Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 2352)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{4} + \beta_{6} ) q^{5} +O(q^{10})\) \( q + ( -\beta_{4} + \beta_{6} ) q^{5} + ( \beta_{2} + \beta_{4} - \beta_{6} ) q^{11} + ( 2 \beta_{2} + \beta_{4} ) q^{13} + ( 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{17} + ( -2 \beta_{1} + \beta_{3} ) q^{19} + ( -\beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{23} + ( -3 - \beta_{1} + 2 \beta_{5} ) q^{25} + ( -2 + \beta_{1} + \beta_{3} ) q^{29} + ( 4 - 2 \beta_{1} - \beta_{5} ) q^{31} + ( \beta_{1} + 2 \beta_{3} ) q^{37} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{41} + ( -2 \beta_{2} + 6 \beta_{4} + \beta_{7} ) q^{43} + ( -2 + 2 \beta_{1} - \beta_{5} ) q^{47} + ( -2 - 2 \beta_{1} - 2 \beta_{5} ) q^{53} + ( 8 + 2 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{55} + ( 6 + 2 \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{59} + ( -\beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{61} + ( 2 + 3 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{65} + ( 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} ) q^{67} + ( 3 \beta_{2} + 3 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{71} + ( 2 \beta_{2} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{73} + ( -4 \beta_{2} - 4 \beta_{4} - \beta_{7} ) q^{79} + ( -2 \beta_{1} - 3 \beta_{3} - \beta_{5} ) q^{83} + ( -8 + \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{85} + ( 2 \beta_{2} + 7 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{89} + ( 8 \beta_{2} + 2 \beta_{4} + 3 \beta_{7} ) q^{95} + ( -6 \beta_{2} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{25} - 16q^{29} + 32q^{31} - 16q^{47} - 16q^{53} + 64q^{55} + 48q^{59} + 16q^{65} - 64q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 20 \)\()/14\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{7} + 7 \nu^{5} - 28 \nu^{3} + 2 \nu \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 7 \nu^{5} - 21 \nu^{3} + 22 \nu \)\()/7\)
\(\beta_{4}\)\(=\)\((\)\( -6 \nu^{7} + 21 \nu^{5} - 70 \nu^{3} + 6 \nu \)\()/14\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 14 \nu^{5} - 49 \nu^{3} + 52 \nu \)\()/7\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{4} - 12 \nu^{2} + 4 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{6} - 14 \nu^{4} + 56 \nu^{2} - 18 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{3} - 2 \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{6} - \beta_{1} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 3 \beta_{2}\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{7} + 4 \beta_{6} + 4 \beta_{1} - 6\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-3 \beta_{5} + 4 \beta_{4} + 7 \beta_{3} - 10 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(14 \beta_{1} - 20\)
\(\nu^{7}\)\(=\)\(-5 \beta_{5} - 7 \beta_{4} + 12 \beta_{3} + 17 \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
−0.662827 0.382683i
−1.60021 + 0.923880i
1.60021 0.923880i
0.662827 + 0.382683i
0.662827 0.382683i
1.60021 + 0.923880i
−1.60021 0.923880i
−0.662827 + 0.382683i
0 0 0 4.29725i 0 0 0 0 0
1567.2 0 0 0 3.21486i 0 0 0 0 0
1567.3 0 0 0 1.68412i 0 0 0 0 0
1567.4 0 0 0 0.601731i 0 0 0 0 0
1567.5 0 0 0 0.601731i 0 0 0 0 0
