Properties

Label 6011.2.a.d
Level 6011
Weight 2
Character orbit 6011.a
Self dual Yes
Analytic conductor 47.998
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6011 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6011.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.998076655\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
−1.41421 2.00000 0 −2.82843 −2.82843 −1.00000 2.82843 1.00000 4.00000
1.2 1.41421 2.00000 0 2.82843 2.82843 −1.00000 −2.82843 1.00000 4.00000
\(n\): 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
\(6011\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6011))\).