LMFDB - Embedded newform 3375.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3375,2,Mod(1,3375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3375, 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("3375.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3375 = 3^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3375.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:	$26.9495106822$
Analytic rank:	$0$
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			1.1
Root			$-1.95630$ of defining polynomial
Character	$\chi$	$=$	3375.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.74724 q^{2} +5.54732 q^{4} -9.74533 q^{8} +O(q^{10})$	$q-2.74724 q^{2} +5.54732 q^{4} -9.74533 q^{8} +15.6781 q^{16} -5.81486 q^{17} +6.80475 q^{19} +8.55922 q^{23} -9.43566 q^{31} -23.5808 q^{32} +15.9748 q^{34} -18.6943 q^{38} -23.5142 q^{46} +13.3445 q^{47} -7.00000 q^{49} -13.3130 q^{53} +15.1980 q^{61} +25.9220 q^{62} +33.4259 q^{64} -32.2569 q^{68} +37.7481 q^{76} +6.03108 q^{79} -1.57526 q^{83} +47.4807 q^{92} -36.6606 q^{94} +19.2307 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 16 q^{4} - 3 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 16 * q^4 - 3 * q^8	$ 4 q - 4 q^{2} + 16 q^{4} - 3 q^{8} + 36 q^{16} - 11 q^{17} + 17 q^{19} - 13 q^{23} + 19 q^{31} + q^{34} + 8 q^{38} + 8 q^{46} + 20 q^{47} - 28 q^{49} - 17 q^{53} + 31 q^{61} + 21 q^{62} + 83 q^{64} - 99 q^{68} + 63 q^{76} + 23 q^{79} - 23 q^{83} - 12 q^{92} + 15 q^{94} + 28 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 16 * q^4 - 3 * q^8 + 36 * q^16 - 11 * q^17 + 17 * q^19 - 13 * q^23 + 19 * q^31 + q^34 + 8 * q^38 + 8 * q^46 + 20 * q^47 - 28 * q^49 - 17 * q^53 + 31 * q^61 + 21 * q^62 + 83 * q^64 - 99 * q^68 + 63 * q^76 + 23 * q^79 - 23 * q^83 - 12 * q^92 + 15 * q^94 + 28 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.74724	−1.94259	−0.971295	−	0.237877i	$-0.923549\pi$
$2$	−2.74724	−1.94259	−0.971295	+	0.237877i	$0.923549\pi$
$3$	0	0
$3$	0	0
$4$	5.54732	2.77366
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−9.74533	−3.44549
$9$	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
$16$	15.6781	3.91953
$17$	−5.81486	−1.41031	−0.705156	−	0.709052i	$-0.749123\pi$
$17$	−5.81486	−1.41031	−0.705156	+	0.709052i	$0.749123\pi$
$18$	0	0
$19$	6.80475	1.56112	0.780558	−	0.625083i	$-0.214935\pi$
$19$	6.80475	1.56112	0.780558	+	0.625083i	$0.214935\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.55922	1.78472	0.892360	−	0.451325i	$-0.149048\pi$
$23$	8.55922	1.78472	0.892360	+	0.451325i	$0.149048\pi$
$24$	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
$31$	−9.43566	−1.69469	−0.847347	−	0.531039i	$-0.821802\pi$
$31$	−9.43566	−1.69469	−0.847347	+	0.531039i	$0.821802\pi$
$32$	−23.5808	−4.16854
$33$	0	0
$34$	15.9748	2.73966
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−18.6943	−3.03261
$39$	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
$46$	−23.5142	−3.46698
$47$	13.3445	1.94650	0.973249	−	0.229755i	$-0.0737924\pi$
$47$	13.3445	1.94650	0.973249	+	0.229755i	$0.0737924\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.3130	−1.82869	−0.914343	−	0.404941i	$-0.867292\pi$
$53$	−13.3130	−1.82869	−0.914343	+	0.404941i	$0.867292\pi$
$54$	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
$61$	15.1980	1.94591	0.972953	−	0.231004i	$-0.0742009\pi$
$61$	15.1980	1.94591	0.972953	+	0.231004i	$0.0742009\pi$
$62$	25.9220	3.29210
$63$	0	0
$64$	33.4259	4.17824
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−32.2569	−3.91172
