# Properties

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

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## 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + 4 \zeta_{12} q^{7} +O(q^{10})$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + 4 \zeta_{12} q^{7} + ( -4 + 4 \zeta_{12}^{2} ) q^{11} + 4 \zeta_{12}^{3} q^{17} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{23} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} + ( 6 - 6 \zeta_{12}^{2} ) q^{29} -4 \zeta_{12}^{2} q^{31} + ( -8 + 4 \zeta_{12}^{3} ) q^{35} + 8 \zeta_{12}^{3} q^{37} -10 \zeta_{12}^{2} q^{41} -4 \zeta_{12} q^{43} + 4 \zeta_{12} q^{47} + 9 \zeta_{12}^{2} q^{49} + 12 \zeta_{12}^{3} q^{53} + ( -4 - 8 \zeta_{12}^{3} ) q^{55} -4 \zeta_{12}^{2} q^{59} + ( -2 + 2 \zeta_{12}^{2} ) q^{61} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{67} -8 \zeta_{12}^{3} q^{73} + ( -16 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{77} + ( -12 + 12 \zeta_{12}^{2} ) q^{79} + 4 \zeta_{12} q^{83} + ( -4 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{85} -10 q^{89} + 8 \zeta_{12} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} + O(q^{10})$$ $$4q + 2q^{5} - 8q^{11} + 6q^{25} + 12q^{29} - 8q^{31} - 32q^{35} - 20q^{41} + 18q^{49} - 16q^{55} - 8q^{59} - 4q^{61} - 24q^{79} - 16q^{85} - 40q^{89} + O(q^{100})$$

