# Properties

 Label 2268.2.i.i Level $2268$ Weight $2$ Character orbit 2268.i Analytic conductor $18.110$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{10})$$ Defining polynomial: $$x^{4} + 10 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 756) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{11} -\beta_{1} q^{17} -7 \beta_{2} q^{19} -\beta_{1} q^{23} + 5 \beta_{2} q^{25} + \beta_{1} q^{29} + 3 q^{31} + ( -2 \beta_{1} + \beta_{3} ) q^{35} -4 \beta_{2} q^{37} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{41} + ( -5 - 5 \beta_{2} ) q^{43} + 3 \beta_{3} q^{47} + ( 3 - 5 \beta_{2} ) q^{49} + 3 \beta_{1} q^{53} + 20 q^{55} + 4 \beta_{3} q^{59} -3 q^{61} + 10 q^{67} -4 \beta_{3} q^{71} + ( 5 + 5 \beta_{2} ) q^{73} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{77} + 12 q^{79} + 2 \beta_{1} q^{83} -10 \beta_{2} q^{85} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{89} -7 \beta_{3} q^{95} + ( 5 + 5 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{7} + O(q^{10})$$ $$4 q - 10 q^{7} + 14 q^{19} - 10 q^{25} + 12 q^{31} + 8 q^{37} - 10 q^{43} + 22 q^{49} + 80 q^{55} - 12 q^{61} + 40 q^{67} + 10 q^{73} + 48 q^{79} + 20 q^{85} + 10 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 10 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/10$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$10 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$10 \beta_{3}$$

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$\beta_{2}$$

## 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
 −1.58114 + 2.73861i 1.58114 − 2.73861i −1.58114 − 2.73861i 1.58114 + 2.73861i
0 0 0 −1.58114 + 2.73861i 0 −2.50000 0.866025i 0 0 0
865.2 0 0 0 1.58114 2.73861i 0 −2.50000 0.866025i 0 0 0
2053.1 0 0 0 −1.58114 2.73861i 0 −2.50000 + 0.866025i 0 0 0
2053.2 0 0 0 1.58114 + 2.73861i 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
3.b odd 2 1 inner
63.h even 3 1 inner
63.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.i.i 4
3.b odd 2 1 inner 2268.2.i.i 4
7.c even 3 1 2268.2.l.i 4
9.c even 3 1 756.2.k.d 4
9.c even 3 1 2268.2.l.i 4
9.d odd 6 1 756.2.k.d 4
9.d odd 6 1 2268.2.l.i 4
21.h odd 6 1 2268.2.l.i 4
63.g even 3 1 756.2.k.d 4
63.h even 3 1 inner 2268.2.i.i 4
63.h even 3 1 5292.2.a.q 2
63.i even 6 1 5292.2.a.r 2
63.j odd 6 1 inner 2268.2.i.i 4
63.j odd 6 1 5292.2.a.q 2
63.n odd 6 1 756.2.k.d 4
63.t odd 6 1 5292.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.d 4 9.c even 3 1
756.2.k.d 4 9.d odd 6 1
756.2.k.d 4 63.g even 3 1
756.2.k.d 4 63.n odd 6 1
2268.2.i.i 4 1.a even 1 1 trivial
2268.2.i.i 4 3.b odd 2 1 inner
2268.2.i.i 4 63.h even 3 1 inner
2268.2.i.i 4 63.j odd 6 1 inner
2268.2.l.i 4 7.c even 3 1
2268.2.l.i 4 9.c even 3 1
2268.2.l.i 4 9.d odd 6 1
2268.2.l.i 4 21.h odd 6 1
5292.2.a.q 2 63.h even 3 1
5292.2.a.q 2 63.j odd 6 1
5292.2.a.r 2 63.i even 6 1
5292.2.a.r 2 63.t odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 10 T_{5}^{2} + 100$$ $$T_{13}$$ $$T_{19}^{2} - 7 T_{19} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$100 + 10 T^{2} + T^{4}$$
$7$ $$( 7 + 5 T + T^{2} )^{2}$$
$11$ $$1600 + 40 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$100 + 10 T^{2} + T^{4}$$
$19$ $$( 49 - 7 T + T^{2} )^{2}$$
$23$ $$100 + 10 T^{2} + T^{4}$$
$29$ $$100 + 10 T^{2} + T^{4}$$
$31$ $$( -3 + T )^{4}$$
$37$ $$( 16 - 4 T + T^{2} )^{2}$$
$41$ $$8100 + 90 T^{2} + T^{4}$$
$43$ $$( 25 + 5 T + T^{2} )^{2}$$
$47$ $$( -90 + T^{2} )^{2}$$
$53$ $$8100 + 90 T^{2} + T^{4}$$
$59$ $$( -160 + T^{2} )^{2}$$
$61$ $$( 3 + T )^{4}$$
$67$ $$( -10 + T )^{4}$$
$71$ $$( -160 + T^{2} )^{2}$$
$73$ $$( 25 - 5 T + T^{2} )^{2}$$
$79$ $$( -12 + T )^{4}$$
$83$ $$1600 + 40 T^{2} + T^{4}$$
$89$ $$8100 + 90 T^{2} + T^{4}$$
$97$ $$( 25 - 5 T + T^{2} )^{2}$$