LMFDB - Embedded newform 189.3.d.b.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,3,Mod(55,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("189.55");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	189.d (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:	$5.14987699641$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	189.55
Dual form			189.3.d.b.55.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{4} +(6.50000 - 2.59808i) q^{7} -25.9808i q^{13} +16.0000 q^{16} -36.3731i q^{19} +25.0000 q^{25} +(-26.0000 + 10.3923i) q^{28} +41.5692i q^{31} -47.0000 q^{37} +22.0000 q^{43} +(35.5000 - 33.7750i) q^{49} +103.923i q^{52} -15.5885i q^{61} -64.0000 q^{64} -109.000 q^{67} +109.119i q^{73} +145.492i q^{76} +131.000 q^{79} +(-67.5000 - 168.875i) q^{91} +98.7269i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 13 q^{7} + 32 q^{16} + 50 q^{25} - 52 q^{28} - 94 q^{37} + 44 q^{43} + 71 q^{49} - 128 q^{64} - 218 q^{67} + 262 q^{79} - 135 q^{91}+O(q^{100}) $ 2 * q - 8 * q^4 + 13 * q^7 + 32 * q^16 + 50 * q^25 - 52 * q^28 - 94 * q^37 + 44 * q^43 + 71 * q^49 - 128 * q^64 - 218 * q^67 + 262 * q^79 - 135 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$1$	$-1$

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			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$		0			0
$3$		0			0
$4$	−4.00000			−1.00000
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	6.50000	−	2.59808i	0.928571	−	0.371154i
$7$	6.50000	−	2.59808i	0.928571	−	0.371154i
$8$		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
$13$		−	25.9808i		−	1.99852i	−0.0384615	−	0.999260i	$-0.512246\pi$
$13$		−	25.9808i		−	1.99852i	0.0384615	−	0.999260i	$-0.487754\pi$
$14$		0			0
$15$		0			0
$16$	16.0000			1.00000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		−	36.3731i		−	1.91437i	−0.289474	−	0.957186i	$-0.593480\pi$
$19$		−	36.3731i		−	1.91437i	0.289474	−	0.957186i	$-0.406520\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	25.0000			1.00000
$26$		0			0
$27$		0			0
$28$	−26.0000	+	10.3923i	−0.928571	+	0.371154i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$			41.5692i			1.34094i	0.741935	+	0.670471i	$0.233908\pi$
$31$			41.5692i			1.34094i	−0.741935	+	0.670471i	$0.766092\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−47.0000			−1.27027			−0.635135	−	0.772401i	$-0.719056\pi$
$37$	−47.0000			−1.27027			−0.635135	+	0.772401i	$0.719056\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	22.0000			0.511628			0.255814	−	0.966726i	$-0.417657\pi$
$43$	22.0000			0.511628			0.255814	+	0.966726i	$0.417657\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	35.5000	−	33.7750i	0.724490	−	0.689286i
$50$		0			0
$51$		0			0
$52$			103.923i			1.99852i
$53$		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		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		−	15.5885i		−	0.255548i	−0.991803	−	0.127774i	$-0.959217\pi$
$61$		−	15.5885i		−	0.255548i	0.991803	−	0.127774i	$-0.0407833\pi$
$62$		0			0
$63$		0			0
$64$	−64.0000			−1.00000
$65$		0			0
$66$		0			0
$67$	−109.000			−1.62687			−0.813433	−	0.581659i	$-0.802404\pi$
$67$	−109.000			−1.62687			−0.813433	+	0.581659i	$0.802404\pi$
$68$		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
$73$			109.119i			1.49478i	0.664384	+	0.747392i	$0.268694\pi$
$73$			109.119i			1.49478i	−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$		0			0
$76$			145.492i			1.91437i
$77$		0			0
$78$		0			0
$79$	131.000			1.65823			0.829114	−	0.559080i	$-0.188845\pi$
$79$	131.000			1.65823			0.829114	+	0.559080i	$0.188845\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−67.5000	−	168.875i	−0.741758	−	1.85577i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			98.7269i			1.01780i	0.860825	+	0.508902i	$0.169948\pi$
$97$			98.7269i			1.01780i	−0.860825	+	0.508902i	$0.830052\pi$
$98$		0			0
$99$		0			0
$100$	−100.000			−1.00000
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		−	202.650i		−	1.96748i	−0.179612	−	0.983738i	$-0.557484\pi$
