LMFDB - Embedded newform 4761.2.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, 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("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.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:	$38.0167764023$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.3
Root			$-0.517638$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} +2.31079 q^{7} +O(q^{10})$	$q-2.00000 q^{4} +2.31079 q^{7} -1.73205 q^{13} +4.00000 q^{16} -3.20736 q^{19} -5.00000 q^{25} -4.62158 q^{28} +10.3923 q^{31} -11.4524 q^{37} +5.03768 q^{43} -1.66025 q^{49} +3.46410 q^{52} -11.7298 q^{61} -8.00000 q^{64} +16.3514 q^{67} -13.8564 q^{73} +6.41473 q^{76} +9.41902 q^{79} -4.00240 q^{91} -19.6975 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} - 32 q^{64}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 28 * q^49 - 32 * q^64

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	2.31079	0.873396	0.436698	−	0.899608i	$-0.356148\pi$
$7$	2.31079	0.873396	0.436698	+	0.899608i	$0.356148\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−1.73205	−0.480384	−0.240192	−	0.970725i	$-0.577210\pi$
$13$	−1.73205	−0.480384	−0.240192	+	0.970725i	$0.577210\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−3.20736	−0.735820	−0.367910	−	0.929861i	$-0.619927\pi$
$19$	−3.20736	−0.735820	−0.367910	+	0.929861i	$0.619927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	−4.62158	−0.873396
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	10.3923	1.86651	0.933257	−	0.359211i	$-0.116954\pi$
$31$	10.3923	1.86651	0.933257	+	0.359211i	$0.116954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.4524	−1.88276	−0.941382	−	0.337342i	$-0.890472\pi$
$37$	−11.4524	−1.88276	−0.941382	+	0.337342i	$0.890472\pi$
$38$	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
$43$	5.03768	0.768239	0.384120	−	0.923283i	$-0.374505\pi$
$43$	5.03768	0.768239	0.384120	+	0.923283i	$0.374505\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−1.66025	−0.237179
$50$	0	0
$51$	0	0
$52$	3.46410	0.480384
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	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
$61$	−11.7298	−1.50185	−0.750924	−	0.660388i	$-0.770392\pi$
$61$	−11.7298	−1.50185	−0.750924	+	0.660388i	$0.770392\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	16.3514	1.99764	0.998820	−	0.0485648i	$-0.0154647\pi$
$67$	16.3514	1.99764	0.998820	+	0.0485648i	$0.0154647\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−13.8564	−1.62177	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	−13.8564	−1.62177	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	6.41473	0.735820
$77$	0	0
$78$	0	0
$79$	9.41902	1.05972	0.529861	−	0.848084i	$-0.322244\pi$
$79$	9.41902	1.05972	0.529861	+	0.848084i	$0.322244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	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
$91$	−4.00240	−0.419566
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−19.6975	−1.99997	−0.999987	−	0.00515471i	$-0.998359\pi$
$97$	−19.6975	−1.99997	−0.999987	+	0.00515471i	$0.998359\pi$
$98$	0	0
$99$	0	0
$100$	10.0000	1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−1.83032	−0.180346	−0.0901732	−	0.995926i	$-0.528742\pi$
$103$	−1.83032	−0.180346	−0.0901732	+	0.995926i	$0.528742\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	13.2827	1.27225	0.636127	−	0.771584i	$-0.280535\pi$
$109$	13.2827	1.27225	0.636127	+	0.771584i	$0.280535\pi$
$110$	0	0
$111$	0	0
$112$	9.24316	0.873396
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	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
$122$	0	0
$123$	0	0
$124$	−20.7846	−1.86651
$125$	0	0
$126$	0	0
$127$	22.5167	1.99803	0.999015	−	0.0443678i	$-0.0141274\pi$
$127$	22.5167	1.99803	0.999015	+	0.0443678i	$0.0141274\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−7.41154	−0.642662
