Properties

Label 72.6.a.f
Level $72$
Weight $6$
Character orbit 72.a
Self dual yes
Analytic conductor $11.548$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 72.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.5476350265\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 74 q^{5} - 24 q^{7} + O(q^{10}) \) \( q + 74 q^{5} - 24 q^{7} - 124 q^{11} + 478 q^{13} + 1198 q^{17} + 3044 q^{19} - 184 q^{23} + 2351 q^{25} + 3282 q^{29} - 5728 q^{31} - 1776 q^{35} + 10326 q^{37} + 8886 q^{41} - 9188 q^{43} - 23664 q^{47} - 16231 q^{49} - 11686 q^{53} - 9176 q^{55} - 16876 q^{59} - 18482 q^{61} + 35372 q^{65} - 15532 q^{67} + 31960 q^{71} - 4886 q^{73} + 2976 q^{77} + 44560 q^{79} - 67364 q^{83} + 88652 q^{85} - 71994 q^{89} - 11472 q^{91} + 225256 q^{95} + 48866 q^{97} + O(q^{100}) \)

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 74.0000 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 72.6.a.f 1
3.b odd 2 1 8.6.a.a 1
4.b odd 2 1 144.6.a.k 1
8.b even 2 1 576.6.a.g 1
8.d odd 2 1 576.6.a.h 1
12.b even 2 1 16.6.a.a 1
15.d odd 2 1 200.6.a.a 1
15.e even 4 2 200.6.c.a 2
21.c even 2 1 392.6.a.b 1
21.g even 6 2 392.6.i.e 2
21.h odd 6 2 392.6.i.b 2
24.f even 2 1 64.6.a.g 1
24.h odd 2 1 64.6.a.a 1
33.d even 2 1 968.6.a.a 1
48.i odd 4 2 256.6.b.f 2
48.k even 4 2 256.6.b.d 2
60.h even 2 1 400.6.a.l 1
60.l odd 4 2 400.6.c.d 2
84.h odd 2 1 784.6.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 3.b odd 2 1
16.6.a.a 1 12.b even 2 1
64.6.a.a 1 24.h odd 2 1
64.6.a.g 1 24.f even 2 1
72.6.a.f 1 1.a even 1 1 trivial
144.6.a.k 1 4.b odd 2 1
200.6.a.a 1 15.d odd 2 1
200.6.c.a 2 15.e even 4 2
256.6.b.d 2 48.k even 4 2
256.6.b.f 2 48.i odd 4 2
392.6.a.b 1 21.c even 2 1
392.6.i.b 2 21.h odd 6 2
392.6.i.e 2 21.g even 6 2
400.6.a.l 1 60.h even 2 1
400.6.c.d 2 60.l odd 4 2
576.6.a.g 1 8.b even 2 1
576.6.a.h 1 8.d odd 2 1
784.6.a.l 1 84.h odd 2 1
968.6.a.a 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 74 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(72))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -74 + T \)
$7$ \( 24 + T \)
$11$ \( 124 + T \)
$13$ \( -478 + T \)
$17$ \( -1198 + T \)
$19$ \( -3044 + T \)
$23$ \( 184 + T \)
$29$ \( -3282 + T \)
$31$ \( 5728 + T \)
$37$ \( -10326 + T \)
$41$ \( -8886 + T \)
$43$ \( 9188 + T \)
$47$ \( 23664 + T \)
$53$ \( 11686 + T \)
$59$ \( 16876 + T \)
$61$ \( 18482 + T \)
$67$ \( 15532 + T \)
$71$ \( -31960 + T \)
$73$ \( 4886 + T \)
$79$ \( -44560 + T \)
$83$ \( 67364 + T \)
$89$ \( 71994 + T \)
$97$ \( -48866 + T \)
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