$103$		−	202.650i		−	1.96748i	0.179612	−	0.983738i	$-0.442516\pi$
$104$		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
$109$	214.000			1.96330			0.981651	−	0.190684i	$-0.0610707\pi$
$109$	214.000			1.96330			0.981651	+	0.190684i	$0.0610707\pi$
$110$		0			0
$111$		0			0
$112$	104.000	−	41.5692i	0.928571	−	0.371154i
$113$		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
$121$	−121.000			−1.00000
$122$		0			0
$123$		0			0
$124$		−	166.277i		−	1.34094i
$125$		0			0
$126$		0			0
$127$	146.000			1.14961			0.574803	−	0.818292i	$-0.305079\pi$
$127$	146.000			1.14961			0.574803	+	0.818292i	$0.305079\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−94.5000	−	236.425i	−0.710526	−	1.77763i
$134$		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
$139$		−	119.512i		−	0.859795i	−0.902878	−	0.429898i	$-0.858550\pi$
$139$		−	119.512i		−	0.859795i	0.902878	−	0.429898i	$-0.141450\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	188.000			1.27027
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−59.0000			−0.390728			−0.195364	−	0.980731i	$-0.562589\pi$
$151$	−59.0000			−0.390728			−0.195364	+	0.980731i	$0.562589\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			290.985i			1.85340i	0.375796	+	0.926702i	$0.377369\pi$
$157$			290.985i			1.85340i	−0.375796	+	0.926702i	$0.622631\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	299.000			1.83436			0.917178	−	0.398478i	$-0.130461\pi$
$163$	299.000			1.83436			0.917178	+	0.398478i	$0.130461\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−506.000			−2.99408
$170$		0			0
$171$		0			0
$172$	−88.0000			−0.511628
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	162.500	−	64.9519i	0.928571	−	0.371154i
$176$		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
$181$			181.865i			1.00478i	0.864641	+	0.502390i	$0.167546\pi$
$181$			181.865i			1.00478i	−0.864641	+	0.502390i	$0.832454\pi$
$182$		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
$193$	−143.000			−0.740933			−0.370466	−	0.928846i	$-0.620802\pi$
$193$	−143.000			−0.740933			−0.370466	+	0.928846i	$0.620802\pi$
$194$		0			0
$195$		0			0
$196$	−142.000	+	135.100i	−0.724490	+	0.689286i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$			285.788i			1.43612i	0.695980	+	0.718061i	$0.254970\pi$
$199$			285.788i			1.43612i	−0.695980	+	0.718061i	$0.745030\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		−	415.692i		−	1.99852i
$209$		0			0
$210$		0			0
$211$	253.000			1.19905			0.599526	−	0.800355i	$-0.295356\pi$
$211$	253.000			1.19905			0.599526	+	0.800355i	$0.295356\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	108.000	+	270.200i	0.497696	+	1.24516i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	290.985i		−	1.30486i	−0.757848	−	0.652432i	$-0.773749\pi$
$223$		−	290.985i		−	1.30486i	0.757848	−	0.652432i	$-0.226251\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$			457.261i			1.99677i	0.0567686	+	0.998387i	$0.481920\pi$
$229$			457.261i			1.99677i	−0.0567686	+	0.998387i	$0.518080\pi$
$230$		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
$241$			441.673i			1.83267i	0.400415	+	0.916334i	$0.368866\pi$
$241$			441.673i			1.83267i	−0.400415	+	0.916334i	$0.631134\pi$
$242$		0			0
$243$		0			0
$244$			62.3538i			0.255548i
$245$		0			0
$246$		0			0
$247$	−945.000			−3.82591
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	256.000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−305.500	+	122.110i	−1.17954	+	0.471466i
$260$		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
$268$	436.000			1.62687
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$			452.065i			1.66814i	0.551661	+	0.834069i	$0.313994\pi$
$271$			452.065i			1.66814i	−0.551661	+	0.834069i	$0.686006\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	122.000			0.440433			0.220217	−	0.975451i	$-0.429324\pi$
$277$	122.000			0.440433			0.220217	+	0.975451i	$0.429324\pi$
$278$		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
$283$		−	332.554i		−	1.17510i	−0.809187	−	0.587551i	$-0.800092\pi$
