Properties

Label 2352.4.a.bc
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

\(f(q)\) \(=\) \( q + 3q^{3} + 2q^{5} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 2q^{5} + 9q^{9} + 18q^{11} + 33q^{13} + 6q^{15} - 68q^{17} - 25q^{19} - 92q^{23} - 121q^{25} + 27q^{27} + 92q^{29} - 25q^{31} + 54q^{33} - 213q^{37} + 99q^{39} + 94q^{41} + 67q^{43} + 18q^{45} - 278q^{47} - 204q^{51} - 400q^{53} + 36q^{55} - 75q^{57} - 744q^{59} - 734q^{61} + 66q^{65} - 555q^{67} - 276q^{69} + 642q^{71} + 973q^{73} - 363q^{75} + 785q^{79} + 81q^{81} + 822q^{83} - 136q^{85} + 276q^{87} + 424q^{89} - 75q^{93} - 50q^{95} - 734q^{97} + 162q^{99} + O(q^{100}) \)

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 2.00000 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.bc 1
4.b odd 2 1 1176.4.a.d 1
7.b odd 2 1 2352.4.a.j 1
7.c even 3 2 336.4.q.b 2
28.d even 2 1 1176.4.a.k 1
28.g odd 6 2 168.4.q.b 2
84.n even 6 2 504.4.s.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.b 2 28.g odd 6 2
336.4.q.b 2 7.c even 3 2
504.4.s.d 2 84.n even 6 2
1176.4.a.d 1 4.b odd 2 1
1176.4.a.k 1 28.d even 2 1
2352.4.a.j 1 7.b odd 2 1
2352.4.a.bc 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} - 2 \)
\( T_{11} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -2 + T \)
$7$ \( T \)
$11$ \( -18 + T \)
$13$ \( -33 + T \)
$17$ \( 68 + T \)
$19$ \( 25 + T \)
$23$ \( 92 + T \)
$29$ \( -92 + T \)
$31$ \( 25 + T \)
$37$ \( 213 + T \)
$41$ \( -94 + T \)
$43$ \( -67 + T \)
$47$ \( 278 + T \)
$53$ \( 400 + T \)
$59$ \( 744 + T \)
$61$ \( 734 + T \)
$67$ \( 555 + T \)
$71$ \( -642 + T \)
$73$ \( -973 + T \)
$79$ \( -785 + T \)
$83$ \( -822 + T \)
$89$ \( -424 + T \)
$97$ \( 734 + T \)
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