LMFDB - Embedded newform 3969.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 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			$0.456850$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.18890 q^{2} +2.79129 q^{4} -1.73205 q^{8} +O(q^{10})$	$q-2.18890 q^{2} +2.79129 q^{4} -1.73205 q^{8} +6.10985 q^{11} -1.79129 q^{16} -13.3739 q^{22} +5.29150 q^{23} -5.00000 q^{25} -10.5830 q^{29} +7.38505 q^{32} -12.1652 q^{37} -10.5826 q^{43} +17.0544 q^{44} -11.5826 q^{46} +10.9445 q^{50} -3.36875 q^{53} +23.1652 q^{58} -12.5826 q^{64} +11.7477 q^{67} -11.2107 q^{71} +26.6283 q^{74} -17.7477 q^{79} +23.1642 q^{86} -10.5826 q^{88} +14.7701 q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4}+O(q^{10}) $ 4 * q + 2 * q^4	$ 4 q + 2 q^{4} + 2 q^{16} - 26 q^{22} - 20 q^{25} - 12 q^{37} - 24 q^{43} - 28 q^{46} + 56 q^{58} - 32 q^{64} - 8 q^{67} - 16 q^{79} - 24 q^{88}+O(q^{100}) $ 4 * q + 2 * q^4 + 2 * q^16 - 26 * q^22 - 20 * q^25 - 12 * q^37 - 24 * q^43 - 28 * q^46 + 56 * q^58 - 32 * q^64 - 8 * q^67 - 16 * q^79 - 24 * q^88

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 $

$2$	−2.18890	−1.54779	−0.773893	−	0.633316i	$-0.781693\pi$
$2$	−2.18890	−1.54779	−0.773893	+	0.633316i	$0.781693\pi$
$3$	0	0
$3$	0	0
$4$	2.79129	1.39564
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.73205	−0.612372
$9$	0	0
$10$	0	0
$11$	6.10985	1.84219	0.921095	−	0.389338i	$-0.127296\pi$
$11$	6.10985	1.84219	0.921095	+	0.389338i	$0.127296\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	−1.79129	−0.447822
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	−13.3739	−2.85132
$23$	5.29150	1.10335	0.551677	−	0.834058i	$-0.313988\pi$
$23$	5.29150	1.10335	0.551677	+	0.834058i	$0.313988\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.5830	−1.96521	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	−10.5830	−1.96521	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	7.38505	1.30551
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−12.1652	−1.99994	−0.999969	−	0.00783774i	$-0.997505\pi$
$37$	−12.1652	−1.99994	−0.999969	+	0.00783774i	$0.997505\pi$
$38$	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
$43$	−10.5826	−1.61383	−0.806914	−	0.590669i	$-0.798864\pi$
$43$	−10.5826	−1.61383	−0.806914	+	0.590669i	$0.798864\pi$
$44$	17.0544	2.57104
$45$	0	0
$46$	−11.5826	−1.70776
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	10.9445	1.54779
$51$	0	0
$52$	0	0
$53$	−3.36875	−0.462734	−0.231367	−	0.972867i	$-0.574320\pi$
$53$	−3.36875	−0.462734	−0.231367	+	0.972867i	$0.574320\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	23.1652	3.04173
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	−12.5826	−1.57282
$65$	0	0
$66$	0	0
$67$	11.7477	1.43521	0.717607	−	0.696449i	$-0.245238\pi$
$67$	11.7477	1.43521	0.717607	+	0.696449i	$0.245238\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.2107	−1.33046	−0.665230	−	0.746639i	$-0.731667\pi$
$71$	−11.2107	−1.33046	−0.665230	+	0.746639i	$0.731667\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	26.6283	3.09548
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−17.7477	−1.99678	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−17.7477	−1.99678	−0.998388	+	0.0567635i	$0.981922\pi$
$80$	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
$86$	23.1642	2.49786
$87$	0	0
$88$	−10.5826	−1.12811
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	14.7701	1.53989
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	−13.9564	−1.39564
$101$	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
$106$	7.37386	0.716213
$107$	19.9663	1.93021	0.965106	−	0.261861i	$-0.0843362\pi$
$107$	19.9663	1.93021	0.965106	+	0.261861i	$0.0843362\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.85095	−0.832628	−0.416314	−	0.909221i	$-0.636678\pi$
