# Properties

 Label 36.9.d Level $36$ Weight $9$ Character orbit 36.d Rep. character $\chi_{36}(19,\cdot)$ Character field $\Q$ Dimension $19$ Newform subspaces $4$ Sturm bound $54$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 36.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$4$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$54$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(36, [\chi])$$.

Total New Old
Modular forms 52 21 31
Cusp forms 44 19 25
Eisenstein series 8 2 6

## Trace form

 $$19 q - 2 q^{2} - 332 q^{4} + 170 q^{5} + 1504 q^{8} + O(q^{10})$$ $$19 q - 2 q^{2} - 332 q^{4} + 170 q^{5} + 1504 q^{8} + 12908 q^{10} + 39974 q^{13} - 17784 q^{14} + 127120 q^{16} + 110378 q^{17} + 2552 q^{20} + 421656 q^{22} + 1295865 q^{25} - 123316 q^{26} + 248016 q^{28} - 399190 q^{29} + 2067808 q^{32} + 787988 q^{34} - 518458 q^{37} - 4146120 q^{38} - 6733120 q^{40} + 993770 q^{41} + 6839280 q^{44} + 5290176 q^{46} - 18765101 q^{49} - 26476998 q^{50} - 19633432 q^{52} - 734614 q^{53} + 50597568 q^{56} + 19674332 q^{58} + 17764742 q^{61} - 82481256 q^{62} - 60826112 q^{64} - 1123820 q^{65} + 141693992 q^{68} + 158548368 q^{70} + 73820102 q^{73} - 203803780 q^{74} - 95408208 q^{76} + 5254272 q^{77} + 305234912 q^{80} + 129868244 q^{82} - 8406740 q^{85} - 388412568 q^{86} - 255189312 q^{88} + 7922090 q^{89} + 506168064 q^{92} + 454893648 q^{94} - 182185210 q^{97} - 707159906 q^{98} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(36, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
36.9.d.a $1$ $14.666$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$-16$$ $$0$$ $$1054$$ $$0$$ $$q-2^{4}q^{2}+2^{8}q^{4}+1054q^{5}-2^{12}q^{8}+\cdots$$
36.9.d.b $2$ $14.666$ $$\Q(\sqrt{-39})$$ None $$20$$ $$0$$ $$-1220$$ $$0$$ $$q+(10-\beta )q^{2}+(-56-20\beta )q^{4}-610q^{5}+\cdots$$
36.9.d.c $8$ $14.666$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-6$$ $$0$$ $$336$$ $$0$$ $$q+(-1-\beta _{1})q^{2}+(-6+\beta _{1}+\beta _{3})q^{4}+\cdots$$
36.9.d.d $8$ $14.666$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-53-\beta _{5})q^{4}+(3\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{9}^{\mathrm{old}}(36, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(36, [\chi]) \simeq$$ $$S_{9}^{\mathrm{new}}(4, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 2}$$