Properties

Label 7056.2.b.g
Level 7056
Weight 2
Character orbit 7056.b
Analytic conductor 56.342
Analytic rank 1
Dimension 2
CM discriminant -3
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.3424436662\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{43}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + ( 3 - 6 \zeta_{6} ) q^{13} - q^{19} + 5 q^{25} -11 q^{31} -11 q^{37} + ( 1 - 2 \zeta_{6} ) q^{43} + ( -4 + 8 \zeta_{6} ) q^{61} + ( -9 + 18 \zeta_{6} ) q^{67} + ( -1 + 2 \zeta_{6} ) q^{73} + ( 3 - 6 \zeta_{6} ) q^{79} + ( 8 - 16 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 2q^{19} + 10q^{25} - 22q^{31} - 22q^{37} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 0 0 0 0
1567.2 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
28.d even 2 1 inner
84.h odd 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.g 2
3.b odd 2 1 CM 7056.2.b.g 2
4.b odd 2 1 7056.2.b.h 2
7.b odd 2 1 7056.2.b.h 2
7.c even 3 1 1008.2.cs.j yes 2
7.d odd 6 1 1008.2.cs.g 2
12.b even 2 1 7056.2.b.h 2
21.c even 2 1 7056.2.b.h 2
21.g even 6 1 1008.2.cs.g 2
21.h odd 6 1 1008.2.cs.j yes 2
28.d even 2 1 inner 7056.2.b.g 2
28.f even 6 1 1008.2.cs.j yes 2
28.g odd 6 1 1008.2.cs.g 2
84.h odd 2 1 inner 7056.2.b.g 2
84.j odd 6 1 1008.2.cs.j yes 2
84.n even 6 1 1008.2.cs.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.cs.g 2 7.d odd 6 1
1008.2.cs.g 2 21.g even 6 1
1008.2.cs.g 2 28.g odd 6 1
1008.2.cs.g 2 84.n even 6 1
1008.2.cs.j yes 2 7.c even 3 1
1008.2.cs.j yes 2 21.h odd 6 1
1008.2.cs.j yes 2 28.f even 6 1
1008.2.cs.j yes 2 84.j odd 6 1
7056.2.b.g 2 1.a even 1 1 trivial
7056.2.b.g 2 3.b odd 2 1 CM
7056.2.b.g 2 28.d even 2 1 inner
7056.2.b.g 2 84.h odd 2 1 inner
7056.2.b.h 2 4.b odd 2 1
7056.2.b.h 2 7.b odd 2 1
7056.2.b.h 2 12.b even 2 1
7056.2.b.h 2 21.c even 2 1

Hecke kernels

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

\( T_{5} \)
\( T_{11} \)
\( T_{13}^{2} + 27 \)
\( T_{17} \)
\( T_{19} + 1 \)
\( T_{31} + 11 \)
\( T_{53} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 5 T^{2} )^{2} \)
$7$ 1
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \)
$17$ \( ( 1 - 17 T^{2} )^{2} \)
$19$ \( ( 1 + T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 11 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 11 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 41 T^{2} )^{2} \)
$43$ \( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \)
$67$ \( ( 1 - 5 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 - 89 T^{2} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \)
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