Properties

Label 2352.2.a.n
Level $2352$
Weight $2$
Character orbit 2352.a
Self dual yes
Analytic conductor $18.781$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.7808145554\)
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

\(f(q)\) \(=\) \( q + q^{3} - 3q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} - 3q^{5} + q^{9} - 3q^{11} + 4q^{13} - 3q^{15} - 4q^{19} + 4q^{25} + q^{27} + 9q^{29} - q^{31} - 3q^{33} + 8q^{37} + 4q^{39} + 10q^{43} - 3q^{45} - 6q^{47} - 3q^{53} + 9q^{55} - 4q^{57} + 3q^{59} + 10q^{61} - 12q^{65} + 10q^{67} + 6q^{71} - 2q^{73} + 4q^{75} + q^{79} + q^{81} - 9q^{83} + 9q^{87} - 6q^{89} - q^{93} + 12q^{95} + q^{97} - 3q^{99} + O(q^{100}) \)

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 1.00000 0 −3.00000 0 0 0 1.00000 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 2352.2.a.n 1
3.b odd 2 1 7056.2.a.bz 1
4.b odd 2 1 294.2.a.a 1
7.b odd 2 1 2352.2.a.m 1
7.c even 3 2 2352.2.q.m 2
7.d odd 6 2 336.2.q.d 2
8.b even 2 1 9408.2.a.bm 1
8.d odd 2 1 9408.2.a.db 1
12.b even 2 1 882.2.a.k 1
20.d odd 2 1 7350.2.a.cw 1
21.c even 2 1 7056.2.a.g 1
21.g even 6 2 1008.2.s.n 2
28.d even 2 1 294.2.a.d 1
28.f even 6 2 42.2.e.b 2
28.g odd 6 2 294.2.e.f 2
56.e even 2 1 9408.2.a.d 1
56.h odd 2 1 9408.2.a.bu 1
56.j odd 6 2 1344.2.q.j 2
56.m even 6 2 1344.2.q.v 2
84.h odd 2 1 882.2.a.g 1
84.j odd 6 2 126.2.g.b 2
84.n even 6 2 882.2.g.b 2
140.c even 2 1 7350.2.a.ce 1
140.s even 6 2 1050.2.i.e 2
140.x odd 12 4 1050.2.o.b 4
252.n even 6 2 1134.2.h.p 2
252.r odd 6 2 1134.2.e.p 2
252.bj even 6 2 1134.2.e.a 2
252.bn odd 6 2 1134.2.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.b 2 28.f even 6 2
126.2.g.b 2 84.j odd 6 2
294.2.a.a 1 4.b odd 2 1
294.2.a.d 1 28.d even 2 1
294.2.e.f 2 28.g odd 6 2
336.2.q.d 2 7.d odd 6 2
882.2.a.g 1 84.h odd 2 1
882.2.a.k 1 12.b even 2 1
882.2.g.b 2 84.n even 6 2
1008.2.s.n 2 21.g even 6 2
1050.2.i.e 2 140.s even 6 2
1050.2.o.b 4 140.x odd 12 4
1134.2.e.a 2 252.bj even 6 2
1134.2.e.p 2 252.r odd 6 2
1134.2.h.a 2 252.bn odd 6 2
1134.2.h.p 2 252.n even 6 2
1344.2.q.j 2 56.j odd 6 2
1344.2.q.v 2 56.m even 6 2
2352.2.a.m 1 7.b odd 2 1
2352.2.a.n 1 1.a even 1 1 trivial
2352.2.q.m 2 7.c even 3 2
7056.2.a.g 1 21.c even 2 1
7056.2.a.bz 1 3.b odd 2 1
7350.2.a.ce 1 140.c even 2 1
7350.2.a.cw 1 20.d odd 2 1
9408.2.a.d 1 56.e even 2 1
9408.2.a.bm 1 8.b even 2 1
9408.2.a.bu 1 56.h odd 2 1
9408.2.a.db 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} + 3 \)
\( T_{11} + 3 \)
\( T_{13} - 4 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T \)
$5$ \( 1 + 3 T + 5 T^{2} \)
$7$ 1
$11$ \( 1 + 3 T + 11 T^{2} \)
$13$ \( 1 - 4 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 9 T + 29 T^{2} \)
$31$ \( 1 + T + 31 T^{2} \)
$37$ \( 1 - 8 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 - 10 T + 43 T^{2} \)
$47$ \( 1 + 6 T + 47 T^{2} \)
$53$ \( 1 + 3 T + 53 T^{2} \)
$59$ \( 1 - 3 T + 59 T^{2} \)
$61$ \( 1 - 10 T + 61 T^{2} \)
$67$ \( 1 - 10 T + 67 T^{2} \)
$71$ \( 1 - 6 T + 71 T^{2} \)
$73$ \( 1 + 2 T + 73 T^{2} \)
$79$ \( 1 - T + 79 T^{2} \)
$83$ \( 1 + 9 T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 - T + 97 T^{2} \)
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