# Properties

 Label 432.2.l.b Level 432 Weight 2 Character orbit 432.l Analytic conductor 3.450 Analytic rank 0 Dimension 32 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 432.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{10} - 8q^{16} - 16q^{19} + 16q^{22} + 24q^{28} + 24q^{34} - 24q^{40} - 16q^{43} + 32q^{46} + 32q^{49} + 48q^{52} - 32q^{55} + 32q^{61} - 24q^{64} - 32q^{67} - 48q^{76} - 80q^{82} + 32q^{85} - 24q^{88} - 48q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −1.41076 0.0987658i 0 1.98049 + 0.278670i 0.0308139 0.0308139i 0 2.10616 −2.76648 0.588741i 0 −0.0465144 + 0.0404277i
107.2 −1.32420 + 0.496490i 0 1.50700 1.31490i −1.75903 + 1.75903i 0 −4.05756 −1.34273 + 2.48940i 0 1.45597 3.20265i
107.3 −1.19933 + 0.749398i 0 0.876807 1.79756i 2.15382 2.15382i 0 4.43758 0.295500 + 2.81295i 0 −0.969085 + 4.19723i
107.4 −1.12975 0.850687i 0 0.552664 + 1.92212i −1.26575 + 1.26575i 0 1.47880 1.01076 2.64166i 0 2.50674 0.353222i
107.5 −0.900149 1.09075i 0 −0.379462 + 1.96367i 1.29039 1.29039i 0 −3.83003 2.48344 1.35370i 0 −2.56904 0.245948i
107.6 −0.762934 + 1.19077i 0 −0.835863 1.81696i −3.06203 + 3.06203i 0 1.75946 2.80128 + 0.390900i 0 −1.31004 5.98230i
107.7 −0.379657 + 1.36230i 0 −1.71172 1.03441i 0.463925 0.463925i 0 −1.85883 2.05905 1.93916i 0 0.455873 + 0.808137i
107.8 −0.0710265 + 1.41243i 0 −1.98991 0.200640i 1.84591 1.84591i 0 −0.0355882 0.424726 2.79636i 0 2.47611 + 2.73833i
107.9 0.0710265 1.41243i 0 −1.98991 0.200640i −1.84591 + 1.84591i 0 −0.0355882 −0.424726 + 2.79636i 0 2.47611 + 2.73833i
107.10 0.379657 1.36230i 0 −1.71172 1.03441i −0.463925 + 0.463925i 0 −1.85883 −2.05905 + 1.93916i 0 0.455873 + 0.808137i
107.11 0.762934 1.19077i 0 −0.835863 1.81696i 3.06203 3.06203i 0 1.75946 −2.80128 0.390900i 0 −1.31004 5.98230i
107.12 0.900149 + 1.09075i 0 −0.379462 + 1.96367i −1.29039 + 1.29039i 0 −3.83003 −2.48344 + 1.35370i 0 −2.56904 0.245948i
107.13 1.12975 + 0.850687i 0 0.552664 + 1.92212i 1.26575 1.26575i 0 1.47880 −1.01076 + 2.64166i 0 2.50674 0.353222i
107.14 1.19933 0.749398i 0 0.876807 1.79756i −2.15382 + 2.15382i 0 4.43758 −0.295500 2.81295i 0 −0.969085 + 4.19723i
107.15 1.32420 0.496490i 0 1.50700 1.31490i 1.75903 1.75903i 0 −4.05756 1.34273 2.48940i 0 1.45597 3.20265i
107.16 1.41076 + 0.0987658i 0 1.98049 + 0.278670i −0.0308139 + 0.0308139i 0 2.10616 2.76648 + 0.588741i 0 −0.0465144 + 0.0404277i
323.1 −1.41076 + 0.0987658i 0 1.98049 0.278670i 0.0308139 + 0.0308139i 0 2.10616 −2.76648 + 0.588741i 0 −0.0465144 0.0404277i
323.2 −1.32420 0.496490i 0 1.50700 + 1.31490i −1.75903 1.75903i 0 −4.05756 −1.34273 2.48940i 0 1.45597 + 3.20265i
323.3 −1.19933 0.749398i 0 0.876807 + 1.79756i 2.15382 + 2.15382i 0 4.43758 0.295500 2.81295i 0 −0.969085 4.19723i
323.4 −1.12975 + 0.850687i 0 0.552664 1.92212i −1.26575 1.26575i 0 1.47880 1.01076 + 2.64166i 0 2.50674 + 0.353222i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.16 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 432.2.l.b 32
3.b odd 2 1 inner 432.2.l.b 32
4.b odd 2 1 1728.2.l.b 32
12.b even 2 1 1728.2.l.b 32
16.e even 4 1 1728.2.l.b 32
16.f odd 4 1 inner 432.2.l.b 32
48.i odd 4 1 1728.2.l.b 32
48.k even 4 1 inner 432.2.l.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.l.b 32 1.a even 1 1 trivial
432.2.l.b 32 3.b odd 2 1 inner
432.2.l.b 32 16.f odd 4 1 inner
432.2.l.b 32 48.k even 4 1 inner
1728.2.l.b 32 4.b odd 2 1
1728.2.l.b 32 12.b even 2 1
1728.2.l.b 32 16.e even 4 1
1728.2.l.b 32 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{32} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database