LMFDB - Embedded newform 225.2.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,2,Mod(1,225)] 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("225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,6,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.79663404548$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} +3.00000 q^{4} +2.23607 q^{8} -1.00000 q^{16} -4.47214 q^{17} +4.00000 q^{19} -8.94427 q^{23} +8.00000 q^{31} -6.70820 q^{32} -10.0000 q^{34} +8.94427 q^{38} -20.0000 q^{46} +8.94427 q^{47} -7.00000 q^{49} +4.47214 q^{53} +2.00000 q^{61} +17.8885 q^{62} -13.0000 q^{64} -13.4164 q^{68} +12.0000 q^{76} +16.0000 q^{79} +17.8885 q^{83} -26.8328 q^{92} +20.0000 q^{94} -15.6525 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} - 2 q^{16} + 8 q^{19} + 16 q^{31} - 20 q^{34} - 40 q^{46} - 14 q^{49} + 4 q^{61} - 26 q^{64} + 24 q^{76} + 32 q^{79} + 40 q^{94}+O(q^{100}) $ 2 * q + 6 * q^4 - 2 * q^16 + 8 * q^19 + 16 * q^31 - 20 * q^34 - 40 * q^46 - 14 * q^49 + 4 * q^61 - 26 * q^64 + 24 * q^76 + 32 * q^79 + 40 * q^94

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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	2.23607	0.790569
$9$	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
$16$	−1.00000	−0.250000
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.94427	−1.86501	−0.932505	−	0.361158i	$-0.882382\pi$
$23$	−8.94427	−1.86501	−0.932505	+	0.361158i	$0.882382\pi$
$24$	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
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	−10.0000	−1.71499
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	8.94427	1.45095
$39$	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
$46$	−20.0000	−2.94884
$47$	8.94427	1.30466	0.652328	−	0.757937i	$-0.273792\pi$
$47$	8.94427	1.30466	0.652328	+	0.757937i	$0.273792\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.47214	0.614295	0.307148	−	0.951662i	$-0.400625\pi$
$53$	4.47214	0.614295	0.307148	+	0.951662i	$0.400625\pi$
$54$	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
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	17.8885	2.27185
$63$	0	0
$64$	−13.0000	−1.62500
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−13.4164	−1.62698
$69$	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
$76$	12.0000	1.37649
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	17.8885	1.96352	0.981761	−	0.190117i	$-0.0608868\pi$
$83$	17.8885	1.96352	0.981761	+	0.190117i	$0.0608868\pi$
$84$	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
$92$	−26.8328	−2.79751
$93$	0	0
$94$	20.0000	2.06284
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−15.6525	−1.58114
$99$	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
$106$	10.0000	0.971286
$107$	−17.8885	−1.72935	−0.864675	−	0.502331i	$-0.832476\pi$
$107$	−17.8885	−1.72935	−0.864675	+	0.502331i	$0.832476\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.47214	0.420703	0.210352	−	0.977626i	$-0.432539\pi$
$113$	4.47214	0.420703	0.210352	+	0.977626i	$0.432539\pi$
$114$	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
$122$	4.47214	0.404888
$123$	0	0
$124$	24.0000	2.15526
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−15.6525	−1.38350
$129$	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
$136$	−10.0000	−0.857493
$137$	22.3607	1.91040	0.955201	−	0.295958i	$-0.0956389\pi$
$137$	22.3607	1.91040	0.955201	+	0.295958i	$0.0956389\pi$
$138$	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	$0.394462\pi$
$152$	8.94427	0.725476
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	35.7771	2.84627
$159$	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
$166$	40.0000	3.10460
$167$	8.94427	0.692129	0.346064	−	0.938211i	$-0.387518\pi$
$167$	8.94427	0.692129	0.346064	+	0.938211i	$0.387518\pi$
$168$	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	$-0.823414\pi$
$173$	−22.3607	−1.70005	−0.850026	+	0.526742i	$0.823414\pi$
