# Properties

 Label 2268.2.l.a Level $2268$ Weight $2$ Character orbit 2268.l 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.l (of order $$3$$, 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 28) 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.

 $$f(q)$$ $$=$$ $$q -3 q^{5} + ( 2 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -3 q^{5} + ( 2 + \zeta_{6} ) q^{7} + 3 q^{11} + ( -2 + 2 \zeta_{6} ) q^{13} + ( 3 - 3 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} -3 q^{23} + 4 q^{25} -6 \zeta_{6} q^{29} + 7 \zeta_{6} q^{31} + ( -6 - 3 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} + ( 6 - 6 \zeta_{6} ) q^{41} + 4 \zeta_{6} q^{43} + ( -9 + 9 \zeta_{6} ) q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( 3 - 3 \zeta_{6} ) q^{53} -9 q^{55} + 9 \zeta_{6} q^{59} + ( 1 - \zeta_{6} ) q^{61} + ( 6 - 6 \zeta_{6} ) q^{65} + 7 \zeta_{6} q^{67} + ( 1 - \zeta_{6} ) q^{73} + ( 6 + 3 \zeta_{6} ) q^{77} + ( 13 - 13 \zeta_{6} ) q^{79} + 12 \zeta_{6} q^{83} + ( -9 + 9 \zeta_{6} ) q^{85} + 15 \zeta_{6} q^{89} + ( -6 + 4 \zeta_{6} ) q^{91} -3 \zeta_{6} q^{95} + 10 \zeta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{5} + 5q^{7} + O(q^{10})$$ $$2q - 6q^{5} + 5q^{7} + 6q^{11} - 2q^{13} + 3q^{17} + q^{19} - 6q^{23} + 8q^{25} - 6q^{29} + 7q^{31} - 15q^{35} + q^{37} + 6q^{41} + 4q^{43} - 9q^{47} + 11q^{49} + 3q^{53} - 18q^{55} + 9q^{59} + q^{61} + 6q^{65} + 7q^{67} + q^{73} + 15q^{77} + 13q^{79} + 12q^{83} - 9q^{85} + 15q^{89} - 8q^{91} - 3q^{95} + 10q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −3.00000 0 2.50000 + 0.866025i 0 0 0
541.1 0 0 0 −3.00000 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.g 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.l.a 2
3.b odd 2 1 2268.2.l.h 2
7.c even 3 1 2268.2.i.h 2
9.c even 3 1 252.2.k.c 2
9.c even 3 1 2268.2.i.h 2
9.d odd 6 1 28.2.e.a 2
9.d odd 6 1 2268.2.i.a 2
21.h odd 6 1 2268.2.i.a 2
36.f odd 6 1 1008.2.s.p 2
36.h even 6 1 112.2.i.b 2
45.h odd 6 1 700.2.i.c 2
45.l even 12 2 700.2.r.b 4
63.g even 3 1 1764.2.a.a 1
63.g even 3 1 inner 2268.2.l.a 2
63.h even 3 1 252.2.k.c 2
63.i even 6 1 196.2.e.a 2
63.j odd 6 1 28.2.e.a 2
63.k odd 6 1 1764.2.a.j 1
63.l odd 6 1 1764.2.k.b 2
63.n odd 6 1 196.2.a.b 1
63.n odd 6 1 2268.2.l.h 2
63.o even 6 1 196.2.e.a 2
63.s even 6 1 196.2.a.a 1
63.t odd 6 1 1764.2.k.b 2
72.j odd 6 1 448.2.i.e 2
72.l even 6 1 448.2.i.c 2
252.n even 6 1 7056.2.a.bw 1
252.o even 6 1 784.2.a.d 1
252.r odd 6 1 784.2.i.d 2
252.s odd 6 1 784.2.i.d 2
252.u odd 6 1 1008.2.s.p 2
252.bb even 6 1 112.2.i.b 2
252.bl odd 6 1 7056.2.a.f 1
252.bn odd 6 1 784.2.a.g 1
315.u even 6 1 4900.2.a.n 1
315.v odd 6 1 4900.2.a.g 1
315.br odd 6 1 700.2.i.c 2
315.bv even 12 2 700.2.r.b 4
315.bw odd 12 2 4900.2.e.h 2
315.bx even 12 2 4900.2.e.i 2
504.u odd 6 1 3136.2.a.k 1
504.y even 6 1 3136.2.a.v 1
504.bi odd 6 1 448.2.i.e 2
504.bt even 6 1 448.2.i.c 2
504.cy even 6 1 3136.2.a.s 1
504.db odd 6 1 3136.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 9.d odd 6 1
28.2.e.a 2 63.j odd 6 1
112.2.i.b 2 36.h even 6 1
112.2.i.b 2 252.bb even 6 1
196.2.a.a 1 63.s even 6 1
196.2.a.b 1 63.n odd 6 1
196.2.e.a 2 63.i even 6 1
196.2.e.a 2 63.o even 6 1
252.2.k.c 2 9.c even 3 1
252.2.k.c 2 63.h even 3 1
448.2.i.c 2 72.l even 6 1
448.2.i.c 2 504.bt even 6 1
448.2.i.e 2 72.j odd 6 1
448.2.i.e 2 504.bi odd 6 1
700.2.i.c 2 45.h odd 6 1
700.2.i.c 2 315.br odd 6 1
700.2.r.b 4 45.l even 12 2
700.2.r.b 4 315.bv even 12 2
784.2.a.d 1 252.o even 6 1
784.2.a.g 1 252.bn odd 6 1
784.2.i.d 2 252.r odd 6 1
784.2.i.d 2 252.s odd 6 1
1008.2.s.p 2 36.f odd 6 1
1008.2.s.p 2 252.u odd 6 1
1764.2.a.a 1 63.g even 3 1
1764.2.a.j 1 63.k odd 6 1
1764.2.k.b 2 63.l odd 6 1
1764.2.k.b 2 63.t odd 6 1
2268.2.i.a 2 9.d odd 6 1
2268.2.i.a 2 21.h odd 6 1
2268.2.i.h 2 7.c even 3 1
2268.2.i.h 2 9.c even 3 1
2268.2.l.a 2 1.a even 1 1 trivial
2268.2.l.a 2 63.g even 3 1 inner
2268.2.l.h 2 3.b odd 2 1
2268.2.l.h 2 63.n odd 6 1
3136.2.a.h 1 504.db odd 6 1
3136.2.a.k 1 504.u odd 6 1
3136.2.a.s 1 504.cy even 6 1
3136.2.a.v 1 504.y even 6 1
4900.2.a.g 1 315.v odd 6 1
4900.2.a.n 1 315.u even 6 1
4900.2.e.h 2 315.bw odd 12 2
4900.2.e.i 2 315.bx even 12 2
7056.2.a.f 1 252.bl odd 6 1
7056.2.a.bw 1 252.n even 6 1

## Hecke kernels

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

 $$T_{5} + 3$$ $$T_{13}^{2} + 2 T_{13} + 4$$ $$T_{19}^{2} - T_{19} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 3 + T )^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$( 3 + T )^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$49 - 7 T + T^{2}$$
$37$ $$1 - T + T^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$16 - 4 T + T^{2}$$
$47$ $$81 + 9 T + T^{2}$$
$53$ $$9 - 3 T + T^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$49 - 7 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$1 - T + T^{2}$$
$79$ $$169 - 13 T + T^{2}$$
$83$ $$144 - 12 T + T^{2}$$
$89$ $$225 - 15 T + T^{2}$$
$97$ $$100 - 10 T + T^{2}$$