$
$587$	11.9978	0.495202	0.247601	+	0.968862i	$0.420358\pi$
$588$	0	0
$589$	11.7660	0.484811
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−47.5058	−1.95083	−0.975414	−	0.220381i	$-0.929270\pi$
$593$	−47.5058	−1.95083	−0.975414	+	0.220381i	$0.929270\pi$
$594$	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
$601$	−48.9545	−1.99690	−0.998448	−	0.0556925i	$-0.982263\pi$
$601$	−48.9545	−1.99690	−0.998448	+	0.0556925i	$0.982263\pi$
$602$	0	0
$603$	0	0
$604$	−26.6736	−1.08534
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−32.3647	−1.31256
$609$	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
$617$	35.7261	1.43828	0.719139	−	0.694866i	$-0.244536\pi$
$617$	35.7261	1.43828	0.719139	+	0.694866i	$0.244536\pi$
$618$	0	0
$619$	21.9379	0.881757	0.440878	−	0.897567i	$-0.354667\pi$
$619$	21.9379	0.881757	0.440878	+	0.897567i	$0.354667\pi$
$620$	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
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	−36.3258	−1.44496
$633$	0	0
$634$	−21.3834	−0.849244
$635$	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
$646$	−2.66326	−0.104785
$647$	44.7214	1.75818	0.879089	−	0.476658i	$-0.158152\pi$
$647$	44.7214	1.75818	0.879089	+	0.476658i	$0.158152\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.3799	1.11059	0.555295	−	0.831654i	$-0.312605\pi$
$653$	28.3799	1.11059	0.555295	+	0.831654i	$0.312605\pi$
$654$	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
$661$	45.1162	1.75482	0.877408	−	0.479745i	$-0.159271\pi$
$661$	45.1162	1.75482	0.877408	+	0.479745i	$0.159271\pi$
$662$	−29.4090	−1.14301
$663$	0	0
$664$	26.0319	1.01023
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−22.0334	−0.852499
$669$	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
$676$	14.1115	0.542750
$677$	−29.8609	−1.14765	−0.573824	−	0.818979i	$-0.694541\pi$
$677$	−29.8609	−1.14765	−0.573824	+	0.818979i	$0.694541\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.54665	−0.250500	−0.125250	−	0.992125i	$-0.539973\pi$
$683$	−6.54665	−0.250500	−0.125250	+	0.992125i	$0.539973\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	46.6218	1.77358	0.886789	−	0.462174i	$-0.152930\pi$
$691$	46.6218	1.77358	0.886789	+	0.462174i	$0.152930\pi$
$692$	−24.2725	−0.922702
$693$	0	0
$694$	−1.12083	−0.0425463
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−35.0124	−1.32524
$699$	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
$706$	35.9023	1.35120
$707$	0	0
$708$	0	0
$709$	−52.2356	−1.96175	−0.980874	−	0.194645i	$-0.937644\pi$
$709$	−52.2356	−1.96175	−0.980874	+	0.194645i	$0.937644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	17.7950	0.666427
$714$	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
$722$	17.7742	0.661487
$723$	0	0
$724$	23.3731	0.868653
$725$	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
$736$	−48.9484	−1.80426
$737$	0	0
$738$	0	0
$739$	−35.1068	−1.29143	−0.645713	−	0.763580i	$-0.723440\pi$
$739$	−35.1068	−1.29143	−0.645713	+	0.763580i	$0.723440\pi$
$740$	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
$751$	−49.0958	−1.79153	−0.895766	−	0.444526i	$-0.853372\pi$
$751$	−49.0958	−1.79153	−0.895766	+	0.444526i	$0.853372\pi$
$752$	8.22969	0.300106
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	3.82518	0.138937
$759$	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
$766$	−28.3386	−1.02391
$767$	0	0
$768$	0	0
$769$	−15.2526	−0.550024	−0.275012	−	0.961441i	$-0.588682\pi$
$769$	−15.2526	−0.550024	−0.275012	+	0.961441i	$0.588682\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	51.3309	1.84625	0.923123	−	0.384505i	$-0.125628\pi$
$773$	51.3309	1.84625	0.923123	+	0.384505i	$0.125628\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−4.02793	−0.144038
$783$	0	0
$784$	4.55484	0.162673
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	21.6093	0.769798
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	28.8777	1.02354
$797$	56.0876	1.98672	0.993362	−	0.115034i	$-0.0366976\pi$
$797$	56.0876	1.98672	0.993362	+	0.115034i	$0.0366976\pi$
$798$	0	0
$799$	−5.74532	−0.203255
$800$	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
$811$	6.03167	0.211801	0.105900	−	0.994377i	$-0.466228\pi$
$811$	6.03167	0.211801	0.105900	+	0.994377i	$0.466228\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	20.4962	0.716634
$819$	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
$827$	31.7735	1.10487	0.552437	−	0.833555i	$-0.313698\pi$
$827$	31.7735	1.10487	0.552437	+	0.833555i	$0.313698\pi$
$828$	0	0
$829$	−33.6945	−1.17026	−0.585130	−	0.810940i	$-0.698956\pi$
