LMFDB - Embedded newform 304.7.e.b.113.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,7,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not 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:	yes
Analytic conductor:	$69.9364414204$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			113.2
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	304.113

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+238.395 q^{5} +33.2060 q^{7} +729.000 q^{9} +O(q^{10})$	$q+238.395 q^{5} +33.2060 q^{7} +729.000 q^{9} +1582.95 q^{11} +6506.16 q^{17} +6859.00 q^{19} -20610.0 q^{23} +41207.3 q^{25} +7916.15 q^{35} -10433.1 q^{43} +173790. q^{45} -205212. q^{47} -116546. q^{49} +377369. q^{55} -361493. q^{61} +24207.1 q^{63} -778013. q^{73} +52563.5 q^{77} +531441. q^{81} +1.13103e6 q^{83} +1.55104e6 q^{85} +1.63515e6 q^{95} +1.15397e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 54 q^{5} + 610 q^{7} + 1458 q^{9}+O(q^{10}) $ 2 * q + 54 * q^5 + 610 * q^7 + 1458 * q^9	$ 2 q + 54 q^{5} + 610 q^{7} + 1458 q^{9} - 1062 q^{11} + 9630 q^{17} + 13718 q^{19} - 41220 q^{23} + 59584 q^{25} - 98442 q^{35} - 142630 q^{43} + 39366 q^{45} - 75150 q^{47} + 98496 q^{49} + 865086 q^{55} + 57062 q^{61} + 444690 q^{63} - 384050 q^{73} - 1473030 q^{77} + 1062882 q^{81} + 2262060 q^{83} + 975018 q^{85} + 370386 q^{95} - 774198 q^{99}+O(q^{100}) $ 2 * q + 54 * q^5 + 610 * q^7 + 1458 * q^9 - 1062 * q^11 + 9630 * q^17 + 13718 * q^19 - 41220 * q^23 + 59584 * q^25 - 98442 * q^35 - 142630 * q^43 + 39366 * q^45 - 75150 * q^47 + 98496 * q^49 + 865086 * q^55 + 57062 * q^61 + 444690 * q^63 - 384050 * q^73 - 1473030 * q^77 + 1062882 * q^81 + 2262060 * q^83 + 975018 * q^85 + 370386 * q^95 - 774198 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	238.395	1.90716	0.953581	−	0.301135i	$-0.0973655\pi$
$5$	238.395	1.90716	0.953581	+	0.301135i	$0.0973655\pi$
$6$	0	0
$7$	33.2060	0.0968104	0.0484052	−	0.998828i	$-0.484586\pi$
$7$	33.2060	0.0968104	0.0484052	+	0.998828i	$0.484586\pi$
$8$	0	0
$9$	729.000	1.00000
$10$	0	0
$11$	1582.95	1.18930	0.594648	−	0.803986i	$-0.297291\pi$
$11$	1582.95	1.18930	0.594648	+	0.803986i	$0.297291\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6506.16	1.32427	0.662137	−	0.749382i	$-0.269649\pi$
$17$	6506.16	1.32427	0.662137	+	0.749382i	$0.269649\pi$
$18$	0	0
$19$	6859.00	1.00000
$19$	6859.00	1.00000
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−20610.0	−1.69393	−0.846963	−	0.531652i	$-0.821572\pi$
$23$	−20610.0	−1.69393	−0.846963	+	0.531652i	$0.821572\pi$
$24$	0	0
$25$	41207.3	2.63727
$26$	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	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7916.15	0.184633
$36$	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	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−10433.1	−0.131223	−0.0656114	−	0.997845i	$-0.520900\pi$
$43$	−10433.1	−0.131223	−0.0656114	+	0.997845i	$0.520900\pi$
$44$	0	0
$45$	173790.	1.90716
$46$	0	0
$47$	−205212.	−1.97655	−0.988276	−	0.152679i	$-0.951210\pi$
$47$	−205212.	−1.97655	−0.988276	+	0.152679i	$0.951210\pi$
$48$	0	0
$49$	−116546.	−0.990628
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	377369.	2.26818
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−361493.	−1.59262	−0.796308	−	0.604892i	$-0.793216\pi$
$61$	−361493.	−1.59262	−0.796308	+	0.604892i	$0.793216\pi$
$62$	0	0
$63$	24207.1	0.0968104
$64$	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
$73$	−778013.	−1.99995	−0.999973	−	0.00735603i	$-0.997658\pi$
$73$	−778013.	−1.99995	−0.999973	+	0.00735603i	$0.997658\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	52563.5	0.115136
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	531441.	1.00000
$82$	0	0
$83$	1.13103e6	1.97806	0.989031	−	0.147709i	$-0.0471899\pi$
$83$	1.13103e6	1.97806	0.989031	+	0.147709i	$0.0471899\pi$
$84$	0	0
$85$	1.55104e6	2.52561
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.63515e6	1.90716
