Properties

Label 144.12.a.d
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 1)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4830q^{5} + 16744q^{7} + O(q^{10}) \) \( q - 4830q^{5} + 16744q^{7} + 534612q^{11} - 577738q^{13} + 6905934q^{17} - 10661420q^{19} + 18643272q^{23} - 25499225q^{25} - 128406630q^{29} + 52843168q^{31} - 80873520q^{35} - 182213314q^{37} - 308120442q^{41} + 17125708q^{43} + 2687348496q^{47} - 1696965207q^{49} + 1596055698q^{53} - 2582175960q^{55} - 5189203740q^{59} + 6956478662q^{61} + 2790474540q^{65} + 15481826884q^{67} + 9791485272q^{71} + 1463791322q^{73} + 8951543328q^{77} - 38116845680q^{79} - 29335099668q^{83} - 33355661220q^{85} + 24992917110q^{89} - 9673645072q^{91} + 51494658600q^{95} + 75013568546q^{97} + O(q^{100}) \)

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 −4830.00 0 16744.0 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.d 1
3.b odd 2 1 16.12.a.a 1
4.b odd 2 1 9.12.a.b 1
12.b even 2 1 1.12.a.a 1
20.d odd 2 1 225.12.a.b 1
20.e even 4 2 225.12.b.d 2
24.f even 2 1 64.12.a.b 1
24.h odd 2 1 64.12.a.f 1
36.f odd 6 2 81.12.c.b 2
36.h even 6 2 81.12.c.d 2
48.i odd 4 2 256.12.b.c 2
48.k even 4 2 256.12.b.e 2
60.h even 2 1 25.12.a.b 1
60.l odd 4 2 25.12.b.b 2
84.h odd 2 1 49.12.a.a 1
84.j odd 6 2 49.12.c.c 2
84.n even 6 2 49.12.c.b 2
132.d odd 2 1 121.12.a.b 1
156.h even 2 1 169.12.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 12.b even 2 1
9.12.a.b 1 4.b odd 2 1
16.12.a.a 1 3.b odd 2 1
25.12.a.b 1 60.h even 2 1
25.12.b.b 2 60.l odd 4 2
49.12.a.a 1 84.h odd 2 1
49.12.c.b 2 84.n even 6 2
49.12.c.c 2 84.j odd 6 2
64.12.a.b 1 24.f even 2 1
64.12.a.f 1 24.h odd 2 1
81.12.c.b 2 36.f odd 6 2
81.12.c.d 2 36.h even 6 2
121.12.a.b 1 132.d odd 2 1
144.12.a.d 1 1.a even 1 1 trivial
169.12.a.a 1 156.h even 2 1
225.12.a.b 1 20.d odd 2 1
225.12.b.d 2 20.e even 4 2
256.12.b.c 2 48.i odd 4 2
256.12.b.e 2 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 4830 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(144))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 4830 + T \)
$7$ \( -16744 + T \)
$11$ \( -534612 + T \)
$13$ \( 577738 + T \)
$17$ \( -6905934 + T \)
$19$ \( 10661420 + T \)
$23$ \( -18643272 + T \)
$29$ \( 128406630 + T \)
$31$ \( -52843168 + T \)
$37$ \( 182213314 + T \)
$41$ \( 308120442 + T \)
$43$ \( -17125708 + T \)
$47$ \( -2687348496 + T \)
$53$ \( -1596055698 + T \)
$59$ \( 5189203740 + T \)
$61$ \( -6956478662 + T \)
$67$ \( -15481826884 + T \)
$71$ \( -9791485272 + T \)
$73$ \( -1463791322 + T \)
$79$ \( 38116845680 + T \)
$83$ \( 29335099668 + T \)
$89$ \( -24992917110 + T \)
$97$ \( -75013568546 + T \)
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