$113$	−8.85095	−0.832628	−0.416314	+	0.909221i	$0.636678\pi$
$114$	0	0
$115$	0	0
$116$	−29.5402	−2.74274
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	26.3303	2.39366
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.74773	0.510028	0.255014	−	0.966937i	$-0.417920\pi$
$127$	5.74773	0.510028	0.255014	+	0.966937i	$0.417920\pi$
$128$	12.7719	1.12889
$129$	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
$134$	−25.7146	−2.22140
$135$	0	0
$136$	0	0
$137$	−1.92275	−0.164272	−0.0821359	−	0.996621i	$-0.526174\pi$
$137$	−1.92275	−0.164272	−0.0821359	+	0.996621i	$0.526174\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	24.5390	2.05927
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−33.9564	−2.79120
$149$	−24.3441	−1.99434	−0.997172	−	0.0751583i	$-0.976054\pi$
$149$	−24.3441	−1.99434	−0.997172	+	0.0751583i	$0.976054\pi$
$150$	0	0
$151$	−7.41742	−0.603621	−0.301811	−	0.953368i	$-0.597591\pi$
$151$	−7.41742	−0.603621	−0.301811	+	0.953368i	$0.597591\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	38.8480	3.09058
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	23.7477	1.86007	0.930033	−	0.367477i	$-0.119778\pi$
$163$	23.7477	1.86007	0.930033	+	0.367477i	$0.119778\pi$
$164$	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
$172$	−29.5390	−2.25233
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	−10.9445	−0.824973
$177$	0	0
$178$	0	0
$179$	26.4575	1.97753	0.988764	−	0.149487i	$-0.0477622\pi$
$179$	26.4575	1.97753	0.988764	+	0.149487i	$0.0477622\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	−9.16515	−0.675664
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.30055	0.455892	0.227946	−	0.973674i	$-0.426799\pi$
$191$	6.30055	0.455892	0.227946	+	0.973674i	$0.426799\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−27.8082	−1.98125	−0.990625	−	0.136611i	$-0.956379\pi$
$197$	−27.8082	−1.98125	−0.990625	+	0.136611i	$0.956379\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	8.66025	0.612372
$201$	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
$211$	16.9129	1.16433	0.582165	−	0.813070i	$-0.302206\pi$
$211$	16.9129	1.16433	0.582165	+	0.813070i	$0.302206\pi$
$212$	−9.40315	−0.645811
$213$	0	0
$214$	−43.7042	−2.98756
$215$	0	0
$216$	0	0
$217$	0	0
$218$	39.4002	2.66852
$219$	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
$226$	19.3739	1.28873
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	18.3303	1.20344
$233$	−21.1660	−1.38663	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	−21.1660	−1.38663	−0.693316	+	0.720634i	$0.743851\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.627650	0.0405993	0.0202996	−	0.999794i	$-0.493538\pi$
$239$	0.627650	0.0405993	0.0202996	+	0.999794i	$0.493538\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−57.6344	−3.70488
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	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
$253$	32.3303	2.03259
$254$	−12.5812	−0.789415
$255$	0	0
$256$	−2.79129	−0.174455
$257$	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
$263$	25.0671	1.54570	0.772851	−	0.634588i	$-0.218830\pi$
$263$	25.0671	1.54570	0.772851	+	0.634588i	$0.218830\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	32.7913	2.00305
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	4.20871	0.254258
$275$	−30.5493	−1.84219
$276$	0	0
$277$	32.4955	1.95246	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	32.4955	1.95246	0.976231	+	0.216731i	$0.0695395\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	33.0997	1.97456	0.987280	−	0.158990i	$-0.0508236\pi$
$281$	33.0997	1.97456	0.987280	+	0.158990i	$0.0508236\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−31.2922	−1.85685
