Properties

Label 1764.1.q.b
Level 1764
Weight 1
Character orbit 1764.q
Analytic conductor 0.880
Analytic rank 0
Dimension 16
Projective image \(D_{8}\)
CM discriminant -4
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Projective image \(D_{8}\)
Projective field Galois closure of 8.0.38423222208.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{48}^{20} q^{2} -\zeta_{48}^{16} q^{4} + ( -\zeta_{48}^{5} - \zeta_{48}^{11} ) q^{5} -\zeta_{48}^{12} q^{8} +O(q^{10})\) \( q -\zeta_{48}^{20} q^{2} -\zeta_{48}^{16} q^{4} + ( -\zeta_{48}^{5} - \zeta_{48}^{11} ) q^{5} -\zeta_{48}^{12} q^{8} + ( -\zeta_{48} - \zeta_{48}^{7} ) q^{10} + ( \zeta_{48}^{3} + \zeta_{48}^{21} ) q^{13} -\zeta_{48}^{8} q^{16} + ( \zeta_{48} - \zeta_{48}^{7} ) q^{17} + ( -\zeta_{48}^{3} + \zeta_{48}^{21} ) q^{20} + ( \zeta_{48}^{10} + \zeta_{48}^{16} + \zeta_{48}^{22} ) q^{25} + ( \zeta_{48}^{17} - \zeta_{48}^{23} ) q^{26} -\zeta_{48}^{4} q^{32} + ( -\zeta_{48}^{3} - \zeta_{48}^{21} ) q^{34} + ( -\zeta_{48}^{2} - \zeta_{48}^{14} ) q^{37} + ( \zeta_{48}^{17} + \zeta_{48}^{23} ) q^{40} + ( -\zeta_{48}^{9} + \zeta_{48}^{15} ) q^{41} + ( \zeta_{48}^{6} + \zeta_{48}^{12} + \zeta_{48}^{18} ) q^{50} + ( \zeta_{48}^{13} - \zeta_{48}^{19} ) q^{52} + ( \zeta_{48}^{10} - \zeta_{48}^{22} ) q^{53} + ( -\zeta_{48}^{17} - \zeta_{48}^{23} ) q^{61} - q^{64} + ( \zeta_{48}^{2} - \zeta_{48}^{14} ) q^{65} + ( -\zeta_{48}^{17} + \zeta_{48}^{23} ) q^{68} + ( \zeta_{48} + \zeta_{48}^{7} ) q^{73} + ( -\zeta_{48}^{10} + \zeta_{48}^{22} ) q^{74} + ( \zeta_{48}^{13} + \zeta_{48}^{19} ) q^{80} + ( -\zeta_{48}^{5} + \zeta_{48}^{11} ) q^{82} + ( -\zeta_{48}^{6} + \zeta_{48}^{18} ) q^{85} + ( \zeta_{48}^{5} + \zeta_{48}^{11} ) q^{89} + ( \zeta_{48}^{9} + \zeta_{48}^{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{4} + O(q^{10}) \) \( 16q + 8q^{4} - 8q^{16} - 8q^{25} - 16q^{64} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{48}^{8}\)

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
−0.608761 + 0.793353i
−0.793353 0.608761i
0.793353 + 0.608761i
0.608761 0.793353i
0.991445 0.130526i
−0.130526 0.991445i
0.130526 + 0.991445i
−0.991445 + 0.130526i
−0.608761 0.793353i
−0.793353 + 0.608761i
0.793353 0.608761i
0.608761 + 0.793353i
0.991445 + 0.130526i
−0.130526 + 0.991445i
0.130526 0.991445i
−0.991445 0.130526i
−0.866025 0.500000i 0 0.500000 + 0.866025i −0.923880 + 1.60021i 0 0 1.00000i 0 1.60021 0.923880i
215.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.382683 + 0.662827i 0 0 1.00000i 0 0.662827 0.382683i
215.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.382683 0.662827i 0 0 1.00000i 0 −0.662827 + 0.382683i
215.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.923880 1.60021i 0 0 1.00000i 0 −1.60021 + 0.923880i
215.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.923880 + 1.60021i 0 0 1.00000i 0 −1.60021 + 0.923880i
215.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.382683 + 0.662827i 0 0 1.00000i 0 −0.662827 + 0.382683i
215.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.382683 0.662827i 0 0 1.00000i 0 0.662827 0.382683i
215.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.923880 1.60021i 0 0 1.00000i 0 1.60021 0.923880i
1403.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.923880 1.60021i 0 0 1.00000i 0 1.60021 + 0.923880i
1403.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.382683 0.662827i 0 0 1.00000i 0 0.662827 + 0.382683i
