Properties

Label 144.4.a.b
Level $144$
Weight $4$
Character orbit 144.a
Self dual yes
Analytic conductor $8.496$
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: \( k \) \(=\) \( 4 \)
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 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 14q^{5} + 24q^{7} + O(q^{10}) \) \( q - 14q^{5} + 24q^{7} - 28q^{11} - 74q^{13} - 82q^{17} - 92q^{19} + 8q^{23} + 71q^{25} + 138q^{29} - 80q^{31} - 336q^{35} + 30q^{37} - 282q^{41} - 4q^{43} + 240q^{47} + 233q^{49} + 130q^{53} + 392q^{55} + 596q^{59} - 218q^{61} + 1036q^{65} + 436q^{67} + 856q^{71} - 998q^{73} - 672q^{77} + 32q^{79} - 1508q^{83} + 1148q^{85} + 246q^{89} - 1776q^{91} + 1288q^{95} + 866q^{97} + O(q^{100}) \)

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 −14.0000 0 24.0000 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 144.4.a.b 1
3.b odd 2 1 48.4.a.b 1
4.b odd 2 1 72.4.a.b 1
8.b even 2 1 576.4.a.v 1
8.d odd 2 1 576.4.a.u 1
12.b even 2 1 24.4.a.a 1
15.d odd 2 1 1200.4.a.u 1
15.e even 4 2 1200.4.f.p 2
20.d odd 2 1 1800.4.a.bg 1
20.e even 4 2 1800.4.f.q 2
21.c even 2 1 2352.4.a.w 1
24.f even 2 1 192.4.a.a 1
24.h odd 2 1 192.4.a.g 1
36.f odd 6 2 648.4.i.k 2
36.h even 6 2 648.4.i.b 2
48.i odd 4 2 768.4.d.b 2
48.k even 4 2 768.4.d.o 2
60.h even 2 1 600.4.a.h 1
60.l odd 4 2 600.4.f.b 2
84.h odd 2 1 1176.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.a.a 1 12.b even 2 1
48.4.a.b 1 3.b odd 2 1
72.4.a.b 1 4.b odd 2 1
144.4.a.b 1 1.a even 1 1 trivial
192.4.a.a 1 24.f even 2 1
192.4.a.g 1 24.h odd 2 1
576.4.a.u 1 8.d odd 2 1
576.4.a.v 1 8.b even 2 1
600.4.a.h 1 60.h even 2 1
600.4.f.b 2 60.l odd 4 2
648.4.i.b 2 36.h even 6 2
648.4.i.k 2 36.f odd 6 2
768.4.d.b 2 48.i odd 4 2
768.4.d.o 2 48.k even 4 2
1176.4.a.a 1 84.h odd 2 1
1200.4.a.u 1 15.d odd 2 1
1200.4.f.p 2 15.e even 4 2
1800.4.a.bg 1 20.d odd 2 1
1800.4.f.q 2 20.e even 4 2
2352.4.a.w 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 14 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(144))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 14 + T \)
$7$ \( -24 + T \)
$11$ \( 28 + T \)
$13$ \( 74 + T \)
$17$ \( 82 + T \)
$19$ \( 92 + T \)
$23$ \( -8 + T \)
$29$ \( -138 + T \)
$31$ \( 80 + T \)
$37$ \( -30 + T \)
$41$ \( 282 + T \)
$43$ \( 4 + T \)
$47$ \( -240 + T \)
$53$ \( -130 + T \)
$59$ \( -596 + T \)
$61$ \( 218 + T \)
$67$ \( -436 + T \)
$71$ \( -856 + T \)
$73$ \( 998 + T \)
$79$ \( -32 + T \)
$83$ \( 1508 + T \)
$89$ \( -246 + T \)
$97$ \( -866 + T \)
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