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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

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