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Properties

Label 21.6.a
Level $21$
Weight $6$
Character orbit 21.a
Rep. character $\chi_{21}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $4$
Sturm bound $16$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(16\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(21))\).

Total New Old
Modular forms 16 4 12
Cusp forms 12 4 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
\(+\)\(+\)$+$\(1\)
\(+\)\(-\)$-$\(1\)
\(-\)\(+\)$-$\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\( 4 q + 10 q^{2} + 34 q^{4} + 32 q^{5} + 180 q^{6} - 98 q^{7} + 270 q^{8} + 324 q^{9} + O(q^{10}) \) \( 4 q + 10 q^{2} + 34 q^{4} + 32 q^{5} + 180 q^{6} - 98 q^{7} + 270 q^{8} + 324 q^{9} - 1092 q^{10} + 248 q^{11} + 792 q^{12} - 88 q^{13} - 1078 q^{14} - 504 q^{15} + 466 q^{16} - 3168 q^{17} + 810 q^{18} + 4400 q^{19} - 6500 q^{20} - 882 q^{21} - 1824 q^{22} - 744 q^{23} + 540 q^{24} + 14812 q^{25} + 5956 q^{26} - 1274 q^{28} - 7048 q^{29} - 792 q^{30} + 7616 q^{31} + 9590 q^{32} + 360 q^{33} - 12372 q^{34} + 6076 q^{35} + 2754 q^{36} + 1208 q^{37} - 15632 q^{38} - 1008 q^{39} - 41244 q^{40} + 17872 q^{41} - 3528 q^{42} - 27040 q^{43} + 18208 q^{44} + 2592 q^{45} - 43032 q^{46} - 11184 q^{47} + 7920 q^{48} + 9604 q^{49} + 89942 q^{50} - 11880 q^{51} + 35108 q^{52} - 42840 q^{53} + 14580 q^{54} + 41328 q^{55} + 3234 q^{56} - 24768 q^{57} + 63180 q^{58} + 6576 q^{59} - 83088 q^{60} + 7832 q^{61} + 65664 q^{62} - 7938 q^{63} + 19018 q^{64} - 193312 q^{65} + 37656 q^{66} + 5360 q^{67} + 23292 q^{68} - 24840 q^{69} + 7644 q^{70} + 4008 q^{71} + 21870 q^{72} + 101816 q^{73} + 47532 q^{74} + 115488 q^{75} - 90784 q^{76} + 31360 q^{77} - 2304 q^{78} - 58720 q^{79} - 341732 q^{80} + 26244 q^{81} - 110868 q^{82} - 6912 q^{83} - 42336 q^{84} - 149232 q^{85} + 42008 q^{86} + 182880 q^{87} + 118992 q^{88} - 16624 q^{89} - 88452 q^{90} - 39004 q^{91} + 39816 q^{92} + 112032 q^{93} + 26208 q^{94} + 438080 q^{95} + 7380 q^{96} - 279592 q^{97} + 24010 q^{98} + 20088 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\) 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
21.6.a.a 21.a 1.a $1$ $3.368$ \(\Q\) None \(-6\) \(-9\) \(78\) \(49\) $+$ $-$ $\mathrm{SU}(2)$ \(q-6q^{2}-9q^{3}+4q^{4}+78q^{5}+54q^{6}+\cdots\)
21.6.a.b 21.a 1.a $1$ $3.368$ \(\Q\) None \(1\) \(-9\) \(-34\) \(-49\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-9q^{3}-31q^{4}-34q^{5}-9q^{6}+\cdots\)
21.6.a.c 21.a 1.a $1$ $3.368$ \(\Q\) None \(5\) \(9\) \(94\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q+5q^{2}+9q^{3}-7q^{4}+94q^{5}+45q^{6}+\cdots\)
21.6.a.d 21.a 1.a $1$ $3.368$ \(\Q\) None \(10\) \(9\) \(-106\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q+10q^{2}+9q^{3}+68q^{4}-106q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)