Properties

Label 1638.2.a.t
Level $1638$
Weight $2$
Character orbit 1638.a
Self dual yes
Analytic conductor $13.079$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.0794958511\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 3 q^{5} + q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + 3 q^{5} + q^{7} + q^{8} + 3 q^{10} + 3 q^{11} + q^{13} + q^{14} + q^{16} - 3 q^{17} + 5 q^{19} + 3 q^{20} + 3 q^{22} - 3 q^{23} + 4 q^{25} + q^{26} + q^{28} - 3 q^{29} - 10 q^{31} + q^{32} - 3 q^{34} + 3 q^{35} - 7 q^{37} + 5 q^{38} + 3 q^{40} + 6 q^{41} + 5 q^{43} + 3 q^{44} - 3 q^{46} + q^{49} + 4 q^{50} + q^{52} + 6 q^{53} + 9 q^{55} + q^{56} - 3 q^{58} + 6 q^{59} - 7 q^{61} - 10 q^{62} + q^{64} + 3 q^{65} - 4 q^{67} - 3 q^{68} + 3 q^{70} - 6 q^{71} - q^{73} - 7 q^{74} + 5 q^{76} + 3 q^{77} - 10 q^{79} + 3 q^{80} + 6 q^{82} + 6 q^{83} - 9 q^{85} + 5 q^{86} + 3 q^{88} - 6 q^{89} + q^{91} - 3 q^{92} + 15 q^{95} + 2 q^{97} + q^{98} + O(q^{100}) \)

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
1.00000 0 1.00000 3.00000 0 1.00000 1.00000 0 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 1638.2.a.t yes 1
3.b odd 2 1 1638.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.a.b 1 3.b odd 2 1
1638.2.a.t yes 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1638))\):

\( T_{5} - 3 \)
\( T_{11} - 3 \)
\( T_{17} + 3 \)
\( T_{19} - 5 \)

Hecke characteristic polynomials

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