Properties

Label 198.4.f
Level $198$
Weight $4$
Character orbit 198.f
Rep. character $\chi_{198}(37,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $60$
Newform subspaces $8$
Sturm bound $144$
Trace bound $5$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 198.f (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 8 \)
Sturm bound: \(144\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(198, [\chi])\).

Total New Old
Modular forms 464 60 404
Cusp forms 400 60 340
Eisenstein series 64 0 64

Trace form

\( 60 q - 2 q^{2} - 60 q^{4} + 24 q^{5} - 8 q^{8} + O(q^{10}) \) \( 60 q - 2 q^{2} - 60 q^{4} + 24 q^{5} - 8 q^{8} - 56 q^{10} - 77 q^{11} - 94 q^{13} + 52 q^{14} - 240 q^{16} + 124 q^{17} - 117 q^{19} + 96 q^{20} + 2 q^{22} - 860 q^{23} - 719 q^{25} - 252 q^{26} + 40 q^{28} + 812 q^{29} - 444 q^{31} + 128 q^{32} + 156 q^{34} + 618 q^{35} + 540 q^{37} - 52 q^{38} - 64 q^{40} - 1408 q^{41} - 2966 q^{43} + 152 q^{44} + 8 q^{46} + 386 q^{47} - 1681 q^{49} - 982 q^{50} + 744 q^{52} - 1658 q^{53} + 2474 q^{55} + 128 q^{56} - 940 q^{58} - 1325 q^{59} + 2182 q^{61} + 1028 q^{62} - 960 q^{64} + 8660 q^{65} + 3898 q^{67} + 496 q^{68} - 1104 q^{70} + 58 q^{71} - 596 q^{73} - 2916 q^{74} - 328 q^{76} - 4438 q^{77} - 548 q^{79} - 256 q^{80} + 974 q^{82} - 6605 q^{83} + 6932 q^{85} - 1050 q^{86} + 568 q^{88} + 3346 q^{89} + 2646 q^{91} + 1320 q^{92} + 3424 q^{94} + 6640 q^{95} - 277 q^{97} + 7368 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(198, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
198.4.f.a 198.f 11.c $4$ $11.682$ \(\Q(\zeta_{10})\) None \(-2\) \(0\) \(-15\) \(-19\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-2+2\zeta_{10}-2\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots\)
198.4.f.b 198.f 11.c $4$ $11.682$ \(\Q(\zeta_{10})\) None \(2\) \(0\) \(-3\) \(25\) $\mathrm{SU}(2)[C_{5}]$ \(q+(2-2\zeta_{10}+2\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}+\cdots\)
198.4.f.c 198.f 11.c $4$ $11.682$ \(\Q(\zeta_{10})\) None \(2\) \(0\) \(15\) \(-31\) $\mathrm{SU}(2)[C_{5}]$ \(q+(2-2\zeta_{10}+2\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}+\cdots\)
198.4.f.d 198.f 11.c $8$ $11.682$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-4\) \(0\) \(-5\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+2\beta _{3}q^{2}+4\beta _{2}q^{4}+(4+8\beta _{2}+8\beta _{3}+\cdots)q^{5}+\cdots\)
198.4.f.e 198.f 11.c $8$ $11.682$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-4\) \(0\) \(27\) \(27\) $\mathrm{SU}(2)[C_{5}]$ \(q-2\beta _{2}q^{2}+4\beta _{3}q^{4}+(-1-\beta _{1}+6\beta _{2}+\cdots)q^{5}+\cdots\)
198.4.f.f 198.f 11.c $8$ $11.682$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(4\) \(0\) \(5\) \(-13\) $\mathrm{SU}(2)[C_{5}]$ \(q+2\beta _{3}q^{2}+(-4+4\beta _{2}+4\beta _{3}+4\beta _{4}+\cdots)q^{4}+\cdots\)
198.4.f.g 198.f 11.c $12$ $11.682$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(-6\) \(0\) \(16\) \(6\) $\mathrm{SU}(2)[C_{5}]$ \(q+2\beta _{3}q^{2}-4\beta _{4}q^{4}+(3-\beta _{1}+2\beta _{3}+\cdots)q^{5}+\cdots\)
198.4.f.h 198.f 11.c $12$ $11.682$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(6\) \(0\) \(-16\) \(6\) $\mathrm{SU}(2)[C_{5}]$ \(q-2\beta _{3}q^{2}-4\beta _{4}q^{4}+(-3+\beta _{1}-2\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(198, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(198, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(11, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(22, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(66, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(99, [\chi])\)\(^{\oplus 2}\)