Properties

Label 1568.2.a.p
Level $1568$
Weight $2$
Character orbit 1568.a
Self dual yes
Analytic conductor $12.521$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \beta q^{5} -3 q^{9} +O(q^{10})\) \( q + 3 \beta q^{5} -3 q^{9} + 5 \beta q^{13} -3 \beta q^{17} + 13 q^{25} + 4 q^{29} + 12 q^{37} -\beta q^{41} -9 \beta q^{45} -14 q^{53} + \beta q^{61} + 30 q^{65} -5 \beta q^{73} + 9 q^{81} -18 q^{85} + 13 \beta q^{89} -13 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{9} + O(q^{10}) \) \( 2q - 6q^{9} + 26q^{25} + 8q^{29} + 24q^{37} - 28q^{53} + 60q^{65} + 18q^{81} - 36q^{85} + O(q^{100}) \)

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 −4.24264 0 0 0 −3.00000 0
1.2 0 0 0 4.24264 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.a.p 2
4.b odd 2 1 CM 1568.2.a.p 2
7.b odd 2 1 inner 1568.2.a.p 2
7.c even 3 2 1568.2.i.s 4
7.d odd 6 2 1568.2.i.s 4
8.b even 2 1 3136.2.a.bj 2
8.d odd 2 1 3136.2.a.bj 2
28.d even 2 1 inner 1568.2.a.p 2
28.f even 6 2 1568.2.i.s 4
28.g odd 6 2 1568.2.i.s 4
56.e even 2 1 3136.2.a.bj 2
56.h odd 2 1 3136.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.p 2 1.a even 1 1 trivial
1568.2.a.p 2 4.b odd 2 1 CM
1568.2.a.p 2 7.b odd 2 1 inner
1568.2.a.p 2 28.d even 2 1 inner
1568.2.i.s 4 7.c even 3 2
1568.2.i.s 4 7.d odd 6 2
1568.2.i.s 4 28.f even 6 2
1568.2.i.s 4 28.g odd 6 2
3136.2.a.bj 2 8.b even 2 1
3136.2.a.bj 2 8.d odd 2 1
3136.2.a.bj 2 56.e even 2 1
3136.2.a.bj 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\):

\( T_{3} \)
\( T_{5}^{2} - 18 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 3 T^{2} )^{2} \)
$5$ \( 1 - 8 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( 1 - 24 T^{2} + 169 T^{4} \)
$17$ \( 1 + 16 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 80 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 14 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( 1 + 120 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 + 96 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 160 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 144 T^{2} + 9409 T^{4} \)
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