Properties

Label 9680.2.a.cg
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4840)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + q^{5} + \beta_{2} q^{7} + ( - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + q^{5} + \beta_{2} q^{7} + ( - \beta_1 + 3) q^{9} + ( - \beta_{2} - \beta_1) q^{13} - \beta_{2} q^{15} + (\beta_{2} - 2) q^{17} + (2 \beta_{2} - \beta_1 + 2) q^{19} + (\beta_1 - 6) q^{21} + (\beta_{2} - \beta_1 + 2) q^{23} + q^{25} + ( - 2 \beta_{2} - \beta_1 + 2) q^{27} + (\beta_1 - 2) q^{29} + ( - \beta_1 + 6) q^{31} + \beta_{2} q^{35} + \beta_1 q^{37} + ( - 2 \beta_{2} - 2 \beta_1 + 8) q^{39} - 2 \beta_{2} q^{41} + (\beta_{2} - \beta_1 + 6) q^{43} + ( - \beta_1 + 3) q^{45} + ( - \beta_{2} - \beta_1 + 2) q^{47} + ( - \beta_1 - 1) q^{49} + (2 \beta_{2} + \beta_1 - 6) q^{51} + \beta_1 q^{53} + ( - 4 \beta_{2} + \beta_1 - 10) q^{57} + ( - 2 \beta_{2} + 2 \beta_1) q^{59} + (4 \beta_{2} - \beta_1 - 2) q^{61} + (5 \beta_{2} + \beta_1 - 2) q^{63} + ( - \beta_{2} - \beta_1) q^{65} + (3 \beta_{2} + \beta_1 + 6) q^{67} + ( - 4 \beta_{2} - 4) q^{69} + ( - 4 \beta_{2} + \beta_1 - 2) q^{71} + (\beta_{2} - \beta_1 + 8) q^{73} - \beta_{2} q^{75} + 12 q^{79} + ( - 4 \beta_{2} + 5) q^{81} + ( - 3 \beta_{2} + 3 \beta_1 - 2) q^{83} + (\beta_{2} - 2) q^{85} + (4 \beta_{2} + \beta_1 - 2) q^{87} + (4 \beta_{2} + \beta_1) q^{89} + (2 \beta_{2} + 2 \beta_1 - 8) q^{91} + ( - 8 \beta_{2} - \beta_1 + 2) q^{93} + (2 \beta_{2} - \beta_1 + 2) q^{95} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 3 q^{5} - q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + 3 q^{5} - q^{7} + 8 q^{9} + q^{15} - 7 q^{17} + 3 q^{19} - 17 q^{21} + 4 q^{23} + 3 q^{25} + 7 q^{27} - 5 q^{29} + 17 q^{31} - q^{35} + q^{37} + 24 q^{39} + 2 q^{41} + 16 q^{43} + 8 q^{45} + 6 q^{47} - 4 q^{49} - 19 q^{51} + q^{53} - 25 q^{57} + 4 q^{59} - 11 q^{61} - 10 q^{63} + 16 q^{67} - 8 q^{69} - q^{71} + 22 q^{73} + q^{75} + 36 q^{79} + 19 q^{81} - 7 q^{85} - 9 q^{87} - 3 q^{89} - 24 q^{91} + 13 q^{93} + 3 q^{95} + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 4 \) 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
−1.76156
3.12489
−0.363328
0 −2.62620 0 1.00000 0 2.62620 0 3.89692 0
1.2 0 0.484862 0 1.00000 0 −0.484862 0 −2.76491 0
1.3 0 3.14134 0 1.00000 0 −3.14134 0 6.86799 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.cg 3
4.b odd 2 1 4840.2.a.r yes 3
11.b odd 2 1 9680.2.a.ch 3
44.c even 2 1 4840.2.a.q 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4840.2.a.q 3 44.c even 2 1
4840.2.a.r yes 3 4.b odd 2 1
9680.2.a.cg 3 1.a even 1 1 trivial
9680.2.a.ch 3 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3}^{3} - T_{3}^{2} - 8T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} + T_{7}^{2} - 8T_{7} - 4 \) Copy content Toggle raw display
\( T_{13}^{3} - 40T_{13} - 64 \) Copy content Toggle raw display
\( T_{17}^{3} + 7T_{17}^{2} + 8T_{17} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 8T + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 8T - 4 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 40T - 64 \) Copy content Toggle raw display
$17$ \( T^{3} + 7 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$23$ \( T^{3} - 4 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{3} + 5 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$31$ \( T^{3} - 17 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$37$ \( T^{3} - T^{2} + \cdots - 20 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$43$ \( T^{3} - 16 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{3} - 6 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$53$ \( T^{3} - T^{2} + \cdots - 20 \) Copy content Toggle raw display
$59$ \( T^{3} - 4 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$61$ \( T^{3} + 11 T^{2} + \cdots - 976 \) Copy content Toggle raw display
$67$ \( T^{3} - 16 T^{2} + \cdots + 976 \) Copy content Toggle raw display
$71$ \( T^{3} + T^{2} + \cdots + 512 \) Copy content Toggle raw display
$73$ \( T^{3} - 22 T^{2} + \cdots - 176 \) Copy content Toggle raw display
$79$ \( (T - 12)^{3} \) Copy content Toggle raw display
$83$ \( T^{3} - 228T - 880 \) Copy content Toggle raw display
$89$ \( T^{3} + 3 T^{2} + \cdots + 604 \) Copy content Toggle raw display
$97$ \( (T - 6)^{3} \) Copy content Toggle raw display
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