LMFDB - Embedded newform 4788.2.i.d.3457.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(3457,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4788.3457");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.i (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2702336256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 $ x^8 + 9x^6 + 56x^4 + 225*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			3457.6
Root			$-0.656712 + 2.13746i$ of defining polynomial
Character	$\chi$	$=$	4788.3457
Dual form			4788.2.i.d.3457.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27492i q^{5} +(-2.63746 + 0.209313i) q^{7} +O(q^{10})$	$q+3.27492i q^{5} +(-2.63746 + 0.209313i) q^{7} -6.50958 q^{11} +7.27492i q^{17} +4.35890i q^{19} -8.71780 q^{23} -5.72508 q^{25} +(-0.685484 - 8.63746i) q^{35} +11.8248 q^{43} -2.72508i q^{47} +(6.91238 - 1.10411i) q^{49} -21.3183i q^{55} -10.8109i q^{61} +16.0646i q^{73} +(17.1687 - 1.36254i) q^{77} -16.0000i q^{83} -23.8248 q^{85} -14.2750 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{7}+O(q^{10}) $ 8 * q - 6 * q^7	$ 8 q - 6 q^{7} - 76 q^{25} + 4 q^{43} + 10 q^{49} - 100 q^{85}+O(q^{100}) $ 8 * q - 6 * q^7 - 76 * q^25 + 4 * q^43 + 10 * q^49 - 100 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4788\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$2395$	$4105$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			3.27492i			1.46459i	0.680989	+	0.732294i	$0.261550\pi$
$5$			3.27492i			1.46459i	−0.680989	+	0.732294i	$0.738450\pi$
$6$		0			0
$7$	−2.63746	+	0.209313i	−0.996866	+	0.0791130i
$7$	−2.63746	+	0.209313i	−0.996866	+	0.0791130i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.50958			−1.96271			−0.981356	−	0.192201i	$-0.938437\pi$
$11$	−6.50958			−1.96271			−0.981356	+	0.192201i	$0.938437\pi$
$12$		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			0
$17$			7.27492i			1.76443i	0.470850	+	0.882213i	$0.343947\pi$
$17$			7.27492i			1.76443i	−0.470850	+	0.882213i	$0.656053\pi$
$18$		0			0
$19$			4.35890i			1.00000i
$19$			4.35890i			1.00000i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.71780			−1.81779			−0.908893	−	0.417029i	$-0.863071\pi$
$23$	−8.71780			−1.81779			−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	−5.72508			−1.14502
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.685484	−	8.63746i	−0.115868	−	1.46000i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	11.8248			1.80326			0.901629	−	0.432511i	$-0.142372\pi$
$43$	11.8248			1.80326			0.901629	+	0.432511i	$0.142372\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.72508i		−	0.397494i	−0.980051	−	0.198747i	$-0.936313\pi$
$47$		−	2.72508i		−	0.397494i	0.980051	−	0.198747i	$-0.0636872\pi$
$48$		0			0
$49$	6.91238	−	1.10411i	0.987482	−	0.157730i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		−	21.3183i		−	2.87456i
$56$		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
$61$		−	10.8109i		−	1.38420i	−0.721803	−	0.692099i	$-0.756686\pi$
$61$		−	10.8109i		−	1.38420i	0.721803	−	0.692099i	$-0.243314\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			16.0646i			1.88022i	0.340868	+	0.940111i	$0.389279\pi$
$73$			16.0646i			1.88022i	−0.340868	+	0.940111i	$0.610721\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	17.1687	−	1.36254i	1.95656	−	0.155276i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	16.0000i		−	1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$		−	16.0000i		−	1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$		0			0
$85$	−23.8248			−2.58416
$86$		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
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−14.2750			−1.46459
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			10.0000i			0.995037i	0.867453	+	0.497519i	$0.165755\pi$
$101$			10.0000i			0.995037i	−0.867453	+	0.497519i	$0.834245\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	28.5501i		−	2.66231i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.52274	−	19.1873i	−0.139589	−	1.75890i
$120$		0			0
$121$	31.3746			2.85224
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	2.37459i		−	0.212389i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			22.3746i			1.95488i	0.211221	+	0.977438i	$0.432256\pi$
$131$			22.3746i			1.95488i	−0.211221	+	0.977438i	$0.567744\pi$
$132$		0			0
$133$	−0.912376	−	11.4964i	−0.0791130	−	0.996866i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−17.7391			−1.51556			−0.757778	−	0.652512i	$-0.773715\pi$
$137$	−17.7391			−1.51556			−0.757778	+	0.652512i	$0.773715\pi$
$138$		0			0
$139$		−	18.6915i		−	1.58539i	−0.609618	−	0.792695i	$-0.708677\pi$