$174$	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
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	0	0
$184$	−20.0000	−1.47442
$185$	0	0
$186$	0	0
$187$	0	0
$188$	26.8328	1.95698
$189$	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
$196$	−21.0000	−1.50000
$197$	−4.47214	−0.318626	−0.159313	−	0.987228i	$-0.550928\pi$
$197$	−4.47214	−0.318626	−0.159313	+	0.987228i	$0.550928\pi$
$198$	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
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	13.4164	0.921443
$213$	0	0
$214$	−40.0000	−2.73434
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−31.3050	−2.12024
$219$	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
$226$	10.0000	0.665190
$227$	−17.8885	−1.18730	−0.593652	−	0.804722i	$-0.702314\pi$
$227$	−17.8885	−1.18730	−0.593652	+	0.804722i	$0.702314\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.3607	−1.46490	−0.732448	−	0.680823i	$-0.761622\pi$
$233$	−22.3607	−1.46490	−0.732448	+	0.680823i	$0.761622\pi$
$234$	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
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−24.5967	−1.58114
$243$	0	0
$244$	6.00000	0.384111
$245$	0	0
$246$	0	0
$247$	0	0
$248$	17.8885	1.13592
$249$	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
$256$	−9.00000	−0.562500
$257$	−31.3050	−1.95275	−0.976375	−	0.216085i	$-0.930671\pi$
$257$	−31.3050	−1.95275	−0.976375	+	0.216085i	$0.930671\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.94427	−0.551527	−0.275764	−	0.961225i	$-0.588931\pi$
$263$	−8.94427	−0.551527	−0.275764	+	0.961225i	$0.588931\pi$
$264$	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
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	4.47214	0.271163
$273$	0	0
$274$	50.0000	3.02061
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	8.94427	0.536442
$279$	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
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	31.3050	1.82885	0.914427	−	0.404750i	$-0.132641\pi$
$293$	31.3050	1.82885	0.914427	+	0.404750i	$0.132641\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	17.8885	1.02937
$303$	0	0
$304$	−4.00000	−0.229416
$305$	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
$316$	48.0000	2.70021
$317$	22.3607	1.25590	0.627950	−	0.778253i	$-0.283894\pi$
$317$	22.3607	1.25590	0.627950	+	0.778253i	$0.283894\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−17.8885	−0.995345
$324$	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	$0.779499\pi$
$332$	53.6656	2.94528
$333$	0	0
$334$	20.0000	1.09435
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	−29.0689	−1.58114
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−50.0000	−2.68802
$347$	35.7771	1.92061	0.960307	−	0.278944i	$-0.0899844\pi$
$347$	35.7771	1.92061	0.960307	+	0.278944i	$0.0899844\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.3050	1.66619	0.833097	−	0.553127i	$-0.186565\pi$
$353$	31.3050	1.66619	0.833097	+	0.553127i	$0.186565\pi$
$354$	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
$361$	−3.00000	−0.157895
$362$	−49.1935	−2.58555
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	8.94427	0.466252
$369$	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
$376$	20.0000	1.03142
$377$	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
$383$	−8.94427	−0.457031	−0.228515	−	0.973540i	$-0.573387\pi$
$383$	−8.94427	−0.457031	−0.228515	+	0.973540i	$0.573387\pi$
$384$	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
$391$	40.0000	2.02289
$392$	−15.6525	−0.790569
$393$	0	0
$394$	−10.0000	−0.503793
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	35.7771	1.79334
$399$	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
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	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
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	−62.6099	−3.04780
$423$	0	0
