# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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