# Properties

 Label 4031.2.a Level 4031 Weight 2 Character orbit a Rep. character $$\chi_{4031}(1,\cdot)$$ Character field $$\Q$$ Dimension 323 Newform subspaces 5 Sturm bound 700 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$4031 = 29 \cdot 139$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4031.a (trivial) Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$700$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(4031))$$.

Total New Old
Modular forms 352 323 29
Cusp forms 349 323 26
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.

$$29$$$$139$$FrickeDim.
$$+$$$$+$$$$+$$$$61$$
$$+$$$$-$$$$-$$$$103$$
$$-$$$$+$$$$-$$$$100$$
$$-$$$$-$$$$+$$$$59$$
Plus space$$+$$$$120$$
Minus space$$-$$$$203$$

## Trace form

 $$323q + 3q^{2} + 329q^{4} + 2q^{5} + 12q^{6} + 8q^{7} + 3q^{8} + 331q^{9} + O(q^{10})$$ $$323q + 3q^{2} + 329q^{4} + 2q^{5} + 12q^{6} + 8q^{7} + 3q^{8} + 331q^{9} + 6q^{10} + 4q^{11} - 8q^{12} + 14q^{13} - 24q^{14} + 4q^{15} + 337q^{16} + 18q^{17} + 19q^{18} + 20q^{19} - 22q^{20} + 32q^{21} + 16q^{22} + 20q^{23} + 40q^{24} + 341q^{25} + 6q^{26} + 12q^{27} + 44q^{28} - 5q^{29} + 60q^{30} + 8q^{31} - 21q^{32} + 12q^{33} + 18q^{34} + 4q^{35} + 357q^{36} + 22q^{37} - 32q^{38} + 20q^{39} + 6q^{40} - 2q^{41} - 36q^{42} + 40q^{43} + 28q^{44} - 58q^{45} + 16q^{46} - 32q^{47} - 48q^{48} + 379q^{49} - 39q^{50} + 8q^{51} + 38q^{52} - 26q^{53} + 36q^{54} + 28q^{55} - 32q^{56} + 40q^{57} + 3q^{58} + 20q^{60} + 50q^{61} + 32q^{62} + 8q^{63} + 405q^{64} + 20q^{65} + 4q^{66} - 16q^{67} + 2q^{68} - 24q^{69} - 36q^{70} - 32q^{71} - 13q^{72} + 34q^{73} + 38q^{74} + 4q^{75} + 60q^{76} + 20q^{77} - 108q^{78} + 40q^{79} + 18q^{80} + 355q^{81} - 90q^{82} - 40q^{83} + 100q^{84} + 60q^{85} + 4q^{86} + 4q^{87} + 72q^{88} + 6q^{89} - 34q^{90} + 48q^{91} - 96q^{92} - 44q^{93} + 28q^{94} + 8q^{95} + 44q^{96} + 70q^{97} + 19q^{98} + 80q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(4031))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 29 139
4031.2.a.a $$2$$ $$32.188$$ $$\Q(\sqrt{2})$$ None $$2$$ $$2$$ $$4$$ $$-2$$ $$-$$ $$+$$ $$q+(1+\beta )q^{2}+(1+\beta )q^{3}+(1+2\beta )q^{4}+\cdots$$
4031.2.a.b $$59$$ $$32.188$$ None $$-5$$ $$-6$$ $$-5$$ $$-10$$ $$-$$ $$-$$
4031.2.a.c $$61$$ $$32.188$$ None $$-1$$ $$-4$$ $$-7$$ $$-10$$ $$+$$ $$+$$
4031.2.a.d $$98$$ $$32.188$$ None $$6$$ $$6$$ $$1$$ $$12$$ $$-$$ $$+$$
4031.2.a.e $$103$$ $$32.188$$ None $$1$$ $$2$$ $$9$$ $$18$$ $$+$$ $$-$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(4031))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(4031)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(29))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(139))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4}$$)
$3$ ($$1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4}$$)
$5$ ($$( 1 - 2 T + 5 T^{2} )^{2}$$)
$7$ ($$1 + 2 T + 7 T^{2} + 14 T^{3} + 49 T^{4}$$)
$11$ ($$( 1 + 11 T^{2} )^{2}$$)
$13$ ($$( 1 - 5 T + 13 T^{2} )^{2}$$)
$17$ ($$1 - 6 T + 41 T^{2} - 102 T^{3} + 289 T^{4}$$)
$19$ ($$1 + 2 T + 21 T^{2} + 38 T^{3} + 361 T^{4}$$)
$23$ ($$1 - 12 T + 74 T^{2} - 276 T^{3} + 529 T^{4}$$)
$29$ ($$( 1 - T )^{2}$$)
$31$ ($$1 - 4 T + 58 T^{2} - 124 T^{3} + 961 T^{4}$$)
$37$ ($$1 + 2 T^{2} + 1369 T^{4}$$)
$41$ ($$( 1 - 2 T + 41 T^{2} )^{2}$$)
$43$ ($$1 + 2 T + 85 T^{2} + 86 T^{3} + 1849 T^{4}$$)
$47$ ($$1 - 4 T + 90 T^{2} - 188 T^{3} + 2209 T^{4}$$)
$53$ ($$1 - 8 T + 114 T^{2} - 424 T^{3} + 2809 T^{4}$$)
$59$ ($$1 + 12 T + 122 T^{2} + 708 T^{3} + 3481 T^{4}$$)
$61$ ($$1 - 2 T + 25 T^{2} - 122 T^{3} + 3721 T^{4}$$)
$67$ ($$1 + 2 T + 7 T^{2} + 134 T^{3} + 4489 T^{4}$$)
$71$ ($$1 + 6 T + 23 T^{2} + 426 T^{3} + 5041 T^{4}$$)
$73$ ($$1 - 6 T + 105 T^{2} - 438 T^{3} + 5329 T^{4}$$)
$79$ ($$1 - 12 T + 122 T^{2} - 948 T^{3} + 6241 T^{4}$$)
$83$ ($$( 1 + 7 T + 83 T^{2} )^{2}$$)
$89$ ($$1 + 28 T + 366 T^{2} + 2492 T^{3} + 7921 T^{4}$$)
$97$ ($$1 - 10 T + 201 T^{2} - 970 T^{3} + 9409 T^{4}$$)