Properties

Label 1764.1.y.a
Level 1764
Weight 1
Character orbit 1764.y
Analytic conductor 0.880
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM discs -4, -7, 28
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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 196)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(i, \sqrt{7})\)
Artin image $C_3\times D_4$
Artin field Galois closure of 12.0.92254156521408.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} + q^{8} +O(q^{10})\) \( q -\zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} + q^{8} -\zeta_{6} q^{16} -\zeta_{6}^{2} q^{25} + 2 q^{29} + \zeta_{6}^{2} q^{32} + 2 \zeta_{6} q^{37} - q^{50} -2 \zeta_{6}^{2} q^{53} -2 \zeta_{6} q^{58} + q^{64} -2 \zeta_{6}^{2} q^{74} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} + 2q^{8} - q^{16} + q^{25} + 4q^{29} - q^{32} + 2q^{37} - 2q^{50} + 2q^{53} - 2q^{58} + 2q^{64} + 2q^{74} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 0 0
1243.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 0 1.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
28.d even 2 1 RM by \(\Q(\sqrt{7}) \)
7.c even 3 1 inner
7.d odd 6 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.a 2
3.b odd 2 1 196.1.g.a 2
4.b odd 2 1 CM 1764.1.y.a 2
7.b odd 2 1 CM 1764.1.y.a 2
7.c even 3 1 1764.1.g.a 1
7.c even 3 1 inner 1764.1.y.a 2
7.d odd 6 1 1764.1.g.a 1
7.d odd 6 1 inner 1764.1.y.a 2
12.b even 2 1 196.1.g.a 2
21.c even 2 1 196.1.g.a 2
21.g even 6 1 196.1.c.a 1
21.g even 6 1 196.1.g.a 2
21.h odd 6 1 196.1.c.a 1
21.h odd 6 1 196.1.g.a 2
24.f even 2 1 3136.1.r.a 2
24.h odd 2 1 3136.1.r.a 2
28.d even 2 1 RM 1764.1.y.a 2
28.f even 6 1 1764.1.g.a 1
28.f even 6 1 inner 1764.1.y.a 2
28.g odd 6 1 1764.1.g.a 1
28.g odd 6 1 inner 1764.1.y.a 2
84.h odd 2 1 196.1.g.a 2
84.j odd 6 1 196.1.c.a 1
84.j odd 6 1 196.1.g.a 2
84.n even 6 1 196.1.c.a 1
84.n even 6 1 196.1.g.a 2
168.e odd 2 1 3136.1.r.a 2
168.i even 2 1 3136.1.r.a 2
168.s odd 6 1 3136.1.d.a 1
168.s odd 6 1 3136.1.r.a 2
168.v even 6 1 3136.1.d.a 1
168.v even 6 1 3136.1.r.a 2
168.ba even 6 1 3136.1.d.a 1
168.ba even 6 1 3136.1.r.a 2
168.be odd 6 1 3136.1.d.a 1
168.be odd 6 1 3136.1.r.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.1.c.a 1 21.g even 6 1
196.1.c.a 1 21.h odd 6 1
196.1.c.a 1 84.j odd 6 1
196.1.c.a 1 84.n even 6 1
196.1.g.a 2 3.b odd 2 1
196.1.g.a 2 12.b even 2 1
196.1.g.a 2 21.c even 2 1
196.1.g.a 2 21.g even 6 1
196.1.g.a 2 21.h odd 6 1
196.1.g.a 2 84.h odd 2 1
196.1.g.a 2 84.j odd 6 1
196.1.g.a 2 84.n even 6 1
1764.1.g.a 1 7.c even 3 1
1764.1.g.a 1 7.d odd 6 1
1764.1.g.a 1 28.f even 6 1
1764.1.g.a 1 28.g odd 6 1
1764.1.y.a 2 1.a even 1 1 trivial
1764.1.y.a 2 4.b odd 2 1 CM
1764.1.y.a 2 7.b odd 2 1 CM
1764.1.y.a 2 7.c even 3 1 inner
1764.1.y.a 2 7.d odd 6 1 inner
1764.1.y.a 2 28.d even 2 1 RM
1764.1.y.a 2 28.f even 6 1 inner
1764.1.y.a 2 28.g odd 6 1 inner
3136.1.d.a 1 168.s odd 6 1
3136.1.d.a 1 168.v even 6 1
3136.1.d.a 1 168.ba even 6 1
3136.1.d.a 1 168.be odd 6 1
3136.1.r.a 2 24.f even 2 1
3136.1.r.a 2 24.h odd 2 1
3136.1.r.a 2 168.e odd 2 1
3136.1.r.a 2 168.i even 2 1
3136.1.r.a 2 168.s odd 6 1
3136.1.r.a 2 168.v even 6 1
3136.1.r.a 2 168.ba even 6 1
3136.1.r.a 2 168.be odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

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