# Properties

 Label 394.2.a Level $394$ Weight $2$ Character orbit 394.a Rep. character $\chi_{394}(1,\cdot)$ Character field $\Q$ Dimension $16$ Newform subspaces $6$ Sturm bound $99$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$394 = 2 \cdot 197$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 394.a (trivial) Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$99$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(394))$$.

Total New Old
Modular forms 51 16 35
Cusp forms 48 16 32
Eisenstein series 3 0 3

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

$$2$$$$197$$FrickeDim
$$+$$$$+$$$$+$$$$4$$
$$+$$$$-$$$$-$$$$4$$
$$-$$$$+$$$$-$$$$6$$
$$-$$$$-$$$$+$$$$2$$
Plus space$$+$$$$6$$
Minus space$$-$$$$10$$

## Trace form

 $$16 q - 4 q^{3} + 16 q^{4} - 2 q^{6} + 6 q^{9} + O(q^{10})$$ $$16 q - 4 q^{3} + 16 q^{4} - 2 q^{6} + 6 q^{9} + 6 q^{10} - 4 q^{12} + 4 q^{13} - 4 q^{14} + 16 q^{16} + 4 q^{17} - 8 q^{18} - 6 q^{19} - 8 q^{21} - 6 q^{22} - 2 q^{24} + 10 q^{25} + 6 q^{26} + 8 q^{27} - 6 q^{29} - 20 q^{31} + 4 q^{33} + 4 q^{34} - 4 q^{35} + 6 q^{36} - 22 q^{37} + 8 q^{38} - 28 q^{39} + 6 q^{40} + 16 q^{41} + 12 q^{42} + 6 q^{43} + 8 q^{45} - 8 q^{46} - 12 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} - 4 q^{51} + 4 q^{52} - 2 q^{53} - 8 q^{54} + 4 q^{55} - 4 q^{56} + 4 q^{57} + 20 q^{58} - 14 q^{59} - 22 q^{61} + 16 q^{62} + 16 q^{63} + 16 q^{64} + 8 q^{65} + 4 q^{68} + 12 q^{69} - 4 q^{70} + 4 q^{71} - 8 q^{72} - 32 q^{73} + 4 q^{74} - 56 q^{75} - 6 q^{76} - 24 q^{77} - 8 q^{78} - 44 q^{79} - 48 q^{81} - 16 q^{82} - 6 q^{83} - 8 q^{84} + 28 q^{85} - 8 q^{86} - 16 q^{87} - 6 q^{88} + 14 q^{90} - 40 q^{91} - 4 q^{93} - 12 q^{94} + 8 q^{95} - 2 q^{96} + 44 q^{97} - 16 q^{98} + 12 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(394))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 197
394.2.a.a $2$ $3.146$ $$\Q(\sqrt{5})$$ None $$2$$ $$-2$$ $$-5$$ $$-4$$ $-$ $-$ $$q+q^{2}-q^{3}+q^{4}+(-2-\beta )q^{5}-q^{6}+\cdots$$
394.2.a.b $2$ $3.146$ $$\Q(\sqrt{21})$$ None $$2$$ $$-1$$ $$0$$ $$4$$ $-$ $+$ $$q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}+2q^{7}+q^{8}+\cdots$$
394.2.a.c $2$ $3.146$ $$\Q(\sqrt{29})$$ None $$2$$ $$0$$ $$3$$ $$4$$ $-$ $+$ $$q+q^{2}+q^{4}+(2-\beta )q^{5}+2q^{7}+q^{8}+\cdots$$
394.2.a.d $2$ $3.146$ $$\Q(\sqrt{5})$$ None $$2$$ $$0$$ $$5$$ $$-6$$ $-$ $+$ $$q+q^{2}+(-1+2\beta )q^{3}+q^{4}+(3-\beta )q^{5}+\cdots$$
394.2.a.e $4$ $3.146$ 4.4.2225.1 None $$-4$$ $$-3$$ $$-5$$ $$2$$ $+$ $+$ $$q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots$$
394.2.a.f $4$ $3.146$ 4.4.4913.1 None $$-4$$ $$2$$ $$2$$ $$0$$ $+$ $-$ $$q-q^{2}+(1-\beta _{2}+\beta _{3})q^{3}+q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(394))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(394)) \simeq$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(197))$$$$^{\oplus 2}$$