Properties

Label 9680.2.a.q
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{5} - 4 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - 4 q^{7} - 3 q^{9} + 2 q^{13} - 2 q^{17} + 4 q^{19} - 4 q^{23} + q^{25} + 2 q^{29} + 8 q^{31} - 4 q^{35} + 6 q^{37} + 6 q^{41} - 8 q^{43} - 3 q^{45} - 4 q^{47} + 9 q^{49} + 6 q^{53} + 4 q^{59} + 2 q^{61} + 12 q^{63} + 2 q^{65} - 8 q^{67} + 6 q^{73} + 9 q^{81} - 16 q^{83} - 2 q^{85} - 6 q^{89} - 8 q^{91} + 4 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 1.00000 0 −4.00000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)
\(11\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9680.2.a.q 1
4.b odd 2 1 4840.2.a.f 1
11.b odd 2 1 80.2.a.a 1
33.d even 2 1 720.2.a.e 1
44.c even 2 1 40.2.a.a 1
55.d odd 2 1 400.2.a.e 1
55.e even 4 2 400.2.c.d 2
77.b even 2 1 3920.2.a.s 1
88.b odd 2 1 320.2.a.d 1
88.g even 2 1 320.2.a.c 1
132.d odd 2 1 360.2.a.a 1
165.d even 2 1 3600.2.a.h 1
165.l odd 4 2 3600.2.f.t 2
176.i even 4 2 1280.2.d.j 2
176.l odd 4 2 1280.2.d.a 2
220.g even 2 1 200.2.a.c 1
220.i odd 4 2 200.2.c.b 2
264.m even 2 1 2880.2.a.bg 1
264.p odd 2 1 2880.2.a.t 1
308.g odd 2 1 1960.2.a.g 1
308.m odd 6 2 1960.2.q.i 2
308.n even 6 2 1960.2.q.h 2
396.k even 6 2 3240.2.q.k 2
396.o odd 6 2 3240.2.q.x 2
440.c even 2 1 1600.2.a.o 1
440.o odd 2 1 1600.2.a.k 1
440.t even 4 2 1600.2.c.m 2
440.w odd 4 2 1600.2.c.k 2
572.b even 2 1 6760.2.a.i 1
660.g odd 2 1 1800.2.a.v 1
660.q even 4 2 1800.2.f.a 2
1540.b odd 2 1 9800.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 44.c even 2 1
80.2.a.a 1 11.b odd 2 1
200.2.a.c 1 220.g even 2 1
200.2.c.b 2 220.i odd 4 2
320.2.a.c 1 88.g even 2 1
320.2.a.d 1 88.b odd 2 1
360.2.a.a 1 132.d odd 2 1
400.2.a.e 1 55.d odd 2 1
400.2.c.d 2 55.e even 4 2
720.2.a.e 1 33.d even 2 1
1280.2.d.a 2 176.l odd 4 2
1280.2.d.j 2 176.i even 4 2
1600.2.a.k 1 440.o odd 2 1
1600.2.a.o 1 440.c even 2 1
1600.2.c.k 2 440.w odd 4 2
1600.2.c.m 2 440.t even 4 2
1800.2.a.v 1 660.g odd 2 1
1800.2.f.a 2 660.q even 4 2
1960.2.a.g 1 308.g odd 2 1
1960.2.q.h 2 308.n even 6 2
1960.2.q.i 2 308.m odd 6 2
2880.2.a.t 1 264.p odd 2 1
2880.2.a.bg 1 264.m even 2 1
3240.2.q.k 2 396.k even 6 2
3240.2.q.x 2 396.o odd 6 2
3600.2.a.h 1 165.d even 2 1
3600.2.f.t 2 165.l odd 4 2
3920.2.a.s 1 77.b even 2 1
4840.2.a.f 1 4.b odd 2 1
6760.2.a.i 1 572.b even 2 1
9680.2.a.q 1 1.a even 1 1 trivial
9800.2.a.x 1 1540.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T + 4 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 8 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 16 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
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