# Properties

 Label 169.8.b Level $169$ Weight $8$ Character orbit 169.b Rep. character $\chi_{169}(168,\cdot)$ Character field $\Q$ Dimension $86$ Newform subspaces $6$ Sturm bound $121$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$121$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(169, [\chi])$$.

Total New Old
Modular forms 112 96 16
Cusp forms 98 86 12
Eisenstein series 14 10 4

## Trace form

 $$86 q + 56 q^{3} - 5374 q^{4} + 59470 q^{9} + O(q^{10})$$ $$86 q + 56 q^{3} - 5374 q^{4} + 59470 q^{9} + 2202 q^{10} - 9566 q^{12} - 7762 q^{14} + 295042 q^{16} - 23486 q^{17} + 61812 q^{22} - 9050 q^{23} - 1061696 q^{25} - 220270 q^{27} - 55844 q^{29} + 80310 q^{30} + 825704 q^{35} - 3249854 q^{36} + 336006 q^{38} - 1258114 q^{40} + 1918636 q^{42} + 749286 q^{43} + 4844616 q^{48} - 9146200 q^{49} + 482814 q^{51} - 2655052 q^{53} + 6892828 q^{55} + 1508720 q^{56} + 10615706 q^{61} - 5982218 q^{62} - 17744804 q^{64} - 11534586 q^{66} + 19682364 q^{68} + 4649884 q^{69} + 34122644 q^{74} - 10402918 q^{75} - 4656712 q^{77} - 17787962 q^{79} + 51153494 q^{81} + 1803098 q^{82} + 43405160 q^{87} - 19485520 q^{88} + 2365264 q^{90} - 51199512 q^{92} + 43334890 q^{94} + 16879398 q^{95} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(169, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
169.8.b.a $2$ $52.793$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-146$$ $$0$$ $$0$$ $$q+10iq^{2}-73q^{3}+28q^{4}-295iq^{5}+\cdots$$
169.8.b.b $4$ $52.793$ $$\Q(i, \sqrt{337})$$ None $$0$$ $$90$$ $$0$$ $$0$$ $$q+(\beta _{1}+5\beta _{2})q^{2}+(21+3\beta _{3})q^{3}+(-37+\cdots)q^{4}+\cdots$$
169.8.b.c $8$ $52.793$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$160$$ $$0$$ $$0$$ $$q+(\beta _{1}+2\beta _{4})q^{2}+(20+\beta _{2}+\beta _{3})q^{3}+\cdots$$
169.8.b.d $14$ $52.793$ $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ None $$0$$ $$-52$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-4-\beta _{5})q^{3}+(-55+\beta _{4}+\cdots)q^{4}+\cdots$$
169.8.b.e $16$ $52.793$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$56$$ $$0$$ $$0$$ $$q+\beta _{5}q^{2}+(4+\beta _{3})q^{3}+(-72+\beta _{1}+\cdots)q^{4}+\cdots$$
169.8.b.f $42$ $52.793$ None $$0$$ $$-52$$ $$0$$ $$0$$

## Decomposition of $$S_{8}^{\mathrm{old}}(169, [\chi])$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(169, [\chi]) \simeq$$ $$S_{8}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 2}$$