Properties

Label 7280.2.a.bq
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: 4.4.1957.1
Defining polynomial: \( x^{4} - 4x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 455)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} + q^{5} + q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{3} + q^{5} + q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{9} + ( - \beta_{3} + \beta_1 + 1) q^{11} - q^{13} + (\beta_{2} - 1) q^{15} + ( - \beta_1 + 4) q^{17} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{19} + (\beta_{2} - 1) q^{21} + ( - 3 \beta_{2} + \beta_1) q^{23} + q^{25} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{27} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{29} + (\beta_{3} - \beta_{2} - \beta_1) q^{31} + (2 \beta_{3} + \beta_{2} - \beta_1) q^{33} + q^{35} + ( - 2 \beta_{3} + 3 \beta_{2} + 1) q^{37} + ( - \beta_{2} + 1) q^{39} + (\beta_{2} + 2 \beta_1 + 2) q^{41} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{43} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{45} + ( - 3 \beta_{3} - 3 \beta_{2} - 1) q^{47} + q^{49} + (5 \beta_{2} + \beta_1 - 5) q^{51} + ( - 3 \beta_{3} - 2 \beta_{2} + \beta_1 + 5) q^{53} + ( - \beta_{3} + \beta_1 + 1) q^{55} + ( - 3 \beta_{3} - \beta_{2} + 4) q^{57} + ( - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{59} + (2 \beta_{3} + 3 \beta_{2} + \beta_1 + 6) q^{61} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{63} - q^{65} + (4 \beta_{2} + \beta_1 + 2) q^{67} + (3 \beta_{3} - \beta_{2} + 2 \beta_1 - 11) q^{69} + ( - 4 \beta_{3} - 2 \beta_1 + 6) q^{71} + ( - \beta_{3} - 3 \beta_1 + 9) q^{73} + (\beta_{2} - 1) q^{75} + ( - \beta_{3} + \beta_1 + 1) q^{77} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{79} + ( - 4 \beta_{3} - 6 \beta_{2} + 4) q^{81} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{83} + ( - \beta_1 + 4) q^{85} + (3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{87} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 7) q^{89} - q^{91} + ( - \beta_{3} + 2 \beta_1 - 5) q^{93} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{95} + (\beta_{3} + 4 \beta_{2} - 3 \beta_1 - 1) q^{97} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 6 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{15} + 16 q^{17} - 2 q^{19} - 4 q^{21} + 4 q^{25} - 10 q^{27} - 2 q^{29} + 2 q^{31} + 4 q^{33} + 4 q^{35} + 4 q^{39} + 8 q^{41} + 10 q^{43} + 6 q^{45} - 10 q^{47} + 4 q^{49} - 20 q^{51} + 14 q^{53} + 2 q^{55} + 10 q^{57} + 14 q^{59} + 28 q^{61} + 6 q^{63} - 4 q^{65} + 8 q^{67} - 38 q^{69} + 16 q^{71} + 34 q^{73} - 4 q^{75} + 2 q^{77} + 28 q^{79} + 8 q^{81} + 6 q^{83} + 16 q^{85} - 14 q^{87} - 24 q^{89} - 4 q^{91} - 22 q^{93} - 2 q^{95} - 2 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} - \nu^{2} - 7\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 3\beta_{2} + 4\beta _1 + 1 ) / 2 \) 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
0.396339
−0.693822
2.06150
−1.76401
0 −3.23925 0 1.00000 0 1.00000 0 7.49277 0
1.2 0 −1.82479 0 1.00000 0 1.00000 0 0.329851 0
1.3 0 −0.811721 0 1.00000 0 1.00000 0 −2.34111 0
1.4 0 1.87576 0 1.00000 0 1.00000 0 0.518489 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.bq 4
4.b odd 2 1 455.2.a.d 4
12.b even 2 1 4095.2.a.bc 4
20.d odd 2 1 2275.2.a.o 4
28.d even 2 1 3185.2.a.q 4
52.b odd 2 1 5915.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
455.2.a.d 4 4.b odd 2 1
2275.2.a.o 4 20.d odd 2 1
3185.2.a.q 4 28.d even 2 1
4095.2.a.bc 4 12.b even 2 1
5915.2.a.m 4 52.b odd 2 1
7280.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7280))\):

\( T_{3}^{4} + 4T_{3}^{3} - T_{3}^{2} - 14T_{3} - 9 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 32T_{11}^{2} + 80T_{11} - 48 \) Copy content Toggle raw display
\( T_{17}^{4} - 16T_{17}^{3} + 83T_{17}^{2} - 130T_{17} - 49 \) Copy content Toggle raw display
\( T_{19}^{4} + 2T_{19}^{3} - 33T_{19}^{2} - 50T_{19} + 173 \) Copy content Toggle raw display
\( T_{23}^{4} - 64T_{23}^{2} + 200T_{23} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} - T^{2} - 14 T - 9 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} - 32 T^{2} + 80 T - 48 \) Copy content Toggle raw display
$13$ \( (T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 16 T^{3} + 83 T^{2} - 130 T - 49 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} - 33 T^{2} - 50 T + 173 \) Copy content Toggle raw display
$23$ \( T^{4} - 64 T^{2} + 200 T - 48 \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} - 25 T^{2} - 78 T - 59 \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} - 45 T^{2} - 22 T + 201 \) Copy content Toggle raw display
$37$ \( T^{4} - 125 T^{2} - 26 T + 479 \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} - 43 T^{2} + 100 T + 393 \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} - 76 T^{2} + \cdots - 1648 \) Copy content Toggle raw display
$47$ \( T^{4} + 10 T^{3} - 120 T^{2} + \cdots - 1136 \) Copy content Toggle raw display
$53$ \( T^{4} - 14 T^{3} - 84 T^{2} + \cdots - 1008 \) Copy content Toggle raw display
$59$ \( T^{4} - 14 T^{3} + 27 T^{2} + \cdots - 479 \) Copy content Toggle raw display
$61$ \( T^{4} - 28 T^{3} + 184 T^{2} + \cdots - 8176 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} - 117 T^{2} + \cdots + 2679 \) Copy content Toggle raw display
$71$ \( T^{4} - 16 T^{3} - 92 T^{2} + \cdots + 2768 \) Copy content Toggle raw display
$73$ \( T^{4} - 34 T^{3} + 328 T^{2} + \cdots - 7312 \) Copy content Toggle raw display
$79$ \( T^{4} - 28 T^{3} + 213 T^{2} + \cdots - 2463 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} - 28 T^{2} + 232 T - 336 \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + 141 T^{2} + \cdots + 213 \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} - 208 T^{2} + \cdots + 8176 \) Copy content Toggle raw display
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