Properties

 Label 1575.1.n.d Level $1575$ Weight $1$ Character orbit 1575.n Analytic conductor $0.786$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24}^{8} - \zeta_{24}^{10} ) q^{2} + ( -\zeta_{24}^{4} - \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{4} + \zeta_{24}^{3} q^{7} + ( -1 - \zeta_{24}^{2} - \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{24}^{8} - \zeta_{24}^{10} ) q^{2} + ( -\zeta_{24}^{4} - \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{4} + \zeta_{24}^{3} q^{7} + ( -1 - \zeta_{24}^{2} - \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{8} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{11} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{14} + ( -1 - \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{16} + ( -\zeta_{24}^{7} + \zeta_{24}^{11} ) q^{22} + ( \zeta_{24}^{8} - \zeta_{24}^{10} ) q^{23} + ( -\zeta_{24}^{7} - \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{28} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{29} + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{32} + \zeta_{24}^{3} q^{37} + \zeta_{24}^{9} q^{43} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{44} + ( \zeta_{24}^{4} - \zeta_{24}^{8} ) q^{46} + \zeta_{24}^{6} q^{49} + ( 1 + \zeta_{24}^{6} ) q^{53} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} - \zeta_{24}^{9} ) q^{56} + ( \zeta_{24}^{7} + 2 \zeta_{24}^{9} + \zeta_{24}^{11} ) q^{58} + ( \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{64} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{67} + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{71} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{74} + ( \zeta_{24}^{2} - \zeta_{24}^{4} ) q^{77} -\zeta_{24}^{6} q^{79} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{86} + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{88} + ( 1 - \zeta_{24}^{6} ) q^{92} + ( \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} - 12q^{8} + O(q^{10})$$ $$8q + 4q^{2} - 12q^{8} - 16q^{16} - 4q^{23} - 8q^{32} + 8q^{46} + 8q^{53} - 4q^{77} + 8q^{92} + 4q^{98} + O(q^{100})$$

Character values

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

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

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
 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i
−0.366025 0.366025i 0 0.732051i 0 0 −0.707107 + 0.707107i −0.633975 + 0.633975i 0 0
818.2 −0.366025 0.366025i 0 0.732051i 0 0 0.707107 0.707107i −0.633975 + 0.633975i 0 0
818.3 1.36603 + 1.36603i 0 2.73205i 0 0 −0.707107 + 0.707107i −2.36603 + 2.36603i 0 0
818.4 1.36603 + 1.36603i 0 2.73205i 0 0 0.707107 0.707107i −2.36603 + 2.36603i 0 0
1007.1 −0.366025 + 0.366025i 0 0.732051i 0 0 −0.707107 0.707107i −0.633975 0.633975i 0 0
1007.2 −0.366025 + 0.366025i 0 0.732051i 0 0 0.707107 + 0.707107i −0.633975 0.633975i 0 0
1007.3 1.36603 1.36603i 0 2.73205i 0 0 −0.707107 0.707107i −2.36603 2.36603i 0 0
1007.4 1.36603 1.36603i 0 2.73205i 0 0 0.707107 + 0.707107i −2.36603 2.36603i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1007.4 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.d yes 8
3.b odd 2 1 1575.1.n.c 8
5.b even 2 1 1575.1.n.c 8
5.c odd 4 1 1575.1.n.c 8
5.c odd 4 1 inner 1575.1.n.d yes 8
7.b odd 2 1 CM 1575.1.n.d yes 8
15.d odd 2 1 inner 1575.1.n.d yes 8
15.e even 4 1 1575.1.n.c 8
15.e even 4 1 inner 1575.1.n.d yes 8
21.c even 2 1 1575.1.n.c 8
35.c odd 2 1 1575.1.n.c 8
35.f even 4 1 1575.1.n.c 8
35.f even 4 1 inner 1575.1.n.d yes 8
105.g even 2 1 inner 1575.1.n.d yes 8
105.k odd 4 1 1575.1.n.c 8
105.k odd 4 1 inner 1575.1.n.d yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.n.c 8 3.b odd 2 1
1575.1.n.c 8 5.b even 2 1
1575.1.n.c 8 5.c odd 4 1
1575.1.n.c 8 15.e even 4 1
1575.1.n.c 8 21.c even 2 1
1575.1.n.c 8 35.c odd 2 1
1575.1.n.c 8 35.f even 4 1
1575.1.n.c 8 105.k odd 4 1
1575.1.n.d yes 8 1.a even 1 1 trivial
1575.1.n.d yes 8 5.c odd 4 1 inner
1575.1.n.d yes 8 7.b odd 2 1 CM
1575.1.n.d yes 8 15.d odd 2 1 inner
1575.1.n.d yes 8 15.e even 4 1 inner
1575.1.n.d yes 8 35.f even 4 1 inner
1575.1.n.d yes 8 105.g even 2 1 inner
1575.1.n.d yes 8 105.k odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2 T_{2}^{3} + 2 T_{2}^{2} + 2 T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1575, [\chi])$$.

Hecke characteristic polynomials

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