Properties

Label 9438.2.a.t
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 78)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} - 4 q^{7} + q^{8} + q^{9} + 2 q^{10} - q^{12} - q^{13} - 4 q^{14} - 2 q^{15} + q^{16} - 2 q^{17} + q^{18} + 8 q^{19} + 2 q^{20} + 4 q^{21} - q^{24} - q^{25} - q^{26} - q^{27} - 4 q^{28} - 6 q^{29} - 2 q^{30} - 4 q^{31} + q^{32} - 2 q^{34} - 8 q^{35} + q^{36} - 2 q^{37} + 8 q^{38} + q^{39} + 2 q^{40} + 10 q^{41} + 4 q^{42} - 4 q^{43} + 2 q^{45} + 8 q^{47} - q^{48} + 9 q^{49} - q^{50} + 2 q^{51} - q^{52} - 10 q^{53} - q^{54} - 4 q^{56} - 8 q^{57} - 6 q^{58} + 4 q^{59} - 2 q^{60} + 2 q^{61} - 4 q^{62} - 4 q^{63} + q^{64} - 2 q^{65} - 16 q^{67} - 2 q^{68} - 8 q^{70} - 8 q^{71} + q^{72} - 2 q^{73} - 2 q^{74} + q^{75} + 8 q^{76} + q^{78} - 8 q^{79} + 2 q^{80} + q^{81} + 10 q^{82} - 12 q^{83} + 4 q^{84} - 4 q^{85} - 4 q^{86} + 6 q^{87} + 14 q^{89} + 2 q^{90} + 4 q^{91} + 4 q^{93} + 8 q^{94} + 16 q^{95} - q^{96} + 10 q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 2.00000 −1.00000 −4.00000 1.00000 1.00000 2.00000
\(n\): 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.t 1
11.b odd 2 1 78.2.a.a 1
33.d even 2 1 234.2.a.c 1
44.c even 2 1 624.2.a.h 1
55.d odd 2 1 1950.2.a.w 1
55.e even 4 2 1950.2.e.i 2
77.b even 2 1 3822.2.a.j 1
88.b odd 2 1 2496.2.a.t 1
88.g even 2 1 2496.2.a.b 1
99.g even 6 2 2106.2.e.j 2
99.h odd 6 2 2106.2.e.q 2
132.d odd 2 1 1872.2.a.c 1
143.d odd 2 1 1014.2.a.d 1
143.g even 4 2 1014.2.b.b 2
143.i odd 6 2 1014.2.e.c 2
143.k odd 6 2 1014.2.e.f 2
143.o even 12 4 1014.2.i.d 4
165.d even 2 1 5850.2.a.d 1
165.l odd 4 2 5850.2.e.bb 2
264.m even 2 1 7488.2.a.bz 1
264.p odd 2 1 7488.2.a.bk 1
429.e even 2 1 3042.2.a.f 1
429.l odd 4 2 3042.2.b.g 2
572.b even 2 1 8112.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 11.b odd 2 1
234.2.a.c 1 33.d even 2 1
624.2.a.h 1 44.c even 2 1
1014.2.a.d 1 143.d odd 2 1
1014.2.b.b 2 143.g even 4 2
1014.2.e.c 2 143.i odd 6 2
1014.2.e.f 2 143.k odd 6 2
1014.2.i.d 4 143.o even 12 4
1872.2.a.c 1 132.d odd 2 1
1950.2.a.w 1 55.d odd 2 1
1950.2.e.i 2 55.e even 4 2
2106.2.e.j 2 99.g even 6 2
2106.2.e.q 2 99.h odd 6 2
2496.2.a.b 1 88.g even 2 1
2496.2.a.t 1 88.b odd 2 1
3042.2.a.f 1 429.e even 2 1
3042.2.b.g 2 429.l odd 4 2
3822.2.a.j 1 77.b even 2 1
5850.2.a.d 1 165.d even 2 1
5850.2.e.bb 2 165.l odd 4 2
7488.2.a.bk 1 264.p odd 2 1
7488.2.a.bz 1 264.m even 2 1
8112.2.a.v 1 572.b even 2 1
9438.2.a.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9438))\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{19} - 8 \) Copy content Toggle raw display
\( T_{29} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T - 8 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 16 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 14 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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