# Properties

 Label 4032.2.v.d Level 4032 Weight 2 Character orbit 4032.v Analytic conductor 32.196 Analytic rank 0 Dimension 36 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: $$36$$ Relative dimension: $$18$$ over $$\Q(i)$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 36q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 36q^{7} - 16q^{13} + 16q^{19} + 20q^{37} - 36q^{43} + 36q^{49} - 32q^{55} + 112q^{61} + 36q^{67} - 96q^{85} - 16q^{91} + 56q^{97} + O(q^{100})$$

## 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 −2.98923 2.98923i 0 1.00000 0 0 0
1583.2 0 0 0 −2.53823 2.53823i 0 1.00000 0 0 0
1583.3 0 0 0 −2.28967 2.28967i 0 1.00000 0 0 0
1583.4 0 0 0 −2.01396 2.01396i 0 1.00000 0 0 0
1583.5 0 0 0 −1.68827 1.68827i 0 1.00000 0 0 0
1583.6 0 0 0 −1.18126 1.18126i 0 1.00000 0 0 0
1583.7 0 0 0 −0.871498 0.871498i 0 1.00000 0 0 0
1583.8 0 0 0 −0.495166 0.495166i 0 1.00000 0 0 0
1583.9 0 0 0 −0.270063 0.270063i 0 1.00000 0 0 0
1583.10 0 0 0 0.270063 + 0.270063i 0 1.00000 0 0 0
1583.11 0 0 0 0.495166 + 0.495166i 0 1.00000 0 0 0
1583.12 0 0 0 0.871498 + 0.871498i 0 1.00000 0 0 0
1583.13 0 0 0 1.18126 + 1.18126i 0 1.00000 0 0 0
1583.14 0 0 0 1.68827 + 1.68827i 0 1.00000 0 0 0
1583.15 0 0 0 2.01396 + 2.01396i 0 1.00000 0 0 0
1583.16 0 0 0 2.28967 + 2.28967i 0 1.00000 0 0 0
1583.17 0 0 0 2.53823 + 2.53823i 0 1.00000 0 0 0
1583.18 0 0 0 2.98923 + 2.98923i 0 1.00000 0 0 0
3599.1 0 0 0 −2.98923 + 2.98923i 0 1.00000 0 0 0
3599.2 0 0 0 −2.53823 + 2.53823i 0 1.00000 0 0 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3599.18 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.d 36
3.b odd 2 1 inner 4032.2.v.d 36
4.b odd 2 1 1008.2.v.d 36
12.b even 2 1 1008.2.v.d 36
16.e even 4 1 1008.2.v.d 36
16.f odd 4 1 inner 4032.2.v.d 36
48.i odd 4 1 1008.2.v.d 36
48.k even 4 1 inner 4032.2.v.d 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.d 36 4.b odd 2 1
1008.2.v.d 36 12.b even 2 1
1008.2.v.d 36 16.e even 4 1
1008.2.v.d 36 48.i odd 4 1
4032.2.v.d 36 1.a even 1 1 trivial
4032.2.v.d 36 3.b odd 2 1 inner
4032.2.v.d 36 16.f odd 4 1 inner
4032.2.v.d 36 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 $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}^{36} + \cdots$$ $$T_{11}^{36} + \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database