Properties

Label 13.7.d
Level 13
Weight 7
Character orbit d
Rep. character \(\chi_{13}(5,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 12
Newforms 1
Sturm bound 8
Trace bound 0

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 13.d (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 13 \)
Character field: \(\Q(i)\)
Newforms: \( 1 \)
Sturm bound: \(8\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(13, [\chi])\).

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

Trace form

\( 12q + 6q^{2} - 4q^{3} + 108q^{5} - 640q^{6} + 398q^{7} - 912q^{8} + 1940q^{9} + O(q^{10}) \) \( 12q + 6q^{2} - 4q^{3} + 108q^{5} - 640q^{6} + 398q^{7} - 912q^{8} + 1940q^{9} + 1686q^{11} - 3926q^{13} + 5484q^{14} - 15268q^{15} + 2132q^{16} + 15254q^{18} + 1766q^{19} - 19044q^{20} + 3428q^{21} + 28832q^{22} + 31608q^{24} - 58266q^{26} - 20464q^{27} + 4092q^{28} - 90108q^{29} + 61014q^{31} - 64932q^{32} - 44452q^{33} + 259896q^{34} + 158772q^{35} - 40212q^{37} + 137852q^{39} - 104196q^{40} - 190416q^{41} - 959204q^{42} + 489372q^{44} - 151444q^{45} - 44412q^{46} + 562446q^{47} + 930308q^{48} + 82422q^{50} - 578500q^{52} + 509136q^{53} - 871432q^{54} - 1264036q^{55} + 939908q^{57} - 1019980q^{58} - 994458q^{59} + 2407804q^{60} + 1013696q^{61} + 865778q^{63} - 1130064q^{65} - 418352q^{66} - 1442386q^{67} - 2313132q^{68} + 2958968q^{70} - 655866q^{71} - 1706508q^{72} + 2588228q^{73} + 3373752q^{74} + 246984q^{76} + 77480q^{78} - 75316q^{79} - 2685408q^{80} - 4016140q^{81} + 894966q^{83} - 3220504q^{84} + 105396q^{85} + 3704832q^{86} + 2109064q^{87} - 977376q^{89} + 1088750q^{91} + 3682872q^{92} - 216268q^{93} - 6238300q^{94} + 896384q^{96} + 983388q^{97} + 1039302q^{98} + 2894714q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(13, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
13.7.d.a \(12\) \(2.991\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(6\) \(-4\) \(108\) \(398\) \(q-\beta _{2}q^{2}+(\beta _{1}+\beta _{2}-\beta _{3}-\beta _{7})q^{3}+\cdots\)