Properties

Label 136.4.s
Level $136$
Weight $4$
Character orbit 136.s
Rep. character $\chi_{136}(3,\cdot)$
Character field $\Q(\zeta_{16})$
Dimension $416$
Newform subspaces $3$
Sturm bound $72$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.s (of order \(16\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 136 \)
Character field: \(\Q(\zeta_{16})\)
Newform subspaces: \( 3 \)
Sturm bound: \(72\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(136, [\chi])\).

Total New Old
Modular forms 448 448 0
Cusp forms 416 416 0
Eisenstein series 32 32 0

Trace form

\( 416 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{6} - 8 q^{8} - 16 q^{9} + O(q^{10}) \) \( 416 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{6} - 8 q^{8} - 16 q^{9} - 8 q^{10} - 16 q^{11} - 8 q^{12} - 8 q^{14} - 16 q^{17} - 16 q^{18} - 16 q^{19} - 8 q^{20} - 8 q^{22} + 904 q^{24} - 16 q^{25} - 744 q^{26} - 16 q^{27} - 152 q^{28} + 1336 q^{30} - 1248 q^{32} + 2104 q^{34} - 32 q^{35} - 1736 q^{36} + 512 q^{38} + 568 q^{40} - 16 q^{41} - 1880 q^{42} - 16 q^{43} + 2456 q^{44} - 1976 q^{46} + 208 q^{48} - 16 q^{49} - 16 q^{51} - 16 q^{52} + 4376 q^{54} + 2592 q^{56} - 1392 q^{57} + 1280 q^{58} - 16 q^{59} - 1152 q^{60} - 2080 q^{62} - 2648 q^{64} - 4896 q^{65} - 6704 q^{66} - 5904 q^{68} - 5440 q^{70} - 4896 q^{72} + 2944 q^{73} - 1208 q^{74} - 16 q^{75} + 272 q^{76} + 4464 q^{78} + 3976 q^{80} + 3664 q^{81} + 5712 q^{82} - 16 q^{83} + 11104 q^{86} - 5192 q^{88} - 16 q^{89} + 4712 q^{90} - 16 q^{91} + 1224 q^{92} - 4328 q^{94} + 10784 q^{96} - 16 q^{97} - 12112 q^{98} + 416 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(136, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
136.4.s.a 136.s 136.s $8$ $8.024$ \(\Q(\zeta_{16})\) \(\Q(\sqrt{-2}) \) \(0\) \(-8\) \(0\) \(0\) $\mathrm{U}(1)[D_{16}]$ \(q+(2\zeta_{16}-2\zeta_{16}^{5})q^{2}+(-1-\zeta_{16}+\cdots)q^{3}+\cdots\)
136.4.s.b 136.s 136.s $8$ $8.024$ \(\Q(\zeta_{16})\) \(\Q(\sqrt{-2}) \) \(0\) \(8\) \(0\) \(0\) $\mathrm{U}(1)[D_{16}]$ \(q+(-2\zeta_{16}+2\zeta_{16}^{5})q^{2}+(1+\zeta_{16}+\cdots)q^{3}+\cdots\)
136.4.s.c 136.s 136.s $400$ $8.024$ None \(-8\) \(-16\) \(0\) \(0\) $\mathrm{SU}(2)[C_{16}]$