Properties

Label 7056.2.k.d
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: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 882)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} - 2 \beta_1 q^{11} + (\beta_{4} - 4 \beta_{2}) q^{13} - 3 \beta_{7} q^{17} - 4 \beta_{4} q^{19} - 4 \beta_{3} q^{23} + (\beta_{6} - 3) q^{25} + (4 \beta_{3} + 2 \beta_1) q^{29} + ( - 4 \beta_{4} - 4 \beta_{2}) q^{31} + ( - 3 \beta_{6} + 4) q^{37} + \beta_{7} q^{41} + ( - 4 \beta_{6} - 4) q^{43} - 4 \beta_{7} q^{47} + ( - 3 \beta_{3} + 2 \beta_1) q^{53} + 4 \beta_{2} q^{55} + (4 \beta_{7} + 4 \beta_{5}) q^{59} + (8 \beta_{4} + \beta_{2}) q^{61} + (3 \beta_{3} + 4 \beta_1) q^{65} - 4 q^{67} + ( - 4 \beta_{3} + 4 \beta_1) q^{71} - 5 \beta_{2} q^{73} + 4 \beta_{6} q^{79} + ( - 4 \beta_{7} + 8 \beta_{5}) q^{83} + 3 \beta_{6} q^{85} + ( - 8 \beta_{7} + 5 \beta_{5}) q^{89} + 4 \beta_{3} q^{95} + 5 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{25} + 32 q^{37} - 32 q^{43} - 32 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{5} + \zeta_{16}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{16}^{6} + \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{16}^{6} + \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{16}^{5} + \zeta_{16}^{3} \) Copy content Toggle raw display
\(\zeta_{16}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( \beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( ( -\beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

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
0.923880 + 0.382683i
0.923880 0.382683i
0.382683 0.923880i
0.382683 + 0.923880i
−0.382683 + 0.923880i
−0.382683 0.923880i
−0.923880 0.382683i
−0.923880 + 0.382683i
0 0 0 −1.84776 0 0 0 0 0
881.2 0 0 0 −1.84776 0 0 0 0 0
881.3 0 0 0 −0.765367 0 0 0 0 0
881.4 0 0 0 −0.765367 0 0 0 0 0
881.5 0 0 0 0.765367 0 0 0 0 0
881.6 0 0 0 0.765367 0 0 0 0 0
881.7 0 0 0 1.84776 0 0 0 0 0
881.8 0 0 0 1.84776 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.d 8
3.b odd 2 1 inner 7056.2.k.d 8
4.b odd 2 1 882.2.d.b 8
7.b odd 2 1 inner 7056.2.k.d 8
12.b even 2 1 882.2.d.b 8
21.c even 2 1 inner 7056.2.k.d 8
28.d even 2 1 882.2.d.b 8
28.f even 6 2 882.2.k.b 16
28.g odd 6 2 882.2.k.b 16
84.h odd 2 1 882.2.d.b 8
84.j odd 6 2 882.2.k.b 16
84.n even 6 2 882.2.k.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.d.b 8 4.b odd 2 1
882.2.d.b 8 12.b even 2 1
882.2.d.b 8 28.d even 2 1
882.2.d.b 8 84.h odd 2 1
882.2.k.b 16 28.f even 6 2
882.2.k.b 16 28.g odd 6 2
882.2.k.b 16 84.j odd 6 2
882.2.k.b 16 84.n even 6 2
7056.2.k.d 8 1.a even 1 1 trivial
7056.2.k.d 8 3.b odd 2 1 inner
7056.2.k.d 8 7.b odd 2 1 inner
7056.2.k.d 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 \(S_{2}^{\mathrm{new}}(7056, [\chi])\):

\( T_{5}^{4} - 4T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} + 68T_{13}^{2} + 1058 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 68 T^{2} + 1058)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 36 T^{2} + 162)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 96 T^{2} + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 128 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 2)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 16)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 68 T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 128 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 260 T^{2} + 12482)^{2} \) Copy content Toggle raw display
$67$ \( (T + 4)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 192 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 100 T^{2} + 1250)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 320 T^{2} + 25088)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 356 T^{2} + 3362)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 100 T^{2} + 1250)^{2} \) Copy content Toggle raw display
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