Properties

Label 136.3.j
Level $136$
Weight $3$
Character orbit 136.j
Rep. character $\chi_{136}(115,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $68$
Newform subspaces $2$
Sturm bound $54$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 136.j (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 136 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(54\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(136, [\chi])\).

Total New Old
Modular forms 76 76 0
Cusp forms 68 68 0
Eisenstein series 8 8 0

Trace form

\( 68 q - 4 q^{3} - 4 q^{4} + 18 q^{6} + O(q^{10}) \) \( 68 q - 4 q^{3} - 4 q^{4} + 18 q^{6} - 30 q^{10} - 4 q^{11} - 30 q^{12} - 32 q^{14} - 60 q^{16} - 12 q^{17} + 12 q^{18} + 70 q^{20} + 18 q^{22} + 86 q^{24} + 32 q^{27} - 76 q^{28} + 56 q^{30} - 8 q^{33} + 62 q^{34} - 8 q^{35} - 88 q^{38} - 62 q^{40} + 4 q^{41} + 194 q^{44} - 280 q^{46} + 146 q^{48} - 364 q^{50} - 164 q^{51} - 204 q^{52} + 204 q^{54} - 328 q^{56} + 112 q^{57} + 194 q^{58} - 72 q^{62} - 52 q^{64} + 112 q^{65} - 8 q^{67} - 226 q^{68} + 252 q^{72} - 124 q^{73} - 250 q^{74} - 68 q^{75} + 508 q^{78} + 674 q^{80} - 108 q^{81} + 300 q^{82} + 1008 q^{84} - 324 q^{86} - 134 q^{88} - 8 q^{89} - 834 q^{90} + 384 q^{91} - 104 q^{92} + 26 q^{96} - 76 q^{97} + 176 q^{98} + 284 q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(136, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
136.3.j.a 136.j 136.j $4$ $3.706$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) \(0\) \(4\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+2\zeta_{8}^{2}q^{2}+(1+\zeta_{8}+\zeta_{8}^{2})q^{3}-4q^{4}+\cdots\)
136.3.j.b 136.j 136.j $64$ $3.706$ None \(0\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$