LMFDB - Embedded newform 9680.2.a.da.1.5

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			1.5
Root			$-0.821728$ 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)

$f(q)$	$=$	$q+2.32476 q^{3} +1.00000 q^{5} -4.78768 q^{7} +2.40452 q^{9} -0.601547 q^{13} +2.32476 q^{15} -2.64953 q^{17} -2.60155 q^{19} -11.1302 q^{21} -5.17083 q^{23} +1.00000 q^{25} -1.38434 q^{27} +5.44643 q^{29} +10.4577 q^{31} -4.78768 q^{35} +9.39816 q^{37} -1.39845 q^{39} -4.54633 q^{41} +3.48547 q^{43} +2.40452 q^{45} -5.07340 q^{47} +15.9218 q^{49} -6.15952 q^{51} +3.76714 q^{53} -6.04798 q^{57} -11.7599 q^{59} +12.6454 q^{61} -11.5121 q^{63} -0.601547 q^{65} +15.0452 q^{67} -12.0210 q^{69} -0.346489 q^{71} +3.68096 q^{73} +2.32476 q^{75} +8.05322 q^{79} -10.4318 q^{81} +13.0094 q^{83} -2.64953 q^{85} +12.6617 q^{87} +0.462692 q^{89} +2.88001 q^{91} +24.3118 q^{93} -2.60155 q^{95} +7.97011 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.32476	1.34220	0.671101	−	0.741366i	$-0.265822\pi$
$3$	2.32476	1.34220	0.671101	+	0.741366i	$0.265822\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.78768	−1.80957	−0.904786	−	0.425867i	$-0.859969\pi$
$7$	−4.78768	−1.80957	−0.904786	+	0.425867i	$0.859969\pi$
$8$	0	0
$9$	2.40452	0.801508
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.601547	−0.166839	−0.0834195	−	0.996515i	$-0.526584\pi$
$13$	−0.601547	−0.166839	−0.0834195	+	0.996515i	$0.526584\pi$
$14$	0	0
$15$	2.32476	0.600251
$16$	0	0
$17$	−2.64953	−0.642605	−0.321302	−	0.946977i	$-0.604121\pi$
$17$	−2.64953	−0.642605	−0.321302	+	0.946977i	$0.604121\pi$
$18$	0	0
$19$	−2.60155	−0.596836	−0.298418	−	0.954435i	$-0.596459\pi$
$19$	−2.60155	−0.596836	−0.298418	+	0.954435i	$0.596459\pi$
$20$	0	0
$21$	−11.1302	−2.42881
$22$	0	0
$23$	−5.17083	−1.07819	−0.539096	−	0.842244i	$-0.681234\pi$
$23$	−5.17083	−1.07819	−0.539096	+	0.842244i	$0.681234\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.38434	−0.266417
$28$	0	0
$29$	5.44643	1.01138	0.505689	−	0.862716i	$-0.331238\pi$
$29$	5.44643	1.01138	0.505689	+	0.862716i	$0.331238\pi$
$30$	0	0
$31$	10.4577	1.87827	0.939133	−	0.343554i	$-0.111631\pi$
$31$	10.4577	1.87827	0.939133	+	0.343554i	$0.111631\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.78768	−0.809265
$36$	0	0
$37$	9.39816	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$37$	9.39816	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$38$	0	0
$39$	−1.39845	−0.223932
$40$	0	0
$41$	−4.54633	−0.710018	−0.355009	−	0.934863i	$-0.615522\pi$
$41$	−4.54633	−0.710018	−0.355009	+	0.934863i	$0.615522\pi$
$42$	0	0
$43$	3.48547	0.531530	0.265765	−	0.964038i	$-0.414376\pi$
$43$	3.48547	0.531530	0.265765	+	0.964038i	$0.414376\pi$
$44$	0	0
$45$	2.40452	0.358445
$46$	0	0
$47$	−5.07340	−0.740031	−0.370016	−	0.929026i	$-0.620648\pi$
$47$	−5.07340	−0.740031	−0.370016	+	0.929026i	$0.620648\pi$
$48$	0	0
$49$	15.9218	2.27455
$50$	0	0
$51$	−6.15952	−0.862506
$52$	0	0
