# Properties

 Label 1620.2.d.c Level $1620$ Weight $2$ Character orbit 1620.d Analytic conductor $12.936$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# 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: $$6$$ Coefficient field: 6.0.301925376.2 Defining polynomial: $$x^{6} + 14 x^{4} + 43 x^{2} + 36$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + ( \beta_{1} + \beta_{3} - \beta_{4} ) q^{7} +O(q^{10})$$ $$q + \beta_{3} q^{5} + ( \beta_{1} + \beta_{3} - \beta_{4} ) q^{7} -\beta_{2} q^{11} + ( -\beta_{1} - \beta_{5} ) q^{13} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{17} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( 3 \beta_{1} + \beta_{3} - \beta_{4} ) q^{23} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{25} + 3 q^{29} + ( -2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{31} + ( -3 + \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{35} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( -3 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{41} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{43} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{47} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{49} + ( -2 \beta_{1} + 4 \beta_{5} ) q^{53} + ( -1 - 3 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{55} + ( 6 + \beta_{3} + \beta_{4} ) q^{59} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{61} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{65} + ( 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{67} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{71} + ( 6 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( -5 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{77} + ( -3 \beta_{3} - 3 \beta_{4} ) q^{79} + ( -\beta_{1} + \beta_{3} - \beta_{4} - 4 \beta_{5} ) q^{83} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{85} + ( 9 - \beta_{3} - \beta_{4} ) q^{89} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{91} + ( -6 - 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{95} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - q^{5} + O(q^{10})$$ $$6q - q^{5} - 2q^{11} + 3q^{25} + 18q^{29} - 6q^{31} - 17q^{35} - 14q^{41} - 3q^{55} + 34q^{59} - 6q^{61} - 15q^{65} + 6q^{79} + 12q^{85} + 56q^{89} + 6q^{91} - 36q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 14 x^{4} + 43 x^{2} + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 6 \nu^{4} + 14 \nu^{3} + 72 \nu^{2} + 31 \nu + 114$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + 6 \nu^{4} - 14 \nu^{3} + 72 \nu^{2} - 31 \nu + 114$$$$)/12$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} - 12 \nu^{3} - 19 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} - 12 \beta_{2} + 41$$ $$\nu^{5}$$ $$=$$ $$-14 \beta_{5} + 36 \beta_{4} - 36 \beta_{3} + 53 \beta_{1}$$

## 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
 − 1.56613i 1.56613i 3.17695i − 3.17695i − 1.20590i 1.20590i
0 0 0 −2.20860 0.349411i 0 2.26496i 0 0 0
649.2 0 0 0 −2.20860 + 0.349411i 0 2.26496i 0 0 0
649.3 0 0 0 −0.123563 2.23265i 0 1.28835i 0 0 0
649.4 0 0 0 −0.123563 + 2.23265i 0 1.28835i 0 0 0
649.5 0 0 0 1.83216 1.28187i 0 3.76963i 0 0 0
649.6 0 0 0 1.83216 + 1.28187i 0 3.76963i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 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.c 6
3.b odd 2 1 1620.2.d.d 6
5.b even 2 1 inner 1620.2.d.c 6
5.c odd 4 2 8100.2.a.bc 6
9.c even 3 2 180.2.r.a 12
9.d odd 6 2 540.2.r.a 12
15.d odd 2 1 1620.2.d.d 6
15.e even 4 2 8100.2.a.bd 6
36.f odd 6 2 720.2.by.e 12
36.h even 6 2 2160.2.by.e 12
45.h odd 6 2 540.2.r.a 12
45.j even 6 2 180.2.r.a 12
45.k odd 12 4 900.2.i.f 12
45.l even 12 4 2700.2.i.f 12
180.n even 6 2 2160.2.by.e 12
180.p odd 6 2 720.2.by.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.r.a 12 9.c even 3 2
180.2.r.a 12 45.j even 6 2
540.2.r.a 12 9.d odd 6 2
540.2.r.a 12 45.h odd 6 2
720.2.by.e 12 36.f odd 6 2
720.2.by.e 12 180.p odd 6 2
900.2.i.f 12 45.k odd 12 4
1620.2.d.c 6 1.a even 1 1 trivial
1620.2.d.c 6 5.b even 2 1 inner
1620.2.d.d 6 3.b odd 2 1
1620.2.d.d 6 15.d odd 2 1
2160.2.by.e 12 36.h even 6 2
2160.2.by.e 12 180.n even 6 2
2700.2.i.f 12 45.l even 12 4
8100.2.a.bc 6 5.c odd 4 2
8100.2.a.bd 6 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 $$S_{2}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{6} + 21 T_{7}^{4} + 105 T_{7}^{2} + 121$$ $$T_{11}^{3} + T_{11}^{2} - 22 T_{11} - 46$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$125 + 25 T - 5 T^{2} + 6 T^{3} - T^{4} + T^{5} + T^{6}$$
$7$ $$121 + 105 T^{2} + 21 T^{4} + T^{6}$$
$11$ $$( -46 - 22 T + T^{2} + T^{3} )^{2}$$
$13$ $$324 + 360 T^{2} + 39 T^{4} + T^{6}$$
$17$ $$64 + 108 T^{2} + 36 T^{4} + T^{6}$$
$19$ $$( 72 - 42 T + T^{3} )^{2}$$
$23$ $$28561 + 2841 T^{2} + 93 T^{4} + T^{6}$$
$29$ $$( -3 + T )^{6}$$
$31$ $$( -358 - 78 T + 3 T^{2} + T^{3} )^{2}$$
$37$ $$256 + 396 T^{2} + 60 T^{4} + T^{6}$$
$41$ $$( -97 - 19 T + 7 T^{2} + T^{3} )^{2}$$
$43$ $$91204 + 14088 T^{2} + 231 T^{4} + T^{6}$$
$47$ $$961 + 957 T^{2} + 105 T^{4} + T^{6}$$
$53$ $$331776 + 20016 T^{2} + 264 T^{4} + T^{6}$$
$59$ $$( -88 + 80 T - 17 T^{2} + T^{3} )^{2}$$
$61$ $$( 31 - 39 T + 3 T^{2} + T^{3} )^{2}$$
$67$ $$14641 + 2541 T^{2} + 105 T^{4} + T^{6}$$
$71$ $$( -72 - 42 T + T^{3} )^{2}$$
$73$ $$369664 + 30144 T^{2} + 384 T^{4} + T^{6}$$
$79$ $$( 108 - 144 T - 3 T^{2} + T^{3} )^{2}$$
$83$ $$1907161 + 49641 T^{2} + 405 T^{4} + T^{6}$$
$89$ $$( -662 + 245 T - 28 T^{2} + T^{3} )^{2}$$
$97$ $$55696 + 8568 T^{2} + 339 T^{4} + T^{6}$$