# Properties

 Label 1620.1.w.b Level $1620$ Weight $1$ Character orbit 1620.w Analytic conductor $0.808$ Analytic rank $0$ Dimension $8$ Projective image $D_{4}$ CM discriminant -4 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1620.w (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.808485320465$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 180) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.13500.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} -\zeta_{24} q^{5} + \zeta_{24}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} -\zeta_{24} q^{5} + \zeta_{24}^{3} q^{8} + \zeta_{24}^{6} q^{10} + ( \zeta_{24}^{4} - \zeta_{24}^{10} ) q^{13} -\zeta_{24}^{8} q^{16} -\zeta_{24}^{11} q^{20} + \zeta_{24}^{2} q^{25} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{26} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{29} -\zeta_{24} q^{32} + ( -1 - \zeta_{24}^{6} ) q^{37} -\zeta_{24}^{4} q^{40} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{41} -\zeta_{24}^{10} q^{49} -\zeta_{24}^{7} q^{50} + ( -\zeta_{24}^{2} + \zeta_{24}^{8} ) q^{52} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{58} + \zeta_{24}^{6} q^{64} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{65} + ( 1 - \zeta_{24}^{6} ) q^{73} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{74} + \zeta_{24}^{9} q^{80} + ( -1 - \zeta_{24}^{6} ) q^{82} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{89} + ( -\zeta_{24}^{2} + \zeta_{24}^{8} ) q^{97} -\zeta_{24}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 4 q^{13} + 4 q^{16} - 8 q^{37} - 4 q^{40} - 4 q^{52} - 4 q^{58} + 8 q^{73} - 8 q^{82} - 4 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$$.

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$-1$$ $$\zeta_{24}^{6}$$ $$\zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 0.965926 + 0.258819i −0.965926 − 0.258819i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.258819 − 0.965926i −0.258819 + 0.965926i
−0.258819 0.965926i 0 −0.866025 + 0.500000i −0.965926 0.258819i 0 0 0.707107 + 0.707107i 0 1.00000i
107.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0.965926 + 0.258819i 0 0 −0.707107 0.707107i 0 1.00000i
863.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i −0.965926 + 0.258819i 0 0 0.707107 0.707107i 0 1.00000i
863.2 0.258819 0.965926i 0 −0.866025 0.500000i 0.965926 0.258819i 0 0 −0.707107 + 0.707107i 0 1.00000i
1187.1 −0.965926 0.258819i 0 0.866025 + 0.500000i −0.258819 0.965926i 0 0 −0.707107 0.707107i 0 1.00000i
1187.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0.258819 + 0.965926i 0 0 0.707107 + 0.707107i 0 1.00000i
1403.1 −0.965926 + 0.258819i 0 0.866025 0.500000i −0.258819 + 0.965926i 0 0 −0.707107 + 0.707107i 0 1.00000i
1403.2 0.965926 0.258819i 0 0.866025 0.500000i 0.258819 0.965926i 0 0 0.707107 0.707107i 0 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1403.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
5.c odd 4 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner
60.l odd 4 1 inner
180.v odd 12 1 inner
180.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.1.w.b 8
3.b odd 2 1 inner 1620.1.w.b 8
4.b odd 2 1 CM 1620.1.w.b 8
5.c odd 4 1 inner 1620.1.w.b 8
9.c even 3 1 180.1.m.a 4
9.c even 3 1 inner 1620.1.w.b 8
9.d odd 6 1 180.1.m.a 4
9.d odd 6 1 inner 1620.1.w.b 8
12.b even 2 1 inner 1620.1.w.b 8
15.e even 4 1 inner 1620.1.w.b 8
20.e even 4 1 inner 1620.1.w.b 8
36.f odd 6 1 180.1.m.a 4
36.f odd 6 1 inner 1620.1.w.b 8
36.h even 6 1 180.1.m.a 4
36.h even 6 1 inner 1620.1.w.b 8
45.h odd 6 1 900.1.m.a 4
45.j even 6 1 900.1.m.a 4
45.k odd 12 1 180.1.m.a 4
45.k odd 12 1 900.1.m.a 4
45.k odd 12 1 inner 1620.1.w.b 8
45.l even 12 1 180.1.m.a 4
45.l even 12 1 900.1.m.a 4
45.l even 12 1 inner 1620.1.w.b 8
60.l odd 4 1 inner 1620.1.w.b 8
72.j odd 6 1 2880.1.bk.b 4
72.l even 6 1 2880.1.bk.b 4
72.n even 6 1 2880.1.bk.b 4
72.p odd 6 1 2880.1.bk.b 4
180.n even 6 1 900.1.m.a 4
180.p odd 6 1 900.1.m.a 4
180.v odd 12 1 180.1.m.a 4
180.v odd 12 1 900.1.m.a 4
180.v odd 12 1 inner 1620.1.w.b 8
180.x even 12 1 180.1.m.a 4
180.x even 12 1 900.1.m.a 4
180.x even 12 1 inner 1620.1.w.b 8
360.bo even 12 1 2880.1.bk.b 4
360.br even 12 1 2880.1.bk.b 4
360.bt odd 12 1 2880.1.bk.b 4
360.bu odd 12 1 2880.1.bk.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.m.a 4 9.c even 3 1
180.1.m.a 4 9.d odd 6 1
180.1.m.a 4 36.f odd 6 1
180.1.m.a 4 36.h even 6 1
180.1.m.a 4 45.k odd 12 1
180.1.m.a 4 45.l even 12 1
180.1.m.a 4 180.v odd 12 1
180.1.m.a 4 180.x even 12 1
900.1.m.a 4 45.h odd 6 1
900.1.m.a 4 45.j even 6 1
900.1.m.a 4 45.k odd 12 1
900.1.m.a 4 45.l even 12 1
900.1.m.a 4 180.n even 6 1
900.1.m.a 4 180.p odd 6 1
900.1.m.a 4 180.v odd 12 1
900.1.m.a 4 180.x even 12 1
1620.1.w.b 8 1.a even 1 1 trivial
1620.1.w.b 8 3.b odd 2 1 inner
1620.1.w.b 8 4.b odd 2 1 CM
1620.1.w.b 8 5.c odd 4 1 inner
1620.1.w.b 8 9.c even 3 1 inner
1620.1.w.b 8 9.d odd 6 1 inner
1620.1.w.b 8 12.b even 2 1 inner
1620.1.w.b 8 15.e even 4 1 inner
1620.1.w.b 8 20.e even 4 1 inner
1620.1.w.b 8 36.f odd 6 1 inner
1620.1.w.b 8 36.h even 6 1 inner
1620.1.w.b 8 45.k odd 12 1 inner
1620.1.w.b 8 45.l even 12 1 inner
1620.1.w.b 8 60.l odd 4 1 inner
1620.1.w.b 8 180.v odd 12 1 inner
1620.1.w.b 8 180.x even 12 1 inner
2880.1.bk.b 4 72.j odd 6 1
2880.1.bk.b 4 72.l even 6 1
2880.1.bk.b 4 72.n even 6 1
2880.1.bk.b 4 72.p odd 6 1
2880.1.bk.b 4 360.bo even 12 1
2880.1.bk.b 4 360.br even 12 1
2880.1.bk.b 4 360.bt odd 12 1
2880.1.bk.b 4 360.bu odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{4} - 2 T_{13}^{3} + 2 T_{13}^{2} - 4 T_{13} + 4$$ acting on $$S_{1}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

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