# Properties

 Label 441.4.c.b Level $441$ Weight $4$ Character orbit 441.c Analytic conductor $26.020$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 96q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 96q^{4} + 144q^{16} + 624q^{22} + 312q^{25} - 864q^{37} + 1248q^{43} - 3888q^{46} - 7440q^{58} - 3360q^{64} - 2688q^{67} + 480q^{79} + 13248q^{85} - 7248q^{88} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
440.1 5.38285i 0 −20.9751 18.3258 0 0 69.8432i 0 98.6449i
440.2 5.38285i 0 −20.9751 18.3258 0 0 69.8432i 0 98.6449i
440.3 3.65356i 0 −5.34851 17.3990 0 0 9.68739i 0 63.5684i
440.4 3.65356i 0 −5.34851 17.3990 0 0 9.68739i 0 63.5684i
440.5 0.641037i 0 7.58907 −12.0729 0 0 9.99318i 0 7.73920i
440.6 0.641037i 0 7.58907 −12.0729 0 0 9.99318i 0 7.73920i
440.7 3.70881i 0 −5.75528 −4.54675 0 0 8.32523i 0 16.8631i
440.8 3.70881i 0 −5.75528 −4.54675 0 0 8.32523i 0 16.8631i
440.9 1.62071i 0 5.37329 4.71986 0 0 21.6743i 0 7.64953i
440.10 1.62071i 0 5.37329 4.71986 0 0 21.6743i 0 7.64953i
440.11 3.58935i 0 −4.88345 −0.857298 0 0 11.1864i 0 3.07715i
440.12 3.58935i 0 −4.88345 −0.857298 0 0 11.1864i 0 3.07715i
440.13 3.58935i 0 −4.88345 0.857298 0 0 11.1864i 0 3.07715i
440.14 3.58935i 0 −4.88345 0.857298 0 0 11.1864i 0 3.07715i
440.15 1.62071i 0 5.37329 −4.71986 0 0 21.6743i 0 7.64953i
440.16 1.62071i 0 5.37329 −4.71986 0 0 21.6743i 0 7.64953i
440.17 3.70881i 0 −5.75528 4.54675 0 0 8.32523i 0 16.8631i
440.18 3.70881i 0 −5.75528 4.54675 0 0 8.32523i 0 16.8631i
440.19 0.641037i 0 7.58907 12.0729 0 0 9.99318i 0 7.73920i
440.20 0.641037i 0 7.58907 12.0729 0 0 9.99318i 0 7.73920i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 440.24 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
21.c even 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 72 T_{2}^{10} + 1902 T_{2}^{8} + 23016 T_{2}^{6} + 124449 T_{2}^{4} + 227424 T_{2}^{2} + 73984$$ acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database