Properties

Label 7056.2.k.g
Level $7056$
Weight $2$
Character orbit 7056.k
Analytic conductor $56.342$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 441)
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 \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{5} +O(q^{10})\) \( q + ( 2 \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{5} + ( 2 \zeta_{16}^{2} + 2 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{11} + ( -\zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{13} + ( -3 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 3 \zeta_{16}^{7} ) q^{17} + ( 2 \zeta_{16} - 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{19} -2 \zeta_{16}^{4} q^{23} + ( 5 - \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{25} + ( 2 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{29} + ( -2 \zeta_{16} + 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{31} + ( -4 - \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{37} + ( \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{41} + ( 4 - 6 \zeta_{16}^{2} + 6 \zeta_{16}^{6} ) q^{43} + ( -4 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{47} + ( 5 \zeta_{16}^{2} + 5 \zeta_{16}^{6} ) q^{53} + ( 4 \zeta_{16} + 6 \zeta_{16}^{3} + 6 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{55} + ( -4 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{59} + ( 4 \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{61} + ( -\zeta_{16}^{2} + 10 \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{65} + ( 6 \zeta_{16}^{2} - 6 \zeta_{16}^{6} ) q^{67} + ( 2 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{71} + ( 2 \zeta_{16} + 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{73} + ( 4 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{79} + ( 8 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 8 \zeta_{16}^{7} ) q^{83} + ( -8 - 9 \zeta_{16}^{2} + 9 \zeta_{16}^{6} ) q^{85} + ( -4 \zeta_{16} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{89} + ( 2 \zeta_{16}^{2} - 20 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{95} + ( -2 \zeta_{16} - 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 40q^{25} - 32q^{37} + 32q^{43} + 32q^{79} - 64q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1765\) \(4609\) \(6175\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−0.382683 + 0.923880i
−0.382683 0.923880i
−0.923880 + 0.382683i
−0.923880 0.382683i
0.923880 0.382683i
0.923880 + 0.382683i
0.382683 0.923880i
0.382683 + 0.923880i
0 0 0 −3.37849 0 0 0 0 0
881.2 0 0 0 −3.37849 0 0 0 0 0
881.3 0 0 0 −2.93015 0 0 0 0 0
881.4 0 0 0 −2.93015 0 0 0 0 0
881.5 0 0 0 2.93015 0 0 0 0 0
881.6 0 0 0 2.93015 0 0 0 0 0
881.7 0 0 0 3.37849 0 0 0 0 0
881.8 0 0 0 3.37849 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.k.g 8
3.b odd 2 1 inner 7056.2.k.g 8
4.b odd 2 1 441.2.c.b 8
7.b odd 2 1 inner 7056.2.k.g 8
12.b even 2 1 441.2.c.b 8
21.c even 2 1 inner 7056.2.k.g 8
28.d even 2 1 441.2.c.b 8
28.f even 6 2 441.2.p.c 16
28.g odd 6 2 441.2.p.c 16
84.h odd 2 1 441.2.c.b 8
84.j odd 6 2 441.2.p.c 16
84.n even 6 2 441.2.p.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.c.b 8 4.b odd 2 1
441.2.c.b 8 12.b even 2 1
441.2.c.b 8 28.d even 2 1
441.2.c.b 8 84.h odd 2 1
441.2.p.c 16 28.f even 6 2
441.2.p.c 16 28.g odd 6 2
441.2.p.c 16 84.j odd 6 2
441.2.p.c 16 84.n even 6 2
7056.2.k.g 8 1.a even 1 1 trivial
7056.2.k.g 8 3.b odd 2 1 inner
7056.2.k.g 8 7.b odd 2 1 inner
7056.2.k.g 8 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(7056, [\chi])\):

\( T_{5}^{4} - 20 T_{5}^{2} + 98 \)
\( T_{11}^{4} + 24 T_{11}^{2} + 16 \)
\( T_{13}^{4} + 20 T_{13}^{2} + 98 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 98 - 20 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 16 + 24 T^{2} + T^{4} )^{2} \)
$13$ \( ( 98 + 20 T^{2} + T^{4} )^{2} \)
$17$ \( ( 98 - 52 T^{2} + T^{4} )^{2} \)
$19$ \( ( 1568 + 80 T^{2} + T^{4} )^{2} \)
$23$ \( ( 4 + T^{2} )^{4} \)
$29$ \( ( 16 + 24 T^{2} + T^{4} )^{2} \)
$31$ \( ( 1568 + 80 T^{2} + T^{4} )^{2} \)
$37$ \( ( 14 + 8 T + T^{2} )^{4} \)
$41$ \( ( 98 - 68 T^{2} + T^{4} )^{2} \)
$43$ \( ( -56 - 8 T + T^{2} )^{4} \)
$47$ \( ( 1568 - 80 T^{2} + T^{4} )^{2} \)
$53$ \( ( 50 + T^{2} )^{4} \)
$59$ \( ( 1568 - 80 T^{2} + T^{4} )^{2} \)
$61$ \( ( 98 + 68 T^{2} + T^{4} )^{2} \)
$67$ \( ( -72 + T^{2} )^{4} \)
$71$ \( ( 16 + 24 T^{2} + T^{4} )^{2} \)
$73$ \( ( 98 + 52 T^{2} + T^{4} )^{2} \)
$79$ \( ( -16 - 8 T + T^{2} )^{4} \)
$83$ \( ( 25088 - 320 T^{2} + T^{4} )^{2} \)
$89$ \( ( 4802 - 164 T^{2} + T^{4} )^{2} \)
$97$ \( ( 98 + 52 T^{2} + T^{4} )^{2} \)
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