Properties

Label 1764.1.d.a
Level $1764$
Weight $1$
Character orbit 1764.d
Analytic conductor $0.880$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.880350682285\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 252)
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.7260624.1

$q$-expansion

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

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{13} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{19} + q^{25} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} + q^{37} + q^{43} - q^{67} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{73} - q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 2q^{25} + 2q^{37} + 2q^{43} - 2q^{67} - 2q^{79} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 0 0 0 0
685.2 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 \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.d.a 2
3.b odd 2 1 CM 1764.1.d.a 2
7.b odd 2 1 inner 1764.1.d.a 2
7.c even 3 1 252.1.z.a 2
7.c even 3 1 1764.1.z.a 2
7.d odd 6 1 252.1.z.a 2
7.d odd 6 1 1764.1.z.a 2
21.c even 2 1 inner 1764.1.d.a 2
21.g even 6 1 252.1.z.a 2
21.g even 6 1 1764.1.z.a 2
21.h odd 6 1 252.1.z.a 2
21.h odd 6 1 1764.1.z.a 2
28.f even 6 1 1008.1.cg.a 2
28.g odd 6 1 1008.1.cg.a 2
63.g even 3 1 2268.1.p.b 2
63.h even 3 1 2268.1.bd.a 2
63.i even 6 1 2268.1.bd.a 2
63.j odd 6 1 2268.1.bd.a 2
63.k odd 6 1 2268.1.p.b 2
63.n odd 6 1 2268.1.p.b 2
63.s even 6 1 2268.1.p.b 2
63.t odd 6 1 2268.1.bd.a 2
84.j odd 6 1 1008.1.cg.a 2
84.n even 6 1 1008.1.cg.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.1.z.a 2 7.c even 3 1
252.1.z.a 2 7.d odd 6 1
252.1.z.a 2 21.g even 6 1
252.1.z.a 2 21.h odd 6 1
1008.1.cg.a 2 28.f even 6 1
1008.1.cg.a 2 28.g odd 6 1
1008.1.cg.a 2 84.j odd 6 1
1008.1.cg.a 2 84.n even 6 1
1764.1.d.a 2 1.a even 1 1 trivial
1764.1.d.a 2 3.b odd 2 1 CM
1764.1.d.a 2 7.b odd 2 1 inner
1764.1.d.a 2 21.c even 2 1 inner
1764.1.z.a 2 7.c even 3 1
1764.1.z.a 2 7.d odd 6 1
1764.1.z.a 2 21.g even 6 1
1764.1.z.a 2 21.h odd 6 1
2268.1.p.b 2 63.g even 3 1
2268.1.p.b 2 63.k odd 6 1
2268.1.p.b 2 63.n odd 6 1
2268.1.p.b 2 63.s even 6 1
2268.1.bd.a 2 63.h even 3 1
2268.1.bd.a 2 63.i even 6 1
2268.1.bd.a 2 63.j odd 6 1
2268.1.bd.a 2 63.t odd 6 1

Hecke kernels

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