# Properties

 Label 3249.1.ba.a Level $3249$ Weight $1$ Character orbit 3249.ba Analytic conductor $1.621$ Analytic rank $0$ Dimension $6$ Projective image $D_{6}$ CM discriminant -3 Inner twists $12$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3249 = 3^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3249.ba (of order $$18$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.62146222604$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 171) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.22284891.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{18}^{7} q^{4} -\zeta_{18}^{3} q^{7} +O(q^{10})$$ $$q -\zeta_{18}^{7} q^{4} -\zeta_{18}^{3} q^{7} + ( \zeta_{18} - \zeta_{18}^{7} ) q^{13} -\zeta_{18}^{5} q^{16} -\zeta_{18}^{4} q^{25} -\zeta_{18} q^{28} + ( -1 + \zeta_{18}^{6} ) q^{31} + ( \zeta_{18}^{3} + \zeta_{18}^{6} ) q^{37} -\zeta_{18}^{2} q^{43} + ( -\zeta_{18}^{5} - \zeta_{18}^{8} ) q^{52} -\zeta_{18}^{7} q^{61} -\zeta_{18}^{3} q^{64} + ( -\zeta_{18}^{4} - \zeta_{18}^{7} ) q^{67} + \zeta_{18}^{5} q^{73} + ( \zeta_{18}^{2} - \zeta_{18}^{8} ) q^{79} + ( -\zeta_{18} - \zeta_{18}^{4} ) q^{91} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 3q^{7} + O(q^{10})$$ $$6q - 3q^{7} - 9q^{31} - 3q^{64} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3249\mathbb{Z}\right)^\times$$.

 $$n$$ $$362$$ $$2890$$ $$\chi(n)$$ $$1$$ $$\zeta_{18}^{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −0.766044 − 0.642788i 0.939693 − 0.342020i −0.766044 + 0.642788i −0.173648 + 0.984808i −0.173648 − 0.984808i 0.939693 + 0.342020i
0 0 0.173648 0.984808i 0 0 −0.500000 + 0.866025i 0 0 0
262.1 0 0 0.766044 + 0.642788i 0 0 −0.500000 + 0.866025i 0 0 0
307.1 0 0 0.173648 + 0.984808i 0 0 −0.500000 0.866025i 0 0 0
694.1 0 0 −0.939693 + 0.342020i 0 0 −0.500000 + 0.866025i 0 0 0
838.1 0 0 −0.939693 0.342020i 0 0 −0.500000 0.866025i 0 0 0
3187.1 0 0 0.766044 0.642788i 0 0 −0.500000 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3187.1 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 $$\Q(\sqrt{-3})$$
19.c even 3 2 inner
19.f odd 18 3 inner
57.h odd 6 2 inner
57.j even 18 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3249.1.ba.a 6
3.b odd 2 1 CM 3249.1.ba.a 6
19.b odd 2 1 3249.1.ba.b 6
19.c even 3 2 inner 3249.1.ba.a 6
19.d odd 6 2 3249.1.ba.b 6
19.e even 9 1 171.1.p.a 2
19.e even 9 1 3249.1.c.a 2
19.e even 9 1 3249.1.p.b 2
19.e even 9 3 3249.1.ba.b 6
19.f odd 18 1 171.1.p.a 2
19.f odd 18 1 3249.1.c.a 2
19.f odd 18 1 3249.1.p.b 2
19.f odd 18 3 inner 3249.1.ba.a 6
57.d even 2 1 3249.1.ba.b 6
57.f even 6 2 3249.1.ba.b 6
57.h odd 6 2 inner 3249.1.ba.a 6
57.j even 18 1 171.1.p.a 2
57.j even 18 1 3249.1.c.a 2
57.j even 18 1 3249.1.p.b 2
57.j even 18 3 inner 3249.1.ba.a 6
57.l odd 18 1 171.1.p.a 2
57.l odd 18 1 3249.1.c.a 2
57.l odd 18 1 3249.1.p.b 2
57.l odd 18 3 3249.1.ba.b 6
76.k even 18 1 2736.1.cd.a 2
76.l odd 18 1 2736.1.cd.a 2
171.v even 9 1 1539.1.s.a 2
171.w even 9 1 1539.1.i.a 2
171.x even 18 1 1539.1.i.a 2
171.z odd 18 1 1539.1.s.a 2
171.bc odd 18 1 1539.1.i.a 2
171.bd even 18 1 1539.1.s.a 2
171.be odd 18 1 1539.1.s.a 2
171.bf odd 18 1 1539.1.i.a 2
228.u odd 18 1 2736.1.cd.a 2
228.v even 18 1 2736.1.cd.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.p.a 2 19.e even 9 1
171.1.p.a 2 19.f odd 18 1
171.1.p.a 2 57.j even 18 1
171.1.p.a 2 57.l odd 18 1
1539.1.i.a 2 171.w even 9 1
1539.1.i.a 2 171.x even 18 1
1539.1.i.a 2 171.bc odd 18 1
1539.1.i.a 2 171.bf odd 18 1
1539.1.s.a 2 171.v even 9 1
1539.1.s.a 2 171.z odd 18 1
1539.1.s.a 2 171.bd even 18 1
1539.1.s.a 2 171.be odd 18 1
2736.1.cd.a 2 76.k even 18 1
2736.1.cd.a 2 76.l odd 18 1
2736.1.cd.a 2 228.u odd 18 1
2736.1.cd.a 2 228.v even 18 1
3249.1.c.a 2 19.e even 9 1
3249.1.c.a 2 19.f odd 18 1
3249.1.c.a 2 57.j even 18 1
3249.1.c.a 2 57.l odd 18 1
3249.1.p.b 2 19.e even 9 1
3249.1.p.b 2 19.f odd 18 1
3249.1.p.b 2 57.j even 18 1
3249.1.p.b 2 57.l odd 18 1
3249.1.ba.a 6 1.a even 1 1 trivial
3249.1.ba.a 6 3.b odd 2 1 CM
3249.1.ba.a 6 19.c even 3 2 inner
3249.1.ba.a 6 19.f odd 18 3 inner
3249.1.ba.a 6 57.h odd 6 2 inner
3249.1.ba.a 6 57.j even 18 3 inner
3249.1.ba.b 6 19.b odd 2 1
3249.1.ba.b 6 19.d odd 6 2
3249.1.ba.b 6 19.e even 9 3
3249.1.ba.b 6 57.d even 2 1
3249.1.ba.b 6 57.f even 6 2
3249.1.ba.b 6 57.l odd 18 3

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3249, [\chi])$$:

 $$T_{7}^{2} + T_{7} + 1$$ $$T_{13}^{6} - 9 T_{13}^{3} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$( 1 + T + T^{2} )^{3}$$
$11$ $$T^{6}$$
$13$ $$27 - 9 T^{3} + T^{6}$$
$17$ $$T^{6}$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$( 3 + 3 T + T^{2} )^{3}$$
$37$ $$( 3 + T^{2} )^{3}$$
$41$ $$T^{6}$$
$43$ $$1 - T^{3} + T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$1 + T^{3} + T^{6}$$
$67$ $$27 + 9 T^{3} + T^{6}$$
$71$ $$T^{6}$$
$73$ $$1 - T^{3} + T^{6}$$
$79$ $$27 - 9 T^{3} + T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$