Properties

Label 7280.2.a.ca
Level $7280$
Weight $2$
Character orbit 7280.a
Self dual yes
Analytic conductor $58.131$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.1310926715\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) 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

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + q^{5} + q^{7} + (\beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + q^{5} + q^{7} + (\beta_{2} + \beta_1) q^{9} + ( - \beta_{3} + 1) q^{11} + q^{13} + \beta_1 q^{15} + ( - \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{19} + \beta_1 q^{21} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{23} + q^{25} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{27} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{29} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{31} + ( - 2 \beta_{2} - 2) q^{33} + q^{35} + ( - \beta_{3} + \beta_1 - 1) q^{37} + \beta_1 q^{39} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{41} + ( - \beta_{3} + 2 \beta_1 + 1) q^{43} + (\beta_{2} + \beta_1) q^{45} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{47} + q^{49} + ( - \beta_{3} + \beta_{2} + 4) q^{51} + (2 \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{3} + 1) q^{55} + (\beta_{3} + 3 \beta_{2} + 4 \beta_1 + 4) q^{57} + ( - 6 \beta_{2} - 3 \beta_1 - 2) q^{59} + 2 \beta_{2} q^{61} + (\beta_{2} + \beta_1) q^{63} + q^{65} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{67} + ( - 2 \beta_{3} - 2) q^{69} + ( - \beta_{3} - 2 \beta_{2} + 3) q^{71} + (4 \beta_{2} + 2 \beta_1 + 2) q^{73} + \beta_1 q^{75} + ( - \beta_{3} + 1) q^{77} + ( - 4 \beta_{3} + \beta_{2} + 5 \beta_1 + 3) q^{79} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 - 5) q^{81} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 2) q^{83} + ( - \beta_{2} + \beta_1 - 1) q^{85} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 7) q^{87} + (3 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 3) q^{89} + q^{91} + (2 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 1) q^{93} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{95} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{97} + (\beta_{3} - 2 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 4 q^{5} + 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + 4 q^{5} + 4 q^{7} - q^{9} + 4 q^{11} + 4 q^{13} + q^{15} - q^{17} + 7 q^{19} + q^{21} + 6 q^{23} + 4 q^{25} + 4 q^{27} + q^{29} + 11 q^{31} - 4 q^{33} + 4 q^{35} - 3 q^{37} + q^{39} - 5 q^{41} + 6 q^{43} - q^{45} + 6 q^{47} + 4 q^{49} + 14 q^{51} - 2 q^{53} + 4 q^{55} + 14 q^{57} + q^{59} - 4 q^{61} - q^{63} + 4 q^{65} + 11 q^{67} - 8 q^{69} + 16 q^{71} + 2 q^{73} + q^{75} + 4 q^{77} + 15 q^{79} - 16 q^{81} + 2 q^{83} - q^{85} + 23 q^{87} - 3 q^{89} + 4 q^{91} - 5 q^{93} + 7 q^{95} + 8 q^{97} - 6 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 2 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.75660
−0.820249
1.13856
2.43828
0 −1.75660 0 1.00000 0 1.00000 0 0.0856374 0
1.2 0 −0.820249 0 1.00000 0 1.00000 0 −2.32719 0
1.3 0 1.13856 0 1.00000 0 1.00000 0 −1.70367 0
1.4 0 2.43828 0 1.00000 0 1.00000 0 2.94523 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.ca 4
4.b odd 2 1 3640.2.a.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.s 4 4.b odd 2 1
7280.2.a.ca 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7280))\):

\( T_{3}^{4} - T_{3}^{3} - 5T_{3}^{2} + 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 4 \) Copy content Toggle raw display
\( T_{17}^{4} + T_{17}^{3} - 15T_{17}^{2} + 8T_{17} + 4 \) Copy content Toggle raw display
\( T_{19}^{4} - 7T_{19}^{3} - 5T_{19}^{2} + 26T_{19} + 4 \) Copy content Toggle raw display
\( T_{23}^{4} - 6T_{23}^{3} - 20T_{23}^{2} + 32T_{23} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 5 T^{2} + 2 T + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} - 15 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} - 7 T^{3} - 5 T^{2} + 26 T + 4 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} - 20 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$29$ \( T^{4} - T^{3} - 53 T^{2} + 236 T - 284 \) Copy content Toggle raw display
$31$ \( T^{4} - 11 T^{3} - 23 T^{2} + \cdots - 1604 \) Copy content Toggle raw display
$37$ \( T^{4} + 3 T^{3} - 7 T^{2} - 12 T - 4 \) Copy content Toggle raw display
$41$ \( T^{4} + 5 T^{3} - 29 T^{2} - 40 T + 124 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} - 8 T^{2} + 56 T + 16 \) Copy content Toggle raw display
$47$ \( T^{4} - 6 T^{3} - 140 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} - 32 T^{2} - 88 T + 16 \) Copy content Toggle raw display
$59$ \( T^{4} - T^{3} - 219 T^{2} + \cdots + 10636 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} - 20 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$67$ \( T^{4} - 11 T^{3} - 9 T^{2} + 44 T + 16 \) Copy content Toggle raw display
$71$ \( T^{4} - 16 T^{3} + 60 T^{2} - 48 T - 16 \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} - 96 T^{2} + \cdots + 2096 \) Copy content Toggle raw display
$79$ \( T^{4} - 15 T^{3} - 99 T^{2} + \cdots - 5776 \) Copy content Toggle raw display
$83$ \( T^{4} - 2 T^{3} - 116 T^{2} + \cdots + 1216 \) Copy content Toggle raw display
$89$ \( T^{4} + 3 T^{3} - 261 T^{2} + \cdots + 6156 \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} - 60 T^{2} - 16 T + 64 \) Copy content Toggle raw display
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