# Properties

 Label 1575.1.ci.a Level $1575$ Weight $1$ Character orbit 1575.ci Analytic conductor $0.786$ Analytic rank $0$ Dimension $8$ Projective image $D_{6}$ CM discriminant -3 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1575.ci (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.786027394897$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.67528125.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{2} q^{4} -\zeta_{24}^{11} q^{7} +O(q^{10})$$ $$q -\zeta_{24}^{2} q^{4} -\zeta_{24}^{11} q^{7} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{13} + \zeta_{24}^{4} q^{16} -\zeta_{24}^{10} q^{19} -\zeta_{24} q^{28} -2 \zeta_{24}^{8} q^{31} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{37} -\zeta_{24}^{10} q^{49} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{52} + \zeta_{24}^{4} q^{61} -\zeta_{24}^{6} q^{64} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} ) q^{67} + ( \zeta_{24} - \zeta_{24}^{9} ) q^{73} - q^{76} + \zeta_{24}^{10} q^{79} + ( -1 - \zeta_{24}^{4} ) q^{91} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 4q^{16} + 8q^{31} + 4q^{61} - 8q^{76} - 12q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$-\zeta_{24}^{6}$$ $$-\zeta_{24}^{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
793.1
 −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
0 0 0.866025 0.500000i 0 0 −0.258819 + 0.965926i 0 0 0
793.2 0 0 0.866025 0.500000i 0 0 0.258819 0.965926i 0 0 0
982.1 0 0 −0.866025 + 0.500000i 0 0 −0.965926 0.258819i 0 0 0
982.2 0 0 −0.866025 + 0.500000i 0 0 0.965926 + 0.258819i 0 0 0
1243.1 0 0 −0.866025 0.500000i 0 0 −0.965926 + 0.258819i 0 0 0
1243.2 0 0 −0.866025 0.500000i 0 0 0.965926 0.258819i 0 0 0
1432.1 0 0 0.866025 + 0.500000i 0 0 −0.258819 0.965926i 0 0 0
1432.2 0 0 0.866025 + 0.500000i 0 0 0.258819 + 0.965926i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1432.2 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
5.c odd 4 2 inner
7.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
21.h odd 6 1 inner
35.j even 6 1 inner
35.l odd 12 2 inner
105.o odd 6 1 inner
105.x even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.ci.a 8
3.b odd 2 1 CM 1575.1.ci.a 8
5.b even 2 1 inner 1575.1.ci.a 8
5.c odd 4 2 inner 1575.1.ci.a 8
7.c even 3 1 inner 1575.1.ci.a 8
15.d odd 2 1 inner 1575.1.ci.a 8
15.e even 4 2 inner 1575.1.ci.a 8
21.h odd 6 1 inner 1575.1.ci.a 8
35.j even 6 1 inner 1575.1.ci.a 8
35.l odd 12 2 inner 1575.1.ci.a 8
105.o odd 6 1 inner 1575.1.ci.a 8
105.x even 12 2 inner 1575.1.ci.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.ci.a 8 1.a even 1 1 trivial
1575.1.ci.a 8 3.b odd 2 1 CM
1575.1.ci.a 8 5.b even 2 1 inner
1575.1.ci.a 8 5.c odd 4 2 inner
1575.1.ci.a 8 7.c even 3 1 inner
1575.1.ci.a 8 15.d odd 2 1 inner
1575.1.ci.a 8 15.e even 4 2 inner
1575.1.ci.a 8 21.h odd 6 1 inner
1575.1.ci.a 8 35.j even 6 1 inner
1575.1.ci.a 8 35.l odd 12 2 inner
1575.1.ci.a 8 105.o odd 6 1 inner
1575.1.ci.a 8 105.x even 12 2 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$1 - T^{4} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$( 9 + T^{4} )^{2}$$
$17$ $$T^{8}$$
$19$ $$( 1 - T^{2} + T^{4} )^{2}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$( 4 - 2 T + T^{2} )^{4}$$
$37$ $$81 - 9 T^{4} + T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$( 1 - T + T^{2} )^{4}$$
$67$ $$81 - 9 T^{4} + T^{8}$$
$71$ $$T^{8}$$
$73$ $$81 - 9 T^{4} + T^{8}$$
$79$ $$( 1 - T^{2} + T^{4} )^{2}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$( 9 + T^{4} )^{2}$$