# Properties

 Label 168.2.ba Level 168 Weight 2 Character orbit ba Rep. character $$\chi_{168}(5,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 56 Newform subspaces 3 Sturm bound 64 Trace bound 3

# Related objects

## Defining parameters

 Level: $$N$$ = $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 168.ba (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$168$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$64$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(168, [\chi])$$.

Total New Old
Modular forms 72 72 0
Cusp forms 56 56 0
Eisenstein series 16 16 0

## Trace form

 $$56q - 2q^{4} - 8q^{7} - 2q^{9} + O(q^{10})$$ $$56q - 2q^{4} - 8q^{7} - 2q^{9} - 6q^{10} - 18q^{12} - 20q^{15} - 10q^{16} - 4q^{22} - 12q^{24} + 8q^{25} - 18q^{28} + 22q^{30} - 12q^{31} - 6q^{33} + 4q^{36} - 8q^{39} - 66q^{40} + 36q^{42} - 8q^{46} - 16q^{49} - 48q^{52} + 60q^{54} + 4q^{57} + 22q^{58} - 8q^{60} - 30q^{63} + 4q^{64} + 54q^{66} + 42q^{70} + 42q^{72} - 36q^{73} - 68q^{78} - 36q^{79} + 6q^{81} + 48q^{82} + 48q^{84} - 24q^{87} - 26q^{88} + 48q^{94} + 90q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(168, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
168.2.ba.a $$4$$ $$1.341$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$-6$$ $$6$$ $$-2$$ $$q+\beta _{1}q^{2}+(-2-\beta _{2})q^{3}+2\beta _{2}q^{4}+\cdots$$
168.2.ba.b $$4$$ $$1.341$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$6$$ $$-6$$ $$-2$$ $$q+\beta _{1}q^{2}+(2+\beta _{2})q^{3}+2\beta _{2}q^{4}+(-1+\cdots)q^{5}+\cdots$$
168.2.ba.c $$48$$ $$1.341$$ None $$0$$ $$0$$ $$0$$ $$-4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T^{2} + 4 T^{4}$$)($$1 + 2 T^{2} + 4 T^{4}$$)
$3$ ($$( 1 + 3 T + 3 T^{2} )^{2}$$)($$( 1 - 3 T + 3 T^{2} )^{2}$$)
$5$ ($$( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 2 T^{2} + 25 T^{4} )$$)($$( 1 + 2 T^{2} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} )$$)
$7$ ($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)
$11$ ($$( 1 - 10 T^{2} + 121 T^{4} )( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} )$$)($$( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} )( 1 - 10 T^{2} + 121 T^{4} )$$)
$13$ ($$( 1 + 13 T^{2} )^{4}$$)($$( 1 + 13 T^{2} )^{4}$$)
$17$ ($$( 1 - 17 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 17 T^{2} + 289 T^{4} )^{2}$$)
$19$ ($$( 1 - 19 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 19 T^{2} + 361 T^{4} )^{2}$$)
$23$ ($$( 1 + 23 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 23 T^{2} + 529 T^{4} )^{2}$$)
$29$ ($$( 1 + 18 T + 137 T^{2} + 522 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 18 T + 137 T^{2} - 522 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 + 10 T + 31 T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )$$)($$( 1 + 10 T + 31 T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )$$)
$37$ ($$( 1 + 37 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 + 37 T^{2} + 1369 T^{4} )^{2}$$)
$41$ ($$( 1 + 41 T^{2} )^{4}$$)($$( 1 + 41 T^{2} )^{4}$$)
$43$ ($$( 1 - 43 T^{2} )^{4}$$)($$( 1 - 43 T^{2} )^{4}$$)
$47$ ($$( 1 - 47 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 47 T^{2} + 2209 T^{4} )^{2}$$)
$53$ ($$( 1 - 94 T^{2} + 2809 T^{4} )( 1 + 6 T + 65 T^{2} + 318 T^{3} + 2809 T^{4} )$$)($$( 1 - 6 T + 65 T^{2} - 318 T^{3} + 2809 T^{4} )( 1 - 94 T^{2} + 2809 T^{4} )$$)
$59$ ($$( 1 - 18 T + 167 T^{2} - 1062 T^{3} + 3481 T^{4} )( 1 - 10 T^{2} + 3481 T^{4} )$$)($$( 1 - 10 T^{2} + 3481 T^{4} )( 1 + 18 T + 167 T^{2} + 1062 T^{3} + 3481 T^{4} )$$)
$61$ ($$( 1 - 61 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 61 T^{2} + 3721 T^{4} )^{2}$$)
$67$ ($$( 1 + 67 T^{2} + 4489 T^{4} )^{2}$$)($$( 1 + 67 T^{2} + 4489 T^{4} )^{2}$$)
$71$ ($$( 1 - 71 T^{2} )^{4}$$)($$( 1 - 71 T^{2} )^{4}$$)
$73$ ($$( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} )( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} )$$)($$( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} )( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} )$$)
$79$ ($$( 1 - 10 T + 79 T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )$$)($$( 1 - 10 T + 79 T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )$$)
$83$ ($$( 1 - 30 T + 383 T^{2} - 2490 T^{3} + 6889 T^{4} )( 1 + 30 T + 383 T^{2} + 2490 T^{3} + 6889 T^{4} )$$)($$( 1 - 30 T + 383 T^{2} - 2490 T^{3} + 6889 T^{4} )( 1 + 30 T + 383 T^{2} + 2490 T^{3} + 6889 T^{4} )$$)
$89$ ($$( 1 - 89 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 89 T^{2} + 7921 T^{4} )^{2}$$)
$97$ ($$( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} )( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )$$)($$( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} )( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )$$)