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−23.0000	−1.95083	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−23.0000	−1.95083	−0.975417	+	0.220366i	$0.929275\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	22.9048	1.88276
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−23.0000	−1.87171	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−23.0000	−1.87171	−0.935857	+	0.352381i	$0.885372\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.79744	−0.382878	−0.191439	−	0.981505i	$-0.561315\pi$
$157$	−4.79744	−0.382878	−0.191439	+	0.981505i	$0.561315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	5.19615	0.406994	0.203497	−	0.979076i	$-0.434769\pi$
$163$	5.19615	0.406994	0.203497	+	0.979076i	$0.434769\pi$
$164$	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
$169$	−10.0000	−0.769231
$170$	0	0
$171$	0	0
$172$	−10.0754	−0.768239
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−11.5539	−0.873396
$176$	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
$181$	23.2838	1.73067	0.865334	−	0.501196i	$-0.167106\pi$
$181$	23.2838	1.73067	0.865334	+	0.501196i	$0.167106\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	0	0
$195$	0	0
$196$	3.32051	0.237179
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−27.9425	−1.98079	−0.990396	−	0.138263i	$-0.955848\pi$
$199$	−27.9425	−1.98079	−0.990396	+	0.138263i	$0.955848\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−6.92820	−0.480384
$209$	0	0
$210$	0	0
$211$	−25.9808	−1.78859	−0.894295	−	0.447478i	$-0.852322\pi$
$211$	−25.9808	−1.78859	−0.894295	+	0.447478i	$0.852322\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	24.0144	1.63021
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−23.0000	−1.54019	−0.770097	−	0.637927i	$-0.779792\pi$
$223$	−23.0000	−1.54019	−0.770097	+	0.637927i	$0.779792\pi$
$224$	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
$229$	−25.7704	−1.70296	−0.851478	−	0.524390i	$-0.824293\pi$
$229$	−25.7704	−1.70296	−0.851478	+	0.524390i	$0.824293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	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
$241$	30.3920	1.95772	0.978860	−	0.204531i	$-0.0655670\pi$
$241$	30.3920	1.95772	0.978860	+	0.204531i	$0.0655670\pi$
$242$	0	0
$243$	0	0
$244$	23.4596	1.50185
$245$	0	0
$246$	0	0
$247$	5.55532	0.353476
$248$	0	0
$249$	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
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−26.4641	−1.64440
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−32.7028	−1.99764
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−15.5885	−0.946931	−0.473466	−	0.880812i	$-0.656997\pi$
$271$	−15.5885	−0.946931	−0.473466	+	0.880812i	$0.656997\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−32.9090	−1.97731	−0.988654	−	0.150210i	$-0.952005\pi$
$277$	−32.9090	−1.97731	−0.988654	+	0.150210i	$0.952005\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−18.3204	−1.08904	−0.544518	−	0.838749i	$-0.683287\pi$
$283$	−18.3204	−1.08904	−0.544518	+	0.838749i	$0.683287\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	27.7128	1.62177
$293$	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
$301$	11.6410	0.670977
$302$	0	0
$303$	0	0
$304$	−12.8295	−0.735820
$305$	0	0
$306$	0	0
$307$	29.4449	1.68051	0.840254	−	0.542194i	$-0.182406\pi$
$307$	29.4449	1.68051	0.840254	+	0.542194i	$0.182406\pi$
$308$	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
$313$	−21.0745	−1.19120	−0.595601	−	0.803281i	$-0.703086\pi$
$313$	−21.0745	−1.19120	−0.595601	+	0.803281i	$0.703086\pi$
$314$	0	0
$315$	0	0
$316$	−18.8380	−1.05972
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	8.66025	0.480384
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−19.0526	−1.04722	−0.523612	−	0.851957i	$-0.675416\pi$
$331$	−19.0526	−1.04722	−0.523612	+	0.851957i	$0.675416\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.58441	−0.249729	−0.124864	−	0.992174i	$-0.539850\pi$
$337$	−4.58441	−0.249729	−0.124864	+	0.992174i	$0.539850\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0120	−1.08055
