Properties

Label 6960.2.a.co
Level $6960$
Weight $2$
Character orbit 6960.a
Self dual yes
Analytic conductor $55.576$
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] = [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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + (\beta_1 - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + (\beta_1 - 1) q^{7} + q^{9} + ( - \beta_1 + 1) q^{11} + ( - \beta_{3} - 2) q^{13} - q^{15} + ( - \beta_{2} - 2) q^{17} + (\beta_{3} - \beta_1 + 1) q^{19} + (\beta_1 - 1) q^{21} + (2 \beta_{2} + 2) q^{23} + q^{25} + q^{27} - q^{29} + ( - 2 \beta_{2} + 2) q^{31} + ( - \beta_1 + 1) q^{33} + ( - \beta_1 + 1) q^{35} + (2 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{37} + ( - \beta_{3} - 2) q^{39} + (2 \beta_{3} + 2 \beta_1 - 4) q^{41} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{43} - q^{45} + ( - \beta_{3} - 2 \beta_{2} + 4) q^{47} + (2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 2) q^{49} + ( - \beta_{2} - 2) q^{51} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{53} + (\beta_1 - 1) q^{55} + (\beta_{3} - \beta_1 + 1) q^{57} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{59} + (\beta_{3} - 3 \beta_1 - 5) q^{61} + (\beta_1 - 1) q^{63} + (\beta_{3} + 2) q^{65} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{67}+ \cdots + ( - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{15} - 10 q^{17} + 2 q^{19} - 2 q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 16 q^{37} - 8 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} + 12 q^{47} + 6 q^{49} - 10 q^{51} - 10 q^{53} - 2 q^{55} + 2 q^{57} - 2 q^{59} - 26 q^{61} - 2 q^{63} + 8 q^{65} - 2 q^{67} + 12 q^{69} + 10 q^{71} + 4 q^{75} - 34 q^{77} - 22 q^{79} + 4 q^{81} + 10 q^{83} + 10 q^{85} - 4 q^{87} - 4 q^{89} + 8 q^{91} + 4 q^{93} - 2 q^{95} - 22 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 2\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 4\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 2\beta_{2} + \beta _1 + 1 \) 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
1.13856
−1.75660
−0.820249
2.43828
0 1.00000 0 −1.00000 0 −5.07830 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −0.393832 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 0.729126 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 2.74301 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.co 4
4.b odd 2 1 435.2.a.j 4
12.b even 2 1 1305.2.a.r 4
20.d odd 2 1 2175.2.a.v 4
20.e even 4 2 2175.2.c.n 8
60.h even 2 1 6525.2.a.bi 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.j 4 4.b odd 2 1
1305.2.a.r 4 12.b even 2 1
2175.2.a.v 4 20.d odd 2 1
2175.2.c.n 8 20.e even 4 2
6525.2.a.bi 4 60.h even 2 1
6960.2.a.co 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6960))\):

\( T_{7}^{4} + 2T_{7}^{3} - 15T_{7}^{2} + 4T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 15T_{11}^{2} - 4T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{4} + 8T_{13}^{3} + 3T_{13}^{2} - 92T_{13} - 164 \) Copy content Toggle raw display
\( T_{17}^{4} + 10T_{17}^{3} + 21T_{17}^{2} - 40T_{17} - 116 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} + \cdots - 164 \) Copy content Toggle raw display
$17$ \( T^{4} + 10 T^{3} + \cdots - 116 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + \cdots - 1024 \) Copy content Toggle raw display
$29$ \( (T + 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( T^{4} + 16 T^{3} + \cdots - 2624 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + \cdots - 1616 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} + \cdots + 1216 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$53$ \( T^{4} + 10 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} + \cdots + 10496 \) Copy content Toggle raw display
$61$ \( T^{4} + 26 T^{3} + \cdots - 11344 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + \cdots - 124 \) Copy content Toggle raw display
$71$ \( T^{4} - 10 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$73$ \( T^{4} - 84 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$79$ \( T^{4} + 22 T^{3} + \cdots + 2416 \) Copy content Toggle raw display
$83$ \( T^{4} - 10 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} + \cdots + 10156 \) Copy content Toggle raw display
$97$ \( T^{4} + 22 T^{3} + \cdots + 2384 \) Copy content Toggle raw display
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