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	1.15397e6	1.18930
$100$	0	0
$101$	2.06030e6	1.99970	0.999852	−	0.0171767i	$-0.00546777\pi$
$101$	2.06030e6	1.99970	0.999852	+	0.0171767i	$0.00546777\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	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
$115$	−4.91333e6	−3.23059
$116$	0	0
$117$	0	0
$118$	0	0
$119$	216043.	0.128204
$120$	0	0
$121$	734181.	0.414426
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.09871e6	3.12254
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.37804e6	1.94745	0.973724	−	0.227730i	$-0.0731305\pi$
$131$	4.37804e6	1.94745	0.973724	+	0.227730i	$0.0731305\pi$
$132$	0	0
$133$	227760.	0.0968104
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.12763e6	−1.99414	−0.997069	−	0.0765040i	$-0.975624\pi$
$137$	−5.12763e6	−1.99414	−0.997069	+	0.0765040i	$0.975624\pi$
$138$	0	0
$139$	−1.42378e6	−0.530151	−0.265075	−	0.964228i	$-0.585397\pi$
$139$	−1.42378e6	−0.530151	−0.265075	+	0.964228i	$0.585397\pi$
$140$	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
$149$	−5.01691e6	−1.51662	−0.758311	−	0.651893i	$-0.773975\pi$
$149$	−5.01691e6	−1.51662	−0.758311	+	0.651893i	$0.773975\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	4.74299e6	1.32427
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−738470.	−0.190824	−0.0954122	−	0.995438i	$-0.530417\pi$
$157$	−738470.	−0.190824	−0.0954122	+	0.995438i	$0.530417\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−684375.	−0.163990
$162$	0	0
$163$	4.30175e6	0.993304	0.496652	−	0.867950i	$-0.334562\pi$
$163$	4.30175e6	0.993304	0.496652	+	0.867950i	$0.334562\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	4.82681e6	1.00000
$170$	0	0
$171$	5.00021e6	1.00000
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	1.36833e6	0.255315
$176$	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	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.02990e7	1.57496
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.36173e7	1.95430	0.977151	−	0.212548i	$-0.0681763\pi$
$191$	1.36173e7	1.95430	0.977151	+	0.212548i	$0.0681763\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.74943e6	−1.27521	−0.637603	−	0.770365i	$-0.720074\pi$
$197$	−9.74943e6	−1.27521	−0.637603	+	0.770365i	$0.720074\pi$
$198$	0	0
$199$	−4.25856e6	−0.540385	−0.270193	−	0.962806i	$-0.587087\pi$
$199$	−4.25856e6	−0.540385	−0.270193	+	0.962806i	$0.587087\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.50247e7	−1.69393
$208$	0	0
$209$	1.08575e7	1.18930
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.48721e6	−0.250263
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	3.00402e7	2.63727
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−2.20061e7	−1.83247	−0.916234	−	0.400644i	$-0.868786\pi$
$229$	−2.20061e7	−1.83247	−0.916234	+	0.400644i	$0.868786\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.12667e6	0.642458	0.321229	−	0.947002i	$-0.395904\pi$
$233$	8.12667e6	0.642458	0.321229	+	0.947002i	$0.395904\pi$
$234$	0	0
$235$	−4.89215e7	−3.76961
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.18996e7	0.871641	0.435821	−	0.900033i	$-0.356458\pi$
$239$	1.18996e7	0.871641	0.435821	+	0.900033i	$0.356458\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.77841e7	−1.88929
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.95749e7	−1.87026	−0.935131	−	0.354301i	$-0.884719\pi$
$251$	−2.95749e7	−1.87026	−0.935131	+	0.354301i	$0.884719\pi$
$252$	0	0
$253$	−3.26247e7	−2.01458
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.87666e7	−1.03161	−0.515807	−	0.856705i	$-0.672508\pi$
$263$	−1.87666e7	−1.03161	−0.515807	+	0.856705i	$0.672508\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−2.84226e7	−1.42809	−0.714045	−	0.700100i	$-0.753139\pi$
$271$	−2.84226e7	−1.42809	−0.714045	+	0.700100i	$0.753139\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.52293e7	3.13650
$276$	0	0
$277$	1.11072e7	0.522596	0.261298	−	0.965258i	$-0.415849\pi$
