Properties

Label 72.12.a.c
Level $72$
Weight $12$
Character orbit 72.a
Self dual yes
Analytic conductor $55.321$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 72.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3207090003\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 3490 q^{5} - 55464 q^{7} + O(q^{10}) \) \( q + 3490 q^{5} - 55464 q^{7} + 597004 q^{11} + 1373878 q^{13} - 10140850 q^{17} - 7297396 q^{19} + 32057464 q^{23} - 36648025 q^{25} + 13605402 q^{29} + 233160800 q^{31} - 193569360 q^{35} - 257786178 q^{37} + 221438598 q^{41} - 1697758892 q^{43} - 527509392 q^{47} + 1098928553 q^{49} - 3277379822 q^{53} + 2083543960 q^{55} + 3001908988 q^{59} - 11630023610 q^{61} + 4794834220 q^{65} - 17189000548 q^{67} - 26169539608 q^{71} - 7039021094 q^{73} - 33112229856 q^{77} - 4199910416 q^{79} + 39739936436 q^{83} - 35391566500 q^{85} - 10565331594 q^{89} - 76200769392 q^{91} - 25467912040 q^{95} - 69851645662 q^{97} + O(q^{100}) \)

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 3490.00 0 −55464.0 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 72.12.a.c 1
3.b odd 2 1 8.12.a.a 1
4.b odd 2 1 144.12.a.j 1
12.b even 2 1 16.12.a.b 1
15.d odd 2 1 200.12.a.b 1
15.e even 4 2 200.12.c.b 2
24.f even 2 1 64.12.a.c 1
24.h odd 2 1 64.12.a.e 1
48.i odd 4 2 256.12.b.g 2
48.k even 4 2 256.12.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.a.a 1 3.b odd 2 1
16.12.a.b 1 12.b even 2 1
64.12.a.c 1 24.f even 2 1
64.12.a.e 1 24.h odd 2 1
72.12.a.c 1 1.a even 1 1 trivial
144.12.a.j 1 4.b odd 2 1
200.12.a.b 1 15.d odd 2 1
200.12.c.b 2 15.e even 4 2
256.12.b.a 2 48.k even 4 2
256.12.b.g 2 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3490 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(72))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -3490 + T \)
$7$ \( 55464 + T \)
$11$ \( -597004 + T \)
$13$ \( -1373878 + T \)
$17$ \( 10140850 + T \)
$19$ \( 7297396 + T \)
$23$ \( -32057464 + T \)
$29$ \( -13605402 + T \)
$31$ \( -233160800 + T \)
$37$ \( 257786178 + T \)
$41$ \( -221438598 + T \)
$43$ \( 1697758892 + T \)
$47$ \( 527509392 + T \)
$53$ \( 3277379822 + T \)
$59$ \( -3001908988 + T \)
$61$ \( 11630023610 + T \)
$67$ \( 17189000548 + T \)
$71$ \( 26169539608 + T \)
$73$ \( 7039021094 + T \)
$79$ \( 4199910416 + T \)
$83$ \( -39739936436 + T \)
$89$ \( 10565331594 + T \)
$97$ \( 69851645662 + T \)
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