# Properties

 Label 1620.2.d.e Level $1620$ Weight $2$ Character orbit 1620.d Analytic conductor $12.936$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.28356903014400.1 Defining polynomial: $$x^{8} - 3 x^{6} + 20 x^{4} - 75 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} -\beta_{2} q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{5} -\beta_{2} q^{7} + ( -\beta_{1} + \beta_{4} + \beta_{6} ) q^{11} + ( -\beta_{3} + \beta_{5} ) q^{13} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{17} + ( -1 + \beta_{3} + \beta_{5} ) q^{19} + ( 2 \beta_{1} + 2 \beta_{6} ) q^{23} + ( 1 + \beta_{2} + \beta_{3} ) q^{25} -3 \beta_{4} q^{29} + ( -1 - \beta_{3} - \beta_{5} ) q^{31} + ( 2 \beta_{6} + \beta_{7} ) q^{35} + ( -\beta_{2} + \beta_{3} - \beta_{5} ) q^{37} + ( -\beta_{1} - \beta_{4} + \beta_{6} ) q^{41} + \beta_{2} q^{43} + ( 2 \beta_{1} + 2 \beta_{6} + \beta_{7} ) q^{47} + ( -7 - 2 \beta_{3} - 2 \beta_{5} ) q^{49} + \beta_{7} q^{53} + ( 5 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{55} + ( -\beta_{1} + 3 \beta_{4} + \beta_{6} ) q^{59} + ( 4 + \beta_{3} + \beta_{5} ) q^{61} + ( 5 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{65} + ( -2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{67} + ( -3 \beta_{1} - \beta_{4} + 3 \beta_{6} ) q^{71} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{73} + ( 2 \beta_{1} + 2 \beta_{6} + \beta_{7} ) q^{77} -6 q^{79} + ( 2 \beta_{1} + 2 \beta_{6} - \beta_{7} ) q^{83} + ( 2 + 2 \beta_{2} - 3 \beta_{3} ) q^{85} + ( -2 \beta_{1} + 3 \beta_{4} + 2 \beta_{6} ) q^{89} + ( 4 + 2 \beta_{3} + 2 \beta_{5} ) q^{91} + ( 3 \beta_{1} - 5 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{95} + ( \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q - 12 q^{19} + 6 q^{25} - 4 q^{31} - 48 q^{49} + 40 q^{55} + 28 q^{61} - 48 q^{79} + 22 q^{85} + 24 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{6} + 20 x^{4} - 75 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} + 30 \nu^{2} - 25$$$$)/50$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 2 \nu^{4} + 20 \nu^{2} - 25$$$$)/50$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{5} + 20 \nu^{3} + 25 \nu$$$$)/250$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 8 \nu^{4} - 10 \nu^{2} + 75$$$$)/50$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{5} + 20 \nu^{3} - 75 \nu$$$$)/125$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 22 \nu^{5} + 70 \nu^{3} + 175 \nu$$$$)/250$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} + 5 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{5} - 2 \beta_{3} + 3 \beta_{2} - 7$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{7} - 9 \beta_{6} - 10 \beta_{4} - 10 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{5} - 26 \beta_{3} + 14 \beta_{2} + 9$$ $$\nu^{7}$$ $$=$$ $$4 \beta_{7} + 58 \beta_{6} - 130 \beta_{4} + 45 \beta_{1}$$

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 2.07237 + 0.839805i 2.07237 − 0.839805i 1.20635 + 1.88274i 1.20635 − 1.88274i −1.20635 + 1.88274i −1.20635 − 1.88274i −2.07237 + 0.839805i −2.07237 − 0.839805i
0 0 0 −2.07237 0.839805i 0 4.93536i 0 0 0
649.2 0 0 0 −2.07237 + 0.839805i 0 4.93536i 0 0 0
649.3 0 0 0 −1.20635 1.88274i 0 1.28148i 0 0 0
649.4 0 0 0 −1.20635 + 1.88274i 0 1.28148i 0 0 0
649.5 0 0 0 1.20635 1.88274i 0 1.28148i 0 0 0
649.6 0 0 0 1.20635 + 1.88274i 0 1.28148i 0 0 0
649.7 0 0 0 2.07237 0.839805i 0 4.93536i 0 0 0
649.8 0 0 0 2.07237 + 0.839805i 0 4.93536i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.8 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
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.d.e 8
3.b odd 2 1 inner 1620.2.d.e 8
5.b even 2 1 inner 1620.2.d.e 8
5.c odd 4 2 8100.2.a.be 8
9.c even 3 2 1620.2.r.h 16
9.d odd 6 2 1620.2.r.h 16
15.d odd 2 1 inner 1620.2.d.e 8
15.e even 4 2 8100.2.a.be 8
45.h odd 6 2 1620.2.r.h 16
45.j even 6 2 1620.2.r.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.d.e 8 1.a even 1 1 trivial
1620.2.d.e 8 3.b odd 2 1 inner
1620.2.d.e 8 5.b even 2 1 inner
1620.2.d.e 8 15.d odd 2 1 inner
1620.2.r.h 16 9.c even 3 2
1620.2.r.h 16 9.d odd 6 2
1620.2.r.h 16 45.h odd 6 2
1620.2.r.h 16 45.j even 6 2
8100.2.a.be 8 5.c odd 4 2
8100.2.a.be 8 15.e even 4 2

## Hecke kernels

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

 $$T_{7}^{4} + 26 T_{7}^{2} + 40$$ $$T_{11}^{4} - 23 T_{11}^{2} + 100$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$625 - 75 T^{2} + 20 T^{4} - 3 T^{6} + T^{8}$$
$7$ $$( 40 + 26 T^{2} + T^{4} )^{2}$$
$11$ $$( 100 - 23 T^{2} + T^{4} )^{2}$$
$13$ $$( 360 + 51 T^{2} + T^{4} )^{2}$$
$17$ $$( 1690 + 83 T^{2} + T^{4} )^{2}$$
$19$ $$( -30 + 3 T + T^{2} )^{4}$$
$23$ $$( 640 + 68 T^{2} + T^{4} )^{2}$$
$29$ $$( -27 + T^{2} )^{4}$$
$31$ $$( -32 + T + T^{2} )^{4}$$
$37$ $$( 1690 + 89 T^{2} + T^{4} )^{2}$$
$41$ $$( 16 - 35 T^{2} + T^{4} )^{2}$$
$43$ $$( 40 + 26 T^{2} + T^{4} )^{2}$$
$47$ $$( 4000 + 170 T^{2} + T^{4} )^{2}$$
$53$ $$( 360 + 78 T^{2} + T^{4} )^{2}$$
$59$ $$( 64 - 59 T^{2} + T^{4} )^{2}$$
$61$ $$( -20 - 7 T + T^{2} )^{4}$$
$67$ $$( 4000 + 260 T^{2} + T^{4} )^{2}$$
$71$ $$( 6084 - 231 T^{2} + T^{4} )^{2}$$
$73$ $$( 250 + 65 T^{2} + T^{4} )^{2}$$
$79$ $$( 6 + T )^{8}$$
$83$ $$( 2560 + 122 T^{2} + T^{4} )^{2}$$
$89$ $$( 961 - 110 T^{2} + T^{4} )^{2}$$
$97$ $$( 160 + 206 T^{2} + T^{4} )^{2}$$