Properties

Label 2352.4.a.bd
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 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + 3q^{5} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 3q^{5} + 9q^{9} + 15q^{11} + 64q^{13} + 9q^{15} - 84q^{17} - 16q^{19} + 84q^{23} - 116q^{25} + 27q^{27} - 297q^{29} - 253q^{31} + 45q^{33} - 316q^{37} + 192q^{39} - 360q^{41} - 26q^{43} + 27q^{45} - 30q^{47} - 252q^{51} + 363q^{53} + 45q^{55} - 48q^{57} - 15q^{59} + 118q^{61} + 192q^{65} + 370q^{67} + 252q^{69} + 342q^{71} - 362q^{73} - 348q^{75} - 467q^{79} + 81q^{81} + 477q^{83} - 252q^{85} - 891q^{87} - 906q^{89} - 759q^{93} - 48q^{95} - 503q^{97} + 135q^{99} + O(q^{100}) \)

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 3.00000 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.bd 1
4.b odd 2 1 147.4.a.a 1
7.b odd 2 1 2352.4.a.i 1
7.d odd 6 2 336.4.q.e 2
12.b even 2 1 441.4.a.k 1
28.d even 2 1 147.4.a.b 1
28.f even 6 2 21.4.e.a 2
28.g odd 6 2 147.4.e.h 2
84.h odd 2 1 441.4.a.l 1
84.j odd 6 2 63.4.e.a 2
84.n even 6 2 441.4.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 28.f even 6 2
63.4.e.a 2 84.j odd 6 2
147.4.a.a 1 4.b odd 2 1
147.4.a.b 1 28.d even 2 1
147.4.e.h 2 28.g odd 6 2
336.4.q.e 2 7.d odd 6 2
441.4.a.k 1 12.b even 2 1
441.4.a.l 1 84.h odd 2 1
441.4.e.c 2 84.n even 6 2
2352.4.a.i 1 7.b odd 2 1
2352.4.a.bd 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} - 3 \)
\( T_{11} - 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -3 + T \)
$7$ \( T \)
$11$ \( -15 + T \)
$13$ \( -64 + T \)
$17$ \( 84 + T \)
$19$ \( 16 + T \)
$23$ \( -84 + T \)
$29$ \( 297 + T \)
$31$ \( 253 + T \)
$37$ \( 316 + T \)
$41$ \( 360 + T \)
$43$ \( 26 + T \)
$47$ \( 30 + T \)
$53$ \( -363 + T \)
$59$ \( 15 + T \)
$61$ \( -118 + T \)
$67$ \( -370 + T \)
$71$ \( -342 + T \)
$73$ \( 362 + T \)
$79$ \( 467 + T \)
$83$ \( -477 + T \)
$89$ \( 906 + T \)
$97$ \( 503 + T \)
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