$53$	3.76714	0.517456	0.258728	−	0.965950i	$-0.416697\pi$
$53$	3.76714	0.517456	0.258728	+	0.965950i	$0.416697\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.04798	−0.801074
$58$	0	0
$59$	−11.7599	−1.53101	−0.765507	−	0.643428i	$-0.777512\pi$
$59$	−11.7599	−1.53101	−0.765507	+	0.643428i	$0.777512\pi$
$60$	0	0
$61$	12.6454	1.61908	0.809542	−	0.587063i	$-0.199716\pi$
$61$	12.6454	1.61908	0.809542	+	0.587063i	$0.199716\pi$
$62$	0	0
$63$	−11.5121	−1.45039
$64$	0	0
$65$	−0.601547	−0.0746127
$66$	0	0
$67$	15.0452	1.83806	0.919030	−	0.394187i	$-0.128974\pi$
$67$	15.0452	1.83806	0.919030	+	0.394187i	$0.128974\pi$
$68$	0	0
$69$	−12.0210	−1.44715
$70$	0	0
$71$	−0.346489	−0.0411207	−0.0205604	−	0.999789i	$-0.506545\pi$
$71$	−0.346489	−0.0411207	−0.0205604	+	0.999789i	$0.506545\pi$
$72$	0	0
$73$	3.68096	0.430823	0.215412	−	0.976523i	$-0.430891\pi$
$73$	3.68096	0.430823	0.215412	+	0.976523i	$0.430891\pi$
$74$	0	0
$75$	2.32476	0.268441
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.05322	0.906058	0.453029	−	0.891496i	$-0.350343\pi$
$79$	8.05322	0.906058	0.453029	+	0.891496i	$0.350343\pi$
$80$	0	0
$81$	−10.4318	−1.15909
$82$	0	0
$83$	13.0094	1.42796	0.713981	−	0.700165i	$-0.246890\pi$
$83$	13.0094	1.42796	0.713981	+	0.700165i	$0.246890\pi$
$84$	0	0
$85$	−2.64953	−0.287381
$86$	0	0
$87$	12.6617	1.35747
$88$	0	0
$89$	0.462692	0.0490453	0.0245226	−	0.999699i	$-0.492193\pi$
$89$	0.462692	0.0490453	0.0245226	+	0.999699i	$0.492193\pi$
$90$	0	0
$91$	2.88001	0.301907
$92$	0	0
$93$	24.3118	2.52101
$94$	0	0
$95$	−2.60155	−0.266913
$96$	0	0
$97$	7.97011	0.809242	0.404621	−	0.914484i	$-0.367403\pi$
$97$	7.97011	0.809242	0.404621	+	0.914484i	$0.367403\pi$
$98$	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.1.5		6
4.3	odd	2		4840.2.a.bd.1.2	yes	6
11.10	odd	2		9680.2.a.db.1.5		6
44.43	even	2		4840.2.a.bc.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.2	✓	6	44.43	even	2
4840.2.a.bd.1.2	yes	6	4.3	odd	2
9680.2.a.da.1.5		6	1.1	even	1	trivial
9680.2.a.db.1.5		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)

\(f(q)\)	\(=\)	\(q+2.32476 q^{3} +1.00000 q^{5} -4.78768 q^{7} +2.40452 q^{9} -0.601547 q^{13} +2.32476 q^{15} -2.64953 q^{17} -2.60155 q^{19} -11.1302 q^{21} -5.17083 q^{23} +1.00000 q^{25} -1.38434 q^{27} +5.44643 q^{29} +10.4577 q^{31} -4.78768 q^{35} +9.39816 q^{37} -1.39845 q^{39} -4.54633 q^{41} +3.48547 q^{43} +2.40452 q^{45} -5.07340 q^{47} +15.9218 q^{49} -6.15952 q^{51} +3.76714 q^{53} -6.04798 q^{57} -11.7599 q^{59} +12.6454 q^{61} -11.5121 q^{63} -0.601547 q^{65} +15.0452 q^{67} -12.0210 q^{69} -0.346489 q^{71} +3.68096 q^{73} +2.32476 q^{75} +8.05322 q^{79} -10.4318 q^{81} +13.0094 q^{83} -2.64953 q^{85} +12.6617 q^{87} +0.462692 q^{89} +2.88001 q^{91} +24.3118 q^{93} -2.60155 q^{95} +7.97011 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) \) 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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