Properties

Label 1638.2.a.o
Level $1638$
Weight $2$
Character orbit 1638.a
Self dual yes
Analytic conductor $13.079$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} - 3q^{11} - q^{13} - q^{14} + q^{16} - 5q^{17} + q^{19} - q^{20} - 3q^{22} - 3q^{23} - 4q^{25} - q^{26} - q^{28} - 5q^{29} + 4q^{31} + q^{32} - 5q^{34} + q^{35} - 5q^{37} + q^{38} - q^{40} + 8q^{41} - q^{43} - 3q^{44} - 3q^{46} - 8q^{47} + q^{49} - 4q^{50} - q^{52} - 6q^{53} + 3q^{55} - q^{56} - 5q^{58} + 13q^{61} + 4q^{62} + q^{64} + q^{65} - 10q^{67} - 5q^{68} + q^{70} - 8q^{71} - 15q^{73} - 5q^{74} + q^{76} + 3q^{77} + 6q^{79} - q^{80} + 8q^{82} + 2q^{83} + 5q^{85} - q^{86} - 3q^{88} + 2q^{89} + q^{91} - 3q^{92} - 8q^{94} - q^{95} - 2q^{97} + q^{98} + O(q^{100}) \)

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.o 1
3.b odd 2 1 546.2.a.c 1
12.b even 2 1 4368.2.a.h 1
21.c even 2 1 3822.2.a.e 1
39.d odd 2 1 7098.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.c 1 3.b odd 2 1
1638.2.a.o 1 1.a even 1 1 trivial
3822.2.a.e 1 21.c even 2 1
4368.2.a.h 1 12.b even 2 1
7098.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1638))\):

\( T_{5} + 1 \)
\( T_{11} + 3 \)
\( T_{17} + 5 \)
\( T_{19} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( 1 + T \)
$11$ \( 3 + T \)
$13$ \( 1 + T \)
$17$ \( 5 + T \)
$19$ \( -1 + T \)
$23$ \( 3 + T \)
$29$ \( 5 + T \)
$31$ \( -4 + T \)
$37$ \( 5 + T \)
$41$ \( -8 + T \)
$43$ \( 1 + T \)
$47$ \( 8 + T \)
$53$ \( 6 + T \)
$59$ \( T \)
$61$ \( -13 + T \)
$67$ \( 10 + T \)
$71$ \( 8 + T \)
$73$ \( 15 + T \)
$79$ \( -6 + T \)
$83$ \( -2 + T \)
$89$ \( -2 + T \)
$97$ \( 2 + T \)
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