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

Related objects

Downloads

Learn more about

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: \(32\)
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.

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

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:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(127\) \(1\)

Hecke kernels

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

\(T_{2}^{32} - \cdots\)
\(T_{5}^{32} - \cdots\)