Properties

Label 6960.2.a.cl
Level $6960$
Weight $2$
Character orbit 6960.a
Self dual yes
Analytic conductor $55.576$
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] = [6960,2,Mod(1,6960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6960.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: \(55.5758798068\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 435)
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.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + (\beta_{2} + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + (\beta_{2} + 1) q^{7} + q^{9} - 3 q^{11} + (\beta_1 + 2) q^{13} + q^{15} + ( - \beta_{2} + 1) q^{17} + (\beta_{2} + \beta_1 + 1) q^{19} + (\beta_{2} + 1) q^{21} + ( - \beta_{2} + 2 \beta_1) q^{23} + q^{25} + q^{27} - q^{29} - 3 q^{33} + (\beta_{2} + 1) q^{35} + (\beta_{2} - 2 \beta_1 + 4) q^{37} + (\beta_1 + 2) q^{39} + (\beta_{2} + 4) q^{41} + ( - 2 \beta_{2} + \beta_1 + 5) q^{43} + q^{45} + ( - 2 \beta_{2} - \beta_1) q^{47} + (3 \beta_{2} - 2 \beta_1 + 2) q^{49} + ( - \beta_{2} + 1) q^{51} + ( - 3 \beta_1 - 1) q^{53} - 3 q^{55} + (\beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{2} + \beta_1 - 7) q^{59} + (\beta_{2} + \beta_1 + 3) q^{61} + (\beta_{2} + 1) q^{63} + (\beta_1 + 2) q^{65} + (\beta_{2} - 2 \beta_1 + 9) q^{67} + ( - \beta_{2} + 2 \beta_1) q^{69} + (3 \beta_{2} - 3 \beta_1 - 1) q^{71} + (3 \beta_{2} + 2 \beta_1) q^{73} + q^{75} + ( - 3 \beta_{2} - 3) q^{77} + ( - \beta_{2} - 3 \beta_1 + 1) q^{79} + q^{81} + ( - \beta_1 + 5) q^{83} + ( - \beta_{2} + 1) q^{85} - q^{87} + ( - 3 \beta_1 + 10) q^{89} + (\beta_1 + 4) q^{91} + (\beta_{2} + \beta_1 + 1) q^{95} + (2 \beta_{2} - 3 \beta_1 - 1) q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9} - 9 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} + 4 q^{19} + 4 q^{21} - q^{23} + 3 q^{25} + 3 q^{27} - 3 q^{29} - 9 q^{33} + 4 q^{35} + 13 q^{37} + 6 q^{39} + 13 q^{41} + 13 q^{43} + 3 q^{45} - 2 q^{47} + 9 q^{49} + 2 q^{51} - 3 q^{53} - 9 q^{55} + 4 q^{57} - 22 q^{59} + 10 q^{61} + 4 q^{63} + 6 q^{65} + 28 q^{67} - q^{69} + 3 q^{73} + 3 q^{75} - 12 q^{77} + 2 q^{79} + 3 q^{81} + 15 q^{83} + 2 q^{85} - 3 q^{87} + 30 q^{89} + 12 q^{91} + 4 q^{95} - q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 7 ) / 2 \) 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
0.772866
2.39138
−2.16425
0 1.00000 0 1.00000 0 −2.17554 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 1.32733 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 4.84822 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(29\) \(1\)

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 6960.2.a.cl 3
4.b odd 2 1 435.2.a.i 3
12.b even 2 1 1305.2.a.q 3
20.d odd 2 1 2175.2.a.u 3
20.e even 4 2 2175.2.c.m 6
60.h even 2 1 6525.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.i 3 4.b odd 2 1
1305.2.a.q 3 12.b even 2 1
2175.2.a.u 3 20.d odd 2 1
2175.2.c.m 6 20.e even 4 2
6525.2.a.bf 3 60.h even 2 1
6960.2.a.cl 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6960))\):

\( T_{7}^{3} - 4T_{7}^{2} - 7T_{7} + 14 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display
\( T_{13}^{3} - 6T_{13}^{2} - T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{3} - 2T_{17}^{2} - 11T_{17} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 4 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$11$ \( (T + 3)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 6T^{2} - T + 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} + \cdots + 88 \) Copy content Toggle raw display
$23$ \( T^{3} + T^{2} + \cdots - 112 \) Copy content Toggle raw display
$29$ \( (T + 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} \) Copy content Toggle raw display
$37$ \( T^{3} - 13T^{2} + 256 \) Copy content Toggle raw display
$41$ \( T^{3} - 13 T^{2} + \cdots - 28 \) Copy content Toggle raw display
$43$ \( T^{3} - 13 T^{2} + \cdots + 308 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} + \cdots - 266 \) Copy content Toggle raw display
$53$ \( T^{3} + 3 T^{2} + \cdots + 316 \) Copy content Toggle raw display
$59$ \( T^{3} + 22 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} + \cdots + 112 \) Copy content Toggle raw display
$67$ \( T^{3} - 28 T^{2} + \cdots - 194 \) Copy content Toggle raw display
$71$ \( T^{3} - 192T - 488 \) Copy content Toggle raw display
$73$ \( T^{3} - 3 T^{2} + \cdots + 1168 \) Copy content Toggle raw display
$79$ \( T^{3} - 2 T^{2} + \cdots + 224 \) Copy content Toggle raw display
$83$ \( T^{3} - 15 T^{2} + \cdots - 44 \) Copy content Toggle raw display
$89$ \( T^{3} - 30 T^{2} + \cdots + 602 \) Copy content Toggle raw display
$97$ \( T^{3} + T^{2} + \cdots + 76 \) Copy content Toggle raw display
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