# Properties

 Label 4140.2.f.d Level $4140$ Weight $2$ Character orbit 4140.f Analytic conductor $33.058$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4140.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.0580664368$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q+O(q^{10})$$ 24 * q $$\operatorname{Tr}(f)(q) =$$ $$24 q + 8 q^{19} - 8 q^{25} - 12 q^{31} - 28 q^{49} - 16 q^{55} - 16 q^{61} + 8 q^{79} + 12 q^{85} - 16 q^{91}+O(q^{100})$$ 24 * q + 8 * q^19 - 8 * q^25 - 12 * q^31 - 28 * q^49 - 16 * q^55 - 16 * q^61 + 8 * q^79 + 12 * q^85 - 16 * q^91

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
829.1 0 0 0 −2.23083 0.152950i 0 4.09652i 0 0 0
829.2 0 0 0 −2.23083 + 0.152950i 0 4.09652i 0 0 0
829.3 0 0 0 −2.09906 0.770673i 0 2.34084i 0 0 0
829.4 0 0 0 −2.09906 + 0.770673i 0 2.34084i 0 0 0
829.5 0 0 0 −1.79970 1.32706i 0 0.476731i 0 0 0
829.6 0 0 0 −1.79970 + 1.32706i 0 0.476731i 0 0 0
829.7 0 0 0 −1.06710 1.96502i 0 3.96013i 0 0 0
829.8 0 0 0 −1.06710 + 1.96502i 0 3.96013i 0 0 0
829.9 0 0 0 −0.405664 2.19896i 0 3.23277i 0 0 0
829.10 0 0 0 −0.405664 + 2.19896i 0 3.23277i 0 0 0
829.11 0 0 0 −0.274117 2.21920i 0 0.615120i 0 0 0
829.12 0 0 0 −0.274117 + 2.21920i 0 0.615120i 0 0 0
829.13 0 0 0 0.274117 2.21920i 0 0.615120i 0 0 0
829.14 0 0 0 0.274117 + 2.21920i 0 0.615120i 0 0 0
829.15 0 0 0 0.405664 2.19896i 0 3.23277i 0 0 0
829.16 0 0 0 0.405664 + 2.19896i 0 3.23277i 0 0 0
829.17 0 0 0 1.06710 1.96502i 0 3.96013i 0 0 0
829.18 0 0 0 1.06710 + 1.96502i 0 3.96013i 0 0 0
829.19 0 0 0 1.79970 1.32706i 0 0.476731i 0 0 0
829.20 0 0 0 1.79970 + 1.32706i 0 0.476731i 0 0 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 829.24 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
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.2.f.d 24
3.b odd 2 1 inner 4140.2.f.d 24
5.b even 2 1 inner 4140.2.f.d 24
15.d odd 2 1 inner 4140.2.f.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.2.f.d 24 1.a even 1 1 trivial
4140.2.f.d 24 3.b odd 2 1 inner
4140.2.f.d 24 5.b even 2 1 inner
4140.2.f.d 24 15.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} + 49T_{7}^{10} + 867T_{7}^{8} + 6563T_{7}^{6} + 18808T_{7}^{4} + 9648T_{7}^{2} + 1296$$ acting on $$S_{2}^{\mathrm{new}}(4140, [\chi])$$.