# Properties

 Label 168.2.u Level 168 Weight 2 Character orbit u Rep. character $$\chi_{168}(17,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 16 Newform subspaces 1 Sturm bound 64 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ = $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 168.u (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$21$$ Character field: $$\Q(\zeta_{6})$$ 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 80 16 64
Cusp forms 48 16 32
Eisenstein series 32 0 32

## Trace form

 $$16q + 4q^{7} + 2q^{9} + O(q^{10})$$ $$16q + 4q^{7} + 2q^{9} + 8q^{15} - 6q^{19} + 14q^{21} - 18q^{25} - 48q^{31} - 12q^{33} - 2q^{37} - 22q^{39} + 20q^{43} - 42q^{45} - 28q^{49} + 6q^{51} - 8q^{57} + 36q^{61} - 32q^{63} + 14q^{67} + 30q^{73} + 54q^{75} + 28q^{79} + 30q^{81} + 16q^{85} + 78q^{87} + 66q^{91} + 16q^{93} + 20q^{99} + O(q^{100})$$

## 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.u.a $$16$$ $$1.341$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+(\beta _{3}+\beta _{9})q^{3}+\beta _{14}q^{5}-\beta _{10}q^{7}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(168, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T^{2} - 7 T^{4} - 24 T^{5} + 22 T^{6} + 48 T^{7} + 10 T^{8} + 144 T^{9} + 198 T^{10} - 648 T^{11} - 567 T^{12} - 729 T^{14} + 6561 T^{16}$$
$5$ $$( 1 - 13 T^{2} + 93 T^{4} - 434 T^{6} + 1886 T^{8} - 10850 T^{10} + 58125 T^{12} - 203125 T^{14} + 390625 T^{16} )( 1 + 2 T^{2} - 39 T^{4} - 38 T^{6} + 836 T^{8} - 950 T^{10} - 24375 T^{12} + 31250 T^{14} + 390625 T^{16} )$$
$7$ $$( 1 - 2 T + 9 T^{2} - 10 T^{3} + 44 T^{4} - 70 T^{5} + 441 T^{6} - 686 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$1 + 49 T^{2} + 1300 T^{4} + 22265 T^{6} + 252641 T^{8} + 1395328 T^{10} - 12534274 T^{12} - 440430082 T^{14} - 6222779240 T^{16} - 53292039922 T^{18} - 183514305634 T^{20} + 2471908667008 T^{22} + 54155842054721 T^{24} + 577496758741265 T^{26} + 4079956889737300 T^{28} + 18607741845578809 T^{30} + 45949729863572161 T^{32}$$
$13$ $$( 1 - 49 T^{2} + 1278 T^{4} - 23495 T^{6} + 341186 T^{8} - 3970655 T^{10} + 36500958 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$1 - 42 T^{2} + 1059 T^{4} - 6894 T^{6} - 204407 T^{8} + 7559412 T^{10} - 49293810 T^{12} - 1407253056 T^{14} + 49740922386 T^{16} - 406696133184 T^{18} - 4117068305010 T^{20} + 182465828749428 T^{22} - 1425893651242487 T^{24} - 13898261949695406 T^{26} + 616996949226316899 T^{28} - 7071868715494839018 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 + 3 T + 62 T^{2} + 177 T^{3} + 2049 T^{4} + 6000 T^{5} + 55390 T^{6} + 152826 T^{7} + 1217108 T^{8} + 2903694 T^{9} + 19995790 T^{10} + 41154000 T^{11} + 267027729 T^{12} + 438269523 T^{13} + 2916844622 T^{14} + 2681615217 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 + 102 T^{2} + 6323 T^{4} + 248898 T^{6} + 6664873 T^{8} + 89132724 T^{10} - 1162854034 T^{12} - 108042764448 T^{14} - 3248548802990 T^{16} - 57154622392992 T^{18} - 325414235728594 T^{20} + 13194842036331636 T^{22} + 521932771402734313 T^{24} + 10310975788054808802 T^{26} +$$$$13\!\cdots\!83$$$$T^{28} +$$$$11\!\cdots\!18$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 - 103 T^{2} + 6606 T^{4} - 293969 T^{6} + 9810626 T^{8} - 247227929 T^{10} + 4672298286 T^{12} - 61266802063 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 + 24 T + 350 T^{2} + 3792 T^{3} + 33057 T^{4} + 243840 T^{5} + 1597150 T^{6} + 9593544 T^{7} + 54548420 T^{8} + 297399864 T^{9} + 1534861150 T^{10} + 7264237440 T^{11} + 30528833697 T^{12} + 108561740592 T^{13} + 310626288350 T^{14} + 660302738664 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 + T - 60 T^{2} + 689 T^{3} + 2741 T^{4} - 33360 T^{5} + 198298 T^{6} + 1145242 T^{7} - 8840520 T^{8} + 42373954 T^{9} + 271469962 T^{10} - 1689784080 T^{11} + 5137075301 T^{12} + 47777986373 T^{13} - 153943584540 T^{14} + 94931877133 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 240 T^{2} + 27996 T^{4} + 2035728 T^{6} + 100303238 T^{8} + 3422058768 T^{10} + 79110004956 T^{12} + 1140025017840 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 - 5 T + 76 T^{2} - 341 T^{3} + 2710 T^{4} - 14663 T^{5} + 140524 T^{6} - 397535 T^{7} + 3418801 T^{8} )^{4}$$