$424$	10.0000	0.485643
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−53.6656	−2.59403
$429$	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
$436$	−42.0000	−2.01144
$437$	−35.7771	−1.71145
$438$	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
$443$	17.8885	0.849910	0.424955	−	0.905214i	$-0.360290\pi$
$443$	17.8885	0.849910	0.424955	+	0.905214i	$0.360290\pi$
$444$	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
$452$	13.4164	0.631055
$453$	0	0
$454$	−40.0000	−1.87729
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	−58.1378	−2.71660
$459$	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
$466$	−50.0000	−2.31621
$467$	35.7771	1.65557	0.827783	−	0.561048i	$-0.189602\pi$
$467$	35.7771	1.65557	0.827783	+	0.561048i	$0.189602\pi$
$468$	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
$482$	4.47214	0.203700
$483$	0	0
$484$	−33.0000	−1.50000
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	4.47214	0.202444
$489$	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
$496$	−8.00000	−0.359211
$497$	0	0
$498$	0	0
$499$	−44.0000	−1.96971	−0.984855	−	0.173379i	$-0.944532\pi$
$499$	−44.0000	−1.96971	−0.984855	+	0.173379i	$0.944532\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	44.7214	1.99403	0.997013	−	0.0772283i	$-0.0246070\pi$
$503$	44.7214	1.99403	0.997013	+	0.0772283i	$0.0246070\pi$
$504$	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
$512$	11.1803	0.494106
$513$	0	0
$514$	−70.0000	−3.08757
$515$	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
$526$	−20.0000	−0.872041
$527$	−35.7771	−1.55847
$528$	0	0
$529$	57.0000	2.47826
$530$	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	$0.486311\pi$
$542$	71.5542	3.07352
$543$	0	0
$544$	30.0000	1.28624
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	67.0820	2.86560
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	12.0000	0.508913
$557$	22.3607	0.947452	0.473726	−	0.880672i	$-0.342909\pi$
$557$	22.3607	0.947452	0.473726	+	0.880672i	$0.342909\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−35.7771	−1.50782	−0.753912	−	0.656975i	$-0.771836\pi$
$563$	−35.7771	−1.50782	−0.753912	+	0.656975i	$0.771836\pi$
$564$	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	$0.5\pi$
$578$	6.70820	0.279024
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	70.0000	2.89167
$587$	−17.8885	−0.738339	−0.369170	−	0.929362i	$-0.620358\pi$
$587$	−17.8885	−0.738339	−0.369170	+	0.929362i	$0.620358\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.47214	0.183649	0.0918243	−	0.995775i	$-0.470730\pi$
$593$	4.47214	0.183649	0.0918243	+	0.995775i	$0.470730\pi$
$594$	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
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	0	0
$604$	24.0000	0.976546
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−26.8328	−1.08821
$609$	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
$617$	49.1935	1.98046	0.990228	−	0.139459i	$-0.0445365\pi$
$617$	49.1935	1.98046	0.990228	+	0.139459i	$0.0445365\pi$
$618$	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
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	35.7771	1.42314
$633$	0	0
$634$	50.0000	1.98575
$635$	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
$646$	−40.0000	−1.57378
$647$	−44.7214	−1.75818	−0.879089	−	0.476658i	$-0.841848\pi$
$647$	−44.7214	−1.75818	−0.879089	+	0.476658i	$0.841848\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−49.1935	−1.92509	−0.962545	−	0.271122i	$-0.912605\pi$
$653$	−49.1935	−1.92509	−0.962545	+	0.271122i	$0.912605\pi$
$654$	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	$0.640729\pi$
$662$	−62.6099	−2.43340
$663$	0	0
$664$	40.0000	1.55230
$665$	0	0
$666$	0	0
$667$	0	0
$668$	26.8328	1.03819
$669$	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
$676$	−39.0000	−1.50000
$677$	−31.3050	−1.20315	−0.601574	−	0.798817i	$-0.705459\pi$
