Properties

Label 2352.4.a.f
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 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} - 6q^{5} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} - 6q^{5} + 9q^{9} + 30q^{11} + 53q^{13} + 18q^{15} - 84q^{17} + 97q^{19} - 84q^{23} - 89q^{25} - 27q^{27} - 180q^{29} - 179q^{31} - 90q^{33} - 145q^{37} - 159q^{39} + 126q^{41} + 325q^{43} - 54q^{45} + 366q^{47} + 252q^{51} - 768q^{53} - 180q^{55} - 291q^{57} + 264q^{59} + 818q^{61} - 318q^{65} + 523q^{67} + 252q^{69} + 342q^{71} - 43q^{73} + 267q^{75} + 1171q^{79} + 81q^{81} + 810q^{83} + 504q^{85} + 540q^{87} - 600q^{89} + 537q^{93} - 582q^{95} + 386q^{97} + 270q^{99} + O(q^{100}) \)

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 −6.00000 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.f 1
4.b odd 2 1 294.4.a.d 1
7.b odd 2 1 2352.4.a.bf 1
7.c even 3 2 336.4.q.f 2
12.b even 2 1 882.4.a.o 1
28.d even 2 1 294.4.a.c 1
28.f even 6 2 294.4.e.i 2
28.g odd 6 2 42.4.e.a 2
84.h odd 2 1 882.4.a.l 1
84.j odd 6 2 882.4.g.g 2
84.n even 6 2 126.4.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 28.g odd 6 2
126.4.g.b 2 84.n even 6 2
294.4.a.c 1 28.d even 2 1
294.4.a.d 1 4.b odd 2 1
294.4.e.i 2 28.f even 6 2
336.4.q.f 2 7.c even 3 2
882.4.a.l 1 84.h odd 2 1
882.4.a.o 1 12.b even 2 1
882.4.g.g 2 84.j odd 6 2
2352.4.a.f 1 1.a even 1 1 trivial
2352.4.a.bf 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} + 6 \)
\( T_{11} - 30 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 6 + T \)
$7$ \( T \)
$11$ \( -30 + T \)
$13$ \( -53 + T \)
$17$ \( 84 + T \)
$19$ \( -97 + T \)
$23$ \( 84 + T \)
$29$ \( 180 + T \)
$31$ \( 179 + T \)
$37$ \( 145 + T \)
$41$ \( -126 + T \)
$43$ \( -325 + T \)
$47$ \( -366 + T \)
$53$ \( 768 + T \)
$59$ \( -264 + T \)
$61$ \( -818 + T \)
$67$ \( -523 + T \)
$71$ \( -342 + T \)
$73$ \( 43 + T \)
$79$ \( -1171 + T \)
$83$ \( -810 + T \)
$89$ \( 600 + T \)
$97$ \( -386 + T \)
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