Properties

Label 1568.2.a.r
Level $1568$
Weight $2$
Character orbit 1568.a
Self dual yes
Analytic conductor $12.521$
Analytic rank $0$
Dimension $2$
CM no
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - q^{9} + 2 q^{11} + 2 \beta q^{13} + 3 \beta q^{17} - 3 \beta q^{19} + 8 q^{23} - 5 q^{25} - 4 \beta q^{27} + 6 q^{29} + 6 \beta q^{31} + 2 \beta q^{33} - 2 q^{37} + 4 q^{39} - 3 \beta q^{41} + 6 q^{43} - 2 \beta q^{47} + 6 q^{51} + 6 q^{53} - 6 q^{57} + 9 \beta q^{59} - 4 \beta q^{61} + 12 q^{67} + 8 \beta q^{69} - 4 q^{71} - \beta q^{73} - 5 \beta q^{75} + 12 q^{79} - 5 q^{81} - 7 \beta q^{83} + 6 \beta q^{87} + 3 \beta q^{89} + 12 q^{93} - 13 \beta q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 4 q^{11} + 16 q^{23} - 10 q^{25} + 12 q^{29} - 4 q^{37} + 8 q^{39} + 12 q^{43} + 12 q^{51} + 12 q^{53} - 12 q^{57} + 24 q^{67} - 8 q^{71} + 24 q^{79} - 10 q^{81} + 24 q^{93} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −1.41421 0 0 0 0 0 −1.00000 0
1.2 0 1.41421 0 0 0 0 0 −1.00000 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
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 1568.2.a.r yes 2
4.b odd 2 1 1568.2.a.q 2
7.b odd 2 1 inner 1568.2.a.r yes 2
7.c even 3 2 1568.2.i.q 4
7.d odd 6 2 1568.2.i.q 4
8.b even 2 1 3136.2.a.bl 2
8.d odd 2 1 3136.2.a.bo 2
28.d even 2 1 1568.2.a.q 2
28.f even 6 2 1568.2.i.r 4
28.g odd 6 2 1568.2.i.r 4
56.e even 2 1 3136.2.a.bo 2
56.h odd 2 1 3136.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.q 2 4.b odd 2 1
1568.2.a.q 2 28.d even 2 1
1568.2.a.r yes 2 1.a even 1 1 trivial
1568.2.a.r yes 2 7.b odd 2 1 inner
1568.2.i.q 4 7.c even 3 2
1568.2.i.q 4 7.d odd 6 2
1568.2.i.r 4 28.f even 6 2
1568.2.i.r 4 28.g odd 6 2
3136.2.a.bl 2 8.b even 2 1
3136.2.a.bl 2 56.h odd 2 1
3136.2.a.bo 2 8.d odd 2 1
3136.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\):

\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 8 \) Copy content Toggle raw display
$17$ \( T^{2} - 18 \) Copy content Toggle raw display
$19$ \( T^{2} - 18 \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 72 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 18 \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 162 \) Copy content Toggle raw display
$61$ \( T^{2} - 32 \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2 \) Copy content Toggle raw display
$79$ \( (T - 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 98 \) Copy content Toggle raw display
$89$ \( T^{2} - 18 \) Copy content Toggle raw display
$97$ \( T^{2} - 338 \) Copy content Toggle raw display
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