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Properties

Label 99.6.f
Level $99$
Weight $6$
Character orbit 99.f
Rep. character $\chi_{99}(37,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $96$
Newform subspaces $4$
Sturm bound $72$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 99.f (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 4 \)
Sturm bound: \(72\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(99, [\chi])\).

Total New Old
Modular forms 256 104 152
Cusp forms 224 96 128
Eisenstein series 32 8 24

Trace form

\( 96 q - 3 q^{2} - 393 q^{4} + 54 q^{5} + 348 q^{7} + 21 q^{8} + O(q^{10}) \) \( 96 q - 3 q^{2} - 393 q^{4} + 54 q^{5} + 348 q^{7} + 21 q^{8} - 816 q^{10} - 762 q^{11} + 114 q^{13} + 438 q^{14} - 8049 q^{16} + 684 q^{17} + 276 q^{19} - 5700 q^{20} + 8697 q^{22} + 1308 q^{23} - 20442 q^{25} - 5730 q^{26} + 3306 q^{28} + 2646 q^{29} + 5916 q^{31} - 5676 q^{32} + 35586 q^{34} + 22104 q^{35} + 22698 q^{37} - 14118 q^{38} - 83028 q^{40} - 32046 q^{41} + 5304 q^{43} + 22074 q^{44} - 49284 q^{46} + 51054 q^{47} + 18822 q^{49} + 45795 q^{50} + 192594 q^{52} + 106488 q^{53} - 14124 q^{55} - 409164 q^{56} - 219612 q^{58} - 64122 q^{59} - 40638 q^{61} + 192660 q^{62} - 84681 q^{64} + 203688 q^{65} + 332472 q^{67} + 127242 q^{68} + 160200 q^{70} - 123672 q^{71} - 5178 q^{73} - 263934 q^{74} - 596514 q^{76} - 73272 q^{77} + 49692 q^{79} + 475866 q^{80} + 133725 q^{82} + 33738 q^{83} + 311838 q^{85} + 14757 q^{86} + 30207 q^{88} - 535488 q^{89} + 91320 q^{91} - 657852 q^{92} + 45606 q^{94} + 247926 q^{95} + 262716 q^{97} + 1703052 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(99, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
99.6.f.a 99.f 11.c $16$ $15.878$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(1\) \(0\) \(10\) \(196\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\beta _{1}+\beta _{2}-\beta _{3}-\beta _{4}-\beta _{7}+\cdots)q^{2}+\cdots\)
99.6.f.b 99.f 11.c $20$ $15.878$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-6\) \(0\) \(11\) \(-139\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\beta _{5}-\beta _{6})q^{2}+(-12+\beta _{1}-12\beta _{2}+\cdots)q^{4}+\cdots\)
99.6.f.c 99.f 11.c $20$ $15.878$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(2\) \(0\) \(33\) \(-335\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\beta _{1}-\beta _{7})q^{2}+(-\beta _{6}+2\beta _{7}-20\beta _{8}+\cdots)q^{4}+\cdots\)
99.6.f.d 99.f 11.c $40$ $15.878$ None \(0\) \(0\) \(0\) \(626\) $\mathrm{SU}(2)[C_{5}]$

Decomposition of \(S_{6}^{\mathrm{old}}(99, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(99, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(11, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 2}\)