Properties

Label 8001.2.a.i
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

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -2 q^{4} \) \( + ( 1 + \beta ) q^{5} \) \(- q^{7}\) \(+O(q^{10})\) \( q\) \( -2 q^{4} \) \( + ( 1 + \beta ) q^{5} \) \(- q^{7}\) \( + ( 2 - 2 \beta ) q^{11} \) \( + ( -1 - \beta ) q^{13} \) \( + 4 q^{16} \) \( + ( -4 - \beta ) q^{17} \) \( + 4 q^{19} \) \( + ( -2 - 2 \beta ) q^{20} \) \( + ( -3 + 3 \beta ) q^{23} \) \( + 3 \beta q^{25} \) \( + 2 q^{28} \) \( + ( 1 - 2 \beta ) q^{29} \) \( + ( -3 + 3 \beta ) q^{31} \) \( + ( -1 - \beta ) q^{35} \) \( + ( -7 + 2 \beta ) q^{37} \) \( + ( -2 + 5 \beta ) q^{41} \) \( + ( 6 - 4 \beta ) q^{43} \) \( + ( -4 + 4 \beta ) q^{44} \) \( + ( 10 - 2 \beta ) q^{47} \) \(+ q^{49}\) \( + ( 2 + 2 \beta ) q^{52} \) \( + 11 q^{53} \) \( + ( -6 - 2 \beta ) q^{55} \) \( + ( 7 - \beta ) q^{59} \) \( + ( -7 - \beta ) q^{61} \) \( -8 q^{64} \) \( + ( -5 - 3 \beta ) q^{65} \) \( -2 q^{67} \) \( + ( 8 + 2 \beta ) q^{68} \) \( + ( 8 + 2 \beta ) q^{71} \) \( + ( -7 + \beta ) q^{73} \) \( -8 q^{76} \) \( + ( -2 + 2 \beta ) q^{77} \) \( + ( -8 + 3 \beta ) q^{79} \) \( + ( 4 + 4 \beta ) q^{80} \) \( + ( -5 - 5 \beta ) q^{83} \) \( + ( -8 - 6 \beta ) q^{85} \) \( + ( 5 - 5 \beta ) q^{89} \) \( + ( 1 + \beta ) q^{91} \) \( + ( 6 - 6 \beta ) q^{92} \) \( + ( 4 + 4 \beta ) q^{95} \) \( + ( 4 - 5 \beta ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 3q^{13} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut -\mathstrut 6q^{20} \) \(\mathstrut -\mathstrut 3q^{23} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut -\mathstrut 3q^{31} \) \(\mathstrut -\mathstrut 3q^{35} \) \(\mathstrut -\mathstrut 12q^{37} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut +\mathstrut 8q^{43} \) \(\mathstrut -\mathstrut 4q^{44} \) \(\mathstrut +\mathstrut 18q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 6q^{52} \) \(\mathstrut +\mathstrut 22q^{53} \) \(\mathstrut -\mathstrut 14q^{55} \) \(\mathstrut +\mathstrut 13q^{59} \) \(\mathstrut -\mathstrut 15q^{61} \) \(\mathstrut -\mathstrut 16q^{64} \) \(\mathstrut -\mathstrut 13q^{65} \) \(\mathstrut -\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 18q^{68} \) \(\mathstrut +\mathstrut 18q^{71} \) \(\mathstrut -\mathstrut 13q^{73} \) \(\mathstrut -\mathstrut 16q^{76} \) \(\mathstrut -\mathstrut 2q^{77} \) \(\mathstrut -\mathstrut 13q^{79} \) \(\mathstrut +\mathstrut 12q^{80} \) \(\mathstrut -\mathstrut 15q^{83} \) \(\mathstrut -\mathstrut 22q^{85} \) \(\mathstrut +\mathstrut 5q^{89} \) \(\mathstrut +\mathstrut 3q^{91} \) \(\mathstrut +\mathstrut 6q^{92} \) \(\mathstrut +\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 3q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−1.56155
2.56155
0 0 −2.00000 −0.561553 0 −1.00000 0 0 0
1.2 0 0 −2.00000 3.56155 0 −1.00000 0 0 0
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2} \)
\(T_{5}^{2} \) \(\mathstrut -\mathstrut 3 T_{5} \) \(\mathstrut -\mathstrut 2 \)