$69$	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
$76$	37.7481	4.33001
$77$	0	0
$78$	0	0
$79$	6.03108	0.678549	0.339275	−	0.940687i	$-0.389818\pi$
$79$	6.03108	0.678549	0.339275	+	0.940687i	$0.389818\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.57526	−0.172908	−0.0864538	−	0.996256i	$-0.527553\pi$
$83$	−1.57526	−0.172908	−0.0864538	+	0.996256i	$0.527553\pi$
$84$	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
$92$	47.4807	4.95020
$93$	0	0
$94$	−36.6606	−3.78125
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	19.2307	1.94259
$99$	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
$106$	36.5741	3.55239
$107$	15.4115	1.48989	0.744945	−	0.667126i	$-0.232476\pi$
$107$	15.4115	1.48989	0.744945	+	0.667126i	$0.232476\pi$
$108$	0	0
$109$	10.4731	1.00314	0.501571	−	0.865116i	$-0.332756\pi$
$109$	10.4731	1.00314	0.501571	+	0.865116i	$0.332756\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.59885	−0.808912	−0.404456	−	0.914557i	$-0.632539\pi$
$113$	−8.59885	−0.808912	−0.404456	+	0.914557i	$0.632539\pi$
$114$	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
$122$	−41.7526	−3.78010
$123$	0	0
$124$	−52.3426	−4.70050
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−44.6674	−3.94808
$129$	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
$136$	56.6678	4.85922
$137$	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
$139$	23.3366	1.97939	0.989693	−	0.143203i	$-0.0457402\pi$
$139$	23.3366	1.97939	0.989693	+	0.143203i	$0.0457402\pi$
$140$	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
$151$	−20.1310	−1.63824	−0.819120	−	0.573622i	$-0.805538\pi$
$151$	−20.1310	−1.63824	−0.819120	+	0.573622i	$0.805538\pi$
$152$	−66.3145	−5.37882
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−16.5688	−1.31814
$159$	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
$166$	4.32762	0.335889
$167$	25.0508	1.93849	0.969245	−	0.246099i	$-0.0791488\pi$
$167$	25.0508	1.93849	0.969245	+	0.246099i	$0.0791488\pi$
$168$	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
$181$	−13.7969	−1.02551	−0.512756	−	0.858534i	$-0.671376\pi$
$181$	−13.7969	−1.02551	−0.512756	+	0.858534i	$0.671376\pi$
$182$	0	0
$183$	0	0
$184$	−83.4124	−6.14924
$185$	0	0
$186$	0	0
$187$	0	0
$188$	74.0263	5.39892
$189$	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
$196$	−38.8312	−2.77366
$197$	23.5871	1.68051	0.840255	−	0.542192i	$-0.182405\pi$
$197$	23.5871	1.68051	0.840255	+	0.542192i	$0.182405\pi$
$198$	0	0
$199$	−17.1563	−1.21618	−0.608088	−	0.793869i	$-0.708063\pi$
$199$	−17.1563	−1.21618	−0.608088	+	0.793869i	$0.708063\pi$
$200$	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
$211$	−12.9793	−0.893531	−0.446765	−	0.894651i	$-0.647424\pi$
$211$	−12.9793	−0.893531	−0.446765	+	0.894651i	$0.647424\pi$
$212$	−73.8516	−5.07215
$213$	0	0
$214$	−42.3392	−2.89424
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−28.7721	−1.94869
$219$	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
$226$	23.6231	1.57139
$227$	−12.4560	−0.826737	−0.413368	−	0.910564i	$-0.635648\pi$
$227$	−12.4560	−0.826737	−0.413368	+	0.910564i	$0.635648\pi$
$228$	0	0
$229$	−5.88465	−0.388868	−0.194434	−	0.980916i	$-0.562287\pi$
$229$	−5.88465	−0.388868	−0.194434	+	0.980916i	$0.562287\pi$
$230$	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
$241$	−8.39820	−0.540976	−0.270488	−	0.962723i	$-0.587185\pi$
$241$	−8.39820	−0.540976	−0.270488	+	0.962723i	$0.587185\pi$
$242$	30.2196	1.94259
$243$	0	0
$244$	84.3082	5.39728
$245$	0	0
$246$	0	0
$247$	0	0
$248$	91.9536	5.83906
