# Properties

 Label 2268.2.k.b Level 2268 Weight 2 Character orbit 2268.k 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.k (of order $$3$$, degree $$2$$, 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 252) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + ( 4 - 4 \zeta_{6} ) q^{11} + 3 q^{13} + ( 7 - 7 \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + q^{29} + ( 3 - 3 \zeta_{6} ) q^{31} + ( -6 + 4 \zeta_{6} ) q^{35} -11 \zeta_{6} q^{37} + 9 q^{41} + 5 q^{43} + 3 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 3 - 3 \zeta_{6} ) q^{53} + 8 q^{55} + ( -7 + 7 \zeta_{6} ) q^{59} -3 \zeta_{6} q^{61} + 6 \zeta_{6} q^{65} + ( -13 + 13 \zeta_{6} ) q^{67} + 8 q^{71} + ( -7 + 7 \zeta_{6} ) q^{73} + ( 8 + 4 \zeta_{6} ) q^{77} + 9 \zeta_{6} q^{79} - q^{83} + 14 q^{85} + 15 \zeta_{6} q^{89} + ( -3 + 9 \zeta_{6} ) q^{91} + ( 10 - 10 \zeta_{6} ) q^{95} -17 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + q^{7} + O(q^{10})$$ $$2q + 2q^{5} + q^{7} + 4q^{11} + 6q^{13} + 7q^{17} - 5q^{19} + 4q^{23} + q^{25} + 2q^{29} + 3q^{31} - 8q^{35} - 11q^{37} + 18q^{41} + 10q^{43} + 3q^{47} - 13q^{49} + 3q^{53} + 16q^{55} - 7q^{59} - 3q^{61} + 6q^{65} - 13q^{67} + 16q^{71} - 7q^{73} + 20q^{77} + 9q^{79} - 2q^{83} + 28q^{85} + 15q^{89} + 3q^{91} + 10q^{95} - 34q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1297.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 1.00000 1.73205i 0 0.500000 2.59808i 0 0 0
1621.1 0 0 0 1.00000 + 1.73205i 0 0.500000 + 2.59808i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.k.b 2
3.b odd 2 1 2268.2.k.a 2
7.c even 3 1 inner 2268.2.k.b 2
9.c even 3 1 756.2.i.a 2
9.c even 3 1 756.2.l.a 2
9.d odd 6 1 252.2.i.a 2
9.d odd 6 1 252.2.l.a yes 2
21.h odd 6 1 2268.2.k.a 2
36.f odd 6 1 3024.2.q.e 2
36.f odd 6 1 3024.2.t.b 2
36.h even 6 1 1008.2.q.f 2
36.h even 6 1 1008.2.t.b 2
63.g even 3 1 756.2.i.a 2
63.g even 3 1 5292.2.j.c 2
63.h even 3 1 756.2.l.a 2
63.h even 3 1 5292.2.j.c 2
63.i even 6 1 1764.2.j.a 2
63.i even 6 1 1764.2.l.b 2
63.j odd 6 1 252.2.l.a yes 2
63.j odd 6 1 1764.2.j.c 2
63.k odd 6 1 5292.2.i.b 2
63.k odd 6 1 5292.2.j.b 2
63.l odd 6 1 5292.2.i.b 2
63.l odd 6 1 5292.2.l.b 2
63.n odd 6 1 252.2.i.a 2
63.n odd 6 1 1764.2.j.c 2
63.o even 6 1 1764.2.i.b 2
63.o even 6 1 1764.2.l.b 2
63.s even 6 1 1764.2.i.b 2
63.s even 6 1 1764.2.j.a 2
63.t odd 6 1 5292.2.j.b 2
63.t odd 6 1 5292.2.l.b 2
252.o even 6 1 1008.2.q.f 2
252.u odd 6 1 3024.2.t.b 2
252.bb even 6 1 1008.2.t.b 2
252.bl odd 6 1 3024.2.q.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.a 2 9.d odd 6 1
252.2.i.a 2 63.n odd 6 1
252.2.l.a yes 2 9.d odd 6 1
252.2.l.a yes 2 63.j odd 6 1
756.2.i.a 2 9.c even 3 1
756.2.i.a 2 63.g even 3 1
756.2.l.a 2 9.c even 3 1
756.2.l.a 2 63.h even 3 1
1008.2.q.f 2 36.h even 6 1
1008.2.q.f 2 252.o even 6 1
1008.2.t.b 2 36.h even 6 1
1008.2.t.b 2 252.bb even 6 1
1764.2.i.b 2 63.o even 6 1
1764.2.i.b 2 63.s even 6 1
1764.2.j.a 2 63.i even 6 1
1764.2.j.a 2 63.s even 6 1
1764.2.j.c 2 63.j odd 6 1
1764.2.j.c 2 63.n odd 6 1
1764.2.l.b 2 63.i even 6 1
1764.2.l.b 2 63.o even 6 1
2268.2.k.a 2 3.b odd 2 1
2268.2.k.a 2 21.h odd 6 1
2268.2.k.b 2 1.a even 1 1 trivial
2268.2.k.b 2 7.c even 3 1 inner
3024.2.q.e 2 36.f odd 6 1
3024.2.q.e 2 252.bl odd 6 1
3024.2.t.b 2 36.f odd 6 1
3024.2.t.b 2 252.u odd 6 1
5292.2.i.b 2 63.k odd 6 1
5292.2.i.b 2 63.l odd 6 1
5292.2.j.b 2 63.k odd 6 1
5292.2.j.b 2 63.t odd 6 1
5292.2.j.c 2 63.g even 3 1
5292.2.j.c 2 63.h even 3 1
5292.2.l.b 2 63.l odd 6 1
5292.2.l.b 2 63.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2 T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(2268, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ $$1 - T + 7 T^{2}$$
$11$ $$1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 3 T + 13 T^{2} )^{2}$$
$17$ $$1 - 7 T + 32 T^{2} - 119 T^{3} + 289 T^{4}$$
$19$ $$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$
$23$ $$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$
$29$ $$( 1 - T + 29 T^{2} )^{2}$$
$31$ $$1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4}$$
$37$ $$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$( 1 - 9 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 5 T + 43 T^{2} )^{2}$$
$47$ $$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$
$53$ $$1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4}$$
$59$ $$1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4}$$
$61$ $$1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4}$$
$67$ $$1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$
$79$ $$1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + T + 83 T^{2} )^{2}$$
$89$ $$1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 17 T + 97 T^{2} )^{2}$$