# Properties

 Label 35.6.a Level $35$ Weight $6$ Character orbit 35.a Rep. character $\chi_{35}(1,\cdot)$ Character field $\Q$ Dimension $10$ Newform subspaces $4$ Sturm bound $24$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 35.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$24$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(35))$$.

Total New Old
Modular forms 22 10 12
Cusp forms 18 10 8
Eisenstein series 4 0 4

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

$$5$$$$7$$FrickeDim
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$3$$
$$-$$$$+$$$$-$$$$4$$
$$-$$$$-$$$$+$$$$1$$
Plus space$$+$$$$3$$
Minus space$$-$$$$7$$

## Trace form

 $$10 q - 6 q^{2} + 44 q^{3} + 162 q^{4} - 64 q^{6} - 98 q^{7} + 354 q^{8} + 832 q^{9} + O(q^{10})$$ $$10 q - 6 q^{2} + 44 q^{3} + 162 q^{4} - 64 q^{6} - 98 q^{7} + 354 q^{8} + 832 q^{9} + 100 q^{10} - 478 q^{11} + 3684 q^{12} + 404 q^{13} - 1078 q^{14} - 350 q^{15} - 686 q^{16} + 3500 q^{17} - 5938 q^{18} + 856 q^{19} + 490 q^{21} + 3868 q^{22} - 5248 q^{23} - 15604 q^{24} + 6250 q^{25} - 9380 q^{26} - 148 q^{27} + 6174 q^{28} - 2366 q^{29} + 8000 q^{30} + 4140 q^{31} - 17398 q^{32} - 27220 q^{33} - 20008 q^{34} - 4900 q^{35} + 4370 q^{36} + 15664 q^{37} - 1420 q^{38} + 63306 q^{39} + 15600 q^{40} + 23644 q^{41} - 7056 q^{42} - 15516 q^{43} + 22488 q^{44} + 5800 q^{45} - 27232 q^{46} + 47204 q^{47} + 15956 q^{48} + 24010 q^{49} - 3750 q^{50} - 9322 q^{51} + 19784 q^{52} - 2156 q^{53} - 25388 q^{54} + 27800 q^{55} - 29106 q^{56} + 2412 q^{57} + 27136 q^{58} + 87212 q^{59} - 48500 q^{60} - 25072 q^{61} - 92424 q^{62} - 16562 q^{63} + 64818 q^{64} - 55650 q^{65} - 109916 q^{66} - 125972 q^{67} - 9084 q^{68} + 141684 q^{69} - 9800 q^{70} - 4200 q^{71} - 308270 q^{72} - 49240 q^{73} - 30388 q^{74} + 27500 q^{75} - 71996 q^{76} - 39984 q^{77} - 195852 q^{78} + 68910 q^{79} - 64800 q^{80} + 148994 q^{81} + 379764 q^{82} - 122256 q^{83} + 112308 q^{84} + 94650 q^{85} + 405096 q^{86} - 254988 q^{87} + 391096 q^{88} - 407896 q^{89} + 174800 q^{90} + 70658 q^{91} + 298024 q^{92} - 101628 q^{93} + 334156 q^{94} - 70700 q^{95} - 235580 q^{96} + 191908 q^{97} - 14406 q^{98} - 343372 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(35))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 7
35.6.a.a $1$ $5.613$ $$\Q$$ None $$-8$$ $$1$$ $$25$$ $$49$$ $-$ $-$ $$q-8q^{2}+q^{3}+2^{5}q^{4}+5^{2}q^{5}-8q^{6}+\cdots$$
35.6.a.b $2$ $5.613$ $$\Q(\sqrt{65})$$ None $$1$$ $$3$$ $$-50$$ $$-98$$ $+$ $+$ $$q+\beta q^{2}+(3-3\beta )q^{3}+(-2^{4}+\beta )q^{4}+\cdots$$
35.6.a.c $3$ $5.613$ 3.3.577880.1 None $$-6$$ $$26$$ $$-75$$ $$147$$ $+$ $-$ $$q+(-2+\beta _{1})q^{2}+(9+\beta _{2})q^{3}+(38+2\beta _{2})q^{4}+\cdots$$
35.6.a.d $4$ $5.613$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$7$$ $$14$$ $$100$$ $$-196$$ $-$ $+$ $$q+(2-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{3})q^{3}+(15+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(35))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(35)) \simeq$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 2}$$