# Properties

 Label 168.2.i Level 168 Weight 2 Character orbit i Rep. character $$\chi_{168}(125,\cdot)$$ Character field $$\Q$$ Dimension 28 Newform subspaces 5 Sturm bound 64 Trace bound 4

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 168.i (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$168$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$64$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$13$$

## Dimensions

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

Total New Old
Modular forms 36 36 0
Cusp forms 28 28 0
Eisenstein series 8 8 0

## Trace form

 $$28q - 4q^{4} - 4q^{7} - 4q^{9} + O(q^{10})$$ $$28q - 4q^{4} - 4q^{7} - 4q^{9} + 8q^{15} + 4q^{16} + 12q^{18} - 32q^{22} - 20q^{25} - 12q^{28} - 4q^{30} - 40q^{36} - 16q^{39} + 12q^{42} + 32q^{46} + 4q^{49} - 16q^{57} - 16q^{58} + 44q^{60} - 36q^{63} - 52q^{64} + 48q^{70} + 12q^{72} + 68q^{78} + 24q^{79} + 12q^{81} + 36q^{84} + 8q^{88} + O(q^{100})$$

## 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.i.a $$4$$ $$1.341$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$-4$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{2}+(-1+\beta _{2})q^{3}+(1+\beta _{3})q^{4}+\cdots$$
168.2.i.b $$4$$ $$1.341$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{1}q^{2}-\beta _{2}q^{3}+2q^{4}+2\beta _{2}q^{5}+\cdots$$
168.2.i.c $$4$$ $$1.341$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$4$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(1+\beta _{3})q^{4}+\cdots$$
168.2.i.d $$8$$ $$1.341$$ 8.0.$$\cdots$$.11 $$\Q(\sqrt{-14})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}-\beta _{1}q^{3}-2q^{4}+(\beta _{6}-\beta _{7})q^{5}+\cdots$$
168.2.i.e $$8$$ $$1.341$$ 8.0.3317760000.1 None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\beta _{5}q^{2}+(-\beta _{4}-\beta _{6})q^{3}+(-1-\beta _{2}+\cdots)q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - 2 T^{2} + 4 T^{4}$$)($$( 1 - 2 T^{2} )^{2}$$)($$1 - 2 T^{2} + 4 T^{4}$$)($$( 1 + 2 T^{2} )^{4}$$)($$( 1 + T^{2} + 4 T^{4} )^{2}$$)
$3$ ($$( 1 + 2 T + 3 T^{2} )^{2}$$)($$( 1 + 3 T^{2} )^{2}$$)($$( 1 - 2 T + 3 T^{2} )^{2}$$)($$1 - 10 T^{4} + 81 T^{8}$$)($$( 1 - 4 T^{2} + 9 T^{4} )^{2}$$)
$5$ ($$( 1 - 8 T^{2} + 25 T^{4} )^{2}$$)($$( 1 + 2 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 8 T^{2} + 25 T^{4} )^{2}$$)($$( 1 + 22 T^{4} + 625 T^{8} )^{2}$$)($$( 1 - 8 T^{2} + 25 T^{4} )^{4}$$)
$7$ ($$( 1 + 4 T + 7 T^{2} )^{2}$$)($$( 1 - 2 T + 7 T^{2} )^{2}$$)($$( 1 + 4 T + 7 T^{2} )^{2}$$)($$( 1 - 7 T^{2} )^{4}$$)($$( 1 - 2 T + 7 T^{2} )^{4}$$)
$11$ ($$( 1 + 16 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 121 T^{4} )^{2}$$)($$( 1 + 16 T^{2} + 121 T^{4} )^{2}$$)($$( 1 + 11 T^{2} )^{8}$$)($$( 1 + 10 T^{2} + 121 T^{4} )^{4}$$)
$13$ ($$( 1 - 2 T + 13 T^{2} )^{4}$$)($$( 1 + 13 T^{2} )^{4}$$)($$( 1 + 2 T + 13 T^{2} )^{4}$$)($$( 1 + 310 T^{4} + 28561 T^{8} )^{2}$$)($$( 1 + 16 T^{2} + 169 T^{4} )^{4}$$)
$17$ ($$( 1 - 20 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 17 T^{2} )^{4}$$)($$( 1 - 20 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 17 T^{2} )^{8}$$)($$( 1 + 17 T^{2} )^{8}$$)
$19$ ($$( 1 + 4 T + 19 T^{2} )^{4}$$)($$( 1 + 19 T^{2} )^{4}$$)($$( 1 - 4 T + 19 T^{2} )^{4}$$)($$( 1 - 650 T^{4} + 130321 T^{8} )^{2}$$)($$( 1 + 28 T^{2} + 361 T^{4} )^{4}$$)
