Properties

Label 1764.1.g.c
Level 1764
Weight 1
Character orbit 1764.g
Analytic conductor 0.880
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM discs -7, -84, 12
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.880350682285\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{3}, \sqrt{-7})\)
Artin image $D_4:C_2$
Artin field Galois closure of 8.0.1372257936.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} + i q^{8} + 2 i q^{11} + q^{16} + 2 q^{22} + 2 i q^{23} - q^{25} -i q^{32} + 2 q^{37} -2 i q^{44} + 2 q^{46} + i q^{50} - q^{64} -2 i q^{71} -2 i q^{74} -2 q^{88} -2 i q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{16} + 4q^{22} - 2q^{25} + 4q^{37} + 4q^{46} - 2q^{64} - 4q^{88} + O(q^{100}) \)

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
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 0 0
883.2 1.00000i 0 −1.00000 0 0 0 1.00000i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
12.b even 2 1 RM by \(\Q(\sqrt{3}) \)
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
21.c even 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.c 2
3.b odd 2 1 inner 1764.1.g.c 2
4.b odd 2 1 inner 1764.1.g.c 2
7.b odd 2 1 CM 1764.1.g.c 2
7.c even 3 2 1764.1.y.c 4
7.d odd 6 2 1764.1.y.c 4
12.b even 2 1 RM 1764.1.g.c 2
21.c even 2 1 inner 1764.1.g.c 2
21.g even 6 2 1764.1.y.c 4
21.h odd 6 2 1764.1.y.c 4
28.d even 2 1 inner 1764.1.g.c 2
28.f even 6 2 1764.1.y.c 4
28.g odd 6 2 1764.1.y.c 4
84.h odd 2 1 CM 1764.1.g.c 2
84.j odd 6 2 1764.1.y.c 4
84.n even 6 2 1764.1.y.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.g.c 2 1.a even 1 1 trivial
1764.1.g.c 2 3.b odd 2 1 inner
1764.1.g.c 2 4.b odd 2 1 inner
1764.1.g.c 2 7.b odd 2 1 CM
1764.1.g.c 2 12.b even 2 1 RM
1764.1.g.c 2 21.c even 2 1 inner
1764.1.g.c 2 28.d even 2 1 inner
1764.1.g.c 2 84.h odd 2 1 CM
1764.1.y.c 4 7.c even 3 2
1764.1.y.c 4 7.d odd 6 2
1764.1.y.c 4 21.g even 6 2
1764.1.y.c 4 21.h odd 6 2
1764.1.y.c 4 28.f even 6 2
1764.1.y.c 4 28.g odd 6 2
1764.1.y.c 4 84.j odd 6 2
1764.1.y.c 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 \(S_{1}^{\mathrm{new}}(1764, [\chi])\):

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

Hecke characteristic polynomials

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