Properties

Label 7056.2.k.f
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: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 126)
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 \( q - \beta_{5} q^{5} + \beta_{7} q^{11} - \beta_{3} q^{13} + \beta_{6} q^{17} + ( - \beta_{3} + 2 \beta_1) q^{19} + \beta_{2} q^{23} + (2 \beta_{4} + 4) q^{25} + ( - \beta_{7} - \beta_{2}) q^{29} + (3 \beta_{3} - \beta_1) q^{31} + (\beta_{4} + 4) q^{37} + 2 \beta_{6} q^{41} + ( - \beta_{4} - 4) q^{43} - \beta_{6} q^{47} + ( - \beta_{7} - \beta_{2}) q^{53} + (3 \beta_{3} + 3 \beta_1) q^{55} + (\beta_{6} - 3 \beta_{5}) q^{59} + (\beta_{3} + 2 \beta_1) q^{61} + ( - 2 \beta_{7} + \beta_{2}) q^{65} + 10 q^{67} + ( - 2 \beta_{7} + \beta_{2}) q^{71} + ( - 2 \beta_{3} - 2 \beta_1) q^{73} + (\beta_{4} + 7) q^{79} + ( - \beta_{6} + \beta_{5}) q^{83} - \beta_{4} q^{85} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{89} - \beta_{2} q^{95} + (2 \beta_{3} - 5 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{25} + 32 q^{37} - 32 q^{43} + 80 q^{67} + 56 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\zeta_{24}^{5} + 3\zeta_{24}^{3} - 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\zeta_{24}^{5} + 3\zeta_{24}^{3} + 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\zeta_{24}^{7} - 2\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 4\zeta_{24}^{2} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\zeta_{24}^{6} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( -\beta_{6} + 2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{6} + 2\beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{7} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{6} - 2\beta_{5} + \beta_{4} + 3\beta_{3} + \beta_{2} ) / 12 \) 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
0.965926 + 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
−0.258819 + 0.965926i
0 0 0 −4.18154 0 0 0 0 0
881.2 0 0 0 −4.18154 0 0 0 0 0
881.3 0 0 0 −0.717439 0 0 0 0 0
881.4 0 0 0 −0.717439 0 0 0 0 0
881.5 0 0 0 0.717439 0 0 0 0 0
881.6 0 0 0 0.717439 0 0 0 0 0
881.7 0 0 0 4.18154 0 0 0 0 0
881.8 0 0 0 4.18154 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.f 8
3.b odd 2 1 inner 7056.2.k.f 8
4.b odd 2 1 882.2.d.a 8
7.b odd 2 1 inner 7056.2.k.f 8
7.c even 3 1 1008.2.bt.c 8
7.d odd 6 1 1008.2.bt.c 8
12.b even 2 1 882.2.d.a 8
21.c even 2 1 inner 7056.2.k.f 8
21.g even 6 1 1008.2.bt.c 8
21.h odd 6 1 1008.2.bt.c 8
28.d even 2 1 882.2.d.a 8
28.f even 6 1 126.2.k.a 8
28.f even 6 1 882.2.k.a 8
28.g odd 6 1 126.2.k.a 8
28.g odd 6 1 882.2.k.a 8
84.h odd 2 1 882.2.d.a 8
84.j odd 6 1 126.2.k.a 8
84.j odd 6 1 882.2.k.a 8
84.n even 6 1 126.2.k.a 8
84.n even 6 1 882.2.k.a 8
140.p odd 6 1 3150.2.bf.a 8
140.s even 6 1 3150.2.bf.a 8
140.w even 12 1 3150.2.bp.b 8
140.w even 12 1 3150.2.bp.e 8
140.x odd 12 1 3150.2.bp.b 8
140.x odd 12 1 3150.2.bp.e 8
252.n even 6 1 1134.2.t.e 8
252.o even 6 1 1134.2.t.e 8
252.r odd 6 1 1134.2.l.f 8
252.u odd 6 1 1134.2.l.f 8
252.bb even 6 1 1134.2.l.f 8
252.bj even 6 1 1134.2.l.f 8
252.bl odd 6 1 1134.2.t.e 8
252.bn odd 6 1 1134.2.t.e 8
420.ba even 6 1 3150.2.bf.a 8
420.be odd 6 1 3150.2.bf.a 8
420.bp odd 12 1 3150.2.bp.b 8
420.bp odd 12 1 3150.2.bp.e 8
420.br even 12 1 3150.2.bp.b 8
420.br even 12 1 3150.2.bp.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.k.a 8 28.f even 6 1
126.2.k.a 8 28.g odd 6 1
126.2.k.a 8 84.j odd 6 1
126.2.k.a 8 84.n even 6 1
882.2.d.a 8 4.b odd 2 1
882.2.d.a 8 12.b even 2 1
882.2.d.a 8 28.d even 2 1
882.2.d.a 8 84.h odd 2 1
882.2.k.a 8 28.f even 6 1
882.2.k.a 8 28.g odd 6 1
882.2.k.a 8 84.j odd 6 1
882.2.k.a 8 84.n even 6 1
1008.2.bt.c 8 7.c even 3 1
1008.2.bt.c 8 7.d odd 6 1
1008.2.bt.c 8 21.g even 6 1
1008.2.bt.c 8 21.h odd 6 1
1134.2.l.f 8 252.r odd 6 1
1134.2.l.f 8 252.u odd 6 1
1134.2.l.f 8 252.bb even 6 1
1134.2.l.f 8 252.bj even 6 1
1134.2.t.e 8 252.n even 6 1
1134.2.t.e 8 252.o even 6 1
1134.2.t.e 8 252.bl odd 6 1
1134.2.t.e 8 252.bn odd 6 1
3150.2.bf.a 8 140.p odd 6 1
3150.2.bf.a 8 140.s even 6 1
3150.2.bf.a 8 420.ba even 6 1
3150.2.bf.a 8 420.be odd 6 1
3150.2.bp.b 8 140.w even 12 1
3150.2.bp.b 8 140.x odd 12 1
3150.2.bp.b 8 420.bp odd 12 1
3150.2.bp.b 8 420.br even 12 1
3150.2.bp.e 8 140.w even 12 1
3150.2.bp.e 8 140.x odd 12 1
3150.2.bp.e 8 420.bp odd 12 1
3150.2.bp.e 8 420.br even 12 1
7056.2.k.f 8 1.a even 1 1 trivial
7056.2.k.f 8 3.b odd 2 1 inner
7056.2.k.f 8 7.b odd 2 1 inner
7056.2.k.f 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 \(S_{2}^{\mathrm{new}}(7056, [\chi])\):

\( T_{5}^{4} - 18T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} + 6 \) 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
$5$ \( (T^{4} - 18 T^{2} + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 36 T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 36 T^{2} + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 54 T^{2} + 81)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 114 T^{2} + 2601)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 2)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 144 T^{2} + 576)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 36 T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 54 T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 198 T^{2} + 8649)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 36 T^{2} + 36)^{2} \) Copy content Toggle raw display
$67$ \( (T - 10)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 108 T^{2} + 324)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 72 T^{2} + 144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 31)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 54 T^{2} + 441)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 198 T^{2} + 2601)^{2} \) Copy content Toggle raw display
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