Properties

Label 31.2.c.a
Level 31
Weight 2
Character orbit 31.c
Analytic conductor 0.248
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 31.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.247536246266\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(2\) \(x^{2}\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{3}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
−2.41421 1.20711 2.09077i 3.82843 −0.500000 0.866025i −2.91421 + 5.04757i −1.20711 + 2.09077i −4.41421 −1.41421 2.44949i 1.20711 + 2.09077i
5.2 0.414214 −0.207107 + 0.358719i −1.82843 −0.500000 0.866025i −0.0857864 + 0.148586i 0.207107 0.358719i −1.58579 1.41421 + 2.44949i −0.207107 0.358719i
25.1 −2.41421 1.20711 + 2.09077i 3.82843 −0.500000 + 0.866025i −2.91421 5.04757i −1.20711 2.09077i −4.41421 −1.41421 + 2.44949i 1.20711 2.09077i
25.2 0.414214 −0.207107 0.358719i −1.82843 −0.500000 + 0.866025i −0.0857864 0.148586i 0.207107 + 0.358719i −1.58579 1.41421 2.44949i −0.207107 + 0.358719i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
31.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(31, [\chi])\).