LMFDB - Embedded newform 441.8.a.ba.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [441,8,Mod(1,441)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("441.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-30,0,458,0,0,0,-4290,0,0,-17760,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$137.761796238$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 574x^{6} - 2240x^{5} + 95697x^{4} + 624960x^{3} - 2980216x^{2} - 17427200x + 41900096 $ x^8 - 574x^6 - 2240x^5 + 95697x^4 + 624960x^3 - 2980216x^2 - 17427200x + 41900096
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6}\cdot 7^{6} $
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.08214$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-20.4398 q^{2} +289.787 q^{4} +267.917 q^{5} -3306.90 q^{8} -5476.19 q^{10} -6858.69 q^{11} -553.232 q^{13} +30499.9 q^{16} +18782.4 q^{17} -26526.1 q^{19} +77639.0 q^{20} +140190. q^{22} +14206.0 q^{23} -6345.24 q^{25} +11308.0 q^{26} +218240. q^{29} +177622. q^{31} -200128. q^{32} -383909. q^{34} +166649. q^{37} +542189. q^{38} -885977. q^{40} -392711. q^{41} -349297. q^{43} -1.98756e6 q^{44} -290369. q^{46} -998513. q^{47} +129696. q^{50} -160319. q^{52} -558894. q^{53} -1.83756e6 q^{55} -4.46080e6 q^{58} +2.86220e6 q^{59} +804192. q^{61} -3.63057e6 q^{62} +186612. q^{64} -148220. q^{65} +1.01152e6 q^{67} +5.44289e6 q^{68} -1.38526e6 q^{71} +584471. q^{73} -3.40629e6 q^{74} -7.68693e6 q^{76} +3.78559e6 q^{79} +8.17144e6 q^{80} +8.02694e6 q^{82} -3.66559e6 q^{83} +5.03213e6 q^{85} +7.13958e6 q^{86} +2.26810e7 q^{88} -9.21773e6 q^{89} +4.11673e6 q^{92} +2.04094e7 q^{94} -7.10681e6 q^{95} +8.21720e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 30 q^{2} + 458 q^{4} - 4290 q^{8} - 17760 q^{11} + 33762 q^{16} + 430660 q^{22} - 96000 q^{23} + 468992 q^{25} - 236280 q^{29} - 1064370 q^{32} + 150040 q^{37} - 253440 q^{43} - 1711260 q^{44} - 4304568 q^{46}+ \cdots - 39561792 q^{95}+O(q^{100}) $ 8 * q - 30 * q^2 + 458 * q^4 - 4290 * q^8 - 17760 * q^11 + 33762 * q^16 + 430660 * q^22 - 96000 * q^23 + 468992 * q^25 - 236280 * q^29 - 1064370 * q^32 + 150040 * q^37 - 253440 * q^43 - 1711260 * q^44 - 4304568 * q^46 + 3585174 * q^50 + 3678240 * q^53 - 10326900 * q^58 - 1466846 * q^64 - 8964984 * q^65 + 2060640 * q^67 - 10702176 * q^71 - 4497372 * q^74 + 24451744 * q^79 + 14525096 * q^85 + 17008260 * q^86 + 38456460 * q^88 - 19789080 * q^92 - 39561792 * q^95

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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 $

$2$	−20.4398	−1.80664	−0.903322	−	0.428963i	$-0.858879\pi$
$2$	−20.4398	−1.80664	−0.903322	+	0.428963i	$0.858879\pi$
$3$	0	0
$3$	0	0
$4$	289.787	2.26396
$5$	267.917	0.958531	0.479265	−	0.877670i	$-0.340903\pi$
$5$	267.917	0.958531	0.479265	+	0.877670i	$0.340903\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3306.90	−2.28353
$9$	0	0
$10$	−5476.19	−1.73172
$11$	−6858.69	−1.55370	−0.776849	−	0.629687i	$-0.783183\pi$
