# Properties

 Label 4032.2.v.e Level 4032 Weight 2 Character orbit 4032.v Analytic conductor 32.196 Analytic rank 0 Dimension 40 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4032.v (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 1008) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 40q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 40q^{7} - 24q^{13} + 32q^{19} - 8q^{37} - 32q^{43} + 40q^{49} - 48q^{55} - 24q^{61} + 64q^{85} + 24q^{91} + 64q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1583.1 0 0 0 −2.96859 2.96859i 0 −1.00000 0 0 0
1583.2 0 0 0 −2.62814 2.62814i 0 −1.00000 0 0 0
1583.3 0 0 0 −2.51504 2.51504i 0 −1.00000 0 0 0
1583.4 0 0 0 −2.12043 2.12043i 0 −1.00000 0 0 0
1583.5 0 0 0 −1.65702 1.65702i 0 −1.00000 0 0 0
1583.6 0 0 0 −1.17902 1.17902i 0 −1.00000 0 0 0
1583.7 0 0 0 −0.925496 0.925496i 0 −1.00000 0 0 0
1583.8 0 0 0 −0.667815 0.667815i 0 −1.00000 0 0 0
1583.9 0 0 0 −0.111394 0.111394i 0 −1.00000 0 0 0
1583.10 0 0 0 −0.0893433 0.0893433i 0 −1.00000 0 0 0
1583.11 0 0 0 0.0893433 + 0.0893433i 0 −1.00000 0 0 0
1583.12 0 0 0 0.111394 + 0.111394i 0 −1.00000 0 0 0
1583.13 0 0 0 0.667815 + 0.667815i 0 −1.00000 0 0 0
1583.14 0 0 0 0.925496 + 0.925496i 0 −1.00000 0 0 0
1583.15 0 0 0 1.17902 + 1.17902i 0 −1.00000 0 0 0
1583.16 0 0 0 1.65702 + 1.65702i 0 −1.00000 0 0 0
1583.17 0 0 0 2.12043 + 2.12043i 0 −1.00000 0 0 0
1583.18 0 0 0 2.51504 + 2.51504i 0 −1.00000 0 0 0
1583.19 0 0 0 2.62814 + 2.62814i 0 −1.00000 0 0 0
1583.20 0 0 0 2.96859 + 2.96859i 0 −1.00000 0 0 0
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3599.20 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
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.v.e 40
3.b odd 2 1 inner 4032.2.v.e 40
4.b odd 2 1 1008.2.v.e 40
12.b even 2 1 1008.2.v.e 40
16.e even 4 1 1008.2.v.e 40
16.f odd 4 1 inner 4032.2.v.e 40
48.i odd 4 1 1008.2.v.e 40
48.k even 4 1 inner 4032.2.v.e 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.e 40 4.b odd 2 1
1008.2.v.e 40 12.b even 2 1
1008.2.v.e 40 16.e even 4 1
1008.2.v.e 40 48.i odd 4 1
4032.2.v.e 40 1.a even 1 1 trivial
4032.2.v.e 40 3.b odd 2 1 inner
4032.2.v.e 40 16.f odd 4 1 inner
4032.2.v.e 40 48.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}^{40} + \cdots$$ $$T_{11}^{40} + \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database