# Properties

 Label 432.2.l.a 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 4q^{10} - 20q^{16} + 8q^{19} + 4q^{22} - 12q^{28} - 36q^{34} - 12q^{40} + 32q^{43} - 16q^{46} + 32q^{49} - 60q^{52} + 64q^{55} - 48q^{58} - 16q^{61} + 48q^{64} - 32q^{67} - 72q^{70} - 96q^{76} + 40q^{82} - 16q^{85} + 36q^{88} + 24q^{91} - 36q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −1.39769 + 0.215558i 0 1.90707 0.602567i −1.33778 + 1.33778i 0 0.400629 −2.53560 + 1.25329i 0 1.58143 2.15816i
107.2 −1.38130 0.303332i 0 1.81598 + 0.837984i 1.39178 1.39178i 0 −1.33599 −2.25423 1.70835i 0 −2.34464 + 1.50029i
107.3 −1.10377 + 0.884132i 0 0.436623 1.95176i −0.800188 + 0.800188i 0 0.180707 1.24368 + 2.54033i 0 0.175753 1.59070i
107.4 −1.06417 0.931414i 0 0.264934 + 1.98237i 2.29090 2.29090i 0 3.92541 1.56448 2.35636i 0 −4.57170 + 0.304140i
107.5 −0.932009 + 1.06365i 0 −0.262720 1.98267i 2.82387 2.82387i 0 −3.49224 2.35373 + 1.56842i 0 0.371751 + 5.63550i
107.6 −0.715232 1.22002i 0 −0.976886 + 1.74519i −2.47363 + 2.47363i 0 −0.311286 2.82786 0.0563981i 0 4.78709 + 1.24865i
107.7 −0.604480 1.27852i 0 −1.26921 + 1.54568i 0.217410 0.217410i 0 −3.81401 2.74338 + 0.688372i 0 −0.409383 0.146542i
107.8 −0.205193 + 1.39925i 0 −1.91579 0.574231i −0.494369 + 0.494369i 0 4.44678 1.19660 2.56284i 0 −0.590304 0.793186i
107.9 0.205193 1.39925i 0 −1.91579 0.574231i 0.494369 0.494369i 0 4.44678 −1.19660 + 2.56284i 0 −0.590304 0.793186i
107.10 0.604480 + 1.27852i 0 −1.26921 + 1.54568i −0.217410 + 0.217410i 0 −3.81401 −2.74338 0.688372i 0 −0.409383 0.146542i
107.11 0.715232 + 1.22002i 0 −0.976886 + 1.74519i 2.47363 2.47363i 0 −0.311286 −2.82786 + 0.0563981i 0 4.78709 + 1.24865i
107.12 0.932009 1.06365i 0 −0.262720 1.98267i −2.82387 + 2.82387i 0 −3.49224 −2.35373 1.56842i 0 0.371751 + 5.63550i
107.13 1.06417 + 0.931414i 0 0.264934 + 1.98237i −2.29090 + 2.29090i 0 3.92541 −1.56448 + 2.35636i 0 −4.57170 + 0.304140i
107.14 1.10377 0.884132i 0 0.436623 1.95176i 0.800188 0.800188i 0 0.180707 −1.24368 2.54033i 0 0.175753 1.59070i
107.15 1.38130 + 0.303332i 0 1.81598 + 0.837984i −1.39178 + 1.39178i 0 −1.33599 2.25423 + 1.70835i 0 −2.34464 + 1.50029i
107.16 1.39769 0.215558i 0 1.90707 0.602567i 1.33778 1.33778i 0 0.400629 2.53560 1.25329i 0 1.58143 2.15816i
323.1 −1.39769 0.215558i 0 1.90707 + 0.602567i −1.33778 1.33778i 0 0.400629 −2.53560 1.25329i 0 1.58143 + 2.15816i
323.2 −1.38130 + 0.303332i 0 1.81598 0.837984i 1.39178 + 1.39178i 0 −1.33599 −2.25423 + 1.70835i 0 −2.34464 1.50029i
323.3 −1.10377 0.884132i 0 0.436623 + 1.95176i −0.800188 0.800188i 0 0.180707 1.24368 2.54033i 0 0.175753 + 1.59070i
323.4 −1.06417 + 0.931414i 0 0.264934 1.98237i 2.29090 + 2.29090i 0 3.92541 1.56448 + 2.35636i 0 −4.57170 0.304140i
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.a 32
3.b odd 2 1 inner 432.2.l.a 32
4.b odd 2 1 1728.2.l.a 32
12.b even 2 1 1728.2.l.a 32
16.e even 4 1 1728.2.l.a 32
16.f odd 4 1 inner 432.2.l.a 32
48.i odd 4 1 1728.2.l.a 32
48.k even 4 1 inner 432.2.l.a 32

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