$11$	−6858.69	−1.55370	−0.776849	+	0.629687i	$0.783183\pi$
$12$	0	0
$13$	−553.232	−0.0698402	−0.0349201	−	0.999390i	$-0.511118\pi$
$13$	−553.232	−0.0698402	−0.0349201	+	0.999390i	$0.511118\pi$
$14$	0	0
$15$	0	0
$16$	30499.9	1.86156
$17$	18782.4	0.927213	0.463606	−	0.886041i	$-0.346555\pi$
$17$	18782.4	0.927213	0.463606	+	0.886041i	$0.346555\pi$
$18$	0	0
$19$	−26526.1	−0.887229	−0.443615	−	0.896218i	$-0.646304\pi$
$19$	−26526.1	−0.887229	−0.443615	+	0.896218i	$0.646304\pi$
$20$	77639.0	2.17008
$21$	0	0
$22$	140190.	2.80698
$23$	14206.0	0.243459	0.121730	−	0.992563i	$-0.461156\pi$
$23$	14206.0	0.243459	0.121730	+	0.992563i	$0.461156\pi$
$24$	0	0
$25$	−6345.24	−0.0812191
$26$	11308.0	0.126176
$27$	0	0
$28$	0	0
$29$	218240.	1.66166	0.830830	−	0.556527i	$-0.187866\pi$
$29$	218240.	1.66166	0.830830	+	0.556527i	$0.187866\pi$
$30$	0	0
$31$	177622.	1.07086	0.535429	−	0.844580i	$-0.320150\pi$
$31$	177622.	1.07086	0.535429	+	0.844580i	$0.320150\pi$
$32$	−200128.	−1.07965
$33$	0	0
$34$	−383909.	−1.67514
$35$	0	0
$36$	0	0
$37$	166649.	0.540876	0.270438	−	0.962737i	$-0.412831\pi$
$37$	166649.	0.540876	0.270438	+	0.962737i	$0.412831\pi$
$38$	542189.	1.60291
$39$	0	0
$40$	−885977.	−2.18883
$41$	−392711.	−0.889875	−0.444938	−	0.895562i	$-0.646774\pi$
$41$	−392711.	−0.889875	−0.444938	+	0.895562i	$0.646774\pi$
$42$	0	0
$43$	−349297.	−0.669971	−0.334985	−	0.942223i	$-0.608731\pi$
$43$	−349297.	−0.669971	−0.334985	+	0.942223i	$0.608731\pi$
$44$	−1.98756e6	−3.51751
$45$	0	0
$46$	−290369.	−0.439844
$47$	−998513.	−1.40285	−0.701425	−	0.712743i	$-0.747452\pi$
$47$	−998513.	−1.40285	−0.701425	+	0.712743i	$0.747452\pi$
$48$	0	0
$49$	0	0
$50$	129696.	0.146734
$51$	0	0
$52$	−160319.	−0.158116
$53$	−558894.	−0.515660	−0.257830	−	0.966190i	$-0.583008\pi$
$53$	−558894.	−0.515660	−0.257830	+	0.966190i	$0.583008\pi$
$54$	0	0
$55$	−1.83756e6	−1.48927
$56$	0	0
$57$	0	0
$58$	−4.46080e6	−3.00203
$59$	2.86220e6	1.81434	0.907169	−	0.420767i	$-0.138239\pi$
$59$	2.86220e6	1.81434	0.907169	+	0.420767i	$0.138239\pi$
$60$	0	0
$61$	804192.	0.453634	0.226817	−	0.973937i	$-0.427168\pi$
$61$	804192.	0.453634	0.226817	+	0.973937i	$0.427168\pi$
$62$	−3.63057e6	−1.93466
$63$	0	0
$64$	186612.	0.0889836
$65$	−148220.	−0.0669439
$66$	0	0
$67$	1.01152e6	0.410876	0.205438	−	0.978670i	$-0.434138\pi$
$67$	1.01152e6	0.410876	0.205438	+	0.978670i	$0.434138\pi$
$68$	5.44289e6	2.09917
$69$	0	0
$70$	0	0
$71$	−1.38526e6	−0.459333	−0.229666	−	0.973269i	$-0.573764\pi$
$71$	−1.38526e6	−0.459333	−0.229666	+	0.973269i	$0.573764\pi$
$72$	0	0
$73$	584471.	0.175846	0.0879231	−	0.996127i	$-0.471977\pi$
$73$	584471.	0.175846	0.0879231	+	0.996127i	$0.471977\pi$
$74$	−3.40629e6	−0.977171
$75$	0	0
$76$	−7.68693e6	−2.00865
$77$	0	0
$78$	0	0
$79$	3.78559e6	0.863850	0.431925	−	0.901910i	$-0.357835\pi$
$79$	3.78559e6	0.863850	0.431925	+	0.901910i	$0.357835\pi$
$80$	8.17144e6	1.78437
$81$	0	0
$82$	8.02694e6	1.60769
