Properties

Label 1638.2.a.j
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 182)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 4q^{5} - q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} + 4q^{5} - q^{7} - q^{8} - 4q^{10} - q^{11} - q^{13} + q^{14} + q^{16} - 6q^{19} + 4q^{20} + q^{22} + 7q^{23} + 11q^{25} + q^{26} - q^{28} + 4q^{29} + 7q^{31} - q^{32} - 4q^{35} + 9q^{37} + 6q^{38} - 4q^{40} + 3q^{41} + 4q^{43} - q^{44} - 7q^{46} - 7q^{47} + q^{49} - 11q^{50} - q^{52} - 4q^{55} + q^{56} - 4q^{58} + 10q^{59} + q^{61} - 7q^{62} + q^{64} - 4q^{65} + q^{67} + 4q^{70} - 16q^{71} + 5q^{73} - 9q^{74} - 6q^{76} + q^{77} + 11q^{79} + 4q^{80} - 3q^{82} - 4q^{86} + q^{88} + 6q^{89} + q^{91} + 7q^{92} + 7q^{94} - 24q^{95} - q^{97} - q^{98} + O(q^{100}) \)

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 4.00000 0 −1.00000 −1.00000 0 −4.00000
\(n\): 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.j 1
3.b odd 2 1 182.2.a.e 1
12.b even 2 1 1456.2.a.a 1
15.d odd 2 1 4550.2.a.a 1
21.c even 2 1 1274.2.a.h 1
21.g even 6 2 1274.2.f.k 2
21.h odd 6 2 1274.2.f.b 2
24.f even 2 1 5824.2.a.bf 1
24.h odd 2 1 5824.2.a.b 1
39.d odd 2 1 2366.2.a.h 1
39.f even 4 2 2366.2.d.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.a.e 1 3.b odd 2 1
1274.2.a.h 1 21.c even 2 1
1274.2.f.b 2 21.h odd 6 2
1274.2.f.k 2 21.g even 6 2
1456.2.a.a 1 12.b even 2 1
1638.2.a.j 1 1.a even 1 1 trivial
2366.2.a.h 1 39.d odd 2 1
2366.2.d.j 2 39.f even 4 2
4550.2.a.a 1 15.d odd 2 1
5824.2.a.b 1 24.h odd 2 1
5824.2.a.bf 1 24.f even 2 1

Hecke kernels

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

\( T_{5} - 4 \)
\( T_{11} + 1 \)
\( T_{17} \)
\( T_{19} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -4 + T \)
$7$ \( 1 + T \)
$11$ \( 1 + T \)
$13$ \( 1 + T \)
$17$ \( T \)
$19$ \( 6 + T \)
$23$ \( -7 + T \)
$29$ \( -4 + T \)
$31$ \( -7 + T \)
$37$ \( -9 + T \)
$41$ \( -3 + T \)
$43$ \( -4 + T \)
$47$ \( 7 + T \)
$53$ \( T \)
$59$ \( -10 + T \)
$61$ \( -1 + T \)
$67$ \( -1 + T \)
$71$ \( 16 + T \)
$73$ \( -5 + T \)
$79$ \( -11 + T \)
$83$ \( T \)
$89$ \( -6 + T \)
$97$ \( 1 + T \)
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