$283$		−	332.554i		−	1.17510i	0.809187	−	0.587551i	$-0.199908\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	289.000			1.00000
$290$		0			0
$291$		0			0
$292$		−	436.477i		−	1.49478i
$293$		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
$301$	143.000	−	57.1577i	0.475083	−	0.189893i
$302$		0			0
$303$		0			0
$304$		−	581.969i		−	1.91437i
$305$		0			0
$306$		0			0
$307$		−	498.831i		−	1.62486i	−0.583062	−	0.812428i	$-0.698145\pi$
$307$		−	498.831i		−	1.62486i	0.583062	−	0.812428i	$-0.301855\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			181.865i			0.581039i	0.956869	+	0.290520i	$0.0938282\pi$
$313$			181.865i			0.581039i	−0.956869	+	0.290520i	$0.906172\pi$
$314$		0			0
$315$		0			0
$316$	−524.000			−1.65823
$317$		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
$325$		−	649.519i		−	1.99852i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	661.000			1.99698			0.998489	−	0.0549442i	$-0.0174981\pi$
$331$	661.000			1.99698			0.998489	+	0.0549442i	$0.0174981\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	649.000			1.92582			0.962908	−	0.269830i	$-0.0869675\pi$
$337$	649.000			1.92582			0.962908	+	0.269830i	$0.0869675\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	143.000	−	311.769i	0.416910	−	0.908948i
$344$		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
$349$			192.258i			0.550881i	0.961318	+	0.275441i	$0.0888238\pi$
$349$			192.258i			0.550881i	−0.961318	+	0.275441i	$0.911176\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		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
$361$	−962.000			−2.66482
$362$		0			0
$363$		0			0
$364$	270.000	+	675.500i	0.741758	+	1.85577i
$365$		0			0
$366$		0			0
$367$		−	545.596i		−	1.48664i	−0.668937	−	0.743319i	$-0.733251\pi$
$367$		−	545.596i		−	1.48664i	0.668937	−	0.743319i	$-0.266749\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−577.000			−1.54692			−0.773458	−	0.633847i	$-0.781475\pi$
$373$	−577.000			−1.54692			−0.773458	+	0.633847i	$0.781475\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−83.0000			−0.218997			−0.109499	−	0.993987i	$-0.534925\pi$
$379$	−83.0000			−0.218997			−0.109499	+	0.993987i	$0.534925\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		−	394.908i		−	1.01780i
$389$		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
$397$			706.677i			1.78004i	0.455919	+	0.890021i	$0.349311\pi$
$397$			706.677i			1.78004i	−0.455919	+	0.890021i	$0.650689\pi$
$398$		0			0
$399$		0			0
$400$	400.000			1.00000
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	1080.00			2.67990
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	805.404i		−	1.96920i	−0.174817	−	0.984601i	$-0.555933\pi$
$409$		−	805.404i		−	1.96920i	0.174817	−	0.984601i	$-0.444067\pi$
$410$		0			0
$411$		0			0
$412$			810.600i			1.96748i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−481.000			−1.14252			−0.571259	−	0.820770i	$-0.693545\pi$
$421$	−481.000			−1.14252			−0.571259	+	0.820770i	$0.693545\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−40.5000	−	101.325i	−0.0948478	−	0.237295i
$428$		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
$433$		−	83.1384i		−	0.192006i	−0.995381	−	0.0960028i	$-0.969394\pi$
$433$		−	83.1384i		−	0.192006i	0.995381	−	0.0960028i	$-0.0306058\pi$
$434$		0			0
$435$		0			0
$436$	−856.000			−1.96330
$437$		0			0
$438$		0			0
$439$			872.954i			1.98850i	0.107062	+	0.994252i	$0.465856\pi$
$439$			872.954i			1.98850i	−0.107062	+	0.994252i	$0.534144\pi$
$440$		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
$448$	−416.000	+	166.277i	−0.928571	+	0.371154i
$449$		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
$457$	814.000			1.78118			0.890591	−	0.454805i	$-0.150291\pi$
$457$	814.000			1.78118			0.890591	+	0.454805i	$0.150291\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−397.000			−0.857451			−0.428726	−	0.903435i	$-0.641037\pi$
