Properties

Label 1568.2.i.k
Level $1568$
Weight $2$
Character orbit 1568.i
Analytic conductor $12.521$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{9} + ( 4 - 4 \zeta_{6} ) q^{11} -4 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} + 6 \zeta_{6} q^{19} -8 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + 4 q^{27} + 2 q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} -8 \zeta_{6} q^{33} -10 \zeta_{6} q^{37} + ( -8 + 8 \zeta_{6} ) q^{39} -10 q^{41} + 4 q^{43} -4 \zeta_{6} q^{47} -4 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{53} + 12 q^{57} + ( -10 + 10 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} + ( 8 - 8 \zeta_{6} ) q^{67} -16 q^{69} + ( 6 - 6 \zeta_{6} ) q^{73} -10 \zeta_{6} q^{75} + 16 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} + 2 q^{83} + ( 4 - 4 \zeta_{6} ) q^{87} -18 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} -2 q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - q^{9} + 4q^{11} - 8q^{13} + 2q^{17} + 6q^{19} - 8q^{23} + 5q^{25} + 8q^{27} + 4q^{29} + 4q^{31} - 8q^{33} - 10q^{37} - 8q^{39} - 20q^{41} + 8q^{43} - 4q^{47} - 4q^{51} + 2q^{53} + 24q^{57} - 10q^{59} + 8q^{61} + 8q^{67} - 32q^{69} + 6q^{73} - 10q^{75} + 16q^{79} + 11q^{81} + 4q^{83} + 4q^{87} - 18q^{89} - 8q^{93} - 4q^{97} - 8q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 + 1.73205i 0 0 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 1.00000 1.73205i 0 0 0 0 0 −0.500000 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.i.k 2
4.b odd 2 1 1568.2.i.b 2
7.b odd 2 1 1568.2.i.c 2
7.c even 3 1 224.2.a.a 1
7.c even 3 1 inner 1568.2.i.k 2
7.d odd 6 1 1568.2.a.h 1
7.d odd 6 1 1568.2.i.c 2
21.h odd 6 1 2016.2.a.e 1
28.d even 2 1 1568.2.i.j 2
28.f even 6 1 1568.2.a.b 1
28.f even 6 1 1568.2.i.j 2
28.g odd 6 1 224.2.a.b yes 1
28.g odd 6 1 1568.2.i.b 2
35.j even 6 1 5600.2.a.t 1
56.j odd 6 1 3136.2.a.f 1
56.k odd 6 1 448.2.a.b 1
56.m even 6 1 3136.2.a.y 1
56.p even 6 1 448.2.a.f 1
84.n even 6 1 2016.2.a.g 1
112.u odd 12 2 1792.2.b.b 2
112.w even 12 2 1792.2.b.f 2
140.p odd 6 1 5600.2.a.c 1
168.s odd 6 1 4032.2.a.p 1
168.v even 6 1 4032.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 7.c even 3 1
224.2.a.b yes 1 28.g odd 6 1
448.2.a.b 1 56.k odd 6 1
448.2.a.f 1 56.p even 6 1
1568.2.a.b 1 28.f even 6 1
1568.2.a.h 1 7.d odd 6 1
1568.2.i.b 2 4.b odd 2 1
1568.2.i.b 2 28.g odd 6 1
1568.2.i.c 2 7.b odd 2 1
1568.2.i.c 2 7.d odd 6 1
1568.2.i.j 2 28.d even 2 1
1568.2.i.j 2 28.f even 6 1
1568.2.i.k 2 1.a even 1 1 trivial
1568.2.i.k 2 7.c even 3 1 inner
1792.2.b.b 2 112.u odd 12 2
1792.2.b.f 2 112.w even 12 2
2016.2.a.e 1 21.h odd 6 1
2016.2.a.g 1 84.n even 6 1
3136.2.a.f 1 56.j odd 6 1
3136.2.a.y 1 56.m even 6 1
4032.2.a.p 1 168.s odd 6 1
4032.2.a.z 1 168.v even 6 1
5600.2.a.c 1 140.p odd 6 1
5600.2.a.t 1 35.j even 6 1

Hecke kernels

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

\( T_{3}^{2} - 2 T_{3} + 4 \)
\( T_{5} \)
\( T_{11}^{2} - 4 T_{11} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 - 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 16 - 4 T + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( 36 - 6 T + T^{2} \)
$23$ \( 64 + 8 T + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( 100 + 10 T + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 16 + 4 T + T^{2} \)
$53$ \( 4 - 2 T + T^{2} \)
$59$ \( 100 + 10 T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 64 - 8 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 36 - 6 T + T^{2} \)
$79$ \( 256 - 16 T + T^{2} \)
$83$ \( ( -2 + T )^{2} \)
$89$ \( 324 + 18 T + T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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