Properties

Label 1568.1.d.a
Level 1568
Weight 1
Character orbit 1568.d
Analytic conductor 0.783
Analytic rank 0
Dimension 2
Projective image \(A_{4}\)
CM/RM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1568.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.782533939809\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.3136.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -i q^{3} - q^{5} +O(q^{10})\) \( q -i q^{3} - q^{5} -i q^{11} + i q^{15} - q^{17} -i q^{19} -i q^{23} -i q^{27} + i q^{31} - q^{33} - q^{37} -i q^{47} + i q^{51} - q^{53} + i q^{55} - q^{57} -i q^{59} + q^{61} + i q^{67} - q^{69} + q^{73} -i q^{79} - q^{81} + q^{85} + q^{89} + q^{93} + i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} - 2q^{17} - 2q^{33} - 2q^{37} - 2q^{53} - 2q^{57} + 2q^{61} - 2q^{69} + 2q^{73} - 2q^{81} + 2q^{85} + 2q^{89} + 2q^{93} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
1.00000i
1.00000i
0 1.00000i 0 −1.00000 0 0 0 0 0
1471.2 0 1.00000i 0 −1.00000 0 0 0 0 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
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 1568.1.d.a 2
4.b odd 2 1 inner 1568.1.d.a 2
7.b odd 2 1 1568.1.d.b 2
7.c even 3 2 1568.1.r.a 4
7.d odd 6 2 224.1.r.a 4
8.b even 2 1 3136.1.d.d 2
8.d odd 2 1 3136.1.d.d 2
21.g even 6 2 2016.1.cd.a 4
28.d even 2 1 1568.1.d.b 2
28.f even 6 2 224.1.r.a 4
28.g odd 6 2 1568.1.r.a 4
56.e even 2 1 3136.1.d.b 2
56.h odd 2 1 3136.1.d.b 2
56.j odd 6 2 448.1.r.a 4
56.k odd 6 2 3136.1.r.b 4
56.m even 6 2 448.1.r.a 4
56.p even 6 2 3136.1.r.b 4
84.j odd 6 2 2016.1.cd.a 4
112.v even 12 2 1792.1.o.a 4
112.v even 12 2 1792.1.o.b 4
112.x odd 12 2 1792.1.o.a 4
112.x odd 12 2 1792.1.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.r.a 4 7.d odd 6 2
224.1.r.a 4 28.f even 6 2
448.1.r.a 4 56.j odd 6 2
448.1.r.a 4 56.m even 6 2
1568.1.d.a 2 1.a even 1 1 trivial
1568.1.d.a 2 4.b odd 2 1 inner
1568.1.d.b 2 7.b odd 2 1
1568.1.d.b 2 28.d even 2 1
1568.1.r.a 4 7.c even 3 2
1568.1.r.a 4 28.g odd 6 2
1792.1.o.a 4 112.v even 12 2
1792.1.o.a 4 112.x odd 12 2
1792.1.o.b 4 112.v even 12 2
1792.1.o.b 4 112.x odd 12 2
2016.1.cd.a 4 21.g even 6 2
2016.1.cd.a 4 84.j odd 6 2
3136.1.d.b 2 56.e even 2 1
3136.1.d.b 2 56.h odd 2 1
3136.1.d.d 2 8.b even 2 1
3136.1.d.d 2 8.d odd 2 1
3136.1.r.b 4 56.k odd 6 2
3136.1.r.b 4 56.p even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1568, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ 1
$11$ \( 1 - T^{2} + T^{4} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( ( 1 + T + T^{2} )^{2} \)
$19$ \( 1 - T^{2} + T^{4} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 1 + T^{2} )^{2} \)
$31$ \( 1 - T^{2} + T^{4} \)
$37$ \( ( 1 + T + T^{2} )^{2} \)
$41$ \( ( 1 + T^{2} )^{2} \)
$43$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$47$ \( 1 - T^{2} + T^{4} \)
$53$ \( ( 1 + T + T^{2} )^{2} \)
$59$ \( 1 - T^{2} + T^{4} \)
$61$ \( ( 1 - T + T^{2} )^{2} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$73$ \( ( 1 - T + T^{2} )^{2} \)
$79$ \( 1 - T^{2} + T^{4} \)
$83$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$89$ \( ( 1 - T + T^{2} )^{2} \)
$97$ \( ( 1 + T^{2} )^{2} \)
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