$463$	−397.000			−0.857451			−0.428726	+	0.903435i	$0.641037\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−708.500	+	283.190i	−1.51066	+	0.603817i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	909.327i		−	1.91437i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$			1221.10i			2.53866i
$482$		0			0
$483$		0			0
$484$	484.000			1.00000
$485$		0			0
$486$		0			0
$487$	349.000			0.716632			0.358316	−	0.933600i	$-0.383351\pi$
$487$	349.000			0.716632			0.358316	+	0.933600i	$0.383351\pi$
$488$		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
$496$			665.108i			1.34094i
$497$		0			0
$498$		0			0
$499$	−26.0000			−0.0521042			−0.0260521	−	0.999661i	$-0.508294\pi$
$499$	−26.0000			−0.0521042			−0.0260521	+	0.999661i	$0.508294\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−584.000			−1.14961
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	283.500	+	709.275i	0.554795	+	1.38801i
$512$		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		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		−	670.304i		−	1.28165i	−0.767686	−	0.640826i	$-0.778592\pi$
$523$		−	670.304i		−	1.28165i	0.767686	−	0.640826i	$-0.221408\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−529.000			−1.00000
$530$		0			0
$531$		0			0
$532$	378.000	+	945.700i	0.710526	+	1.77763i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−241.000			−0.445471			−0.222736	−	0.974879i	$-0.571499\pi$
$541$	−241.000			−0.445471			−0.222736	+	0.974879i	$0.571499\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1093.00			−1.99817			−0.999086	−	0.0427471i	$-0.986389\pi$
$547$	−1093.00			−1.99817			−0.999086	+	0.0427471i	$0.986389\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	851.500	−	340.348i	1.53978	−	0.615457i
$554$		0			0
$555$		0			0
$556$			478.046i			0.859795i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		−	571.577i		−	1.02250i
$560$		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
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−181.000			−0.316988			−0.158494	−	0.987360i	$-0.550664\pi$
$571$	−181.000			−0.316988			−0.158494	+	0.987360i	$0.550664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	514.419i		−	0.891541i	−0.895147	−	0.445770i	$-0.852930\pi$
$577$		−	514.419i		−	0.891541i	0.895147	−	0.445770i	$-0.147070\pi$
$578$		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		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	1512.00			2.56706
$590$		0			0
$591$		0			0
$592$	−752.000			−1.27027
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		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
$601$		−	1080.80i		−	1.79834i	−0.437604	−	0.899168i	$-0.644173\pi$
$601$		−	1080.80i		−	1.79834i	0.437604	−	0.899168i	$-0.355827\pi$
$602$		0			0
$603$		0			0
$604$	236.000			0.390728
$605$		0			0
$606$		0			0
$607$		−	254.611i		−	0.419459i	−0.977759	−	0.209729i	$-0.932742\pi$
$607$		−	254.611i		−	0.419459i	0.977759	−	0.209729i	$-0.0672583\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	983.000			1.60359			0.801794	−	0.597600i	$-0.203879\pi$
$613$	983.000			1.60359			0.801794	+	0.597600i	$0.203879\pi$
$614$		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
$619$		−	795.011i		−	1.28435i	−0.766559	−	0.642174i	$-0.778033\pi$
$619$		−	795.011i		−	1.28435i	0.766559	−	0.642174i	$-0.221967\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	625.000			1.00000
$626$		0			0
$627$		0			0
$628$		−	1163.94i		−	1.85340i
$629$		0			0
$630$		0			0
$631$	587.000			0.930269			0.465135	−	0.885240i	$-0.346006\pi$
$631$	587.000			0.930269			0.465135	+	0.885240i	$0.346006\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−877.500	−	922.317i	−1.37755	−	1.44791i
$638$		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
$643$		−	1247.08i		−	1.93947i	−0.244168	−	0.969733i	$-0.578515\pi$
$643$		−	1247.08i		−	1.93947i	0.244168	−	0.969733i	$-0.421485\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1196.00			−1.83436
$653$		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
$661$			763.834i			1.15557i	0.816188	+	0.577787i	$0.196084\pi$
