Properties

Label 8040.2.a.m
Level 8040
Weight 2
Character orbit 8040.a
Self dual Yes
Analytic conductor 64.200
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8040.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1997232251\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \(+ q^{5}\) \( + \beta q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \(+ q^{5}\) \( + \beta q^{7} \) \(+ q^{9}\) \( + ( 4 - \beta ) q^{11} \) \( + 2 \beta q^{13} \) \(+ q^{15}\) \( + ( -4 + 2 \beta ) q^{17} \) \( + ( 4 - 2 \beta ) q^{19} \) \( + \beta q^{21} \) \( -4 q^{23} \) \(+ q^{25}\) \(+ q^{27}\) \( + ( 2 + 2 \beta ) q^{29} \) \( + ( 2 + 2 \beta ) q^{31} \) \( + ( 4 - \beta ) q^{33} \) \( + \beta q^{35} \) \( + ( 2 - 3 \beta ) q^{37} \) \( + 2 \beta q^{39} \) \( + 6 q^{41} \) \( + ( 8 - 2 \beta ) q^{43} \) \(+ q^{45}\) \( -2 \beta q^{47} \) \( + ( -3 + \beta ) q^{49} \) \( + ( -4 + 2 \beta ) q^{51} \) \( + ( -2 + 2 \beta ) q^{53} \) \( + ( 4 - \beta ) q^{55} \) \( + ( 4 - 2 \beta ) q^{57} \) \( + ( -2 - 4 \beta ) q^{59} \) \( + ( -2 + 3 \beta ) q^{61} \) \( + \beta q^{63} \) \( + 2 \beta q^{65} \) \(+ q^{67}\) \( -4 q^{69} \) \( + ( 10 - \beta ) q^{71} \) \( + ( -6 - 2 \beta ) q^{73} \) \(+ q^{75}\) \( + ( -4 + 3 \beta ) q^{77} \) \( -2 q^{79} \) \(+ q^{81}\) \( + ( -12 + \beta ) q^{83} \) \( + ( -4 + 2 \beta ) q^{85} \) \( + ( 2 + 2 \beta ) q^{87} \) \( + ( 10 - 5 \beta ) q^{89} \) \( + ( 8 + 2 \beta ) q^{91} \) \( + ( 2 + 2 \beta ) q^{93} \) \( + ( 4 - 2 \beta ) q^{95} \) \( + ( 12 + \beta ) q^{97} \) \( + ( 4 - \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 7q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut 6q^{19} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut +\mathstrut 7q^{33} \) \(\mathstrut +\mathstrut q^{35} \) \(\mathstrut +\mathstrut q^{37} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 12q^{41} \) \(\mathstrut +\mathstrut 14q^{43} \) \(\mathstrut +\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut 2q^{47} \) \(\mathstrut -\mathstrut 5q^{49} \) \(\mathstrut -\mathstrut 6q^{51} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut +\mathstrut 7q^{55} \) \(\mathstrut +\mathstrut 6q^{57} \) \(\mathstrut -\mathstrut 8q^{59} \) \(\mathstrut -\mathstrut q^{61} \) \(\mathstrut +\mathstrut q^{63} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 2q^{67} \) \(\mathstrut -\mathstrut 8q^{69} \) \(\mathstrut +\mathstrut 19q^{71} \) \(\mathstrut -\mathstrut 14q^{73} \) \(\mathstrut +\mathstrut 2q^{75} \) \(\mathstrut -\mathstrut 5q^{77} \) \(\mathstrut -\mathstrut 4q^{79} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 23q^{83} \) \(\mathstrut -\mathstrut 6q^{85} \) \(\mathstrut +\mathstrut 6q^{87} \) \(\mathstrut +\mathstrut 15q^{89} \) \(\mathstrut +\mathstrut 18q^{91} \) \(\mathstrut +\mathstrut 6q^{93} \) \(\mathstrut +\mathstrut 6q^{95} \) \(\mathstrut +\mathstrut 25q^{97} \) \(\mathstrut +\mathstrut 7q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 1.00000 0 1.00000 0 −1.56155 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 2.56155 0 1.00000 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
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(67\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\(T_{7}^{2} \) \(\mathstrut -\mathstrut T_{7} \) \(\mathstrut -\mathstrut 4 \)
\(T_{11}^{2} \) \(\mathstrut -\mathstrut 7 T_{11} \) \(\mathstrut +\mathstrut 8 \)
\(T_{13}^{2} \) \(\mathstrut -\mathstrut 2 T_{13} \) \(\mathstrut -\mathstrut 16 \)
\(T_{17}^{2} \) \(\mathstrut +\mathstrut 6 T_{17} \) \(\mathstrut -\mathstrut 8 \)