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Properties

Label 9.8.c
Level $9$
Weight $8$
Character orbit 9.c
Rep. character $\chi_{9}(4,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $1$
Sturm bound $8$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(8\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(9, [\chi])\).

Total New Old
Modular forms 16 16 0
Cusp forms 12 12 0
Eisenstein series 4 4 0

Trace form

\( 12 q - 9 q^{2} + 24 q^{3} - 321 q^{4} - 180 q^{5} - 1233 q^{6} - 84 q^{7} + 5922 q^{8} + 990 q^{9} + O(q^{10}) \) \( 12 q - 9 q^{2} + 24 q^{3} - 321 q^{4} - 180 q^{5} - 1233 q^{6} - 84 q^{7} + 5922 q^{8} + 990 q^{9} + 252 q^{10} - 8460 q^{11} + 8052 q^{12} - 1848 q^{13} - 16272 q^{14} - 1188 q^{15} - 12417 q^{16} + 30564 q^{17} + 42876 q^{18} + 24432 q^{19} - 40788 q^{20} - 187224 q^{21} - 35001 q^{22} - 51588 q^{23} + 215469 q^{24} + 4746 q^{25} + 536472 q^{26} + 322272 q^{27} + 75516 q^{28} - 414648 q^{29} - 1112112 q^{30} + 8196 q^{31} - 1048977 q^{32} - 148518 q^{33} - 106623 q^{34} + 2210616 q^{35} + 2501811 q^{36} + 139344 q^{37} - 1952685 q^{38} - 2057316 q^{39} + 305496 q^{40} - 1731582 q^{41} + 538866 q^{42} + 408372 q^{43} + 5169114 q^{44} + 2687580 q^{45} - 1684008 q^{46} - 1631484 q^{47} - 7434525 q^{48} - 179010 q^{49} - 1654461 q^{50} + 2525688 q^{51} + 681594 q^{52} + 2835648 q^{53} + 5816529 q^{54} - 16056 q^{55} - 1784466 q^{56} - 3071850 q^{57} - 948384 q^{58} - 2055636 q^{59} + 371484 q^{60} - 2723196 q^{61} - 1026828 q^{62} - 2238804 q^{63} + 7178178 q^{64} - 1387620 q^{65} + 5754762 q^{66} + 3806556 q^{67} + 2142639 q^{68} - 3002292 q^{69} + 953442 q^{70} + 2408400 q^{71} - 9638325 q^{72} - 10670052 q^{73} + 9846504 q^{74} + 19174632 q^{75} - 6727827 q^{76} + 3478824 q^{77} - 13339962 q^{78} + 6020916 q^{79} - 38072448 q^{80} - 28538730 q^{81} + 9403002 q^{82} + 9605052 q^{83} + 30090090 q^{84} - 1698624 q^{85} + 34278561 q^{86} + 10290708 q^{87} - 16459029 q^{88} - 24630264 q^{89} - 13660596 q^{90} + 13570104 q^{91} + 39143394 q^{92} + 27331212 q^{93} + 12602808 q^{94} + 10422072 q^{95} - 3404376 q^{96} + 9977226 q^{97} - 95833314 q^{98} - 49382676 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(9, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
9.8.c.a 9.c 9.c $12$ $2.811$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-9\) \(24\) \(-180\) \(-84\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{2}-\beta _{3})q^{2}+(1-\beta _{1}-\beta _{4}+\beta _{6}+\cdots)q^{3}+\cdots\)