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	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
$296$	21.0707	1.22471
$297$	0	0
$298$	53.2867	3.08682
$299$	0	0
$300$	0	0
$301$	0	0
$302$	16.2360	0.934277
$303$	0	0
$304$	0	0
$305$	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
$316$	−49.5390	−2.78679
$317$	−10.5830	−0.594401	−0.297200	−	0.954815i	$-0.596053\pi$
$317$	−10.5830	−0.594401	−0.297200	+	0.954815i	$0.596053\pi$
$318$	0	0
$319$	−64.6606	−3.62030
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−51.9814	−2.87898
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−36.0000	−1.97874	−0.989369	−	0.145424i	$-0.953545\pi$
$331$	−36.0000	−1.97874	−0.989369	+	0.145424i	$0.953545\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	33.3303	1.81562	0.907809	−	0.419385i	$-0.137754\pi$
$337$	33.3303	1.81562	0.907809	+	0.419385i	$0.137754\pi$
$338$	28.4557	1.54779
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	18.3296	0.988264
$345$	0	0
$346$	0	0
$347$	15.0562	0.808257	0.404128	−	0.914702i	$-0.367575\pi$
$347$	15.0562	0.808257	0.404128	+	0.914702i	$0.367575\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	45.1216	2.40499
$353$	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
$358$	−57.9129	−3.06079
$359$	−11.5921	−0.611805	−0.305903	−	0.952063i	$-0.598958\pi$
$359$	−11.5921	−0.611805	−0.305903	+	0.952063i	$0.598958\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−9.47860	−0.494106
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−38.4955	−1.99322	−0.996610	−	0.0822766i	$-0.973781\pi$
$373$	−38.4955	−1.99322	−0.996610	+	0.0822766i	$0.973781\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−38.0780	−1.95594	−0.977969	−	0.208752i	$-0.933060\pi$
$379$	−38.0780	−1.95594	−0.977969	+	0.208752i	$0.933060\pi$
$380$	0	0
$381$	0	0
$382$	−13.7913	−0.705624
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	39.4002	2.00542
$387$	0	0
$388$	0	0
$389$	10.5830	0.536580	0.268290	−	0.963338i	$-0.413542\pi$
$389$	10.5830	0.536580	0.268290	+	0.963338i	$0.413542\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	60.8693	3.06655
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	8.95644	0.447822
$401$	−40.0279	−1.99890	−0.999448	−	0.0332161i	$-0.989425\pi$
$401$	−40.0279	−1.99890	−0.999448	+	0.0332161i	$0.989425\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−74.3273	−3.68427
$408$	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
$415$	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
$421$	−14.4955	−0.706465	−0.353233	−	0.935536i	$-0.614918\pi$
$421$	−14.4955	−0.706465	−0.353233	+	0.935536i	$0.614918\pi$
$422$	−37.0206	−1.80214
$423$	0	0
$424$	5.83485	0.283365
$425$	0	0
$426$	0	0
$427$	0	0
$428$	55.7316	2.69389
$429$	0	0
$430$	0	0
$431$	−26.4575	−1.27441	−0.637207	−	0.770693i	$-0.719910\pi$
$431$	−26.4575	−1.27441	−0.637207	+	0.770693i	$0.719910\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−50.2432	−2.40621
$437$	0	0
$438$	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
$443$	−37.0405	−1.75985	−0.879924	−	0.475114i	$-0.842407\pi$
$443$	−37.0405	−1.75985	−0.879924	+	0.475114i	$0.842407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.4340	0.917145	0.458573	−	0.888657i	$-0.348361\pi$
$449$	19.4340	0.917145	0.458573	+	0.888657i	$0.348361\pi$
$450$	0	0
$451$	0	0
$452$	−24.7056	−1.16205
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−39.6606	−1.85524	−0.927622	−	0.373519i	$-0.878151\pi$
$457$	−39.6606	−1.85524	−0.927622	+	0.373519i	$0.878151\pi$
$458$	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
$463$	6.25227	0.290568	0.145284	−	0.989390i	$-0.453590\pi$
$463$	6.25227	0.290568	0.145284	+	0.989390i	$0.453590\pi$
$464$	18.9572	0.880066