$677$	−31.3050	−1.20315	−0.601574	+	0.798817i	$0.705459\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−35.7771	−1.36897	−0.684486	−	0.729026i	$-0.739973\pi$
$683$	−35.7771	−1.36897	−0.684486	+	0.729026i	$0.739973\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−52.0000	−1.97817	−0.989087	−	0.147335i	$-0.952930\pi$
$691$	−52.0000	−1.97817	−0.989087	+	0.147335i	$0.952930\pi$
$692$	−67.0820	−2.55008
$693$	0	0
$694$	80.0000	3.03676
$695$	0	0
$696$	0	0
$697$	0	0
$698$	76.0263	2.87764
$699$	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
$706$	70.0000	2.63448
$707$	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
$713$	−71.5542	−2.67972
$714$	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
$722$	−6.70820	−0.249653
$723$	0	0
$724$	−66.0000	−2.45287
$725$	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
$736$	60.0000	2.21163
$737$	0	0
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.7214	1.64067	0.820334	−	0.571885i	$-0.193788\pi$
$743$	44.7214	1.64067	0.820334	+	0.571885i	$0.193788\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−8.94427	−0.326164
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	8.94427	0.324871
$759$	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
$766$	−20.0000	−0.722629
$767$	0	0
$768$	0	0
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4.47214	0.160852	0.0804258	−	0.996761i	$-0.474372\pi$
$773$	4.47214	0.160852	0.0804258	+	0.996761i	$0.474372\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	89.4427	3.19847
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	−13.4164	−0.477940
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	48.0000	1.70131
$797$	49.1935	1.74252	0.871262	−	0.490819i	$-0.163302\pi$
$797$	49.1935	1.74252	0.871262	+	0.490819i	$0.163302\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	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
$818$	−58.1378	−2.03274
$819$	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
$827$	35.7771	1.24409	0.622046	−	0.782981i	$-0.286302\pi$
$827$	35.7771	1.24409	0.622046	+	0.782981i	$0.286302\pi$
$828$	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
$833$	31.3050	1.08465
$834$	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
$842$	84.9706	2.92828
$843$	0	0
$844$	−84.0000	−2.89140
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−4.47214	−0.153574
$849$	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
$856$	−40.0000	−1.36717
$857$	−58.1378	−1.98595	−0.992974	−	0.118331i	$-0.962245\pi$
$857$	−58.1378	−1.98595	−0.992974	+	0.118331i	$0.962245\pi$
$858$	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	$-0.224627\pi$
$863$	44.7214	1.52233	0.761166	+	0.648557i	$0.224627\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−31.3050	−1.06012
$873$	0	0
$874$	−80.0000	−2.70604
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	35.7771	1.20742
$879$	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
$886$	40.0000	1.34383
$887$	8.94427	0.300319	0.150160	−	0.988662i	$-0.452021\pi$
$887$	8.94427	0.300319	0.150160	+	0.988662i	$0.452021\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	35.7771	1.19723
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−20.0000	−0.666297
$902$	0	0
$903$	0	0
$904$	10.0000	0.332595
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	−53.6656	−1.78096
$909$	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
$916$	−78.0000	−2.57719
$917$	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
$931$	−28.0000	−0.917663
$932$	−67.0820	−2.19735
$933$	0	0
$934$	80.0000	2.61768
$935$	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
$947$	−17.8885	−0.581300	−0.290650	−	0.956830i	$-0.593871\pi$
$947$	−17.8885	−0.581300	−0.290650	+	0.956830i	$0.593871\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	58.1378	1.88327	0.941634	−	0.336640i	$-0.109290\pi$
$953$	58.1378	1.88327	0.941634	+	0.336640i	$0.109290\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	6.00000	0.193247
