Properties

Label 2352.4.a.a
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} - 18q^{5} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} - 18q^{5} + 9q^{9} + 72q^{11} + 34q^{13} + 54q^{15} - 6q^{17} + 92q^{19} + 180q^{23} + 199q^{25} - 27q^{27} - 114q^{29} + 56q^{31} - 216q^{33} - 34q^{37} - 102q^{39} - 6q^{41} - 164q^{43} - 162q^{45} + 168q^{47} + 18q^{51} + 654q^{53} - 1296q^{55} - 276q^{57} - 492q^{59} + 250q^{61} - 612q^{65} + 124q^{67} - 540q^{69} - 36q^{71} - 1010q^{73} - 597q^{75} - 56q^{79} + 81q^{81} + 228q^{83} + 108q^{85} + 342q^{87} - 390q^{89} - 168q^{93} - 1656q^{95} + 70q^{97} + 648q^{99} + O(q^{100}) \)

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 −18.0000 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.a 1
4.b odd 2 1 294.4.a.i 1
7.b odd 2 1 336.4.a.l 1
12.b even 2 1 882.4.a.g 1
21.c even 2 1 1008.4.a.b 1
28.d even 2 1 42.4.a.a 1
28.f even 6 2 294.4.e.c 2
28.g odd 6 2 294.4.e.b 2
56.e even 2 1 1344.4.a.o 1
56.h odd 2 1 1344.4.a.a 1
84.h odd 2 1 126.4.a.a 1
84.j odd 6 2 882.4.g.w 2
84.n even 6 2 882.4.g.o 2
140.c even 2 1 1050.4.a.g 1
140.j odd 4 2 1050.4.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 28.d even 2 1
126.4.a.a 1 84.h odd 2 1
294.4.a.i 1 4.b odd 2 1
294.4.e.b 2 28.g odd 6 2
294.4.e.c 2 28.f even 6 2
336.4.a.l 1 7.b odd 2 1
882.4.a.g 1 12.b even 2 1
882.4.g.o 2 84.n even 6 2
882.4.g.w 2 84.j odd 6 2
1008.4.a.b 1 21.c even 2 1
1050.4.a.g 1 140.c even 2 1
1050.4.g.a 2 140.j odd 4 2
1344.4.a.a 1 56.h odd 2 1
1344.4.a.o 1 56.e even 2 1
2352.4.a.a 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} + 18 \)
\( T_{11} - 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 18 + T \)
$7$ \( T \)
$11$ \( -72 + T \)
$13$ \( -34 + T \)
$17$ \( 6 + T \)
$19$ \( -92 + T \)
$23$ \( -180 + T \)
$29$ \( 114 + T \)
$31$ \( -56 + T \)
$37$ \( 34 + T \)
$41$ \( 6 + T \)
$43$ \( 164 + T \)
$47$ \( -168 + T \)
$53$ \( -654 + T \)
$59$ \( 492 + T \)
$61$ \( -250 + T \)
$67$ \( -124 + T \)
$71$ \( 36 + T \)
$73$ \( 1010 + T \)
$79$ \( 56 + T \)
$83$ \( -228 + T \)
$89$ \( 390 + T \)
$97$ \( -70 + T \)
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