# Properties

 Label 4032.2.i.c Level 4032 Weight 2 Character orbit 4032.i Analytic conductor 32.196 Analytic rank 0 Dimension 48 CM no Inner twists 16

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4032.i (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$48$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 80q^{25} - 16q^{49} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1889.1 0 0 0 1.77767i 0 −2.39248 1.12962i 0 0 0
1889.2 0 0 0 1.77767i 0 −2.39248 + 1.12962i 0 0 0
1889.3 0 0 0 1.21807i 0 0.355387 2.62177i 0 0 0
1889.4 0 0 0 1.21807i 0 0.355387 + 2.62177i 0 0 0
1889.5 0 0 0 1.77767i 0 −2.39248 1.12962i 0 0 0
1889.6 0 0 0 1.77767i 0 −2.39248 + 1.12962i 0 0 0
1889.7 0 0 0 3.91870i 0 2.03709 1.68827i 0 0 0
1889.8 0 0 0 3.91870i 0 2.03709 + 1.68827i 0 0 0
1889.9 0 0 0 1.21807i 0 0.355387 + 2.62177i 0 0 0
1889.10 0 0 0 1.21807i 0 0.355387 2.62177i 0 0 0
1889.11 0 0 0 1.77767i 0 2.39248 1.12962i 0 0 0
1889.12 0 0 0 1.77767i 0 2.39248 + 1.12962i 0 0 0
1889.13 0 0 0 3.91870i 0 2.03709 1.68827i 0 0 0
1889.14 0 0 0 3.91870i 0 2.03709 + 1.68827i 0 0 0
1889.15 0 0 0 3.91870i 0 2.03709 + 1.68827i 0 0 0
1889.16 0 0 0 3.91870i 0 2.03709 1.68827i 0 0 0
1889.17 0 0 0 3.91870i 0 −2.03709 1.68827i 0 0 0
1889.18 0 0 0 3.91870i 0 −2.03709 + 1.68827i 0 0 0
1889.19 0 0 0 1.77767i 0 −2.39248 + 1.12962i 0 0 0
1889.20 0 0 0 1.77767i 0 −2.39248 1.12962i 0 0 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1889.48 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
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner
84.h odd 2 1 inner
168.e odd 2 1 inner
168.i even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.i.c 48
3.b odd 2 1 inner 4032.2.i.c 48
4.b odd 2 1 inner 4032.2.i.c 48
7.b odd 2 1 inner 4032.2.i.c 48
8.b even 2 1 inner 4032.2.i.c 48
8.d odd 2 1 inner 4032.2.i.c 48
12.b even 2 1 inner 4032.2.i.c 48
21.c even 2 1 inner 4032.2.i.c 48
24.f even 2 1 inner 4032.2.i.c 48
24.h odd 2 1 inner 4032.2.i.c 48
28.d even 2 1 inner 4032.2.i.c 48
56.e even 2 1 inner 4032.2.i.c 48
56.h odd 2 1 inner 4032.2.i.c 48
84.h odd 2 1 inner 4032.2.i.c 48
168.e odd 2 1 inner 4032.2.i.c 48
168.i even 2 1 inner 4032.2.i.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4032.2.i.c 48 1.a even 1 1 trivial
4032.2.i.c 48 3.b odd 2 1 inner
4032.2.i.c 48 4.b odd 2 1 inner
4032.2.i.c 48 7.b odd 2 1 inner
4032.2.i.c 48 8.b even 2 1 inner
4032.2.i.c 48 8.d odd 2 1 inner
4032.2.i.c 48 12.b even 2 1 inner
4032.2.i.c 48 21.c even 2 1 inner
4032.2.i.c 48 24.f even 2 1 inner
4032.2.i.c 48 24.h odd 2 1 inner
4032.2.i.c 48 28.d even 2 1 inner
4032.2.i.c 48 56.e even 2 1 inner
4032.2.i.c 48 56.h odd 2 1 inner
4032.2.i.c 48 84.h odd 2 1 inner
4032.2.i.c 48 168.e odd 2 1 inner
4032.2.i.c 48 168.i even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{6} + 20 T_{5}^{4} + 76 T_{5}^{2} + 72$$ $$T_{47}^{6} - 128 T_{47}^{4} + 3520 T_{47}^{2} - 4608$$