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Properties

Label 42.9.g
Level $42$
Weight $9$
Character orbit 42.g
Rep. character $\chi_{42}(19,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $20$
Newform subspaces $2$
Sturm bound $72$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 42.g (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(72\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(42, [\chi])\).

Total New Old
Modular forms 136 20 116
Cusp forms 120 20 100
Eisenstein series 16 0 16

Trace form

\( 20 q - 162 q^{3} - 1280 q^{4} - 3348 q^{5} - 4574 q^{7} + 21870 q^{9} + O(q^{10}) \) \( 20 q - 162 q^{3} - 1280 q^{4} - 3348 q^{5} - 4574 q^{7} + 21870 q^{9} - 17664 q^{10} + 2172 q^{11} + 20736 q^{12} + 12288 q^{14} - 59616 q^{15} - 163840 q^{16} - 219456 q^{17} - 457674 q^{19} + 285768 q^{21} + 1115136 q^{22} - 272280 q^{23} + 462122 q^{25} - 1391616 q^{26} + 1052672 q^{28} + 3626688 q^{29} - 705024 q^{30} - 6426822 q^{31} - 1380240 q^{33} + 2321148 q^{35} - 5598720 q^{36} + 2152562 q^{37} - 2557440 q^{38} + 2245158 q^{39} + 2260992 q^{40} + 2716416 q^{42} + 13179932 q^{43} + 278016 q^{44} - 7322076 q^{45} + 3270144 q^{46} - 23565492 q^{47} + 15889046 q^{49} + 8835072 q^{50} + 6353640 q^{51} + 2075904 q^{52} - 22265136 q^{53} - 16711680 q^{56} - 58778460 q^{57} - 2582784 q^{58} + 34182936 q^{59} + 3815424 q^{60} + 101452272 q^{61} + 7982550 q^{63} + 41943040 q^{64} + 21790572 q^{65} + 56804734 q^{67} + 28090368 q^{68} - 48721920 q^{70} - 47313480 q^{71} - 19756614 q^{73} - 111375360 q^{74} - 59931414 q^{75} - 140431080 q^{77} - 11860992 q^{78} + 16730794 q^{79} + 54853632 q^{80} - 47829690 q^{81} + 120317952 q^{82} - 7941888 q^{84} + 226017024 q^{85} - 109036032 q^{86} - 334654740 q^{87} - 71368704 q^{88} - 113541192 q^{89} + 445651866 q^{91} + 69703680 q^{92} + 46818 q^{93} + 283819008 q^{94} + 9921756 q^{95} - 432961536 q^{98} + 9500328 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(42, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
42.9.g.a 42.g 7.d $8$ $17.110$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(324\) \(-2226\) \(-140\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{2}-\beta _{3})q^{2}+(54+3^{3}\beta _{1})q^{3}+2^{7}\beta _{1}q^{4}+\cdots\)
42.9.g.b 42.g 7.d $12$ $17.110$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-486\) \(-1122\) \(-4434\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{2}+\beta _{3})q^{2}+(-3^{3}+3^{3}\beta _{1})q^{3}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)