Properties

Label 7280.2.a.bc
Level $7280$
Weight $2$
Character orbit 7280.a
Self dual yes
Analytic conductor $58.131$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) 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 910)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} - q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + q^{5} - q^{7} + 5 q^{9} - 4 q^{11} + q^{13} + \beta q^{15} + ( - \beta - 2) q^{17} + ( - \beta + 4) q^{19} - \beta q^{21} + ( - \beta + 4) q^{23} + q^{25} + 2 \beta q^{27} + 6 q^{29} + (2 \beta + 4) q^{31} - 4 \beta q^{33} - q^{35} + (2 \beta + 2) q^{37} + \beta q^{39} + (3 \beta - 2) q^{41} + 4 \beta q^{43} + 5 q^{45} - 4 \beta q^{47} + q^{49} + ( - 2 \beta - 8) q^{51} + ( - \beta + 6) q^{53} - 4 q^{55} + (4 \beta - 8) q^{57} + (\beta - 4) q^{59} + (2 \beta - 6) q^{61} - 5 q^{63} + q^{65} + ( - 2 \beta + 4) q^{67} + (4 \beta - 8) q^{69} + (\beta - 8) q^{71} + (2 \beta + 6) q^{73} + \beta q^{75} + 4 q^{77} - 4 \beta q^{79} + q^{81} + (2 \beta + 8) q^{83} + ( - \beta - 2) q^{85} + 6 \beta q^{87} + ( - \beta + 14) q^{89} - q^{91} + (4 \beta + 16) q^{93} + ( - \beta + 4) q^{95} + (4 \beta + 6) q^{97} - 20 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} + 10 q^{9} - 8 q^{11} + 2 q^{13} - 4 q^{17} + 8 q^{19} + 8 q^{23} + 2 q^{25} + 12 q^{29} + 8 q^{31} - 2 q^{35} + 4 q^{37} - 4 q^{41} + 10 q^{45} + 2 q^{49} - 16 q^{51} + 12 q^{53} - 8 q^{55} - 16 q^{57} - 8 q^{59} - 12 q^{61} - 10 q^{63} + 2 q^{65} + 8 q^{67} - 16 q^{69} - 16 q^{71} + 12 q^{73} + 8 q^{77} + 2 q^{81} + 16 q^{83} - 4 q^{85} + 28 q^{89} - 2 q^{91} + 32 q^{93} + 8 q^{95} + 12 q^{97} - 40 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
0 −2.82843 0 1.00000 0 −1.00000 0 5.00000 0
1.2 0 2.82843 0 1.00000 0 −1.00000 0 5.00000 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.bc 2
4.b odd 2 1 910.2.a.m 2
12.b even 2 1 8190.2.a.ck 2
20.d odd 2 1 4550.2.a.bn 2
28.d even 2 1 6370.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
910.2.a.m 2 4.b odd 2 1
4550.2.a.bn 2 20.d odd 2 1
6370.2.a.bd 2 28.d even 2 1
7280.2.a.bc 2 1.a even 1 1 trivial
8190.2.a.ck 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7280))\):

\( T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 4 \) Copy content Toggle raw display
\( T_{19}^{2} - 8T_{19} + 8 \) Copy content Toggle raw display
\( T_{23}^{2} - 8T_{23} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$43$ \( T^{2} - 128 \) Copy content Toggle raw display
$47$ \( T^{2} - 128 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 16T + 56 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 128 \) Copy content Toggle raw display
$83$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$89$ \( T^{2} - 28T + 188 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T - 92 \) Copy content Toggle raw display
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