Properties

Label 560.6.a.g
Level $560$
Weight $6$
Character orbit 560.a
Self dual yes
Analytic conductor $89.815$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.8149390953\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 12 q^{3} + 25 q^{5} + 49 q^{7} - 99 q^{9} + O(q^{10}) \) \( q + 12 q^{3} + 25 q^{5} + 49 q^{7} - 99 q^{9} - 556 q^{11} - 354 q^{13} + 300 q^{15} + 770 q^{17} + 2684 q^{19} + 588 q^{21} + 1528 q^{23} + 625 q^{25} - 4104 q^{27} - 2418 q^{29} - 7840 q^{31} - 6672 q^{33} + 1225 q^{35} - 314 q^{37} - 4248 q^{39} - 17878 q^{41} - 16476 q^{43} - 2475 q^{45} - 5376 q^{47} + 2401 q^{49} + 9240 q^{51} + 1654 q^{53} - 13900 q^{55} + 32208 q^{57} + 29492 q^{59} + 27630 q^{61} - 4851 q^{63} - 8850 q^{65} - 57716 q^{67} + 18336 q^{69} - 70648 q^{71} + 74202 q^{73} + 7500 q^{75} - 27244 q^{77} - 74336 q^{79} - 25191 q^{81} - 44068 q^{83} + 19250 q^{85} - 29016 q^{87} + 129306 q^{89} - 17346 q^{91} - 94080 q^{93} + 67100 q^{95} - 137646 q^{97} + 55044 q^{99} + O(q^{100}) \)

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
0 12.0000 0 25.0000 0 49.0000 0 −99.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 560.6.a.g 1
4.b odd 2 1 280.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.a 1 4.b odd 2 1
560.6.a.g 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 12 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(560))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -12 + T \)
$5$ \( -25 + T \)
$7$ \( -49 + T \)
$11$ \( 556 + T \)
$13$ \( 354 + T \)
$17$ \( -770 + T \)
$19$ \( -2684 + T \)
$23$ \( -1528 + T \)
$29$ \( 2418 + T \)
$31$ \( 7840 + T \)
$37$ \( 314 + T \)
$41$ \( 17878 + T \)
$43$ \( 16476 + T \)
$47$ \( 5376 + T \)
$53$ \( -1654 + T \)
$59$ \( -29492 + T \)
$61$ \( -27630 + T \)
$67$ \( 57716 + T \)
$71$ \( 70648 + T \)
$73$ \( -74202 + T \)
$79$ \( 74336 + T \)
$83$ \( 44068 + T \)
$89$ \( -129306 + T \)
$97$ \( 137646 + T \)
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