# Properties

 Label 272.10.a Level $272$ Weight $10$ Character orbit 272.a Rep. character $\chi_{272}(1,\cdot)$ Character field $\Q$ Dimension $72$ Newform subspaces $12$ Sturm bound $360$ Trace bound $5$

# Learn more about

## Defining parameters

 Level: $$N$$ $$=$$ $$272 = 2^{4} \cdot 17$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 272.a (trivial) Character field: $$\Q$$ Newform subspaces: $$12$$ Sturm bound: $$360$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_0(272))$$.

Total New Old
Modular forms 330 72 258
Cusp forms 318 72 246
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.

$$2$$$$17$$FrickeDim.
$$+$$$$+$$$$+$$$$17$$
$$+$$$$-$$$$-$$$$19$$
$$-$$$$+$$$$-$$$$19$$
$$-$$$$-$$$$+$$$$17$$
Plus space$$+$$$$34$$
Minus space$$-$$$$38$$

## Trace form

 $$72q + 162q^{3} - 2750q^{7} + 472392q^{9} + O(q^{10})$$ $$72q + 162q^{3} - 2750q^{7} + 472392q^{9} + 21986q^{11} + 120972q^{15} - 670140q^{19} + 634168q^{21} + 53942q^{23} + 28887552q^{25} + 10049544q^{27} - 4511232q^{29} - 21158646q^{31} - 2566728q^{33} + 19120308q^{35} - 1214776q^{37} - 35468688q^{39} - 3633992q^{41} + 137083580q^{43} - 13805000q^{45} - 175570560q^{47} + 419740672q^{49} + 40591206q^{51} + 74907848q^{53} - 158432004q^{55} - 141556552q^{57} + 49534260q^{59} - 103676922q^{63} + 119062824q^{65} + 423033072q^{67} + 224559728q^{69} - 441183954q^{71} + 502617600q^{73} - 58649842q^{75} + 93212160q^{77} - 471543298q^{79} + 2825893888q^{81} + 397761580q^{83} + 3395836388q^{87} - 1049140040q^{89} - 2514971976q^{91} + 937580944q^{93} + 774164712q^{95} + 1809005128q^{97} + 346948030q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_0(272))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 17
272.10.a.a $$2$$ $$140.090$$ $$\Q(\sqrt{43})$$ None $$0$$ $$-64$$ $$-2788$$ $$728$$ $$-$$ $$-$$ $$q+(-2^{5}+\beta )q^{3}+(-1394-30\beta )q^{5}+\cdots$$
272.10.a.b $$3$$ $$140.090$$ $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ None $$0$$ $$-84$$ $$-1304$$ $$-690$$ $$-$$ $$+$$ $$q+(-28+\beta _{1})q^{3}+(-435-3\beta _{1}-\beta _{2})q^{5}+\cdots$$
272.10.a.c $$3$$ $$140.090$$ 3.3.3262740.1 None $$0$$ $$-84$$ $$1314$$ $$-6912$$ $$-$$ $$-$$ $$q+(-28+\beta _{2})q^{3}+(438+13\beta _{1}-4\beta _{2})q^{5}+\cdots$$
272.10.a.d $$4$$ $$140.090$$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$0$$ $$-226$$ $$-1656$$ $$-2654$$ $$-$$ $$+$$ $$q+(-57-\beta _{1})q^{3}+(-412+4\beta _{1}-3\beta _{2}+\cdots)q^{5}+\cdots$$
272.10.a.e $$5$$ $$140.090$$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$0$$ $$236$$ $$-1138$$ $$-3712$$ $$-$$ $$-$$ $$q+(47-\beta _{1})q^{3}+(-227+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$
272.10.a.f $$5$$ $$140.090$$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$0$$ $$236$$ $$1480$$ $$13202$$ $$-$$ $$+$$ $$q+(47-\beta _{3})q^{3}+(295-2\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots$$
272.10.a.g $$7$$ $$140.090$$ $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ None $$0$$ $$-88$$ $$1362$$ $$-9388$$ $$-$$ $$-$$ $$q+(-13+\beta _{1})q^{3}+(195-\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots$$
272.10.a.h $$7$$ $$140.090$$ $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ None $$0$$ $$74$$ $$1480$$ $$-5132$$ $$-$$ $$+$$ $$q+(11-\beta _{1})q^{3}+(212-\beta _{1}-\beta _{4})q^{5}+\cdots$$
272.10.a.i $$8$$ $$140.090$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-152$$ $$176$$ $$-48$$ $$+$$ $$+$$ $$q+(-19+\beta _{1})q^{3}+(22+\beta _{2})q^{5}+(-6+\cdots)q^{7}+\cdots$$
272.10.a.j $$9$$ $$140.090$$ $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ None $$0$$ $$-10$$ $$-176$$ $$5952$$ $$+$$ $$+$$ $$q+(-1-\beta _{1})q^{3}+(-19-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$
272.10.a.k $$9$$ $$140.090$$ $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ None $$0$$ $$314$$ $$-1426$$ $$10754$$ $$+$$ $$-$$ $$q+(35+\beta _{1})q^{3}+(-158+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$
272.10.a.l $$10$$ $$140.090$$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$0$$ $$10$$ $$2676$$ $$-4850$$ $$+$$ $$-$$ $$q+(1+\beta _{1})q^{3}+(268+\beta _{2})q^{5}+(-485+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_0(272))$$ into lower level spaces

$$S_{10}^{\mathrm{old}}(\Gamma_0(272)) \cong$$ $$S_{10}^{\mathrm{new}}(\Gamma_0(2))$$$$^{\oplus 8}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(17))$$$$^{\oplus 5}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(34))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(68))$$$$^{\oplus 3}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(136))$$$$^{\oplus 2}$$