Properties

Label 4011.2.a.e
Level 4011
Weight 2
Character orbit 4011.a
Self dual Yes
Analytic conductor 32.028
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4011 = 3 \cdot 7 \cdot 191 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4011.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0279962507\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(4\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)

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
−1.86081
−0.254102
2.11491
−1.86081 −1.00000 1.46260 1.46260 1.86081 −1.00000 1.00000 1.00000 −2.72161
1.2 −0.254102 −1.00000 −1.93543 −1.93543 0.254102 −1.00000 1.00000 1.00000 0.491797
1.3 2.11491 −1.00000 2.47283 2.47283 −2.11491 −1.00000 1.00000 1.00000 5.22982
\(n\): 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
\(3\) \(1\)
\(7\) \(1\)
\(191\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{3} \) \(\mathstrut -\mathstrut 4 T_{2} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4011))\).