$661$			763.834i			1.15557i	−0.816188	+	0.577787i	$0.803916\pi$
$662$		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
$673$	−23.0000			−0.0341753			−0.0170877	−	0.999854i	$-0.505439\pi$
$673$	−23.0000			−0.0341753			−0.0170877	+	0.999854i	$0.505439\pi$
$674$		0			0
$675$		0			0
$676$	2024.00			2.99408
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	256.500	+	641.725i	0.377761	+	0.945103i
$680$		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
$688$	352.000			0.511628
$689$		0			0
$690$		0			0
$691$			415.692i			0.601581i	0.953690	+	0.300790i	$0.0972504\pi$
$691$			415.692i			0.601581i	−0.953690	+	0.300790i	$0.902750\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−650.000	+	259.808i	−0.928571	+	0.371154i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$			1709.53i			2.43177i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1391.00			−1.96192			−0.980959	−	0.194214i	$-0.937784\pi$
$709$	−1391.00			−1.96192			−0.980959	+	0.194214i	$0.937784\pi$
$710$		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		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	−526.500	−	1317.22i	−0.730236	−	1.82694i
$722$		0			0
$723$		0			0
$724$		−	727.461i		−	1.00478i
$725$		0			0
$726$		0			0
$727$			1371.78i			1.88691i	0.331499	+	0.943455i	$0.392446\pi$
$727$			1371.78i			1.88691i	−0.331499	+	0.943455i	$0.607554\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	1039.23i		−	1.41778i	−0.705321	−	0.708888i	$-0.749197\pi$
$733$		−	1039.23i		−	1.41778i	0.705321	−	0.708888i	$-0.250803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1222.00			−1.65359			−0.826793	−	0.562506i	$-0.809837\pi$
$739$	−1222.00			−1.65359			−0.826793	+	0.562506i	$0.809837\pi$
$740$		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
$751$	−179.000			−0.238349			−0.119174	−	0.992873i	$-0.538025\pi$
$751$	−179.000			−0.238349			−0.119174	+	0.992873i	$0.538025\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	673.000			0.889036			0.444518	−	0.895770i	$-0.353375\pi$
$757$	673.000			0.889036			0.444518	+	0.895770i	$0.353375\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	1391.00	−	555.988i	1.82307	−	0.728687i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			1273.06i			1.65547i	0.561118	+	0.827736i	$0.310371\pi$
$769$			1273.06i			1.65547i	−0.561118	+	0.827736i	$0.689629\pi$
$770$		0			0
$771$		0			0
$772$	572.000			0.740933
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$			1039.23i			1.34094i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	568.000	−	540.400i	0.724490	−	0.689286i
$785$		0			0
$786$		0			0
$787$			1449.73i			1.84209i	0.389454	+	0.921046i	$0.372664\pi$
$787$			1449.73i			1.84209i	−0.389454	+	0.921046i	$0.627336\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−405.000			−0.510719
$794$		0			0
$795$		0			0
$796$		−	1143.15i		−	1.43612i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		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
$811$			581.969i			0.717594i	0.933416	+	0.358797i	$0.116813\pi$
$811$			581.969i			0.717594i	−0.933416	+	0.358797i	$0.883187\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	800.207i		−	0.979446i
$818$		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
$823$	−1621.00			−1.96962			−0.984812	−	0.173626i	$-0.944452\pi$
$823$	−1621.00			−1.96962			−0.984812	+	0.173626i	$0.944452\pi$
$824$		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
$829$		−	400.104i		−	0.482634i	−0.970446	−	0.241317i	$-0.922421\pi$
$829$		−	400.104i		−	0.482634i	0.970446	−	0.241317i	$-0.0775794\pi$
$830$		0			0
$831$		0			0
$832$			1662.77i			1.99852i
$833$		0			0
$834$		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
$841$	−841.000			−1.00000
$842$		0			0
$843$		0			0
$844$	−1012.00			−1.19905
$845$		0			0
$846$		0			0
$847$	−786.500	+	314.367i	−0.928571	+	0.371154i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			1636.79i			1.91886i	0.281946	+	0.959430i	$0.409020\pi$
