# Properties

 Label 4020.2.g.c Level 4020 Weight 2 Character orbit 4020.g Analytic conductor 32.100 Analytic rank 0 Dimension 38 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4020.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.0998616126$$ Analytic rank: $$0$$ Dimension: $$38$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$38q - 2q^{5} - 38q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$38q - 2q^{5} - 38q^{9} + 24q^{11} + 2q^{15} - 16q^{19} + 12q^{21} + 4q^{25} - 56q^{29} - 4q^{31} - 2q^{35} + 60q^{41} + 2q^{45} - 70q^{49} + 12q^{55} - 52q^{59} + 48q^{61} + 16q^{65} - 12q^{69} + 12q^{75} - 24q^{79} + 38q^{81} + 16q^{85} - 76q^{89} + 44q^{91} + 36q^{95} - 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1609.1 0 1.00000i 0 −2.23543 + 0.0533841i 0 4.50874i 0 −1.00000 0
1609.2 0 1.00000i 0 −2.22744 + 0.196195i 0 0.128273i 0 −1.00000 0
1609.3 0 1.00000i 0 −2.06455 + 0.858848i 0 3.58783i 0 −1.00000 0
1609.4 0 1.00000i 0 −2.02556 0.947153i 0 2.95752i 0 −1.00000 0
1609.5 0 1.00000i 0 −1.48830 + 1.66882i 0 2.27974i 0 −1.00000 0
1609.6 0 1.00000i 0 −0.953985 2.02235i 0 5.11769i 0 −1.00000 0
1609.7 0 1.00000i 0 −0.847850 + 2.06909i 0 3.67313i 0 −1.00000 0
1609.8 0 1.00000i 0 −0.676225 2.13137i 0 4.23231i 0 −1.00000 0
1609.9 0 1.00000i 0 −0.599682 2.15415i 0 0.0778518i 0 −1.00000 0
1609.10 0 1.00000i 0 −0.504884 2.17832i 0 3.91807i 0 −1.00000 0
1609.11 0 1.00000i 0 −0.471887 + 2.18571i 0 1.05400i 0 −1.00000 0
1609.12 0 1.00000i 0 0.145309 + 2.23134i 0 1.12842i 0 −1.00000 0
1609.13 0 1.00000i 0 1.17292 1.90375i 0 2.37828i 0 −1.00000 0
1609.14 0 1.00000i 0 1.34274 + 1.78803i 0 0.521655i 0 −1.00000 0
1609.15 0 1.00000i 0 1.96436 + 1.06831i 0 3.03191i 0 −1.00000 0
1609.16 0 1.00000i 0 1.96624 + 1.06486i 0 4.32230i 0 −1.00000 0
1609.17 0 1.00000i 0 2.08125 0.817560i 0 3.41849i 0 −1.00000 0
1609.18 0 1.00000i 0 2.20926 0.345224i 0 0.0620642i 0 −1.00000 0
1609.19 0 1.00000i 0 2.21373 + 0.315287i 0 0.0918930i 0 −1.00000 0
1609.20 0 1.00000i 0 −2.23543 0.0533841i 0 4.50874i 0 −1.00000 0
See all 38 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1609.38 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 4020.2.g.c 38
5.b even 2 1 inner 4020.2.g.c 38

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.g.c 38 1.a even 1 1 trivial
4020.2.g.c 38 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{38} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(4020, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database