Properties

Label 7056.2.a.ch
Level $7056$
Weight $2$
Character orbit 7056.a
Self dual yes
Analytic conductor $56.342$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.3424436662\)
Analytic rank: \(1\)
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.

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

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
4.27492
−3.27492
0 0 0 −4.27492 0 0 0 0 0
1.2 0 0 0 3.27492 0 0 0 0 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 7056.2.a.ch 2
3.b odd 2 1 2352.2.a.ba 2
4.b odd 2 1 3528.2.a.bd 2
7.b odd 2 1 7056.2.a.cu 2
7.d odd 6 2 1008.2.s.r 4
12.b even 2 1 1176.2.a.n 2
21.c even 2 1 2352.2.a.bf 2
21.g even 6 2 336.2.q.g 4
21.h odd 6 2 2352.2.q.bf 4
24.f even 2 1 9408.2.a.dj 2
24.h odd 2 1 9408.2.a.dw 2
28.d even 2 1 3528.2.a.bk 2
28.f even 6 2 504.2.s.i 4
28.g odd 6 2 3528.2.s.bk 4
84.h odd 2 1 1176.2.a.k 2
84.j odd 6 2 168.2.q.c 4
84.n even 6 2 1176.2.q.l 4
168.e odd 2 1 9408.2.a.ec 2
168.i even 2 1 9408.2.a.dp 2
168.ba even 6 2 1344.2.q.x 4
168.be odd 6 2 1344.2.q.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 84.j odd 6 2
336.2.q.g 4 21.g even 6 2
504.2.s.i 4 28.f even 6 2
1008.2.s.r 4 7.d odd 6 2
1176.2.a.k 2 84.h odd 2 1
1176.2.a.n 2 12.b even 2 1
1176.2.q.l 4 84.n even 6 2
1344.2.q.w 4 168.be odd 6 2
1344.2.q.x 4 168.ba even 6 2
2352.2.a.ba 2 3.b odd 2 1
2352.2.a.bf 2 21.c even 2 1
2352.2.q.bf 4 21.h odd 6 2
3528.2.a.bd 2 4.b odd 2 1
3528.2.a.bk 2 28.d even 2 1
3528.2.s.bk 4 28.g odd 6 2
7056.2.a.ch 2 1.a even 1 1 trivial
7056.2.a.cu 2 7.b odd 2 1
9408.2.a.dj 2 24.f even 2 1
9408.2.a.dp 2 168.i even 2 1
9408.2.a.dw 2 24.h odd 2 1
9408.2.a.ec 2 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7056))\):

\( T_{5}^{2} + T_{5} - 14 \)
\( T_{11}^{2} + T_{11} - 14 \)
\( T_{13}^{2} + 5 T_{13} - 8 \)
\( T_{17} + 4 \)
\( T_{23} - 4 \)