$465$	0	0
$466$	46.3303	2.14621
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−64.6580	−2.97298
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−1.37386	−0.0628391
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	73.4955	3.34070
$485$	0	0
$486$	0	0
$487$	20.0780	0.909822	0.454911	−	0.890537i	$-0.349671\pi$
$487$	20.0780	0.909822	0.454911	+	0.890537i	$0.349671\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.29150	0.238802	0.119401	−	0.992846i	$-0.461903\pi$
$491$	5.29150	0.238802	0.119401	+	0.992846i	$0.461903\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	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
$506$	−70.7678	−3.14601
$507$	0	0
$508$	16.0436	0.711818
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	−19.4340	−0.858868
$513$	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
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	−54.8693	−2.39242
$527$	0	0
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−20.3477	−0.878885
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	44.4955	1.91301	0.956504	−	0.291718i	$-0.0942267\pi$
$541$	44.4955	1.91301	0.956504	+	0.291718i	$0.0942267\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−35.7477	−1.52846	−0.764231	−	0.644942i	$-0.776881\pi$
$547$	−35.7477	−1.52846	−0.764231	+	0.644942i	$0.776881\pi$
$548$	−5.36695	−0.229265
$549$	0	0
$550$	66.8693	2.85132
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−71.1293	−3.02200
$555$	0	0
$556$	0	0
$557$	34.5457	1.46375	0.731873	−	0.681441i	$-0.238646\pi$
$557$	34.5457	1.46375	0.731873	+	0.681441i	$0.238646\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−72.4519	−3.05620
$563$	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
$568$	19.4174	0.814737
$569$	−2.11345	−0.0886005	−0.0443003	−	0.999018i	$-0.514106\pi$
$569$	−2.11345	−0.0886005	−0.0443003	+	0.999018i	$0.514106\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−26.4575	−1.10335
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	37.2113	1.54779
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−20.5826	−0.852443
$584$	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
$589$	0	0
$590$	0	0
$591$	0	0
$592$	21.7913	0.895616
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−67.9513	−2.78339
$597$	0	0
$598$	0	0
$599$	−46.2331	−1.88903	−0.944516	−	0.328465i	$-0.893469\pi$
$599$	−46.2331	−1.88903	−0.944516	+	0.328465i	$0.893469\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	−20.7042	−0.842441
$605$	0	0
$606$	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
$613$	−8.49545	−0.343128	−0.171564	−	0.985173i	$-0.554882\pi$
$613$	−8.49545	−0.343128	−0.171564	+	0.985173i	$0.554882\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.3320	1.70422	0.852111	−	0.523360i	$-0.175322\pi$
$617$	42.3320	1.70422	0.852111	+	0.523360i	$0.175322\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	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
$628$	0	0
$629$	0	0
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	30.7400	1.22277
$633$	0	0
$634$	23.1652	0.920006
$635$	0	0
$636$	0	0
$637$	0	0
$638$	141.536	5.60345
$639$	0	0
$640$	0	0
$641$	−29.2542	−1.15547	−0.577735	−	0.816224i	$-0.696063\pi$
$641$	−29.2542	−1.15547	−0.577735	+	0.816224i	$0.696063\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	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
$652$	66.2867	2.59599
$653$	−38.0098	−1.48744	−0.743719	−	0.668493i	$-0.766940\pi$
$653$	−38.0098	−1.48744	−0.743719	+	0.668493i	$0.766940\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.8764	−0.969045	−0.484523	−	0.874779i	$-0.661007\pi$
$659$	−24.8764	−0.969045	−0.484523	+	0.874779i	$0.661007\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	78.8004	3.06267
