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Properties

Label 309.2.m
Level $309$
Weight $2$
Character orbit 309.m
Rep. character $\chi_{309}(4,\cdot)$
Character field $\Q(\zeta_{51})$
Dimension $544$
Newform subspaces $2$
Sturm bound $69$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 309.m (of order \(51\) and degree \(32\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 103 \)
Character field: \(\Q(\zeta_{51})\)
Newform subspaces: \( 2 \)
Sturm bound: \(69\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(309, [\chi])\).

Total New Old
Modular forms 1184 544 640
Cusp forms 1056 544 512
Eisenstein series 128 0 128

Trace form

\( 544 q - 2 q^{3} + 16 q^{4} + 2 q^{6} - 3 q^{7} + 12 q^{8} - 34 q^{9} + O(q^{10}) \) \( 544 q - 2 q^{3} + 16 q^{4} + 2 q^{6} - 3 q^{7} + 12 q^{8} - 34 q^{9} - 100 q^{10} - 6 q^{11} - 58 q^{12} + 2 q^{13} + 20 q^{14} + 2 q^{15} + 22 q^{16} + 2 q^{17} - 21 q^{19} - 6 q^{20} + 5 q^{21} + 24 q^{22} - 54 q^{23} - 47 q^{25} + 14 q^{26} - 2 q^{27} - 76 q^{28} - 52 q^{30} - 26 q^{31} + 18 q^{32} - 10 q^{33} - 70 q^{34} - 20 q^{35} + 16 q^{36} + 18 q^{37} - 112 q^{38} - 18 q^{39} + 22 q^{40} + 2 q^{41} + 8 q^{42} + q^{43} - 44 q^{44} - 184 q^{46} + 12 q^{47} + 36 q^{48} - 82 q^{49} - 98 q^{50} + 2 q^{51} - 14 q^{52} - 16 q^{53} + 2 q^{54} + 20 q^{55} - 202 q^{56} - 59 q^{57} - 80 q^{58} - 36 q^{59} + 4 q^{60} - 8 q^{61} - 32 q^{62} - 3 q^{63} - 130 q^{64} + 16 q^{65} - 16 q^{66} + 24 q^{67} - 18 q^{68} - 12 q^{69} + 30 q^{70} - 108 q^{71} + 12 q^{72} - 90 q^{73} + 4 q^{74} + 7 q^{75} - 128 q^{76} - 126 q^{77} - 116 q^{78} - 34 q^{79} - 12 q^{80} - 34 q^{81} - 384 q^{82} - 16 q^{83} - 216 q^{84} + 40 q^{85} - 252 q^{86} + 64 q^{87} + 124 q^{88} + 146 q^{89} + 36 q^{90} - 118 q^{91} + 380 q^{92} + 158 q^{93} + 52 q^{94} + 84 q^{95} + 410 q^{96} + 23 q^{97} + 502 q^{98} + 62 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(309, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
309.2.m.a 309.m 103.g $256$ $2.467$ None \(-1\) \(16\) \(-1\) \(-4\) $\mathrm{SU}(2)[C_{51}]$
309.2.m.b 309.m 103.g $288$ $2.467$ None \(1\) \(-18\) \(1\) \(1\) $\mathrm{SU}(2)[C_{51}]$

Decomposition of \(S_{2}^{\mathrm{old}}(309, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(309, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(103, [\chi])\)\(^{\oplus 2}\)