$277$	1.11072e7	0.522596	0.261298	+	0.965258i	$0.415849\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−4.32964e7	−1.91026	−0.955131	−	0.296184i	$-0.904286\pi$
$283$	−4.32964e7	−1.91026	−0.955131	+	0.296184i	$0.904286\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.81926e7	0.753704
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−346442.	−0.0127037
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.61784e7	−3.03738
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.74058e7	1.90842	0.954211	−	0.299134i	$-0.0966979\pi$
$311$	5.74058e7	1.90842	0.954211	+	0.299134i	$0.0966979\pi$
$312$	0	0
$313$	−3.19739e7	−1.04271	−0.521353	−	0.853341i	$-0.674573\pi$
$313$	−3.19739e7	−1.04271	−0.521353	+	0.853341i	$0.674573\pi$
$314$	0	0
$315$	5.77087e6	0.184633
$316$	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
$323$	4.46258e7	1.32427
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.81425e6	−0.191351
$330$	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
$343$	−7.77668e6	−0.192713
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.49609e7	−0.358072	−0.179036	−	0.983843i	$-0.557298\pi$
$347$	−1.49609e7	−0.358072	−0.179036	+	0.983843i	$0.557298\pi$
$348$	0	0
$349$	8.49067e7	1.99740	0.998702	−	0.0509439i	$-0.0162230\pi$
$349$	8.49067e7	1.99740	0.998702	+	0.0509439i	$0.0162230\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.80008e7	1.31859	0.659295	−	0.751885i	$-0.270855\pi$
$353$	5.80008e7	1.31859	0.659295	+	0.751885i	$0.270855\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.06622e7	−1.74336	−0.871680	−	0.490076i	$-0.836969\pi$
$359$	−8.06622e7	−1.74336	−0.871680	+	0.490076i	$0.836969\pi$
$360$	0	0
$361$	4.70459e7	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.85475e8	−3.81422
$366$	0	0
$367$	2.00784e7	0.406191	0.203095	−	0.979159i	$-0.434900\pi$
$367$	2.00784e7	0.406191	0.203095	+	0.979159i	$0.434900\pi$
$368$	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	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	1.25309e7	0.219584
$386$	0	0
$387$	−7.60576e6	−0.131223
$388$	0	0
$389$	5.15626e7	0.875964	0.437982	−	0.898984i	$-0.355693\pi$
$389$	5.15626e7	0.875964	0.437982	+	0.898984i	$0.355693\pi$
$390$	0	0
$391$	−1.34092e8	−2.24322
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.38946e7	−1.02116	−0.510579	−	0.859831i	$-0.670569\pi$
$397$	−6.38946e7	−1.02116	−0.510579	+	0.859831i	$0.670569\pi$
$398$	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
$405$	1.26693e8	1.90716
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.69632e8	3.77249
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.11183e7	−0.558976	−0.279488	−	0.960149i	$-0.590165\pi$
$419$	−4.11183e7	−0.558976	−0.279488	+	0.960149i	$0.590165\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	−1.49599e8	−1.97655
$424$	0	0
$425$	2.68102e8	3.49247
$426$	0	0
$427$	−1.20037e7	−0.154182
$428$	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	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.41364e8	−1.69393
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−8.49623e7	−0.990628
$442$	0	0
$443$	−1.35634e8	−1.56012	−0.780060	−	0.625705i	$-0.784812\pi$
$443$	−1.35634e8	−1.56012	−0.780060	+	0.625705i	$0.784812\pi$
$444$	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
$457$	3.12613e7	0.327536	0.163768	−	0.986499i	$-0.447635\pi$
$457$	3.12613e7	0.327536	0.163768	+	0.986499i	$0.447635\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.89605e7	0.908018	0.454009	−	0.890997i	$-0.349993\pi$
$461$	8.89605e7	0.908018	0.454009	+	0.890997i	$0.349993\pi$
$462$	0	0
$463$	1.37548e8	1.38583	0.692917	−	0.721018i	$-0.256325\pi$
$463$	1.37548e8	1.38583	0.692917	+	0.721018i	$0.256325\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.84202e8	1.80861	0.904304	−	0.426889i	$-0.140390\pi$
$467$	1.84202e8	1.80861	0.904304	+	0.426889i	$0.140390\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.65152e7	−0.156063
$474$	0	0
$475$	2.82641e8	2.63727
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.87497e8	1.70603	0.853015	−	0.521886i	$-0.174771\pi$
