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Properties

Label 36.3.f
Level $36$
Weight $3$
Character orbit 36.f
Rep. character $\chi_{36}(7,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $20$
Newform subspaces $3$
Sturm bound $18$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 36 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(18\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(36, [\chi])\).

Total New Old
Modular forms 28 28 0
Cusp forms 20 20 0
Eisenstein series 8 8 0

Trace form

\( 20 q - q^{2} - q^{4} - 2 q^{5} - 9 q^{6} - 22 q^{8} + O(q^{10}) \) \( 20 q - q^{2} - q^{4} - 2 q^{5} - 9 q^{6} - 22 q^{8} + 4 q^{10} - 36 q^{12} - 2 q^{13} - 24 q^{14} - q^{16} - 32 q^{17} + 12 q^{18} + 52 q^{20} - 30 q^{21} - 9 q^{22} + 129 q^{24} - 12 q^{25} + 160 q^{26} - 36 q^{28} - 26 q^{29} + 156 q^{30} + 119 q^{32} - 42 q^{33} - 11 q^{34} - 45 q^{36} - 8 q^{37} - 153 q^{38} + 4 q^{40} + 58 q^{41} - 354 q^{42} - 390 q^{44} + 102 q^{45} - 96 q^{46} - 333 q^{48} - 16 q^{49} - 237 q^{50} + 22 q^{52} + 136 q^{53} + 57 q^{54} + 270 q^{56} + 204 q^{57} + 52 q^{58} + 576 q^{60} - 2 q^{61} + 564 q^{62} + 2 q^{64} + 146 q^{65} + 510 q^{66} + 331 q^{68} - 54 q^{69} + 102 q^{70} - 189 q^{72} + 16 q^{73} - 404 q^{74} + 93 q^{76} - 246 q^{77} - 690 q^{78} - 848 q^{80} - 264 q^{81} + 202 q^{82} - 642 q^{84} - 52 q^{85} - 447 q^{86} + 75 q^{88} - 392 q^{89} + 180 q^{90} + 426 q^{92} - 486 q^{93} + 48 q^{94} + 900 q^{96} - 62 q^{97} + 1022 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(36, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
36.3.f.a 36.f 36.f $2$ $0.981$ \(\Q(\sqrt{-3}) \) None \(-2\) \(3\) \(-4\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2+2\zeta_{6})q^{2}+3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots\)
36.3.f.b 36.f 36.f $2$ $0.981$ \(\Q(\sqrt{-3}) \) None \(4\) \(-3\) \(-4\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+2q^{2}-3\zeta_{6}q^{3}+4q^{4}+(-4+4\zeta_{6})q^{5}+\cdots\)
36.3.f.c 36.f 36.f $16$ $0.981$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-3\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{1}q^{2}+\beta _{14}q^{3}+(-\beta _{2}+\beta _{3})q^{4}+\cdots\)