# Properties

 Label 1620.2.r.g Level $1620$ Weight $2$ Character orbit 1620.r 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.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.3317760000.2 Defining polynomial: $$x^{8} - 25 x^{4} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 540) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} + \beta_{7} ) q^{5} + ( \beta_{2} - \beta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{3} + \beta_{7} ) q^{5} + ( \beta_{2} - \beta_{6} ) q^{7} + ( -\beta_{1} - \beta_{7} ) q^{11} + 3 \beta_{2} q^{13} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{17} -3 q^{19} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{23} + ( 5 \beta_{2} - 5 \beta_{6} ) q^{25} + ( -3 \beta_{1} - 3 \beta_{7} ) q^{29} + ( 2 - 2 \beta_{4} ) q^{31} + ( -\beta_{1} + \beta_{5} ) q^{35} + \beta_{6} q^{37} + ( -\beta_{3} + \beta_{5} + \beta_{7} ) q^{41} + ( -10 \beta_{2} + 10 \beta_{6} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{7} ) q^{47} + ( -6 + 6 \beta_{4} ) q^{49} + ( -3 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} ) q^{53} + ( 5 + 5 \beta_{6} ) q^{55} + ( 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{59} + \beta_{4} q^{61} -3 \beta_{1} q^{65} -11 \beta_{2} q^{67} + ( -3 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} ) q^{71} -13 \beta_{6} q^{73} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{77} -3 \beta_{4} q^{79} + ( -5 \beta_{1} + 5 \beta_{7} ) q^{83} + ( 10 - 10 \beta_{2} - 10 \beta_{4} ) q^{85} + ( -4 \beta_{1} - 4 \beta_{3} + 4 \beta_{5} ) q^{89} + 3 q^{91} + ( 3 \beta_{3} - 3 \beta_{7} ) q^{95} + ( -\beta_{2} + \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q - 24 q^{19} + 8 q^{31} - 24 q^{49} + 40 q^{55} + 4 q^{61} - 12 q^{79} + 40 q^{85} + 24 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 25 x^{4} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/25$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/25$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/125$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/125$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$25 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$25 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$125 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$125 \beta_{7}$$

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 2.15988 + 0.578737i 0.578737 − 2.15988i −0.578737 + 2.15988i −2.15988 − 0.578737i 2.15988 − 0.578737i 0.578737 + 2.15988i −0.578737 − 2.15988i −2.15988 + 0.578737i
0 0 0 −2.15988 + 0.578737i 0 0.866025 0.500000i 0 0 0
109.2 0 0 0 −0.578737 2.15988i 0 −0.866025 + 0.500000i 0 0 0
109.3 0 0 0 0.578737 + 2.15988i 0 −0.866025 + 0.500000i 0 0 0
109.4 0 0 0 2.15988 0.578737i 0 0.866025 0.500000i 0 0 0
1189.1 0 0 0 −2.15988 0.578737i 0 0.866025 + 0.500000i 0 0 0
1189.2 0 0 0 −0.578737 + 2.15988i 0 −0.866025 0.500000i 0 0 0
1189.3 0 0 0 0.578737 2.15988i 0 −0.866025 0.500000i 0 0 0
1189.4 0 0 0 2.15988 + 0.578737i 0 0.866025 + 0.500000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1189.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.r.g 8
3.b odd 2 1 inner 1620.2.r.g 8
5.b even 2 1 inner 1620.2.r.g 8
9.c even 3 1 540.2.d.c 4
9.c even 3 1 inner 1620.2.r.g 8
9.d odd 6 1 540.2.d.c 4
9.d odd 6 1 inner 1620.2.r.g 8
15.d odd 2 1 inner 1620.2.r.g 8
36.f odd 6 1 2160.2.f.l 4
36.h even 6 1 2160.2.f.l 4
45.h odd 6 1 540.2.d.c 4
45.h odd 6 1 inner 1620.2.r.g 8
45.j even 6 1 540.2.d.c 4
45.j even 6 1 inner 1620.2.r.g 8
45.k odd 12 1 2700.2.a.v 2
45.k odd 12 1 2700.2.a.w 2
45.l even 12 1 2700.2.a.v 2
45.l even 12 1 2700.2.a.w 2
180.n even 6 1 2160.2.f.l 4
180.p odd 6 1 2160.2.f.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.d.c 4 9.c even 3 1
540.2.d.c 4 9.d odd 6 1
540.2.d.c 4 45.h odd 6 1
540.2.d.c 4 45.j even 6 1
1620.2.r.g 8 1.a even 1 1 trivial
1620.2.r.g 8 3.b odd 2 1 inner
1620.2.r.g 8 5.b even 2 1 inner
1620.2.r.g 8 9.c even 3 1 inner
1620.2.r.g 8 9.d odd 6 1 inner
1620.2.r.g 8 15.d odd 2 1 inner
1620.2.r.g 8 45.h odd 6 1 inner
1620.2.r.g 8 45.j even 6 1 inner
2160.2.f.l 4 36.f odd 6 1
2160.2.f.l 4 36.h even 6 1
2160.2.f.l 4 180.n even 6 1
2160.2.f.l 4 180.p odd 6 1
2700.2.a.v 2 45.k odd 12 1
2700.2.a.v 2 45.l even 12 1
2700.2.a.w 2 45.k odd 12 1
2700.2.a.w 2 45.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{4} - T_{7}^{2} + 1$$ $$T_{11}^{4} + 10 T_{11}^{2} + 100$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$625 - 25 T^{4} + T^{8}$$
$7$ $$( 1 - T^{2} + T^{4} )^{2}$$
$11$ $$( 100 + 10 T^{2} + T^{4} )^{2}$$
$13$ $$( 81 - 9 T^{2} + T^{4} )^{2}$$
$17$ $$( 40 + T^{2} )^{4}$$
$19$ $$( 3 + T )^{8}$$
$23$ $$( 100 - 10 T^{2} + T^{4} )^{2}$$
$29$ $$( 8100 + 90 T^{2} + T^{4} )^{2}$$
$31$ $$( 4 - 2 T + T^{2} )^{4}$$
$37$ $$( 1 + T^{2} )^{4}$$
$41$ $$( 100 + 10 T^{2} + T^{4} )^{2}$$
$43$ $$( 10000 - 100 T^{2} + T^{4} )^{2}$$
$47$ $$( 1600 - 40 T^{2} + T^{4} )^{2}$$
$53$ $$( 90 + T^{2} )^{4}$$
$59$ $$( 1600 + 40 T^{2} + T^{4} )^{2}$$
$61$ $$( 1 - T + T^{2} )^{4}$$
$67$ $$( 14641 - 121 T^{2} + T^{4} )^{2}$$
$71$ $$( -90 + T^{2} )^{4}$$
$73$ $$( 169 + T^{2} )^{4}$$
$79$ $$( 9 + 3 T + T^{2} )^{4}$$
$83$ $$( 62500 - 250 T^{2} + T^{4} )^{2}$$
$89$ $$( -160 + T^{2} )^{4}$$
$97$ $$( 1 - T^{2} + T^{4} )^{2}$$