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−23.0000	−1.23116	−0.615581	−	0.788074i	$-0.711079\pi$
$349$	−23.0000	−1.23116	−0.615581	+	0.788074i	$0.711079\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	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
$361$	−8.71281	−0.458569
$362$	0	0
$363$	0	0
$364$	8.00481	0.419566
$365$	0	0
$366$	0	0
$367$	29.7728	1.55413	0.777064	−	0.629421i	$-0.216708\pi$
$367$	29.7728	1.55413	0.777064	+	0.629421i	$0.216708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.5273	0.855749	0.427874	−	0.903838i	$-0.359263\pi$
$373$	16.5273	0.855749	0.427874	+	0.903838i	$0.359263\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.13492	0.109664	0.0548318	−	0.998496i	$-0.482538\pi$
$379$	2.13492	0.109664	0.0548318	+	0.998496i	$0.482538\pi$
$380$	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
$385$	0	0
$386$	0	0
$387$	0	0
$388$	39.3949	1.99997
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−18.0000	−0.896644
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	0	0
$412$	3.66063	0.180346
$413$	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
$421$	30.2161	1.47264	0.736322	−	0.676631i	$-0.236561\pi$
$421$	30.2161	1.47264	0.736322	+	0.676631i	$0.236561\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−27.1051	−1.31171
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−39.6351	−1.90474	−0.952372	−	0.304939i	$-0.901364\pi$
$433$	−39.6351	−1.90474	−0.952372	+	0.304939i	$0.901364\pi$
$434$	0	0
$435$	0	0
$436$	−26.5654	−1.27225
$437$	0	0
$438$	0	0
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−18.4863	−0.873396
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.5646	−1.75720	−0.878599	−	0.477561i	$-0.841521\pi$
$457$	−37.5646	−1.75720	−0.878599	+	0.477561i	$0.841521\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−23.0000	−1.06890	−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000	−1.06890	−0.534450	+	0.845200i	$0.679481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	37.7846	1.74473
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	16.0368	0.735820
$476$	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
$481$	19.8362	0.904451
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	−19.0000	−0.860972	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	−19.0000	−0.860972	−0.430486	+	0.902597i	$0.641658\pi$
$488$	0	0
$489$	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
$496$	41.5692	1.86651
$497$	0	0
$498$	0	0
$499$	12.1244	0.542761	0.271380	−	0.962472i	$-0.412520\pi$
$499$	12.1244	0.542761	0.271380	+	0.962472i	$0.412520\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−45.0333	−1.99803
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−32.0192	−1.41645
$512$	0	0
$513$	0	0
$514$	0	0
$515$	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
$523$	37.5002	1.63977	0.819885	−	0.572528i	$-0.194037\pi$
$523$	37.5002	1.63977	0.819885	+	0.572528i	$0.194037\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	0	0
$530$	0	0
$531$	0	0
$532$	14.8231	0.642662
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	46.0000	1.97769	0.988847	−	0.148933i	$-0.0475840\pi$
$541$	46.0000	1.97769	0.988847	+	0.148933i	$0.0475840\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.2487	1.03680	0.518400	−	0.855138i	$-0.326528\pi$
$547$	24.2487	1.03680	0.518400	+	0.855138i	$0.326528\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	21.7654	0.925558
$554$	0	0
$555$	0	0
$556$	46.0000	1.95083
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−8.72552	−0.369050
$560$	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		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	47.6400	1.99367	0.996835	−	0.0794985i	$-0.0253319\pi$
$571$	47.6400	1.99367	0.996835	+	0.0794985i	$0.0253319\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	46.0000	1.91501	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000	1.91501	0.957503	+	0.288425i	$0.0931316\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−33.3319	−1.37342