$479$	1.87497e8	1.70603	0.853015	+	0.521886i	$0.174771\pi$
$480$	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
$491$	1.09684e8	0.926610	0.463305	−	0.886199i	$-0.346663\pi$
$491$	1.09684e8	0.926610	0.463305	+	0.886199i	$0.346663\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	2.75102e8	2.26818
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.05758e8	−0.851162	−0.425581	−	0.904920i	$-0.639930\pi$
$499$	−1.05758e8	−0.851162	−0.425581	+	0.904920i	$0.639930\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.84619e7	−0.773685	−0.386843	−	0.922146i	$-0.626434\pi$
$503$	−9.84619e7	−0.773685	−0.386843	+	0.922146i	$0.626434\pi$
$504$	0	0
$505$	4.91165e8	3.81376
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−2.58347e7	−0.193616
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.24840e8	−2.35071
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	2.76736e8	1.86939
$530$	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
$539$	−1.84487e8	−1.17815
$540$	0	0
$541$	−2.24114e8	−1.41539	−0.707697	−	0.706516i	$-0.750266\pi$
$541$	−2.24114e8	−1.41539	−0.707697	+	0.706516i	$0.750266\pi$
$542$	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
$549$	−2.63529e8	−1.59262
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.04570e8	−1.76247	−0.881234	−	0.472680i	$-0.843287\pi$
$557$	−3.04570e8	−1.76247	−0.881234	+	0.472680i	$0.843287\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.76470e7	0.0968104
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	3.51908e8	1.89026	0.945130	−	0.326696i	$-0.105935\pi$
$571$	3.51908e8	1.89026	0.945130	+	0.326696i	$0.105935\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.49283e8	−4.46734
$576$	0	0
$577$	3.00600e8	1.56481	0.782405	−	0.622770i	$-0.213993\pi$
$577$	3.00600e8	1.56481	0.782405	+	0.622770i	$0.213993\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.75569e7	0.191497
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.98176e8	−1.96861	−0.984307	−	0.176463i	$-0.943535\pi$
$587$	−3.98176e8	−1.96861	−0.984307	+	0.176463i	$0.943535\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.16214e7	0.151641	0.0758206	−	0.997121i	$-0.475842\pi$
$593$	3.16214e7	0.151641	0.0758206	+	0.997121i	$0.475842\pi$
$594$	0	0
$595$	5.15037e7	0.244505
$596$	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	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.75025e8	0.790378
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.44124e8	−0.625685	−0.312842	−	0.949805i	$-0.601281\pi$
$613$	−1.44124e8	−0.625685	−0.312842	+	0.949805i	$0.601281\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.10945e8	−1.74956	−0.874778	−	0.484524i	$-0.838993\pi$
$617$	−4.10945e8	−1.74956	−0.874778	+	0.484524i	$0.838993\pi$
$618$	0	0
$619$	−4.70840e8	−1.98519	−0.992594	−	0.121480i	$-0.961236\pi$
$619$	−4.70840e8	−1.98519	−0.992594	+	0.121480i	$0.961236\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.10040e8	3.31792
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−3.60046e8	−1.43308	−0.716539	−	0.697547i	$-0.754275\pi$
$631$	−3.60046e8	−1.43308	−0.716539	+	0.697547i	$0.754275\pi$
$632$	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
$643$	4.62058e8	1.73805	0.869027	−	0.494764i	$-0.164746\pi$
$643$	4.62058e8	1.73805	0.869027	+	0.494764i	$0.164746\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.41002e8	1.99750	0.998748	−	0.0500307i	$-0.0159319\pi$
$647$	5.41002e8	1.99750	0.998748	+	0.0500307i	$0.0159319\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.23107e8	1.87867	0.939335	−	0.343000i	$-0.111443\pi$
$653$	5.23107e8	1.87867	0.939335	+	0.343000i	$0.111443\pi$
$654$	0	0
$655$	1.04370e9	3.71410
$656$	0	0
$657$	−5.67171e8	−1.99995
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.42969e7	0.184633
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.72227e8	−1.89409
$672$	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	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	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
