Properties

Label 1764.1.q.a
Level 1764
Weight 1
Character orbit 1764.q
Analytic conductor 0.880
Analytic rank 0
Dimension 8
Projective image \(D_{4}\)
CM discriminant -7
Inner twists 16

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1764.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.880350682285\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.21168.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{9} q^{8} +O(q^{10})\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{9} q^{8} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{11} + \zeta_{24}^{4} q^{16} + ( -1 - \zeta_{24}^{6} ) q^{22} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{8} q^{25} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{29} -\zeta_{24}^{11} q^{32} + 2 \zeta_{24}^{6} q^{43} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{44} + ( -\zeta_{24}^{2} + \zeta_{24}^{8} ) q^{46} -\zeta_{24}^{3} q^{50} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{53} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{58} -\zeta_{24}^{6} q^{64} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{71} + 2 \zeta_{24} q^{86} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{88} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 4q^{16} - 8q^{22} + 4q^{25} - 4q^{46} - 4q^{58} - 4q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−0.258819 0.965926i
−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
−0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 0 −0.707107 + 0.707107i 0 0
215.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 0 0.707107 + 0.707107i 0 0
215.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 0 −0.707107 0.707107i 0 0
215.4 0.965926 0.258819i 0 0.866025 0.500000i 0 0 0 0.707107 0.707107i 0 0
1403.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 0 −0.707107 0.707107i 0 0
1403.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 0 0.707107 0.707107i 0 0
1403.3 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 0 −0.707107 + 0.707107i 0 0
1403.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 0 0.707107 + 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1403.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
84.h odd 2 1 inner
84.j odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.q.a 8
3.b odd 2 1 inner 1764.1.q.a 8
4.b odd 2 1 inner 1764.1.q.a 8
7.b odd 2 1 CM 1764.1.q.a 8
7.c even 3 1 252.1.h.a 4
7.c even 3 1 inner 1764.1.q.a 8
7.d odd 6 1 252.1.h.a 4
7.d odd 6 1 inner 1764.1.q.a 8
12.b even 2 1 inner 1764.1.q.a 8
21.c even 2 1 inner 1764.1.q.a 8
21.g even 6 1 252.1.h.a 4
21.g even 6 1 inner 1764.1.q.a 8
21.h odd 6 1 252.1.h.a 4
21.h odd 6 1 inner 1764.1.q.a 8
28.d even 2 1 inner 1764.1.q.a 8
28.f even 6 1 252.1.h.a 4
28.f even 6 1 inner 1764.1.q.a 8
28.g odd 6 1 252.1.h.a 4
28.g odd 6 1 inner 1764.1.q.a 8
63.g even 3 1 2268.1.s.g 8
63.h even 3 1 2268.1.s.g 8
63.i even 6 1 2268.1.s.g 8
63.j odd 6 1 2268.1.s.g 8
63.k odd 6 1 2268.1.s.g 8
63.n odd 6 1 2268.1.s.g 8
63.s even 6 1 2268.1.s.g 8
63.t odd 6 1 2268.1.s.g 8
84.h odd 2 1 inner 1764.1.q.a 8
84.j odd 6 1 252.1.h.a 4
84.j odd 6 1 inner 1764.1.q.a 8
84.n even 6 1 252.1.h.a 4
84.n even 6 1 inner 1764.1.q.a 8
252.n even 6 1 2268.1.s.g 8
252.o even 6 1 2268.1.s.g 8
252.r odd 6 1 2268.1.s.g 8
252.u odd 6 1 2268.1.s.g 8
252.bb even 6 1 2268.1.s.g 8
252.bj even 6 1 2268.1.s.g 8
252.bl odd 6 1 2268.1.s.g 8
252.bn odd 6 1 2268.1.s.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.1.h.a 4 7.c even 3 1
252.1.h.a 4 7.d odd 6 1
252.1.h.a 4 21.g even 6 1
252.1.h.a 4 21.h odd 6 1
252.1.h.a 4 28.f even 6 1
252.1.h.a 4 28.g odd 6 1
252.1.h.a 4 84.j odd 6 1
252.1.h.a 4 84.n even 6 1
1764.1.q.a 8 1.a even 1 1 trivial
1764.1.q.a 8 3.b odd 2 1 inner
1764.1.q.a 8 4.b odd 2 1 inner
1764.1.q.a 8 7.b odd 2 1 CM
1764.1.q.a 8 7.c even 3 1 inner
1764.1.q.a 8 7.d odd 6 1 inner
1764.1.q.a 8 12.b even 2 1 inner
1764.1.q.a 8 21.c even 2 1 inner
1764.1.q.a 8 21.g even 6 1 inner
1764.1.q.a 8 21.h odd 6 1 inner
1764.1.q.a 8 28.d even 2 1 inner
1764.1.q.a 8 28.f even 6 1 inner
1764.1.q.a 8 28.g odd 6 1 inner
1764.1.q.a 8 84.h odd 2 1 inner
1764.1.q.a 8 84.j odd 6 1 inner
1764.1.q.a 8 84.n even 6 1 inner
2268.1.s.g 8 63.g even 3 1
2268.1.s.g 8 63.h even 3 1
2268.1.s.g 8 63.i even 6 1
2268.1.s.g 8 63.j odd 6 1
2268.1.s.g 8 63.k odd 6 1
2268.1.s.g 8 63.n odd 6 1
2268.1.s.g 8 63.s even 6 1
2268.1.s.g 8 63.t odd 6 1
2268.1.s.g 8 252.n even 6 1
2268.1.s.g 8 252.o even 6 1
2268.1.s.g 8 252.r odd 6 1
2268.1.s.g 8 252.u odd 6 1
2268.1.s.g 8 252.bb even 6 1
2268.1.s.g 8 252.bj even 6 1
2268.1.s.g 8 252.bl odd 6 1
2268.1.s.g 8 252.bn odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{1}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ 1
$5$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$7$ 1
$11$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$13$ \( ( 1 - T )^{8}( 1 + T )^{8} \)
$17$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$19$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$23$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$29$ \( ( 1 + T^{4} )^{4} \)
$31$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$37$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$41$ \( ( 1 + T^{2} )^{8} \)
$43$ \( ( 1 + T^{2} )^{8} \)
$47$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$53$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$59$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$61$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$67$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$71$ \( ( 1 + T^{4} )^{4} \)
$73$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$79$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$83$ \( ( 1 - T )^{8}( 1 + T )^{8} \)
$89$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$97$ \( ( 1 - T )^{8}( 1 + T )^{8} \)
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