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Properties

Label 21.10.a
Level $21$
Weight $10$
Character orbit 21.a
Rep. character $\chi_{21}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $4$
Sturm bound $26$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(26\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(21))\).

Total New Old
Modular forms 26 8 18
Cusp forms 22 8 14
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
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(3\)
Minus space\(-\)\(5\)

Trace form

\( 8 q + 2 q^{2} + 2050 q^{4} + 4552 q^{5} - 6156 q^{6} - 4802 q^{7} + 9390 q^{8} + 52488 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{2} + 2050 q^{4} + 4552 q^{5} - 6156 q^{6} - 4802 q^{7} + 9390 q^{8} + 52488 q^{9} + 16108 q^{10} - 78140 q^{11} + 115992 q^{12} + 279176 q^{13} + 24010 q^{14} - 3564 q^{15} + 214546 q^{16} - 378912 q^{17} + 13122 q^{18} + 586184 q^{19} + 1411100 q^{20} - 388962 q^{21} + 2396416 q^{22} + 725724 q^{23} - 1495908 q^{24} - 2577248 q^{25} + 957428 q^{26} - 4057690 q^{28} + 8576440 q^{29} + 3227688 q^{30} - 8356584 q^{31} - 45299498 q^{32} + 5409504 q^{33} + 28069404 q^{34} - 6204184 q^{35} + 13450050 q^{36} - 37232136 q^{37} - 30025936 q^{38} - 14262480 q^{39} + 52787556 q^{40} + 13951376 q^{41} - 6223392 q^{42} + 50242720 q^{43} - 42073024 q^{44} + 29865672 q^{45} - 98920632 q^{46} + 62344392 q^{47} + 92730096 q^{48} + 46118408 q^{49} + 73994302 q^{50} + 23031540 q^{51} + 231982372 q^{52} - 217936680 q^{53} - 40389516 q^{54} + 130559768 q^{55} + 63083874 q^{56} + 5276664 q^{57} - 443463044 q^{58} + 69912216 q^{59} - 29254608 q^{60} - 85792600 q^{61} - 308107200 q^{62} - 31505922 q^{63} - 56923702 q^{64} + 246963088 q^{65} + 35066520 q^{66} + 179326184 q^{67} - 853640772 q^{68} + 547698672 q^{69} + 291106844 q^{70} + 151882212 q^{71} + 61607790 q^{72} + 781362720 q^{73} - 758102628 q^{74} - 638703792 q^{75} - 219853280 q^{76} + 467349848 q^{77} - 408004128 q^{78} + 75820360 q^{79} - 225047332 q^{80} + 344373768 q^{81} - 1050294052 q^{82} - 167217024 q^{83} - 298722816 q^{84} - 2054101752 q^{85} + 3203864920 q^{86} - 472571496 q^{87} + 1195436784 q^{88} + 473386912 q^{89} + 105684588 q^{90} - 720175148 q^{91} + 3191894664 q^{92} + 89676072 q^{93} + 130819488 q^{94} + 1028816560 q^{95} - 374635692 q^{96} + 2488240800 q^{97} + 11529602 q^{98} - 512676540 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(21))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7
21.10.a.a 21.a 1.a $1$ $10.816$ \(\Q\) None 21.10.a.a \(-24\) \(81\) \(-144\) \(2401\) $-$ $-$ $\mathrm{SU}(2)$ \(q-24q^{2}+3^{4}q^{3}+2^{6}q^{4}-12^{2}q^{5}+\cdots\)
21.10.a.b 21.a 1.a $2$ $10.816$ \(\Q(\sqrt{2353}) \) None 21.10.a.b \(9\) \(-162\) \(1170\) \(-4802\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(5-\beta )q^{2}-3^{4}q^{3}+(101-9\beta )q^{4}+\cdots\)
21.10.a.c 21.a 1.a $2$ $10.816$ \(\Q(\sqrt{345}) \) None 21.10.a.c \(30\) \(-162\) \(1128\) \(4802\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(15-\beta )q^{2}-3^{4}q^{3}+(58-30\beta )q^{4}+\cdots\)
21.10.a.d 21.a 1.a $3$ $10.816$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 21.10.a.d \(-13\) \(243\) \(2398\) \(-7203\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-4-\beta _{1})q^{2}+3^{4}q^{3}+(555+7\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(21)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)