# Properties

 Label 1620.2.x.a Level $1620$ Weight $2$ Character orbit 1620.x Analytic conductor $12.936$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$12.9357651274$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} - 2 \zeta_{24}^{7} ) q^{5} + ( -2 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} ) q^{7} +O(q^{10})$$ $$q + ( \zeta_{24} - 2 \zeta_{24}^{7} ) q^{5} + ( -2 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} ) q^{7} + ( -2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{11} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{13} -2 \zeta_{24}^{3} q^{17} -4 \zeta_{24}^{6} q^{19} -8 \zeta_{24} q^{23} + ( 4 - 3 \zeta_{24}^{2} - 4 \zeta_{24}^{4} ) q^{25} + ( 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{29} + 8 \zeta_{24}^{4} q^{31} + ( 2 \zeta_{24} + 6 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{35} + ( -3 + 3 \zeta_{24}^{6} ) q^{37} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{41} + ( 12 \zeta_{24}^{3} - 12 \zeta_{24}^{7} ) q^{47} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{49} + ( 10 \zeta_{24} - 10 \zeta_{24}^{5} ) q^{53} + ( -6 + 2 \zeta_{24}^{6} ) q^{55} + ( -2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{59} + ( -3 \zeta_{24}^{3} - 9 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{65} + ( -8 \zeta_{24}^{2} - 8 \zeta_{24}^{4} + 8 \zeta_{24}^{6} ) q^{67} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{71} + ( 7 + 7 \zeta_{24}^{6} ) q^{73} -8 \zeta_{24}^{7} q^{77} -12 \zeta_{24}^{5} q^{83} + ( -4 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{85} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{89} + 12 q^{91} + ( -8 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{95} + ( -3 + 3 \zeta_{24}^{2} + 3 \zeta_{24}^{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{7} + O(q^{10})$$ $$8 q - 8 q^{7} - 12 q^{13} + 16 q^{25} + 32 q^{31} - 24 q^{37} - 48 q^{55} - 32 q^{67} + 56 q^{73} - 8 q^{85} + 96 q^{91} - 12 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −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 −2.19067 0.448288i 0 −2.73205 0.732051i 0 0 0
53.2 0 0 0 2.19067 + 0.448288i 0 −2.73205 0.732051i 0 0 0
377.1 0 0 0 −1.48356 1.67303i 0 0.732051 2.73205i 0 0 0
377.2 0 0 0 1.48356 + 1.67303i 0 0.732051 2.73205i 0 0 0
593.1 0 0 0 −1.48356 + 1.67303i 0 0.732051 + 2.73205i 0 0 0
593.2 0 0 0 1.48356 1.67303i 0 0.732051 + 2.73205i 0 0 0
917.1 0 0 0 −2.19067 + 0.448288i 0 −2.73205 + 0.732051i 0 0 0
917.2 0 0 0 2.19067 0.448288i 0 −2.73205 + 0.732051i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 917.2 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 inner
5.c odd 4 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.x.a 8
3.b odd 2 1 inner 1620.2.x.a 8
5.c odd 4 1 inner 1620.2.x.a 8
9.c even 3 1 180.2.j.a 4
9.c even 3 1 inner 1620.2.x.a 8
