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Properties

Label 21.9.h
Level $21$
Weight $9$
Character orbit 21.h
Rep. character $\chi_{21}(2,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $38$
Newform subspaces $2$
Sturm bound $24$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(21, [\chi])\).

Total New Old
Modular forms 46 46 0
Cusp forms 38 38 0
Eisenstein series 8 8 0

Trace form

\( 38 q - q^{3} + 2046 q^{4} - 1028 q^{6} - 2967 q^{7} - 6847 q^{9} + O(q^{10}) \) \( 38 q - q^{3} + 2046 q^{4} - 1028 q^{6} - 2967 q^{7} - 6847 q^{9} + 4862 q^{10} + 41092 q^{12} + 6030 q^{13} - 132676 q^{15} - 51978 q^{16} + 127516 q^{18} + 172857 q^{19} + 113674 q^{21} - 428116 q^{22} - 238362 q^{24} + 1130019 q^{25} + 111746 q^{27} - 1233238 q^{28} - 55820 q^{30} - 1778149 q^{31} + 2140034 q^{33} + 5992464 q^{34} - 7910060 q^{36} - 3019 q^{37} - 4572859 q^{39} - 11393286 q^{40} + 12825218 q^{42} + 17095966 q^{43} + 8604500 q^{45} + 2834700 q^{46} - 7136888 q^{48} - 11961403 q^{49} - 8525928 q^{51} - 6258320 q^{52} + 36951838 q^{54} - 17482820 q^{55} - 82203922 q^{57} + 26032034 q^{58} - 38452490 q^{60} - 59642262 q^{61} + 86701987 q^{63} + 161372396 q^{64} + 72740152 q^{66} + 77626377 q^{67} - 26793528 q^{69} - 228954530 q^{70} - 38169168 q^{72} - 109819999 q^{73} + 139694431 q^{75} + 65498296 q^{76} - 277758680 q^{78} + 65947915 q^{79} - 60427831 q^{81} - 109600652 q^{82} + 153770072 q^{84} + 208839936 q^{85} + 89390378 q^{87} + 96328590 q^{88} + 229037236 q^{90} - 18892571 q^{91} - 6074667 q^{93} - 109682268 q^{94} - 23819614 q^{96} - 309025336 q^{97} - 330294424 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(21, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
21.9.h.a 21.h 21.h $2$ $8.555$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-81\) \(0\) \(239\) $\mathrm{U}(1)[D_{6}]$ \(q-3^{4}\zeta_{6}q^{3}-2^{8}\zeta_{6}q^{4}+(-1265+2769\zeta_{6})q^{7}+\cdots\)
21.9.h.b 21.h 21.h $36$ $8.555$ None \(0\) \(80\) \(0\) \(-3206\) $\mathrm{SU}(2)[C_{6}]$