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Properties

Label 21.7.f
Level $21$
Weight $7$
Character orbit 21.f
Rep. character $\chi_{21}(10,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $16$
Newform subspaces $2$
Sturm bound $18$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 21.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(18\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(21, [\chi])\).

Total New Old
Modular forms 36 16 20
Cusp forms 28 16 12
Eisenstein series 8 0 8

Trace form

\( 16 q - 10 q^{2} - 346 q^{4} - 336 q^{5} + 92 q^{7} + 2872 q^{8} + 1944 q^{9} + O(q^{10}) \) \( 16 q - 10 q^{2} - 346 q^{4} - 336 q^{5} + 92 q^{7} + 2872 q^{8} + 1944 q^{9} - 3150 q^{10} - 1384 q^{11} + 710 q^{14} + 4536 q^{15} - 8810 q^{16} + 1680 q^{17} + 2430 q^{18} - 42840 q^{19} + 17496 q^{21} + 86028 q^{22} - 11296 q^{23} - 51030 q^{24} + 5236 q^{25} - 6678 q^{26} + 81622 q^{28} + 24224 q^{29} + 33048 q^{30} - 49308 q^{31} - 116004 q^{32} - 20412 q^{33} - 268296 q^{35} - 168156 q^{36} + 19036 q^{37} + 426174 q^{38} + 122472 q^{39} + 126882 q^{40} + 228744 q^{42} - 399920 q^{43} - 96300 q^{44} - 81648 q^{45} + 414300 q^{46} + 566160 q^{47} - 286568 q^{49} - 251668 q^{50} + 114696 q^{51} - 951132 q^{52} - 554008 q^{53} - 114824 q^{56} - 114696 q^{57} + 380298 q^{58} + 1628592 q^{59} + 283338 q^{60} + 25368 q^{61} - 54432 q^{63} + 968180 q^{64} - 803208 q^{65} - 1163484 q^{66} - 774808 q^{67} - 2437596 q^{68} + 1493922 q^{70} + 1668032 q^{71} + 348948 q^{72} + 524412 q^{73} + 752590 q^{74} + 1061424 q^{75} + 738320 q^{77} - 565380 q^{78} + 51860 q^{79} - 1247232 q^{80} - 472392 q^{81} - 1213632 q^{82} - 2459160 q^{84} + 1678464 q^{85} - 1486178 q^{86} + 551124 q^{87} - 61554 q^{88} - 2759232 q^{89} - 2656272 q^{91} + 565656 q^{92} + 1167372 q^{93} - 2553768 q^{94} + 568320 q^{95} + 6923070 q^{96} + 9242096 q^{98} - 672624 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(21, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
21.7.f.a 21.f 7.d $8$ $4.831$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-5\) \(-108\) \(-294\) \(-656\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\beta _{1}+\beta _{2})q^{2}+(-18+9\beta _{2}+\cdots)q^{3}+\cdots\)
21.7.f.b 21.f 7.d $8$ $4.831$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-5\) \(108\) \(-42\) \(748\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}-\beta _{2})q^{2}+(9+9\beta _{2})q^{3}+(-43+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(21, [\chi])\) into lower level spaces

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