Properties

Label 31.2.a.a
Level 31
Weight 2
Character orbit 31.a
Self dual Yes
Analytic conductor 0.248
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 31.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.247536246266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
−0.618034 1.23607 −1.61803 1.00000 −0.763932 −4.23607 2.23607 −1.47214 −0.618034
1.2 1.61803 −3.23607 0.618034 1.00000 −5.23607 0.236068 −2.23607 7.47214 1.61803
\(n\): 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
\(31\) \(-1\)

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\).