Properties

Label 1764.4.a.i
Level $1764$
Weight $4$
Character orbit 1764.a
Self dual yes
Analytic conductor $104.079$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 588)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{5} + O(q^{10}) \) \( q + 4q^{5} + 20q^{11} + 4q^{13} + 24q^{17} - 44q^{19} - 72q^{23} - 109q^{25} + 38q^{29} - 184q^{31} - 30q^{37} - 216q^{41} - 164q^{43} + 520q^{47} + 146q^{53} + 80q^{55} + 460q^{59} - 628q^{61} + 16q^{65} + 556q^{67} - 592q^{71} - 1024q^{73} - 104q^{79} - 324q^{83} + 96q^{85} + 896q^{89} - 176q^{95} + 920q^{97} + O(q^{100}) \)

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 4.00000 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

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 1764.4.a.i 1
3.b odd 2 1 588.4.a.e yes 1
7.b odd 2 1 1764.4.a.d 1
7.c even 3 2 1764.4.k.g 2
7.d odd 6 2 1764.4.k.j 2
12.b even 2 1 2352.4.a.g 1
21.c even 2 1 588.4.a.b 1
21.g even 6 2 588.4.i.g 2
21.h odd 6 2 588.4.i.b 2
84.h odd 2 1 2352.4.a.be 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.b 1 21.c even 2 1
588.4.a.e yes 1 3.b odd 2 1
588.4.i.b 2 21.h odd 6 2
588.4.i.g 2 21.g even 6 2
1764.4.a.d 1 7.b odd 2 1
1764.4.a.i 1 1.a even 1 1 trivial
1764.4.k.g 2 7.c even 3 2
1764.4.k.j 2 7.d odd 6 2
2352.4.a.g 1 12.b even 2 1
2352.4.a.be 1 84.h odd 2 1

Hecke kernels

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

\( T_{5} - 4 \)
\( T_{11} - 20 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -4 + T \)
$7$ \( T \)
$11$ \( -20 + T \)
$13$ \( -4 + T \)
$17$ \( -24 + T \)
$19$ \( 44 + T \)
$23$ \( 72 + T \)
$29$ \( -38 + T \)
$31$ \( 184 + T \)
$37$ \( 30 + T \)
$41$ \( 216 + T \)
$43$ \( 164 + T \)
$47$ \( -520 + T \)
$53$ \( -146 + T \)
$59$ \( -460 + T \)
$61$ \( 628 + T \)
$67$ \( -556 + T \)
$71$ \( 592 + T \)
$73$ \( 1024 + T \)
$79$ \( 104 + T \)
$83$ \( 324 + T \)
$89$ \( -896 + T \)
$97$ \( -920 + T \)
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