Properties

Label 144.16.a.j
Level $144$
Weight $16$
Character orbit 144.a
Self dual yes
Analytic conductor $205.479$
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: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(205.478647344\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 114810 q^{5} + 3034528 q^{7} + O(q^{10}) \) \( q + 114810 q^{5} + 3034528 q^{7} - 103451700 q^{11} - 104365834 q^{13} - 997689762 q^{17} - 4934015444 q^{19} + 8324920200 q^{23} - 17336242025 q^{25} - 104128242846 q^{29} + 296696681512 q^{31} + 348394159680 q^{35} - 178337455666 q^{37} + 1790882416086 q^{41} + 2863459422772 q^{43} + 4332907521600 q^{47} + 4460798672841 q^{49} - 9732317104422 q^{53} - 11877289677000 q^{55} - 13514837176500 q^{59} + 5352663511190 q^{61} - 11982241401540 q^{65} + 53233909720108 q^{67} - 20229661643400 q^{71} + 26264166466106 q^{73} - 313927080297600 q^{77} + 339031361615128 q^{79} + 131684771045076 q^{83} - 114544761575220 q^{85} + 39352148322678 q^{89} - 316701045516352 q^{91} - 566474313125640 q^{95} + 1128750908801474 q^{97} + O(q^{100}) \)

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 114810. 0 3.03453e6 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 144.16.a.j 1
3.b odd 2 1 48.16.a.d 1
4.b odd 2 1 18.16.a.b 1
12.b even 2 1 6.16.a.b 1
60.h even 2 1 150.16.a.f 1
60.l odd 4 2 150.16.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.16.a.b 1 12.b even 2 1
18.16.a.b 1 4.b odd 2 1
48.16.a.d 1 3.b odd 2 1
144.16.a.j 1 1.a even 1 1 trivial
150.16.a.f 1 60.h even 2 1
150.16.c.a 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 114810 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(144))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -114810 + T \)
$7$ \( -3034528 + T \)
$11$ \( 103451700 + T \)
$13$ \( 104365834 + T \)
$17$ \( 997689762 + T \)
$19$ \( 4934015444 + T \)
$23$ \( -8324920200 + T \)
$29$ \( 104128242846 + T \)
$31$ \( -296696681512 + T \)
$37$ \( 178337455666 + T \)
$41$ \( -1790882416086 + T \)
$43$ \( -2863459422772 + T \)
$47$ \( -4332907521600 + T \)
$53$ \( 9732317104422 + T \)
$59$ \( 13514837176500 + T \)
$61$ \( -5352663511190 + T \)
$67$ \( -53233909720108 + T \)
$71$ \( 20229661643400 + T \)
$73$ \( -26264166466106 + T \)
$79$ \( -339031361615128 + T \)
$83$ \( -131684771045076 + T \)
$89$ \( -39352148322678 + T \)
$97$ \( -1128750908801474 + T \)
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