Properties

Label 4010.2.a.g
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
−1.00000 −1.41421 1.00000 1.00000 1.41421 −2.82843 −1.00000 −1.00000 −1.00000
1.2 −1.00000 1.41421 1.00000 1.00000 −1.41421 2.82843 −1.00000 −1.00000 −1.00000
\(n\): 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
\(2\) \(1\)
\(5\) \(-1\)
\(401\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{2} \) \(\mathstrut -\mathstrut 2 \)
\(T_{7}^{2} \) \(\mathstrut -\mathstrut 8 \)
\(T_{11} \) \(\mathstrut +\mathstrut 2 \)