LMFDB - Embedded newform 9680.2.a.da.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9680,2,Mod(1,9680)] 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2,0,6,0,-4,0,4,0,0,0,0,0,2,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.22733568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4840)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.45825$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.873518 q^{3} +1.00000 q^{5} +3.64048 q^{7} -2.23697 q^{9} -1.06751 q^{13} +0.873518 q^{15} +0.252963 q^{17} -3.06751 q^{19} +3.18003 q^{21} +7.04399 q^{23} +1.00000 q^{25} -4.57459 q^{27} +1.61202 q^{29} +0.788991 q^{31} +3.64048 q^{35} -4.91208 q^{37} -0.932490 q^{39} +6.39545 q^{41} +2.63065 q^{43} -2.23697 q^{45} +7.78559 q^{47} +6.25309 q^{49} +0.220968 q^{51} -6.32291 q^{53} -2.67953 q^{57} +5.48214 q^{59} -6.80160 q^{61} -8.14363 q^{63} -1.06751 q^{65} +9.49860 q^{67} +6.15306 q^{69} -7.53405 q^{71} +16.4419 q^{73} +0.873518 q^{75} +3.02596 q^{79} +2.71491 q^{81} -13.1746 q^{83} +0.252963 q^{85} +1.40813 q^{87} +12.4460 q^{89} -3.88625 q^{91} +0.689198 q^{93} -3.06751 q^{95} +8.77719 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) $ 6 * q + 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^15 + 8 * q^17 - 12 * q^19 - 8 * q^21 + 8 * q^23 + 6 * q^25 + 14 * q^27 + 16 * q^29 + 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 + 32 * q^41 + 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 - 16 * q^57 - 4 * q^59 + 16 * q^61 - 28 * q^63 + 2 * q^67 + 8 * q^69 + 28 * q^71 + 16 * q^73 + 2 * q^75 - 10 * q^81 - 12 * q^83 + 8 * q^85 + 24 * q^87 + 18 * q^89 + 24 * q^91 + 20 * q^93 - 12 * 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 $

$2$	0	0
$2$	0	0
$3$	0.873518	0.504326	0.252163	−	0.967685i	$-0.418858\pi$
$3$	0.873518	0.504326	0.252163	+	0.967685i	$0.418858\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.64048	1.37597	0.687986	−	0.725724i	$-0.258495\pi$
$7$	3.64048	1.37597	0.687986	+	0.725724i	$0.258495\pi$
$8$	0	0
$9$	−2.23697	−0.745655
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.06751	−0.296074	−0.148037	−	0.988982i	$-0.547295\pi$
$13$	−1.06751	−0.296074	−0.148037	+	0.988982i	$0.547295\pi$
$14$	0	0
$15$	0.873518	0.225541
$16$	0	0
$17$	0.252963	0.0613526	0.0306763	−	0.999529i	$-0.490234\pi$
$17$	0.252963	0.0613526	0.0306763	+	0.999529i	$0.490234\pi$
$18$	0	0
$19$	−3.06751	−0.703735	−0.351868	−	0.936050i	$-0.614453\pi$
$19$	−3.06751	−0.703735	−0.351868	+	0.936050i	$0.614453\pi$
$20$	0	0
$21$	3.18003	0.693939
$22$	0	0
$23$	7.04399	1.46877	0.734387	−	0.678731i	$-0.237470\pi$
$23$	7.04399	1.46877	0.734387	+	0.678731i	$0.237470\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.57459	−0.880379
$28$	0	0
$29$	1.61202	0.299344	0.149672	−	0.988736i	$-0.452178\pi$
$29$	1.61202	0.299344	0.149672	+	0.988736i	$0.452178\pi$
$30$	0	0
$31$	0.788991	0.141707	0.0708535	−	0.997487i	$-0.477428\pi$
$31$	0.788991	0.141707	0.0708535	+	0.997487i	$0.477428\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.64048	0.615353
$36$	0	0
$37$	−4.91208	−0.807540	−0.403770	−	0.914860i	$-0.632300\pi$
$37$	−4.91208	−0.807540	−0.403770	+	0.914860i	$0.632300\pi$
$38$	0	0
$39$	−0.932490	−0.149318
$40$	0	0
$41$	6.39545	0.998802	0.499401	−	0.866371i	$-0.333554\pi$
$41$	6.39545	0.998802	0.499401	+	0.866371i	$0.333554\pi$
$42$	0	0
$43$	2.63065	0.401170	0.200585	−	0.979676i	$-0.435716\pi$
$43$	2.63065	0.401170	0.200585	+	0.979676i	$0.435716\pi$
$44$	0	0
$45$	−2.23697	−0.333467
$46$	0	0
$47$	7.78559	1.13565	0.567823	−	0.823151i	$-0.307786\pi$
