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Properties

Label 31.6.a
Level $31$
Weight $6$
Character orbit 31.a
Rep. character $\chi_{31}(1,\cdot)$
Character field $\Q$
Dimension $13$
Newform subspaces $2$
Sturm bound $16$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 31.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(31))\).

Total New Old
Modular forms 15 13 2
Cusp forms 13 13 0
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(31\)Dim
\(+\)\(5\)
\(-\)\(8\)

Trace form

\( 13 q - 2 q^{2} - 22 q^{3} + 202 q^{4} + 56 q^{5} - 72 q^{6} - 20 q^{7} + 237 q^{8} + 989 q^{9} + O(q^{10}) \) \( 13 q - 2 q^{2} - 22 q^{3} + 202 q^{4} + 56 q^{5} - 72 q^{6} - 20 q^{7} + 237 q^{8} + 989 q^{9} + 797 q^{10} - 664 q^{11} - 758 q^{12} - 288 q^{13} - 795 q^{14} + 78 q^{15} + 2930 q^{16} + 648 q^{17} - 2018 q^{18} - 1140 q^{19} + 2405 q^{20} + 446 q^{21} - 1672 q^{22} - 8130 q^{23} - 9164 q^{24} + 3153 q^{25} + 10766 q^{26} - 1228 q^{27} - 8089 q^{28} - 2708 q^{29} - 21534 q^{30} + 2883 q^{31} + 6262 q^{32} + 25284 q^{33} - 32460 q^{34} + 5932 q^{35} + 8646 q^{36} + 6258 q^{37} + 2429 q^{38} + 17028 q^{39} + 46764 q^{40} + 17344 q^{41} - 3326 q^{42} + 32658 q^{43} - 24288 q^{44} - 23988 q^{45} + 37314 q^{46} - 8628 q^{47} - 54788 q^{48} + 18235 q^{49} + 61573 q^{50} + 51276 q^{51} - 46472 q^{52} - 16366 q^{53} - 16292 q^{54} - 27732 q^{55} - 169716 q^{56} + 45118 q^{57} + 30148 q^{58} + 55172 q^{59} - 90652 q^{60} - 123612 q^{61} + 15376 q^{62} - 94408 q^{63} + 104973 q^{64} - 7778 q^{65} - 104084 q^{66} + 10588 q^{67} - 15546 q^{68} + 45324 q^{69} - 104263 q^{70} + 145812 q^{71} + 180589 q^{72} + 104382 q^{73} + 171566 q^{74} - 260452 q^{75} - 211861 q^{76} + 42800 q^{77} - 88660 q^{78} - 64738 q^{79} + 520557 q^{80} + 90369 q^{81} + 228711 q^{82} - 90784 q^{83} + 177240 q^{84} - 149618 q^{85} + 155546 q^{86} - 406168 q^{87} + 30034 q^{88} - 46428 q^{89} + 398961 q^{90} + 93978 q^{91} - 376500 q^{92} + 17298 q^{93} - 260948 q^{94} + 51424 q^{95} - 248242 q^{96} - 209068 q^{97} + 127745 q^{98} - 49280 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(31))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 31
31.6.a.a 31.a 1.a $5$ $4.972$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-9\) \(-20\) \(-72\) \(-108\) $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}+(-4-\beta _{1}-\beta _{3})q^{3}+\cdots\)
31.6.a.b 31.a 1.a $8$ $4.972$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(7\) \(-2\) \(128\) \(88\) $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+\beta _{6}q^{3}+(19-3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)