$83$	−3.66559e6	−0.703673	−0.351837	−	0.936061i	$-0.614443\pi$
$83$	−3.66559e6	−0.703673	−0.351837	+	0.936061i	$0.614443\pi$
$84$	0	0
$85$	5.03213e6	0.888762
$86$	7.13958e6	1.21040
$87$	0	0
$88$	2.26810e7	3.54792
$89$	−9.21773e6	−1.38599	−0.692993	−	0.720944i	$-0.743709\pi$
$89$	−9.21773e6	−1.38599	−0.692993	+	0.720944i	$0.743709\pi$
$90$	0	0
$91$	0	0
$92$	4.11673e6	0.551182
$93$	0	0
$94$	2.04094e7	2.53445
$95$	−7.10681e6	−0.850437
$96$	0	0
$97$	8.21720e6	0.914161	0.457081	−	0.889425i	$-0.348895\pi$
$97$	8.21720e6	0.914161	0.457081	+	0.889425i	$0.348895\pi$
$98$	0	0
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.8.a.ba.1.2		8
3.2	odd	2		49.8.a.g.1.8	yes	8
7.6	odd	2	inner	441.8.a.ba.1.1		8
21.2	odd	6		49.8.c.h.18.1		16
21.5	even	6		49.8.c.h.18.2		16
21.11	odd	6		49.8.c.h.30.1		16
21.17	even	6		49.8.c.h.30.2		16
21.20	even	2		49.8.a.g.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.8.a.g.1.7	✓	8	21.20	even	2
49.8.a.g.1.8	yes	8	3.2	odd	2
49.8.c.h.18.1		16	21.2	odd	6
49.8.c.h.18.2		16	21.5	even	6
49.8.c.h.30.1		16	21.11	odd	6
49.8.c.h.30.2		16	21.17	even	6
441.8.a.ba.1.1		8	7.6	odd	2	inner
441.8.a.ba.1.2		8	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-20.4398 q^{2} +289.787 q^{4} +267.917 q^{5} -3306.90 q^{8} -5476.19 q^{10} -6858.69 q^{11} -553.232 q^{13} +30499.9 q^{16} +18782.4 q^{17} -26526.1 q^{19} +77639.0 q^{20} +140190. q^{22} +14206.0 q^{23} -6345.24 q^{25} +11308.0 q^{26} +218240. q^{29} +177622. q^{31} -200128. q^{32} -383909. q^{34} +166649. q^{37} +542189. q^{38} -885977. q^{40} -392711. q^{41} -349297. q^{43} -1.98756e6 q^{44} -290369. q^{46} -998513. q^{47} +129696. q^{50} -160319. q^{52} -558894. q^{53} -1.83756e6 q^{55} -4.46080e6 q^{58} +2.86220e6 q^{59} +804192. q^{61} -3.63057e6 q^{62} +186612. q^{64} -148220. q^{65} +1.01152e6 q^{67} +5.44289e6 q^{68} -1.38526e6 q^{71} +584471. q^{73} -3.40629e6 q^{74} -7.68693e6 q^{76} +3.78559e6 q^{79} +8.17144e6 q^{80} +8.02694e6 q^{82} -3.66559e6 q^{83} +5.03213e6 q^{85} +7.13958e6 q^{86} +2.26810e7 q^{88} -9.21773e6 q^{89} +4.11673e6 q^{92} +2.04094e7 q^{94} -7.10681e6 q^{95} +8.21720e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 30 q^{2} + 458 q^{4} - 4290 q^{8} - 17760 q^{11} + 33762 q^{16} + 430660 q^{22} - 96000 q^{23} + 468992 q^{25} - 236280 q^{29} - 1064370 q^{32} + 150040 q^{37} - 253440 q^{43} - 1711260 q^{44} - 4304568 q^{46}+ \cdots - 39561792 q^{95}+O(q^{100}) \) 8 * q - 30 * q^2 + 458 * q^4 - 4290 * q^8 - 17760 * q^11 + 33762 * q^16 + 430660 * q^22 - 96000 * q^23 + 468992 * q^25 - 236280 * q^29 - 1064370 * q^32 + 150040 * q^37 - 253440 * q^43 - 1711260 * q^44 - 4304568 * q^46 + 3585174 * q^50 + 3678240 * q^53 - 10326900 * q^58 - 1466846 * q^64 - 8964984 * q^65 + 2060640 * q^67 - 10702176 * q^71 - 4497372 * q^74 + 24451744 * q^79 + 14525096 * q^85 + 17008260 * q^86 + 38456460 * q^88 - 19789080 * q^92 - 39561792 * q^95

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