$47$	7.78559	1.13565	0.567823	+	0.823151i	$0.307786\pi$
$48$	0	0
$49$	6.25309	0.893299
$50$	0	0
$51$	0.220968	0.0309417
$52$	0	0
$53$	−6.32291	−0.868519	−0.434259	−	0.900788i	$-0.642990\pi$
$53$	−6.32291	−0.868519	−0.434259	+	0.900788i	$0.642990\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.67953	−0.354912
$58$	0	0
$59$	5.48214	0.713714	0.356857	−	0.934159i	$-0.383848\pi$
$59$	5.48214	0.713714	0.356857	+	0.934159i	$0.383848\pi$
$60$	0	0
$61$	−6.80160	−0.870856	−0.435428	−	0.900224i	$-0.643403\pi$
$61$	−6.80160	−0.870856	−0.435428	+	0.900224i	$0.643403\pi$
$62$	0	0
$63$	−8.14363	−1.02600
$64$	0	0
$65$	−1.06751	−0.132408
$66$	0	0
$67$	9.49860	1.16044	0.580219	−	0.814460i	$-0.302967\pi$
$67$	9.49860	1.16044	0.580219	+	0.814460i	$0.302967\pi$
$68$	0	0
$69$	6.15306	0.740741
$70$	0	0
$71$	−7.53405	−0.894127	−0.447064	−	0.894502i	$-0.647530\pi$
$71$	−7.53405	−0.894127	−0.447064	+	0.894502i	$0.647530\pi$
$72$	0	0
$73$	16.4419	1.92438	0.962192	−	0.272374i	$-0.0878087\pi$
$73$	16.4419	1.92438	0.962192	+	0.272374i	$0.0878087\pi$
$74$	0	0
$75$	0.873518	0.100865
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.02596	0.340447	0.170223	−	0.985406i	$-0.445551\pi$
$79$	3.02596	0.340447	0.170223	+	0.985406i	$0.445551\pi$
$80$	0	0
$81$	2.71491	0.301657
$82$	0	0
$83$	−13.1746	−1.44610	−0.723050	−	0.690796i	$-0.757260\pi$
$83$	−13.1746	−1.44610	−0.723050	+	0.690796i	$0.757260\pi$
$84$	0	0
$85$	0.252963	0.0274377
$86$	0	0
$87$	1.40813	0.150967
$88$	0	0
$89$	12.4460	1.31927	0.659634	−	0.751587i	$-0.270711\pi$
$89$	12.4460	1.31927	0.659634	+	0.751587i	$0.270711\pi$
$90$	0	0
$91$	−3.88625	−0.407390
$92$	0	0
$93$	0.689198	0.0714665
$94$	0	0
$95$	−3.06751	−0.314720
$96$	0	0
$97$	8.77719	0.891188	0.445594	−	0.895235i	$-0.352992\pi$
$97$	8.77719	0.891188	0.445594	+	0.895235i	$0.352992\pi$
$98$	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	9680.2.a.da.1.4		6
4.3	odd	2		4840.2.a.bd.1.3	yes	6
11.10	odd	2		9680.2.a.db.1.4		6
44.43	even	2		4840.2.a.bc.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.3	✓	6	44.43	even	2
4840.2.a.bd.1.3	yes	6	4.3	odd	2
9680.2.a.da.1.4		6	1.1	even	1	trivial
9680.2.a.db.1.4		6	11.10	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q+0.873518 q^{3} +1.00000 q^{5} +3.64048 q^{7} -2.23697 q^{9} -1.06751 q^{13} +0.873518 q^{15} +0.252963 q^{17} -3.06751 q^{19} +3.18003 q^{21} +7.04399 q^{23} +1.00000 q^{25} -4.57459 q^{27} +1.61202 q^{29} +0.788991 q^{31} +3.64048 q^{35} -4.91208 q^{37} -0.932490 q^{39} +6.39545 q^{41} +2.63065 q^{43} -2.23697 q^{45} +7.78559 q^{47} +6.25309 q^{49} +0.220968 q^{51} -6.32291 q^{53} -2.67953 q^{57} +5.48214 q^{59} -6.80160 q^{61} -8.14363 q^{63} -1.06751 q^{65} +9.49860 q^{67} +6.15306 q^{69} -7.53405 q^{71} +16.4419 q^{73} +0.873518 q^{75} +3.02596 q^{79} +2.71491 q^{81} -13.1746 q^{83} +0.252963 q^{85} +1.40813 q^{87} +12.4460 q^{89} -3.88625 q^{91} +0.689198 q^{93} -3.06751 q^{95} +8.77719 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) \) 6 * q + 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^15 + 8 * q^17 - 12 * q^19 - 8 * q^21 + 8 * q^23 + 6 * q^25 + 14 * q^27 + 16 * q^29 + 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 + 32 * q^41 + 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 - 16 * q^57 - 4 * q^59 + 16 * q^61 - 28 * q^63 + 2 * q^67 + 8 * q^69 + 28 * q^71 + 16 * q^73 + 2 * q^75 - 10 * q^81 - 12 * q^83 + 8 * q^85 + 24 * q^87 + 18 * q^89 + 24 * q^91 + 20 * q^93 - 12 * q^95

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