Properties

Label 144.4.a.d
Level $144$
Weight $4$
Character orbit 144.a
Self dual yes
Analytic conductor $8.496$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(8.49627504083\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 20q^{7} + O(q^{10}) \) \( q - 20q^{7} - 70q^{13} - 56q^{19} - 125q^{25} - 308q^{31} + 110q^{37} + 520q^{43} + 57q^{49} + 182q^{61} + 880q^{67} + 1190q^{73} - 884q^{79} + 1400q^{91} - 1330q^{97} + O(q^{100}) \)

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 0 0 −20.0000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.4.a.d 1
3.b odd 2 1 CM 144.4.a.d 1
4.b odd 2 1 9.4.a.a 1
8.b even 2 1 576.4.a.l 1
8.d odd 2 1 576.4.a.m 1
12.b even 2 1 9.4.a.a 1
20.d odd 2 1 225.4.a.d 1
20.e even 4 2 225.4.b.g 2
24.f even 2 1 576.4.a.m 1
24.h odd 2 1 576.4.a.l 1
28.d even 2 1 441.4.a.f 1
28.f even 6 2 441.4.e.j 2
28.g odd 6 2 441.4.e.i 2
36.f odd 6 2 81.4.c.b 2
36.h even 6 2 81.4.c.b 2
44.c even 2 1 1089.4.a.g 1
52.b odd 2 1 1521.4.a.g 1
60.h even 2 1 225.4.a.d 1
60.l odd 4 2 225.4.b.g 2
84.h odd 2 1 441.4.a.f 1
84.j odd 6 2 441.4.e.j 2
84.n even 6 2 441.4.e.i 2
132.d odd 2 1 1089.4.a.g 1
156.h even 2 1 1521.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 4.b odd 2 1
9.4.a.a 1 12.b even 2 1
81.4.c.b 2 36.f odd 6 2
81.4.c.b 2 36.h even 6 2
144.4.a.d 1 1.a even 1 1 trivial
144.4.a.d 1 3.b odd 2 1 CM
225.4.a.d 1 20.d odd 2 1
225.4.a.d 1 60.h even 2 1
225.4.b.g 2 20.e even 4 2
225.4.b.g 2 60.l odd 4 2
441.4.a.f 1 28.d even 2 1
441.4.a.f 1 84.h odd 2 1
441.4.e.i 2 28.g odd 6 2
441.4.e.i 2 84.n even 6 2
441.4.e.j 2 28.f even 6 2
441.4.e.j 2 84.j odd 6 2
576.4.a.l 1 8.b even 2 1
576.4.a.l 1 24.h odd 2 1
576.4.a.m 1 8.d odd 2 1
576.4.a.m 1 24.f even 2 1
1089.4.a.g 1 44.c even 2 1
1089.4.a.g 1 132.d odd 2 1
1521.4.a.g 1 52.b odd 2 1
1521.4.a.g 1 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(144))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 20 + T \)
$11$ \( T \)
$13$ \( 70 + T \)
$17$ \( T \)
$19$ \( 56 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( 308 + T \)
$37$ \( -110 + T \)
$41$ \( T \)
$43$ \( -520 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( -182 + T \)
$67$ \( -880 + T \)
$71$ \( T \)
$73$ \( -1190 + T \)
$79$ \( 884 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( 1330 + T \)
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