Properties

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

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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 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} + 11q^{5} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} + 11q^{5} + 9q^{9} - 39q^{11} + 32q^{13} - 33q^{15} - 12q^{17} - 88q^{19} + 92q^{23} - 4q^{25} - 27q^{27} + 255q^{29} - 35q^{31} + 117q^{33} - 4q^{37} - 96q^{39} - 16q^{41} + 330q^{43} + 99q^{45} - 298q^{47} + 36q^{51} - 717q^{53} - 429q^{55} + 264q^{57} - 217q^{59} - 386q^{61} + 352q^{65} - 906q^{67} - 276q^{69} + 34q^{71} + 838q^{73} + 12q^{75} - 1325q^{79} + 81q^{81} + 1163q^{83} - 132q^{85} - 765q^{87} + 54q^{89} + 105q^{93} - 968q^{95} - 7q^{97} - 351q^{99} + O(q^{100}) \)

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 11.0000 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.o 1
4.b odd 2 1 1176.4.a.n 1
7.b odd 2 1 2352.4.a.y 1
7.d odd 6 2 336.4.q.c 2
28.d even 2 1 1176.4.a.c 1
28.f even 6 2 168.4.q.c 2
84.j odd 6 2 504.4.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.c 2 28.f even 6 2
336.4.q.c 2 7.d odd 6 2
504.4.s.a 2 84.j odd 6 2
1176.4.a.c 1 28.d even 2 1
1176.4.a.n 1 4.b odd 2 1
2352.4.a.o 1 1.a even 1 1 trivial
2352.4.a.y 1 7.b odd 2 1

Hecke kernels

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

\( T_{5} - 11 \)
\( T_{11} + 39 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( -11 + T \)
$7$ \( T \)
$11$ \( 39 + T \)
$13$ \( -32 + T \)
$17$ \( 12 + T \)
$19$ \( 88 + T \)
$23$ \( -92 + T \)
$29$ \( -255 + T \)
$31$ \( 35 + T \)
$37$ \( 4 + T \)
$41$ \( 16 + T \)
$43$ \( -330 + T \)
$47$ \( 298 + T \)
$53$ \( 717 + T \)
$59$ \( 217 + T \)
$61$ \( 386 + T \)
$67$ \( 906 + T \)
$71$ \( -34 + T \)
$73$ \( -838 + T \)
$79$ \( 1325 + T \)
$83$ \( -1163 + T \)
$89$ \( -54 + T \)
$97$ \( 7 + T \)
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