# Properties

 Label 2268.2.x.f Level $2268$ Weight $2$ Character orbit 2268.x Analytic conductor $18.110$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 756) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 3 - 2 \zeta_{6} ) q^{7} + ( -2 + \zeta_{6} ) q^{13} + ( 5 - 10 \zeta_{6} ) q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -12 + 6 \zeta_{6} ) q^{31} + q^{37} + ( 8 - 8 \zeta_{6} ) q^{43} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 5 + 5 \zeta_{6} ) q^{61} -11 \zeta_{6} q^{67} + ( -1 + 2 \zeta_{6} ) q^{73} + ( 13 - 13 \zeta_{6} ) q^{79} + ( -4 + 5 \zeta_{6} ) q^{91} + ( -11 - 11 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} + O(q^{10})$$ $$2q + 4q^{7} - 3q^{13} + 5q^{25} - 18q^{31} + 2q^{37} + 8q^{43} + 2q^{49} + 15q^{61} - 11q^{67} + 13q^{79} - 3q^{91} - 33q^{97} + O(q^{100})$$

## 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 0 0 2.00000 + 1.73205i 0 0 0
1889.1 0 0 0 0 0 2.00000 1.73205i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
63.l odd 6 1 inner
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.f 2
3.b odd 2 1 CM 2268.2.x.f 2
7.b odd 2 1 2268.2.x.d 2
9.c even 3 1 756.2.f.b 2
9.c even 3 1 2268.2.x.d 2
9.d odd 6 1 756.2.f.b 2
9.d odd 6 1 2268.2.x.d 2
21.c even 2 1 2268.2.x.d 2
36.f odd 6 1 3024.2.k.c 2
36.h even 6 1 3024.2.k.c 2
63.l odd 6 1 756.2.f.b 2
63.l odd 6 1 inner 2268.2.x.f 2
63.o even 6 1 756.2.f.b 2
63.o even 6 1 inner 2268.2.x.f 2
252.s odd 6 1 3024.2.k.c 2
252.bi even 6 1 3024.2.k.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.f.b 2 9.c even 3 1
756.2.f.b 2 9.d odd 6 1
756.2.f.b 2 63.l odd 6 1
756.2.f.b 2 63.o even 6 1
2268.2.x.d 2 7.b odd 2 1
2268.2.x.d 2 9.c even 3 1
2268.2.x.d 2 9.d odd 6 1
2268.2.x.d 2 21.c even 2 1
2268.2.x.f 2 1.a even 1 1 trivial
2268.2.x.f 2 3.b odd 2 1 CM
2268.2.x.f 2 63.l odd 6 1 inner
2268.2.x.f 2 63.o even 6 1 inner
3024.2.k.c 2 36.f odd 6 1
3024.2.k.c 2 36.h even 6 1
3024.2.k.c 2 252.s odd 6 1
3024.2.k.c 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 $$S_{2}^{\mathrm{new}}(2268, [\chi])$$:

 $$T_{5}$$ $$T_{11}$$ $$T_{13}^{2} + 3 T_{13} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} )$$
$23$ $$1 + 23 T^{2} + 529 T^{4}$$
$29$ $$1 + 29 T^{2} + 841 T^{4}$$
$31$ $$( 1 + 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$( 1 - T + 37 T^{2} )^{2}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} )$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 53 T^{2} )^{2}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} )$$
$67$ $$( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$( 1 - 71 T^{2} )^{2}$$
$73$ $$( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$
$79$ $$( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$
$83$ $$1 - 83 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$( 1 + 14 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$