Properties

Label 1728.1.e.a
Level 1728
Weight 1
Character orbit 1728.e
Self dual yes
Analytic conductor 0.862
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1728.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.862384341830\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.108.1
Artin image $D_6$
Artin field Galois closure of 6.2.1492992.4

$q$-expansion

\(f(q)\) \(=\) \( q - q^{7} + O(q^{10}) \) \( q - q^{7} + q^{13} + q^{19} + q^{25} + 2q^{31} + q^{37} - 2q^{43} + q^{61} + q^{67} - q^{73} - q^{79} - q^{91} - q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
0
0 0 0 0 0 −1.00000 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.1.e.a 1
3.b odd 2 1 CM 1728.1.e.a 1
4.b odd 2 1 1728.1.e.b 1
8.b even 2 1 108.1.c.a 1
8.d odd 2 1 432.1.e.a 1
12.b even 2 1 1728.1.e.b 1
24.f even 2 1 432.1.e.a 1
24.h odd 2 1 108.1.c.a 1
40.f even 2 1 2700.1.g.b 1
40.i odd 4 2 2700.1.b.b 2
72.j odd 6 2 324.1.g.a 2
72.l even 6 2 1296.1.q.a 2
72.n even 6 2 324.1.g.a 2
72.p odd 6 2 1296.1.q.a 2
120.i odd 2 1 2700.1.g.b 1
120.w even 4 2 2700.1.b.b 2
216.t even 18 6 2916.1.k.c 6
216.x odd 18 6 2916.1.k.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.1.c.a 1 8.b even 2 1
108.1.c.a 1 24.h odd 2 1
324.1.g.a 2 72.j odd 6 2
324.1.g.a 2 72.n even 6 2
432.1.e.a 1 8.d odd 2 1
432.1.e.a 1 24.f even 2 1
1296.1.q.a 2 72.l even 6 2
1296.1.q.a 2 72.p odd 6 2
1728.1.e.a 1 1.a even 1 1 trivial
1728.1.e.a 1 3.b odd 2 1 CM
1728.1.e.b 1 4.b odd 2 1
1728.1.e.b 1 12.b even 2 1
2700.1.b.b 2 40.i odd 4 2
2700.1.b.b 2 120.w even 4 2
2700.1.g.b 1 40.f even 2 1
2700.1.g.b 1 120.i odd 2 1
2916.1.k.c 6 216.t even 18 6
2916.1.k.c 6 216.x odd 18 6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1728, [\chi])\).

Hecke characteristic polynomials

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