Properties

Label 2352.4.a.be
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
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

\(f(q)\) \(=\) \( q + 3q^{3} + 4q^{5} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 4q^{5} + 9q^{9} + 20q^{11} - 4q^{13} + 12q^{15} + 24q^{17} - 44q^{19} - 72q^{23} - 109q^{25} + 27q^{27} - 38q^{29} - 184q^{31} + 60q^{33} - 30q^{37} - 12q^{39} - 216q^{41} + 164q^{43} + 36q^{45} - 520q^{47} + 72q^{51} - 146q^{53} + 80q^{55} - 132q^{57} - 460q^{59} + 628q^{61} - 16q^{65} - 556q^{67} - 216q^{69} - 592q^{71} + 1024q^{73} - 327q^{75} + 104q^{79} + 81q^{81} + 324q^{83} + 96q^{85} - 114q^{87} + 896q^{89} - 552q^{93} - 176q^{95} - 920q^{97} + 180q^{99} + O(q^{100}) \)

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 3.00000 0 4.00000 0 0 0 9.00000 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 2352.4.a.be 1
4.b odd 2 1 588.4.a.b 1
7.b odd 2 1 2352.4.a.g 1
12.b even 2 1 1764.4.a.d 1
28.d even 2 1 588.4.a.e yes 1
28.f even 6 2 588.4.i.b 2
28.g odd 6 2 588.4.i.g 2
84.h odd 2 1 1764.4.a.i 1
84.j odd 6 2 1764.4.k.g 2
84.n even 6 2 1764.4.k.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.b 1 4.b odd 2 1
588.4.a.e yes 1 28.d even 2 1
588.4.i.b 2 28.f even 6 2
588.4.i.g 2 28.g odd 6 2
1764.4.a.d 1 12.b even 2 1
1764.4.a.i 1 84.h odd 2 1
1764.4.k.g 2 84.j odd 6 2
1764.4.k.j 2 84.n even 6 2
2352.4.a.g 1 7.b odd 2 1
2352.4.a.be 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

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

Hecke characteristic polynomials

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