Properties

Label 1568.2.a.m
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: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} -3 q^{5} + 4 q^{9} +O(q^{10})\) \( q + \beta q^{3} -3 q^{5} + 4 q^{9} -\beta q^{11} -4 q^{13} -3 \beta q^{15} + q^{17} -3 \beta q^{19} -\beta q^{23} + 4 q^{25} + \beta q^{27} -4 q^{29} -\beta q^{31} -7 q^{33} -5 q^{37} -4 \beta q^{39} + 8 q^{41} + 4 \beta q^{43} -12 q^{45} + \beta q^{47} + \beta q^{51} + 7 q^{53} + 3 \beta q^{55} -21 q^{57} + \beta q^{59} -5 q^{61} + 12 q^{65} + \beta q^{67} -7 q^{69} -9 q^{73} + 4 \beta q^{75} -\beta q^{79} -5 q^{81} -4 \beta q^{83} -3 q^{85} -4 \beta q^{87} -9 q^{89} -7 q^{93} + 9 \beta q^{95} -8 q^{97} -4 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{5} + 8q^{9} + O(q^{10}) \) \( 2q - 6q^{5} + 8q^{9} - 8q^{13} + 2q^{17} + 8q^{25} - 8q^{29} - 14q^{33} - 10q^{37} + 16q^{41} - 24q^{45} + 14q^{53} - 42q^{57} - 10q^{61} + 24q^{65} - 14q^{69} - 18q^{73} - 10q^{81} - 6q^{85} - 18q^{89} - 14q^{93} - 16q^{97} + O(q^{100}) \)

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
−2.64575
2.64575
0 −2.64575 0 −3.00000 0 0 0 4.00000 0
1.2 0 2.64575 0 −3.00000 0 0 0 4.00000 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.a.m 2
4.b odd 2 1 inner 1568.2.a.m 2
7.b odd 2 1 1568.2.a.t 2
7.c even 3 2 224.2.i.c 4
7.d odd 6 2 1568.2.i.p 4
8.b even 2 1 3136.2.a.bv 2
8.d odd 2 1 3136.2.a.bv 2
21.h odd 6 2 2016.2.s.o 4
28.d even 2 1 1568.2.a.t 2
28.f even 6 2 1568.2.i.p 4
28.g odd 6 2 224.2.i.c 4
56.e even 2 1 3136.2.a.bg 2
56.h odd 2 1 3136.2.a.bg 2
56.k odd 6 2 448.2.i.h 4
56.p even 6 2 448.2.i.h 4
84.n even 6 2 2016.2.s.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.c 4 7.c even 3 2
224.2.i.c 4 28.g odd 6 2
448.2.i.h 4 56.k odd 6 2
448.2.i.h 4 56.p even 6 2
1568.2.a.m 2 1.a even 1 1 trivial
1568.2.a.m 2 4.b odd 2 1 inner
1568.2.a.t 2 7.b odd 2 1
1568.2.a.t 2 28.d even 2 1
1568.2.i.p 4 7.d odd 6 2
1568.2.i.p 4 28.f even 6 2
2016.2.s.o 4 21.h odd 6 2
2016.2.s.o 4 84.n even 6 2
3136.2.a.bg 2 56.e even 2 1
3136.2.a.bg 2 56.h odd 2 1
3136.2.a.bv 2 8.b even 2 1
3136.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\):

\( T_{3}^{2} - 7 \)
\( T_{5} + 3 \)
\( T_{11}^{2} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T^{2} + 9 T^{4} \)
$5$ \( ( 1 + 3 T + 5 T^{2} )^{2} \)
$7$ 1
$11$ \( 1 + 15 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 4 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - T + 17 T^{2} )^{2} \)
$19$ \( 1 - 25 T^{2} + 361 T^{4} \)
$23$ \( 1 + 39 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 55 T^{2} + 961 T^{4} \)
$37$ \( ( 1 + 5 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 8 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 26 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 87 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 7 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 111 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 + 5 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 127 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 9 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 151 T^{2} + 6241 T^{4} \)
$83$ \( 1 + 54 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 9 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 8 T + 97 T^{2} )^{2} \)
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