1403.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.382683 + 0.662827i 0 0 1.00000i 0 −0.662827 0.382683i
1403.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.923880 + 1.60021i 0 0 1.00000i 0 −1.60021 0.923880i
1403.5 0.866025 0.500000i 0 0.500000 0.866025i −0.923880 1.60021i 0 0 1.00000i 0 −1.60021 0.923880i
1403.6 0.866025 0.500000i 0 0.500000 0.866025i −0.382683 0.662827i 0 0 1.00000i 0 −0.662827 0.382683i
1403.7 0.866025 0.500000i 0 0.500000 0.866025i 0.382683 + 0.662827i 0 0 1.00000i 0 0.662827 + 0.382683i
1403.8 0.866025 0.500000i 0 0.500000 0.866025i 0.923880 + 1.60021i 0 0 1.00000i 0 1.60021 + 0.923880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1403.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
7.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.b 16
3.b odd 2 1 inner 1764.1.q.b 16
4.b odd 2 1 CM 1764.1.q.b 16
7.b odd 2 1 inner 1764.1.q.b 16
7.c even 3 1 1764.1.h.a 8
7.c even 3 1 inner 1764.1.q.b 16
7.d odd 6 1 1764.1.h.a 8
7.d odd 6 1 inner 1764.1.q.b 16
12.b even 2 1 inner 1764.1.q.b 16
21.c even 2 1 inner 1764.1.q.b 16
21.g even 6 1 1764.1.h.a 8
21.g even 6 1 inner 1764.1.q.b 16
21.h odd 6 1 1764.1.h.a 8
21.h odd 6 1 inner 1764.1.q.b 16
28.d even 2 1 inner 1764.1.q.b 16
28.f even 6 1 1764.1.h.a 8
28.f even 6 1 inner 1764.1.q.b 16
28.g odd 6 1 1764.1.h.a 8
28.g odd 6 1 inner 1764.1.q.b 16
84.h odd 2 1 inner 1764.1.q.b 16
84.j odd 6 1 1764.1.h.a 8
84.j odd 6 1 inner 1764.1.q.b 16
84.n even 6 1 1764.1.h.a 8
84.n even 6 1 inner 1764.1.q.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.h.a 8 7.c even 3 1
1764.1.h.a 8 7.d odd 6 1
1764.1.h.a 8 21.g even 6 1
1764.1.h.a 8 21.h odd 6 1
1764.1.h.a 8 28.f even 6 1
1764.1.h.a 8 28.g odd 6 1
1764.1.h.a 8 84.j odd 6 1
1764.1.h.a 8 84.n even 6 1
1764.1.q.b 16 1.a even 1 1 trivial
1764.1.q.b 16 3.b odd 2 1 inner
1764.1.q.b 16 4.b odd 2 1 CM
1764.1.q.b 16 7.b odd 2 1 inner
1764.1.q.b 16 7.c even 3 1 inner
1764.1.q.b 16 7.d odd 6 1 inner
1764.1.q.b 16 12.b even 2 1 inner
1764.1.q.b 16 21.c even 2 1 inner
1764.1.q.b 16 21.g even 6 1 inner
1764.1.q.b 16 21.h odd 6 1 inner
1764.1.q.b 16 28.d even 2 1 inner
1764.1.q.b 16 28.f even 6 1 inner
1764.1.q.b 16 28.g odd 6 1 inner
1764.1.q.b 16 84.h odd 2 1 inner
1764.1.q.b 16 84.j odd 6 1 inner
1764.1.q.b 16 84.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 4 T_{5}^{6} + 14 T_{5}^{4} + 8 T_{5}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ 1
$5$ \( ( 1 - T^{8} + T^{16} )^{2} \)
$7$ 1
$11$ \( ( 1 - T^{2} + T^{4} )^{8} \)
$13$ \( ( 1 + T^{8} )^{4} \)
$17$ \( ( 1 - T^{8} + T^{16} )^{2} \)
$19$ \( ( 1 - T^{2} + T^{4} )^{8} \)
$23$ \( ( 1 - T^{2} + T^{4} )^{8} \)
$29$ \( ( 1 - T )^{16}( 1 + T )^{16} \)
$31$ \( ( 1 - T^{2} + T^{4} )^{8} \)
$37$ \( ( 1 - T^{4} + T^{8} )^{4} \)
$41$ \( ( 1 + T^{8} )^{4} \)
$43$ \( ( 1 - T )^{16}( 1 + T )^{16} \)
$47$ \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
$53$ \( ( 1 - T^{4} + T^{8} )^{4} \)
$59$ \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
$61$ \( ( 1 - T^{8} + T^{16} )^{2} \)
$67$ \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
$71$ \( ( 1 + T^{2} )^{16} \)
$73$ \( ( 1 - T^{8} + T^{16} )^{2} \)
$79$ \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
$83$ \( ( 1 - T )^{16}( 1 + T )^{16} \)
$89$ \( ( 1 - T^{8} + T^{16} )^{2} \)
$97$ \( ( 1 + T^{8} )^{4} \)
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