# Properties

 Label 168.2.p Level 168 Weight 2 Character orbit p Rep. character $$\chi_{168}(139,\cdot)$$ Character field $$\Q$$ Dimension 16 Newform subspaces 1 Sturm bound 64 Trace bound 0

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## Defining parameters

 Level: $$N$$ = $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 168.p (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$56$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$64$$ Trace bound: $$0$$

## Dimensions

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

Total New Old
Modular forms 36 16 20
Cusp forms 28 16 12
Eisenstein series 8 0 8

## Trace form

 $$16q + 2q^{2} + 2q^{4} - 10q^{8} - 16q^{9} + O(q^{10})$$ $$16q + 2q^{2} + 2q^{4} - 10q^{8} - 16q^{9} - 8q^{11} - 14q^{14} + 18q^{16} - 2q^{18} + 8q^{22} + 16q^{25} - 10q^{28} - 16q^{30} - 18q^{32} + 24q^{35} - 2q^{36} - 4q^{42} - 8q^{43} + 52q^{46} - 8q^{49} - 34q^{50} + 50q^{56} - 16q^{57} + 24q^{58} + 32q^{60} + 2q^{64} - 40q^{67} - 24q^{70} + 10q^{72} - 32q^{74} - 24q^{78} + 16q^{81} + 8q^{84} - 32q^{86} - 88q^{88} - 56q^{91} + 44q^{92} + 66q^{98} + 8q^{99} + O(q^{100})$$

## 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.p.a $$16$$ $$1.341$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{5}q^{2}-\beta _{3}q^{3}+\beta _{1}q^{4}+(\beta _{6}-\beta _{11}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(168, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(168, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + 2 T^{3} - 4 T^{4} + 4 T^{5} - 8 T^{7} + 16 T^{8} )^{2}$$
$3$ $$( 1 + T^{2} )^{8}$$
$5$ $$( 1 + 16 T^{2} + 140 T^{4} + 832 T^{6} + 4326 T^{8} + 20800 T^{10} + 87500 T^{12} + 250000 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$1 + 4 T^{2} + 68 T^{4} + 572 T^{6} + 3382 T^{8} + 28028 T^{10} + 163268 T^{12} + 470596 T^{14} + 5764801 T^{16}$$
$11$ $$( 1 + 2 T + 20 T^{2} + 6 T^{3} + 182 T^{4} + 66 T^{5} + 2420 T^{6} + 2662 T^{7} + 14641 T^{8} )^{4}$$
$13$ $$( 1 + 36 T^{2} + 692 T^{4} + 12604 T^{6} + 195894 T^{8} + 2130076 T^{10} + 19764212 T^{12} + 173765124 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 - 72 T^{2} + 2652 T^{4} - 66024 T^{6} + 1253414 T^{8} - 19080936 T^{10} + 221497692 T^{12} - 1737904968 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 56 T^{2} + 1532 T^{4} - 36808 T^{6} + 804262 T^{8} - 13287688 T^{10} + 199651772 T^{12} - 2634569336 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 88 T^{2} + 3548 T^{4} - 96952 T^{6} + 2287782 T^{8} - 51287608 T^{10} + 992875868 T^{12} - 13027158232 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 164 T^{2} + 12980 T^{4} - 650812 T^{6} + 22500790 T^{8} - 547332892 T^{10} + 9180507380 T^{12} - 97551024644 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 + 68 T^{2} + 5252 T^{4} + 204988 T^{6} + 8264182 T^{8} + 196993468 T^{10} + 4850332292 T^{12} + 60350250308 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 - 232 T^{2} + 25148 T^{4} - 1670040 T^{6} + 74486502 T^{8} - 2286284760 T^{10} + 47131400828 T^{12} - 595248526888 T^{14} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 - 120 T^{2} + 10204 T^{4} - 627512 T^{6} + 28679654 T^{8} - 1054847672 T^{10} + 28834065244 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 + 2 T + 120 T^{2} + 250 T^{3} + 6782 T^{4} + 10750 T^{5} + 221880 T^{6} + 159014 T^{7} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 120 T^{2} + 10396 T^{4} + 591944 T^{6} + 31424966 T^{8} + 1307604296 T^{10} + 50729163676 T^{12} + 1293505839480 T^{14} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 260 T^{2} + 33908 T^{4} - 2877468 T^{6} + 176791542 T^{8} - 8082807612 T^{10} + 267550429748 T^{12} - 5762733893540 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 - 264 T^{2} + 34076 T^{4} - 3023352 T^{6} + 203740198 T^{8} - 10524288312 T^{10} + 412911193436 T^{12} - 11135660881224 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 + 324 T^{2} + 46708 T^{4} + 4167580 T^{6} + 280236854 T^{8} + 15507565180 T^{10} + 646711541428 T^{12} + 16692601292964 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 + 10 T + 208 T^{2} + 1122 T^{3} + 16622 T^{4} + 75174 T^{5} + 933712 T^{6} + 3007630 T^{7} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 456 T^{2} + 94300 T^{4} - 11817416 T^{6} + 1002933542 T^{8} - 59571594056 T^{10} + 2396321518300 T^{12} - 58413729467976 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 328 T^{2} + 54492 T^{4} - 6043256 T^{6} + 500200454 T^{8} - 32204511224 T^{10} + 1547476948572 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 444 T^{2} + 95492 T^{4} - 12990660 T^{6} + 1221053302 T^{8} - 81074709060 T^{10} + 3719421134852 T^{12} - 107930830251324 T^{14} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 - 120 T^{2} + 20284 T^{4} - 1531784 T^{6} + 164981030 T^{8} - 10552459976 T^{10} + 962644583164 T^{12} - 39232844804280 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 - 312 T^{2} + 51484 T^{4} - 5696504 T^{6} + 530357222 T^{8} - 45122008184 T^{10} + 3230221535644 T^{12} - 155058162779832 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 232 T^{2} + 17244 T^{4} + 409768 T^{6} - 157308730 T^{8} + 3855507112 T^{10} + 1526598921564 T^{12} - 193249505143528 T^{14} + 7837433594376961 T^{16} )^{2}$$
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