Properties

Label 663.2.a.a
Level $663$
Weight $2$
Character orbit 663.a
Self dual yes
Analytic conductor $5.294$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.29408165401\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + 3 q^{8} + q^{9} + 2 q^{10} + 4 q^{11} + q^{12} + q^{13} + 2 q^{15} - q^{16} + q^{17} - q^{18} - 4 q^{19} + 2 q^{20} - 4 q^{22} - 3 q^{24} - q^{25} - q^{26} - q^{27} - 2 q^{29} - 2 q^{30} - 8 q^{31} - 5 q^{32} - 4 q^{33} - q^{34} - q^{36} - 2 q^{37} + 4 q^{38} - q^{39} - 6 q^{40} + 2 q^{41} - 4 q^{43} - 4 q^{44} - 2 q^{45} + 8 q^{47} + q^{48} - 7 q^{49} + q^{50} - q^{51} - q^{52} - 10 q^{53} + q^{54} - 8 q^{55} + 4 q^{57} + 2 q^{58} + 4 q^{59} - 2 q^{60} + 14 q^{61} + 8 q^{62} + 7 q^{64} - 2 q^{65} + 4 q^{66} - 4 q^{67} - q^{68} + 3 q^{72} - 14 q^{73} + 2 q^{74} + q^{75} + 4 q^{76} + q^{78} - 8 q^{79} + 2 q^{80} + q^{81} - 2 q^{82} - 4 q^{83} - 2 q^{85} + 4 q^{86} + 2 q^{87} + 12 q^{88} - 6 q^{89} + 2 q^{90} + 8 q^{93} - 8 q^{94} + 8 q^{95} + 5 q^{96} - 6 q^{97} + 7 q^{98} + 4 q^{99}+O(q^{100}) \) 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
−1.00000 −1.00000 −1.00000 −2.00000 1.00000 0 3.00000 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(-1\)
\(17\) \(-1\)

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 663.2.a.a 1
3.b odd 2 1 1989.2.a.e 1
13.b even 2 1 8619.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.a.a 1 1.a even 1 1 trivial
1989.2.a.e 1 3.b odd 2 1
8619.2.a.i 1 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(663))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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