Properties

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

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.h (of order \(2\), degree \(1\), 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: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} - 3 \beta_{5} q^{11} + (\beta_{5} + 2 \beta_{3}) q^{13} - \beta_{4} q^{17} + (\beta_{7} - 3 \beta_{2}) q^{19} + (3 \beta_{5} + 2 \beta_{3}) q^{23} + ( - \beta_{6} + 1) q^{25} - \beta_{7} q^{29} + ( - \beta_{7} - 3 \beta_{2}) q^{31} - \beta_{6} q^{37} + (2 \beta_{4} + \beta_1) q^{41} + ( - 2 \beta_{4} + 4 \beta_1) q^{43} + \beta_{6} q^{47} + \beta_{2} q^{53} + (3 \beta_{7} - 3 \beta_{2}) q^{55} - 3 \beta_{6} q^{59} + (4 \beta_{5} + \beta_{3}) q^{61} + ( - 3 \beta_{7} + 7 \beta_{2}) q^{65} + ( - \beta_{4} + 5 \beta_1) q^{67} - 3 \beta_{5} q^{71} + ( - 4 \beta_{5} - \beta_{3}) q^{73} + (7 \beta_{4} - 5 \beta_1) q^{79} + (\beta_{6} - 6) q^{83} - 2 q^{85} + (2 \beta_{4} + 3 \beta_1) q^{89} + (6 \beta_{5} + 4 \beta_{3}) q^{95} + (2 \beta_{5} - 3 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{25} - 48 q^{83} - 16 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} + 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{5} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} ) / 4 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4607.1
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0 0 0 2.73205i 0 0 0 0 0
4607.2 0 0 0 2.73205i 0 0 0 0 0
4607.3 0 0 0 0.732051i 0 0 0 0 0
4607.4 0 0 0 0.732051i 0 0 0 0 0
4607.5 0 0 0 0.732051i 0 0 0 0 0
4607.6 0 0 0 0.732051i 0 0 0 0 0
4607.7 0 0 0 2.73205i 0 0 0 0 0
4607.8 0 0 0 2.73205i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4607.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
21.c even 2 1 inner
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.h.l 8
3.b odd 2 1 7056.2.h.m yes 8
4.b odd 2 1 7056.2.h.m yes 8
7.b odd 2 1 7056.2.h.m yes 8
12.b even 2 1 inner 7056.2.h.l 8
21.c even 2 1 inner 7056.2.h.l 8
28.d even 2 1 inner 7056.2.h.l 8
84.h odd 2 1 7056.2.h.m yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7056.2.h.l 8 1.a even 1 1 trivial
7056.2.h.l 8 12.b even 2 1 inner
7056.2.h.l 8 21.c even 2 1 inner
7056.2.h.l 8 28.d even 2 1 inner
7056.2.h.m yes 8 3.b odd 2 1
7056.2.h.m yes 8 4.b odd 2 1
7056.2.h.m yes 8 7.b odd 2 1
7056.2.h.m yes 8 84.h odd 2 1

Hecke kernels

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

\( T_{5}^{4} + 8T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 18 \) Copy content Toggle raw display
\( T_{13}^{4} - 52T_{13}^{2} + 484 \) Copy content Toggle raw display
\( T_{61}^{4} - 76T_{61}^{2} + 676 \) Copy content Toggle raw display
\( T_{83}^{2} + 12T_{83} + 24 \) 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
$5$ \( (T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 52 T^{2} + 484)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 84 T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 56 T^{2} + 676)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 76 T^{2} + 676)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 168 T^{2} + 144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 76 T^{2} + 676)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 312 T^{2} + 17424)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T + 24)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 152 T^{2} + 5476)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 124 T^{2} + 2116)^{2} \) Copy content Toggle raw display
show more
show less