Properties

Label 8024.2.a.p
Level 8024
Weight 2
Character orbit 8024.a
Self dual Yes
Analytic conductor 64.072
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8024.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0719625819\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q\) \( + \beta q^{3} \) \( + ( -2 + \beta ) q^{5} \) \( + ( -2 + \beta ) q^{7} \) \( + ( 1 + \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{3} \) \( + ( -2 + \beta ) q^{5} \) \( + ( -2 + \beta ) q^{7} \) \( + ( 1 + \beta ) q^{9} \) \( + ( -3 - \beta ) q^{11} \) \( + ( -4 + 2 \beta ) q^{13} \) \( + ( 4 - \beta ) q^{15} \) \(+ q^{17}\) \( -3 q^{19} \) \( + ( 4 - \beta ) q^{21} \) \( + ( -3 + \beta ) q^{23} \) \( + ( 3 - 3 \beta ) q^{25} \) \( + ( 4 - \beta ) q^{27} \) \( + ( 6 - \beta ) q^{29} \) \( -4 \beta q^{31} \) \( + ( -4 - 4 \beta ) q^{33} \) \( + ( 8 - 3 \beta ) q^{35} \) \( + ( 2 - 4 \beta ) q^{37} \) \( + ( 8 - 2 \beta ) q^{39} \) \( + ( -8 + 3 \beta ) q^{41} \) \( + 2 q^{45} \) \( + ( 8 - 2 \beta ) q^{47} \) \( + ( 1 - 3 \beta ) q^{49} \) \( + \beta q^{51} \) \( + ( 1 - 6 \beta ) q^{53} \) \( + ( 2 - 2 \beta ) q^{55} \) \( -3 \beta q^{57} \) \(- q^{59}\) \( + ( -5 - 3 \beta ) q^{61} \) \( + 2 q^{63} \) \( + ( 16 - 6 \beta ) q^{65} \) \( + ( 10 + 2 \beta ) q^{67} \) \( + ( 4 - 2 \beta ) q^{69} \) \( -8 q^{71} \) \( + ( -3 + 5 \beta ) q^{73} \) \( -12 q^{75} \) \( + ( 2 - 2 \beta ) q^{77} \) \( + ( -2 + 3 \beta ) q^{79} \) \( -7 q^{81} \) \( + ( -2 + 4 \beta ) q^{83} \) \( + ( -2 + \beta ) q^{85} \) \( + ( -4 + 5 \beta ) q^{87} \) \( + ( -2 - 2 \beta ) q^{89} \) \( + ( 16 - 6 \beta ) q^{91} \) \( + ( -16 - 4 \beta ) q^{93} \) \( + ( 6 - 3 \beta ) q^{95} \) \( + ( 7 - 5 \beta ) q^{97} \) \( + ( -7 - 5 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut -\mathstrut 7q^{11} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 7q^{15} \) \(\mathstrut +\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut +\mathstrut 7q^{21} \) \(\mathstrut -\mathstrut 5q^{23} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 7q^{27} \) \(\mathstrut +\mathstrut 11q^{29} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut 12q^{33} \) \(\mathstrut +\mathstrut 13q^{35} \) \(\mathstrut +\mathstrut 14q^{39} \) \(\mathstrut -\mathstrut 13q^{41} \) \(\mathstrut +\mathstrut 4q^{45} \) \(\mathstrut +\mathstrut 14q^{47} \) \(\mathstrut -\mathstrut q^{49} \) \(\mathstrut +\mathstrut q^{51} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut -\mathstrut 3q^{57} \) \(\mathstrut -\mathstrut 2q^{59} \) \(\mathstrut -\mathstrut 13q^{61} \) \(\mathstrut +\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 26q^{65} \) \(\mathstrut +\mathstrut 22q^{67} \) \(\mathstrut +\mathstrut 6q^{69} \) \(\mathstrut -\mathstrut 16q^{71} \) \(\mathstrut -\mathstrut q^{73} \) \(\mathstrut -\mathstrut 24q^{75} \) \(\mathstrut +\mathstrut 2q^{77} \) \(\mathstrut -\mathstrut q^{79} \) \(\mathstrut -\mathstrut 14q^{81} \) \(\mathstrut -\mathstrut 3q^{85} \) \(\mathstrut -\mathstrut 3q^{87} \) \(\mathstrut -\mathstrut 6q^{89} \) \(\mathstrut +\mathstrut 26q^{91} \) \(\mathstrut -\mathstrut 36q^{93} \) \(\mathstrut +\mathstrut 9q^{95} \) \(\mathstrut +\mathstrut 9q^{97} \) \(\mathstrut -\mathstrut 19q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −1.56155 0 −3.56155 0 −3.56155 0 −0.561553 0
1.2 0 2.56155 0 0.561553 0 0.561553 0 3.56155 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
\(2\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8024))\):

\(T_{3}^{2} \) \(\mathstrut -\mathstrut T_{3} \) \(\mathstrut -\mathstrut 4 \)
\(T_{5}^{2} \) \(\mathstrut +\mathstrut 3 T_{5} \) \(\mathstrut -\mathstrut 2 \)
\(T_{7}^{2} \) \(\mathstrut +\mathstrut 3 T_{7} \) \(\mathstrut -\mathstrut 2 \)