# Properties

 Label 432.2.k.d Level 432 Weight 2 Character orbit 432.k 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.k (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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 8q^{10} + 24q^{16} + 16q^{19} + 32q^{22} + 24q^{28} - 8q^{34} + 56q^{40} - 16q^{43} - 32q^{49} - 16q^{52} - 32q^{61} + 24q^{64} + 32q^{67} - 96q^{70} - 48q^{76} - 32q^{79} + 32q^{85} - 88q^{88} - 48q^{91} - 96q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 −1.41384 0.0323775i 0 1.99790 + 0.0915534i −1.82760 + 1.82760i 0 1.81881i −2.82176 0.194129i 0 2.64311 2.52477i
109.2 −1.36223 + 0.379905i 0 1.71134 1.03504i 2.51567 2.51567i 0 4.16864i −1.93803 + 2.06011i 0 −2.47121 + 4.38264i
109.3 −1.27588 0.610024i 0 1.25574 + 1.55664i −0.486359 + 0.486359i 0 0.822162i −0.652590 2.75211i 0 0.917226 0.323845i
109.4 −1.24081 + 0.678517i 0 1.07923 1.68382i 0.920303 0.920303i 0 4.02115i −0.196619 + 2.82158i 0 −0.517482 + 1.76636i
109.5 −0.636726 + 1.26277i 0 −1.18916 1.60807i −1.27540 + 1.27540i 0 0.0285435i 2.78779 0.477727i 0 −0.798453 2.42262i
109.6 −0.524142 + 1.31350i 0 −1.45055 1.37692i 1.14284 1.14284i 0 4.40079i 2.56887 1.18359i 0 0.902107 + 2.10013i
109.7 −0.451925 1.34006i 0 −1.59153 + 1.21121i 2.86290 2.86290i 0 1.20181i 2.34235 + 1.58537i 0 −5.13028 2.54265i
109.8 −0.305792 1.38076i 0 −1.81298 + 0.844450i −1.45562 + 1.45562i 0 2.37837i 1.72038 + 2.24506i 0 2.45498 + 1.56474i
109.9 0.305792 + 1.38076i 0 −1.81298 + 0.844450i 1.45562 1.45562i 0 2.37837i −1.72038 2.24506i 0 2.45498 + 1.56474i
109.10 0.451925 + 1.34006i 0 −1.59153 + 1.21121i −2.86290 + 2.86290i 0 1.20181i −2.34235 1.58537i 0 −5.13028 2.54265i
109.11 0.524142 1.31350i 0 −1.45055 1.37692i −1.14284 + 1.14284i 0 4.40079i −2.56887 + 1.18359i 0 0.902107 + 2.10013i
109.12 0.636726 1.26277i 0 −1.18916 1.60807i 1.27540 1.27540i 0 0.0285435i −2.78779 + 0.477727i 0 −0.798453 2.42262i
109.13 1.24081 0.678517i 0 1.07923 1.68382i −0.920303 + 0.920303i 0 4.02115i 0.196619 2.82158i 0 −0.517482 + 1.76636i
109.14 1.27588 + 0.610024i 0 1.25574 + 1.55664i 0.486359 0.486359i 0 0.822162i 0.652590 + 2.75211i 0 0.917226 0.323845i
109.15 1.36223 0.379905i 0 1.71134 1.03504i −2.51567 + 2.51567i 0 4.16864i 1.93803 2.06011i 0 −2.47121 + 4.38264i
109.16 1.41384 + 0.0323775i 0 1.99790 + 0.0915534i 1.82760 1.82760i 0 1.81881i 2.82176 + 0.194129i 0 2.64311 2.52477i
325.1 −1.41384 + 0.0323775i 0 1.99790 0.0915534i −1.82760 1.82760i 0 1.81881i −2.82176 + 0.194129i 0 2.64311 + 2.52477i
325.2 −1.36223 0.379905i 0 1.71134 + 1.03504i 2.51567 + 2.51567i 0 4.16864i −1.93803 2.06011i 0 −2.47121 4.38264i
325.3 −1.27588 + 0.610024i 0 1.25574 1.55664i −0.486359 0.486359i 0 0.822162i −0.652590 + 2.75211i 0 0.917226 + 0.323845i
325.4 −1.24081 0.678517i 0 1.07923 + 1.68382i 0.920303 + 0.920303i 0 4.02115i −0.196619 2.82158i 0 −0.517482 1.76636i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 325.16 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.e even 4 1 inner
48.i odd 4 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$:

 $$T_{5}^{32} + \cdots$$ $$T_{11}^{32} + \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database