$685$	−1.22240e9	−3.80315
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−4.27950e8	−1.29706	−0.648529	−	0.761190i	$-0.724615\pi$
$691$	−4.27950e8	−1.29706	−0.648529	+	0.761190i	$0.724615\pi$
$692$	0	0
$693$	3.83188e7	0.115136
$694$	0	0
$695$	−3.39423e8	−1.01108
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.94922e8	−0.856156	−0.428078	−	0.903742i	$-0.640809\pi$
$701$	−2.94922e8	−0.856156	−0.428078	+	0.903742i	$0.640809\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.84142e7	0.193592
$708$	0	0
$709$	−3.01929e8	−0.847161	−0.423580	−	0.905859i	$-0.639227\pi$
$709$	−3.01929e8	−0.847161	−0.423580	+	0.905859i	$0.639227\pi$
$710$	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
$719$	−6.79790e8	−1.82889	−0.914446	−	0.404708i	$-0.867373\pi$
$719$	−6.79790e8	−1.82889	−0.914446	+	0.404708i	$0.867373\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.22384e8	1.88003	0.940016	−	0.341132i	$-0.110810\pi$
$727$	7.22384e8	1.88003	0.940016	+	0.341132i	$0.110810\pi$
$728$	0	0
$729$	3.87420e8	1.00000
$730$	0	0
$731$	−6.78797e7	−0.173775
$732$	0	0
$733$	−1.31191e6	−0.00333113	−0.00166557	−	0.999999i	$-0.500530\pi$
$733$	−1.31191e6	−0.00333113	−0.00166557	+	0.999999i	$0.500530\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−6.59861e8	−1.63501	−0.817503	−	0.575924i	$-0.804642\pi$
$739$	−6.59861e8	−1.63501	−0.817503	+	0.575924i	$0.804642\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−1.19601e9	−2.89245
$746$	0	0
$747$	8.24521e8	1.97806
$748$	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
$757$	−3.84490e8	−0.886334	−0.443167	−	0.896439i	$-0.646145\pi$
$757$	−3.84490e8	−0.886334	−0.443167	+	0.896439i	$0.646145\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.40796e8	−1.68091	−0.840455	−	0.541881i	$-0.817712\pi$
$761$	−7.40796e8	−1.68091	−0.840455	+	0.541881i	$0.817712\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.13071e9	2.52561
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1.76120e8	−0.387283	−0.193642	−	0.981072i	$-0.562030\pi$
$769$	−1.76120e8	−0.387283	−0.193642	+	0.981072i	$0.562030\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	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
$785$	−1.76048e8	−0.363933
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	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	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	−1.33514e9	−2.61750
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.23156e9	−2.37853
$804$	0	0
$805$	−1.63152e8	−0.312755
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.19427e8	1.73649	0.868244	−	0.496137i	$-0.165249\pi$
$809$	9.19427e8	1.73649	0.868244	+	0.496137i	$0.165249\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.02552e9	1.89439
$816$	0	0
$817$	−7.15609e7	−0.131223
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.07334e9	1.93958	0.969788	−	0.243949i	$-0.0784431\pi$
$821$	1.07334e9	1.93958	0.969788	+	0.243949i	$0.0784431\pi$
$822$	0	0
$823$	−1.10907e9	−1.98958	−0.994789	−	0.101956i	$-0.967490\pi$
$823$	−1.10907e9	−1.98958	−0.994789	+	0.101956i	$0.967490\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.58270e8	−1.31186
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	5.94823e8	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.15069e9	1.90716
$846$	0	0
$847$	2.43792e7	0.0401208
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1.15558e9	1.86189	0.930947	−	0.365156i	$-0.118984\pi$
$853$	1.15558e9	1.86189	0.930947	+	0.365156i	$0.118984\pi$
$854$	0	0
$855$	1.19203e9	1.90716
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1.08616e9	1.71362	0.856811	−	0.515631i	$-0.172442\pi$
$859$	1.08616e9	1.71362	0.856811	+	0.515631i	$0.172442\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.02514e8	0.302294
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.25670e9	−1.83782	−0.918912	−	0.394463i	$-0.870931\pi$
