# Properties

 Label 1620.2.i.h Level $1620$ Weight $2$ Character orbit 1620.i Analytic conductor $12.936$ Analytic rank $1$ Dimension $2$ 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.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.9357651274$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
541.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 0 0
1081.1 0 0 0 0.500000 0.866025i 0 −1.00000 1.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.i.h 2
3.b odd 2 1 1620.2.i.b 2
9.c even 3 1 20.2.a.a 1
9.c even 3 1 inner 1620.2.i.h 2
9.d odd 6 1 180.2.a.a 1
9.d odd 6 1 1620.2.i.b 2
36.f odd 6 1 80.2.a.b 1
36.h even 6 1 720.2.a.h 1
45.h odd 6 1 900.2.a.b 1
45.j even 6 1 100.2.a.a 1
45.k odd 12 2 100.2.c.a 2
45.l even 12 2 900.2.d.c 2
63.g even 3 1 980.2.i.i 2
63.h even 3 1 980.2.i.i 2
63.k odd 6 1 980.2.i.c 2
63.l odd 6 1 980.2.a.h 1
63.o even 6 1 8820.2.a.g 1
63.t odd 6 1 980.2.i.c 2
72.j odd 6 1 2880.2.a.m 1
72.l even 6 1 2880.2.a.f 1
72.n even 6 1 320.2.a.f 1
72.p odd 6 1 320.2.a.a 1
99.h odd 6 1 2420.2.a.a 1
117.t even 6 1 3380.2.a.c 1
117.y odd 12 2 3380.2.f.b 2
144.v odd 12 2 1280.2.d.g 2
144.x even 12 2 1280.2.d.c 2
153.h even 6 1 5780.2.a.f 1
153.n even 12 2 5780.2.c.a 2
171.o odd 6 1 7220.2.a.f 1
180.n even 6 1 3600.2.a.be 1
180.p odd 6 1 400.2.a.c 1
180.v odd 12 2 3600.2.f.j 2
180.x even 12 2 400.2.c.b 2
252.bi even 6 1 3920.2.a.h 1
315.bg odd 6 1 4900.2.a.e 1
315.cb even 12 2 4900.2.e.f 2
360.z odd 6 1 1600.2.a.w 1
360.bk even 6 1 1600.2.a.c 1
360.bo even 12 2 1600.2.c.e 2
360.bu odd 12 2 1600.2.c.d 2
396.k even 6 1 9680.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 9.c even 3 1
80.2.a.b 1 36.f odd 6 1
100.2.a.a 1 45.j even 6 1
100.2.c.a 2 45.k odd 12 2
180.2.a.a 1 9.d odd 6 1
320.2.a.a 1 72.p odd 6 1
320.2.a.f 1 72.n even 6 1
400.2.a.c 1 180.p odd 6 1
400.2.c.b 2 180.x even 12 2
720.2.a.h 1 36.h even 6 1
900.2.a.b 1 45.h odd 6 1
900.2.d.c 2 45.l even 12 2
980.2.a.h 1 63.l odd 6 1
980.2.i.c 2 63.k odd 6 1
980.2.i.c 2 63.t odd 6 1
980.2.i.i 2 63.g even 3 1
980.2.i.i 2 63.h even 3 1
1280.2.d.c 2 144.x even 12 2
1280.2.d.g 2 144.v odd 12 2
1600.2.a.c 1 360.bk even 6 1
1600.2.a.w 1 360.z odd 6 1
1600.2.c.d 2 360.bu odd 12 2
1600.2.c.e 2 360.bo even 12 2
1620.2.i.b 2 3.b odd 2 1
1620.2.i.b 2 9.d odd 6 1
1620.2.i.h 2 1.a even 1 1 trivial
1620.2.i.h 2 9.c even 3 1 inner
2420.2.a.a 1 99.h odd 6 1
2880.2.a.f 1 72.l even 6 1
2880.2.a.m 1 72.j odd 6 1
3380.2.a.c 1 117.t even 6 1
3380.2.f.b 2 117.y odd 12 2
3600.2.a.be 1 180.n even 6 1
3600.2.f.j 2 180.v odd 12 2
3920.2.a.h 1 252.bi even 6 1
4900.2.a.e 1 315.bg odd 6 1
4900.2.e.f 2 315.cb even 12 2
5780.2.a.f 1 153.h even 6 1
5780.2.c.a 2 153.n even 12 2
7220.2.a.f 1 171.o odd 6 1
8820.2.a.g 1 63.o even 6 1
9680.2.a.ba 1 396.k even 6 1

## Hecke kernels

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

 $$T_{7}^{2} + 2 T_{7} + 4$$ $$T_{11}$$ $$T_{17} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$100 - 10 T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$36 + 6 T + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$4 + 2 T + T^{2}$$