# Properties

 Label 1911.1.bc.b Level $1911$ Weight $1$ Character orbit 1911.bc Analytic conductor $0.954$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1911.bc (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953713239142$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.24069811311.4

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} -\zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{3} -\zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{9} + q^{12} + \zeta_{6} q^{13} -\zeta_{6} q^{16} + ( 1 + \zeta_{6} ) q^{19} - q^{25} - q^{27} + \zeta_{6} q^{36} + ( 1 - \zeta_{6}^{2} ) q^{37} + \zeta_{6}^{2} q^{39} -\zeta_{6}^{2} q^{43} -\zeta_{6}^{2} q^{48} + q^{52} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{57} + \zeta_{6}^{2} q^{61} - q^{64} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{73} -\zeta_{6} q^{75} + ( 1 - \zeta_{6}^{2} ) q^{76} -2 q^{79} -\zeta_{6} q^{81} + ( -1 - \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{4} - q^{9} + O(q^{10})$$ $$2 q + q^{3} + q^{4} - q^{9} + 2 q^{12} + q^{13} - q^{16} + 3 q^{19} - 2 q^{25} - 2 q^{27} + q^{36} + 3 q^{37} - q^{39} + q^{43} + q^{48} + 2 q^{52} - q^{61} - 2 q^{64} - q^{75} + 3 q^{76} - 4 q^{79} - q^{81} - 3 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times$$.

 $$n$$ $$638$$ $$1471$$ $$1522$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{6}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
491.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 + 0.866025i 0.500000 0.866025i 0 0 0 0 −0.500000 + 0.866025i 0
1226.1 0 0.500000 0.866025i 0.500000 + 0.866025i 0 0 0 0 −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 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 $$\Q(\sqrt{-3})$$
13.e even 6 1 inner
39.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.bc.b 2
3.b odd 2 1 CM 1911.1.bc.b 2
7.b odd 2 1 1911.1.bc.a 2
7.c even 3 1 273.1.x.a 2
7.c even 3 1 273.1.bp.a yes 2
7.d odd 6 1 1911.1.x.a 2
7.d odd 6 1 1911.1.bp.a 2
13.e even 6 1 inner 1911.1.bc.b 2
21.c even 2 1 1911.1.bc.a 2
21.g even 6 1 1911.1.x.a 2
21.g even 6 1 1911.1.bp.a 2
21.h odd 6 1 273.1.x.a 2
21.h odd 6 1 273.1.bp.a yes 2
39.h odd 6 1 inner 1911.1.bc.b 2
91.g even 3 1 3549.1.w.a 2
91.g even 3 1 3549.1.bp.a 2
91.h even 3 1 3549.1.w.b 2
91.h even 3 1 3549.1.x.a 2
91.k even 6 1 273.1.x.a 2
91.k even 6 1 3549.1.w.a 2
91.l odd 6 1 1911.1.x.a 2
91.p odd 6 1 1911.1.bp.a 2
91.r even 6 1 3549.1.x.a 2
91.r even 6 1 3549.1.bp.a 2
91.t odd 6 1 1911.1.bc.a 2
91.u even 6 1 273.1.bp.a yes 2
91.u even 6 1 3549.1.w.b 2
91.x odd 12 2 3549.1.bk.e 4
91.x odd 12 2 3549.1.bm.b 4
91.z odd 12 2 3549.1.s.c 4
91.z odd 12 2 3549.1.bm.b 4
91.bd odd 12 2 3549.1.s.c 4
91.bd odd 12 2 3549.1.bk.e 4
273.s odd 6 1 3549.1.w.b 2
273.s odd 6 1 3549.1.x.a 2
273.u even 6 1 1911.1.bc.a 2
273.w odd 6 1 3549.1.x.a 2
273.w odd 6 1 3549.1.bp.a 2
273.x odd 6 1 273.1.bp.a yes 2
273.x odd 6 1 3549.1.w.b 2
273.y even 6 1 1911.1.bp.a 2
273.bm odd 6 1 3549.1.w.a 2
273.bm odd 6 1 3549.1.bp.a 2
273.bp odd 6 1 273.1.x.a 2
273.bp odd 6 1 3549.1.w.a 2
273.br even 6 1 1911.1.x.a 2
273.bv even 12 2 3549.1.bk.e 4
273.bv even 12 2 3549.1.bm.b 4
273.bw even 12 2 3549.1.s.c 4
273.bw even 12 2 3549.1.bk.e 4
273.cd even 12 2 3549.1.s.c 4
273.cd even 12 2 3549.1.bm.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.x.a 2 7.c even 3 1
273.1.x.a 2 21.h odd 6 1
273.1.x.a 2 91.k even 6 1
273.1.x.a 2 273.bp odd 6 1
273.1.bp.a yes 2 7.c even 3 1
273.1.bp.a yes 2 21.h odd 6 1
273.1.bp.a yes 2 91.u even 6 1
273.1.bp.a yes 2 273.x odd 6 1
1911.1.x.a 2 7.d odd 6 1
1911.1.x.a 2 21.g even 6 1
1911.1.x.a 2 91.l odd 6 1
1911.1.x.a 2 273.br even 6 1
1911.1.bc.a 2 7.b odd 2 1
1911.1.bc.a 2 21.c even 2 1
1911.1.bc.a 2 91.t odd 6 1
1911.1.bc.a 2 273.u even 6 1
1911.1.bc.b 2 1.a even 1 1 trivial
1911.1.bc.b 2 3.b odd 2 1 CM
1911.1.bc.b 2 13.e even 6 1 inner
1911.1.bc.b 2 39.h odd 6 1 inner
1911.1.bp.a 2 7.d odd 6 1
1911.1.bp.a 2 21.g even 6 1
1911.1.bp.a 2 91.p odd 6 1
1911.1.bp.a 2 273.y even 6 1
3549.1.s.c 4 91.z odd 12 2
3549.1.s.c 4 91.bd odd 12 2
3549.1.s.c 4 273.bw even 12 2
3549.1.s.c 4 273.cd even 12 2
3549.1.w.a 2 91.g even 3 1
3549.1.w.a 2 91.k even 6 1
3549.1.w.a 2 273.bm odd 6 1
3549.1.w.a 2 273.bp odd 6 1
3549.1.w.b 2 91.h even 3 1
3549.1.w.b 2 91.u even 6 1
3549.1.w.b 2 273.s odd 6 1
3549.1.w.b 2 273.x odd 6 1
3549.1.x.a 2 91.h even 3 1
3549.1.x.a 2 91.r even 6 1
3549.1.x.a 2 273.s odd 6 1
3549.1.x.a 2 273.w odd 6 1
3549.1.bk.e 4 91.x odd 12 2
3549.1.bk.e 4 91.bd odd 12 2
3549.1.bk.e 4 273.bv even 12 2
3549.1.bk.e 4 273.bw even 12 2
3549.1.bm.b 4 91.x odd 12 2
3549.1.bm.b 4 91.z odd 12 2
3549.1.bm.b 4 273.bv even 12 2
3549.1.bm.b 4 273.cd even 12 2
3549.1.bp.a 2 91.g even 3 1
3549.1.bp.a 2 91.r even 6 1
3549.1.bp.a 2 273.w odd 6 1
3549.1.bp.a 2 273.bm odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{19}^{2} - 3 T_{19} + 3$$ acting on $$S_{1}^{\mathrm{new}}(1911, [\chi])$$.

## Hecke characteristic polynomials

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