# Properties

 Label 1575.1.n.a Level 1575 Weight 1 Character orbit 1575.n Analytic conductor 0.786 Analytic rank 0 Dimension 4 Projective image $$D_{4}$$ 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.n (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.786027394897$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{4}$$ Projective field Galois closure of 4.2.33075.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{8}^{2} ) q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{7} +O(q^{10})$$ $$q + ( -1 - \zeta_{8}^{2} ) q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{7} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{11} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{14} + q^{16} -2 \zeta_{8}^{3} q^{22} + ( 1 - \zeta_{8}^{2} ) q^{23} + \zeta_{8} q^{28} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{29} + ( -1 - \zeta_{8}^{2} ) q^{32} + 2 \zeta_{8}^{3} q^{37} + 2 \zeta_{8} q^{43} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{44} -2 q^{46} -\zeta_{8}^{2} q^{49} + ( 1 - \zeta_{8}^{2} ) q^{53} -2 \zeta_{8} q^{58} + \zeta_{8}^{2} q^{64} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{71} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{74} + ( 1 + \zeta_{8}^{2} ) q^{77} -2 \zeta_{8}^{2} q^{79} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{86} + ( 1 + \zeta_{8}^{2} ) q^{92} + ( -1 + \zeta_{8}^{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + O(q^{10})$$ $$4q - 4q^{2} + 4q^{16} + 4q^{23} - 4q^{32} - 8q^{46} + 4q^{53} + 4q^{77} + 4q^{92} - 4q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
818.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−1.00000 1.00000i 0 1.00000i 0 0 −0.707107 + 0.707107i 0 0 0
818.2 −1.00000 1.00000i 0 1.00000i 0 0 0.707107 0.707107i 0 0 0
1007.1 −1.00000 + 1.00000i 0 1.00000i 0 0 −0.707107 0.707107i 0 0 0
1007.2 −1.00000 + 1.00000i 0 1.00000i 0 0 0.707107 + 0.707107i 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
5.c odd 4 1 inner
15.d odd 2 1 inner
15.e even 4 1 inner
35.f even 4 1 inner
105.g even 2 1 inner
105.k odd 4 1 inner

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

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