Properties

Label 9680.2.a.ct
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: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 2\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 3\beta_1 \) 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
0.737640
−0.477260
2.09529
−1.35567
0 −1.19353 0 1.00000 0 −1.83785 0 −1.57549 0
1.2 0 −0.294963 0 1.00000 0 −4.39026 0 −2.91300 0
1.3 0 1.29496 0 1.00000 0 −0.227777 0 −1.32307 0
1.4 0 2.19353 0 1.00000 0 −0.544113 0 1.81156 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.ct 4
4.b odd 2 1 4840.2.a.z 4
11.b odd 2 1 9680.2.a.cu 4
11.d odd 10 2 880.2.bo.d 8
44.c even 2 1 4840.2.a.y 4
44.g even 10 2 440.2.y.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.a 8 44.g even 10 2
880.2.bo.d 8 11.d odd 10 2
4840.2.a.y 4 44.c even 2 1
4840.2.a.z 4 4.b odd 2 1
9680.2.a.ct 4 1.a even 1 1 trivial
9680.2.a.cu 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3}^{4} - 2T_{3}^{3} - 2T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 7T_{7}^{3} + 13T_{7}^{2} + 7T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + 3T_{13}^{3} - 21T_{13}^{2} - 13T_{13} + 41 \) Copy content Toggle raw display
\( T_{17}^{4} + 11T_{17}^{3} + 20T_{17}^{2} - 62T_{17} - 11 \) 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
$7$ \( T^{4} + 7 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 3 T^{3} + \cdots + 41 \) Copy content Toggle raw display
$17$ \( T^{4} + 11 T^{3} + \cdots - 11 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + \cdots + 101 \) Copy content Toggle raw display
$23$ \( T^{4} - 11 T^{3} + \cdots + 31 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{4} - 17 T^{3} + \cdots - 379 \) Copy content Toggle raw display
$37$ \( T^{4} + 3 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 13 T^{3} + \cdots + 541 \) Copy content Toggle raw display
$43$ \( T^{4} + 7 T^{3} + \cdots + 61 \) Copy content Toggle raw display
$47$ \( T^{4} - T^{3} + \cdots - 149 \) Copy content Toggle raw display
$53$ \( T^{4} + 15 T^{3} + \cdots - 1361 \) Copy content Toggle raw display
$59$ \( T^{4} - 17 T^{3} + \cdots - 1681 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + \cdots + 811 \) Copy content Toggle raw display
$67$ \( T^{4} + 7 T^{3} + \cdots + 431 \) Copy content Toggle raw display
$71$ \( T^{4} - 15 T^{3} + \cdots - 9871 \) Copy content Toggle raw display
$73$ \( T^{4} + 7 T^{3} + \cdots + 3751 \) Copy content Toggle raw display
$79$ \( T^{4} + 12 T^{3} + \cdots + 2381 \) Copy content Toggle raw display
$83$ \( T^{4} - 9 T^{3} + \cdots + 11 \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + \cdots - 479 \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} + \cdots + 2179 \) Copy content Toggle raw display
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