Properties

Label 6768.2.a.h
Level $6768$
Weight $2$
Character orbit 6768.a
Self dual yes
Analytic conductor $54.043$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6768 = 2^{4} \cdot 3^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6768.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.0427520880\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 141)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{7} + O(q^{10}) \) \( q - 4q^{7} + 6q^{13} + 6q^{17} - 2q^{19} + 4q^{23} - 5q^{25} - 8q^{29} - 6q^{31} - 6q^{37} + 8q^{41} + 6q^{43} + q^{47} + 9q^{49} - 2q^{53} + 12q^{59} + 2q^{61} + 2q^{67} - 10q^{73} + 4q^{79} + 4q^{83} + 10q^{89} - 24q^{91} - 18q^{97} + O(q^{100}) \)

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 0 0 −4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(47\) \(-1\)

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 6768.2.a.h 1
3.b odd 2 1 2256.2.a.l 1
4.b odd 2 1 423.2.a.e 1
12.b even 2 1 141.2.a.b 1
24.f even 2 1 9024.2.a.bk 1
24.h odd 2 1 9024.2.a.i 1
60.h even 2 1 3525.2.a.k 1
84.h odd 2 1 6909.2.a.e 1
564.f odd 2 1 6627.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.2.a.b 1 12.b even 2 1
423.2.a.e 1 4.b odd 2 1
2256.2.a.l 1 3.b odd 2 1
3525.2.a.k 1 60.h even 2 1
6627.2.a.b 1 564.f odd 2 1
6768.2.a.h 1 1.a even 1 1 trivial
6909.2.a.e 1 84.h odd 2 1
9024.2.a.i 1 24.h odd 2 1
9024.2.a.bk 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6768))\):

\( T_{5} \)
\( T_{7} + 4 \)
\( T_{11} \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 4 + T \)
$11$ \( T \)
$13$ \( -6 + T \)
$17$ \( -6 + T \)
$19$ \( 2 + T \)
$23$ \( -4 + T \)
$29$ \( 8 + T \)
$31$ \( 6 + T \)
$37$ \( 6 + T \)
$41$ \( -8 + T \)
$43$ \( -6 + T \)
$47$ \( -1 + T \)
$53$ \( 2 + T \)
$59$ \( -12 + T \)
$61$ \( -2 + T \)
$67$ \( -2 + T \)
$71$ \( T \)
$73$ \( 10 + T \)
$79$ \( -4 + T \)
$83$ \( -4 + T \)
$89$ \( -10 + T \)
$97$ \( 18 + T \)
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