Properties

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

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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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 2 \beta q^{5} +O(q^{10})\) \( q + 2 \beta q^{5} -2 q^{11} -\beta q^{17} -5 \beta q^{19} -4 q^{23} + 3 q^{25} -2 q^{29} + 6 \beta q^{31} + 10 q^{37} -7 \beta q^{41} -2 q^{43} -2 \beta q^{47} + 2 q^{53} -4 \beta q^{55} + \beta q^{59} -2 \beta q^{61} -12 q^{67} -12 q^{71} + \beta q^{73} + 4 q^{79} -7 \beta q^{83} -4 q^{85} -5 \beta q^{89} -20 q^{95} -7 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + O(q^{10}) \) \( 2 q - 4 q^{11} - 8 q^{23} + 6 q^{25} - 4 q^{29} + 20 q^{37} - 4 q^{43} + 4 q^{53} - 24 q^{67} - 24 q^{71} + 8 q^{79} - 8 q^{85} - 40 q^{95} + O(q^{100}) \)

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
−1.41421
1.41421
0 0 0 −2.82843 0 0 0 0 0
1.2 0 0 0 2.82843 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.cl 2
3.b odd 2 1 784.2.a.l 2
4.b odd 2 1 882.2.a.n 2
7.b odd 2 1 inner 7056.2.a.cl 2
12.b even 2 1 98.2.a.b 2
21.c even 2 1 784.2.a.l 2
21.g even 6 2 784.2.i.m 4
21.h odd 6 2 784.2.i.m 4
24.f even 2 1 3136.2.a.bn 2
24.h odd 2 1 3136.2.a.bm 2
28.d even 2 1 882.2.a.n 2
28.f even 6 2 882.2.g.l 4
28.g odd 6 2 882.2.g.l 4
60.h even 2 1 2450.2.a.bj 2
60.l odd 4 2 2450.2.c.v 4
84.h odd 2 1 98.2.a.b 2
84.j odd 6 2 98.2.c.c 4
84.n even 6 2 98.2.c.c 4
168.e odd 2 1 3136.2.a.bn 2
168.i even 2 1 3136.2.a.bm 2
420.o odd 2 1 2450.2.a.bj 2
420.w even 4 2 2450.2.c.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.a.b 2 12.b even 2 1
98.2.a.b 2 84.h odd 2 1
98.2.c.c 4 84.j odd 6 2
98.2.c.c 4 84.n even 6 2
784.2.a.l 2 3.b odd 2 1
784.2.a.l 2 21.c even 2 1
784.2.i.m 4 21.g even 6 2
784.2.i.m 4 21.h odd 6 2
882.2.a.n 2 4.b odd 2 1
882.2.a.n 2 28.d even 2 1
882.2.g.l 4 28.f even 6 2
882.2.g.l 4 28.g odd 6 2
2450.2.a.bj 2 60.h even 2 1
2450.2.a.bj 2 420.o odd 2 1
2450.2.c.v 4 60.l odd 4 2
2450.2.c.v 4 420.w even 4 2
3136.2.a.bm 2 24.h odd 2 1
3136.2.a.bm 2 168.i even 2 1
3136.2.a.bn 2 24.f even 2 1
3136.2.a.bn 2 168.e odd 2 1
7056.2.a.cl 2 1.a even 1 1 trivial
7056.2.a.cl 2 7.b odd 2 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( -50 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( -72 + T^{2} \)
$37$ \( ( -10 + T )^{2} \)
$41$ \( -98 + T^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( -2 + T^{2} \)
$61$ \( -8 + T^{2} \)
$67$ \( ( 12 + T )^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( -2 + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( -98 + T^{2} \)
$89$ \( -50 + T^{2} \)
$97$ \( -98 + T^{2} \)
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