$139$		−	18.6915i		−	1.58539i	0.609618	−	0.792695i	$-0.291323\pi$
$140$		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
$149$	−1.37097			−0.112314			−0.0561570	−	0.998422i	$-0.517885\pi$
$149$	−1.37097			−0.112314			−0.0561570	+	0.998422i	$0.517885\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	17.4356i		−	1.39151i	−0.718278	−	0.695756i	$-0.755069\pi$
$157$		−	17.4356i		−	1.39151i	0.718278	−	0.695756i	$-0.244931\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	22.9928	−	1.82475i	1.81209	−	0.143811i
$162$		0			0
$163$	−24.0000			−1.87983			−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000			−1.87983			−0.939913	+	0.341415i	$0.889094\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	15.0997	−	1.19834i	1.14143	−	0.0905857i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	47.3566i		−	3.46306i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	25.6197			1.85377			0.926887	−	0.375339i	$-0.122474\pi$
$191$	25.6197			1.85377			0.926887	+	0.375339i	$0.122474\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.4356			1.24223			0.621117	−	0.783718i	$-0.286679\pi$
$197$	17.4356			1.24223			0.621117	+	0.783718i	$0.286679\pi$
$198$		0			0
$199$		−	15.1123i		−	1.07128i	−0.844446	−	0.535641i	$-0.820070\pi$
$199$		−	15.1123i		−	1.07128i	0.844446	−	0.535641i	$-0.179930\pi$
$200$		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
$209$		−	28.3746i		−	1.96271i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			38.7251i			2.64103i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		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
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$			29.0838i			1.92191i	0.276704	+	0.960955i	$0.410758\pi$
$229$			29.0838i			1.92191i	−0.276704	+	0.960955i	$0.589242\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.3901			0.942728			0.471364	−	0.881939i	$-0.343762\pi$
$233$	14.3901			0.942728			0.471364	+	0.881939i	$0.343762\pi$
$234$		0			0
$235$	8.92442			0.582165
$236$		0			0
$237$		0			0
$238$		0			0
$239$	10.9260			0.706745			0.353373	−	0.935483i	$-0.385035\pi$
$239$	10.9260			0.706745			0.353373	+	0.935483i	$0.385035\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.61587	+	22.6375i	0.231010	+	1.44625i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			30.3746i			1.91723i	0.284711	+	0.958613i	$0.408102\pi$
$251$			30.3746i			1.91723i	−0.284711	+	0.958613i	$0.591898\pi$
$252$		0			0
$253$	56.7492			3.56779
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−24.7824			−1.52815			−0.764075	−	0.645128i	$-0.776804\pi$
$263$	−24.7824			−1.52815			−0.764075	+	0.645128i	$0.776804\pi$
$264$		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
$271$		−	26.1534i		−	1.58871i	−0.607457	−	0.794353i	$-0.707810\pi$
$271$		−	26.1534i		−	1.58871i	0.607457	−	0.794353i	$-0.292190\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	37.2679			2.24734
$276$		0			0
$277$	12.7251			0.764576			0.382288	−	0.924043i	$-0.375136\pi$
$277$	12.7251			0.764576			0.382288	+	0.924043i	$0.375136\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$			33.3851i			1.98454i	0.124096	+	0.992270i	$0.460397\pi$
$283$			33.3851i			1.98454i	−0.124096	+	0.992270i	$0.539603\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−35.9244			−2.11320
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−31.1873	+	2.47508i	−1.79761	+	0.142661i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	35.4049			2.02728
$306$		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
$311$		−	21.2749i		−	1.20639i	−0.797594	−	0.603195i	$-0.793894\pi$
$311$		−	21.2749i		−	1.20639i	0.797594	−	0.603195i	$-0.206106\pi$
$312$		0			0
$313$		−	34.8712i		−	1.97104i	−0.169570	−	0.985518i	$-0.554238\pi$
$313$		−	34.8712i		−	1.97104i	0.169570	−	0.985518i	$-0.445762\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−31.7106			−1.76443
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.570396	+	7.18729i	0.0314470	+	0.396248i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.0000	+	4.35890i	−0.971909	+	0.235358i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−34.2224			−1.83715			−0.918577	−	0.395242i	$-0.870661\pi$
$347$	−34.2224			−1.83715			−0.918577	+	0.395242i	$0.870661\pi$
$348$		0			0
$349$		−	36.8492i		−	1.97249i	−0.165277	−	0.986247i	$-0.552852\pi$
$349$		−	36.8492i		−	1.97249i	0.165277	−	0.986247i	$-0.447148\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	14.0000i		−	0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$		−	14.0000i		−	0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.9495			0.841785			0.420892	−	0.907111i	$-0.361717\pi$
