Properties

Label 224.2.p
Level 224
Weight 2
Character orbit p
Rep. character \(\chi_{224}(31,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 16
Newform subspaces 1
Sturm bound 64
Trace bound 0

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Defining parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 224.p (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
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 \(M_{2}(224, [\chi])\).

Total New Old
Modular forms 80 16 64
Cusp forms 48 16 32
Eisenstein series 32 0 32

Trace form

\( 16q - 8q^{9} + O(q^{10}) \) \( 16q - 8q^{9} - 24q^{21} + 16q^{25} + 16q^{29} + 24q^{33} - 8q^{37} - 24q^{45} - 32q^{49} - 8q^{53} - 16q^{57} - 24q^{61} + 8q^{65} - 24q^{73} + 64q^{77} - 48q^{81} - 16q^{85} - 72q^{89} + 8q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(224, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
224.2.p.a \(16\) \(1.789\) 16.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{14}q^{3}-\beta _{13}q^{5}-\beta _{9}q^{7}+(-\beta _{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(224, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(224, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 8 T^{2} + 38 T^{4} - 80 T^{6} - 23 T^{8} + 752 T^{10} - 1162 T^{12} - 4696 T^{14} + 30100 T^{16} - 42264 T^{18} - 94122 T^{20} + 548208 T^{22} - 150903 T^{24} - 4723920 T^{26} + 20194758 T^{28} - 38263752 T^{30} + 43046721 T^{32} \)
$5$ \( ( 1 + 6 T^{2} + 9 T^{4} - 138 T^{6} - 796 T^{8} - 3450 T^{10} + 5625 T^{12} + 93750 T^{14} + 390625 T^{16} )^{2} \)
$7$ \( 1 + 16 T^{2} + 60 T^{4} - 784 T^{6} - 10426 T^{8} - 38416 T^{10} + 144060 T^{12} + 1882384 T^{14} + 5764801 T^{16} \)
$11$ \( 1 + 24 T^{2} + 86 T^{4} - 2064 T^{6} - 12647 T^{8} + 54000 T^{10} - 509434 T^{12} - 4559928 T^{14} + 74796148 T^{16} - 551751288 T^{18} - 7458623194 T^{20} + 95664294000 T^{22} - 2710996768007 T^{24} - 53534844376464 T^{26} + 269904840398006 T^{28} + 9113996005997784 T^{30} + 45949729863572161 T^{32} \)
$13$ \( ( 1 - 48 T^{2} + 1308 T^{4} - 24144 T^{6} + 351974 T^{8} - 4080336 T^{10} + 37357788 T^{12} - 231686832 T^{14} + 815730721 T^{16} )^{2} \)
$17$ \( ( 1 + 46 T^{2} + 1137 T^{4} - 1392 T^{5} + 19214 T^{6} - 46992 T^{7} + 276452 T^{8} - 798864 T^{9} + 5552846 T^{10} - 6838896 T^{11} + 94963377 T^{12} + 1110328174 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( 1 - 120 T^{2} + 7878 T^{4} - 351600 T^{6} + 11807689 T^{8} - 315489360 T^{10} + 7073822934 T^{12} - 141586703880 T^{14} + 2708298760020 T^{16} - 51112800100680 T^{18} + 921867678581814 T^{20} - 14842474887326160 T^{22} + 200536630500022249 T^{24} - 2155682896242831600 T^{26} + 17436494932403216358 T^{28} - 95880802293946094520 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 + 136 T^{2} + 9622 T^{4} + 484880 T^{6} + 19684601 T^{8} + 683013040 T^{10} + 20851380422 T^{12} + 567119503160 T^{14} + 13792455622804 T^{16} + 300006217171640 T^{18} + 5835071148672902 T^{20} + 101110442574992560 T^{22} + 1541520499173357881 T^{24} + 20086886757274127120 T^{26} + \)\(21\!\cdots\!62\)\( T^{28} + \)\(15\!\cdots\!24\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( ( 1 - 4 T + 88 T^{2} - 332 T^{3} + 3534 T^{4} - 9628 T^{5} + 74008 T^{6} - 97556 T^{7} + 707281 T^{8} )^{4} \)
$31$ \( 1 - 120 T^{2} + 6726 T^{4} - 224880 T^{6} + 4887337 T^{8} - 66674640 T^{10} + 1194526422 T^{12} - 84694288200 T^{14} + 3829419751284 T^{16} - 81391210960200 T^{18} + 1103170235771862 T^{20} - 59173988429349840 T^{22} + 4168365924253784617 T^{24} - \)\(18\!\cdots\!80\)\( T^{26} + \)\(52\!\cdots\!86\)\( T^{28} - \)\(90\!\cdots\!20\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( ( 1 + 4 T - 74 T^{2} - 88 T^{3} + 2945 T^{4} - 5624 T^{5} - 119866 T^{6} + 157628 T^{7} + 4940404 T^{8} + 5832236 T^{9} - 164096554 T^{10} - 284872472 T^{11} + 5519404145 T^{12} - 6102268216 T^{13} - 189863754266 T^{14} + 379727508532 T^{15} + 3512479453921 T^{16} )^{2} \)
