Properties

Label 1568.2.a.n
Level 1568
Weight 2
Character orbit 1568.a
Self dual yes
Analytic conductor 12.521
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{5} +O(q^{10})\) \( q + \beta q^{3} - q^{5} -3 \beta q^{11} -\beta q^{15} -5 q^{17} + \beta q^{19} + \beta q^{23} -4 q^{25} -3 \beta q^{27} + 8 q^{29} -5 \beta q^{31} -9 q^{33} -5 q^{37} -4 q^{41} -4 \beta q^{43} + 5 \beta q^{47} -5 \beta q^{51} - q^{53} + 3 \beta q^{55} + 3 q^{57} + \beta q^{59} -11 q^{61} + 7 \beta q^{67} + 3 q^{69} + 8 \beta q^{71} -15 q^{73} -4 \beta q^{75} + \beta q^{79} -9 q^{81} + 4 \beta q^{83} + 5 q^{85} + 8 \beta q^{87} -7 q^{89} -15 q^{93} -\beta q^{95} -12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} - 10q^{17} - 8q^{25} + 16q^{29} - 18q^{33} - 10q^{37} - 8q^{41} - 2q^{53} + 6q^{57} - 22q^{61} + 6q^{69} - 30q^{73} - 18q^{81} + 10q^{85} - 14q^{89} - 30q^{93} - 24q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
0 −1.73205 0 −1.00000 0 0 0 0 0
1.2 0 1.73205 0 −1.00000 0 0 0 0 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.a.n 2
4.b odd 2 1 inner 1568.2.a.n 2
7.b odd 2 1 1568.2.a.s 2
7.c even 3 2 1568.2.i.u 4
7.d odd 6 2 224.2.i.b 4
8.b even 2 1 3136.2.a.bu 2
8.d odd 2 1 3136.2.a.bu 2
21.g even 6 2 2016.2.s.r 4
28.d even 2 1 1568.2.a.s 2
28.f even 6 2 224.2.i.b 4
28.g odd 6 2 1568.2.i.u 4
56.e even 2 1 3136.2.a.bh 2
56.h odd 2 1 3136.2.a.bh 2
56.j odd 6 2 448.2.i.i 4
56.m even 6 2 448.2.i.i 4
84.j odd 6 2 2016.2.s.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.b 4 7.d odd 6 2
224.2.i.b 4 28.f even 6 2
448.2.i.i 4 56.j odd 6 2
448.2.i.i 4 56.m even 6 2
1568.2.a.n 2 1.a even 1 1 trivial
1568.2.a.n 2 4.b odd 2 1 inner
1568.2.a.s 2 7.b odd 2 1
1568.2.a.s 2 28.d even 2 1
1568.2.i.u 4 7.c even 3 2
1568.2.i.u 4 28.g odd 6 2
2016.2.s.r 4 21.g even 6 2
2016.2.s.r 4 84.j odd 6 2
3136.2.a.bh 2 56.e even 2 1
3136.2.a.bh 2 56.h odd 2 1
3136.2.a.bu 2 8.b even 2 1
3136.2.a.bu 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\):

\( T_{3}^{2} - 3 \)
\( T_{5} + 1 \)
\( T_{11}^{2} - 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} + 9 T^{4} \)
$5$ \( ( 1 + T + 5 T^{2} )^{2} \)
$7$ 1
$11$ \( 1 - 5 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( ( 1 + 5 T + 17 T^{2} )^{2} \)
$19$ \( 1 + 35 T^{2} + 361 T^{4} \)
$23$ \( 1 + 43 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 8 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 13 T^{2} + 961 T^{4} \)
$37$ \( ( 1 + 5 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 4 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 38 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 19 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 + T + 53 T^{2} )^{2} \)
$59$ \( 1 + 115 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 + 11 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 13 T^{2} + 4489 T^{4} \)
$71$ \( 1 - 50 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 + 15 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 155 T^{2} + 6241 T^{4} \)
$83$ \( 1 + 118 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 7 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 12 T + 97 T^{2} )^{2} \)
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