# Properties

 Label 2268.1.h.a Level $2268$ Weight $1$ Character orbit 2268.h Analytic conductor $1.132$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2268.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.13187944865$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{6} q^{7} -\zeta_{24}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{6} q^{7} -\zeta_{24}^{3} q^{8} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{11} -\zeta_{24}^{7} q^{14} + \zeta_{24}^{4} q^{16} + ( \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{22} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{23} - q^{25} + \zeta_{24}^{8} q^{28} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{29} -\zeta_{24}^{5} q^{32} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{37} + \zeta_{24}^{6} q^{43} + ( -\zeta_{24}^{7} + \zeta_{24}^{9} ) q^{44} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{46} - q^{49} + \zeta_{24} q^{50} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{53} -\zeta_{24}^{9} q^{56} + ( -\zeta_{24}^{4} - \zeta_{24}^{10} ) q^{58} + \zeta_{24}^{6} q^{64} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{67} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{71} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{74} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{77} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{79} -\zeta_{24}^{7} q^{86} + ( \zeta_{24}^{8} - \zeta_{24}^{10} ) q^{88} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{92} + \zeta_{24} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 4q^{16} + 4q^{22} - 8q^{25} - 4q^{28} - 4q^{46} - 8q^{49} - 4q^{58} - 4q^{88} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2267.1
 0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i −0.965926 − 0.258819i
−0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 1.00000i −0.707107 0.707107i 0 0
2267.2 −0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 1.00000i −0.707107 + 0.707107i 0 0
2267.3 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 1.00000i 0.707107 + 0.707107i 0 0
2267.4 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 1.00000i 0.707107 0.707107i 0 0
2267.5 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 1.00000i −0.707107 + 0.707107i 0 0
2267.6 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 1.00000i −0.707107 0.707107i 0 0
2267.7 0.965926 0.258819i 0 0.866025 0.500000i 0 0 1.00000i 0.707107 0.707107i 0 0
2267.8 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 1.00000i 0.707107 + 0.707107i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2267.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.1.h.a 8
3.b odd 2 1 inner 2268.1.h.a 8
4.b odd 2 1 inner 2268.1.h.a 8
7.b odd 2 1 CM 2268.1.h.a 8
9.c even 3 1 2268.1.s.e 8
9.c even 3 1 2268.1.s.f 8
9.d odd 6 1 2268.1.s.e 8
9.d odd 6 1 2268.1.s.f 8
12.b even 2 1 inner 2268.1.h.a 8
21.c even 2 1 inner 2268.1.h.a 8
28.d even 2 1 inner 2268.1.h.a 8
36.f odd 6 1 2268.1.s.e 8
36.f odd 6 1 2268.1.s.f 8
36.h even 6 1 2268.1.s.e 8
36.h even 6 1 2268.1.s.f 8
63.l odd 6 1 2268.1.s.e 8
63.l odd 6 1 2268.1.s.f 8
63.o even 6 1 2268.1.s.e 8
63.o even 6 1 2268.1.s.f 8
84.h odd 2 1 inner 2268.1.h.a 8
252.s odd 6 1 2268.1.s.e 8
252.s odd 6 1 2268.1.s.f 8
252.bi even 6 1 2268.1.s.e 8
252.bi even 6 1 2268.1.s.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.1.h.a 8 1.a even 1 1 trivial
2268.1.h.a 8 3.b odd 2 1 inner
2268.1.h.a 8 4.b odd 2 1 inner
2268.1.h.a 8 7.b odd 2 1 CM
2268.1.h.a 8 12.b even 2 1 inner
2268.1.h.a 8 21.c even 2 1 inner
2268.1.h.a 8 28.d even 2 1 inner
2268.1.h.a 8 84.h odd 2 1 inner
2268.1.s.e 8 9.c even 3 1
2268.1.s.e 8 9.d odd 6 1
2268.1.s.e 8 36.f odd 6 1
2268.1.s.e 8 36.h even 6 1
2268.1.s.e 8 63.l odd 6 1
2268.1.s.e 8 63.o even 6 1
2268.1.s.e 8 252.s odd 6 1
2268.1.s.e 8 252.bi even 6 1
2268.1.s.f 8 9.c even 3 1
2268.1.s.f 8 9.d odd 6 1
2268.1.s.f 8 36.f odd 6 1
2268.1.s.f 8 36.h even 6 1
2268.1.s.f 8 63.l odd 6 1
2268.1.s.f 8 63.o even 6 1
2268.1.s.f 8 252.s odd 6 1
2268.1.s.f 8 252.bi even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2268, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$( 1 - 4 T^{2} + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( -2 + T^{2} )^{4}$$
$29$ $$( 2 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$( -3 + T^{2} )^{4}$$
$41$ $$T^{8}$$
$43$ $$( 1 + T^{2} )^{4}$$
$47$ $$T^{8}$$
$53$ $$( 1 + 4 T^{2} + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 3 + T^{2} )^{4}$$
$71$ $$( 1 - 4 T^{2} + T^{4} )^{2}$$
$73$ $$T^{8}$$
$79$ $$( 3 + T^{2} )^{4}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$