# Properties

 Label 1575.1.bj.a Level $1575$ Weight $1$ Character orbit 1575.bj Analytic conductor $0.786$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1575.bj (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.786027394897$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.283618125.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{7} +O(q^{10})$$ $$q -\zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{7} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{13} + \zeta_{12}^{4} q^{16} + ( -1 - \zeta_{12}^{2} ) q^{19} + \zeta_{12} q^{28} -\zeta_{12} q^{37} + 2 \zeta_{12}^{3} q^{43} -\zeta_{12}^{4} q^{49} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{52} + ( -1 - \zeta_{12}^{2} ) q^{61} + q^{64} + \zeta_{12}^{5} q^{67} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{73} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{76} + \zeta_{12}^{4} q^{79} + ( 1 - \zeta_{12}^{4} ) q^{91} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{4} + O(q^{10})$$ $$4q - 2q^{4} - 2q^{16} - 6q^{19} + 2q^{49} - 6q^{61} + 4q^{64} - 2q^{79} + 6q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 −0.500000 + 0.866025i 0 0 −0.866025 0.500000i 0 0 0
199.2 0 0 −0.500000 + 0.866025i 0 0 0.866025 + 0.500000i 0 0 0
649.1 0 0 −0.500000 0.866025i 0 0 −0.866025 + 0.500000i 0 0 0
649.2 0 0 −0.500000 0.866025i 0 0 0.866025 0.500000i 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 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.bj.a 4
3.b odd 2 1 CM 1575.1.bj.a 4
5.b even 2 1 inner 1575.1.bj.a 4
5.c odd 4 1 1575.1.x.a 2
5.c odd 4 1 1575.1.x.b yes 2
7.d odd 6 1 inner 1575.1.bj.a 4
15.d odd 2 1 inner 1575.1.bj.a 4
15.e even 4 1 1575.1.x.a 2
15.e even 4 1 1575.1.x.b yes 2
21.g even 6 1 inner 1575.1.bj.a 4
35.i odd 6 1 inner 1575.1.bj.a 4
35.k even 12 1 1575.1.x.a 2
35.k even 12 1 1575.1.x.b yes 2
105.p even 6 1 inner 1575.1.bj.a 4
105.w odd 12 1 1575.1.x.a 2
105.w odd 12 1 1575.1.x.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.x.a 2 5.c odd 4 1
1575.1.x.a 2 15.e even 4 1
1575.1.x.a 2 35.k even 12 1
1575.1.x.a 2 105.w odd 12 1
1575.1.x.b yes 2 5.c odd 4 1
1575.1.x.b yes 2 15.e even 4 1
1575.1.x.b yes 2 35.k even 12 1
1575.1.x.b yes 2 105.w odd 12 1
1575.1.bj.a 4 1.a even 1 1 trivial
1575.1.bj.a 4 3.b odd 2 1 CM
1575.1.bj.a 4 5.b even 2 1 inner
1575.1.bj.a 4 7.d odd 6 1 inner
1575.1.bj.a 4 15.d odd 2 1 inner
1575.1.bj.a 4 21.g even 6 1 inner
1575.1.bj.a 4 35.i odd 6 1 inner
1575.1.bj.a 4 105.p even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1575, [\chi])$$.

## Hecke characteristic polynomials

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