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−56.0000	−2.16833
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−21.6606	−0.834955	−0.417477	−	0.908687i	$-0.637086\pi$
$673$	−21.6606	−0.834955	−0.417477	+	0.908687i	$0.637086\pi$
$674$	−72.9567	−2.81019
$675$	0	0
$676$	−36.2867	−1.39564
$677$	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
$683$	42.3876	1.62192	0.810958	−	0.585105i	$-0.198947\pi$
$683$	42.3876	1.62192	0.810958	+	0.585105i	$0.198947\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	18.9564	0.722707
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	−32.9564	−1.25101
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−28.1896	−1.06471	−0.532353	−	0.846523i	$-0.678692\pi$
$701$	−28.1896	−1.06471	−0.532353	+	0.846523i	$0.678692\pi$
$702$	0	0
$703$	0	0
$704$	−76.8777	−2.89744
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	42.8258	1.60836	0.804178	−	0.594389i	$-0.202606\pi$
$709$	42.8258	1.60836	0.804178	+	0.594389i	$0.202606\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	73.8505	2.75992
$717$	0	0
$718$	25.3739	0.946944
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	41.5891	1.54779
$723$	0	0
$724$	0	0
$725$	52.9150	1.96521
$726$	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
$736$	39.0780	1.44044
$737$	71.7769	2.64394
$738$	0	0
$739$	12.2523	0.450707	0.225354	−	0.974277i	$-0.427646\pi$
$739$	12.2523	0.450707	0.225354	+	0.974277i	$0.427646\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.1208	0.591413	0.295707	−	0.955279i	$-0.404445\pi$
$743$	16.1208	0.591413	0.295707	+	0.955279i	$0.404445\pi$
$744$	0	0
$745$	0	0
$746$	84.2627	3.08508
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.08712	−0.0396696	−0.0198348	−	0.999803i	$-0.506314\pi$
$751$	−1.08712	−0.0396696	−0.0198348	+	0.999803i	$0.506314\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	54.0000	1.96266	0.981332	−	0.192323i	$-0.0616021\pi$
$757$	54.0000	1.96266	0.981332	+	0.192323i	$0.0616021\pi$
$758$	83.3490	3.02737
$759$	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
$764$	17.5867	0.636263
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	−50.2432	−1.80829
$773$	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
$778$	−23.1652	−0.830511
$779$	0	0
$780$	0	0
$781$	−68.4955	−2.45096
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	−77.6206	−2.76512
$789$	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
$799$	0	0
$800$	−36.9253	−1.30551
$801$	0	0
$802$	87.6170	3.09387
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	11.7430	0.412860	0.206430	−	0.978461i	$-0.433815\pi$
$809$	11.7430	0.412860	0.206430	+	0.978461i	$0.433815\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	162.695	5.70246
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.5101	1.58831	0.794156	−	0.607714i	$-0.207913\pi$
$821$	45.5101	1.58831	0.794156	+	0.607714i	$0.207913\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.5849	0.681032	0.340516	−	0.940239i	$-0.389398\pi$
$827$	19.5849	0.681032	0.340516	+	0.940239i	$0.389398\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	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
$841$	83.0000	2.86207
$842$	31.7291	1.09346
$843$	0	0
$844$	47.2087	1.62499
$845$	0	0
$846$	0	0
$847$	0	0
$848$	6.03440	0.207222
$849$	0	0
$850$	0	0
$851$	−64.3719	−2.20664
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−34.5826	−1.18201
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	57.9129	1.97252
$863$	36.0315	1.22653	0.613263	−	0.789879i	$-0.289857\pi$
$863$	36.0315	1.22653	0.613263	+	0.789879i	$0.289857\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−108.436	−3.67844
$870$	0	0
$871$	0	0
$872$	31.1769	1.05578