$881$	−1.25670e9	−1.83782	−0.918912	+	0.394463i	$0.870931\pi$
$882$	0	0
$883$	−5.74337e8	−0.834228	−0.417114	−	0.908854i	$-0.636958\pi$
$883$	−5.74337e8	−0.834228	−0.417114	+	0.908854i	$0.636958\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	8.41246e8	1.18930
$892$	0	0
$893$	−1.40755e9	−1.97655
$894$	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
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	1.50196e9	1.99970
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	1.79037e9	2.35250
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.45377e8	0.188533
$918$	0	0
$919$	−1.00603e9	−1.29618	−0.648091	−	0.761563i	$-0.724432\pi$
$919$	−1.00603e9	−1.29618	−0.648091	+	0.761563i	$0.724432\pi$
$920$	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
$929$	−1.39761e9	−1.74316	−0.871582	−	0.490250i	$-0.836905\pi$
$929$	−1.39761e9	−1.74316	−0.871582	+	0.490250i	$0.836905\pi$
$930$	0	0
$931$	−7.99392e8	−0.990628
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.45522e9	3.00370
$936$	0	0
$937$	1.64511e9	1.99976	0.999879	−	0.0155847i	$-0.00496097\pi$
$937$	1.64511e9	1.99976	0.999879	+	0.0155847i	$0.00496097\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.20501e9	1.41886	0.709429	−	0.704777i	$-0.248953\pi$
$947$	1.20501e9	1.41886	0.709429	+	0.704777i	$0.248953\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	3.24631e9	3.72717
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.70268e8	−0.193053
$960$	0	0
$961$	8.87504e8	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.13911e8	0.789523	0.394761	−	0.918784i	$-0.370827\pi$
$967$	7.13911e8	0.789523	0.394761	+	0.918784i	$0.370827\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	−4.72781e7	−0.0513241
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−2.32422e9	−2.43203
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.15027e8	0.222282
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.01522e9	−1.03060
$996$	0	0
$997$	−1.59837e9	−1.61284	−0.806419	−	0.591344i	$-0.798597\pi$
$997$	−1.59837e9	−1.61284	−0.806419	+	0.591344i	$0.798597\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.b.113.2		2
4.3	odd	2		76.7.c.a.37.2	✓	2
12.11	even	2		684.7.h.a.37.1		2
19.18	odd	2	CM	304.7.e.b.113.2		2
76.75	even	2		76.7.c.a.37.2	✓	2
228.227	odd	2		684.7.h.a.37.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.c.a.37.2	✓	2	4.3	odd	2
76.7.c.a.37.2	✓	2	76.75	even	2
304.7.e.b.113.2		2	1.1	even	1	trivial
304.7.e.b.113.2		2	19.18	odd	2	CM
684.7.h.a.37.1		2	12.11	even	2
684.7.h.a.37.1		2	228.227	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+238.395 q^{5} +33.2060 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+238.395 q^{5} +33.2060 q^{7} +729.000 q^{9} +1582.95 q^{11} +6506.16 q^{17} +6859.00 q^{19} -20610.0 q^{23} +41207.3 q^{25} +7916.15 q^{35} -10433.1 q^{43} +173790. q^{45} -205212. q^{47} -116546. q^{49} +377369. q^{55} -361493. q^{61} +24207.1 q^{63} -778013. q^{73} +52563.5 q^{77} +531441. q^{81} +1.13103e6 q^{83} +1.55104e6 q^{85} +1.63515e6 q^{95} +1.15397e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 54 q^{5} + 610 q^{7} + 1458 q^{9}+O(q^{10}) \) 2 * q + 54 * q^5 + 610 * q^7 + 1458 * q^9	\( 2 q + 54 q^{5} + 610 q^{7} + 1458 q^{9} - 1062 q^{11} + 9630 q^{17} + 13718 q^{19} - 41220 q^{23} + 59584 q^{25} - 98442 q^{35} - 142630 q^{43} + 39366 q^{45} - 75150 q^{47} + 98496 q^{49} + 865086 q^{55} + 57062 q^{61} + 444690 q^{63} - 384050 q^{73} - 1473030 q^{77} + 1062882 q^{81} + 2262060 q^{83} + 975018 q^{85} + 370386 q^{95} - 774198 q^{99}+O(q^{100}) \) 2 * q + 54 * q^5 + 610 * q^7 + 1458 * q^9 - 1062 * q^11 + 9630 * q^17 + 13718 * q^19 - 41220 * q^23 + 59584 * q^25 - 98442 * q^35 - 142630 * q^43 + 39366 * q^45 - 75150 * q^47 + 98496 * q^49 + 865086 * q^55 + 57062 * q^61 + 444690 * q^63 - 384050 * q^73 - 1473030 * q^77 + 1062882 * q^81 + 2262060 * q^83 + 975018 * q^85 + 370386 * q^95 - 774198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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