# Properties

 Label 1620.3.t.f Level $1620$ Weight $3$ Character orbit 1620.t Analytic conductor $44.142$ Analytic rank $0$ Dimension $48$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$44.1418028264$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q+O(q^{10})$$ 48 * q $$\operatorname{Tr}(f)(q) =$$ $$48 q + 48 q^{25} + 288 q^{49} - 72 q^{55} - 120 q^{61} - 480 q^{79} + 24 q^{85} + 96 q^{91}+O(q^{100})$$ 48 * q + 48 * q^25 + 288 * q^49 - 72 * q^55 - 120 * q^61 - 480 * q^79 + 24 * q^85 + 96 * q^91

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1 0 0 0 −4.98895 0.332280i 0 −0.607949 0.351000i 0 0 0
269.2 0 0 0 −4.98514 0.385228i 0 −6.51924 3.76389i 0 0 0
269.3 0 0 0 −4.91882 0.897339i 0 10.9644 + 6.33031i 0 0 0
269.4 0 0 0 −4.58175 + 2.00188i 0 9.28893 + 5.36297i 0 0 0
269.5 0 0 0 −4.45514 2.26973i 0 −2.97789 1.71928i 0 0 0
269.6 0 0 0 −4.02456 + 2.96697i 0 −9.28893 5.36297i 0 0 0
269.7 0 0 0 −2.76464 4.16614i 0 −4.03229 2.32805i 0 0 0
269.8 0 0 0 −2.22566 4.47732i 0 4.03229 + 2.32805i 0 0 0
269.9 0 0 0 −2.20671 + 4.48669i 0 0.607949 + 0.351000i 0 0 0
269.10 0 0 0 −2.15895 + 4.50987i 0 6.51924 + 3.76389i 0 0 0
269.11 0 0 0 −1.68229 + 4.70849i 0 −10.9644 6.33031i 0 0 0
269.12 0 0 0 −0.261925 + 4.99313i 0 2.97789 + 1.71928i 0 0 0
269.13 0 0 0 0.261925 4.99313i 0 2.97789 + 1.71928i 0 0 0
269.14 0 0 0 1.68229 4.70849i 0 −10.9644 6.33031i 0 0 0
269.15 0 0 0 2.15895 4.50987i 0 6.51924 + 3.76389i 0 0 0
269.16 0 0 0 2.20671 4.48669i 0 0.607949 + 0.351000i 0 0 0
269.17 0 0 0 2.22566 + 4.47732i 0 4.03229 + 2.32805i 0 0 0
269.18 0 0 0 2.76464 + 4.16614i 0 −4.03229 2.32805i 0 0 0
269.19 0 0 0 4.02456 2.96697i 0 −9.28893 5.36297i 0 0 0
269.20 0 0 0 4.45514 + 2.26973i 0 −2.97789 1.71928i 0 0 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1349.24 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
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 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 1620.3.t.f 48
3.b odd 2 1 inner 1620.3.t.f 48
5.b even 2 1 inner 1620.3.t.f 48
9.c even 3 1 1620.3.b.a 24
9.c even 3 1 inner 1620.3.t.f 48
9.d odd 6 1 1620.3.b.a 24
9.d odd 6 1 inner 1620.3.t.f 48
15.d odd 2 1 inner 1620.3.t.f 48
45.h odd 6 1 1620.3.b.a 24
45.h odd 6 1 inner 1620.3.t.f 48
45.j even 6 1 1620.3.b.a 24
45.j even 6 1 inner 1620.3.t.f 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.3.b.a 24 9.c even 3 1
1620.3.b.a 24 9.d odd 6 1
1620.3.b.a 24 45.h odd 6 1
1620.3.b.a 24 45.j even 6 1
1620.3.t.f 48 1.a even 1 1 trivial
1620.3.t.f 48 3.b odd 2 1 inner
1620.3.t.f 48 5.b even 2 1 inner
1620.3.t.f 48 9.c even 3 1 inner
1620.3.t.f 48 9.d odd 6 1 inner
1620.3.t.f 48 15.d odd 2 1 inner
1620.3.t.f 48 45.h odd 6 1 inner
1620.3.t.f 48 45.j even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{24} - 366 T_{7}^{22} + 88353 T_{7}^{20} - 12104630 T_{7}^{18} + 1195527555 T_{7}^{16} - 72024338430 T_{7}^{14} + 3106792494870 T_{7}^{12} - 76507326662148 T_{7}^{10} + \cdots + 17\!\cdots\!36$$ T7^24 - 366*T7^22 + 88353*T7^20 - 12104630*T7^18 + 1195527555*T7^16 - 72024338430*T7^14 + 3106792494870*T7^12 - 76507326662148*T7^10 + 1342366541599500*T7^8 - 12376530555225872*T7^6 + 77840214971174544*T7^4 - 38203781472698688*T7^2 + 17424859594601536 $$T_{17}^{12} - 1662 T_{17}^{10} + 827391 T_{17}^{8} - 157133482 T_{17}^{6} + 10180375374 T_{17}^{4} - 42366144912 T_{17}^{2} + 41987187424$$ T17^12 - 1662*T17^10 + 827391*T17^8 - 157133482*T17^6 + 10180375374*T17^4 - 42366144912*T17^2 + 41987187424