# Properties

 Label 308.2.i Level $308$ Weight $2$ Character orbit 308.i Rep. character $\chi_{308}(177,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $12$ Newform subspaces $2$ Sturm bound $96$ Trace bound $3$

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## Defining parameters

 Level: $$N$$ $$=$$ $$308 = 2^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 308.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$2$$ Sturm bound: $$96$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(308, [\chi])$$.

Total New Old
Modular forms 108 12 96
Cusp forms 84 12 72
Eisenstein series 24 0 24

## Trace form

 $$12 q + 2 q^{3} + 4 q^{5} + 6 q^{7} - 8 q^{9} + O(q^{10})$$ $$12 q + 2 q^{3} + 4 q^{5} + 6 q^{7} - 8 q^{9} - 28 q^{15} + 6 q^{17} - 2 q^{19} - 18 q^{21} + 4 q^{23} - 6 q^{25} + 32 q^{27} + 8 q^{29} - 10 q^{31} + 4 q^{33} - 26 q^{35} - 12 q^{37} - 10 q^{39} + 24 q^{41} + 36 q^{43} + 14 q^{45} - 10 q^{47} - 12 q^{51} + 6 q^{53} + 8 q^{57} + 4 q^{59} + 4 q^{61} + 54 q^{63} + 34 q^{65} - 60 q^{69} - 20 q^{71} - 34 q^{73} - 10 q^{75} - 2 q^{77} - 12 q^{79} - 6 q^{81} - 44 q^{83} - 32 q^{85} + 42 q^{87} + 30 q^{89} - 54 q^{91} - 20 q^{93} + 16 q^{95} + 28 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(308, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
308.2.i.a $6$ $2.459$ 6.0.309123.1 None $$0$$ $$-1$$ $$2$$ $$4$$ $$q-\beta _{2}q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3}+\beta _{5})q^{5}+\cdots$$
308.2.i.b $6$ $2.459$ 6.0.1783323.2 None $$0$$ $$3$$ $$2$$ $$2$$ $$q+(1-\beta _{4}+\beta _{5})q^{3}+(\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(308, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(308, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(77, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(154, [\chi])$$$$^{\oplus 2}$$