Properties

Label 644.2.a.b
Level $644$
Weight $2$
Character orbit 644.a
Self dual yes
Analytic conductor $5.142$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.14236589017\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + q^{3} - 2 q^{5} - q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - 2 q^{5} - q^{7} - 2 q^{9} - 2 q^{11} - q^{13} - 2 q^{15} - 6 q^{19} - q^{21} + q^{23} - q^{25} - 5 q^{27} + q^{29} + q^{31} - 2 q^{33} + 2 q^{35} - 6 q^{37} - q^{39} + 3 q^{41} + 4 q^{45} - 3 q^{47} + q^{49} + 6 q^{53} + 4 q^{55} - 6 q^{57} + 8 q^{59} - 10 q^{61} + 2 q^{63} + 2 q^{65} - 2 q^{67} + q^{69} + 5 q^{71} - 7 q^{73} - q^{75} + 2 q^{77} + 14 q^{79} + q^{81} - 4 q^{83} + q^{87} - 12 q^{89} + q^{91} + q^{93} + 12 q^{95} - 8 q^{97} + 4 q^{99}+O(q^{100}) \) 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
0
0 1.00000 0 −2.00000 0 −1.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 644.2.a.b 1
3.b odd 2 1 5796.2.a.f 1
4.b odd 2 1 2576.2.a.g 1
7.b odd 2 1 4508.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.a.b 1 1.a even 1 1 trivial
2576.2.a.g 1 4.b odd 2 1
4508.2.a.b 1 7.b odd 2 1
5796.2.a.f 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(644))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T - 1 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T - 3 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 3 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 8 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T - 5 \) Copy content Toggle raw display
$73$ \( T + 7 \) Copy content Toggle raw display
$79$ \( T - 14 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 12 \) Copy content Toggle raw display
$97$ \( T + 8 \) Copy content Toggle raw display
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