# Properties

 Label 2268.2.x.a Level $2268$ Weight $2$ Character orbit 2268.x Analytic conductor $18.110$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$18.1100711784$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 756) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -3 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} + ( -3 - 3 \zeta_{6} ) q^{11} + ( 4 - 2 \zeta_{6} ) q^{13} -6 q^{17} + ( -1 + 2 \zeta_{6} ) q^{19} + ( -6 + 3 \zeta_{6} ) q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( 6 + 6 \zeta_{6} ) q^{29} + ( 6 - 3 \zeta_{6} ) q^{31} + ( 3 + 6 \zeta_{6} ) q^{35} + q^{37} + 3 \zeta_{6} q^{41} + ( -10 + 10 \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( -9 + 18 \zeta_{6} ) q^{55} -6 \zeta_{6} q^{59} + ( 8 + 8 \zeta_{6} ) q^{61} + ( -6 - 6 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} + ( 3 - 6 \zeta_{6} ) q^{71} + ( 2 - 4 \zeta_{6} ) q^{73} + ( 12 + 3 \zeta_{6} ) q^{77} + ( -14 + 14 \zeta_{6} ) q^{79} + ( -6 + 6 \zeta_{6} ) q^{83} + 18 \zeta_{6} q^{85} + 9 q^{89} + ( -10 + 8 \zeta_{6} ) q^{91} + ( 6 - 3 \zeta_{6} ) q^{95} + ( 4 + 4 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{5} - 5q^{7} + O(q^{10})$$ $$2q - 3q^{5} - 5q^{7} - 9q^{11} + 6q^{13} - 12q^{17} - 9q^{23} - 4q^{25} + 18q^{29} + 9q^{31} + 12q^{35} + 2q^{37} + 3q^{41} - 10q^{43} + 6q^{47} + 11q^{49} - 6q^{59} + 24q^{61} - 18q^{65} - 2q^{67} + 27q^{77} - 14q^{79} - 6q^{83} + 18q^{85} + 18q^{89} - 12q^{91} + 9q^{95} + 12q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
377.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.50000 + 2.59808i 0 −2.50000 0.866025i 0 0 0
1889.1 0 0 0 −1.50000 2.59808i 0 −2.50000 + 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
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.x.a 2
3.b odd 2 1 2268.2.x.g 2
7.b odd 2 1 2268.2.x.h 2
9.c even 3 1 756.2.f.c yes 2
9.c even 3 1 2268.2.x.b 2
9.d odd 6 1 756.2.f.a 2
9.d odd 6 1 2268.2.x.h 2
21.c even 2 1 2268.2.x.b 2
36.f odd 6 1 3024.2.k.d 2
36.h even 6 1 3024.2.k.a 2
63.l odd 6 1 756.2.f.a 2
63.l odd 6 1 2268.2.x.g 2
63.o even 6 1 756.2.f.c yes 2
63.o even 6 1 inner 2268.2.x.a 2
252.s odd 6 1 3024.2.k.d 2
252.bi even 6 1 3024.2.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.f.a 2 9.d odd 6 1
756.2.f.a 2 63.l odd 6 1
756.2.f.c yes 2 9.c even 3 1
756.2.f.c yes 2 63.o even 6 1
2268.2.x.a 2 1.a even 1 1 trivial
2268.2.x.a 2 63.o even 6 1 inner
2268.2.x.b 2 9.c even 3 1
2268.2.x.b 2 21.c even 2 1
2268.2.x.g 2 3.b odd 2 1
2268.2.x.g 2 63.l odd 6 1
2268.2.x.h 2 7.b odd 2 1
2268.2.x.h 2 9.d odd 6 1
3024.2.k.a 2 36.h even 6 1
3024.2.k.a 2 252.bi even 6 1
3024.2.k.d 2 36.f odd 6 1
3024.2.k.d 2 252.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2268, [\chi])$$:

 $$T_{5}^{2} + 3 T_{5} + 9$$ $$T_{11}^{2} + 9 T_{11} + 27$$ $$T_{13}^{2} - 6 T_{13} + 12$$