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Properties

Label 2960.2.a.bc
Level $2960$
Weight $2$
Character orbit 2960.a
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $6$
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] = [2960,2,Mod(1,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, 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("2960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.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: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.693982032.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 16x^{4} + 12x^{3} + 60x^{2} - 18x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1480)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 2) q^{9} + ( - \beta_{4} - \beta_{2}) q^{11} + (\beta_{5} + \beta_{4} - \beta_1 + 2) q^{13} - \beta_1 q^{15} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{17} + (\beta_{3} + 1) q^{19} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{21}+ \cdots + (2 \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9} + 10 q^{13} - q^{15} + 3 q^{17} + 6 q^{19} + 9 q^{21} - 4 q^{23} + 6 q^{25} - 7 q^{27} + 3 q^{29} + 3 q^{31} + 9 q^{33} - 8 q^{35} - 6 q^{37} + 16 q^{39} + 10 q^{41} + 11 q^{43} + 15 q^{45} - 23 q^{47} + 8 q^{49} + 16 q^{51} + 10 q^{53} + 4 q^{57} - 6 q^{59} + q^{61} - 23 q^{63} + 10 q^{65} + 4 q^{67} - 2 q^{69} - 9 q^{71} + 11 q^{73} - q^{75} + 32 q^{77} + 14 q^{79} + 46 q^{81} - 11 q^{83} + 3 q^{85} - 32 q^{87} + 30 q^{89} + 4 q^{91} + 2 q^{93} + 6 q^{95} + 29 q^{97} + 39 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 16x^{4} + 12x^{3} + 60x^{2} - 18x - 44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 25\nu^{4} + 4\nu^{3} - 304\nu^{2} + 82\nu + 500 ) / 98 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - \nu^{4} - 8\nu^{3} + 20\nu^{2} - 115\nu - 69 ) / 49 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} + 2\nu^{4} + 65\nu^{3} - 40\nu^{2} - 211\nu + 89 ) / 49 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 3\nu^{4} + 73\nu^{3} - 11\nu^{2} - 96\nu - 87 ) / 49 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{3} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{5} - 12\beta_{4} + 13\beta_{3} + 4\beta_{2} - \beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{5} + 8\beta_{4} + 29\beta_{3} + 2\beta_{2} + 93\beta _1 + 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
3.36175
2.45040
1.07986
−0.854190
−1.77441
−3.26341
0 −3.36175 0 1.00000 0 −3.60894 0 8.30138 0
1.2 0 −2.45040 0 1.00000 0 1.57732 0 3.00447 0
1.3 0 −1.07986 0 1.00000 0 −3.77160 0 −1.83390 0
1.4 0 0.854190 0 1.00000 0 −3.23895 0 −2.27036 0
1.5 0 1.77441 0 1.00000 0 2.66921 0 0.148539 0
1.6 0 3.26341 0 1.00000 0 −1.62704 0 7.64987 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(37\) \(1\)

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 2960.2.a.bc 6
4.b odd 2 1 1480.2.a.k ✓ 6
20.d odd 2 1 7400.2.a.s 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.a.k ✓ 6 4.b odd 2 1
2960.2.a.bc 6 1.a even 1 1 trivial
7400.2.a.s 6 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{6} + T_{3}^{5} - 16T_{3}^{4} - 12T_{3}^{3} + 60T_{3}^{2} + 18T_{3} - 44 \) Copy content Toggle raw display
\( T_{7}^{6} + 8T_{7}^{5} + 7T_{7}^{4} - 76T_{7}^{3} - 144T_{7}^{2} + 138T_{7} + 302 \) Copy content Toggle raw display
\( T_{13}^{6} - 10T_{13}^{5} + 4T_{13}^{4} + 180T_{13}^{3} - 260T_{13}^{2} - 648T_{13} + 384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + \cdots - 44 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 8 T^{5} + \cdots + 302 \) Copy content Toggle raw display
$11$ \( T^{6} - 33 T^{4} + \cdots - 96 \) Copy content Toggle raw display
$13$ \( T^{6} - 10 T^{5} + \cdots + 384 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + \cdots - 464 \) Copy content Toggle raw display
$19$ \( T^{6} - 6 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$23$ \( T^{6} + 4 T^{5} + \cdots - 576 \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} + \cdots + 32 \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + \cdots + 5496 \) Copy content Toggle raw display
$37$ \( (T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 10 T^{5} + \cdots + 81504 \) Copy content Toggle raw display
$43$ \( T^{6} - 11 T^{5} + \cdots - 51488 \) Copy content Toggle raw display
$47$ \( T^{6} + 23 T^{5} + \cdots - 37784 \) Copy content Toggle raw display
$53$ \( T^{6} - 10 T^{5} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + \cdots - 128 \) Copy content Toggle raw display
$61$ \( T^{6} - T^{5} + \cdots + 576 \) Copy content Toggle raw display
$67$ \( T^{6} - 4 T^{5} + \cdots + 44544 \) Copy content Toggle raw display
$71$ \( T^{6} + 9 T^{5} + \cdots + 4224 \) Copy content Toggle raw display
$73$ \( T^{6} - 11 T^{5} + \cdots - 300672 \) Copy content Toggle raw display
$79$ \( T^{6} - 14 T^{5} + \cdots + 5632 \) Copy content Toggle raw display
$83$ \( T^{6} + 11 T^{5} + \cdots - 9624 \) Copy content Toggle raw display
$89$ \( T^{6} - 30 T^{5} + \cdots + 1147008 \) Copy content Toggle raw display
$97$ \( T^{6} - 29 T^{5} + \cdots + 1463296 \) Copy content Toggle raw display
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