1567.6 0 0 0 1.68412i 0 0 0 0 0
1567.7 0 0 0 3.21486i 0 0 0 0 0
1567.8 0 0 0 4.29725i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d 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.b.x 8
3.b odd 2 1 2352.2.b.k 8
4.b odd 2 1 7056.2.b.w 8
7.b odd 2 1 7056.2.b.w 8
12.b even 2 1 2352.2.b.l yes 8
21.c even 2 1 2352.2.b.l yes 8
21.g even 6 1 2352.2.bl.o 8
21.g even 6 1 2352.2.bl.q 8
21.h odd 6 1 2352.2.bl.r 8
21.h odd 6 1 2352.2.bl.t 8
28.d even 2 1 inner 7056.2.b.x 8
84.h odd 2 1 2352.2.b.k 8
84.j odd 6 1 2352.2.bl.r 8
84.j odd 6 1 2352.2.bl.t 8
84.n even 6 1 2352.2.bl.o 8
84.n even 6 1 2352.2.bl.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.b.k 8 3.b odd 2 1
2352.2.b.k 8 84.h odd 2 1
2352.2.b.l yes 8 12.b even 2 1
2352.2.b.l yes 8 21.c even 2 1
2352.2.bl.o 8 21.g even 6 1
2352.2.bl.o 8 84.n even 6 1
2352.2.bl.q 8 21.g even 6 1
2352.2.bl.q 8 84.n even 6 1
2352.2.bl.r 8 21.h odd 6 1
2352.2.bl.r 8 84.j odd 6 1
2352.2.bl.t 8 21.h odd 6 1
2352.2.bl.t 8 84.j odd 6 1
7056.2.b.w 8 4.b odd 2 1
7056.2.b.w 8 7.b odd 2 1
7056.2.b.x 8 1.a even 1 1 trivial
7056.2.b.x 8 28.d 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 \(S_{2}^{\mathrm{new}}(7056, [\chi])\):

\( T_{5}^{8} + 32 T_{5}^{6} + 284 T_{5}^{4} + 640 T_{5}^{2} + 196 \)
\( T_{11}^{8} + 40 T_{11}^{6} + 392 T_{11}^{4} + 608 T_{11}^{2} + 16 \)
\( T_{13}^{4} + 20 T_{13}^{2} + 98 \)
\( T_{17}^{8} + 64 T_{17}^{6} + 860 T_{17}^{4} + 4160 T_{17}^{2} + 6724 \)
\( T_{19}^{4} - 40 T_{19}^{2} - 96 T_{19} - 56 \)
\( T_{31}^{4} - 16 T_{31}^{3} + 56 T_{31}^{2} + 160 T_{31} - 824 \)
\( T_{53}^{4} + 8 T_{53}^{3} - 88 T_{53}^{2} - 32 T_{53} + 784 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 196 + 640 T^{2} + 284 T^{4} + 32 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 16 + 608 T^{2} + 392 T^{4} + 40 T^{6} + T^{8} \)
$13$ \( ( 98 + 20 T^{2} + T^{4} )^{2} \)
$17$ \( 6724 + 4160 T^{2} + 860 T^{4} + 64 T^{6} + T^{8} \)
$19$ \( ( -56 - 96 T - 40 T^{2} + T^{4} )^{2} \)
$23$ \( 38416 + 20384 T^{2} + 2312 T^{4} + 88 T^{6} + T^{8} \)
$29$ \( ( 28 - 32 T - 4 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$31$ \( ( -824 + 160 T + 56 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$37$ \( ( 964 + 192 T - 100 T^{2} + T^{4} )^{2} \)
$41$ \( 6724 + 10912 T^{2} + 1820 T^{4} + 80 T^{6} + T^{8} \)
$43$ \( 20214016 + 1679104 T^{2} + 42848 T^{4} + 368 T^{6} + T^{8} \)
$47$ \( ( -392 - 224 T - 16 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$53$ \( ( 784 - 32 T - 88 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$59$ \( ( -8456 + 1440 T + 80 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$61$ \( 9604 + 292432 T^{2} + 23636 T^{4} + 296 T^{6} + T^{8} \)
$67$ \( 802816 + 327680 T^{2} + 18176 T^{4} + 256 T^{6} + T^{8} \)
$71$ \( 10265616 + 1314144 T^{2} + 39816 T^{4} + 360 T^{6} + T^{8} \)
$73$ \( 4866436 + 1008272 T^{2} + 33236 T^{4} + 328 T^{6} + T^{8} \)
$79$ \( 430336 + 592640 T^{2} + 24416 T^{4} + 304 T^{6} + T^{8} \)
$83$ \( ( 12256 - 192 T - 256 T^{2} + T^{4} )^{2} \)
$89$ \( 1766857156 + 34989632 T^{2} + 257180 T^{4} + 832 T^{6} + T^{8} \)
$97$ \( 325658116 + 10546576 T^{2} + 121364 T^{4} + 584 T^{6} + T^{8} \)
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