# Properties

 Label 336.7.bh Level $336$ Weight $7$ Character orbit 336.bh Rep. character $\chi_{336}(145,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $96$ Newform subspaces $8$ Sturm bound $448$ Trace bound $5$

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## Defining parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 336.bh (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$8$$ Sturm bound: $$448$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(336, [\chi])$$.

Total New Old
Modular forms 792 96 696
Cusp forms 744 96 648
Eisenstein series 48 0 48

## Trace form

 $$96 q + 360 q^{7} + 11664 q^{9} + O(q^{10})$$ $$96 q + 360 q^{7} + 11664 q^{9} - 1360 q^{11} - 15120 q^{19} - 24368 q^{23} + 161928 q^{25} + 66400 q^{29} + 27720 q^{31} + 13608 q^{33} - 147120 q^{35} + 3600 q^{37} + 68040 q^{39} + 101760 q^{43} - 376992 q^{47} + 182928 q^{49} - 100320 q^{53} - 136080 q^{57} - 1329216 q^{59} - 529200 q^{61} - 87480 q^{63} + 266000 q^{65} + 450336 q^{67} - 1285248 q^{71} + 385560 q^{73} + 241040 q^{77} - 696648 q^{79} - 2834352 q^{81} - 650592 q^{85} + 3591696 q^{91} - 664848 q^{93} - 2936544 q^{95} - 660960 q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(336, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
336.7.bh.a $8$ $77.298$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-108$$ $$-294$$ $$-232$$ $$q+(-9-9\beta _{1})q^{3}+(-7^{2}+5^{2}\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$
336.7.bh.b $8$ $77.298$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-108$$ $$-42$$ $$-748$$ $$q+(-9+9\beta _{2})q^{3}+(-7-3\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots$$
336.7.bh.c $8$ $77.298$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-108$$ $$210$$ $$608$$ $$q+(-18+9\beta _{2})q^{3}+(15-5\beta _{1}+20\beta _{2}+\cdots)q^{5}+\cdots$$
336.7.bh.d $8$ $77.298$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$108$$ $$-294$$ $$656$$ $$q+(18-9\beta _{1})q^{3}+(-24-5^{2}\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$
336.7.bh.e $8$ $77.298$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$108$$ $$-42$$ $$92$$ $$q+(9-9\beta _{1})q^{3}+(-7-3\beta _{1}+\beta _{4})q^{5}+\cdots$$
336.7.bh.f $8$ $77.298$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$108$$ $$462$$ $$-580$$ $$q+(9-9\beta _{1})q^{3}+(77+39\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{5}+\cdots$$
336.7.bh.g $24$ $77.298$ None $$0$$ $$-324$$ $$126$$ $$552$$
336.7.bh.h $24$ $77.298$ None $$0$$ $$324$$ $$-126$$ $$12$$

## Decomposition of $$S_{7}^{\mathrm{old}}(336, [\chi])$$ into lower level spaces

$$S_{7}^{\mathrm{old}}(336, [\chi]) \cong$$ $$S_{7}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 2}$$