$590$	0	0
$591$	0	0
$592$	−45.8096	−1.88276
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	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
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	0	0
$603$	0	0
$604$	46.0000	1.87171
$605$	0	0
$606$	0	0
$607$	29.0000	1.17707	0.588537	−	0.808470i	$-0.299704\pi$
$607$	29.0000	1.17707	0.588537	+	0.808470i	$0.299704\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	44.2567	1.78751	0.893756	−	0.448553i	$-0.148061\pi$
$613$	44.2567	1.78751	0.893756	+	0.448553i	$0.148061\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	40.7720	1.63876	0.819382	−	0.573248i	$-0.194317\pi$
$619$	40.7720	1.63876	0.819382	+	0.573248i	$0.194317\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	9.59489	0.382878
$629$	0	0
$630$	0	0
$631$	−34.8105	−1.38578	−0.692892	−	0.721041i	$-0.743664\pi$
$631$	−34.8105	−1.38578	−0.692892	+	0.721041i	$0.743664\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.87564	0.113937
$638$	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
$643$	30.5679	1.20548	0.602739	−	0.797938i	$-0.294076\pi$
$643$	30.5679	1.20548	0.602739	+	0.797938i	$0.294076\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−10.3923	−0.406994
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	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
$661$	51.3650	1.99787	0.998933	−	0.0461915i	$-0.0147084\pi$
$661$	51.3650	1.99787	0.998933	+	0.0461915i	$0.0147084\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	36.3731	1.40208	0.701039	−	0.713123i	$-0.252720\pi$
$673$	36.3731	1.40208	0.701039	+	0.713123i	$0.252720\pi$
$674$	0	0
$675$	0	0
$676$	20.0000	0.769231
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−45.5167	−1.74677
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	20.1507	0.768239
$689$	0	0
$690$	0	0
$691$	32.9090	1.25192	0.625958	−	0.779857i	$-0.284708\pi$
$691$	32.9090	1.25192	0.625958	+	0.779857i	$0.284708\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	23.1079	0.873396
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	36.7321	1.38538
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8.69831	−0.326672	−0.163336	−	0.986571i	$-0.552225\pi$
$709$	−8.69831	−0.326672	−0.163336	+	0.986571i	$0.552225\pi$
$710$	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
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−4.22947	−0.157514
$722$	0	0
$723$	0	0
$724$	−46.5675	−1.73067
$725$	0	0
$726$	0	0
$727$	9.06729	0.336287	0.168144	−	0.985763i	$-0.446223\pi$
$727$	9.06729	0.336287	0.168144	+	0.985763i	$0.446223\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−53.6757	−1.98256	−0.991279	−	0.131777i	$-0.957932\pi$
$733$	−53.6757	−1.98256	−0.991279	+	0.131777i	$0.957932\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−37.0000	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	49.0170	1.78866	0.894328	−	0.447412i	$-0.147654\pi$
$751$	49.0170	1.78866	0.894328	+	0.447412i	$0.147654\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−52.6776	−1.91460	−0.957301	−	0.289095i	$-0.906646\pi$
$757$	−52.6776	−1.91460	−0.957301	+	0.289095i	$0.906646\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	30.6936	1.11118
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−54.0547	−1.94926	−0.974631	−	0.223819i	$-0.928148\pi$
$769$	−54.0547	−1.94926	−0.974631	+	0.223819i	$0.928148\pi$
$770$	0	0
$771$	0	0
$772$	46.0000	1.65558
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−51.9615	−1.86651
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−6.64102	−0.237179
$785$	0	0
$786$	0	0
$787$	37.1485	1.32420	0.662100	−	0.749415i	$-0.269665\pi$
$787$	37.1485	1.32420	0.662100	+	0.749415i	$0.269665\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	20.3166	0.721465
$794$	0	0
$795$	0	0
$796$	55.8850	1.98079
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	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
$811$	−43.3013	−1.52051	−0.760257	−	0.649623i	$-0.774927\pi$
