# Properties

 Label 441.4.p.d Level $441$ Weight $4$ Character orbit 441.p Analytic conductor $26.020$ Analytic rank $0$ Dimension $48$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 96q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 96q^{4} - 144q^{16} + 1248q^{22} - 312q^{25} + 864q^{37} + 2496q^{43} + 3888q^{46} + 7440q^{58} - 6720q^{64} + 2688q^{67} - 480q^{79} + 26496q^{85} + 7248q^{88} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1 −4.66169 + 2.69143i 0 10.4876 18.1650i −9.16288 15.8706i 0 0 69.8432i 0 85.4290 + 49.3225i
80.2 −4.66169 + 2.69143i 0 10.4876 18.1650i 9.16288 + 15.8706i 0 0 69.8432i 0 −85.4290 49.3225i
80.3 −3.21193 + 1.85441i 0 2.87764 4.98422i −2.27338 3.93760i 0 0 8.32523i 0 14.6038 + 8.43153i
80.4 −3.21193 + 1.85441i 0 2.87764 4.98422i 2.27338 + 3.93760i 0 0 8.32523i 0 −14.6038 8.43153i
80.5 −3.16408 + 1.82678i 0 2.67425 4.63194i −8.69951 15.0680i 0 0 9.68739i 0 55.0518 + 31.7842i
80.6 −3.16408 + 1.82678i 0 2.67425 4.63194i 8.69951 + 15.0680i 0 0 9.68739i 0 −55.0518 31.7842i
80.7 −3.10847 + 1.79468i 0 2.44173 4.22919i 0.428649 + 0.742442i 0 0 11.1864i 0 −2.66489 1.53857i
80.8 −3.10847 + 1.79468i 0 2.44173 4.22919i −0.428649 0.742442i 0 0 11.1864i 0 2.66489 + 1.53857i
80.9 −1.40358 + 0.810356i 0 −2.68665 + 4.65341i −2.35993 4.08752i 0 0 21.6743i 0 6.62469 + 3.82477i
80.10 −1.40358 + 0.810356i 0 −2.68665 + 4.65341i 2.35993 + 4.08752i 0 0 21.6743i 0 −6.62469 3.82477i
80.11 −0.555155 + 0.320519i 0 −3.79454 + 6.57233i −6.03646 10.4555i 0 0 9.99318i 0 6.70234 + 3.86960i
80.12 −0.555155 + 0.320519i 0 −3.79454 + 6.57233i 6.03646 + 10.4555i 0 0 9.99318i 0 −6.70234 3.86960i
80.13 0.555155 0.320519i 0 −3.79454 + 6.57233i −6.03646 10.4555i 0 0 9.99318i 0 −6.70234 3.86960i
80.14 0.555155 0.320519i 0 −3.79454 + 6.57233i 6.03646 + 10.4555i 0 0 9.99318i 0 6.70234 + 3.86960i
80.15 1.40358 0.810356i 0 −2.68665 + 4.65341i −2.35993 4.08752i 0 0 21.6743i 0 −6.62469 3.82477i
80.16 1.40358 0.810356i 0 −2.68665 + 4.65341i 2.35993 + 4.08752i 0 0 21.6743i 0 6.62469 + 3.82477i
80.17 3.10847 1.79468i 0 2.44173 4.22919i 0.428649 + 0.742442i 0 0 11.1864i 0 2.66489 + 1.53857i
80.18 3.10847 1.79468i 0 2.44173 4.22919i −0.428649 0.742442i 0 0 11.1864i 0 −2.66489 1.53857i
80.19 3.16408 1.82678i 0 2.67425 4.63194i −8.69951 15.0680i 0 0 9.68739i 0 −55.0518 31.7842i
80.20 3.16408 1.82678i 0 2.67425 4.63194i 8.69951 + 15.0680i 0 0 9.68739i 0 55.0518 + 31.7842i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 215.24 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.p.d 48
3.b odd 2 1 inner 441.4.p.d 48
7.b odd 2 1 inner 441.4.p.d 48
7.c even 3 1 441.4.c.b 24
7.c even 3 1 inner 441.4.p.d 48
7.d odd 6 1 441.4.c.b 24
7.d odd 6 1 inner 441.4.p.d 48
21.c even 2 1 inner 441.4.p.d 48
21.g even 6 1 441.4.c.b 24
21.g even 6 1 inner 441.4.p.d 48
21.h odd 6 1 441.4.c.b 24
21.h odd 6 1 inner 441.4.p.d 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.4.c.b 24 7.c even 3 1
441.4.c.b 24 7.d odd 6 1
441.4.c.b 24 21.g even 6 1
441.4.c.b 24 21.h odd 6 1
441.4.p.d 48 1.a even 1 1 trivial
441.4.p.d 48 3.b odd 2 1 inner
441.4.p.d 48 7.b odd 2 1 inner
441.4.p.d 48 7.c even 3 1 inner
441.4.p.d 48 7.d odd 6 1 inner
441.4.p.d 48 21.c even 2 1 inner
441.4.p.d 48 21.g even 6 1 inner
441.4.p.d 48 21.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database