# Properties

 Label 2268.2.i.g Level $2268$ Weight $2$ Character orbit 2268.i 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.i (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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -2 \zeta_{6} q^{11} + 3 \zeta_{6} q^{13} + ( -8 + 8 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( -8 + 8 \zeta_{6} ) q^{23} + \zeta_{6} q^{25} + ( -4 + 4 \zeta_{6} ) q^{29} + 3 q^{31} + ( 6 - 2 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} -6 \zeta_{6} q^{41} + ( -11 + 11 \zeta_{6} ) q^{43} + 6 q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 12 - 12 \zeta_{6} ) q^{53} -4 q^{55} + 4 q^{59} -6 q^{61} + 6 q^{65} + 13 q^{67} -10 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} + ( 4 - 6 \zeta_{6} ) q^{77} -3 q^{79} + ( -2 + 2 \zeta_{6} ) q^{83} + 16 \zeta_{6} q^{85} + ( -6 + 9 \zeta_{6} ) q^{91} + 2 q^{95} + ( -10 + 10 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + 4q^{7} + O(q^{10})$$ $$2q + 2q^{5} + 4q^{7} - 2q^{11} + 3q^{13} - 8q^{17} + q^{19} - 8q^{23} + q^{25} - 4q^{29} + 6q^{31} + 10q^{35} + q^{37} - 6q^{41} - 11q^{43} + 12q^{47} + 2q^{49} + 12q^{53} - 8q^{55} + 8q^{59} - 12q^{61} + 12q^{65} + 26q^{67} - 20q^{71} + 11q^{73} + 2q^{77} - 6q^{79} - 2q^{83} + 16q^{85} - 3q^{91} + 4q^{95} - 10q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.00000 1.73205i 0 2.00000 + 1.73205i 0 0 0
2053.1 0 0 0 1.00000 + 1.73205i 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
63.h 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.i.g 2
3.b odd 2 1 2268.2.i.b 2
7.c even 3 1 2268.2.l.b 2
9.c even 3 1 84.2.i.a 2
9.c even 3 1 2268.2.l.b 2
9.d odd 6 1 252.2.k.a 2
9.d odd 6 1 2268.2.l.g 2
21.h odd 6 1 2268.2.l.g 2
36.f odd 6 1 336.2.q.c 2
36.h even 6 1 1008.2.s.c 2
45.j even 6 1 2100.2.q.b 2
45.k odd 12 2 2100.2.bc.a 4
63.g even 3 1 84.2.i.a 2
63.h even 3 1 588.2.a.a 1
63.h even 3 1 inner 2268.2.i.g 2
63.i even 6 1 1764.2.a.c 1
63.j odd 6 1 1764.2.a.h 1
63.j odd 6 1 2268.2.i.b 2
63.k odd 6 1 588.2.i.b 2
63.l odd 6 1 588.2.i.b 2
63.n odd 6 1 252.2.k.a 2
63.o even 6 1 1764.2.k.j 2
63.s even 6 1 1764.2.k.j 2
63.t odd 6 1 588.2.a.f 1
72.n even 6 1 1344.2.q.b 2
72.p odd 6 1 1344.2.q.n 2
252.n even 6 1 2352.2.q.q 2
252.o even 6 1 1008.2.s.c 2
252.r odd 6 1 7056.2.a.o 1
252.u odd 6 1 2352.2.a.o 1
252.bb even 6 1 7056.2.a.bs 1
252.bi even 6 1 2352.2.q.q 2
252.bj even 6 1 2352.2.a.k 1
252.bl odd 6 1 336.2.q.c 2
315.bo even 6 1 2100.2.q.b 2
315.ch odd 12 2 2100.2.bc.a 4
504.w even 6 1 1344.2.q.b 2
504.ba odd 6 1 1344.2.q.n 2
504.bf even 6 1 9408.2.a.bx 1
504.bp odd 6 1 9408.2.a.i 1
504.ce odd 6 1 9408.2.a.bi 1
504.cq even 6 1 9408.2.a.cx 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 9.c even 3 1
84.2.i.a 2 63.g even 3 1
252.2.k.a 2 9.d odd 6 1
252.2.k.a 2 63.n odd 6 1
336.2.q.c 2 36.f odd 6 1
336.2.q.c 2 252.bl odd 6 1
588.2.a.a 1 63.h even 3 1
588.2.a.f 1 63.t odd 6 1
588.2.i.b 2 63.k odd 6 1
588.2.i.b 2 63.l odd 6 1
1008.2.s.c 2 36.h even 6 1
1008.2.s.c 2 252.o even 6 1
1344.2.q.b 2 72.n even 6 1
1344.2.q.b 2 504.w even 6 1
1344.2.q.n 2 72.p odd 6 1
1344.2.q.n 2 504.ba odd 6 1
1764.2.a.c 1 63.i even 6 1
1764.2.a.h 1 63.j odd 6 1
1764.2.k.j 2 63.o even 6 1
1764.2.k.j 2 63.s even 6 1
2100.2.q.b 2 45.j even 6 1
2100.2.q.b 2 315.bo even 6 1
2100.2.bc.a 4 45.k odd 12 2
2100.2.bc.a 4 315.ch odd 12 2
2268.2.i.b 2 3.b odd 2 1
2268.2.i.b 2 63.j odd 6 1
2268.2.i.g 2 1.a even 1 1 trivial
2268.2.i.g 2 63.h even 3 1 inner
2268.2.l.b 2 7.c even 3 1
2268.2.l.b 2 9.c even 3 1
2268.2.l.g 2 9.d odd 6 1
2268.2.l.g 2 21.h odd 6 1
2352.2.a.k 1 252.bj even 6 1
2352.2.a.o 1 252.u odd 6 1
2352.2.q.q 2 252.n even 6 1
2352.2.q.q 2 252.bi even 6 1
7056.2.a.o 1 252.r odd 6 1
7056.2.a.bs 1 252.bb even 6 1
9408.2.a.i 1 504.bp odd 6 1
9408.2.a.bi 1 504.ce odd 6 1
9408.2.a.bx 1 504.bf even 6 1
9408.2.a.cx 1 504.cq even 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{13}^{2} - 3 T_{13} + 9$$ $$T_{19}^{2} - T_{19} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$4 + 2 T + T^{2}$$
$13$ $$9 - 3 T + T^{2}$$
$17$ $$64 + 8 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$64 + 8 T + T^{2}$$
$29$ $$16 + 4 T + T^{2}$$
$31$ $$( -3 + T )^{2}$$
$37$ $$1 - T + T^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$121 + 11 T + T^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$( -13 + T )^{2}$$
$71$ $$( 10 + T )^{2}$$
$73$ $$121 - 11 T + T^{2}$$
$79$ $$( 3 + T )^{2}$$
$83$ $$4 + 2 T + T^{2}$$
$89$ $$T^{2}$$
$97$ $$100 + 10 T + T^{2}$$