# Properties

 Label 2268.1.bc.a Level $2268$ Weight $1$ Character orbit 2268.bc Analytic conductor $1.132$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ RM discriminant 21 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2268.bc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.13187944865$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 756) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.4000752.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6}^{2} ) q^{5} -\zeta_{6}^{2} q^{7} +O(q^{10})$$ $$q + ( -1 + \zeta_{6}^{2} ) q^{5} -\zeta_{6}^{2} q^{7} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{17} + ( 1 - \zeta_{6} - \zeta_{6}^{2} ) q^{25} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{35} + q^{37} + ( 1 - \zeta_{6}^{2} ) q^{41} + \zeta_{6}^{2} q^{43} + ( 1 + \zeta_{6} ) q^{47} -\zeta_{6} q^{49} + ( 1 - \zeta_{6}^{2} ) q^{59} -2 \zeta_{6} q^{67} -\zeta_{6}^{2} q^{79} + ( -1 - \zeta_{6} ) q^{83} + ( 1 + 2 \zeta_{6} + \zeta_{6}^{2} ) q^{85} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{5} + q^{7} + O(q^{10})$$ $$2q - 3q^{5} + q^{7} + 2q^{25} + 2q^{37} + 3q^{41} - q^{43} + 3q^{47} - q^{49} + 3q^{59} - 2q^{67} + q^{79} - 3q^{83} + 3q^{85} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.50000 0.866025i 0 0.500000 + 0.866025i 0 0 0
1189.1 0 0 0 −1.50000 + 0.866025i 0 0.500000 0.866025i 0 0 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
21.c even 2 1 RM by $$\Q(\sqrt{21})$$
9.d odd 6 1 inner
63.l odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.1.bc.a 2
3.b odd 2 1 2268.1.bc.d 2
7.b odd 2 1 2268.1.bc.d 2
9.c even 3 1 756.1.d.a 2
9.c even 3 1 2268.1.bc.d 2
9.d odd 6 1 756.1.d.a 2
9.d odd 6 1 inner 2268.1.bc.a 2
21.c even 2 1 RM 2268.1.bc.a 2
36.f odd 6 1 3024.1.f.b 2
36.h even 6 1 3024.1.f.b 2
63.l odd 6 1 756.1.d.a 2
63.l odd 6 1 inner 2268.1.bc.a 2
63.o even 6 1 756.1.d.a 2
63.o even 6 1 2268.1.bc.d 2
252.s odd 6 1 3024.1.f.b 2
252.bi even 6 1 3024.1.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.1.d.a 2 9.c even 3 1
756.1.d.a 2 9.d odd 6 1
756.1.d.a 2 63.l odd 6 1
756.1.d.a 2 63.o even 6 1
2268.1.bc.a 2 1.a even 1 1 trivial
2268.1.bc.a 2 9.d odd 6 1 inner
2268.1.bc.a 2 21.c even 2 1 RM
2268.1.bc.a 2 63.l odd 6 1 inner
2268.1.bc.d 2 3.b odd 2 1
2268.1.bc.d 2 7.b odd 2 1
2268.1.bc.d 2 9.c even 3 1
2268.1.bc.d 2 63.o even 6 1
3024.1.f.b 2 36.f odd 6 1
3024.1.f.b 2 36.h even 6 1
3024.1.f.b 2 252.s odd 6 1
3024.1.f.b 2 252.bi even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2268, [\chi])$$:

 $$T_{5}^{2} + 3 T_{5} + 3$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$3 + 3 T + T^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$3 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$( -1 + T )^{2}$$
$41$ $$3 - 3 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$3 - 3 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$3 - 3 T + T^{2}$$
$61$ $$T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$3 + 3 T + T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$