Properties

Label 8001.2.a.k
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + 4 q^{4} \) \( + \beta q^{5} \) \(+ q^{7}\) \( + 2 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + 4 q^{4} \) \( + \beta q^{5} \) \(+ q^{7}\) \( + 2 \beta q^{8} \) \( + 6 q^{10} \) \( + ( 2 + 2 \beta ) q^{13} \) \( + \beta q^{14} \) \( + 4 q^{16} \) \( + ( 3 - \beta ) q^{17} \) \( + ( 2 - \beta ) q^{19} \) \( + 4 \beta q^{20} \) \( -6 q^{23} \) \(+ q^{25}\) \( + ( 12 + 2 \beta ) q^{26} \) \( + 4 q^{28} \) \( + ( 3 - 2 \beta ) q^{29} \) \( + 8 q^{31} \) \( + ( -6 + 3 \beta ) q^{34} \) \( + \beta q^{35} \) \( -7 q^{37} \) \( + ( -6 + 2 \beta ) q^{38} \) \( + 12 q^{40} \) \( + ( -3 + \beta ) q^{41} \) \( + 2 q^{43} \) \( -6 \beta q^{46} \) \( + 12 q^{47} \) \(+ q^{49}\) \( + \beta q^{50} \) \( + ( 8 + 8 \beta ) q^{52} \) \( + ( 3 + 2 \beta ) q^{53} \) \( + 2 \beta q^{56} \) \( + ( -12 + 3 \beta ) q^{58} \) \( -\beta q^{59} \) \( + ( 2 - \beta ) q^{61} \) \( + 8 \beta q^{62} \) \( -8 q^{64} \) \( + ( 12 + 2 \beta ) q^{65} \) \( + ( 2 + 5 \beta ) q^{67} \) \( + ( 12 - 4 \beta ) q^{68} \) \( + 6 q^{70} \) \( + ( -6 - \beta ) q^{71} \) \( + ( -4 - 3 \beta ) q^{73} \) \( -7 \beta q^{74} \) \( + ( 8 - 4 \beta ) q^{76} \) \( + ( -1 - 2 \beta ) q^{79} \) \( + 4 \beta q^{80} \) \( + ( 6 - 3 \beta ) q^{82} \) \( -6 q^{83} \) \( + ( -6 + 3 \beta ) q^{85} \) \( + 2 \beta q^{86} \) \( -4 \beta q^{89} \) \( + ( 2 + 2 \beta ) q^{91} \) \( -24 q^{92} \) \( + 12 \beta q^{94} \) \( + ( -6 + 2 \beta ) q^{95} \) \( + ( 5 + 3 \beta ) q^{97} \) \( + \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 12q^{10} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 24q^{26} \) \(\mathstrut +\mathstrut 8q^{28} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut 16q^{31} \) \(\mathstrut -\mathstrut 12q^{34} \) \(\mathstrut -\mathstrut 14q^{37} \) \(\mathstrut -\mathstrut 12q^{38} \) \(\mathstrut +\mathstrut 24q^{40} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut +\mathstrut 4q^{43} \) \(\mathstrut +\mathstrut 24q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 16q^{52} \) \(\mathstrut +\mathstrut 6q^{53} \) \(\mathstrut -\mathstrut 24q^{58} \) \(\mathstrut +\mathstrut 4q^{61} \) \(\mathstrut -\mathstrut 16q^{64} \) \(\mathstrut +\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 24q^{68} \) \(\mathstrut +\mathstrut 12q^{70} \) \(\mathstrut -\mathstrut 12q^{71} \) \(\mathstrut -\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 16q^{76} \) \(\mathstrut -\mathstrut 2q^{79} \) \(\mathstrut +\mathstrut 12q^{82} \) \(\mathstrut -\mathstrut 12q^{83} \) \(\mathstrut -\mathstrut 12q^{85} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 48q^{92} \) \(\mathstrut -\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 10q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−2.44949
2.44949
−2.44949 0 4.00000 −2.44949 0 1.00000 −4.89898 0 6.00000
1.2 2.44949 0 4.00000 2.44949 0 1.00000 4.89898 0 6.00000
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{2} \) \(\mathstrut -\mathstrut 6 \)
\(T_{5}^{2} \) \(\mathstrut -\mathstrut 6 \)