$41$ \( ( 1 - 272 T^{2} + 34236 T^{4} - 2596336 T^{6} + 129652166 T^{8} - 4364440816 T^{10} + 96742753596 T^{12} - 1292028353552 T^{14} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 - 152 T^{2} + 8604 T^{4} - 204136 T^{6} + 3083174 T^{8} - 377447464 T^{10} + 29415363804 T^{12} - 960847183448 T^{14} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 - 200 T^{2} + 18934 T^{4} - 1138000 T^{6} + 52923737 T^{8} - 2454592400 T^{10} + 135007848038 T^{12} - 7743351520600 T^{14} + 394816080594964 T^{16} - 17105063509005400 T^{18} + 658795230921915878 T^{20} - 26458580024526899600 T^{22} + \)\(12\!\cdots\!57\)\( T^{24} - \)\(59\!\cdots\!00\)\( T^{26} + \)\(21\!\cdots\!94\)\( T^{28} - \)\(51\!\cdots\!00\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( ( 1 + 4 T - 106 T^{2} - 984 T^{3} + 4577 T^{4} + 69896 T^{5} + 172774 T^{6} - 2132484 T^{7} - 19431692 T^{8} - 113021652 T^{9} + 485322166 T^{10} + 10405906792 T^{11} + 36114731537 T^{12} - 411504365112 T^{13} - 2349422279674 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} )^{2} \)
$59$ \( 1 - 216 T^{2} + 25302 T^{4} - 1976880 T^{6} + 99819481 T^{8} - 1826074704 T^{10} - 264226437882 T^{12} + 36647033732760 T^{14} - 2664686254766220 T^{16} + 127568324423737560 T^{18} - 3201727133560269402 T^{20} - 77024805483051117264 T^{22} + \)\(14\!\cdots\!01\)\( T^{24} - \)\(10\!\cdots\!80\)\( T^{26} + \)\(45\!\cdots\!62\)\( T^{28} - \)\(13\!\cdots\!76\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( ( 1 + 12 T + 182 T^{2} + 1608 T^{3} + 12321 T^{4} + 99864 T^{5} + 781126 T^{6} + 6487236 T^{7} + 58967444 T^{8} + 395721396 T^{9} + 2906569846 T^{10} + 22667230584 T^{11} + 170594606961 T^{12} + 1358110852008 T^{13} + 9376708133702 T^{14} + 37712914032252 T^{15} + 191707312997281 T^{16} )^{2} \)
$67$ \( 1 + 248 T^{2} + 30214 T^{4} + 1955248 T^{6} + 42292553 T^{8} - 3785460880 T^{10} - 336164526826 T^{12} - 6739836332504 T^{14} + 264085897769428 T^{16} - 30255125296610456 T^{18} - 6774092055978471946 T^{20} - \)\(34\!\cdots\!20\)\( T^{22} + \)\(17\!\cdots\!73\)\( T^{24} + \)\(35\!\cdots\!52\)\( T^{26} + \)\(24\!\cdots\!54\)\( T^{28} + \)\(91\!\cdots\!92\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( ( 1 - 248 T^{2} + 34492 T^{4} - 3463880 T^{6} + 277458886 T^{8} - 17461419080 T^{10} + 876499701052 T^{12} - 31768870412408 T^{14} + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 + 12 T + 270 T^{2} + 2664 T^{3} + 35289 T^{4} + 261048 T^{5} + 2808174 T^{6} + 17932596 T^{7} + 187438964 T^{8} + 1309079508 T^{9} + 14964759246 T^{10} + 101552109816 T^{11} + 1002145526649 T^{12} + 5522662723752 T^{13} + 40860241098030 T^{14} + 132568782229164 T^{15} + 806460091894081 T^{16} )^{2} \)
$79$ \( 1 + 296 T^{2} + 48502 T^{4} + 5442256 T^{6} + 407298713 T^{8} + 12121526192 T^{10} - 1601181207706 T^{12} - 316632003296936 T^{14} - 31211270417534060 T^{16} - 1976100332576177576 T^{18} - 62366137735826524186 T^{20} + \)\(29\!\cdots\!32\)\( T^{22} + \)\(61\!\cdots\!93\)\( T^{24} + \)\(51\!\cdots\!56\)\( T^{26} + \)\(28\!\cdots\!82\)\( T^{28} + \)\(10\!\cdots\!76\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( ( 1 + 248 T^{2} + 25404 T^{4} + 1379080 T^{6} + 70811942 T^{8} + 9500482120 T^{10} + 1205631186684 T^{12} + 81081212595512 T^{14} + 2252292232139041 T^{16} )^{2} \)
$89$ \( ( 1 + 36 T + 822 T^{2} + 14040 T^{3} + 193953 T^{4} + 2261112 T^{5} + 23442342 T^{6} + 226172268 T^{7} + 2129495396 T^{8} + 20129331852 T^{9} + 185686790982 T^{10} + 1594013865528 T^{11} + 12169045868673 T^{12} + 78400194663960 T^{13} + 408518621169942 T^{14} + 1592328056239044 T^{15} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 - 304 T^{2} + 67644 T^{4} - 9727184 T^{6} + 1110022022 T^{8} - 91523074256 T^{10} + 5988474683964 T^{12} - 253223489498416 T^{14} + 7837433594376961 T^{16} )^{2} \)
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