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 \(3\), degree \(2\), minimal)

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

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.2.c.a 4
3.b odd 2 1 279.2.h.c 4
4.b odd 2 1 496.2.i.h 4
5.b even 2 1 775.2.e.e 4
5.c odd 4 2 775.2.o.d 8
31.b odd 2 1 961.2.c.a 4
31.c even 3 1 inner 31.2.c.a 4
31.c even 3 1 961.2.a.a 2
31.d even 5 4 961.2.g.o 16
31.e odd 6 1 961.2.a.c 2
31.e odd 6 1 961.2.c.a 4
31.f odd 10 4 961.2.g.r 16
31.g even 15 4 961.2.d.l 8
31.g even 15 4 961.2.g.o 16
31.h odd 30 4 961.2.d.i 8
31.h odd 30 4 961.2.g.r 16
93.g even 6 1 8649.2.a.k 2
93.h odd 6 1 279.2.h.c 4
93.h odd 6 1 8649.2.a.l 2
124.i odd 6 1 496.2.i.h 4
155.j even 6 1 775.2.e.e 4
155.o odd 12 2 775.2.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.c.a 4 1.a even 1 1 trivial
31.2.c.a 4 31.c even 3 1 inner
279.2.h.c 4 3.b odd 2 1
279.2.h.c 4 93.h odd 6 1
496.2.i.h 4 4.b odd 2 1
496.2.i.h 4 124.i odd 6 1
775.2.e.e 4 5.b even 2 1
775.2.e.e 4 155.j even 6 1
775.2.o.d 8 5.c odd 4 2
775.2.o.d 8 155.o odd 12 2
961.2.a.a 2 31.c even 3 1
961.2.a.c 2 31.e odd 6 1
961.2.c.a 4 31.b odd 2 1
961.2.c.a 4 31.e odd 6 1
961.2.d.i 8 31.h odd 30 4
961.2.d.l 8 31.g even 15 4
961.2.g.o 16 31.d even 5 4
961.2.g.o 16 31.g even 15 4
961.2.g.r 16 31.f odd 10 4
961.2.g.r 16 31.h odd 30 4
8649.2.a.k 2 93.g even 6 1
8649.2.a.l 2 93.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(31, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} )^{2} \)
$3$ \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} - 9 T^{6} - 54 T^{7} + 81 T^{8} \)
$5$ \( ( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} )^{2} \)
$7$ \( 1 + 2 T - 9 T^{2} - 2 T^{3} + 92 T^{4} - 14 T^{5} - 441 T^{6} + 686 T^{7} + 2401 T^{8} \)
$11$ \( 1 - 2 T - T^{2} + 34 T^{3} - 140 T^{4} + 374 T^{5} - 121 T^{6} - 2662 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 2 T - 15 T^{2} + 14 T^{3} + 140 T^{4} + 182 T^{5} - 2535 T^{6} - 4394 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 102 T^{5} + 289 T^{6} - 29478 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 6 T - 9 T^{2} + 42 T^{3} + 980 T^{4} + 798 T^{5} - 3249 T^{6} + 41154 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 + 4 T + 23 T^{2} )^{4} \)
$29$ \( ( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 10 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 10 T + 37 T^{2} )^{2}( 1 + 11 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 2 T - 7 T^{2} - 142 T^{3} - 1724 T^{4} - 5822 T^{5} - 11767 T^{6} + 137842 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 2 T + 15 T^{2} + 194 T^{3} - 1900 T^{4} + 8342 T^{5} + 27735 T^{6} - 159014 T^{7} + 3418801 T^{8} \)
$47$ \( ( 1 - 8 T + 78 T^{2} - 376 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( 1 + 6 T - 71 T^{2} + 6 T^{3} + 6732 T^{4} + 318 T^{5} - 199439 T^{6} + 893262 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 6 T - 41 T^{2} + 246 T^{3} + 324 T^{4} + 14514 T^{5} - 142721 T^{6} - 1232274 T^{7} + 12117361 T^{8} \)
$61$ \( ( 1 + 114 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( 1 - 2 T - 113 T^{2} + 34 T^{3} + 8932 T^{4} + 2278 T^{5} - 507257 T^{6} - 601526 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 14 T + 55 T^{2} + 14 T^{3} + 924 T^{4} + 994 T^{5} + 277255 T^{6} - 5010754 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 2 T - 135 T^{2} - 14 T^{3} + 13700 T^{4} - 1022 T^{5} - 719415 T^{6} + 778034 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 22 T + 223 T^{2} - 2266 T^{3} + 23644 T^{4} - 179014 T^{5} + 1391743 T^{6} - 10846858 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 6 T - 89 T^{2} + 246 T^{3} + 5748 T^{4} + 20418 T^{5} - 613121 T^{6} - 3430722 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 + 8 T + 122 T^{2} + 712 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 16 T + 250 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \)
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