Properties

Label 6042.2.a.o
Level 6042
Weight 2
Character orbit 6042.a
Self dual Yes
Analytic conductor 48.246
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6042.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2456129013\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(- q^{3}\) \(+ q^{4}\) \( + ( -1 - \beta ) q^{5} \) \(- q^{6}\) \(+ q^{7}\) \(+ q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(- q^{3}\) \(+ q^{4}\) \( + ( -1 - \beta ) q^{5} \) \(- q^{6}\) \(+ q^{7}\) \(+ q^{8}\) \(+ q^{9}\) \( + ( -1 - \beta ) q^{10} \) \( -4 q^{11} \) \(- q^{12}\) \( + ( -2 + \beta ) q^{13} \) \(+ q^{14}\) \( + ( 1 + \beta ) q^{15} \) \(+ q^{16}\) \( + ( 4 - \beta ) q^{17} \) \(+ q^{18}\) \(+ q^{19}\) \( + ( -1 - \beta ) q^{20} \) \(- q^{21}\) \( -4 q^{22} \) \( + ( 1 + \beta ) q^{23} \) \(- q^{24}\) \( + ( 4 + 3 \beta ) q^{25} \) \( + ( -2 + \beta ) q^{26} \) \(- q^{27}\) \(+ q^{28}\) \( + ( -5 + 2 \beta ) q^{29} \) \( + ( 1 + \beta ) q^{30} \) \( -7 q^{31} \) \(+ q^{32}\) \( + 4 q^{33} \) \( + ( 4 - \beta ) q^{34} \) \( + ( -1 - \beta ) q^{35} \) \(+ q^{36}\) \( + ( 4 + \beta ) q^{37} \) \(+ q^{38}\) \( + ( 2 - \beta ) q^{39} \) \( + ( -1 - \beta ) q^{40} \) \( -6 q^{41} \) \(- q^{42}\) \( + ( 7 + \beta ) q^{43} \) \( -4 q^{44} \) \( + ( -1 - \beta ) q^{45} \) \( + ( 1 + \beta ) q^{46} \) \( -3 \beta q^{47} \) \(- q^{48}\) \( -6 q^{49} \) \( + ( 4 + 3 \beta ) q^{50} \) \( + ( -4 + \beta ) q^{51} \) \( + ( -2 + \beta ) q^{52} \) \(- q^{53}\) \(- q^{54}\) \( + ( 4 + 4 \beta ) q^{55} \) \(+ q^{56}\) \(- q^{57}\) \( + ( -5 + 2 \beta ) q^{58} \) \( + ( 7 + 2 \beta ) q^{59} \) \( + ( 1 + \beta ) q^{60} \) \( + ( -2 - \beta ) q^{61} \) \( -7 q^{62} \) \(+ q^{63}\) \(+ q^{64}\) \( -6 q^{65} \) \( + 4 q^{66} \) \( + ( -1 + \beta ) q^{67} \) \( + ( 4 - \beta ) q^{68} \) \( + ( -1 - \beta ) q^{69} \) \( + ( -1 - \beta ) q^{70} \) \( + ( -10 + \beta ) q^{71} \) \(+ q^{72}\) \( + 4 q^{73} \) \( + ( 4 + \beta ) q^{74} \) \( + ( -4 - 3 \beta ) q^{75} \) \(+ q^{76}\) \( -4 q^{77} \) \( + ( 2 - \beta ) q^{78} \) \( -8 q^{79} \) \( + ( -1 - \beta ) q^{80} \) \(+ q^{81}\) \( -6 q^{82} \) \( + ( 12 + \beta ) q^{83} \) \(- q^{84}\) \( + ( 4 - 2 \beta ) q^{85} \) \( + ( 7 + \beta ) q^{86} \) \( + ( 5 - 2 \beta ) q^{87} \) \( -4 q^{88} \) \( + ( -11 + \beta ) q^{89} \) \( + ( -1 - \beta ) q^{90} \) \( + ( -2 + \beta ) q^{91} \) \( + ( 1 + \beta ) q^{92} \) \( + 7 q^{93} \) \( -3 \beta q^{94} \) \( + ( -1 - \beta ) q^{95} \) \(- q^{96}\) \( -14 q^{97} \) \( -6 q^{98} \) \( -4 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut -\mathstrut 8q^{11} \) \(\mathstrut -\mathstrut 2q^{12} \) \(\mathstrut -\mathstrut 3q^{13} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 3q^{20} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut 8q^{22} \) \(\mathstrut +\mathstrut 3q^{23} \) \(\mathstrut -\mathstrut 2q^{24} \) \(\mathstrut +\mathstrut 11q^{25} \) \(\mathstrut -\mathstrut 3q^{26} \) \(\mathstrut -\mathstrut 2q^{27} \) \(\mathstrut +\mathstrut 2q^{28} \) \(\mathstrut -\mathstrut 8q^{29} \) \(\mathstrut +\mathstrut 3q^{30} \) \(\mathstrut -\mathstrut 14q^{31} \) \(\mathstrut +\mathstrut 2q^{32} \) \(\mathstrut +\mathstrut 8q^{33} \) \(\mathstrut +\mathstrut 7q^{34} \) \(\mathstrut -\mathstrut 3q^{35} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut +\mathstrut 9q^{37} \) \(\mathstrut +\mathstrut 2q^{38} \) \(\mathstrut +\mathstrut 3q^{39} \) \(\mathstrut -\mathstrut 3q^{40} \) \(\mathstrut -\mathstrut 12q^{41} \) \(\mathstrut -\mathstrut 2q^{42} \) \(\mathstrut +\mathstrut 15q^{43} \) \(\mathstrut -\mathstrut 8q^{44} \) \(\mathstrut -\mathstrut 3q^{45} \) \(\mathstrut +\mathstrut 3q^{46} \) \(\mathstrut -\mathstrut 3q^{47} \) \(\mathstrut -\mathstrut 2q^{48} \) \(\mathstrut -\mathstrut 12q^{49} \) \(\mathstrut +\mathstrut 11q^{50} \) \(\mathstrut -\mathstrut 7q^{51} \) \(\mathstrut -\mathstrut 3q^{52} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut -\mathstrut 2q^{54} \) \(\mathstrut +\mathstrut 12q^{55} \) \(\mathstrut +\mathstrut 2q^{56} \) \(\mathstrut -\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut 8q^{58} \) \(\mathstrut +\mathstrut 16q^{59} \) \(\mathstrut +\mathstrut 3q^{60} \) \(\mathstrut -\mathstrut 5q^{61} \) \(\mathstrut -\mathstrut 14q^{62} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 12q^{65} \) \(\mathstrut +\mathstrut 8q^{66} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut +\mathstrut 7q^{68} \) \(\mathstrut -\mathstrut 3q^{69} \) \(\mathstrut -\mathstrut 3q^{70} \) \(\mathstrut -\mathstrut 19q^{71} \) \(\mathstrut +\mathstrut 2q^{72} \) \(\mathstrut +\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 9q^{74} \) \(\mathstrut -\mathstrut 11q^{75} \) \(\mathstrut +\mathstrut 2q^{76} \) \(\mathstrut -\mathstrut 8q^{77} \) \(\mathstrut +\mathstrut 3q^{78} \) \(\mathstrut -\mathstrut 16q^{79} \) \(\mathstrut -\mathstrut 3q^{80} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 12q^{82} \) \(\mathstrut +\mathstrut 25q^{83} \) \(\mathstrut -\mathstrut 2q^{84} \) \(\mathstrut +\mathstrut 6q^{85} \) \(\mathstrut +\mathstrut 15q^{86} \) \(\mathstrut +\mathstrut 8q^{87} \) \(\mathstrut -\mathstrut 8q^{88} \) \(\mathstrut -\mathstrut 21q^{89} \) \(\mathstrut -\mathstrut 3q^{90} \) \(\mathstrut -\mathstrut 3q^{91} \) \(\mathstrut +\mathstrut 3q^{92} \) \(\mathstrut +\mathstrut 14q^{93} \) \(\mathstrut -\mathstrut 3q^{94} \) \(\mathstrut -\mathstrut 3q^{95} \) \(\mathstrut -\mathstrut 2q^{96} \) \(\mathstrut -\mathstrut 28q^{97} \) \(\mathstrut -\mathstrut 12q^{98} \) \(\mathstrut -\mathstrut 8q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
3.37228
−2.37228
1.00000 −1.00000 1.00000 −4.37228 −1.00000 1.00000 1.00000 1.00000 −4.37228
1.2 1.00000 −1.00000 1.00000 1.37228 −1.00000 1.00000 1.00000 1.00000 1.37228
\(n\): 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
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(-1\)
\(53\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\(T_{5}^{2} \) \(\mathstrut +\mathstrut 3 T_{5} \) \(\mathstrut -\mathstrut 6 \)
\(T_{7} \) \(\mathstrut -\mathstrut 1 \)
\(T_{11} \) \(\mathstrut +\mathstrut 4 \)