Properties

Label 72.8.a.e
Level $72$
Weight $8$
Character orbit 72.a
Self dual yes
Analytic conductor $22.492$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.4917218349\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 530 q^{5} + 120 q^{7} + O(q^{10}) \) \( q + 530 q^{5} + 120 q^{7} + 7196 q^{11} - 9626 q^{13} - 18674 q^{17} + 7004 q^{19} + 63704 q^{23} + 202775 q^{25} - 29334 q^{29} + 87968 q^{31} + 63600 q^{35} + 227982 q^{37} + 160806 q^{41} + 136132 q^{43} + 1206960 q^{47} - 809143 q^{49} + 398786 q^{53} + 3813880 q^{55} - 1152436 q^{59} - 2070602 q^{61} - 5101780 q^{65} - 4073428 q^{67} + 383752 q^{71} + 3006010 q^{73} + 863520 q^{77} - 4948112 q^{79} + 9163492 q^{83} - 9897220 q^{85} - 7304106 q^{89} - 1155120 q^{91} + 3712120 q^{95} - 690526 q^{97} + O(q^{100}) \)

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 0 530.000 0 120.000 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 72.8.a.e 1
3.b odd 2 1 24.8.a.b 1
4.b odd 2 1 144.8.a.k 1
8.b even 2 1 576.8.a.c 1
8.d odd 2 1 576.8.a.b 1
12.b even 2 1 48.8.a.a 1
15.d odd 2 1 600.8.a.b 1
15.e even 4 2 600.8.f.a 2
24.f even 2 1 192.8.a.p 1
24.h odd 2 1 192.8.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.a.b 1 3.b odd 2 1
48.8.a.a 1 12.b even 2 1
72.8.a.e 1 1.a even 1 1 trivial
144.8.a.k 1 4.b odd 2 1
192.8.a.h 1 24.h odd 2 1
192.8.a.p 1 24.f even 2 1
576.8.a.b 1 8.d odd 2 1
576.8.a.c 1 8.b even 2 1
600.8.a.b 1 15.d odd 2 1
600.8.f.a 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 530 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(72))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -530 + T \)
$7$ \( -120 + T \)
$11$ \( -7196 + T \)
$13$ \( 9626 + T \)
$17$ \( 18674 + T \)
$19$ \( -7004 + T \)
$23$ \( -63704 + T \)
$29$ \( 29334 + T \)
$31$ \( -87968 + T \)
$37$ \( -227982 + T \)
$41$ \( -160806 + T \)
$43$ \( -136132 + T \)
$47$ \( -1206960 + T \)
$53$ \( -398786 + T \)
$59$ \( 1152436 + T \)
$61$ \( 2070602 + T \)
$67$ \( 4073428 + T \)
$71$ \( -383752 + T \)
$73$ \( -3006010 + T \)
$79$ \( 4948112 + T \)
$83$ \( -9163492 + T \)
$89$ \( 7304106 + T \)
$97$ \( 690526 + T \)
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