# 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 $1$

# Related objects

## 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

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

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

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

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

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