Properties

Label 2916.1.k.c
Level 2916
Weight 1
Character orbit 2916.k
Analytic conductor 1.455
Analytic rank 0
Dimension 6
Projective image \(D_{3}\)
CM discriminant -3
Inner twists 12

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Newspace parameters

Level: \( N \) \(=\) \( 2916 = 2^{2} \cdot 3^{6} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2916.k (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.45527357684\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 $C_9\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{18}^{8} q^{7} +O(q^{10})\) \( q -\zeta_{18}^{8} q^{7} -\zeta_{18}^{4} q^{13} -\zeta_{18}^{6} q^{19} -\zeta_{18}^{5} q^{25} -2 \zeta_{18} q^{31} + \zeta_{18}^{3} q^{37} + 2 \zeta_{18}^{2} q^{43} -\zeta_{18}^{8} q^{61} -\zeta_{18}^{4} q^{67} -\zeta_{18}^{6} q^{73} + \zeta_{18}^{5} q^{79} -\zeta_{18}^{3} q^{91} -\zeta_{18}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q + 3q^{19} + 3q^{37} + 3q^{73} - 3q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2916\mathbb{Z}\right)^\times\).

\(n\) \(1459\) \(2189\)
\(\chi(n)\) \(1\) \(\zeta_{18}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 + 0.984808i
−0.173648 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
0 0 0 0 0 −0.766044 + 0.642788i 0 0 0
809.1 0 0 0 0 0 0.939693 0.342020i 0 0 0
1133.1 0 0 0 0 0 −0.173648 0.984808i 0 0 0
1781.1 0 0 0 0 0 −0.173648 + 0.984808i 0 0 0
2105.1 0 0 0 0 0 0.939693 + 0.342020i 0 0 0
2753.1 0 0 0 0 0 −0.766044 0.642788i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2753.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
9.c even 3 2 inner
9.d odd 6 2 inner
27.e even 9 3 inner
27.f odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2916.1.k.c 6
3.b odd 2 1 CM 2916.1.k.c 6
9.c even 3 2 inner 2916.1.k.c 6
9.d odd 6 2 inner 2916.1.k.c 6
27.e even 9 1 108.1.c.a 1
27.e even 9 2 324.1.g.a 2
27.e even 9 3 inner 2916.1.k.c 6
27.f odd 18 1 108.1.c.a 1
27.f odd 18 2 324.1.g.a 2
27.f odd 18 3 inner 2916.1.k.c 6
108.j odd 18 1 432.1.e.a 1
108.j odd 18 2 1296.1.q.a 2
108.l even 18 1 432.1.e.a 1
108.l even 18 2 1296.1.q.a 2
135.n odd 18 1 2700.1.g.b 1
135.p even 18 1 2700.1.g.b 1
135.q even 36 2 2700.1.b.b 2
135.r odd 36 2 2700.1.b.b 2
216.r odd 18 1 1728.1.e.b 1
216.t even 18 1 1728.1.e.a 1
216.v even 18 1 1728.1.e.b 1
216.x odd 18 1 1728.1.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.1.c.a 1 27.e even 9 1
108.1.c.a 1 27.f odd 18 1
324.1.g.a 2 27.e even 9 2
324.1.g.a 2 27.f odd 18 2
432.1.e.a 1 108.j odd 18 1
432.1.e.a 1 108.l even 18 1
1296.1.q.a 2 108.j odd 18 2
1296.1.q.a 2 108.l even 18 2
1728.1.e.a 1 216.t even 18 1
1728.1.e.a 1 216.x odd 18 1
1728.1.e.b 1 216.r odd 18 1
1728.1.e.b 1 216.v even 18 1
2700.1.b.b 2 135.q even 36 2
2700.1.b.b 2 135.r odd 36 2
2700.1.g.b 1 135.n odd 18 1
2700.1.g.b 1 135.p even 18 1
2916.1.k.c 6 1.a even 1 1 trivial
2916.1.k.c 6 3.b odd 2 1 CM
2916.1.k.c 6 9.c even 3 2 inner
2916.1.k.c 6 9.d odd 6 2 inner
2916.1.k.c 6 27.e even 9 3 inner
2916.1.k.c 6 27.f odd 18 3 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2916, [\chi])\):

\( T_{7}^{6} - T_{7}^{3} + 1 \)
\( T_{13}^{6} - T_{13}^{3} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$7$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$11$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$13$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$17$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$19$ \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
$23$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$29$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$31$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$37$ \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
$41$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$43$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$47$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$53$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$59$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$61$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$67$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$71$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$73$ \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
$79$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$83$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$89$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$97$ \( ( 1 + T^{3} + T^{6} )^{2} \)
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