Properties

Label 1764.1.g.a
Level 1764
Weight 1
Character orbit 1764.g
Self dual yes
Analytic conductor 0.880
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -4, -7, 28
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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 $D_4$
Artin field Galois closure of 4.0.12348.1

$q$-expansion

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

Character values

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

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

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
0
1.00000 0 1.00000 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 \(\Q(\sqrt{7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.g.a 1
3.b odd 2 1 196.1.c.a 1
4.b odd 2 1 CM 1764.1.g.a 1
7.b odd 2 1 CM 1764.1.g.a 1
7.c even 3 2 1764.1.y.a 2
7.d odd 6 2 1764.1.y.a 2
12.b even 2 1 196.1.c.a 1
21.c even 2 1 196.1.c.a 1
21.g even 6 2 196.1.g.a 2
21.h odd 6 2 196.1.g.a 2
24.f even 2 1 3136.1.d.a 1
24.h odd 2 1 3136.1.d.a 1
28.d even 2 1 RM 1764.1.g.a 1
28.f even 6 2 1764.1.y.a 2
28.g odd 6 2 1764.1.y.a 2
84.h odd 2 1 196.1.c.a 1
84.j odd 6 2 196.1.g.a 2
84.n even 6 2 196.1.g.a 2
168.e odd 2 1 3136.1.d.a 1
168.i even 2 1 3136.1.d.a 1
168.s odd 6 2 3136.1.r.a 2
168.v even 6 2 3136.1.r.a 2
168.ba even 6 2 3136.1.r.a 2
168.be odd 6 2 3136.1.r.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.1.c.a 1 3.b odd 2 1
196.1.c.a 1 12.b even 2 1
196.1.c.a 1 21.c even 2 1
196.1.c.a 1 84.h odd 2 1
196.1.g.a 2 21.g even 6 2
196.1.g.a 2 21.h odd 6 2
196.1.g.a 2 84.j odd 6 2
196.1.g.a 2 84.n even 6 2
1764.1.g.a 1 1.a even 1 1 trivial
1764.1.g.a 1 4.b odd 2 1 CM
1764.1.g.a 1 7.b odd 2 1 CM
1764.1.g.a 1 28.d even 2 1 RM
1764.1.y.a 2 7.c even 3 2
1764.1.y.a 2 7.d odd 6 2
1764.1.y.a 2 28.f even 6 2
1764.1.y.a 2 28.g odd 6 2
3136.1.d.a 1 24.f even 2 1
3136.1.d.a 1 24.h odd 2 1
3136.1.d.a 1 168.e odd 2 1
3136.1.d.a 1 168.i even 2 1
3136.1.r.a 2 168.s odd 6 2
3136.1.r.a 2 168.v even 6 2
3136.1.r.a 2 168.ba even 6 2
3136.1.r.a 2 168.be odd 6 2

Hecke kernels

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

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

Hecke characteristic polynomials

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