Properties

Label 1764.4.a.a
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 196)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 20q^{5} + O(q^{10}) \) \( q - 20q^{5} - 44q^{11} + 44q^{13} + 72q^{17} - 100q^{19} + 120q^{23} + 275q^{25} - 218q^{29} + 280q^{31} - 30q^{37} + 120q^{41} + 220q^{43} + 88q^{47} - 110q^{53} + 880q^{55} + 580q^{59} - 380q^{61} - 880q^{65} - 980q^{67} + 112q^{71} + 640q^{73} - 488q^{79} + 660q^{83} - 1440q^{85} + 320q^{89} + 2000q^{95} - 248q^{97} + O(q^{100}) \)

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 −20.0000 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.a 1
3.b odd 2 1 196.4.a.c yes 1
7.b odd 2 1 1764.4.a.m 1
7.c even 3 2 1764.4.k.p 2
7.d odd 6 2 1764.4.k.a 2
12.b even 2 1 784.4.a.f 1
21.c even 2 1 196.4.a.a 1
21.g even 6 2 196.4.e.e 2
21.h odd 6 2 196.4.e.b 2
84.h odd 2 1 784.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.4.a.a 1 21.c even 2 1
196.4.a.c yes 1 3.b odd 2 1
196.4.e.b 2 21.h odd 6 2
196.4.e.e 2 21.g even 6 2
784.4.a.f 1 12.b even 2 1
784.4.a.m 1 84.h odd 2 1
1764.4.a.a 1 1.a even 1 1 trivial
1764.4.a.m 1 7.b odd 2 1
1764.4.k.a 2 7.d odd 6 2
1764.4.k.p 2 7.c even 3 2

Hecke kernels

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

\( T_{5} + 20 \)
\( T_{11} + 44 \)
\( T_{13} - 44 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 20 + T \)
$7$ \( T \)
$11$ \( 44 + T \)
$13$ \( -44 + T \)
$17$ \( -72 + T \)
$19$ \( 100 + T \)
$23$ \( -120 + T \)
$29$ \( 218 + T \)
$31$ \( -280 + T \)
$37$ \( 30 + T \)
$41$ \( -120 + T \)
$43$ \( -220 + T \)
$47$ \( -88 + T \)
$53$ \( 110 + T \)
$59$ \( -580 + T \)
$61$ \( 380 + T \)
$67$ \( 980 + T \)
$71$ \( -112 + T \)
$73$ \( -640 + T \)
$79$ \( 488 + T \)
$83$ \( -660 + T \)
$89$ \( -320 + T \)
$97$ \( 248 + T \)
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