Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1764.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.880350682285\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q -\zeta_{16}^{4} q^{2} - q^{4} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{5} + \zeta_{16}^{4} q^{8} +O(q^{10})\) \( q -\zeta_{16}^{4} q^{2} - q^{4} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{5} + \zeta_{16}^{4} q^{8} + ( -\zeta_{16} - \zeta_{16}^{7} ) q^{10} + ( \zeta_{16}^{3} + \zeta_{16}^{5} ) q^{13} + q^{16} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{17} + ( -\zeta_{16}^{3} + \zeta_{16}^{5} ) q^{20} + ( 1 - \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{25} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{26} -\zeta_{16}^{4} q^{32} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{34} + ( -\zeta_{16}^{2} + \zeta_{16}^{6} ) q^{37} + ( \zeta_{16} + \zeta_{16}^{7} ) q^{40} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{41} + ( \zeta_{16}^{2} - \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{50} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{52} + ( -\zeta_{16}^{2} - \zeta_{16}^{6} ) q^{53} + ( -\zeta_{16} - \zeta_{16}^{7} ) q^{61} - q^{64} + ( \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{65} + ( -\zeta_{16} + \zeta_{16}^{7} ) q^{68} + ( \zeta_{16} + \zeta_{16}^{7} ) q^{73} + ( \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{74} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{80} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{82} + ( \zeta_{16}^{2} - \zeta_{16}^{6} ) q^{85} + ( -\zeta_{16}^{3} + \zeta_{16}^{5} ) q^{89} + ( -\zeta_{16} - \zeta_{16}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} + 8q^{16} + 8q^{25} - 8q^{64} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1763.1
0.382683 0.923880i
−0.923880 0.382683i
0.923880 + 0.382683i
−0.382683 + 0.923880i
0.382683 + 0.923880i
−0.923880 + 0.382683i
0.923880 0.382683i
−0.382683 0.923880i
1.00000i 0 −1.00000 −1.84776 0 0 1.00000i 0 1.84776i
1763.2 1.00000i 0 −1.00000 −0.765367 0 0 1.00000i 0 0.765367i
1763.3 1.00000i 0 −1.00000 0.765367 0 0 1.00000i 0 0.765367i
1763.4 1.00000i 0 −1.00000 1.84776 0 0 1.00000i 0 1.84776i
1763.5 1.00000i 0 −1.00000 −1.84776 0 0 1.00000i 0 1.84776i
1763.6 1.00000i 0 −1.00000 −0.765367 0 0 1.00000i 0 0.765367i
1763.7 1.00000i 0 −1.00000 0.765367 0 0 1.00000i 0 0.765367i
1763.8 1.00000i 0 −1.00000 1.84776 0 0 1.00000i 0 1.84776i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1763.8
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
12.b even 2 1 inner
21.c even 2 1 inner
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 1764.1.h.a 8
3.b odd 2 1 inner 1764.1.h.a 8
4.b odd 2 1 CM 1764.1.h.a 8
7.b odd 2 1 inner 1764.1.h.a 8
7.c even 3 2 1764.1.q.b 16
7.d odd 6 2 1764.1.q.b 16
12.b even 2 1 inner 1764.1.h.a 8
21.c even 2 1 inner 1764.1.h.a 8
21.g even 6 2 1764.1.q.b 16
21.h odd 6 2 1764.1.q.b 16
28.d even 2 1 inner 1764.1.h.a 8
28.f even 6 2 1764.1.q.b 16
28.g odd 6 2 1764.1.q.b 16
84.h odd 2 1 inner 1764.1.h.a 8
84.j odd 6 2 1764.1.q.b 16
84.n even 6 2 1764.1.q.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.h.a 8 1.a even 1 1 trivial
1764.1.h.a 8 3.b odd 2 1 inner
1764.1.h.a 8 4.b odd 2 1 CM
1764.1.h.a 8 7.b odd 2 1 inner
1764.1.h.a 8 12.b even 2 1 inner
1764.1.h.a 8 21.c even 2 1 inner
1764.1.h.a 8 28.d even 2 1 inner
1764.1.h.a 8 84.h odd 2 1 inner
1764.1.q.b 16 7.c even 3 2
1764.1.q.b 16 7.d odd 6 2
1764.1.q.b 16 21.g even 6 2
1764.1.q.b 16 21.h odd 6 2
1764.1.q.b 16 28.f even 6 2
1764.1.q.b 16 28.g odd 6 2
1764.1.q.b 16 84.j odd 6 2
1764.1.q.b 16 84.n even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

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