Properties

Label 2352.2.a.ba
Level $2352$
Weight $2$
Character orbit 2352.a
Self dual yes
Analytic conductor $18.781$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta q^{5} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta q^{5} + q^{9} + \beta q^{11} + ( -3 + \beta ) q^{13} -\beta q^{15} + 4 q^{17} + ( -3 + \beta ) q^{19} -4 q^{23} + ( 9 + \beta ) q^{25} - q^{27} + ( 2 - \beta ) q^{29} - q^{31} -\beta q^{33} + ( 1 + \beta ) q^{37} + ( 3 - \beta ) q^{39} + ( -2 - 2 \beta ) q^{41} + ( 3 + \beta ) q^{43} + \beta q^{45} -6 q^{47} -4 q^{51} + ( 6 - \beta ) q^{53} + ( 14 + \beta ) q^{55} + ( 3 - \beta ) q^{57} + ( 2 + \beta ) q^{59} -10 q^{61} + ( 14 - 2 \beta ) q^{65} + ( -3 - \beta ) q^{67} + 4 q^{69} -2 q^{71} + ( 1 - \beta ) q^{73} + ( -9 - \beta ) q^{75} + ( -5 + 2 \beta ) q^{79} + q^{81} + ( 4 - \beta ) q^{83} + 4 \beta q^{85} + ( -2 + \beta ) q^{87} + ( 4 - 2 \beta ) q^{89} + q^{93} + ( 14 - 2 \beta ) q^{95} + ( -12 - \beta ) q^{97} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + q^{5} + 2q^{9} + q^{11} - 5q^{13} - q^{15} + 8q^{17} - 5q^{19} - 8q^{23} + 19q^{25} - 2q^{27} + 3q^{29} - 2q^{31} - q^{33} + 3q^{37} + 5q^{39} - 6q^{41} + 7q^{43} + q^{45} - 12q^{47} - 8q^{51} + 11q^{53} + 29q^{55} + 5q^{57} + 5q^{59} - 20q^{61} + 26q^{65} - 7q^{67} + 8q^{69} - 4q^{71} + q^{73} - 19q^{75} - 8q^{79} + 2q^{81} + 7q^{83} + 4q^{85} - 3q^{87} + 6q^{89} + 2q^{93} + 26q^{95} - 25q^{97} + q^{99} + O(q^{100}) \)

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
−3.27492
4.27492
0 −1.00000 0 −3.27492 0 0 0 1.00000 0
1.2 0 −1.00000 0 4.27492 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.ba 2
3.b odd 2 1 7056.2.a.ch 2
4.b odd 2 1 1176.2.a.n 2
7.b odd 2 1 2352.2.a.bf 2
7.c even 3 2 2352.2.q.bf 4
7.d odd 6 2 336.2.q.g 4
8.b even 2 1 9408.2.a.dw 2
8.d odd 2 1 9408.2.a.dj 2
12.b even 2 1 3528.2.a.bd 2
21.c even 2 1 7056.2.a.cu 2
21.g even 6 2 1008.2.s.r 4
28.d even 2 1 1176.2.a.k 2
28.f even 6 2 168.2.q.c 4
28.g odd 6 2 1176.2.q.l 4
56.e even 2 1 9408.2.a.ec 2
56.h odd 2 1 9408.2.a.dp 2
56.j odd 6 2 1344.2.q.x 4
56.m even 6 2 1344.2.q.w 4
84.h odd 2 1 3528.2.a.bk 2
84.j odd 6 2 504.2.s.i 4
84.n even 6 2 3528.2.s.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 28.f even 6 2
336.2.q.g 4 7.d odd 6 2
504.2.s.i 4 84.j odd 6 2
1008.2.s.r 4 21.g even 6 2
1176.2.a.k 2 28.d even 2 1
1176.2.a.n 2 4.b odd 2 1
1176.2.q.l 4 28.g odd 6 2
1344.2.q.w 4 56.m even 6 2
1344.2.q.x 4 56.j odd 6 2
2352.2.a.ba 2 1.a even 1 1 trivial
2352.2.a.bf 2 7.b odd 2 1
2352.2.q.bf 4 7.c even 3 2
3528.2.a.bd 2 12.b even 2 1
3528.2.a.bk 2 84.h odd 2 1
3528.2.s.bk 4 84.n even 6 2
7056.2.a.ch 2 3.b odd 2 1
7056.2.a.cu 2 21.c even 2 1
9408.2.a.dj 2 8.d odd 2 1
9408.2.a.dp 2 56.h odd 2 1
9408.2.a.dw 2 8.b even 2 1
9408.2.a.ec 2 56.e even 2 1

Hecke kernels

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

\( T_{5}^{2} - T_{5} - 14 \)
\( T_{11}^{2} - T_{11} - 14 \)
\( T_{13}^{2} + 5 T_{13} - 8 \)
\( T_{17} - 4 \)