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Properties

Label 21.5.f
Level $21$
Weight $5$
Character orbit 21.f
Rep. character $\chi_{21}(10,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $10$
Newform subspaces $3$
Sturm bound $13$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 21.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(13\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(21, [\chi])\).

Total New Old
Modular forms 26 10 16
Cusp forms 18 10 8
Eisenstein series 8 0 8

Trace form

\( 10 q + 6 q^{2} - 9 q^{3} - 26 q^{4} + 54 q^{5} + 9 q^{7} - 552 q^{8} + 135 q^{9} + O(q^{10}) \) \( 10 q + 6 q^{2} - 9 q^{3} - 26 q^{4} + 54 q^{5} + 9 q^{7} - 552 q^{8} + 135 q^{9} + 498 q^{10} + 270 q^{11} + 288 q^{12} - 138 q^{14} - 144 q^{15} - 522 q^{16} - 864 q^{17} - 162 q^{18} - 717 q^{19} - 108 q^{21} + 1580 q^{22} + 564 q^{23} + 2106 q^{24} + 421 q^{25} + 378 q^{26} - 538 q^{28} - 480 q^{29} - 1800 q^{30} - 459 q^{31} + 1212 q^{32} - 3240 q^{33} + 1230 q^{35} - 1404 q^{36} - 711 q^{37} + 270 q^{38} + 27 q^{39} - 4830 q^{40} + 1656 q^{42} - 1122 q^{43} + 6372 q^{44} + 1458 q^{45} - 2452 q^{46} + 15606 q^{47} + 7291 q^{49} - 14580 q^{50} - 2700 q^{51} + 11172 q^{52} + 5448 q^{53} - 26376 q^{56} + 2322 q^{57} - 11302 q^{58} - 14868 q^{59} + 1242 q^{60} - 23448 q^{61} + 4131 q^{63} + 19092 q^{64} + 13974 q^{65} + 13284 q^{66} + 839 q^{67} + 42516 q^{68} + 24162 q^{70} - 4932 q^{71} - 7452 q^{72} + 957 q^{73} - 12642 q^{74} - 24291 q^{75} - 21876 q^{77} + 8892 q^{78} - 2371 q^{79} - 68400 q^{80} - 3645 q^{81} - 18864 q^{82} + 26280 q^{84} + 23424 q^{85} + 31614 q^{86} + 25110 q^{87} - 8354 q^{88} + 24732 q^{89} - 10155 q^{91} + 3192 q^{92} - 24111 q^{93} + 13800 q^{94} + 9918 q^{95} - 48546 q^{96} + 18624 q^{98} + 14580 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{5}^{\mathrm{new}}(21, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
21.5.f.a 21.f 7.d $2$ $2.171$ \(\Q(\sqrt{-3}) \) None \(-2\) \(9\) \(18\) \(77\) $\mathrm{SU}(2)[C_{6}]$ \(q-2\zeta_{6}q^{2}+(3+3\zeta_{6})q^{3}+(12-12\zeta_{6})q^{4}+\cdots\)
21.5.f.b 21.f 7.d $2$ $2.171$ \(\Q(\sqrt{-3}) \) None \(5\) \(9\) \(-3\) \(-91\) $\mathrm{SU}(2)[C_{6}]$ \(q+5\zeta_{6}q^{2}+(3+3\zeta_{6})q^{3}+(-9+9\zeta_{6})q^{4}+\cdots\)
21.5.f.c 21.f 7.d $6$ $2.171$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(3\) \(-27\) \(39\) \(23\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{2})q^{2}+(-3-3\beta _{2})q^{3}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(21, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(21, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)