Properties

Label 9680.2.a.cr
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.4752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) Copy content Toggle raw display
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: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} - q^{5} + ( - \beta_{3} + \beta_{2} + 2) q^{7} + (\beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} - q^{5} + ( - \beta_{3} + \beta_{2} + 2) q^{7} + (\beta_1 + 1) q^{9} + ( - \beta_1 - 1) q^{13} + ( - \beta_{2} - 1) q^{15} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_{3} + 2 \beta_{2} + 4) q^{21} + ( - 2 \beta_{3} - \beta_1 - 1) q^{23} + q^{25} + (2 \beta_{3} - \beta_{2} + \beta_1) q^{27} - 4 q^{29} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 - 5) q^{31} + (\beta_{3} - \beta_{2} - 2) q^{35} + ( - 2 \beta_1 - 2) q^{37} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 3) q^{39}+ \cdots + (4 \beta_{3} - 6 \beta_{2} + \beta_1 - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{5} + 6 q^{7} + 4 q^{9} - 4 q^{13} - 2 q^{15} + 12 q^{21} - 4 q^{23} + 4 q^{25} + 2 q^{27} - 16 q^{29} - 16 q^{31} - 6 q^{35} - 8 q^{37} - 8 q^{39} - 8 q^{41} - 10 q^{43} - 4 q^{45} - 14 q^{47} - 4 q^{49} + 8 q^{53} - 12 q^{57} + 4 q^{61} + 12 q^{63} + 4 q^{65} + 2 q^{67} - 20 q^{69} - 8 q^{71} - 32 q^{73} + 2 q^{75} - 4 q^{79} - 8 q^{81} + 12 q^{83} - 8 q^{87} + 12 q^{89} - 12 q^{91} - 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 + 4 \) 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
1.21969
−1.49551
−0.219687
2.49551
0 −2.33225 0 −1.00000 0 0.399804 0 2.43937 0
1.2 0 −0.0947876 0 −1.00000 0 −0.826838 0 −2.99102 0
1.3 0 1.60020 0 −1.00000 0 4.33225 0 −0.439374 0
1.4 0 2.82684 0 −1.00000 0 2.09479 0 4.99102 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.cr 4
4.b odd 2 1 4840.2.a.w 4
11.b odd 2 1 9680.2.a.cq 4
44.c even 2 1 4840.2.a.x yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4840.2.a.w 4 4.b odd 2 1
4840.2.a.x yes 4 44.c even 2 1
9680.2.a.cq 4 11.b odd 2 1
9680.2.a.cr 4 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 10T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{3} + 6T_{7}^{2} + 6T_{7} - 3 \) Copy content Toggle raw display
\( T_{13}^{4} + 4T_{13}^{3} - 12T_{13}^{2} - 32T_{13} + 16 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + \cdots - 3 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 36 T^{2} + \cdots - 48 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 16 T^{3} + \cdots - 1136 \) Copy content Toggle raw display
$37$ \( T^{4} + 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$47$ \( T^{4} + 14 T^{3} + \cdots + 157 \) Copy content Toggle raw display
$53$ \( T^{4} - 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{4} - 108 T^{2} + \cdots - 432 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + \cdots + 229 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + \cdots + 577 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} + \cdots + 3664 \) Copy content Toggle raw display
$73$ \( (T + 8)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} + \cdots - 752 \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} + \cdots - 4656 \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + \cdots - 4287 \) Copy content Toggle raw display
$97$ \( T^{4} - 204 T^{2} + \cdots + 3408 \) Copy content Toggle raw display
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