$249$	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
$256$	55.8600	3.49125
$257$	−10.1625	−0.633920	−0.316960	−	0.948439i	$-0.602662\pi$
$257$	−10.1625	−0.633920	−0.316960	+	0.948439i	$0.602662\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	32.4149	1.99879	0.999395	−	0.0347681i	$-0.0110693\pi$
$263$	32.4149	1.99879	0.999395	+	0.0347681i	$0.0110693\pi$
$264$	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
$271$	27.1686	1.65037	0.825186	−	0.564861i	$-0.191070\pi$
$271$	27.1686	1.65037	0.825186	+	0.564861i	$0.191070\pi$
$272$	−91.1660	−5.52775
$273$	0	0
$274$	−47.1985	−2.85136
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	−64.1113	−3.84514
$279$	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
$289$	16.8126	0.988979
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.6498	−0.622167	−0.311084	−	0.950383i	$-0.600692\pi$
$293$	−10.6498	−0.622167	−0.311084	+	0.950383i	$0.600692\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	55.3047	3.18243
$303$	0	0
$304$	106.686	6.11884
$305$	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
$316$	33.4563	1.88206
$317$	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
$323$	−39.5687	−2.20166
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	36.3114	1.99585	0.997927	−	0.0643604i	$-0.0205007\pi$
$331$	36.3114	1.99585	0.997927	+	0.0643604i	$0.0205007\pi$
$332$	−8.73848	−0.479587
$333$	0	0
$334$	−68.8205	−3.76569
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	35.7141	1.94259
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−2.25180	−0.121058
$347$	22.8358	1.22589	0.612946	−	0.790125i	$-0.289984\pi$
$347$	22.8358	1.22589	0.612946	+	0.790125i	$0.289984\pi$
$348$	0	0
$349$	34.2632	1.83407	0.917033	−	0.398811i	$-0.130577\pi$
$349$	34.2632	1.83407	0.917033	+	0.398811i	$0.130577\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.0936	0.537227	0.268613	−	0.963248i	$-0.413435\pi$
$353$	10.0936	0.537227	0.268613	+	0.963248i	$0.413435\pi$
$354$	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
$361$	27.3046	1.43709
$362$	37.9033	1.99215
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	134.192	6.99525
$369$	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
$376$	−130.047	−6.70664
$377$	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
$383$	−0.826319	−0.0422229	−0.0211115	−	0.999777i	$-0.506720\pi$
$383$	−0.826319	−0.0422229	−0.0211115	+	0.999777i	$0.506720\pi$
$384$	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
$391$	−49.7707	−2.51701
$392$	68.2173	3.44549
$393$	0	0
$394$	−64.7993	−3.26454
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	47.1324	2.36253
$399$	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
$409$	33.5319	1.65804	0.829022	−	0.559216i	$-0.188898\pi$
$409$	33.5319	1.65804	0.829022	+	0.559216i	$0.188898\pi$
$410$	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
$421$	41.0159	1.99899	0.999495	−	0.0317621i	$-0.0101119\pi$
$421$	41.0159	1.99899	0.999495	+	0.0317621i	$0.0101119\pi$
$422$	35.6572	1.73576
$423$	0	0
$424$	129.740	6.30073
$425$	0	0
$426$	0	0
$427$	0	0
$428$	85.4927	4.13244
$429$	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
$436$	58.0977	2.78237
$437$	58.2433	2.78616
$438$	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
$443$	−0.843250	−0.0400640	−0.0200320	−	0.999799i	$-0.506377\pi$
$443$	−0.843250	−0.0400640	−0.0200320	+	0.999799i	$0.506377\pi$
$444$	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
$452$	−47.7006	−2.24365
$453$	0	0
$454$	34.2197	1.60601
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	16.1665	0.755412