$359$	15.9495			0.841785			0.420892	+	0.907111i	$0.361717\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−52.6103			−2.75375
$366$		0			0
$367$			26.1534i			1.36520i	0.730794	+	0.682598i	$0.239150\pi$
$367$			26.1534i			1.36520i	−0.730794	+	0.682598i	$0.760850\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	4.46221	+	56.2262i	0.227415	+	2.86555i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.39449			−0.324213			−0.162107	−	0.986773i	$-0.551829\pi$
$389$	−6.39449			−0.324213			−0.162107	+	0.986773i	$0.551829\pi$
$390$		0			0
$391$		−	63.4213i		−	3.20735i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			13.5529i			0.680199i	0.940389	+	0.340099i	$0.110461\pi$
$397$			13.5529i			0.680199i	−0.940389	+	0.340099i	$0.889539\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	52.3987			2.57215
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	40.0000i		−	1.95413i	−0.212946	−	0.977064i	$-0.568306\pi$
$419$		−	40.0000i		−	1.95413i	0.212946	−	0.977064i	$-0.431694\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	41.6495i		−	2.02030i
$426$		0			0
$427$	2.26287	+	28.5134i	0.109508	+	1.37986i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	38.0000i		−	1.81779i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.85859			0.468396			0.234198	−	0.972189i	$-0.424754\pi$
$443$	9.85859			0.468396			0.234198	+	0.972189i	$0.424754\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	25.4743			1.19164			0.595818	−	0.803120i	$-0.296828\pi$
$457$	25.4743			1.19164			0.595818	+	0.803120i	$0.296828\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			0.374586i			0.0174462i	0.999962	+	0.00872311i	$0.00277669\pi$
$461$			0.374586i			0.0174462i	−0.999962	+	0.00872311i	$0.997223\pi$
$462$		0			0
$463$	−31.8248			−1.47902			−0.739511	−	0.673145i	$-0.764943\pi$
$463$	−31.8248			−1.47902			−0.739511	+	0.673145i	$0.764943\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	25.2749i		−	1.16958i	−0.811183	−	0.584792i	$-0.801176\pi$
$467$		−	25.2749i		−	1.16958i	0.811183	−	0.584792i	$-0.198824\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−76.9741			−3.53927
$474$		0			0
$475$		−	24.9551i		−	1.14502i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	4.00000i		−	0.182765i	−0.995816	−	0.0913823i	$-0.970871\pi$
$479$		−	4.00000i		−	0.182765i	0.995816	−	0.0913823i	$-0.0291285\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−43.5890			−1.96714			−0.983572	−	0.180517i	$-0.942223\pi$
$491$	−43.5890			−1.96714			−0.983572	+	0.180517i	$0.942223\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−38.3746			−1.71788			−0.858941	−	0.512074i	$-0.828877\pi$
$499$	−38.3746			−1.71788			−0.858941	+	0.512074i	$0.828877\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			44.0000i			1.96186i	0.194354	+	0.980932i	$0.437739\pi$
$503$			44.0000i			1.96186i	−0.194354	+	0.980932i	$0.562261\pi$
$504$		0			0
$505$	−32.7492			−1.45732
$506$		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
$511$	−3.36254	−	42.3698i	−0.148750	−	1.87433i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			17.7391i			0.780166i
$518$		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
$526$		0			0
$527$		0			0
$528$		0			0
$529$	53.0000			2.30435
$530$		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
$539$	−44.9966	+	7.18729i	−1.93814	+	0.309579i
$540$		0			0
$541$	−21.4743			−0.923250			−0.461625	−	0.887075i	$-0.652733\pi$
$541$	−21.4743			−0.923250			−0.461625	+	0.887075i	$0.652733\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	38.5237			1.63230			0.816152	−	0.577838i	$-0.196103\pi$
$557$	38.5237			1.63230			0.816152	+	0.577838i	$0.196103\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	40.0000			1.67395			0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000			1.67395			0.836974	+	0.547243i	$0.184323\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	49.9101			2.08140
$576$		0			0
$577$			26.5720i			1.10621i	0.833112	+	0.553104i	$0.186557\pi$
$577$			26.5720i			1.10621i	−0.833112	+	0.553104i	$0.813443\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.34901	+	42.1993i	0.138941	+	1.75072i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			38.0241i			1.56942i	0.619862	+	0.784711i	$0.287189\pi$
$587$			38.0241i			1.56942i	−0.619862	+	0.784711i	$0.712811\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	34.0000i		−	1.39621i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$		−	34.0000i		−	1.39621i	0.715994	−	0.698106i	$-0.245974\pi$
$594$		0			0
