Properties

Label 9.4.c
Level $9$
Weight $4$
Character orbit 9.c
Rep. character $\chi_{9}(4,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $4$
Newform subspaces $1$
Sturm bound $4$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 9.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(9, [\chi])\).

Total New Old
Modular forms 8 8 0
Cusp forms 4 4 0
Eisenstein series 4 4 0

Trace form

\( 4 q - 3 q^{2} - 3 q^{3} - 5 q^{4} - 15 q^{5} + 9 q^{6} - 7 q^{7} + 66 q^{8} + 45 q^{9} + O(q^{10}) \) \( 4 q - 3 q^{2} - 3 q^{3} - 5 q^{4} - 15 q^{5} + 9 q^{6} - 7 q^{7} + 66 q^{8} + 45 q^{9} + 12 q^{10} - 66 q^{11} - 156 q^{12} + 11 q^{13} - 60 q^{14} + 27 q^{15} + 7 q^{16} + 198 q^{17} + 216 q^{18} - 154 q^{19} + 12 q^{20} + 21 q^{21} + 33 q^{22} - 33 q^{23} - 99 q^{24} + 121 q^{25} - 528 q^{26} - 432 q^{27} + 332 q^{28} + 51 q^{29} + 288 q^{30} - 43 q^{31} + 423 q^{32} + 198 q^{33} - 297 q^{34} + 6 q^{35} - 225 q^{36} - 100 q^{37} + 561 q^{38} + 759 q^{39} - 264 q^{40} - 132 q^{41} - 486 q^{42} - 88 q^{43} - 462 q^{44} - 675 q^{45} - 528 q^{46} - 399 q^{47} - 21 q^{48} + 513 q^{49} + 429 q^{50} + 297 q^{51} + 770 q^{52} + 108 q^{53} + 1215 q^{54} + 1254 q^{55} - 66 q^{56} - 1221 q^{57} + 60 q^{58} - 798 q^{59} - 36 q^{60} - 439 q^{61} + 228 q^{62} + 603 q^{63} - 1454 q^{64} - 165 q^{65} - 990 q^{66} - 988 q^{67} - 693 q^{68} + 891 q^{69} - 318 q^{70} + 2736 q^{71} + 891 q^{72} - 910 q^{73} - 816 q^{74} - 363 q^{75} + 1529 q^{76} + 165 q^{77} - 990 q^{78} + 803 q^{79} + 192 q^{80} - 567 q^{81} + 3630 q^{82} - 813 q^{83} + 642 q^{84} - 594 q^{85} - 33 q^{86} - 153 q^{87} - 1221 q^{88} - 792 q^{89} - 756 q^{90} - 1562 q^{91} + 858 q^{92} - 213 q^{93} - 2100 q^{94} + 132 q^{95} + 1080 q^{96} - 736 q^{97} - 846 q^{98} + 297 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(9, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
9.4.c.a 9.c 9.c $4$ $0.531$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(-3\) \(-3\) \(-15\) \(-7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}-\beta _{3})q^{2}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)