Properties

Label 21.12.e
Level $21$
Weight $12$
Character orbit 21.e
Rep. character $\chi_{21}(4,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $30$
Newform subspaces $2$
Sturm bound $32$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 21.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(21, [\chi])\).

Total New Old
Modular forms 62 30 32
Cusp forms 54 30 24
Eisenstein series 8 0 8

Trace form

\( 30 q - 46 q^{2} + 243 q^{3} - 19466 q^{4} + 5062 q^{5} - 31104 q^{6} + 2659 q^{7} - 185832 q^{8} - 885735 q^{9} + O(q^{10}) \) \( 30 q - 46 q^{2} + 243 q^{3} - 19466 q^{4} + 5062 q^{5} - 31104 q^{6} + 2659 q^{7} - 185832 q^{8} - 885735 q^{9} - 133174 q^{10} + 1688986 q^{11} + 497664 q^{12} - 4790266 q^{13} - 1341122 q^{14} - 4555764 q^{15} - 20571674 q^{16} - 3237696 q^{17} - 2716254 q^{18} + 14567843 q^{19} + 128878952 q^{20} - 15076692 q^{21} + 50603732 q^{22} + 2754144 q^{23} + 95987430 q^{24} - 221448929 q^{25} - 106714042 q^{26} - 28697814 q^{27} - 370640094 q^{28} - 102532136 q^{29} + 316982808 q^{30} + 486154231 q^{31} - 77245688 q^{32} - 110970810 q^{33} - 2767284408 q^{34} - 390827050 q^{35} + 2298895668 q^{36} + 1452836479 q^{37} + 1742379110 q^{38} - 109282689 q^{39} + 1330701162 q^{40} - 3085840876 q^{41} - 1759576608 q^{42} - 180394062 q^{43} + 5935945352 q^{44} + 298906038 q^{45} - 4602839460 q^{46} - 1116048714 q^{47} - 1623236112 q^{48} - 8844432267 q^{49} + 29155502260 q^{50} + 1055323728 q^{51} + 20712162516 q^{52} - 10523248404 q^{53} + 918330048 q^{54} - 32032320872 q^{55} - 60728413392 q^{56} - 4398118722 q^{57} + 12814898906 q^{58} + 31253182908 q^{59} - 6373539594 q^{60} + 29668188482 q^{61} - 74371895100 q^{62} + 921223449 q^{63} + 78653417508 q^{64} + 28886837230 q^{65} + 16970045940 q^{66} + 11838470033 q^{67} - 8505810684 q^{68} - 25600567104 q^{69} - 35249043734 q^{70} + 44570311284 q^{71} + 5486596884 q^{72} + 7142829469 q^{73} + 27659961690 q^{74} + 31424816133 q^{75} - 836608440 q^{76} + 27905048552 q^{77} - 90736547004 q^{78} - 19693427343 q^{79} - 227475960868 q^{80} - 52301766015 q^{81} + 8362080604 q^{82} + 297427177500 q^{83} + 131141254392 q^{84} - 303292920384 q^{85} - 127389436418 q^{86} + 74236145220 q^{87} - 49559291946 q^{88} - 258781695824 q^{89} + 15727583052 q^{90} + 271466601303 q^{91} + 628134797304 q^{92} - 176994402075 q^{93} - 255894080460 q^{94} + 210756726418 q^{95} + 337889987082 q^{96} + 452474006812 q^{97} - 321274913548 q^{98} - 199465868628 q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(21, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
21.12.e.a 21.e 7.c $14$ $16.135$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(9\) \(-1701\) \(7218\) \(9219\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+3^{5}\beta _{2}q^{3}+(1241\beta _{2}+\cdots)q^{4}+\cdots\)
21.12.e.b 21.e 7.c $16$ $16.135$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-55\) \(1944\) \(-2156\) \(-6560\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-7-7\beta _{2}-\beta _{4})q^{2}-3^{5}\beta _{2}q^{3}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(21, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(21, [\chi]) \cong \) \(S_{12}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)