$829$	−33.6945	−1.17026	−0.585130	+	0.810940i	$0.698956\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.17983	−0.110175
$834$	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
$842$	−2.86033	−0.0985736
$843$	0	0
$844$	−19.6470	−0.676278
$845$	0	0
$846$	0	0
$847$	0	0
$848$	9.47420	0.325345
$849$	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
$856$	24.6783	0.843487
$857$	24.5547	0.838772	0.419386	−	0.907808i	$-0.362245\pi$
$857$	24.5547	0.838772	0.419386	+	0.907808i	$0.362245\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44.7214	−1.52233	−0.761166	−	0.648557i	$-0.775373\pi$
$863$	−44.7214	−1.52233	−0.761166	+	0.648557i	$0.775373\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−60.2881	−2.04161
$873$	0	0
$874$	54.3615	1.83881
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	15.3007	0.516374
$879$	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
$886$	−39.9425	−1.34190
$887$	53.2435	1.78774	0.893871	−	0.448324i	$-0.147979\pi$
$887$	53.2435	1.78774	0.893871	+	0.448324i	$0.147979\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	77.5397	2.59477
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−6.61414	−0.220349
$902$	0	0
$903$	0	0
$904$	−54.2488	−1.80429
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	31.1812	1.03478
$909$	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
$916$	−12.9484	−0.427827
$917$	0	0
$918$	0	0
$919$	58.9638	1.94504	0.972519	−	0.232824i	$-0.0747966\pi$
$919$	58.9638	1.94504	0.972519	+	0.232824i	$0.0747966\pi$
$920$	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
$931$	42.9155	1.40650
$932$	−24.2725	−0.795072
$933$	0	0
$934$	−32.6265	−1.06757
$935$	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
$947$	−49.0866	−1.59510	−0.797551	−	0.603252i	$-0.793871\pi$
$947$	−49.0866	−1.59510	−0.797551	+	0.603252i	$0.793871\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	34.8176	1.12785	0.563926	−	0.825826i	$-0.309290\pi$
$953$	34.8176	1.12785	0.563926	+	0.825826i	$0.309290\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.3168	−0.881187
$962$	0	0
$963$	0	0
$964$	−18.0126	−0.580146
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	32.4571	1.04321
$969$	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
$976$	−9.98925	−0.319748
$977$	−40.2687	−1.28831	−0.644155	−	0.764895i	$-0.722791\pi$
$977$	−40.2687	−1.28831	−0.644155	+	0.764895i	$0.722791\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.6421	0.722170	0.361085	−	0.932533i	$-0.382406\pi$
$983$	22.6421	0.722170	0.361085	+	0.932533i	$0.382406\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	61.4565	1.95223	0.976115	−	0.217254i	$-0.0697099\pi$
$991$	61.4565	1.95223	0.976115	+	0.217254i	$0.0697099\pi$
$992$	−10.1314	−0.321672
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−5.95762	−0.188585
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1125.2.a.g.1.4	✓	4
3.2	odd	2		1125.2.a.m.1.1	yes	4
5.2	odd	4		1125.2.b.h.874.6		8
5.3	odd	4		1125.2.b.h.874.3		8
5.4	even	2		1125.2.a.m.1.1	yes	4
15.2	even	4		1125.2.b.h.874.3		8
15.8	even	4		1125.2.b.h.874.6		8
15.14	odd	2	CM	1125.2.a.g.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1125.2.a.g.1.4	✓	4	1.1	even	1	trivial
1125.2.a.g.1.4	✓	4	15.14	odd	2	CM
1125.2.a.m.1.1	yes	4	3.2	odd	2
1125.2.a.m.1.1	yes	4	5.4	even	2
1125.2.b.h.874.3		8	5.3	odd	4
1125.2.b.h.874.3		8	15.2	even	4
1125.2.b.h.874.6		8	5.2	odd	4
1125.2.b.h.874.6		8	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1125 = 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1125.a (trivial)

\(f(q)\)	\(=\)	\(q+0.956295 q^{2} -1.08550 q^{4} -2.95065 q^{8} -0.650692 q^{16} +0.454262 q^{17} -6.13078 q^{19} -9.27222 q^{23} -1.91917 q^{31} +5.27904 q^{32} +0.434408 q^{34} -5.86284 q^{38} -8.86698 q^{46} -12.6476 q^{47} -7.00000 q^{49} -14.5602 q^{53} +15.3517 q^{61} -1.83529 q^{62} +6.34971 q^{64} -0.493101 q^{68} +6.65496 q^{76} +12.3111 q^{79} -8.82242 q^{83} +10.0650 q^{92} -12.0948 q^{94} -6.69407 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} + 7 q^{4} - 15 q^{8} + 21 q^{16} - 10 q^{17} - 4 q^{19} - 20 q^{23} - 8 q^{31} - 30 q^{32} + 10 q^{34} - 5 q^{38} + 20 q^{46} - 20 q^{47} - 28 q^{49} - 10 q^{53} - 2 q^{61} + 30 q^{62} + 53 q^{64}+ \cdots + 35 q^{98}+O(q^{100}) \) 4 * q - 5 * q^2 + 7 * q^4 - 15 * q^8 + 21 * q^16 - 10 * q^17 - 4 * q^19 - 20 * q^23 - 8 * q^31 - 30 * q^32 + 10 * q^34 - 5 * q^38 + 20 * q^46 - 20 * q^47 - 28 * q^49 - 10 * q^53 - 2 * q^61 + 30 * q^62 + 53 * q^64 + 15 * q^68 - 12 * q^76 - 16 * q^79 - 40 * q^83 - 45 * q^92 - 15 * q^94 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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