# Properties

 Label 208.10.a Level $208$ Weight $10$ Character orbit 208.a Rep. character $\chi_{208}(1,\cdot)$ Character field $\Q$ Dimension $54$ Newform subspaces $13$ Sturm bound $280$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 208.a (trivial) Character field: $$\Q$$ Newform subspaces: $$13$$ Sturm bound: $$280$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_0(208))$$.

Total New Old
Modular forms 258 54 204
Cusp forms 246 54 192
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$13$$FrickeDim
$$+$$$$+$$$$+$$$$13$$
$$+$$$$-$$$$-$$$$14$$
$$-$$$$+$$$$-$$$$14$$
$$-$$$$-$$$$+$$$$13$$
Plus space$$+$$$$26$$
Minus space$$-$$$$28$$

## Trace form

 $$54 q - 4802 q^{7} + 382034 q^{9} + O(q^{10})$$ $$54 q - 4802 q^{7} + 382034 q^{9} + 21986 q^{11} - 81528 q^{15} - 242212 q^{17} + 1262814 q^{19} + 634168 q^{21} - 1020280 q^{23} + 21856302 q^{25} - 527028 q^{27} - 4511232 q^{29} - 11885838 q^{31} + 7018680 q^{33} + 41639040 q^{35} - 1214776 q^{37} - 9253764 q^{39} - 4071260 q^{41} + 45987380 q^{43} - 13805000 q^{45} - 224367370 q^{47} + 269172846 q^{49} + 207371428 q^{51} + 64563556 q^{53} - 144893436 q^{55} + 89709312 q^{57} + 218112498 q^{59} + 145215404 q^{61} - 559687034 q^{63} + 218286534 q^{67} - 72690488 q^{69} - 614785734 q^{71} + 111758796 q^{73} - 702852148 q^{75} - 321924380 q^{77} + 994007268 q^{79} + 3011851446 q^{81} - 1915074314 q^{83} - 40098816 q^{85} + 403827080 q^{87} - 842440028 q^{89} + 411449766 q^{91} + 1205123184 q^{93} + 4493399076 q^{95} + 335423196 q^{97} - 3586752702 q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_0(208))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 13
208.10.a.a $1$ $107.127$ $$\Q$$ None $$0$$ $$-192$$ $$-1310$$ $$5810$$ $-$ $+$ $$q-192q^{3}-1310q^{5}+5810q^{7}+17181q^{9}+\cdots$$
208.10.a.b $1$ $107.127$ $$\Q$$ None $$0$$ $$-75$$ $$-1979$$ $$10115$$ $-$ $-$ $$q-75q^{3}-1979q^{5}+10115q^{7}-14058q^{9}+\cdots$$
208.10.a.c $1$ $107.127$ $$\Q$$ None $$0$$ $$273$$ $$1015$$ $$-3955$$ $-$ $+$ $$q+273q^{3}+1015q^{5}-3955q^{7}+54846q^{9}+\cdots$$
208.10.a.d $3$ $107.127$ 3.3.2119705.1 None $$0$$ $$-156$$ $$-1272$$ $$-17058$$ $-$ $+$ $$q+(-52+\beta _{1}+\beta _{2})q^{3}+(-424-18\beta _{1}+\cdots)q^{5}+\cdots$$
208.10.a.e $3$ $107.127$ $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ None $$0$$ $$0$$ $$248$$ $$2956$$ $-$ $-$ $$q+\beta _{2}q^{3}+(79-11\beta _{1}-9\beta _{2})q^{5}+(972+\cdots)q^{7}+\cdots$$
208.10.a.f $4$ $107.127$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$0$$ $$147$$ $$-1947$$ $$-10251$$ $-$ $-$ $$q+(37+\beta _{1})q^{3}+(-486+2\beta _{1}+\beta _{3})q^{5}+\cdots$$
208.10.a.g $4$ $107.127$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$0$$ $$163$$ $$471$$ $$11241$$ $-$ $+$ $$q+(41+\beta _{3})q^{3}+(118-2\beta _{1}+7\beta _{2}+\cdots)q^{5}+\cdots$$
208.10.a.h $5$ $107.127$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$0$$ $$-161$$ $$1803$$ $$-10099$$ $-$ $-$ $$q+(-2^{5}+\beta _{3})q^{3}+(19^{2}+3\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$
208.10.a.i $5$ $107.127$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$0$$ $$147$$ $$1721$$ $$-3317$$ $-$ $+$ $$q+(29-\beta _{1})q^{3}+(345+\beta _{1}+\beta _{2})q^{5}+\cdots$$
208.10.a.j $6$ $107.127$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$-60$$ $$176$$ $$-3416$$ $+$ $+$ $$q+(-10+\beta _{1})q^{3}+(29-2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$
208.10.a.k $6$ $107.127$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$68$$ $$-176$$ $$9664$$ $+$ $-$ $$q+(11+\beta _{1})q^{3}+(-29+\beta _{2})q^{5}+(1610+\cdots)q^{7}+\cdots$$
208.10.a.l $7$ $107.127$ $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ None $$0$$ $$-13$$ $$-801$$ $$1091$$ $+$ $+$ $$q+(-2+\beta _{1})q^{3}+(-114+\beta _{2})q^{5}+\cdots$$
208.10.a.m $8$ $107.127$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-141$$ $$2051$$ $$2417$$ $+$ $-$ $$q+(-18+\beta _{1})q^{3}+(2^{8}+2\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_0(208))$$ into lower level spaces

$$S_{10}^{\mathrm{old}}(\Gamma_0(208)) \simeq$$ $$S_{10}^{\mathrm{new}}(\Gamma_0(2))$$$$^{\oplus 8}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(13))$$$$^{\oplus 5}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(26))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(52))$$$$^{\oplus 3}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(104))$$$$^{\oplus 2}$$