$853$			1636.79i			1.91886i	−0.281946	+	0.959430i	$0.590980\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$			743.050i			0.865017i	0.901630	+	0.432509i	$0.142371\pi$
$859$			743.050i			0.865017i	−0.901630	+	0.432509i	$0.857629\pi$
$860$		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
$868$	−432.000	−	1080.80i	−0.497696	−	1.24516i
$869$		0			0
$870$		0			0
$871$			2831.90i			3.25132i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1727.00			1.96921			0.984607	−	0.174785i	$-0.0559231\pi$
$877$	1727.00			1.96921			0.984607	+	0.174785i	$0.0559231\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−443.000			−0.501699			−0.250849	−	0.968026i	$-0.580710\pi$
$883$	−443.000			−0.501699			−0.250849	+	0.968026i	$0.580710\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	949.000	−	379.319i	1.06749	−	0.426681i
$890$		0			0
$891$		0			0
$892$			1163.94i			1.30486i
$893$		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
$907$	1453.00			1.60198			0.800992	−	0.598675i	$-0.204306\pi$
$907$	1453.00			1.60198			0.800992	+	0.598675i	$0.204306\pi$
$908$		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
$916$		−	1829.05i		−	1.99677i
$917$		0			0
$918$		0			0
$919$	−866.000			−0.942329			−0.471164	−	0.882045i	$-0.656166\pi$
$919$	−866.000			−0.942329			−0.471164	+	0.882045i	$0.656166\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1175.00			−1.27027
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	−1228.50	−	1291.24i	−1.31955	−	1.38694i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			316.965i			0.338277i	0.985592	+	0.169138i	$0.0540985\pi$
$937$			316.965i			0.338277i	−0.985592	+	0.169138i	$0.945902\pi$
$938$		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
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	2835.00			2.98736
$950$		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
$961$	−767.000			−0.798127
$962$		0			0
$963$		0			0
$964$		−	1766.69i		−	1.83267i
$965$		0			0
$966$		0			0
$967$	−253.000			−0.261634			−0.130817	−	0.991407i	$-0.541760\pi$
$967$	−253.000			−0.261634			−0.130817	+	0.991407i	$0.541760\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	−310.500	−	776.825i	−0.319116	−	0.798381i
$974$		0			0
$975$		0			0
$976$		−	249.415i		−	0.255548i
$977$		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		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	3780.00			3.82591
$989$		0			0
$990$		0			0
$991$	−1739.00			−1.75479			−0.877397	−	0.479766i	$-0.840722\pi$
$991$	−1739.00			−1.75479			−0.877397	+	0.479766i	$0.840722\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	623.538i		−	0.625415i	−0.949850	−	0.312707i	$-0.898764\pi$
$997$		−	623.538i		−	0.625415i	0.949850	−	0.312707i	$-0.101236\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.d.b.55.1	✓	2
3.2	odd	2	CM	189.3.d.b.55.1	✓	2
7.6	odd	2	inner	189.3.d.b.55.2	yes	2
21.20	even	2	inner	189.3.d.b.55.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.d.b.55.1	✓	2	1.1	even	1	trivial
189.3.d.b.55.1	✓	2	3.2	odd	2	CM
189.3.d.b.55.2	yes	2	7.6	odd	2	inner
189.3.d.b.55.2	yes	2	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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	189.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000 q^{4} +(6.50000 - 2.59808i) q^{7} -25.9808i q^{13} +16.0000 q^{16} -36.3731i q^{19} +25.0000 q^{25} +(-26.0000 + 10.3923i) q^{28} +41.5692i q^{31} -47.0000 q^{37} +22.0000 q^{43} +(35.5000 - 33.7750i) q^{49} +103.923i q^{52} -15.5885i q^{61} -64.0000 q^{64} -109.000 q^{67} +109.119i q^{73} +145.492i q^{76} +131.000 q^{79} +(-67.5000 - 168.875i) q^{91} +98.7269i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 13 q^{7} + 32 q^{16} + 50 q^{25} - 52 q^{28} - 94 q^{37} + 44 q^{43} + 71 q^{49} - 128 q^{64} - 218 q^{67} + 262 q^{79} - 135 q^{91}+O(q^{100}) \) 2 * q - 8 * q^4 + 13 * q^7 + 32 * q^16 + 50 * q^25 - 52 * q^28 - 94 * q^37 + 44 * q^43 + 71 * q^49 - 128 * q^64 - 218 * q^67 + 262 * q^79 - 135 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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