# Properties

 Label 196.6.a Level $196$ Weight $6$ Character orbit 196.a Rep. character $\chi_{196}(1,\cdot)$ Character field $\Q$ Dimension $17$ Newform subspaces $11$ Sturm bound $168$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 196.a (trivial) Character field: $$\Q$$ Newform subspaces: $$11$$ Sturm bound: $$168$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(196))$$.

Total New Old
Modular forms 152 17 135
Cusp forms 128 17 111
Eisenstein series 24 0 24

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

$$2$$$$7$$FrickeDim
$$-$$$$+$$$$-$$$$9$$
$$-$$$$-$$$$+$$$$8$$
Plus space$$+$$$$8$$
Minus space$$-$$$$9$$

## Trace form

 $$17 q - 12 q^{3} + 26 q^{5} + 1635 q^{9} + O(q^{10})$$ $$17 q - 12 q^{3} + 26 q^{5} + 1635 q^{9} - 18 q^{11} - 838 q^{13} - 754 q^{15} + 370 q^{17} + 1956 q^{19} + 4222 q^{23} + 10825 q^{25} - 10008 q^{27} + 7514 q^{29} - 2800 q^{31} + 4832 q^{33} + 11496 q^{37} - 24236 q^{39} - 42582 q^{41} + 23228 q^{43} - 24526 q^{45} + 25920 q^{47} + 106534 q^{51} - 20940 q^{53} - 98408 q^{55} + 127194 q^{57} - 28276 q^{59} + 28178 q^{61} + 27092 q^{65} + 33110 q^{67} - 136000 q^{69} + 3304 q^{71} - 35198 q^{73} + 84268 q^{75} + 179870 q^{79} + 118201 q^{81} - 131420 q^{83} - 124818 q^{85} + 68728 q^{87} - 4350 q^{89} + 31598 q^{93} - 49214 q^{95} - 1886 q^{97} + 269504 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(196))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 7
196.6.a.a $1$ $31.435$ $$\Q$$ None $$0$$ $$-26$$ $$-16$$ $$0$$ $-$ $-$ $$q-26q^{3}-2^{4}q^{5}+433q^{9}+8q^{11}+\cdots$$
196.6.a.b $1$ $31.435$ $$\Q$$ None $$0$$ $$-19$$ $$19$$ $$0$$ $-$ $+$ $$q-19q^{3}+19q^{5}+118q^{9}-559q^{11}+\cdots$$
196.6.a.c $1$ $31.435$ $$\Q$$ None $$0$$ $$-16$$ $$16$$ $$0$$ $-$ $-$ $$q-2^{4}q^{3}+2^{4}q^{5}+13q^{9}-76q^{11}+\cdots$$
196.6.a.d $1$ $31.435$ $$\Q$$ None $$0$$ $$2$$ $$96$$ $$0$$ $-$ $-$ $$q+2q^{3}+96q^{5}-239q^{9}-720q^{11}+\cdots$$
196.6.a.e $1$ $31.435$ $$\Q$$ None $$0$$ $$12$$ $$-54$$ $$0$$ $-$ $-$ $$q+12q^{3}-54q^{5}-99q^{9}+540q^{11}+\cdots$$
196.6.a.f $1$ $31.435$ $$\Q$$ None $$0$$ $$16$$ $$-16$$ $$0$$ $-$ $-$ $$q+2^{4}q^{3}-2^{4}q^{5}+13q^{9}-76q^{11}+\cdots$$
196.6.a.g $1$ $31.435$ $$\Q$$ None $$0$$ $$19$$ $$-19$$ $$0$$ $-$ $-$ $$q+19q^{3}-19q^{5}+118q^{9}-559q^{11}+\cdots$$
196.6.a.h $2$ $31.435$ $$\Q(\sqrt{109})$$ None $$0$$ $$-28$$ $$-42$$ $$0$$ $-$ $-$ $$q+(-14-\beta )q^{3}+(-21-6\beta )q^{5}+(62+\cdots)q^{9}+\cdots$$
196.6.a.i $2$ $31.435$ $$\Q(\sqrt{2})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $-$ $+$ $$q+3\beta q^{3}+46\beta q^{5}-15^{2}q^{9}+274q^{11}+\cdots$$
196.6.a.j $2$ $31.435$ $$\Q(\sqrt{109})$$ None $$0$$ $$28$$ $$42$$ $$0$$ $-$ $+$ $$q+(14-\beta )q^{3}+(21-6\beta )q^{5}+(62-28\beta )q^{9}+\cdots$$
196.6.a.k $4$ $31.435$ $$\Q(\sqrt{2}, \sqrt{1177})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $-$ $+$ $$q+(\beta _{1}+\beta _{3})q^{3}+(-8\beta _{1}-\beta _{3})q^{5}+(370+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(196))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(196)) \simeq$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(28))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(49))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(98))$$$$^{\oplus 2}$$