Properties

Label 1568.2.a.f
Level $1568$
Weight $2$
Character orbit 1568.a
Self dual yes
Analytic conductor $12.521$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{5} - 3q^{9} + O(q^{10}) \) \( q + 4q^{5} - 3q^{9} - 4q^{13} + 8q^{17} + 11q^{25} + 10q^{29} + 2q^{37} + 8q^{41} - 12q^{45} + 14q^{53} - 12q^{61} - 16q^{65} - 16q^{73} + 9q^{81} + 32q^{85} - 16q^{89} - 8q^{97} + O(q^{100}) \)

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
0
0 0 0 4.00000 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 \(\Q(\sqrt{-1}) \)

Twists

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

\( T_{3} \)
\( T_{5} - 4 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ 1
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 + 4 T + 13 T^{2} \)
$17$ \( 1 - 8 T + 17 T^{2} \)
$19$ \( 1 + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 10 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 - 8 T + 41 T^{2} \)
$43$ \( 1 + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 - 14 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 + 12 T + 61 T^{2} \)
$67$ \( 1 + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 16 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 + 16 T + 89 T^{2} \)
$97$ \( 1 + 8 T + 97 T^{2} \)
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