Properties

Label 8001.2.a.j
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -\beta q^{5} \) \(+ q^{7}\) \( -2 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -\beta q^{5} \) \(+ q^{7}\) \( -2 \beta q^{8} \) \( -2 q^{10} \) \( + 2 \beta q^{11} \) \( -2 q^{13} \) \( + \beta q^{14} \) \( -4 q^{16} \) \( + ( 3 - 2 \beta ) q^{17} \) \( + ( -4 + 3 \beta ) q^{19} \) \( + 4 q^{22} \) \( + 2 \beta q^{23} \) \( -3 q^{25} \) \( -2 \beta q^{26} \) \( + ( 3 + \beta ) q^{29} \) \( + 4 q^{31} \) \( + ( -4 + 3 \beta ) q^{34} \) \( -\beta q^{35} \) \( + ( 3 - 6 \beta ) q^{37} \) \( + ( 6 - 4 \beta ) q^{38} \) \( + 4 q^{40} \) \( + ( 3 - 4 \beta ) q^{41} \) \( -10 q^{43} \) \( + 4 q^{46} \) \( + ( 6 + 2 \beta ) q^{47} \) \(+ q^{49}\) \( -3 \beta q^{50} \) \( + ( 9 + \beta ) q^{53} \) \( -4 q^{55} \) \( -2 \beta q^{56} \) \( + ( 2 + 3 \beta ) q^{58} \) \( -5 \beta q^{59} \) \( + ( -8 - 3 \beta ) q^{61} \) \( + 4 \beta q^{62} \) \( + 8 q^{64} \) \( + 2 \beta q^{65} \) \( -3 \beta q^{67} \) \( -2 q^{70} \) \( + ( -6 + \beta ) q^{71} \) \( + ( 2 + 9 \beta ) q^{73} \) \( + ( -12 + 3 \beta ) q^{74} \) \( + 2 \beta q^{77} \) \( -11 q^{79} \) \( + 4 \beta q^{80} \) \( + ( -8 + 3 \beta ) q^{82} \) \( + ( -6 + 4 \beta ) q^{83} \) \( + ( 4 - 3 \beta ) q^{85} \) \( -10 \beta q^{86} \) \( -8 q^{88} \) \( + ( -12 - 2 \beta ) q^{89} \) \( -2 q^{91} \) \( + ( 4 + 6 \beta ) q^{94} \) \( + ( -6 + 4 \beta ) q^{95} \) \( + ( 7 - 3 \beta ) q^{97} \) \( + \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 4q^{10} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 8q^{22} \) \(\mathstrut -\mathstrut 6q^{25} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut 8q^{31} \) \(\mathstrut -\mathstrut 8q^{34} \) \(\mathstrut +\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut 12q^{38} \) \(\mathstrut +\mathstrut 8q^{40} \) \(\mathstrut +\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 20q^{43} \) \(\mathstrut +\mathstrut 8q^{46} \) \(\mathstrut +\mathstrut 12q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 18q^{53} \) \(\mathstrut -\mathstrut 8q^{55} \) \(\mathstrut +\mathstrut 4q^{58} \) \(\mathstrut -\mathstrut 16q^{61} \) \(\mathstrut +\mathstrut 16q^{64} \) \(\mathstrut -\mathstrut 4q^{70} \) \(\mathstrut -\mathstrut 12q^{71} \) \(\mathstrut +\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 24q^{74} \) \(\mathstrut -\mathstrut 22q^{79} \) \(\mathstrut -\mathstrut 16q^{82} \) \(\mathstrut -\mathstrut 12q^{83} \) \(\mathstrut +\mathstrut 8q^{85} \) \(\mathstrut -\mathstrut 16q^{88} \) \(\mathstrut -\mathstrut 24q^{89} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut 8q^{94} \) \(\mathstrut -\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 14q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−1.41421 0 0 1.41421 0 1.00000 2.82843 0 −2.00000
1.2 1.41421 0 0 −1.41421 0 1.00000 −2.82843 0 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{2} \) \(\mathstrut -\mathstrut 2 \)
\(T_{5}^{2} \) \(\mathstrut -\mathstrut 2 \)