Properties

Label 21.8.a
Level $21$
Weight $8$
Character orbit 21.a
Rep. character $\chi_{21}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $4$
Sturm bound $21$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(21\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(21))\).

Total New Old
Modular forms 20 8 12
Cusp forms 16 8 8
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(7\)FrickeDim
\(+\)\(+\)$+$\(2\)
\(+\)\(-\)$-$\(2\)
\(-\)\(+\)$-$\(1\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(3\)

Trace form

\( 8 q + 2 q^{2} + 770 q^{4} - 776 q^{5} - 108 q^{6} + 686 q^{7} + 1854 q^{8} + 5832 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{2} + 770 q^{4} - 776 q^{5} - 108 q^{6} + 686 q^{7} + 1854 q^{8} + 5832 q^{9} + 3852 q^{10} + 1432 q^{11} - 15336 q^{12} + 12112 q^{13} - 8918 q^{14} - 216 q^{15} + 92882 q^{16} - 40968 q^{17} + 1458 q^{18} - 97424 q^{19} + 572 q^{20} + 18522 q^{21} - 68400 q^{22} + 134808 q^{23} - 54756 q^{24} + 138632 q^{25} - 651436 q^{26} + 107702 q^{28} - 439328 q^{29} + 367848 q^{30} + 442432 q^{31} + 523654 q^{32} + 190296 q^{33} - 935460 q^{34} - 58996 q^{35} + 561330 q^{36} - 518624 q^{37} + 444368 q^{38} - 30672 q^{39} + 1448676 q^{40} + 1751048 q^{41} + 148176 q^{42} + 271216 q^{43} - 3091024 q^{44} - 565704 q^{45} + 37512 q^{46} + 1356144 q^{47} - 1884816 q^{48} + 941192 q^{49} - 2099378 q^{50} + 2017224 q^{51} - 696572 q^{52} - 1085904 q^{53} - 78732 q^{54} - 3055728 q^{55} - 3825822 q^{56} + 2204928 q^{57} - 2019924 q^{58} - 1151088 q^{59} - 4629744 q^{60} + 5183200 q^{61} + 12950208 q^{62} + 500094 q^{63} + 8451050 q^{64} + 3820624 q^{65} - 4485672 q^{66} - 6863648 q^{67} - 6501060 q^{68} - 4491288 q^{69} - 1140132 q^{70} - 3994680 q^{71} + 1351566 q^{72} - 10351520 q^{73} + 12127020 q^{74} + 3748896 q^{75} + 832192 q^{76} + 2118368 q^{77} + 1252800 q^{78} + 13872064 q^{79} + 28655804 q^{80} + 4251528 q^{81} - 21988356 q^{82} - 9046368 q^{83} + 3556224 q^{84} - 16470384 q^{85} - 4645208 q^{86} + 271296 q^{87} + 14529696 q^{88} - 353816 q^{89} + 2808108 q^{90} + 9888004 q^{91} - 15951096 q^{92} - 10010736 q^{93} + 7381056 q^{94} - 21731744 q^{95} - 16331436 q^{96} - 5445920 q^{97} + 235298 q^{98} + 1043928 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(21))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7
21.8.a.a 21.a 1.a $1$ $6.560$ \(\Q\) None \(2\) \(27\) \(-278\) \(-343\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3^{3}q^{3}-124q^{4}-278q^{5}+\cdots\)
21.8.a.b 21.a 1.a $2$ $6.560$ \(\Q(\sqrt{1065}) \) None \(-9\) \(-54\) \(-360\) \(686\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{2}-3^{3}q^{3}+(154+9\beta )q^{4}+\cdots\)
21.8.a.c 21.a 1.a $2$ $6.560$ \(\Q(\sqrt{67}) \) None \(12\) \(-54\) \(-24\) \(-686\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(6+\beta )q^{2}-3^{3}q^{3}+(176+12\beta )q^{4}+\cdots\)
21.8.a.d 21.a 1.a $3$ $6.560$ 3.3.2910828.1 None \(-3\) \(81\) \(-114\) \(1029\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+3^{3}q^{3}+(75+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)