$595$	62.8368	−	4.98684i	2.57606	−	0.204440i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			102.749i			4.17735i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−6.92442			−0.279675			−0.139837	−	0.990174i	$-0.544658\pi$
$613$	−6.92442			−0.279675			−0.139837	+	0.990174i	$0.544658\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−12.7156			−0.511911			−0.255956	−	0.966689i	$-0.582390\pi$
$617$	−12.7156			−0.511911			−0.255956	+	0.966689i	$0.582390\pi$
$618$		0			0
$619$			43.5890i			1.75199i	0.482321	+	0.875995i	$0.339794\pi$
$619$			43.5890i			1.75199i	−0.482321	+	0.875995i	$0.660206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.8488			−0.833954
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	34.0241			1.35448			0.677239	−	0.735763i	$-0.263176\pi$
$631$	34.0241			1.35448			0.677239	+	0.735763i	$0.263176\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$			35.8969i			1.41564i	0.706395	+	0.707818i	$0.250320\pi$
$643$			35.8969i			1.41564i	−0.706395	+	0.707818i	$0.749680\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	50.0241i		−	1.96665i	−0.181857	−	0.983325i	$-0.558211\pi$
$647$		−	50.0241i		−	1.96665i	0.181857	−	0.983325i	$-0.441789\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−20.2509			−0.792479			−0.396239	−	0.918147i	$-0.629685\pi$
$653$	−20.2509			−0.792479			−0.396239	+	0.918147i	$0.629685\pi$
$654$		0			0
$655$	−73.2749			−2.86309
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	37.6498	−	2.98796i	1.46000	−	0.115868i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$			70.3746i			2.71678i
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		−	58.0942i		−	2.21967i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	10.6958i		−	0.406889i	−0.979086	−	0.203445i	$-0.934786\pi$
$691$		−	10.6958i		−	0.406889i	0.979086	−	0.203445i	$-0.0652137\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	61.2130			2.32194
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−17.4356			−0.658533			−0.329267	−	0.944237i	$-0.606802\pi$
$701$	−17.4356			−0.658533			−0.329267	+	0.944237i	$0.606802\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.09313	−	26.3746i	−0.0787204	−	0.991918i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			5.62541i			0.209793i	0.994483	+	0.104896i	$0.0334511\pi$
$719$			5.62541i			0.209793i	−0.994483	+	0.104896i	$0.966549\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	51.6580i		−	1.91589i	−0.286954	−	0.957944i	$-0.592643\pi$
$727$		−	51.6580i		−	1.91589i	0.286954	−	0.957944i	$-0.407357\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			86.0241i			3.18172i
$732$		0			0
$733$			52.3068i			1.93200i	0.258551	+	0.965998i	$0.416755\pi$
$733$			52.3068i			1.93200i	−0.258551	+	0.965998i	$0.583245\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−48.9244			−1.79971			−0.899857	−	0.436185i	$-0.856329\pi$
$739$	−48.9244			−1.79971			−0.899857	+	0.436185i	$0.856329\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		−	4.48981i		−	0.164494i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−28.0241			−1.01855			−0.509276	−	0.860603i	$-0.670087\pi$
$757$	−28.0241			−1.01855			−0.509276	+	0.860603i	$0.670087\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	31.2749i		−	1.13371i	−0.823816	−	0.566857i	$-0.808159\pi$
$761$		−	31.2749i		−	1.13371i	0.823816	−	0.566857i	$-0.191841\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			33.2700i			1.19975i	0.800094	+	0.599874i	$0.204783\pi$
$769$			33.2700i			1.19975i	−0.800094	+	0.599874i	$0.795217\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	57.1001			2.03799
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	19.8248			0.701349
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	104.574i		−	3.69033i
$804$		0			0
$805$	5.97591	+	75.2996i	0.210623	+	2.65396i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−32.6630

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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.27492i q^{5} +(-2.63746 + 0.209313i) q^{7} +O(q^{10})\)	\(q+3.27492i q^{5} +(-2.63746 + 0.209313i) q^{7} -6.50958 q^{11} +7.27492i q^{17} +4.35890i q^{19} -8.71780 q^{23} -5.72508 q^{25} +(-0.685484 - 8.63746i) q^{35} +11.8248 q^{43} -2.72508i q^{47} +(6.91238 - 1.10411i) q^{49} -21.3183i q^{55} -10.8109i q^{61} +16.0646i q^{73} +(17.1687 - 1.36254i) q^{77} -16.0000i q^{83} -23.8248 q^{85} -14.2750 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{7}+O(q^{10}) \) 8 * q - 6 * q^7	\( 8 q - 6 q^{7} - 76 q^{25} + 4 q^{43} + 10 q^{49} - 100 q^{85}+O(q^{100}) \) 8 * q - 6 * q^7 - 76 * q^25 + 4 * q^43 + 10 * q^49 - 100 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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