Properties

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

Newform invariants

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

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} + q^{5} + q^{9} - 5q^{11} + q^{15} - 4q^{17} - 8q^{19} + 4q^{23} - 4q^{25} + q^{27} - 5q^{29} - 3q^{31} - 5q^{33} - 4q^{37} - 2q^{43} + q^{45} + 6q^{47} - 4q^{51} - 9q^{53} - 5q^{55} - 8q^{57} + 11q^{59} - 6q^{61} + 2q^{67} + 4q^{69} - 2q^{71} + 10q^{73} - 4q^{75} - 3q^{79} + q^{81} + 7q^{83} - 4q^{85} - 5q^{87} - 6q^{89} - 3q^{93} - 8q^{95} + 7q^{97} - 5q^{99} + O(q^{100}) \)

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 1.00000 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.t 1
3.b odd 2 1 7056.2.a.w 1
4.b odd 2 1 294.2.a.e 1
7.b odd 2 1 2352.2.a.f 1
7.c even 3 2 336.2.q.b 2
7.d odd 6 2 2352.2.q.u 2
8.b even 2 1 9408.2.a.q 1
8.d odd 2 1 9408.2.a.ce 1
12.b even 2 1 882.2.a.c 1
20.d odd 2 1 7350.2.a.bl 1
21.c even 2 1 7056.2.a.bl 1
21.h odd 6 2 1008.2.s.k 2
28.d even 2 1 294.2.a.f 1
28.f even 6 2 294.2.e.b 2
28.g odd 6 2 42.2.e.a 2
56.e even 2 1 9408.2.a.z 1
56.h odd 2 1 9408.2.a.cr 1
56.k odd 6 2 1344.2.q.g 2
56.p even 6 2 1344.2.q.s 2
84.h odd 2 1 882.2.a.d 1
84.j odd 6 2 882.2.g.i 2
84.n even 6 2 126.2.g.c 2
140.c even 2 1 7350.2.a.q 1
140.p odd 6 2 1050.2.i.l 2
140.w even 12 4 1050.2.o.a 4
252.o even 6 2 1134.2.h.l 2
252.u odd 6 2 1134.2.e.l 2
252.bb even 6 2 1134.2.e.e 2
252.bl odd 6 2 1134.2.h.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 28.g odd 6 2
126.2.g.c 2 84.n even 6 2
294.2.a.e 1 4.b odd 2 1
294.2.a.f 1 28.d even 2 1
294.2.e.b 2 28.f even 6 2
336.2.q.b 2 7.c even 3 2
882.2.a.c 1 12.b even 2 1
882.2.a.d 1 84.h odd 2 1
882.2.g.i 2 84.j odd 6 2
1008.2.s.k 2 21.h odd 6 2
1050.2.i.l 2 140.p odd 6 2
1050.2.o.a 4 140.w even 12 4
1134.2.e.e 2 252.bb even 6 2
1134.2.e.l 2 252.u odd 6 2
1134.2.h.e 2 252.bl odd 6 2
1134.2.h.l 2 252.o even 6 2
1344.2.q.g 2 56.k odd 6 2
1344.2.q.s 2 56.p even 6 2
2352.2.a.f 1 7.b odd 2 1
2352.2.a.t 1 1.a even 1 1 trivial
2352.2.q.u 2 7.d odd 6 2
7056.2.a.w 1 3.b odd 2 1
7056.2.a.bl 1 21.c even 2 1
7350.2.a.q 1 140.c even 2 1
7350.2.a.bl 1 20.d odd 2 1
9408.2.a.q 1 8.b even 2 1
9408.2.a.z 1 56.e even 2 1
9408.2.a.ce 1 8.d odd 2 1
9408.2.a.cr 1 56.h odd 2 1

Hecke kernels

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

\( T_{5} - 1 \)
\( T_{11} + 5 \)
\( T_{13} \)
\( T_{17} + 4 \)

Hecke characteristic polynomials

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