Properties

Label 1764.1.y.d
Level $1764$
Weight $1$
Character orbit 1764.y
Analytic conductor $0.880$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
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.y (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.880350682285\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
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 $C_3\times D_8$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} - q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} - q^{8} -\beta_{1} q^{10} + \beta_{3} q^{13} + \beta_{2} q^{16} -\beta_{1} q^{17} + \beta_{3} q^{20} + ( -1 - \beta_{2} ) q^{25} + ( \beta_{1} + \beta_{3} ) q^{26} + 2 q^{29} + ( 1 + \beta_{2} ) q^{32} + \beta_{3} q^{34} + ( \beta_{1} + \beta_{3} ) q^{40} -\beta_{3} q^{41} - q^{50} + \beta_{1} q^{52} -2 \beta_{2} q^{58} + ( \beta_{1} + \beta_{3} ) q^{61} + q^{64} + 2 \beta_{2} q^{65} + ( \beta_{1} + \beta_{3} ) q^{68} -\beta_{1} q^{73} + \beta_{1} q^{80} + ( -\beta_{1} - \beta_{3} ) q^{82} -2 q^{85} + ( -\beta_{1} - \beta_{3} ) q^{89} -\beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{4} - 4q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{4} - 4q^{8} - 2q^{16} - 2q^{25} + 8q^{29} + 2q^{32} - 4q^{50} + 4q^{58} + 4q^{64} - 4q^{65} - 8q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
667.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0.500000 0.866025i 0 −0.500000 0.866025i −0.707107 + 1.22474i 0 0 −1.00000 0 0.707107 + 1.22474i
667.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.707107 1.22474i 0 0 −1.00000 0 −0.707107 1.22474i
1243.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.707107 1.22474i 0 0 −1.00000 0 0.707107 1.22474i
1243.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.707107 + 1.22474i 0 0 −1.00000 0 −0.707107 + 1.22474i
\(n\): 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
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.y.d 4
3.b odd 2 1 1764.1.y.b 4
4.b odd 2 1 CM 1764.1.y.d 4
7.b odd 2 1 inner 1764.1.y.d 4
7.c even 3 1 1764.1.g.b 2
7.c even 3 1 inner 1764.1.y.d 4
7.d odd 6 1 1764.1.g.b 2
7.d odd 6 1 inner 1764.1.y.d 4
12.b even 2 1 1764.1.y.b 4
21.c even 2 1 1764.1.y.b 4
21.g even 6 1 1764.1.g.d yes 2
21.g even 6 1 1764.1.y.b 4
21.h odd 6 1 1764.1.g.d yes 2
21.h odd 6 1 1764.1.y.b 4
28.d even 2 1 inner 1764.1.y.d 4
28.f even 6 1 1764.1.g.b 2
28.f even 6 1 inner 1764.1.y.d 4
28.g odd 6 1 1764.1.g.b 2
28.g odd 6 1 inner 1764.1.y.d 4
84.h odd 2 1 1764.1.y.b 4
84.j odd 6 1 1764.1.g.d yes 2
84.j odd 6 1 1764.1.y.b 4
84.n even 6 1 1764.1.g.d yes 2
84.n even 6 1 1764.1.y.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.g.b 2 7.c even 3 1
1764.1.g.b 2 7.d odd 6 1
1764.1.g.b 2 28.f even 6 1
1764.1.g.b 2 28.g odd 6 1
1764.1.g.d yes 2 21.g even 6 1
1764.1.g.d yes 2 21.h odd 6 1
1764.1.g.d yes 2 84.j odd 6 1
1764.1.g.d yes 2 84.n even 6 1
1764.1.y.b 4 3.b odd 2 1
1764.1.y.b 4 12.b even 2 1
1764.1.y.b 4 21.c even 2 1
1764.1.y.b 4 21.g even 6 1
1764.1.y.b 4 21.h odd 6 1
1764.1.y.b 4 84.h odd 2 1
1764.1.y.b 4 84.j odd 6 1
1764.1.y.b 4 84.n even 6 1
1764.1.y.d 4 1.a even 1 1 trivial
1764.1.y.d 4 4.b odd 2 1 CM
1764.1.y.d 4 7.b odd 2 1 inner
1764.1.y.d 4 7.c even 3 1 inner
1764.1.y.d 4 7.d odd 6 1 inner
1764.1.y.d 4 28.d even 2 1 inner
1764.1.y.d 4 28.f even 6 1 inner
1764.1.y.d 4 28.g odd 6 1 inner

Hecke kernels

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

\( T_{5}^{4} + 2 T_{5}^{2} + 4 \)
\( T_{11} \)
\( T_{29} - 2 \)