# Properties

 Label 8001.2.a.y Level $8001$ Weight $2$ Character orbit 8001.a Self dual yes Analytic conductor $63.888$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8001 = 3^{2} \cdot 7 \cdot 127$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.8883066572$$ Analytic rank: $$0$$ Dimension: $$28$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 30q^{4} - 28q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 30q^{4} - 28q^{7} + 4q^{10} + 8q^{13} + 42q^{16} + 34q^{19} - 10q^{22} + 14q^{25} - 30q^{28} + 56q^{31} - 6q^{37} + 38q^{40} + 18q^{43} + 16q^{46} + 28q^{49} + 18q^{52} + 48q^{55} + 2q^{58} + 36q^{61} + 76q^{64} - 4q^{70} + 50q^{73} + 132q^{76} + 66q^{79} - 36q^{82} + 20q^{85} + 6q^{88} - 8q^{91} + 54q^{94} - 8q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.72816 0 5.44285 0.297532 0 −1.00000 −9.39264 0 −0.811714
1.2 −2.71687 0 5.38141 −1.23671 0 −1.00000 −9.18686 0 3.35997
1.3 −2.47474 0 4.12433 −3.91787 0 −1.00000 −5.25716 0 9.69570
1.4 −2.27499 0 3.17556 3.81133 0 −1.00000 −2.67439 0 −8.67073
1.5 −2.02489 0 2.10018 2.65037 0 −1.00000 −0.202857 0 −5.36671
1.6 −1.99802 0 1.99208 −1.55092 0 −1.00000 0.0158280 0 3.09877
1.7 −1.55816 0 0.427866 −3.44623 0 −1.00000 2.44964 0 5.36979
1.8 −1.42730 0 0.0371861 2.54057 0 −1.00000 2.80152 0 −3.62615
1.9 −1.34710 0 −0.185322 0.124848 0 −1.00000 2.94385 0 −0.168183
1.10 −1.24740 0 −0.443996 1.34151 0 −1.00000 3.04864 0 −1.67339
1.11 −0.662421 0 −1.56120 −3.26980 0 −1.00000 2.35901 0 2.16599
1.12 −0.512239 0 −1.73761 1.76990 0 −1.00000 1.91455 0 −0.906610
1.13 −0.487315 0 −1.76252 0.708639 0 −1.00000 1.83353 0 −0.345330
1.14 −0.0958522 0 −1.99081 1.26637 0 −1.00000 0.382528 0 −0.121384
1.15 0.0958522 0 −1.99081 −1.26637 0 −1.00000 −0.382528 0 −0.121384
1.16 0.487315 0 −1.76252 −0.708639 0 −1.00000 −1.83353 0 −0.345330
1.17 0.512239 0 −1.73761 −1.76990 0 −1.00000 −1.91455 0 −0.906610
1.18 0.662421 0 −1.56120 3.26980 0 −1.00000 −2.35901 0 2.16599
1.19 1.24740 0 −0.443996 −1.34151 0 −1.00000 −3.04864 0 −1.67339
1.20 1.34710 0 −0.185322 −0.124848 0 −1.00000 −2.94385 0 −0.168183
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$1$$
$$127$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8001.2.a.y 28
3.b odd 2 1 inner 8001.2.a.y 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8001.2.a.y 28 1.a even 1 1 trivial
8001.2.a.y 28 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8001))$$:

 $$T_{2}^{28} - \cdots$$ $$T_{5}^{28} - \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database