# Properties

 Label 1575.1.x.b Level $1575$ Weight $1$ Character orbit 1575.x Analytic conductor $0.786$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1575.x (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.786027394897$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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.

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

## Character values

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

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0.500000 0.866025i 0 0 0.500000 0.866025i 0 0 0
901.1 0 0 0.500000 + 0.866025i 0 0 0.500000 + 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 CM by $$\Q(\sqrt{-3})$$
7.d odd 6 1 inner
21.g 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.x.b yes 2
3.b odd 2 1 CM 1575.1.x.b yes 2
5.b even 2 1 1575.1.x.a 2
5.c odd 4 2 1575.1.bj.a 4
7.d odd 6 1 inner 1575.1.x.b yes 2
15.d odd 2 1 1575.1.x.a 2
15.e even 4 2 1575.1.bj.a 4
21.g even 6 1 inner 1575.1.x.b yes 2
35.i odd 6 1 1575.1.x.a 2
35.k even 12 2 1575.1.bj.a 4
105.p even 6 1 1575.1.x.a 2
105.w odd 12 2 1575.1.bj.a 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{37}^{2} + T_{37} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1575, [\chi])$$.

## Hecke characteristic polynomials

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