Properties

Label 1638.2.a.r
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: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} + q^{11} + q^{13} - q^{14} + q^{16} + q^{17} + 7q^{19} + q^{20} + q^{22} - 3q^{23} - 4q^{25} + q^{26} - q^{28} + 3q^{29} + 8q^{31} + q^{32} + q^{34} - q^{35} + 7q^{37} + 7q^{38} + q^{40} - 8q^{41} + 7q^{43} + q^{44} - 3q^{46} - 8q^{47} + q^{49} - 4q^{50} + q^{52} + 10q^{53} + q^{55} - q^{56} + 3q^{58} - 4q^{59} + 7q^{61} + 8q^{62} + q^{64} + q^{65} + 2q^{67} + q^{68} - q^{70} - 4q^{71} - q^{73} + 7q^{74} + 7q^{76} - q^{77} + 2q^{79} + q^{80} - 8q^{82} + 6q^{83} + q^{85} + 7q^{86} + q^{88} - 14q^{89} - q^{91} - 3q^{92} - 8q^{94} + 7q^{95} - 14q^{97} + q^{98} + O(q^{100}) \)

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 1.00000 0 −1.00000 1.00000 0 1.00000
\(n\): 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.r 1
3.b odd 2 1 546.2.a.a 1
12.b even 2 1 4368.2.a.s 1
21.c even 2 1 3822.2.a.n 1
39.d odd 2 1 7098.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.a 1 3.b odd 2 1
1638.2.a.r 1 1.a even 1 1 trivial
3822.2.a.n 1 21.c even 2 1
4368.2.a.s 1 12.b even 2 1
7098.2.a.t 1 39.d odd 2 1

Hecke kernels

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

\( T_{5} - 1 \)
\( T_{11} - 1 \)
\( T_{17} - 1 \)
\( T_{19} - 7 \)

Hecke characteristic polynomials

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