# Properties

 Label 1620.3.o.g Level $1620$ Weight $3$ Character orbit 1620.o Analytic conductor $44.142$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1620.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$44.1418028264$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 8 q^{7}+O(q^{10})$$ 32 * q - 8 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100})$$ 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1 0 0 0 −1.93649 1.11803i 0 −2.26486 3.92285i 0 0 0
701.2 0 0 0 −1.93649 1.11803i 0 5.78429 + 10.0187i 0 0 0
701.3 0 0 0 −1.93649 1.11803i 0 −0.973755 1.68659i 0 0 0
701.4 0 0 0 −1.93649 1.11803i 0 3.57283 + 6.18832i 0 0 0
701.5 0 0 0 −1.93649 1.11803i 0 −3.54568 6.14130i 0 0 0
701.6 0 0 0 −1.93649 1.11803i 0 −6.81144 11.7978i 0 0 0
701.7 0 0 0 −1.93649 1.11803i 0 3.48846 + 6.04219i 0 0 0
701.8 0 0 0 −1.93649 1.11803i 0 −1.24985 2.16480i 0 0 0
701.9 0 0 0 1.93649 + 1.11803i 0 3.48846 + 6.04219i 0 0 0
701.10 0 0 0 1.93649 + 1.11803i 0 −1.24985 2.16480i 0 0 0
701.11 0 0 0 1.93649 + 1.11803i 0 −6.81144 11.7978i 0 0 0
701.12 0 0 0 1.93649 + 1.11803i 0 3.57283 + 6.18832i 0 0 0
701.13 0 0 0 1.93649 + 1.11803i 0 −3.54568 6.14130i 0 0 0
701.14 0 0 0 1.93649 + 1.11803i 0 5.78429 + 10.0187i 0 0 0
701.15 0 0 0 1.93649 + 1.11803i 0 −0.973755 1.68659i 0 0 0
701.16 0 0 0 1.93649 + 1.11803i 0 −2.26486 3.92285i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −2.26486 + 3.92285i 0 0 0
1241.2 0 0 0 −1.93649 + 1.11803i 0 5.78429 10.0187i 0 0 0
1241.3 0 0 0 −1.93649 + 1.11803i 0 −0.973755 + 1.68659i 0 0 0
1241.4 0 0 0 −1.93649 + 1.11803i 0 3.57283 6.18832i 0 0 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1241.16 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
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.o.g 32
3.b odd 2 1 inner 1620.3.o.g 32
9.c even 3 1 1620.3.g.c 16
9.c even 3 1 inner 1620.3.o.g 32
9.d odd 6 1 1620.3.g.c 16
9.d odd 6 1 inner 1620.3.o.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.3.g.c 16 9.c even 3 1
1620.3.g.c 16 9.d odd 6 1
1620.3.o.g 32 1.a even 1 1 trivial
1620.3.o.g 32 3.b odd 2 1 inner
1620.3.o.g 32 9.c even 3 1 inner
1620.3.o.g 32 9.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{16} + 4 T_{7}^{15} + 258 T_{7}^{14} + 512 T_{7}^{13} + 46349 T_{7}^{12} + 108708 T_{7}^{11} + 3508258 T_{7}^{10} + 11250064 T_{7}^{9} + 198230175 T_{7}^{8} + 684910540 T_{7}^{7} + \cdots + 1509541505956$$ acting on $$S_{3}^{\mathrm{new}}(1620, [\chi])$$.