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 −1.23205 1.86603i 0 3.46410 2.00000i 0 0 0
109.2 0 0 0 2.23205 + 0.133975i 0 −3.46410 + 2.00000i 0 0 0
1189.1 0 0 0 −1.23205 + 1.86603i 0 3.46410 + 2.00000i 0 0 0
1189.2 0 0 0 2.23205 0.133975i 0 −3.46410 2.00000i 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
5.b even 2 1 inner
9.c even 3 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.d 4
3.b odd 2 1 1620.2.r.c 4
5.b even 2 1 inner 1620.2.r.d 4
9.c even 3 1 180.2.d.a 2
9.c even 3 1 inner 1620.2.r.d 4
9.d odd 6 1 60.2.d.a 2
9.d odd 6 1 1620.2.r.c 4
15.d odd 2 1 1620.2.r.c 4
36.f odd 6 1 720.2.f.c 2
36.h even 6 1 240.2.f.b 2
45.h odd 6 1 60.2.d.a 2
45.h odd 6 1 1620.2.r.c 4
45.j even 6 1 180.2.d.a 2
45.j even 6 1 inner 1620.2.r.d 4
45.k odd 12 1 900.2.a.a 1
45.k odd 12 1 900.2.a.h 1
45.l even 12 1 300.2.a.a 1
45.l even 12 1 300.2.a.d 1
63.i even 6 1 2940.2.bb.e 4
63.j odd 6 1 2940.2.bb.d 4
63.n odd 6 1 2940.2.bb.d 4
63.o even 6 1 2940.2.k.c 2
63.s even 6 1 2940.2.bb.e 4
72.j odd 6 1 960.2.f.f 2
72.l even 6 1 960.2.f.c 2
72.n even 6 1 2880.2.f.l 2
72.p odd 6 1 2880.2.f.p 2
144.u even 12 1 3840.2.d.b 2
144.u even 12 1 3840.2.d.be 2
144.w odd 12 1 3840.2.d.o 2
144.w odd 12 1 3840.2.d.r 2
180.n even 6 1 240.2.f.b 2
180.p odd 6 1 720.2.f.c 2
180.v odd 12 1 1200.2.a.a 1
180.v odd 12 1 1200.2.a.s 1
180.x even 12 1 3600.2.a.d 1
180.x even 12 1 3600.2.a.bm 1
315.u even 6 1 2940.2.bb.e 4
315.v odd 6 1 2940.2.bb.d 4
315.z even 6 1 2940.2.k.c 2
315.bq even 6 1 2940.2.bb.e 4
315.br odd 6 1 2940.2.bb.d 4
360.z odd 6 1 2880.2.f.p 2
360.bd even 6 1 960.2.f.c 2
360.bh odd 6 1 960.2.f.f 2
360.bk even 6 1 2880.2.f.l 2
360.br even 12 1 4800.2.a.bj 1
360.br even 12 1 4800.2.a.bn 1
360.bt odd 12 1 4800.2.a.bf 1
360.bt odd 12 1 4800.2.a.bk 1
720.ch odd 12 1 3840.2.d.o 2
720.ch odd 12 1 3840.2.d.r 2
720.da even 12 1 3840.2.d.b 2
720.da even 12 1 3840.2.d.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 9.d odd 6 1
60.2.d.a 2 45.h odd 6 1
180.2.d.a 2 9.c even 3 1
180.2.d.a 2 45.j even 6 1
240.2.f.b 2 36.h even 6 1
240.2.f.b 2 180.n even 6 1
300.2.a.a 1 45.l even 12 1
300.2.a.d 1 45.l even 12 1
720.2.f.c 2 36.f odd 6 1
720.2.f.c 2 180.p odd 6 1
900.2.a.a 1 45.k odd 12 1
900.2.a.h 1 45.k odd 12 1
960.2.f.c 2 72.l even 6 1
960.2.f.c 2 360.bd even 6 1
960.2.f.f 2 72.j odd 6 1
960.2.f.f 2 360.bh odd 6 1
1200.2.a.a 1 180.v odd 12 1
1200.2.a.s 1 180.v odd 12 1
1620.2.r.c 4 3.b odd 2 1
1620.2.r.c 4 9.d odd 6 1
1620.2.r.c 4 15.d odd 2 1
1620.2.r.c 4 45.h odd 6 1
1620.2.r.d 4 1.a even 1 1 trivial
1620.2.r.d 4 5.b even 2 1 inner
1620.2.r.d 4 9.c even 3 1 inner
1620.2.r.d 4 45.j even 6 1 inner
2880.2.f.l 2 72.n even 6 1
2880.2.f.l 2 360.bk even 6 1
2880.2.f.p 2 72.p odd 6 1
2880.2.f.p 2 360.z odd 6 1
2940.2.k.c 2 63.o even 6 1
2940.2.k.c 2 315.z even 6 1
2940.2.bb.d 4 63.j odd 6 1
2940.2.bb.d 4 63.n odd 6 1
2940.2.bb.d 4 315.v odd 6 1
2940.2.bb.d 4 315.br odd 6 1
2940.2.bb.e 4 63.i even 6 1
2940.2.bb.e 4 63.s even 6 1
2940.2.bb.e 4 315.u even 6 1
2940.2.bb.e 4 315.bq even 6 1
3600.2.a.d 1 180.x even 12 1
3600.2.a.bm 1 180.x even 12 1
3840.2.d.b 2 144.u even 12 1
3840.2.d.b 2 720.da even 12 1
3840.2.d.o 2 144.w odd 12 1
3840.2.d.o 2 720.ch odd 12 1
3840.2.d.r 2 144.w odd 12 1
3840.2.d.r 2 720.ch odd 12 1
3840.2.d.be 2 144.u even 12 1
3840.2.d.be 2 720.da even 12 1
4800.2.a.bf 1 360.bt odd 12 1
4800.2.a.bj 1 360.br even 12 1
4800.2.a.bk 1 360.bt odd 12 1
4800.2.a.bn 1 360.br even 12 1

## Hecke kernels

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

 $$T_{7}^{4} - 16 T_{7}^{2} + 256$$ $$T_{11}^{2} + 4 T_{11} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 - 10 T - T^{2} - 2 T^{3} + T^{4}$$
$7$ $$256 - 16 T^{2} + T^{4}$$
$11$ $$( 16 + 4 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 16 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$256 - 16 T^{2} + T^{4}$$
$29$ $$( 36 - 6 T + T^{2} )^{2}$$
$31$ $$( 16 + 4 T + T^{2} )^{2}$$
$37$ $$( 64 + T^{2} )^{2}$$
$41$ $$( 100 + 10 T + T^{2} )^{2}$$
$43$ $$256 - 16 T^{2} + T^{4}$$
$47$ $$256 - 16 T^{2} + T^{4}$$
$53$ $$( 144 + T^{2} )^{2}$$
$59$ $$( 16 + 4 T + T^{2} )^{2}$$
$61$ $$( 4 + 2 T + T^{2} )^{2}$$
$67$ $$256 - 16 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 64 + T^{2} )^{2}$$
$79$ $$( 144 + 12 T + T^{2} )^{2}$$
$83$ $$256 - 16 T^{2} + T^{4}$$
$89$ $$( 10 + T )^{4}$$
$97$ $$4096 - 64 T^{2} + T^{4}$$
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