$459$	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
$466$	−61.4301	−2.84570
$467$	−22.8212	−1.05604	−0.528019	−	0.849233i	$-0.677065\pi$
$467$	−22.8212	−1.05604	−0.528019	+	0.849233i	$0.677065\pi$
$468$	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
$482$	23.0719	1.05089
$483$	0	0
$484$	−61.0205	−2.77366
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−148.110	−6.70461
$489$	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
$496$	−147.933	−6.64240
$497$	0	0
$498$	0	0
$499$	12.3028	0.550748	0.275374	−	0.961337i	$-0.411198\pi$
$499$	12.3028	0.550748	0.275374	+	0.961337i	$0.411198\pi$
$500$	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
$512$	−64.1261	−2.83400
$513$	0	0
$514$	27.9188	1.23145
$515$	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
$526$	−89.0516	−3.88283
$527$	54.8671	2.39005
$528$	0	0
$529$	50.2602	2.18522
$530$	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
$541$	−43.5831	−1.87378	−0.936891	−	0.349620i	$-0.886311\pi$
$541$	−43.5831	−1.87378	−0.936891	+	0.349620i	$0.886311\pi$
$542$	−74.6385	−3.20600
$543$	0	0
$544$	137.119	5.87894
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	95.3048	4.07122
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	129.456	5.49014
$557$	−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
$563$	−18.5953	−0.783698	−0.391849	−	0.920030i	$-0.628164\pi$
$563$	−18.5953	−0.783698	−0.391849	+	0.920030i	$0.628164\pi$
$564$	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	$0.5\pi$
$578$	−46.1883	−1.92118
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	29.2575	1.20862
$587$	−21.4965	−0.887256	−0.443628	−	0.896211i	$-0.646309\pi$
$587$	−21.4965	−0.887256	−0.443628	+	0.896211i	$0.646309\pi$
$588$	0	0
$589$	−64.2073	−2.64562
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.6402	0.642265	0.321133	−	0.947034i	$-0.395936\pi$
$593$	15.6402	0.642265	0.321133	+	0.947034i	$0.395936\pi$
$594$	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
$601$	48.4525	1.97642	0.988209	−	0.153114i	$-0.0489300\pi$
$601$	48.4525	1.97642	0.988209	+	0.153114i	$0.0489300\pi$
$602$	0	0
$603$	0	0
$604$	−111.673	−4.54392
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−160.462	−6.50758
$609$	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
$617$	−27.7682	−1.11791	−0.558953	−	0.829199i	$-0.688797\pi$
$617$	−27.7682	−1.11791	−0.558953	+	0.829199i	$0.688797\pi$
$618$	0	0
$619$	−35.6973	−1.43480	−0.717398	−	0.696664i	$-0.754667\pi$
$619$	−35.6973	−1.43480	−0.717398	+	0.696664i	$0.754667\pi$
$620$	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	$0.946862\pi$
$632$	−58.7748	−2.33794
$633$	0	0
$634$	−96.6488	−3.83841
$635$	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
$646$	108.705	4.27693
$647$	−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
$653$	39.3046	1.53811	0.769054	−	0.639184i	$-0.220728\pi$
$653$	39.3046	1.53811	0.769054	+	0.639184i	$0.220728\pi$
$654$	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
$661$	37.4027	1.45480	0.727399	−	0.686215i	$-0.240729\pi$
$661$	37.4027	1.45480	0.727399	+	0.686215i	$0.240729\pi$
$662$	−99.7560	−3.87713
$663$	0	0
$664$	15.3515	0.595752
$665$	0	0
$666$	0	0
$667$	0	0
$668$	138.965	5.37671
$669$	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
$676$	−72.1151	−2.77366
$677$	−0.892464	−0.0343002	−0.0171501	−	0.999853i	$-0.505459\pi$
$677$	−0.892464	−0.0343002	−0.0171501	+	0.999853i	$0.505459\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.37808	−0.167522	−0.0837612	−	0.996486i	$-0.526693\pi$