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.49545	0.0842655	0.0421327	−	0.999112i	$-0.486585\pi$
$877$	2.49545	0.0842655	0.0421327	+	0.999112i	$0.486585\pi$
$878$	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
$883$	44.4083	1.49446	0.747230	−	0.664566i	$-0.231383\pi$
$883$	44.4083	1.49446	0.747230	+	0.664566i	$0.231383\pi$
$884$	0	0
$885$	0	0
$886$	81.0780	2.72387
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−42.5390	−1.41955
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	15.3303	0.509878
$905$	0	0
$906$	0	0
$907$	25.4174	0.843972	0.421986	−	0.906602i	$-0.361333\pi$
$907$	25.4174	0.843972	0.421986	+	0.906602i	$0.361333\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	58.2065	1.92847	0.964234	−	0.265052i	$-0.0853891\pi$
$911$	58.2065	1.92847	0.964234	+	0.265052i	$0.0853891\pi$
$912$	0	0
$913$	0	0
$914$	86.8131	2.87152
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−56.0780	−1.84984	−0.924922	−	0.380158i	$-0.875870\pi$
$919$	−56.0780	−1.84984	−0.924922	+	0.380158i	$0.875870\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	60.8258	1.99994
$926$	−13.6856	−0.449737
$927$	0	0
$928$	−78.1561	−2.56560
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	−59.0804	−1.93524
$933$	0	0
$934$	0	0
$935$	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
$946$	141.530	4.60153
$947$	58.2065	1.89146	0.945729	−	0.324956i	$-0.105350\pi$
$947$	58.2065	1.89146	0.945729	+	0.324956i	$0.105350\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21.1660	−0.685634	−0.342817	−	0.939402i	$-0.611381\pi$
$953$	−21.1660	−0.685634	−0.342817	+	0.939402i	$0.611381\pi$
$954$	0	0
$955$	0	0
$956$	1.75195	0.0566622
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−45.6054	−1.46581
$969$	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
$974$	−43.9488	−1.40821
$975$	0	0
$976$	0	0
$977$	42.3320	1.35432	0.677161	−	0.735835i	$-0.263210\pi$
$977$	42.3320	1.35432	0.677161	+	0.735835i	$0.263210\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−11.5826	−0.369615
$983$	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
$988$	0	0
$989$	−55.9977	−1.78062
$990$	0	0
$991$	−62.4083	−1.98247	−0.991233	−	0.132125i	$-0.957820\pi$
$991$	−62.4083	−1.98247	−0.991233	+	0.132125i	$0.957820\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	78.8004	2.49438
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.v.1.1	✓	4
3.2	odd	2	inner	3969.2.a.v.1.4	yes	4
7.6	odd	2	CM	3969.2.a.v.1.1	✓	4
21.20	even	2	inner	3969.2.a.v.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3969.2.a.v.1.1	✓	4	1.1	even	1	trivial
3969.2.a.v.1.1	✓	4	7.6	odd	2	CM
3969.2.a.v.1.4	yes	4	3.2	odd	2	inner
3969.2.a.v.1.4	yes	4	21.20	even	2	inner

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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18890 q^{2} +2.79129 q^{4} -1.73205 q^{8} +O(q^{10})\)	\(q-2.18890 q^{2} +2.79129 q^{4} -1.73205 q^{8} +6.10985 q^{11} -1.79129 q^{16} -13.3739 q^{22} +5.29150 q^{23} -5.00000 q^{25} -10.5830 q^{29} +7.38505 q^{32} -12.1652 q^{37} -10.5826 q^{43} +17.0544 q^{44} -11.5826 q^{46} +10.9445 q^{50} -3.36875 q^{53} +23.1652 q^{58} -12.5826 q^{64} +11.7477 q^{67} -11.2107 q^{71} +26.6283 q^{74} -17.7477 q^{79} +23.1642 q^{86} -10.5826 q^{88} +14.7701 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4}+O(q^{10}) \) 4 * q + 2 * q^4	\( 4 q + 2 q^{4} + 2 q^{16} - 26 q^{22} - 20 q^{25} - 12 q^{37} - 24 q^{43} - 28 q^{46} + 56 q^{58} - 32 q^{64} - 8 q^{67} - 16 q^{79} - 24 q^{88}+O(q^{100}) \) 4 * q + 2 * q^4 + 2 * q^16 - 26 * q^22 - 20 * q^25 - 12 * q^37 - 24 * q^43 - 28 * q^46 + 56 * q^58 - 32 * q^64 - 8 * q^67 - 16 * q^79 - 24 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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