$23$ ($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 23 T^{2} )^{4}$$)($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 6 T + 23 T^{2} )^{4}( 1 + 6 T + 23 T^{2} )^{4}$$)($$( 1 - 26 T^{2} + 529 T^{4} )^{4}$$)
$29$ ($$( 1 + 34 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 50 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 34 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 29 T^{2} )^{8}$$)($$( 1 + 10 T^{2} + 841 T^{4} )^{4}$$)
$31$ ($$( 1 - 14 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 10 T + 31 T^{2} )^{2}( 1 + 10 T + 31 T^{2} )^{2}$$)($$( 1 - 14 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 31 T^{2} )^{8}$$)($$( 1 - 10 T + 31 T^{2} )^{4}( 1 + 10 T + 31 T^{2} )^{4}$$)
$37$ ($$( 1 + 34 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 37 T^{2} )^{4}$$)($$( 1 + 34 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 37 T^{2} )^{8}$$)($$( 1 - 37 T^{2} )^{8}$$)
$41$ ($$( 1 + 76 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 41 T^{2} )^{4}$$)($$( 1 + 76 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 41 T^{2} )^{8}$$)($$( 1 - 38 T^{2} + 1681 T^{4} )^{4}$$)
$43$ ($$( 1 - 74 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 43 T^{2} )^{4}$$)($$( 1 - 74 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 43 T^{2} )^{8}$$)($$( 1 - 26 T^{2} + 1849 T^{4} )^{4}$$)
$47$ ($$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 47 T^{2} )^{8}$$)($$( 1 - 26 T^{2} + 2209 T^{4} )^{4}$$)
$53$ ($$( 1 + 53 T^{2} )^{4}$$)($$( 1 - 94 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 + 53 T^{2} )^{4}$$)($$( 1 + 53 T^{2} )^{8}$$)($$( 1 + 53 T^{2} )^{8}$$)
$59$ ($$( 1 - 86 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 86 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 1130 T^{4} + 12117361 T^{8} )^{2}$$)($$( 1 - 20 T^{2} + 3481 T^{4} )^{4}$$)
$61$ ($$( 1 + 10 T + 61 T^{2} )^{4}$$)($$( 1 + 61 T^{2} )^{4}$$)($$( 1 - 10 T + 61 T^{2} )^{4}$$)($$( 1 - 7370 T^{4} + 13845841 T^{8} )^{2}$$)($$( 1 + 112 T^{2} + 3721 T^{4} )^{4}$$)
$67$ ($$( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2}$$)($$( 1 - 67 T^{2} )^{4}$$)($$( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2}$$)($$( 1 - 67 T^{2} )^{8}$$)($$( 1 - 74 T^{2} + 4489 T^{4} )^{4}$$)
$71$ ($$( 1 - 140 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 71 T^{2} )^{4}$$)($$( 1 - 140 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 110 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 62 T^{2} + 5041 T^{4} )^{4}$$)
$73$ ($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 73 T^{2} )^{8}$$)($$( 1 + 70 T^{2} + 5329 T^{4} )^{4}$$)
$79$ ($$( 1 - 8 T + 79 T^{2} )^{4}$$)($$( 1 - 10 T + 79 T^{2} )^{4}$$)($$( 1 - 8 T + 79 T^{2} )^{4}$$)($$( 1 + 130 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 + 10 T + 79 T^{2} )^{8}$$)
$83$ ($$( 1 - 38 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 134 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 13130 T^{4} + 47458321 T^{8} )^{2}$$)($$( 1 - 116 T^{2} + 6889 T^{4} )^{4}$$)
$89$ ($$( 1 + 124 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 124 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 89 T^{2} )^{8}$$)($$( 1 + 89 T^{2} )^{8}$$)
$97$ ($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 2 T + 97 T^{2} )^{2}( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 97 T^{2} )^{8}$$)($$( 1 - 170 T^{2} + 9409 T^{4} )^{4}$$)