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−24.5967	−0.790569
$969$	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
$976$	−2.00000	−0.0640184
$977$	−4.47214	−0.143076	−0.0715382	−	0.997438i	$-0.522791\pi$
$977$	−4.47214	−0.143076	−0.0715382	+	0.997438i	$0.522791\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−62.6099	−1.99695	−0.998473	−	0.0552438i	$-0.982406\pi$
$983$	−62.6099	−1.99695	−0.998473	+	0.0552438i	$0.982406\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−53.6656	−1.70389
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−98.3870	−3.11439
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.2.a.f.1.2		2
3.2	odd	2	inner	225.2.a.f.1.1		2
4.3	odd	2		3600.2.a.bs.1.1		2
5.2	odd	4		45.2.b.a.19.2	yes	2
5.3	odd	4		45.2.b.a.19.1	✓	2
5.4	even	2	inner	225.2.a.f.1.1		2
12.11	even	2		3600.2.a.bs.1.2		2
15.2	even	4		45.2.b.a.19.1	✓	2
15.8	even	4		45.2.b.a.19.2	yes	2
15.14	odd	2	CM	225.2.a.f.1.2		2
20.3	even	4		720.2.f.d.289.2		2
20.7	even	4		720.2.f.d.289.1		2
20.19	odd	2		3600.2.a.bs.1.2		2
35.13	even	4		2205.2.d.a.1324.1		2
35.27	even	4		2205.2.d.a.1324.2		2
40.3	even	4		2880.2.f.j.1729.1		2
40.13	odd	4		2880.2.f.k.1729.1		2
40.27	even	4		2880.2.f.j.1729.2		2
40.37	odd	4		2880.2.f.k.1729.2		2
45.2	even	12		405.2.j.c.109.1		4
45.7	odd	12		405.2.j.c.109.2		4
45.13	odd	12		405.2.j.c.379.2		4
45.22	odd	12		405.2.j.c.379.1		4
45.23	even	12		405.2.j.c.379.1		4
45.32	even	12		405.2.j.c.379.2		4
45.38	even	12		405.2.j.c.109.2		4
45.43	odd	12		405.2.j.c.109.1		4
60.23	odd	4		720.2.f.d.289.1		2
60.47	odd	4		720.2.f.d.289.2		2
60.59	even	2		3600.2.a.bs.1.1		2
105.62	odd	4		2205.2.d.a.1324.1		2
105.83	odd	4		2205.2.d.a.1324.2		2
120.53	even	4		2880.2.f.k.1729.2		2
120.77	even	4		2880.2.f.k.1729.1		2
120.83	odd	4		2880.2.f.j.1729.2		2
120.107	odd	4		2880.2.f.j.1729.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.b.a.19.1	✓	2	5.3	odd	4
45.2.b.a.19.1	✓	2	15.2	even	4
45.2.b.a.19.2	yes	2	5.2	odd	4
45.2.b.a.19.2	yes	2	15.8	even	4
225.2.a.f.1.1		2	3.2	odd	2	inner
225.2.a.f.1.1		2	5.4	even	2	inner
225.2.a.f.1.2		2	1.1	even	1	trivial
225.2.a.f.1.2		2	15.14	odd	2	CM
405.2.j.c.109.1		4	45.2	even	12
405.2.j.c.109.1		4	45.43	odd	12
405.2.j.c.109.2		4	45.7	odd	12
405.2.j.c.109.2		4	45.38	even	12
405.2.j.c.379.1		4	45.22	odd	12
405.2.j.c.379.1		4	45.23	even	12
405.2.j.c.379.2		4	45.13	odd	12
405.2.j.c.379.2		4	45.32	even	12
720.2.f.d.289.1		2	20.7	even	4
720.2.f.d.289.1		2	60.23	odd	4
720.2.f.d.289.2		2	20.3	even	4
720.2.f.d.289.2		2	60.47	odd	4
2205.2.d.a.1324.1		2	35.13	even	4
2205.2.d.a.1324.1		2	105.62	odd	4
2205.2.d.a.1324.2		2	35.27	even	4
2205.2.d.a.1324.2		2	105.83	odd	4
2880.2.f.j.1729.1		2	40.3	even	4
2880.2.f.j.1729.1		2	120.107	odd	4
2880.2.f.j.1729.2		2	40.27	even	4
2880.2.f.j.1729.2		2	120.83	odd	4
2880.2.f.k.1729.1		2	40.13	odd	4
2880.2.f.k.1729.1		2	120.77	even	4
2880.2.f.k.1729.2		2	40.37	odd	4
2880.2.f.k.1729.2		2	120.53	even	4
3600.2.a.bs.1.1		2	4.3	odd	2
3600.2.a.bs.1.1		2	60.59	even	2
3600.2.a.bs.1.2		2	12.11	even	2
3600.2.a.bs.1.2		2	20.19	odd	2

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +3.00000 q^{4} +2.23607 q^{8} -1.00000 q^{16} -4.47214 q^{17} +4.00000 q^{19} -8.94427 q^{23} +8.00000 q^{31} -6.70820 q^{32} -10.0000 q^{34} +8.94427 q^{38} -20.0000 q^{46} +8.94427 q^{47} -7.00000 q^{49} +4.47214 q^{53} +2.00000 q^{61} +17.8885 q^{62} -13.0000 q^{64} -13.4164 q^{68} +12.0000 q^{76} +16.0000 q^{79} +17.8885 q^{83} -26.8328 q^{92} +20.0000 q^{94} -15.6525 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} - 2 q^{16} + 8 q^{19} + 16 q^{31} - 20 q^{34} - 40 q^{46} - 14 q^{49} + 4 q^{61} - 26 q^{64} + 24 q^{76} + 32 q^{79} + 40 q^{94}+O(q^{100}) \) 2 * q + 6 * q^4 - 2 * q^16 + 8 * q^19 + 16 * q^31 - 20 * q^34 - 40 * q^46 - 14 * q^49 + 4 * q^61 - 26 * q^64 + 24 * q^76 + 32 * q^79 + 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more