Properties

Label 1568.3.g.f
Level 1568
Weight 3
Character orbit 1568.g
Analytic conductor 42.725
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -3 + 6 \zeta_{6} ) q^{5} -8 q^{9} +O(q^{10})\) \( q + q^{3} + ( -3 + 6 \zeta_{6} ) q^{5} -8 q^{9} -17 q^{11} + ( -8 + 16 \zeta_{6} ) q^{13} + ( -3 + 6 \zeta_{6} ) q^{15} + 25 q^{17} -7 q^{19} + ( 3 - 6 \zeta_{6} ) q^{23} -2 q^{25} -17 q^{27} + ( -8 + 16 \zeta_{6} ) q^{29} + ( 19 - 38 \zeta_{6} ) q^{31} -17 q^{33} + ( -5 + 10 \zeta_{6} ) q^{37} + ( -8 + 16 \zeta_{6} ) q^{39} -26 q^{41} -14 q^{43} + ( 24 - 48 \zeta_{6} ) q^{45} + ( 29 - 58 \zeta_{6} ) q^{47} + 25 q^{51} + ( 53 - 106 \zeta_{6} ) q^{53} + ( 51 - 102 \zeta_{6} ) q^{55} -7 q^{57} -55 q^{59} + ( 13 - 26 \zeta_{6} ) q^{61} -72 q^{65} -17 q^{67} + ( 3 - 6 \zeta_{6} ) q^{69} -119 q^{73} -2 q^{75} + ( 43 - 86 \zeta_{6} ) q^{79} + 55 q^{81} + 110 q^{83} + ( -75 + 150 \zeta_{6} ) q^{85} + ( -8 + 16 \zeta_{6} ) q^{87} -71 q^{89} + ( 19 - 38 \zeta_{6} ) q^{93} + ( 21 - 42 \zeta_{6} ) q^{95} + 22 q^{97} + 136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 16q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 16q^{9} - 34q^{11} + 50q^{17} - 14q^{19} - 4q^{25} - 34q^{27} - 34q^{33} - 52q^{41} - 28q^{43} + 50q^{51} - 14q^{57} - 110q^{59} - 144q^{65} - 34q^{67} - 238q^{73} - 4q^{75} + 110q^{81} + 220q^{83} - 142q^{89} + 44q^{97} + 272q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
687.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 0 5.19615i 0 0 0 −8.00000 0
687.2 0 1.00000 0 5.19615i 0 0 0 −8.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.3.g.f 2
4.b odd 2 1 392.3.g.d 2
7.b odd 2 1 1568.3.g.c 2
7.d odd 6 1 224.3.o.a 2
7.d odd 6 1 224.3.o.b 2
8.b even 2 1 392.3.g.d 2
8.d odd 2 1 inner 1568.3.g.f 2
28.d even 2 1 392.3.g.e 2
28.f even 6 1 56.3.k.a 2
28.f even 6 1 56.3.k.b yes 2
28.g odd 6 1 392.3.k.a 2
28.g odd 6 1 392.3.k.c 2
56.e even 2 1 1568.3.g.c 2
56.h odd 2 1 392.3.g.e 2
56.j odd 6 1 56.3.k.a 2
56.j odd 6 1 56.3.k.b yes 2
56.m even 6 1 224.3.o.a 2
56.m even 6 1 224.3.o.b 2
56.p even 6 1 392.3.k.a 2
56.p even 6 1 392.3.k.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.a 2 28.f even 6 1
56.3.k.a 2 56.j odd 6 1
56.3.k.b yes 2 28.f even 6 1
56.3.k.b yes 2 56.j odd 6 1
224.3.o.a 2 7.d odd 6 1
224.3.o.a 2 56.m even 6 1
224.3.o.b 2 7.d odd 6 1
224.3.o.b 2 56.m even 6 1
392.3.g.d 2 4.b odd 2 1
392.3.g.d 2 8.b even 2 1
392.3.g.e 2 28.d even 2 1
392.3.g.e 2 56.h odd 2 1
392.3.k.a 2 28.g odd 6 1
392.3.k.a 2 56.p even 6 1
392.3.k.c 2 28.g odd 6 1
392.3.k.c 2 56.p even 6 1
1568.3.g.c 2 7.b odd 2 1
1568.3.g.c 2 56.e even 2 1
1568.3.g.f 2 1.a even 1 1 trivial
1568.3.g.f 2 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{3}^{\mathrm{new}}(1568, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T + 9 T^{2} )^{2} \)
$5$ \( 1 - 23 T^{2} + 625 T^{4} \)
$7$ 1
$11$ \( ( 1 + 17 T + 121 T^{2} )^{2} \)
$13$ \( ( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} ) \)
$17$ \( ( 1 - 25 T + 289 T^{2} )^{2} \)
$19$ \( ( 1 + 7 T + 361 T^{2} )^{2} \)
$23$ \( 1 - 1031 T^{2} + 279841 T^{4} \)
$29$ \( 1 - 1490 T^{2} + 707281 T^{4} \)
$31$ \( 1 - 839 T^{2} + 923521 T^{4} \)
$37$ \( 1 - 2663 T^{2} + 1874161 T^{4} \)
$41$ \( ( 1 + 26 T + 1681 T^{2} )^{2} \)
$43$ \( ( 1 + 14 T + 1849 T^{2} )^{2} \)
$47$ \( 1 - 1895 T^{2} + 4879681 T^{4} \)
$53$ \( ( 1 - 53 T + 2809 T^{2} )( 1 + 53 T + 2809 T^{2} ) \)
$59$ \( ( 1 + 55 T + 3481 T^{2} )^{2} \)
$61$ \( 1 - 6935 T^{2} + 13845841 T^{4} \)
$67$ \( ( 1 + 17 T + 4489 T^{2} )^{2} \)
$71$ \( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \)
$73$ \( ( 1 + 119 T + 5329 T^{2} )^{2} \)
$79$ \( 1 - 6935 T^{2} + 38950081 T^{4} \)
$83$ \( ( 1 - 110 T + 6889 T^{2} )^{2} \)
$89$ \( ( 1 + 71 T + 7921 T^{2} )^{2} \)
$97$ \( ( 1 - 22 T + 9409 T^{2} )^{2} \)
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