# Properties

 Label 1575.1.e.c Level $1575$ Weight $1$ Character orbit 1575.e Analytic conductor $0.786$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.786027394897$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.5788125.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{2} + ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{4} -\zeta_{12}^{3} q^{7} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{2} + ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{4} -\zeta_{12}^{3} q^{7} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{8} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{11} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{14} + q^{16} + ( \zeta_{12} + 2 \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{22} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{23} + ( \zeta_{12} + \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{28} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{29} + \zeta_{12}^{3} q^{37} -\zeta_{12}^{3} q^{43} + ( 2 \zeta_{12} - 2 \zeta_{12}^{5} ) q^{44} + ( -2 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{46} - q^{49} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{56} + ( \zeta_{12} + 2 \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{58} + q^{64} -\zeta_{12}^{3} q^{67} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{71} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{74} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{77} + q^{79} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{86} + ( -\zeta_{12} - 2 \zeta_{12}^{3} - \zeta_{12}^{5} ) q^{88} + ( 2 \zeta_{12}^{2} + 2 \zeta_{12}^{4} ) q^{92} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} + O(q^{10})$$ $$4 q - 8 q^{4} + 4 q^{16} - 12 q^{46} - 4 q^{49} + 4 q^{64} + 4 q^{79} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
874.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
1.73205i 0 −2.00000 0 0 1.00000i 1.73205i 0 0
874.2 1.73205i 0 −2.00000 0 0 1.00000i 1.73205i 0 0
874.3 1.73205i 0 −2.00000 0 0 1.00000i 1.73205i 0 0
874.4 1.73205i 0 −2.00000 0 0 1.00000i 1.73205i 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3$$ acting on $$S_{1}^{\mathrm{new}}(1575, [\chi])$$.

## Hecke characteristic polynomials

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