Properties

Label 144.12.a.l
Level $144$
Weight $12$
Character orbit 144.a
Self dual yes
Analytic conductor $110.641$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.641418001\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 5370 q^{5} + 27760 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5370 q^{5} + 27760 q^{7} + 637836 q^{11} + 766214 q^{13} - 3084354 q^{17} + 19511404 q^{19} + 15312360 q^{23} - 19991225 q^{25} - 10751262 q^{29} + 50937400 q^{31} + 149071200 q^{35} + 664740830 q^{37} - 898833450 q^{41} + 957947188 q^{43} - 1555741344 q^{47} - 1206709143 q^{49} - 3792417030 q^{53} + 3425179320 q^{55} + 555306924 q^{59} + 4950420998 q^{61} + 4114569180 q^{65} - 5292399284 q^{67} - 14831086248 q^{71} + 13971005210 q^{73} + 17706327360 q^{77} - 3720542360 q^{79} + 8768454036 q^{83} - 16562980980 q^{85} + 25472769174 q^{89} + 21270100640 q^{91} + 104776239480 q^{95} - 39092494846 q^{97}+O(q^{100}) \) 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
0 0 0 5370.00 0 27760.0 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 144.12.a.l 1
3.b odd 2 1 48.12.a.f 1
4.b odd 2 1 9.12.a.a 1
12.b even 2 1 3.12.a.a 1
20.d odd 2 1 225.12.a.f 1
20.e even 4 2 225.12.b.a 2
24.f even 2 1 192.12.a.q 1
24.h odd 2 1 192.12.a.g 1
36.f odd 6 2 81.12.c.e 2
36.h even 6 2 81.12.c.a 2
60.h even 2 1 75.12.a.a 1
60.l odd 4 2 75.12.b.a 2
84.h odd 2 1 147.12.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 12.b even 2 1
9.12.a.a 1 4.b odd 2 1
48.12.a.f 1 3.b odd 2 1
75.12.a.a 1 60.h even 2 1
75.12.b.a 2 60.l odd 4 2
81.12.c.a 2 36.h even 6 2
81.12.c.e 2 36.f odd 6 2
144.12.a.l 1 1.a even 1 1 trivial
147.12.a.c 1 84.h odd 2 1
192.12.a.g 1 24.h odd 2 1
192.12.a.q 1 24.f even 2 1
225.12.a.f 1 20.d odd 2 1
225.12.b.a 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 5370 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(144))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 5370 \) Copy content Toggle raw display
$7$ \( T - 27760 \) Copy content Toggle raw display
$11$ \( T - 637836 \) Copy content Toggle raw display
$13$ \( T - 766214 \) Copy content Toggle raw display
$17$ \( T + 3084354 \) Copy content Toggle raw display
$19$ \( T - 19511404 \) Copy content Toggle raw display
$23$ \( T - 15312360 \) Copy content Toggle raw display
$29$ \( T + 10751262 \) Copy content Toggle raw display
$31$ \( T - 50937400 \) Copy content Toggle raw display
$37$ \( T - 664740830 \) Copy content Toggle raw display
$41$ \( T + 898833450 \) Copy content Toggle raw display
$43$ \( T - 957947188 \) Copy content Toggle raw display
$47$ \( T + 1555741344 \) Copy content Toggle raw display
$53$ \( T + 3792417030 \) Copy content Toggle raw display
$59$ \( T - 555306924 \) Copy content Toggle raw display
$61$ \( T - 4950420998 \) Copy content Toggle raw display
$67$ \( T + 5292399284 \) Copy content Toggle raw display
$71$ \( T + 14831086248 \) Copy content Toggle raw display
$73$ \( T - 13971005210 \) Copy content Toggle raw display
$79$ \( T + 3720542360 \) Copy content Toggle raw display
$83$ \( T - 8768454036 \) Copy content Toggle raw display
$89$ \( T - 25472769174 \) Copy content Toggle raw display
$97$ \( T + 39092494846 \) Copy content Toggle raw display
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