$683$	−4.37808	−0.167522	−0.0837612	+	0.996486i	$0.526693\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	49.2532	1.87368	0.936840	−	0.349758i	$-0.113736\pi$
$691$	49.2532	1.87368	0.936840	+	0.349758i	$0.113736\pi$
$692$	4.54692	0.172848
$693$	0	0
$694$	−62.7355	−2.38141
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−94.1291	−3.56284
$699$	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
$706$	−27.7295	−1.04361
$707$	0	0
$708$	0	0
$709$	48.3522	1.81591	0.907953	−	0.419072i	$-0.137644\pi$
$709$	48.3522	1.81591	0.907953	+	0.419072i	$0.137644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−80.7618	−3.02455
$714$	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
$722$	−75.0123	−2.79167
$723$	0	0
$724$	−76.5356	−2.84442
$725$	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
$736$	−201.833	−7.43968
$737$	0	0
$738$	0	0
$739$	7.36075	0.270770	0.135385	−	0.990793i	$-0.456773\pi$
$739$	7.36075	0.270770	0.135385	+	0.990793i	$0.456773\pi$
$740$	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
$751$	−54.7610	−1.99826	−0.999128	−	0.0417535i	$-0.986706\pi$
$751$	−54.7610	−1.99826	−0.999128	+	0.0417535i	$0.986706\pi$
$752$	209.217	7.62934
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	−86.6507	−3.14730
$759$	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
$766$	2.27009	0.0820218
$767$	0	0
$768$	0	0
$769$	54.6254	1.96984	0.984921	−	0.173007i	$-0.0553485\pi$
$769$	54.6254	1.96984	0.984921	+	0.173007i	$0.0553485\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−26.6291	−0.957783	−0.478892	−	0.877874i	$-0.658961\pi$
$773$	−26.6291	−0.957783	−0.478892	+	0.877874i	$0.658961\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	136.732	4.88952
$783$	0	0
$784$	−109.747	−3.91953
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	130.845	4.66116
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−95.1714	−3.37326
$797$	42.3567	1.50035	0.750175	−	0.661239i	$-0.229969\pi$
$797$	42.3567	1.50035	0.750175	+	0.661239i	$0.229969\pi$
$798$	0	0
$799$	−77.5965	−2.74517
$800$	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
$811$	28.4100	0.997610	0.498805	−	0.866714i	$-0.333772\pi$
$811$	28.4100	0.997610	0.498805	+	0.866714i	$0.333772\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−92.1200	−3.22090
$819$	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
$827$	48.5263	1.68743	0.843713	−	0.536794i	$-0.180365\pi$
$827$	48.5263	1.68743	0.843713	+	0.536794i	$0.180365\pi$
$828$	0	0
$829$	−0.188788	−0.00655687	−0.00327844	−	0.999995i	$-0.501044\pi$
$829$	−0.188788	−0.00655687	−0.00327844	+	0.999995i	$0.501044\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	40.7040	1.41031
$834$	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
$842$	−112.680	−3.88322
$843$	0	0
$844$	−72.0002	−2.47835
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−208.723	−7.16758
$849$	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
$856$	−150.190	−5.13340
$857$	50.2935	1.71799	0.858997	−	0.511980i	$-0.171088\pi$
$857$	50.2935	1.71799	0.858997	+	0.511980i	$0.171088\pi$
$858$	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
$872$	−102.064	−3.45632
$873$	0	0
$874$	−160.008	−5.41236
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−70.1673	−2.36803
$879$	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
$886$	2.31661	0.0778280
$887$	−32.1236	−1.07861	−0.539303	−	0.842112i	$-0.681312\pi$
$887$	−32.1236	−1.07861	−0.539303	+	0.842112i	$0.681312\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	90.8061	3.03871