$811$	−43.3013	−1.52051	−0.760257	+	0.649623i	$0.774927\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−16.1577	−0.565286
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	−24.2487	−0.845257	−0.422628	−	0.906303i	$-0.638892\pi$
$823$	−24.2487	−0.845257	−0.422628	+	0.906303i	$0.638892\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	0	0
$832$	13.8564	0.480384
$833$	0	0
$834$	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
$842$	0	0
$843$	0	0
$844$	51.9615	1.78859
$845$	0	0
$846$	0	0
$847$	−25.4187	−0.873396
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−23.0000	−0.787505	−0.393753	−	0.919216i	$-0.628823\pi$
$853$	−23.0000	−0.787505	−0.393753	+	0.919216i	$0.628823\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−43.0000	−1.46714	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−43.0000	−1.46714	−0.733571	+	0.679613i	$0.762148\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	−48.0288	−1.63021
$869$	0	0
$870$	0	0
$871$	−28.3214	−0.959635
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	48.4974	1.63764	0.818821	−	0.574049i	$-0.194628\pi$
$877$	48.4974	1.63764	0.818821	+	0.574049i	$0.194628\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	58.8897	1.98180	0.990899	−	0.134611i	$-0.0429784\pi$
$883$	58.8897	1.98180	0.990899	+	0.134611i	$0.0429784\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	52.0313	1.74507
$890$	0	0
$891$	0	0
$892$	46.0000	1.54019
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−53.8516	−1.78811	−0.894057	−	0.447954i	$-0.852153\pi$
$907$	−53.8516	−1.78811	−0.894057	+	0.447954i	$0.852153\pi$
$908$	0	0
$909$	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
$916$	51.5408	1.70296
$917$	0	0
$918$	0	0
$919$	58.2973	1.92305	0.961526	−	0.274715i	$-0.0885836\pi$
$919$	58.2973	1.92305	0.961526	+	0.274715i	$0.0885836\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	57.2620	1.88276
$926$	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
$931$	5.32504	0.174521
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−39.4593	−1.28908	−0.644539	−	0.764571i	$-0.722951\pi$
$937$	−39.4593	−1.28908	−0.644539	+	0.764571i	$0.722951\pi$
$938$	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
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	77.0000	2.48387
$962$	0	0
$963$	0	0
$964$	−60.7840	−1.95772
$965$	0	0
$966$	0	0
$967$	−46.7654	−1.50387	−0.751936	−	0.659236i	$-0.770880\pi$
$967$	−46.7654	−1.50387	−0.751936	+	0.659236i	$0.770880\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−53.1481	−1.70385
$974$	0	0
$975$	0	0
$976$	−46.9192	−1.50185
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−11.1106	−0.353476
$989$	0	0
$990$	0	0
$991$	45.0333	1.43053	0.715265	−	0.698853i	$-0.246306\pi$
$991$	45.0333	1.43053	0.715265	+	0.698853i	$0.246306\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−49.0000	−1.55185	−0.775923	−	0.630828i	$-0.782715\pi$
$997$	−49.0000	−1.55185	−0.775923	+	0.630828i	$0.782715\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bc.1.3	yes	4
3.2	odd	2	CM	4761.2.a.bc.1.3	yes	4
23.22	odd	2	inner	4761.2.a.bc.1.2	✓	4
69.68	even	2	inner	4761.2.a.bc.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4761.2.a.bc.1.2	✓	4	23.22	odd	2	inner
4761.2.a.bc.1.2	✓	4	69.68	even	2	inner
4761.2.a.bc.1.3	yes	4	1.1	even	1	trivial
4761.2.a.bc.1.3	yes	4	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} +2.31079 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} +2.31079 q^{7} -1.73205 q^{13} +4.00000 q^{16} -3.20736 q^{19} -5.00000 q^{25} -4.62158 q^{28} +10.3923 q^{31} -11.4524 q^{37} +5.03768 q^{43} -1.66025 q^{49} +3.46410 q^{52} -11.7298 q^{61} -8.00000 q^{64} +16.3514 q^{67} -13.8564 q^{73} +6.41473 q^{76} +9.41902 q^{79} -4.00240 q^{91} -19.6975 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} - 32 q^{64}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 28 * q^49 - 32 * q^64

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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