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Properties

Label 7280.2.a.ce
Level $7280$
Weight $2$
Character orbit 7280.a
Self dual yes
Analytic conductor $58.131$
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] = [7280,2,Mod(1,7280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7280, 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("7280.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7280.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: \(58.1310926715\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.45853772.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 14x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 455)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + q^{5} - q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + q^{5} - q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + \beta_1 q^{11} + q^{13} - \beta_{5} q^{15} + ( - \beta_{5} + \beta_{3} + \beta_{2} + \cdots + 1) q^{17}+ \cdots + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{5} - 6 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{5} - 6 q^{7} + 8 q^{9} + 2 q^{11} + 6 q^{13} + 8 q^{17} - 6 q^{19} - 4 q^{23} + 6 q^{25} + 6 q^{27} - 14 q^{29} + 6 q^{31} + 12 q^{33} - 6 q^{35} + 20 q^{37} - 4 q^{41} - 10 q^{43} + 8 q^{45} + 6 q^{47} + 6 q^{49} + 36 q^{51} + 2 q^{53} + 2 q^{55} + 18 q^{57} + 18 q^{59} - 8 q^{63} + 6 q^{65} - 20 q^{67} - 10 q^{69} + 4 q^{71} + 6 q^{73} - 2 q^{77} + 4 q^{79} + 18 q^{81} + 6 q^{83} + 8 q^{85} + 26 q^{87} - 12 q^{89} - 6 q^{91} + 2 q^{93} - 6 q^{95} + 26 q^{97} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 14x - 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{5} + 8\nu^{3} - 2\nu^{2} - 10\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} + \nu^{4} - 9\nu^{3} - 6\nu^{2} + 16\nu + 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{4} - 16\nu^{2} + 4\nu + 19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + \nu^{4} - 9\nu^{3} - 8\nu^{2} + 16\nu + 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{3} + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} - 2\beta_{3} - 2\beta_{2} + 4\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{5} + \beta_{4} + 8\beta_{3} - 2\beta _1 + 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 4\beta_{4} - 9\beta_{3} - 9\beta_{2} + 11\beta _1 - 11 \) 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
1.55061
−0.851902
−1.16074
−2.73570
2.43655
1.76118
0 −2.76555 0 1.00000 0 −1.00000 0 4.64827 0
1.2 0 −2.20599 0 1.00000 0 −1.00000 0 1.86638 0
1.3 0 −0.432805 0 1.00000 0 −1.00000 0 −2.81268 0
1.4 0 0.594815 0 1.00000 0 −1.00000 0 −2.64619 0
1.5 0 1.57457 0 1.00000 0 −1.00000 0 −0.520730 0
1.6 0 3.23496 0 1.00000 0 −1.00000 0 7.46495 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
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)
\(13\) \(-1\)

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 7280.2.a.ce 6
4.b odd 2 1 455.2.a.e ✓ 6
12.b even 2 1 4095.2.a.bl 6
20.d odd 2 1 2275.2.a.s 6
28.d even 2 1 3185.2.a.r 6
52.b odd 2 1 5915.2.a.u 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
455.2.a.e ✓ 6 4.b odd 2 1
2275.2.a.s 6 20.d odd 2 1
3185.2.a.r 6 28.d even 2 1
4095.2.a.bl 6 12.b even 2 1
5915.2.a.u 6 52.b odd 2 1
7280.2.a.ce 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7280))\):

\( T_{3}^{6} - 13T_{3}^{4} - 2T_{3}^{3} + 35T_{3}^{2} - 4T_{3} - 8 \) Copy content Toggle raw display
\( T_{11}^{6} - 2T_{11}^{5} - 40T_{11}^{4} + 80T_{11}^{3} + 368T_{11}^{2} - 448T_{11} - 1152 \) Copy content Toggle raw display
\( T_{17}^{6} - 8T_{17}^{5} - 49T_{17}^{4} + 518T_{17}^{3} - 265T_{17}^{2} - 5596T_{17} + 9948 \) Copy content Toggle raw display
\( T_{19}^{6} + 6T_{19}^{5} - 33T_{19}^{4} - 70T_{19}^{3} + 173T_{19}^{2} - 76T_{19} + 8 \) Copy content Toggle raw display
\( T_{23}^{6} + 4T_{23}^{5} - 64T_{23}^{4} - 232T_{23}^{3} + 1040T_{23}^{2} + 3456T_{23} - 1152 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 13 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( (T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + \cdots - 1152 \) Copy content Toggle raw display
$13$ \( (T - 1)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 8 T^{5} + \cdots + 9948 \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$23$ \( T^{6} + 4 T^{5} + \cdots - 1152 \) Copy content Toggle raw display
$29$ \( T^{6} + 14 T^{5} + \cdots - 5004 \) Copy content Toggle raw display
$31$ \( T^{6} - 6 T^{5} + \cdots - 24536 \) Copy content Toggle raw display
$37$ \( T^{6} - 20 T^{5} + \cdots - 4564 \) Copy content Toggle raw display
$41$ \( T^{6} + 4 T^{5} + \cdots + 89748 \) Copy content Toggle raw display
$43$ \( T^{6} + 10 T^{5} + \cdots - 105856 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + \cdots + 36864 \) Copy content Toggle raw display
$53$ \( T^{6} - 2 T^{5} + \cdots + 4032 \) Copy content Toggle raw display
$59$ \( T^{6} - 18 T^{5} + \cdots - 8856 \) Copy content Toggle raw display
$61$ \( T^{6} - 92 T^{4} + \cdots - 64 \) Copy content Toggle raw display
$67$ \( T^{6} + 20 T^{5} + \cdots + 15472 \) Copy content Toggle raw display
$71$ \( T^{6} - 4 T^{5} + \cdots - 169344 \) Copy content Toggle raw display
$73$ \( T^{6} - 6 T^{5} + \cdots - 10432 \) Copy content Toggle raw display
$79$ \( T^{6} - 4 T^{5} + \cdots - 1545312 \) Copy content Toggle raw display
$83$ \( T^{6} - 6 T^{5} + \cdots + 2304 \) Copy content Toggle raw display
$89$ \( T^{6} + 12 T^{5} + \cdots - 692316 \) Copy content Toggle raw display
$97$ \( T^{6} - 26 T^{5} + \cdots + 152512 \) Copy content Toggle raw display
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