9.d odd 6 1 180.2.j.a 4
9.d odd 6 1 inner 1620.2.x.a 8
15.e even 4 1 inner 1620.2.x.a 8
36.f odd 6 1 720.2.w.b 4
36.h even 6 1 720.2.w.b 4
45.h odd 6 1 900.2.j.a 4
45.j even 6 1 900.2.j.a 4
45.k odd 12 1 180.2.j.a 4
45.k odd 12 1 900.2.j.a 4
45.k odd 12 1 inner 1620.2.x.a 8
45.l even 12 1 180.2.j.a 4
45.l even 12 1 900.2.j.a 4
45.l even 12 1 inner 1620.2.x.a 8
72.j odd 6 1 2880.2.w.j 4
72.l even 6 1 2880.2.w.a 4
72.n even 6 1 2880.2.w.j 4
72.p odd 6 1 2880.2.w.a 4
180.n even 6 1 3600.2.w.f 4
180.p odd 6 1 3600.2.w.f 4
180.v odd 12 1 720.2.w.b 4
180.v odd 12 1 3600.2.w.f 4
180.x even 12 1 720.2.w.b 4
180.x even 12 1 3600.2.w.f 4
360.bo even 12 1 2880.2.w.a 4
360.br even 12 1 2880.2.w.j 4
360.bt odd 12 1 2880.2.w.a 4
360.bu odd 12 1 2880.2.w.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.j.a 4 9.c even 3 1
180.2.j.a 4 9.d odd 6 1
180.2.j.a 4 45.k odd 12 1
180.2.j.a 4 45.l even 12 1
720.2.w.b 4 36.f odd 6 1
720.2.w.b 4 36.h even 6 1
720.2.w.b 4 180.v odd 12 1
720.2.w.b 4 180.x even 12 1
900.2.j.a 4 45.h odd 6 1
900.2.j.a 4 45.j even 6 1
900.2.j.a 4 45.k odd 12 1
900.2.j.a 4 45.l even 12 1
1620.2.x.a 8 1.a even 1 1 trivial
1620.2.x.a 8 3.b odd 2 1 inner
1620.2.x.a 8 5.c odd 4 1 inner
1620.2.x.a 8 9.c even 3 1 inner
1620.2.x.a 8 9.d odd 6 1 inner
1620.2.x.a 8 15.e even 4 1 inner
1620.2.x.a 8 45.k odd 12 1 inner
1620.2.x.a 8 45.l even 12 1 inner
2880.2.w.a 4 72.l even 6 1
2880.2.w.a 4 72.p odd 6 1
2880.2.w.a 4 360.bo even 12 1
2880.2.w.a 4 360.bt odd 12 1
2880.2.w.j 4 72.j odd 6 1
2880.2.w.j 4 72.n even 6 1
2880.2.w.j 4 360.br even 12 1
2880.2.w.j 4 360.bu odd 12 1
3600.2.w.f 4 180.n even 6 1
3600.2.w.f 4 180.p odd 6 1
3600.2.w.f 4 180.v odd 12 1
3600.2.w.f 4 180.x even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 4 T_{7}^{3} + 8 T_{7}^{2} + 32 T_{7} + 64$$ acting on $$S_{2}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$625 - 200 T^{2} + 39 T^{4} - 8 T^{6} + T^{8}$$
$7$ $$( 64 + 32 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$11$ $$( 64 - 8 T^{2} + T^{4} )^{2}$$
$13$ $$( 324 + 108 T + 18 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$17$ $$( 16 + T^{4} )^{2}$$
$19$ $$( 16 + T^{2} )^{4}$$
$23$ $$16777216 - 4096 T^{4} + T^{8}$$
$29$ $$( 9604 + 98 T^{2} + T^{4} )^{2}$$
$31$ $$( 64 - 8 T + T^{2} )^{4}$$
$37$ $$( 18 + 6 T + T^{2} )^{4}$$
$41$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$43$ $$T^{8}$$
$47$ $$429981696 - 20736 T^{4} + T^{8}$$
$53$ $$( 10000 + T^{4} )^{2}$$
$59$ $$( 64 + 8 T^{2} + T^{4} )^{2}$$
$61$ $$T^{8}$$
$67$ $$( 16384 + 2048 T + 128 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$71$ $$( 32 + T^{2} )^{4}$$
$73$ $$( 98 - 14 T + T^{2} )^{4}$$
$79$ $$T^{8}$$
$83$ $$429981696 - 20736 T^{4} + T^{8}$$
$89$ $$( -2 + T^{2} )^{4}$$
$97$ $$( 324 + 108 T + 18 T^{2} + 6 T^{3} + T^{4} )^{2}$$