Properties

Label 144.6.a.c
Level $144$
Weight $6$
Character orbit 144.a
Self dual yes
Analytic conductor $23.095$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.0952700531\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 54 q^{5} + 88 q^{7} + O(q^{10}) \) \( q - 54 q^{5} + 88 q^{7} + 540 q^{11} - 418 q^{13} - 594 q^{17} - 836 q^{19} - 4104 q^{23} - 209 q^{25} + 594 q^{29} - 4256 q^{31} - 4752 q^{35} - 298 q^{37} - 17226 q^{41} + 12100 q^{43} - 1296 q^{47} - 9063 q^{49} - 19494 q^{53} - 29160 q^{55} - 7668 q^{59} - 34738 q^{61} + 22572 q^{65} - 21812 q^{67} - 46872 q^{71} + 67562 q^{73} + 47520 q^{77} + 76912 q^{79} + 67716 q^{83} + 32076 q^{85} - 29754 q^{89} - 36784 q^{91} + 45144 q^{95} - 122398 q^{97} + O(q^{100}) \)

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
0 0 0 −54.0000 0 88.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 144.6.a.c 1
3.b odd 2 1 16.6.a.b 1
4.b odd 2 1 36.6.a.a 1
8.b even 2 1 576.6.a.bd 1
8.d odd 2 1 576.6.a.bc 1
12.b even 2 1 4.6.a.a 1
15.d odd 2 1 400.6.a.d 1
15.e even 4 2 400.6.c.f 2
20.d odd 2 1 900.6.a.h 1
20.e even 4 2 900.6.d.a 2
21.c even 2 1 784.6.a.d 1
24.f even 2 1 64.6.a.f 1
24.h odd 2 1 64.6.a.b 1
36.f odd 6 2 324.6.e.d 2
36.h even 6 2 324.6.e.a 2
48.i odd 4 2 256.6.b.c 2
48.k even 4 2 256.6.b.g 2
60.h even 2 1 100.6.a.b 1
60.l odd 4 2 100.6.c.b 2
84.h odd 2 1 196.6.a.e 1
84.j odd 6 2 196.6.e.d 2
84.n even 6 2 196.6.e.g 2
132.d odd 2 1 484.6.a.a 1
156.h even 2 1 676.6.a.a 1
156.l odd 4 2 676.6.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 12.b even 2 1
16.6.a.b 1 3.b odd 2 1
36.6.a.a 1 4.b odd 2 1
64.6.a.b 1 24.h odd 2 1
64.6.a.f 1 24.f even 2 1
100.6.a.b 1 60.h even 2 1
100.6.c.b 2 60.l odd 4 2
144.6.a.c 1 1.a even 1 1 trivial
196.6.a.e 1 84.h odd 2 1
196.6.e.d 2 84.j odd 6 2
196.6.e.g 2 84.n even 6 2
256.6.b.c 2 48.i odd 4 2
256.6.b.g 2 48.k even 4 2
324.6.e.a 2 36.h even 6 2
324.6.e.d 2 36.f odd 6 2
400.6.a.d 1 15.d odd 2 1
400.6.c.f 2 15.e even 4 2
484.6.a.a 1 132.d odd 2 1
576.6.a.bc 1 8.d odd 2 1
576.6.a.bd 1 8.b even 2 1
676.6.a.a 1 156.h even 2 1
676.6.d.a 2 156.l odd 4 2
784.6.a.d 1 21.c even 2 1
900.6.a.h 1 20.d odd 2 1
900.6.d.a 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 54 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(144))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 54 + T \)
$7$ \( -88 + T \)
$11$ \( -540 + T \)
$13$ \( 418 + T \)
$17$ \( 594 + T \)
$19$ \( 836 + T \)
$23$ \( 4104 + T \)
$29$ \( -594 + T \)
$31$ \( 4256 + T \)
$37$ \( 298 + T \)
$41$ \( 17226 + T \)
$43$ \( -12100 + T \)
$47$ \( 1296 + T \)
$53$ \( 19494 + T \)
$59$ \( 7668 + T \)
$61$ \( 34738 + T \)
$67$ \( 21812 + T \)
$71$ \( 46872 + T \)
$73$ \( -67562 + T \)
$79$ \( -76912 + T \)
$83$ \( -67716 + T \)
$89$ \( 29754 + T \)
$97$ \( 122398 + T \)
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