Properties

Label 9438.2.a.d
Level $9438$
Weight $2$
Character orbit 9438.a
Self dual yes
Analytic conductor $75.363$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9438.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.3628094277\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 858)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 3q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 3q^{7} - q^{8} + q^{9} + q^{10} - q^{12} + q^{13} - 3q^{14} + q^{15} + q^{16} + 4q^{17} - q^{18} + 2q^{19} - q^{20} - 3q^{21} - q^{23} + q^{24} - 4q^{25} - q^{26} - q^{27} + 3q^{28} + 9q^{29} - q^{30} - 4q^{31} - q^{32} - 4q^{34} - 3q^{35} + q^{36} - 6q^{37} - 2q^{38} - q^{39} + q^{40} - q^{41} + 3q^{42} - 11q^{43} - q^{45} + q^{46} - q^{48} + 2q^{49} + 4q^{50} - 4q^{51} + q^{52} - 10q^{53} + q^{54} - 3q^{56} - 2q^{57} - 9q^{58} - 3q^{59} + q^{60} - 5q^{61} + 4q^{62} + 3q^{63} + q^{64} - q^{65} + 3q^{67} + 4q^{68} + q^{69} + 3q^{70} + 10q^{71} - q^{72} - 9q^{73} + 6q^{74} + 4q^{75} + 2q^{76} + q^{78} - 10q^{79} - q^{80} + q^{81} + q^{82} + 6q^{83} - 3q^{84} - 4q^{85} + 11q^{86} - 9q^{87} - 8q^{89} + q^{90} + 3q^{91} - q^{92} + 4q^{93} - 2q^{95} + q^{96} + 2q^{97} - 2q^{98} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(11\) \(-1\)
\(13\) \(-1\)

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 9438.2.a.d 1
11.b odd 2 1 858.2.a.g 1
33.d even 2 1 2574.2.a.i 1
44.c even 2 1 6864.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
858.2.a.g 1 11.b odd 2 1
2574.2.a.i 1 33.d even 2 1
6864.2.a.u 1 44.c even 2 1
9438.2.a.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9438))\):

\( T_{5} + 1 \)
\( T_{7} - 3 \)
\( T_{17} - 4 \)
\( T_{19} - 2 \)
\( T_{29} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( 1 + T \)
$7$ \( -3 + T \)
$11$ \( T \)
$13$ \( -1 + T \)
$17$ \( -4 + T \)
$19$ \( -2 + T \)
$23$ \( 1 + T \)
$29$ \( -9 + T \)
$31$ \( 4 + T \)
$37$ \( 6 + T \)
$41$ \( 1 + T \)
$43$ \( 11 + T \)
$47$ \( T \)
$53$ \( 10 + T \)
$59$ \( 3 + T \)
$61$ \( 5 + T \)
$67$ \( -3 + T \)
$71$ \( -10 + T \)
$73$ \( 9 + T \)
$79$ \( 10 + T \)
$83$ \( -6 + T \)
$89$ \( 8 + T \)
$97$ \( -2 + T \)
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