# Properties

 Label 176.4.a Level $176$ Weight $4$ Character orbit 176.a Rep. character $\chi_{176}(1,\cdot)$ Character field $\Q$ Dimension $15$ Newform subspaces $10$ Sturm bound $96$ Trace bound $5$

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## Defining parameters

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 176.a (trivial) Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(176))$$.

Total New Old
Modular forms 78 15 63
Cusp forms 66 15 51
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.

$$2$$$$11$$FrickeDim
$$+$$$$+$$$$+$$$$5$$
$$+$$$$-$$$$-$$$$2$$
$$-$$$$+$$$$-$$$$4$$
$$-$$$$-$$$$+$$$$4$$
Plus space$$+$$$$9$$
Minus space$$-$$$$6$$

## Trace form

 $$15 q - 6 q^{3} + 2 q^{5} + 36 q^{7} + 135 q^{9} + O(q^{10})$$ $$15 q - 6 q^{3} + 2 q^{5} + 36 q^{7} + 135 q^{9} - 33 q^{11} - 46 q^{13} + 66 q^{15} - 26 q^{17} + 204 q^{19} + 136 q^{21} + 138 q^{23} + 397 q^{25} + 138 q^{27} - 198 q^{29} - 318 q^{31} + 228 q^{35} + 330 q^{37} + 120 q^{39} + 118 q^{41} + 336 q^{43} - 350 q^{45} + 848 q^{47} + 375 q^{49} - 44 q^{51} - 286 q^{53} + 220 q^{55} + 680 q^{57} - 1738 q^{59} - 1302 q^{61} - 416 q^{63} + 116 q^{65} - 570 q^{67} - 1544 q^{69} + 390 q^{71} - 154 q^{73} + 1292 q^{75} - 3180 q^{79} + 295 q^{81} - 1448 q^{83} - 1348 q^{85} - 1384 q^{87} + 26 q^{89} + 368 q^{91} - 2136 q^{93} - 2064 q^{95} + 1490 q^{97} - 891 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(176))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 11
176.4.a.a $1$ $10.384$ $$\Q$$ None $$0$$ $$-7$$ $$9$$ $$-2$$ $+$ $-$ $$q-7q^{3}+9q^{5}-2q^{7}+22q^{9}+11q^{11}+\cdots$$
176.4.a.b $1$ $10.384$ $$\Q$$ None $$0$$ $$-4$$ $$14$$ $$8$$ $-$ $-$ $$q-4q^{3}+14q^{5}+8q^{7}-11q^{9}+11q^{11}+\cdots$$
176.4.a.c $1$ $10.384$ $$\Q$$ None $$0$$ $$-1$$ $$-3$$ $$10$$ $-$ $+$ $$q-q^{3}-3q^{5}+10q^{7}-26q^{9}-11q^{11}+\cdots$$
176.4.a.d $1$ $10.384$ $$\Q$$ None $$0$$ $$1$$ $$-7$$ $$6$$ $+$ $-$ $$q+q^{3}-7q^{5}+6q^{7}-26q^{9}+11q^{11}+\cdots$$
176.4.a.e $1$ $10.384$ $$\Q$$ None $$0$$ $$5$$ $$-7$$ $$26$$ $-$ $-$ $$q+5q^{3}-7q^{5}+26q^{7}-2q^{9}+11q^{11}+\cdots$$
176.4.a.f $1$ $10.384$ $$\Q$$ None $$0$$ $$7$$ $$-19$$ $$-14$$ $-$ $+$ $$q+7q^{3}-19q^{5}-14q^{7}+22q^{9}-11q^{11}+\cdots$$
176.4.a.g $2$ $10.384$ $$\Q(\sqrt{97})$$ None $$0$$ $$-9$$ $$11$$ $$-10$$ $-$ $+$ $$q+(-4-\beta )q^{3}+(6-\beta )q^{5}+(-8+6\beta )q^{7}+\cdots$$
176.4.a.h $2$ $10.384$ $$\Q(\sqrt{5})$$ None $$0$$ $$2$$ $$-6$$ $$56$$ $+$ $+$ $$q+(1+\beta )q^{3}+(-3+4\beta )q^{5}+(28+\beta )q^{7}+\cdots$$
176.4.a.i $2$ $10.384$ $$\Q(\sqrt{3})$$ None $$0$$ $$2$$ $$2$$ $$-20$$ $-$ $-$ $$q+(1+\beta )q^{3}+(1+2\beta )q^{5}+(-10+\beta )q^{7}+\cdots$$
176.4.a.j $3$ $10.384$ 3.3.11109.1 None $$0$$ $$-2$$ $$8$$ $$-24$$ $+$ $+$ $$q+(-1+\beta _{1})q^{3}+(3+\beta _{2})q^{5}+(-8+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(176))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(176)) \simeq$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(22))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(44))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(88))$$$$^{\oplus 2}$$