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Properties

Label 13.9.d
Level $13$
Weight $9$
Character orbit 13.d
Rep. character $\chi_{13}(5,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $18$
Newform subspaces $1$
Sturm bound $10$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 13.d (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(10\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(13, [\chi])\).

Total New Old
Modular forms 22 22 0
Cusp forms 18 18 0
Eisenstein series 4 4 0

Trace form

\( 18 q - 2 q^{2} - 4 q^{3} + 166 q^{5} + 2720 q^{6} + 5308 q^{7} + 10464 q^{8} + 39362 q^{9} + O(q^{10}) \) \( 18 q - 2 q^{2} - 4 q^{3} + 166 q^{5} + 2720 q^{6} + 5308 q^{7} + 10464 q^{8} + 39362 q^{9} - 31556 q^{11} + 71300 q^{13} - 110260 q^{14} + 121124 q^{15} - 522860 q^{16} - 308962 q^{18} + 100288 q^{19} + 736268 q^{20} + 279416 q^{21} - 977312 q^{22} - 986232 q^{24} + 2952238 q^{26} + 364916 q^{27} + 4497084 q^{28} - 2479024 q^{29} - 1892664 q^{31} + 947212 q^{32} + 1368644 q^{33} - 531576 q^{34} - 2918284 q^{35} - 8343978 q^{37} - 1101268 q^{39} - 12691908 q^{40} + 1140178 q^{41} + 15137212 q^{42} - 3867188 q^{44} + 6565370 q^{45} + 2006148 q^{46} - 13368572 q^{47} - 10106716 q^{48} + 37369598 q^{50} - 14821220 q^{52} + 50561348 q^{53} - 4995928 q^{54} + 76994128 q^{55} - 34755760 q^{57} + 22505716 q^{58} + 2127976 q^{59} - 161134196 q^{60} - 52016516 q^{61} - 15210112 q^{63} + 10413082 q^{65} - 51097328 q^{66} + 960292 q^{67} + 283187508 q^{68} - 166635032 q^{70} + 67412140 q^{71} + 96846900 q^{72} - 145213226 q^{73} - 233620024 q^{74} - 150533640 q^{76} + 212071160 q^{78} - 76829120 q^{79} + 524889520 q^{80} + 293280674 q^{81} - 241951556 q^{83} + 345496184 q^{84} + 260737764 q^{85} - 579480384 q^{86} - 618300656 q^{87} - 89187110 q^{89} + 232660948 q^{91} - 122690376 q^{92} + 669553400 q^{93} + 1069637380 q^{94} - 1324092400 q^{96} + 331183146 q^{97} + 588677614 q^{98} - 1125076336 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.9.d.a 13.d 13.d $18$ $5.296$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(-2\) \(-4\) \(166\) \(5308\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{2}+\beta _{6}q^{3}+(-\beta _{1}-142\beta _{2}+\cdots)q^{4}+\cdots\)