$47$ $$1 - 158 T^{2} + 13171 T^{4} - 396034 T^{6} - 16747399 T^{8} + 2338864468 T^{10} - 73391270338 T^{12} - 1850417430616 T^{14} + 239524986298546 T^{16} - 4087572104230744 T^{18} - 358125987434202178 T^{20} + 25211123725919029972 T^{22} -$$$$39\!\cdots\!39$$$$T^{24} -$$$$20\!\cdots\!66$$$$T^{26} +$$$$15\!\cdots\!11$$$$T^{28} -$$$$40\!\cdots\!02$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 265 T^{2} + 32968 T^{4} + 3168029 T^{6} + 288393293 T^{8} + 22350648928 T^{10} + 1460664537746 T^{12} + 90342579430370 T^{14} + 5160533915433520 T^{16} + 253772305619909330 T^{18} + 11525345782458595826 T^{20} +$$$$49\!\cdots\!12$$$$T^{22} +$$$$17\!\cdots\!73$$$$T^{24} +$$$$55\!\cdots\!21$$$$T^{26} +$$$$16\!\cdots\!88$$$$T^{28} +$$$$36\!\cdots\!85$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 - 187 T^{2} + 11904 T^{4} - 267479 T^{6} + 18518765 T^{8} - 3131679552 T^{10} + 215589458578 T^{12} - 8824777678534 T^{14} + 393339242196864 T^{16} - 30719051098976854 T^{18} + 2612375297384172658 T^{20} -$$$$13\!\cdots\!32$$$$T^{22} +$$$$27\!\cdots\!65$$$$T^{24} -$$$$13\!\cdots\!79$$$$T^{26} +$$$$21\!\cdots\!24$$$$T^{28} -$$$$11\!\cdots\!07$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 - 18 T + 331 T^{2} - 4014 T^{3} + 46777 T^{4} - 446148 T^{5} + 4203790 T^{6} - 34615152 T^{7} + 285617722 T^{8} - 2111524272 T^{9} + 15642302590 T^{10} - 101267119188 T^{11} + 647666904457 T^{12} - 3390209552214 T^{13} + 17053243913491 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 - 7 T - 200 T^{2} + 865 T^{3} + 28259 T^{4} - 67468 T^{5} - 2804524 T^{6} + 1482940 T^{7} + 222922288 T^{8} + 99356980 T^{9} - 12589508236 T^{10} - 20291878084 T^{11} + 569450528339 T^{12} + 1167858217555 T^{13} - 18091676433800 T^{14} - 42424981237261 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 224 T^{2} + 21244 T^{4} - 988448 T^{6} + 37865158 T^{8} - 4982766368 T^{10} + 539845751164 T^{12} - 28694463598304 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 15 T + 312 T^{2} - 3555 T^{3} + 48909 T^{4} - 547104 T^{5} + 5639826 T^{6} - 55083990 T^{7} + 457851344 T^{8} - 4021131270 T^{9} + 30054632754 T^{10} - 212832756768 T^{11} + 1388929569069 T^{12} - 7369769513115 T^{13} + 47216278602168 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 14 T - 152 T^{2} + 1448 T^{3} + 34763 T^{4} - 170492 T^{5} - 3816772 T^{6} + 3401582 T^{7} + 382142944 T^{8} + 268724978 T^{9} - 23820474052 T^{10} - 84059205188 T^{11} + 1354021665803 T^{12} + 4455577665752 T^{13} - 36949293239192 T^{14} - 268854725806226 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 + 141 T^{2} + 22278 T^{4} + 1798779 T^{6} + 189218258 T^{8} + 12391788531 T^{10} + 1057276475238 T^{12} + 46098592645029 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$1 - 378 T^{2} + 69795 T^{4} - 8481534 T^{6} + 768803689 T^{8} - 53847736620 T^{10} + 2382276614382 T^{12} + 24847322779392 T^{14} - 11761219221673230 T^{16} + 196815643735564032 T^{18} +$$$$14\!\cdots\!62$$$$T^{20} -$$$$26\!\cdots\!20$$$$T^{22} +$$$$30\!\cdots\!09$$$$T^{24} -$$$$26\!\cdots\!34$$$$T^{26} +$$$$17\!\cdots\!95$$$$T^{28} -$$$$73\!\cdots\!98$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$( 1 - 429 T^{2} + 87258 T^{4} - 11450019 T^{6} + 1191212138 T^{8} - 107733228771 T^{10} + 7724888001498 T^{12} - 357344990114541 T^{14} + 7837433594376961 T^{16} )^{2}$$