Properties

Label 6040.2.a.k
Level 6040
Weight 2
Character orbit 6040.a
Self dual Yes
Analytic conductor 48.230
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6040.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2296428209\)
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.

\(f(q)\) \(=\) \( q\) \( -\beta q^{3} \) \(+ q^{5}\) \( + ( -1 + \beta ) q^{7} \) \( + ( 1 + \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta q^{3} \) \(+ q^{5}\) \( + ( -1 + \beta ) q^{7} \) \( + ( 1 + \beta ) q^{9} \) \(+ q^{11}\) \(- q^{13}\) \( -\beta q^{15} \) \( -2 q^{17} \) \( + ( 4 - \beta ) q^{19} \) \( -4 q^{21} \) \( + ( -1 + 2 \beta ) q^{23} \) \(+ q^{25}\) \( + ( -4 + \beta ) q^{27} \) \( -3 q^{29} \) \( + ( -1 - 2 \beta ) q^{31} \) \( -\beta q^{33} \) \( + ( -1 + \beta ) q^{35} \) \( + ( 2 - 4 \beta ) q^{37} \) \( + \beta q^{39} \) \( + ( -6 + 2 \beta ) q^{41} \) \( -2 q^{43} \) \( + ( 1 + \beta ) q^{45} \) \( + ( -4 + 2 \beta ) q^{47} \) \( + ( -2 - \beta ) q^{49} \) \( + 2 \beta q^{51} \) \( + 2 q^{53} \) \(+ q^{55}\) \( + ( 4 - 3 \beta ) q^{57} \) \( -9 q^{59} \) \( + ( 4 + 2 \beta ) q^{61} \) \( + ( 3 + \beta ) q^{63} \) \(- q^{65}\) \( + ( -1 + 4 \beta ) q^{67} \) \( + ( -8 - \beta ) q^{69} \) \( -2 \beta q^{71} \) \( + ( 3 - 4 \beta ) q^{73} \) \( -\beta q^{75} \) \( + ( -1 + \beta ) q^{77} \) \( -8 q^{79} \) \( -7 q^{81} \) \( + ( 3 + 4 \beta ) q^{83} \) \( -2 q^{85} \) \( + 3 \beta q^{87} \) \( + ( -10 - 2 \beta ) q^{89} \) \( + ( 1 - \beta ) q^{91} \) \( + ( 8 + 3 \beta ) q^{93} \) \( + ( 4 - \beta ) q^{95} \) \( + ( -4 + 4 \beta ) q^{97} \) \( + ( 1 + \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut q^{15} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 7q^{19} \) \(\mathstrut -\mathstrut 8q^{21} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 7q^{27} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut q^{33} \) \(\mathstrut -\mathstrut q^{35} \) \(\mathstrut +\mathstrut q^{39} \) \(\mathstrut -\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 4q^{43} \) \(\mathstrut +\mathstrut 3q^{45} \) \(\mathstrut -\mathstrut 6q^{47} \) \(\mathstrut -\mathstrut 5q^{49} \) \(\mathstrut +\mathstrut 2q^{51} \) \(\mathstrut +\mathstrut 4q^{53} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut 5q^{57} \) \(\mathstrut -\mathstrut 18q^{59} \) \(\mathstrut +\mathstrut 10q^{61} \) \(\mathstrut +\mathstrut 7q^{63} \) \(\mathstrut -\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 2q^{67} \) \(\mathstrut -\mathstrut 17q^{69} \) \(\mathstrut -\mathstrut 2q^{71} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut q^{75} \) \(\mathstrut -\mathstrut q^{77} \) \(\mathstrut -\mathstrut 16q^{79} \) \(\mathstrut -\mathstrut 14q^{81} \) \(\mathstrut +\mathstrut 10q^{83} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut +\mathstrut 3q^{87} \) \(\mathstrut -\mathstrut 22q^{89} \) \(\mathstrut +\mathstrut q^{91} \) \(\mathstrut +\mathstrut 19q^{93} \) \(\mathstrut +\mathstrut 7q^{95} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut 3q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
2.56155
−1.56155
0 −2.56155 0 1.00000 0 1.56155 0 3.56155 0
1.2 0 1.56155 0 1.00000 0 −2.56155 0 −0.561553 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
\(2\) \(-1\)
\(5\) \(-1\)
\(151\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\(T_{3}^{2} \) \(\mathstrut +\mathstrut T_{3} \) \(\mathstrut -\mathstrut 4 \)
\(T_{7}^{2} \) \(\mathstrut +\mathstrut T_{7} \) \(\mathstrut -\mathstrut 4 \)
\(T_{11} \) \(\mathstrut -\mathstrut 1 \)