Properties

Label 1568.2.a.j
Level 1568
Weight 2
Character orbit 1568.a
Self dual yes
Analytic conductor 12.521
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} + ( -1 + 2 \beta ) q^{5} -2 \beta q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} + ( -1 + 2 \beta ) q^{5} -2 \beta q^{9} + ( -1 - \beta ) q^{11} -2 \beta q^{13} + ( 5 - 3 \beta ) q^{15} + ( 3 - 2 \beta ) q^{17} + ( -5 - \beta ) q^{19} + ( -1 - 3 \beta ) q^{23} + ( 4 - 4 \beta ) q^{25} + ( -1 - \beta ) q^{27} -2 \beta q^{29} + ( -7 + \beta ) q^{31} - q^{33} + ( 3 + 4 \beta ) q^{37} + ( -4 + 2 \beta ) q^{39} + ( 4 + 2 \beta ) q^{41} + ( 4 + 4 \beta ) q^{43} + ( -8 + 2 \beta ) q^{45} + ( -9 - \beta ) q^{47} + ( -7 + 5 \beta ) q^{51} - q^{53} + ( -3 - \beta ) q^{55} + ( 3 - 4 \beta ) q^{57} + ( -1 - 7 \beta ) q^{59} + ( -3 - 4 \beta ) q^{61} + ( -8 + 2 \beta ) q^{65} + ( -7 + 3 \beta ) q^{67} + ( -5 + 2 \beta ) q^{69} + ( 8 + 4 \beta ) q^{71} + ( 9 + 4 \beta ) q^{73} + ( -12 + 8 \beta ) q^{75} + ( -1 + 5 \beta ) q^{79} + ( -1 + 6 \beta ) q^{81} + ( -4 + 8 \beta ) q^{83} + ( -11 + 8 \beta ) q^{85} + ( -4 + 2 \beta ) q^{87} + 9 q^{89} + ( 9 - 8 \beta ) q^{93} + ( 1 - 9 \beta ) q^{95} + ( 4 - 2 \beta ) q^{97} + ( 4 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} - 2q^{11} + 10q^{15} + 6q^{17} - 10q^{19} - 2q^{23} + 8q^{25} - 2q^{27} - 14q^{31} - 2q^{33} + 6q^{37} - 8q^{39} + 8q^{41} + 8q^{43} - 16q^{45} - 18q^{47} - 14q^{51} - 2q^{53} - 6q^{55} + 6q^{57} - 2q^{59} - 6q^{61} - 16q^{65} - 14q^{67} - 10q^{69} + 16q^{71} + 18q^{73} - 24q^{75} - 2q^{79} - 2q^{81} - 8q^{83} - 22q^{85} - 8q^{87} + 18q^{89} + 18q^{93} + 2q^{95} + 8q^{97} + 8q^{99} + O(q^{100}) \)

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 −2.41421 0 −3.82843 0 0 0 2.82843 0
1.2 0 0.414214 0 1.82843 0 0 0 −2.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.a.j 2
4.b odd 2 1 1568.2.a.u 2
7.b odd 2 1 1568.2.a.w 2
7.c even 3 2 1568.2.i.x 4
7.d odd 6 2 224.2.i.a 4
8.b even 2 1 3136.2.a.bx 2
8.d odd 2 1 3136.2.a.be 2
21.g even 6 2 2016.2.s.q 4
28.d even 2 1 1568.2.a.l 2
28.f even 6 2 224.2.i.d yes 4
28.g odd 6 2 1568.2.i.o 4
56.e even 2 1 3136.2.a.bw 2
56.h odd 2 1 3136.2.a.bd 2
56.j odd 6 2 448.2.i.j 4
56.m even 6 2 448.2.i.g 4
84.j odd 6 2 2016.2.s.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.a 4 7.d odd 6 2
224.2.i.d yes 4 28.f even 6 2
448.2.i.g 4 56.m even 6 2
448.2.i.j 4 56.j odd 6 2
1568.2.a.j 2 1.a even 1 1 trivial
1568.2.a.l 2 28.d even 2 1
1568.2.a.u 2 4.b odd 2 1
1568.2.a.w 2 7.b odd 2 1
1568.2.i.o 4 28.g odd 6 2
1568.2.i.x 4 7.c even 3 2
2016.2.s.q 4 21.g even 6 2
2016.2.s.s 4 84.j odd 6 2
3136.2.a.bd 2 56.h odd 2 1
3136.2.a.be 2 8.d odd 2 1
3136.2.a.bw 2 56.e even 2 1
3136.2.a.bx 2 8.b even 2 1

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} - 1 \)
\( T_{5}^{2} + 2 T_{5} - 7 \)
\( T_{11}^{2} + 2 T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ \( 1 + 2 T + 3 T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 + 2 T + 21 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( 1 + 18 T^{2} + 169 T^{4} \)
$17$ \( 1 - 6 T + 35 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 10 T + 61 T^{2} + 190 T^{3} + 361 T^{4} \)
$23$ \( 1 + 2 T + 29 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( 1 + 50 T^{2} + 841 T^{4} \)
$31$ \( 1 + 14 T + 109 T^{2} + 434 T^{3} + 961 T^{4} \)
$37$ \( 1 - 6 T + 51 T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 8 T + 90 T^{2} - 328 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 18 T + 173 T^{2} + 846 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + T + 53 T^{2} )^{2} \)
$59$ \( 1 + 2 T + 21 T^{2} + 118 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 6 T + 99 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 14 T + 165 T^{2} + 938 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 16 T + 174 T^{2} - 1136 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 18 T + 195 T^{2} - 1314 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 2 T + 109 T^{2} + 158 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 9 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 8 T + 202 T^{2} - 776 T^{3} + 9409 T^{4} \)
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