Properties

Label 4031.2.a.a
Level 4031
Weight 2
Character orbit 4031.a
Self dual Yes
Analytic conductor 32.188
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4031 = 29 \cdot 139 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4031.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1876970548\)
Analytic rank: \(0\)
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.

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

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
−0.414214 −0.414214 −1.82843 2.00000 0.171573 1.82843 1.58579 −2.82843 −0.828427
1.2 2.41421 2.41421 3.82843 2.00000 5.82843 −3.82843 4.41421 2.82843 4.82843
\(n\): 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
\(29\) \(-1\)
\(139\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 2 T_{2} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4031))\).