Properties

 Label 8001.2.a.z Level $8001$ Weight $2$ Character orbit 8001.a Self dual yes Analytic conductor $63.888$ Analytic rank $1$ Dimension $32$ 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: $$1$$ Dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 30q^{4} - 32q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100})$$

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 −2.69601 0 5.26849 3.35934 0 −1.00000 −8.81189 0 −9.05682
1.2 −2.49464 0 4.22324 −0.373762 0 −1.00000 −5.54618 0 0.932401
1.3 −2.44637 0 3.98471 1.74324 0 −1.00000 −4.85534 0 −4.26460
1.4 −2.41110 0 3.81340 −2.91376 0 −1.00000 −4.37229 0 7.02536
1.5 −2.23843 0 3.01059 −0.715002 0 −1.00000 −2.26213 0 1.60049
1.6 −1.96743 0 1.87078 2.35101 0 −1.00000 0.254227 0 −4.62544
1.7 −1.66842 0 0.783626 −3.76494 0 −1.00000 2.02942 0 6.28150
1.8 −1.63487 0 0.672794 0.107826 0 −1.00000 2.16981 0 −0.176281
1.9 −1.56027 0 0.434445 1.69832 0 −1.00000 2.44269 0 −2.64985
1.10 −1.18013 0 −0.607291 4.43826 0 −1.00000 3.07694 0 −5.23773
1.11 −0.959867 0 −1.07866 −2.40466 0 −1.00000 2.95510 0 2.30815
1.12 −0.942656 0 −1.11140 −2.02754 0 −1.00000 2.93298 0 1.91127
1.13 −0.904420 0 −1.18202 −0.974726 0 −1.00000 2.87789 0 0.881562
1.14 −0.903971 0 −1.18284 2.97865 0 −1.00000 2.87719 0 −2.69261
1.15 −0.242500 0 −1.94119 2.79671 0 −1.00000 0.955741 0 −0.678204
1.16 −0.203292 0 −1.95867 −2.16833 0 −1.00000 0.804765 0 0.440803
1.17 0.203292 0 −1.95867 2.16833 0 −1.00000 −0.804765 0 0.440803
1.18 0.242500 0 −1.94119 −2.79671 0 −1.00000 −0.955741 0 −0.678204
1.19 0.903971 0 −1.18284 −2.97865 0 −1.00000 −2.87719 0 −2.69261
1.20 0.904420 0 −1.18202 0.974726 0 −1.00000 −2.87789 0 0.881562
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.32 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.z 32
3.b odd 2 1 inner 8001.2.a.z 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8001.2.a.z 32 1.a even 1 1 trivial
8001.2.a.z 32 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8001))$$:

 $$T_{2}^{32} - \cdots$$ $$T_{5}^{32} - \cdots$$

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database