# Properties

 Label 441.4.e Level $441$ Weight $4$ Character orbit 441.e Rep. character $\chi_{441}(226,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $96$ Newform subspaces $26$ Sturm bound $224$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$26$$ Sturm bound: $$224$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$2$$, $$5$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(441, [\chi])$$.

Total New Old
Modular forms 368 104 264
Cusp forms 304 96 208
Eisenstein series 64 8 56

## Trace form

 $$96 q + q^{2} - 193 q^{4} - 15 q^{5} + 102 q^{8} + O(q^{10})$$ $$96 q + q^{2} - 193 q^{4} - 15 q^{5} + 102 q^{8} - 10 q^{10} - 29 q^{11} - 172 q^{13} - 741 q^{16} - 111 q^{17} + 53 q^{19} + 816 q^{20} + 372 q^{22} + 41 q^{23} - 825 q^{25} - 438 q^{26} - 784 q^{29} + 19 q^{31} - 1327 q^{32} - 1404 q^{34} + 407 q^{37} + 252 q^{38} + 612 q^{40} + 1908 q^{41} + 1504 q^{43} + 548 q^{44} + 922 q^{46} - 285 q^{47} - 2234 q^{50} + 276 q^{52} - 2299 q^{53} - 2978 q^{55} - 3314 q^{58} - 1023 q^{59} + 731 q^{61} + 4224 q^{62} + 6578 q^{64} + 378 q^{65} + 1017 q^{67} - 2112 q^{68} - 2312 q^{71} + 1987 q^{73} - 3138 q^{74} + 192 q^{76} + 351 q^{79} - 276 q^{80} + 2224 q^{82} + 4176 q^{83} - 3118 q^{85} + 546 q^{86} - 2760 q^{88} - 2343 q^{89} + 4768 q^{92} - 3570 q^{94} - 57 q^{95} - 5300 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(441, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
441.4.e.a $2$ $26.020$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-7})$$ $$-5$$ $$0$$ $$0$$ $$0$$ $$q-5\zeta_{6}q^{2}+(-17+17\zeta_{6})q^{4}+45q^{8}+\cdots$$
441.4.e.b $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$0$$ $$-18$$ $$0$$ $$q-3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-18\zeta_{6}q^{5}+\cdots$$
441.4.e.c $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$0$$ $$3$$ $$0$$ $$q-3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots$$
441.4.e.d $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$0$$ $$18$$ $$0$$ $$q-3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+18\zeta_{6}q^{5}+\cdots$$
441.4.e.e $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-16$$ $$0$$ $$q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}-2^{4}\zeta_{6}q^{5}+\cdots$$
441.4.e.f $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-12$$ $$0$$ $$q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}-12\zeta_{6}q^{5}+\cdots$$
441.4.e.g $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$12$$ $$0$$ $$q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}+12\zeta_{6}q^{5}+\cdots$$
441.4.e.h $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$16$$ $$0$$ $$q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}+2^{4}\zeta_{6}q^{5}+\cdots$$
441.4.e.i $2$ $26.020$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(8-8\zeta_{6})q^{4}-70q^{13}-2^{6}\zeta_{6}q^{16}+\cdots$$
441.4.e.j $2$ $26.020$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(8-8\zeta_{6})q^{4}+70q^{13}-2^{6}\zeta_{6}q^{16}+\cdots$$
441.4.e.k $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-7$$ $$0$$ $$q+2\zeta_{6}q^{2}+(4-4\zeta_{6})q^{4}-7\zeta_{6}q^{5}+\cdots$$
441.4.e.l $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$4$$ $$0$$ $$-18$$ $$0$$ $$q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}-18\zeta_{6}q^{5}+\cdots$$
441.4.e.m $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$4$$ $$0$$ $$-4$$ $$0$$ $$q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}-4\zeta_{6}q^{5}+\cdots$$
441.4.e.n $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$4$$ $$0$$ $$4$$ $$0$$ $$q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}+4\zeta_{6}q^{5}+\cdots$$
441.4.e.o $2$ $26.020$ $$\Q(\sqrt{-3})$$ None $$4$$ $$0$$ $$18$$ $$0$$ $$q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}+18\zeta_{6}q^{5}+\cdots$$
441.4.e.p $4$ $26.020$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$-3$$ $$0$$ $$-6$$ $$0$$ $$q+(-1-\beta _{1}-\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots$$
441.4.e.q $4$ $26.020$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$-3$$ $$0$$ $$6$$ $$0$$ $$q+(-1-\beta _{1}-\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots$$
441.4.e.r $4$ $26.020$ $$\Q(\sqrt{-3}, \sqrt{19})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+11\beta _{2}q^{4}+2\beta _{1}q^{5}+3\beta _{3}q^{8}+\cdots$$
441.4.e.s $4$ $26.020$ $$\Q(\sqrt{-3}, \sqrt{19})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+11\beta _{2}q^{4}-2\beta _{1}q^{5}+3\beta _{3}q^{8}+\cdots$$
441.4.e.t $4$ $26.020$ $$\Q(\sqrt{-3}, \sqrt{7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-\beta _{2}q^{4}-9\beta _{3}q^{8}+(-10\beta _{1}+\cdots)q^{11}+\cdots$$
441.4.e.u $4$ $26.020$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$-20$$ $$0$$ $$q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}-5\beta _{2}+2\beta _{3})q^{4}+\cdots$$
441.4.e.v $4$ $26.020$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$20$$ $$0$$ $$q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}-5\beta _{2}+2\beta _{3})q^{4}+\cdots$$
441.4.e.w $6$ $26.020$ 6.0.9924270768.1 None $$1$$ $$0$$ $$-11$$ $$0$$ $$q+\beta _{1}q^{2}+(-8+\beta _{1}+\beta _{2}+8\beta _{4}+\beta _{5})q^{4}+\cdots$$
441.4.e.x $8$ $26.020$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{3})q^{2}+(-2-2\beta _{2}-\beta _{6})q^{4}+\cdots$$
441.4.e.y $8$ $26.020$ 8.0.$$\cdots$$.19 None $$2$$ $$0$$ $$0$$ $$0$$ $$q+(1-\beta _{2}+\beta _{6})q^{2}+(-8-8\beta _{1}+\beta _{6}+\cdots)q^{4}+\cdots$$
441.4.e.z $16$ $26.020$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{2}+(-9+9\beta _{2}+\beta _{7})q^{4}-\beta _{6}q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(441, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(441, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(147, [\chi])$$$$^{\oplus 2}$$