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: \(\Q(\zeta_{24})\)
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 primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{5} +O(q^{10})\) \( q + ( -\zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{5} -3 \zeta_{24}^{6} q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + ( -\zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{17} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{19} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{23} + ( 4 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{25} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{29} + ( 1 + 3 \zeta_{24} - 3 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{31} + ( 4 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{37} + ( -2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{41} + ( -4 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{43} + ( \zeta_{24} - 4 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{47} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{53} + ( -3 + 3 \zeta_{24} - 3 \zeta_{24}^{3} + 6 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{55} + ( -4 \zeta_{24} - 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{59} + ( -2 + \zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{61} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{65} + 10 q^{67} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{71} + ( 2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{73} + ( 7 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{79} + ( 2 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{83} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{85} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{89} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{95} + ( 5 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 32q^{25} + 32q^{37} - 32q^{43} + 80q^{67} + 56q^{79} + O(q^{100}) \)

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
0.965926 + 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 \(S_{2}^{\mathrm{new}}(7056, [\chi])\):

\( T_{5}^{4} - 18 T_{5}^{2} + 9 \)
\( T_{11}^{2} + 9 \)
\( T_{13}^{2} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} )^{2} \)
$7$ \( \)
$11$ \( ( 1 - 13 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 - 20 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 32 T^{2} + 546 T^{4} + 9248 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 40 T^{2} + 834 T^{4} - 14440 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 28 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 62 T^{2} + 1995 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 10 T^{2} + 1299 T^{4} - 9610 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 8 T + 72 T^{2} - 296 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 20 T^{2} - 1146 T^{4} + 33620 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 + 8 T + 84 T^{2} + 344 T^{3} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 152 T^{2} + 9906 T^{4} + 335768 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 158 T^{2} + 11211 T^{4} - 443822 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 38 T^{2} + 6171 T^{4} + 132278 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 208 T^{2} + 17970 T^{4} - 773968 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 10 T + 67 T^{2} )^{8} \)
$71$ \( ( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 220 T^{2} + 21606 T^{4} - 1172380 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 14 T + 189 T^{2} - 1106 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 278 T^{2} + 32811 T^{4} + 1915142 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 70 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 190 T^{2} + 20643 T^{4} - 1787710 T^{6} + 88529281 T^{8} )^{2} \)
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