# Properties

 Label 2268.2.x.j Level 2268 Weight 2 Character orbit 2268.x Analytic conductor 18.110 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2268.x (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.1100711784$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 756) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} +O(q^{10})$$ $$q + ( \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( 3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{11} + ( -2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{13} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{17} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{19} + ( -6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{23} + 2 \zeta_{24}^{4} q^{25} + ( 3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{29} + ( -6 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{31} + ( -3 \zeta_{24} + 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{35} + q^{37} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{41} -7 \zeta_{24}^{4} q^{43} + ( 7 \zeta_{24}^{2} + 7 \zeta_{24}^{6} ) q^{47} + ( 5 - 4 \zeta_{24} - 2 \zeta_{24}^{3} - 5 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{49} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{55} + ( 5 \zeta_{24}^{2} - 10 \zeta_{24}^{6} ) q^{59} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{61} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{65} + ( 10 - 10 \zeta_{24}^{4} ) q^{67} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{71} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{73} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 12 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{77} -5 \zeta_{24}^{4} q^{79} + ( -7 \zeta_{24}^{2} - 7 \zeta_{24}^{6} ) q^{83} + ( -3 + 3 \zeta_{24}^{4} ) q^{85} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{89} + ( 6 - \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{91} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{95} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{7} + O(q^{10})$$ $$8q + 4q^{7} + 8q^{25} + 8q^{37} - 28q^{43} + 20q^{49} + 40q^{67} - 20q^{79} - 12q^{85} + 48q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1 - \zeta_{24}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
377.1
 −0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i
0 0 0 −0.866025 + 1.50000i 0 −1.62132 + 2.09077i 0 0 0
377.2 0 0 0 −0.866025 + 1.50000i 0 2.62132 0.358719i 0 0 0
377.3 0 0 0 0.866025 1.50000i 0 −1.62132 + 2.09077i 0 0 0
377.4 0 0 0 0.866025 1.50000i 0 2.62132 0.358719i 0 0 0
1889.1 0 0 0 −0.866025 1.50000i 0 −1.62132 2.09077i 0 0 0
1889.2 0 0 0 −0.866025 1.50000i 0 2.62132 + 0.358719i 0 0 0
1889.3 0 0 0 0.866025 + 1.50000i 0 −1.62132 2.09077i 0 0 0
1889.4 0 0 0 0.866025 + 1.50000i 0 2.62132 + 0.358719i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1889.4 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
7.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
21.c even 2 1 inner
63.l odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.x.j 8
3.b odd 2 1 inner 2268.2.x.j 8
7.b odd 2 1 inner 2268.2.x.j 8
9.c even 3 1 756.2.f.d 4
9.c even 3 1 inner 2268.2.x.j 8
9.d odd 6 1 756.2.f.d 4
9.d odd 6 1 inner 2268.2.x.j 8
21.c even 2 1 inner 2268.2.x.j 8
36.f odd 6 1 3024.2.k.h 4
36.h even 6 1 3024.2.k.h 4
63.l odd 6 1 756.2.f.d 4
63.l odd 6 1 inner 2268.2.x.j 8
63.o even 6 1 756.2.f.d 4
63.o even 6 1 inner 2268.2.x.j 8
252.s odd 6 1 3024.2.k.h 4
252.bi even 6 1 3024.2.k.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.f.d 4 9.c even 3 1
756.2.f.d 4 9.d odd 6 1
756.2.f.d 4 63.l odd 6 1
756.2.f.d 4 63.o even 6 1
2268.2.x.j 8 1.a even 1 1 trivial
2268.2.x.j 8 3.b odd 2 1 inner
2268.2.x.j 8 7.b odd 2 1 inner
2268.2.x.j 8 9.c even 3 1 inner
2268.2.x.j 8 9.d odd 6 1 inner
2268.2.x.j 8 21.c even 2 1 inner
2268.2.x.j 8 63.l odd 6 1 inner
2268.2.x.j 8 63.o even 6 1 inner
3024.2.k.h 4 36.f odd 6 1
3024.2.k.h 4 36.h even 6 1
3024.2.k.h 4 252.s odd 6 1
3024.2.k.h 4 252.bi even 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 3 T_{5}^{2} + 9$$ $$T_{11}^{4} - 18 T_{11}^{2} + 324$$ $$T_{13}^{4} - 6 T_{13}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 + 31 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 - 32 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 26 T^{2} + 147 T^{4} - 13754 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 40 T^{2} + 759 T^{4} + 33640 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 8 T^{2} - 897 T^{4} + 7688 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - T + 37 T^{2} )^{8}$$
$41$ $$( 1 - 79 T^{2} + 4560 T^{4} - 132799 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 7 T + 6 T^{2} + 301 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 53 T^{2} + 600 T^{4} + 117077 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{8}$$
$59$ $$( 1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 116 T^{2} + 9735 T^{4} + 431636 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 70 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 14 T + 73 T^{2} )^{4}( 1 + 14 T + 73 T^{2} )^{4}$$
$79$ $$( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 - 19 T^{2} - 6528 T^{4} - 130891 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 70 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 + 188 T^{2} + 25935 T^{4} + 1768892 T^{6} + 88529281 T^{8} )^{2}$$