Properties

Label 21.4.a
Level $21$
Weight $4$
Character orbit 21.a
Rep. character $\chi_{21}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $3$
Sturm bound $10$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(10\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(21))\).

Total New Old
Modular forms 10 4 6
Cusp forms 6 4 2
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\(+\)\(3\)
Minus space\(-\)\(1\)

Trace form

\( 4 q - 2 q^{2} + 26 q^{4} - 16 q^{5} - 12 q^{6} + 14 q^{7} - 66 q^{8} + 36 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{2} + 26 q^{4} - 16 q^{5} - 12 q^{6} + 14 q^{7} - 66 q^{8} + 36 q^{9} - 28 q^{10} + 20 q^{11} + 24 q^{12} - 80 q^{13} - 70 q^{14} + 84 q^{15} + 2 q^{16} + 120 q^{17} - 18 q^{18} + 40 q^{19} + 172 q^{20} + 42 q^{21} + 80 q^{22} - 36 q^{23} - 324 q^{24} - 28 q^{25} + 172 q^{26} + 70 q^{28} - 160 q^{29} - 312 q^{30} - 168 q^{31} - 490 q^{32} - 96 q^{33} + 276 q^{34} - 56 q^{35} + 234 q^{36} - 96 q^{37} + 1360 q^{38} + 336 q^{39} - 924 q^{40} - 1016 q^{41} + 84 q^{42} + 176 q^{43} + 1264 q^{44} - 144 q^{45} + 792 q^{46} + 552 q^{47} + 816 q^{48} + 196 q^{49} - 1198 q^{50} - 396 q^{51} - 1420 q^{52} - 1344 q^{53} - 108 q^{54} + 952 q^{55} - 462 q^{56} + 264 q^{57} - 652 q^{58} + 792 q^{59} + 816 q^{60} + 592 q^{61} - 1824 q^{62} + 126 q^{63} + 1210 q^{64} + 224 q^{65} - 1896 q^{66} + 1144 q^{67} + 492 q^{68} + 144 q^{69} + 28 q^{70} - 444 q^{71} - 594 q^{72} - 72 q^{73} - 780 q^{74} - 624 q^{75} - 832 q^{76} - 728 q^{77} + 1392 q^{78} - 856 q^{79} + 3484 q^{80} + 324 q^{81} + 1492 q^{82} + 2208 q^{83} + 504 q^{84} - 1224 q^{85} + 1880 q^{86} - 1032 q^{87} - 1920 q^{88} + 920 q^{89} - 252 q^{90} + 308 q^{91} - 3192 q^{92} + 744 q^{93} - 1344 q^{94} + 656 q^{95} - 204 q^{96} - 744 q^{97} - 98 q^{98} + 180 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(21))\) 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
21.4.a.a 21.a 1.a $1$ $1.239$ \(\Q\) None \(-3\) \(-3\) \(-18\) \(7\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}-3q^{3}+q^{4}-18q^{5}+9q^{6}+\cdots\)
21.4.a.b 21.a 1.a $1$ $1.239$ \(\Q\) None \(4\) \(-3\) \(-4\) \(-7\) $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-3q^{3}+8q^{4}-4q^{5}-12q^{6}+\cdots\)
21.4.a.c 21.a 1.a $2$ $1.239$ \(\Q(\sqrt{57}) \) None \(-3\) \(6\) \(6\) \(14\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+3q^{3}+(7+3\beta )q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)