Properties

Label 1764.1.g.d
Level $1764$
Weight $1$
Character orbit 1764.g
Self dual yes
Analytic conductor $0.880$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.880350682285\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.49392.1
Artin image $D_8$
Artin field Galois closure of 8.0.38423222208.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} -\beta q^{5} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} -\beta q^{5} + q^{8} -\beta q^{10} + \beta q^{13} + q^{16} + \beta q^{17} -\beta q^{20} + q^{25} + \beta q^{26} -2 q^{29} + q^{32} + \beta q^{34} -\beta q^{40} + \beta q^{41} + q^{50} + \beta q^{52} -2 q^{58} -\beta q^{61} + q^{64} -2 q^{65} + \beta q^{68} -\beta q^{73} -\beta q^{80} + \beta q^{82} -2 q^{85} -\beta q^{89} -\beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + 2q^{16} + 2q^{25} - 4q^{29} + 2q^{32} + 2q^{50} - 4q^{58} + 2q^{64} - 4q^{65} - 4q^{85} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
1.41421
−1.41421
1.00000 0 1.00000 −1.41421 0 0 1.00000 0 −1.41421
883.2 1.00000 0 1.00000 1.41421 0 0 1.00000 0 1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} - 2 \)
\( T_{11} \)
\( T_{29} + 2 \)