Properties

Label 1764.4.a.e
Level $1764$
Weight $4$
Character orbit 1764.a
Self dual yes
Analytic conductor $104.079$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + O(q^{10}) \) \( q - 89q^{13} + 163q^{19} - 125q^{25} + 19q^{31} - 433q^{37} + 449q^{43} - 182q^{61} + 1007q^{67} + 919q^{73} + 503q^{79} + 1330q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.a.e 1
3.b odd 2 1 CM 1764.4.a.e 1
7.b odd 2 1 1764.4.a.h 1
7.c even 3 2 1764.4.k.h 2
7.d odd 6 2 252.4.k.a 2
21.c even 2 1 1764.4.a.h 1
21.g even 6 2 252.4.k.a 2
21.h odd 6 2 1764.4.k.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.k.a 2 7.d odd 6 2
252.4.k.a 2 21.g even 6 2
1764.4.a.e 1 1.a even 1 1 trivial
1764.4.a.e 1 3.b odd 2 1 CM
1764.4.a.h 1 7.b odd 2 1
1764.4.a.h 1 21.c even 2 1
1764.4.k.h 2 7.c even 3 2
1764.4.k.h 2 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1764))\):

\( T_{5} \)
\( T_{11} \)
\( T_{13} + 89 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( 89 + T \)
$17$ \( T \)
$19$ \( -163 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( -19 + T \)
$37$ \( 433 + T \)
$41$ \( T \)
$43$ \( -449 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 182 + T \)
$67$ \( -1007 + T \)
$71$ \( T \)
$73$ \( -919 + T \)
$79$ \( -503 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( -1330 + T \)
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