# Properties

 Label 1200.1.bj.b Level $1200$ Weight $1$ Character orbit 1200.bj Analytic conductor $0.599$ Analytic rank $0$ Dimension $8$ Projective image $D_{6}$ CM discriminant -3 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1200.bj (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.598878015160$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.21600000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{3} q^{3} + \zeta_{24}^{9} q^{7} + \zeta_{24}^{6} q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{3} q^{3} + \zeta_{24}^{9} q^{7} + \zeta_{24}^{6} q^{9} + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{13} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{19} + q^{21} -\zeta_{24}^{9} q^{27} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{31} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{39} + \zeta_{24}^{3} q^{43} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{57} - q^{61} -\zeta_{24}^{3} q^{63} -\zeta_{24}^{9} q^{67} - q^{81} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{91} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{93} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{21} - 8q^{61} - 8q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$\zeta_{24}^{6}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 0.965926 − 0.258819i −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 −0.707107 + 0.707107i 0 0 0 −0.707107 0.707107i 0 1.00000i 0
143.2 0 −0.707107 + 0.707107i 0 0 0 −0.707107 0.707107i 0 1.00000i 0
143.3 0 0.707107 0.707107i 0 0 0 0.707107 + 0.707107i 0 1.00000i 0
143.4 0 0.707107 0.707107i 0 0 0 0.707107 + 0.707107i 0 1.00000i 0
1007.1 0 −0.707107 0.707107i 0 0 0 −0.707107 + 0.707107i 0 1.00000i 0
1007.2 0 −0.707107 0.707107i 0 0 0 −0.707107 + 0.707107i 0 1.00000i 0
1007.3 0 0.707107 + 0.707107i 0 0 0 0.707107 0.707107i 0 1.00000i 0
1007.4 0 0.707107 + 0.707107i 0 0 0 0.707107 0.707107i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1007.4 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 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
12.b even 2 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner
60.h even 2 1 inner
60.l odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.1.bj.b 8
3.b odd 2 1 CM 1200.1.bj.b 8
4.b odd 2 1 inner 1200.1.bj.b 8
5.b even 2 1 inner 1200.1.bj.b 8
5.c odd 4 2 inner 1200.1.bj.b 8
12.b even 2 1 inner 1200.1.bj.b 8
15.d odd 2 1 inner 1200.1.bj.b 8
15.e even 4 2 inner 1200.1.bj.b 8
20.d odd 2 1 inner 1200.1.bj.b 8
20.e even 4 2 inner 1200.1.bj.b 8
60.h even 2 1 inner 1200.1.bj.b 8
60.l odd 4 2 inner 1200.1.bj.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.1.bj.b 8 1.a even 1 1 trivial
1200.1.bj.b 8 3.b odd 2 1 CM
1200.1.bj.b 8 4.b odd 2 1 inner
1200.1.bj.b 8 5.b even 2 1 inner
1200.1.bj.b 8 5.c odd 4 2 inner
1200.1.bj.b 8 12.b even 2 1 inner
1200.1.bj.b 8 15.d odd 2 1 inner
1200.1.bj.b 8 15.e even 4 2 inner
1200.1.bj.b 8 20.d odd 2 1 inner
1200.1.bj.b 8 20.e even 4 2 inner
1200.1.bj.b 8 60.h even 2 1 inner
1200.1.bj.b 8 60.l odd 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

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