Properties

Label 7280.2.a.br
Level $7280$
Weight $2$
Character orbit 7280.a
Self dual yes
Analytic conductor $58.131$
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] = [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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3640)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - q^{5} + q^{7} + ( - \beta_{3} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} - q^{5} + q^{7} + ( - \beta_{3} - \beta_1 + 1) q^{9} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{11} + q^{13} + ( - \beta_1 + 1) q^{15} + ( - \beta_{3} + \beta_1 + 2) q^{17} + (\beta_{2} - 1) q^{19} + (\beta_1 - 1) q^{21} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{23} + q^{25} + (2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{27} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{29} + ( - 2 \beta_{2} + \beta_1 - 1) q^{31} - 4 \beta_1 q^{33} - q^{35} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 4) q^{37} + (\beta_1 - 1) q^{39} + (\beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{41} + (\beta_{3} + \beta_{2} + \beta_1 - 3) q^{43} + (\beta_{3} + \beta_1 - 1) q^{45} + (2 \beta_{2} - 2 \beta_1) q^{47} + q^{49} + ( - \beta_{2} + 5 \beta_1) q^{51} + (2 \beta_{2} - 2 \beta_1 + 4) q^{53} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{55} + ( - \beta_{2} - \beta_1 + 2) q^{57} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{59} + (4 \beta_{2} - 4 \beta_1 + 2) q^{61} + ( - \beta_{3} - \beta_1 + 1) q^{63} - q^{65} + (\beta_{3} - 3 \beta_1 - 4) q^{67} + (2 \beta_1 + 2) q^{69} + (\beta_{3} - 5 \beta_{2} + 7 \beta_1 - 1) q^{71} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{73} + (\beta_1 - 1) q^{75} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{77} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{79} + (3 \beta_{2} - 5 \beta_1 + 3) q^{81} + (6 \beta_{2} - 2 \beta_1) q^{83} + (\beta_{3} - \beta_1 - 2) q^{85} + (\beta_{3} - 3 \beta_1 - 4) q^{87} + (\beta_{3} + 3 \beta_{2} + 4 \beta_1 + 6) q^{89} + q^{91} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{93} + ( - \beta_{2} + 1) q^{95} + ( - 2 \beta_{3} + 6 \beta_{2} + 6) q^{97} + (\beta_{3} - 3 \beta_{2} + 3 \beta_1 - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9} - 6 q^{11} + 4 q^{13} + 3 q^{15} + 9 q^{17} - 5 q^{19} - 3 q^{21} - 2 q^{23} + 4 q^{25} - 6 q^{27} - 3 q^{29} - q^{31} - 4 q^{33} - 4 q^{35} - 11 q^{37} - 3 q^{39} - 5 q^{41} - 12 q^{43} - 3 q^{45} - 4 q^{47} + 4 q^{49} + 6 q^{51} + 12 q^{53} + 6 q^{55} + 8 q^{57} - 11 q^{59} + 3 q^{63} - 4 q^{65} - 19 q^{67} + 10 q^{69} + 8 q^{71} + 8 q^{73} - 3 q^{75} - 6 q^{77} - 13 q^{79} + 4 q^{81} - 8 q^{83} - 9 q^{85} - 19 q^{87} + 25 q^{89} + 4 q^{91} + 5 q^{93} + 5 q^{95} + 18 q^{97} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} + 4\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 6\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 7\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} + \beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + 5\beta_{2} - 3\beta _1 + 6 \) 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
−0.344151
0.487928
2.90570
−2.04948
0 −3.04948 0 −1.00000 0 1.00000 0 6.29934 0
1.2 0 −1.34415 0 −1.00000 0 1.00000 0 −1.19326 0
1.3 0 −0.512072 0 −1.00000 0 1.00000 0 −2.73778 0
1.4 0 1.90570 0 −1.00000 0 1.00000 0 0.631706 0
\(n\): e.g. 2-40 or 990-1000
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.br 4
4.b odd 2 1 3640.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.x 4 4.b odd 2 1
7280.2.a.br 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7280))\):

\( T_{3}^{4} + 3T_{3}^{3} - 3T_{3}^{2} - 10T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 6T_{11}^{3} - 12T_{11}^{2} - 80T_{11} - 64 \) Copy content Toggle raw display
\( T_{17}^{4} - 9T_{17}^{3} + 7T_{17}^{2} + 32T_{17} + 16 \) Copy content Toggle raw display
\( T_{19}^{4} + 5T_{19}^{3} + 3T_{19}^{2} - 6T_{19} - 4 \) Copy content Toggle raw display
\( T_{23}^{4} + 2T_{23}^{3} - 24T_{23}^{2} - 8T_{23} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} - 3 T^{2} - 10 T - 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} - 12 T^{2} - 80 T - 64 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 9 T^{3} + 7 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + 3 T^{2} - 6 T - 4 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} - 24 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$29$ \( T^{4} + 3 T^{3} - 37 T^{2} + 24 T - 4 \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} - 23 T^{2} - 18 T + 52 \) Copy content Toggle raw display
$37$ \( T^{4} + 11 T^{3} - 29 T^{2} + \cdots - 676 \) Copy content Toggle raw display
$41$ \( T^{4} + 5 T^{3} - 133 T^{2} + \cdots - 1024 \) Copy content Toggle raw display
$43$ \( T^{4} + 12 T^{3} + 20 T^{2} + \cdots - 208 \) Copy content Toggle raw display
$47$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 11 T^{3} - 63 T^{2} + \cdots - 268 \) Copy content Toggle raw display
$61$ \( (T^{2} - 68)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 19 T^{3} + 61 T^{2} + 60 T + 16 \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} - 316 T^{2} + \cdots + 15248 \) Copy content Toggle raw display
$73$ \( T^{4} - 8 T^{3} - 44 T^{2} + 240 T + 832 \) Copy content Toggle raw display
$79$ \( T^{4} + 13 T^{3} - 147 T^{2} + \cdots - 4096 \) Copy content Toggle raw display
$83$ \( T^{4} + 8 T^{3} - 180 T^{2} + \cdots + 832 \) Copy content Toggle raw display
$89$ \( T^{4} - 25 T^{3} + 7 T^{2} + \cdots + 3008 \) Copy content Toggle raw display
$97$ \( T^{4} - 18 T^{3} - 176 T^{2} + \cdots - 24496 \) Copy content Toggle raw display
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