# Properties

 Label 2268.2.j.a Level $2268$ Weight $2$ Character orbit 2268.j 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.j (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 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.

 $$f(q)$$ $$=$$ $$q -4 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -4 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + ( -2 + 2 \zeta_{6} ) q^{11} + 6 \zeta_{6} q^{13} -4 q^{17} -4 q^{19} -2 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + ( 2 - 2 \zeta_{6} ) q^{29} -4 q^{35} + 2 q^{37} + ( 4 - 4 \zeta_{6} ) q^{43} + ( -12 + 12 \zeta_{6} ) q^{47} -\zeta_{6} q^{49} -6 q^{53} + 8 q^{55} + 8 \zeta_{6} q^{59} + ( -6 + 6 \zeta_{6} ) q^{61} + ( 24 - 24 \zeta_{6} ) q^{65} + 8 \zeta_{6} q^{67} + 14 q^{71} -2 q^{73} + 2 \zeta_{6} q^{77} + ( -12 + 12 \zeta_{6} ) q^{79} + ( 4 - 4 \zeta_{6} ) q^{83} + 16 \zeta_{6} q^{85} + 6 q^{91} + 16 \zeta_{6} q^{95} + ( 2 - 2 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} + q^{7} + O(q^{10})$$ $$2q - 4q^{5} + q^{7} - 2q^{11} + 6q^{13} - 8q^{17} - 8q^{19} - 2q^{23} - 11q^{25} + 2q^{29} - 8q^{35} + 4q^{37} + 4q^{43} - 12q^{47} - q^{49} - 12q^{53} + 16q^{55} + 8q^{59} - 6q^{61} + 24q^{65} + 8q^{67} + 28q^{71} - 4q^{73} + 2q^{77} - 12q^{79} + 4q^{83} + 16q^{85} + 12q^{91} + 16q^{95} + 2q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
757.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −2.00000 + 3.46410i 0 0.500000 + 0.866025i 0 0 0
1513.1 0 0 0 −2.00000 3.46410i 0 0.500000 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
9.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.j.a 2
3.b odd 2 1 2268.2.j.n 2
9.c even 3 1 84.2.a.a 1
9.c even 3 1 inner 2268.2.j.a 2
9.d odd 6 1 252.2.a.a 1
9.d odd 6 1 2268.2.j.n 2
36.f odd 6 1 336.2.a.f 1
36.h even 6 1 1008.2.a.a 1
45.h odd 6 1 6300.2.a.w 1
45.j even 6 1 2100.2.a.r 1
45.k odd 12 2 2100.2.k.i 2
45.l even 12 2 6300.2.k.g 2
63.g even 3 1 588.2.i.e 2
63.h even 3 1 588.2.i.e 2
63.i even 6 1 1764.2.k.a 2
63.j odd 6 1 1764.2.k.k 2
63.k odd 6 1 588.2.i.d 2
63.l odd 6 1 588.2.a.d 1
63.n odd 6 1 1764.2.k.k 2
63.o even 6 1 1764.2.a.k 1
63.s even 6 1 1764.2.k.a 2
63.t odd 6 1 588.2.i.d 2
72.j odd 6 1 4032.2.a.bm 1
72.l even 6 1 4032.2.a.bn 1
72.n even 6 1 1344.2.a.k 1
72.p odd 6 1 1344.2.a.a 1
144.v odd 12 2 5376.2.c.p 2
144.x even 12 2 5376.2.c.q 2
180.p odd 6 1 8400.2.a.e 1
252.n even 6 1 2352.2.q.z 2
252.s odd 6 1 7056.2.a.cd 1
252.u odd 6 1 2352.2.q.b 2
252.bi even 6 1 2352.2.a.a 1
252.bj even 6 1 2352.2.q.z 2
252.bl odd 6 1 2352.2.q.b 2
504.be even 6 1 9408.2.a.df 1
504.bn odd 6 1 9408.2.a.bn 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.a 1 9.c even 3 1
252.2.a.a 1 9.d odd 6 1
336.2.a.f 1 36.f odd 6 1
588.2.a.d 1 63.l odd 6 1
588.2.i.d 2 63.k odd 6 1
588.2.i.d 2 63.t odd 6 1
588.2.i.e 2 63.g even 3 1
588.2.i.e 2 63.h even 3 1
1008.2.a.a 1 36.h even 6 1
1344.2.a.a 1 72.p odd 6 1
1344.2.a.k 1 72.n even 6 1
1764.2.a.k 1 63.o even 6 1
1764.2.k.a 2 63.i even 6 1
1764.2.k.a 2 63.s even 6 1
1764.2.k.k 2 63.j odd 6 1
1764.2.k.k 2 63.n odd 6 1
2100.2.a.r 1 45.j even 6 1
2100.2.k.i 2 45.k odd 12 2
2268.2.j.a 2 1.a even 1 1 trivial
2268.2.j.a 2 9.c even 3 1 inner
2268.2.j.n 2 3.b odd 2 1
2268.2.j.n 2 9.d odd 6 1
2352.2.a.a 1 252.bi even 6 1
2352.2.q.b 2 252.u odd 6 1
2352.2.q.b 2 252.bl odd 6 1
2352.2.q.z 2 252.n even 6 1
2352.2.q.z 2 252.bj even 6 1
4032.2.a.bm 1 72.j odd 6 1
4032.2.a.bn 1 72.l even 6 1
5376.2.c.p 2 144.v odd 12 2
5376.2.c.q 2 144.x even 12 2
6300.2.a.w 1 45.h odd 6 1
6300.2.k.g 2 45.l even 12 2
7056.2.a.cd 1 252.s odd 6 1
8400.2.a.e 1 180.p odd 6 1
9408.2.a.bn 1 504.bn odd 6 1
9408.2.a.df 1 504.be even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{11}^{2} + 2 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$
$7$ $$1 - T + T^{2}$$
$11$ $$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 4 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4}$$
$31$ $$1 - 31 T^{2} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4}$$
$47$ $$1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 6 T + 53 T^{2} )^{2}$$
$59$ $$1 - 8 T + 5 T^{2} - 472 T^{3} + 3481 T^{4}$$
$61$ $$1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4}$$
$67$ $$1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 14 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 2 T + 73 T^{2} )^{2}$$
$79$ $$1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4}$$
$83$ $$1 - 4 T - 67 T^{2} - 332 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4}$$