Properties

Label 1764.4.a.k
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 28)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 6q^{5} + O(q^{10}) \) \( q + 6q^{5} + 12q^{11} + 82q^{13} - 30q^{17} - 68q^{19} - 216q^{23} - 89q^{25} - 246q^{29} + 112q^{31} + 110q^{37} - 246q^{41} - 172q^{43} + 192q^{47} - 558q^{53} + 72q^{55} + 540q^{59} - 110q^{61} + 492q^{65} + 140q^{67} + 840q^{71} + 550q^{73} - 208q^{79} + 516q^{83} - 180q^{85} - 1398q^{89} - 408q^{95} - 1586q^{97} + O(q^{100}) \)

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 6.00000 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.k 1
3.b odd 2 1 196.4.a.b 1
7.b odd 2 1 252.4.a.c 1
7.c even 3 2 1764.4.k.e 2
7.d odd 6 2 1764.4.k.k 2
12.b even 2 1 784.4.a.n 1
21.c even 2 1 28.4.a.b 1
21.g even 6 2 196.4.e.c 2
21.h odd 6 2 196.4.e.d 2
28.d even 2 1 1008.4.a.f 1
84.h odd 2 1 112.4.a.c 1
105.g even 2 1 700.4.a.e 1
105.k odd 4 2 700.4.e.f 2
168.e odd 2 1 448.4.a.m 1
168.i even 2 1 448.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 21.c even 2 1
112.4.a.c 1 84.h odd 2 1
196.4.a.b 1 3.b odd 2 1
196.4.e.c 2 21.g even 6 2
196.4.e.d 2 21.h odd 6 2
252.4.a.c 1 7.b odd 2 1
448.4.a.d 1 168.i even 2 1
448.4.a.m 1 168.e odd 2 1
700.4.a.e 1 105.g even 2 1
700.4.e.f 2 105.k odd 4 2
784.4.a.n 1 12.b even 2 1
1008.4.a.f 1 28.d even 2 1
1764.4.a.k 1 1.a even 1 1 trivial
1764.4.k.e 2 7.c even 3 2
1764.4.k.k 2 7.d odd 6 2

Hecke kernels

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

\( T_{5} - 6 \)
\( T_{11} - 12 \)
\( T_{13} - 82 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -6 + T \)
$7$ \( T \)
$11$ \( -12 + T \)
$13$ \( -82 + T \)
$17$ \( 30 + T \)
$19$ \( 68 + T \)
$23$ \( 216 + T \)
$29$ \( 246 + T \)
$31$ \( -112 + T \)
$37$ \( -110 + T \)
$41$ \( 246 + T \)
$43$ \( 172 + T \)
$47$ \( -192 + T \)
$53$ \( 558 + T \)
$59$ \( -540 + T \)
$61$ \( 110 + T \)
$67$ \( -140 + T \)
$71$ \( -840 + T \)
$73$ \( -550 + T \)
$79$ \( 208 + T \)
$83$ \( -516 + T \)
$89$ \( 1398 + T \)
$97$ \( 1586 + T \)
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