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	77.4135	2.57902
$902$	0	0
$903$	0	0
$904$	83.7986	2.78710
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	−69.0976	−2.29309
$909$	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
$916$	−32.6440	−1.07859
$917$	0	0
$918$	0	0
$919$	4.79560	0.158192	0.0790962	−	0.996867i	$-0.474797\pi$
$919$	4.79560	0.158192	0.0790962	+	0.996867i	$0.474797\pi$
$920$	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
$931$	−47.6333	−1.56112
$932$	124.042	4.06312
$933$	0	0
$934$	62.6952	2.05145
$935$	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
$947$	55.7334	1.81109	0.905546	−	0.424249i	$-0.139462\pi$
$947$	55.7334	1.81109	0.905546	+	0.424249i	$0.139462\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−37.7329	−1.22229	−0.611144	−	0.791519i	$-0.709290\pi$
$953$	−37.7329	−1.22229	−0.611144	+	0.791519i	$0.709290\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	58.0316	1.87199
$962$	0	0
$963$	0	0
$964$	−46.5875	−1.50048
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	107.199	3.44549
$969$	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
$976$	238.276	7.62703
$977$	−57.9200	−1.85303	−0.926513	−	0.376262i	$-0.877209\pi$
$977$	−57.9200	−1.85303	−0.926513	+	0.376262i	$0.877209\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	55.7880	1.77936	0.889681	−	0.456583i	$-0.150927\pi$
$983$	55.7880	1.77936	0.889681	+	0.456583i	$0.150927\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	7.17948	0.228064	0.114032	−	0.993477i	$-0.463623\pi$
$991$	7.17948	0.228064	0.114032	+	0.993477i	$0.463623\pi$
$992$	222.501	7.06440
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−33.7987	−1.06988
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3375.2.a.a.1.1	✓	4
3.2	odd	2		3375.2.a.d.1.4	yes	4
5.4	even	2		3375.2.a.d.1.4	yes	4
15.14	odd	2	CM	3375.2.a.a.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3375.2.a.a.1.1	✓	4	1.1	even	1	trivial
3375.2.a.a.1.1	✓	4	15.14	odd	2	CM
3375.2.a.d.1.4	yes	4	3.2	odd	2
3375.2.a.d.1.4	yes	4	5.4	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3375 = 3^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3375.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74724 q^{2} +5.54732 q^{4} -9.74533 q^{8} +O(q^{10})\)	\(q-2.74724 q^{2} +5.54732 q^{4} -9.74533 q^{8} +15.6781 q^{16} -5.81486 q^{17} +6.80475 q^{19} +8.55922 q^{23} -9.43566 q^{31} -23.5808 q^{32} +15.9748 q^{34} -18.6943 q^{38} -23.5142 q^{46} +13.3445 q^{47} -7.00000 q^{49} -13.3130 q^{53} +15.1980 q^{61} +25.9220 q^{62} +33.4259 q^{64} -32.2569 q^{68} +37.7481 q^{76} +6.03108 q^{79} -1.57526 q^{83} +47.4807 q^{92} -36.6606 q^{94} +19.2307 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 16 q^{4} - 3 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 16 * q^4 - 3 * q^8	\( 4 q - 4 q^{2} + 16 q^{4} - 3 q^{8} + 36 q^{16} - 11 q^{17} + 17 q^{19} - 13 q^{23} + 19 q^{31} + q^{34} + 8 q^{38} + 8 q^{46} + 20 q^{47} - 28 q^{49} - 17 q^{53} + 31 q^{61} + 21 q^{62} + 83 q^{64} - 99 q^{68} + 63 q^{76} + 23 q^{79} - 23 q^{83} - 12 q^{92} + 15 q^{94} + 28 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 16 * q^4 - 3 * q^8 + 36 * q^16 - 11 * q^17 + 17 * q^19 - 13 * q^23 + 19 * q^31 + q^34 + 8 * q^38 + 8 * q^46 + 20 * q^47 - 28 * q^49 - 17 * q^53 + 31 * q^61 + 21 * q^62 + 83 * q^64 - 99 * q^68 + 63 * q^76 + 23 * q^79 - 23 * q^83 - 12 * q^92 + 15 * q^94 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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