Properties

Label 660.6.a.a
Level $660$
Weight $6$
Character orbit 660.a
Self dual yes
Analytic conductor $105.853$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 660.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(105.853321077\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 9 q^{3} - 25 q^{5} + 188 q^{7} + 81 q^{9} + O(q^{10}) \) \( q + 9 q^{3} - 25 q^{5} + 188 q^{7} + 81 q^{9} + 121 q^{11} + 698 q^{13} - 225 q^{15} + 1890 q^{17} - 2428 q^{19} + 1692 q^{21} + 2340 q^{23} + 625 q^{25} + 729 q^{27} + 990 q^{29} + 128 q^{31} + 1089 q^{33} - 4700 q^{35} - 3202 q^{37} + 6282 q^{39} + 17370 q^{41} - 6652 q^{43} - 2025 q^{45} - 25020 q^{47} + 18537 q^{49} + 17010 q^{51} - 18246 q^{53} - 3025 q^{55} - 21852 q^{57} - 8652 q^{59} + 37682 q^{61} + 15228 q^{63} - 17450 q^{65} - 18676 q^{67} + 21060 q^{69} - 2340 q^{71} + 43058 q^{73} + 5625 q^{75} + 22748 q^{77} + 65300 q^{79} + 6561 q^{81} - 55308 q^{83} - 47250 q^{85} + 8910 q^{87} - 64806 q^{89} + 131224 q^{91} + 1152 q^{93} + 60700 q^{95} + 38306 q^{97} + 9801 q^{99} + O(q^{100}) \)

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 9.00000 0 −25.0000 0 188.000 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

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 660.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.6.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 188 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(660))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -9 + T \)
$5$ \( 25 + T \)
$7$ \( -188 + T \)
$11$ \( -121 + T \)
$13$ \( -698 + T \)
$17$ \( -1890 + T \)
$19$ \( 2428 + T \)
$23$ \( -2340 + T \)
$29$ \( -990 + T \)
$31$ \( -128 + T \)
$37$ \( 3202 + T \)
$41$ \( -17370 + T \)
$43$ \( 6652 + T \)
$47$ \( 25020 + T \)
$53$ \( 18246 + T \)
$59$ \( 8652 + T \)
$61$ \( -37682 + T \)
$67$ \( 18676 + T \)
$71$ \( 2340 + T \)
$73$ \( -43058 + T \)
$79$ \( -65300 + T \)
$83$ \( 55308 + T \)
$89$ \( 64806 + T \)
$97$ \( -38306 + T \)
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