Properties

Label 229.2.b
Level 229
Weight 2
Character orbit b
Rep. character \(\chi_{229}(228,\cdot)\)
Character field \(\Q\)
Dimension 18
Newform subspaces 2
Sturm bound 38
Trace bound 1

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Defining parameters

Level: \( N \) = \( 229 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 229.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 229 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(38\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(229, [\chi])\).

Total New Old
Modular forms 20 20 0
Cusp forms 18 18 0
Eisenstein series 2 2 0

Trace form

\( 18q + 2q^{3} - 22q^{4} - 6q^{5} + 16q^{9} + O(q^{10}) \) \( 18q + 2q^{3} - 22q^{4} - 6q^{5} + 16q^{9} - 2q^{11} - 18q^{12} + 2q^{14} - 14q^{15} + 10q^{16} + 8q^{17} - 4q^{19} + 26q^{20} - 2q^{26} + 20q^{27} - 20q^{33} - 58q^{36} - 30q^{37} - 58q^{42} + 36q^{43} + 32q^{44} - 40q^{45} + 56q^{46} + 74q^{48} - 30q^{49} + 10q^{51} - 16q^{53} + 26q^{56} + 36q^{57} - 34q^{58} + 28q^{60} + 36q^{61} + 4q^{62} - 42q^{64} - 76q^{68} + 28q^{70} - 18q^{71} - 16q^{75} + 80q^{76} + 22q^{78} - 36q^{80} + 18q^{81} + 52q^{82} - 10q^{83} - 42q^{85} + 24q^{91} + 34q^{94} - 2q^{95} - 44q^{97} - 10q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(229, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
229.2.b.a \(2\) \(1.829\) \(\Q(\sqrt{-5}) \) None \(0\) \(2\) \(6\) \(0\) \(q+\beta q^{2}+q^{3}-3q^{4}+3q^{5}+\beta q^{6}+\cdots\)
229.2.b.b \(16\) \(1.829\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(-12\) \(0\) \(q+\beta _{1}q^{2}-\beta _{7}q^{3}+(-1-\beta _{4}+\beta _{6}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} + 4 T^{4} \))(\( 1 - 8 T^{2} + 37 T^{4} - 118 T^{6} + 311 T^{8} - 757 T^{10} + 1858 T^{12} - 4373 T^{14} + 9365 T^{16} - 17492 T^{18} + 29728 T^{20} - 48448 T^{22} + 79616 T^{24} - 120832 T^{26} + 151552 T^{28} - 131072 T^{30} + 65536 T^{32} \))
$3$ (\( ( 1 - T + 3 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} - 5 T^{3} + 30 T^{4} - 31 T^{5} + 126 T^{6} - 114 T^{7} + 436 T^{8} - 342 T^{9} + 1134 T^{10} - 837 T^{11} + 2430 T^{12} - 1215 T^{13} + 5103 T^{14} + 6561 T^{16} )^{2} \))
$5$ (\( ( 1 - 3 T + 5 T^{2} )^{2} \))(\( ( 1 + 6 T + 40 T^{2} + 160 T^{3} + 648 T^{4} + 1993 T^{5} + 6089 T^{6} + 15169 T^{7} + 37231 T^{8} + 75845 T^{9} + 152225 T^{10} + 249125 T^{11} + 405000 T^{12} + 500000 T^{13} + 625000 T^{14} + 468750 T^{15} + 390625 T^{16} )^{2} \))
$7$ (\( ( 1 - 7 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 716 T^{4} - 10574 T^{6} + 124720 T^{8} - 1214049 T^{10} + 10353144 T^{12} - 79384418 T^{14} + 572635904 T^{16} - 3889836482 T^{18} + 24857898744 T^{20} - 142831650801 T^{22} + 718985980720 T^{24} - 2986893282926 T^{26} + 9910361635916 T^{28} - 23059584476866 T^{30} + 33232930569601 T^{32} \))
$11$ (\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 50 T^{2} + 185 T^{3} + 1295 T^{4} + 4371 T^{5} + 22737 T^{6} + 69250 T^{7} + 290786 T^{8} + 761750 T^{9} + 2751177 T^{10} + 5817801 T^{11} + 18960095 T^{12} + 29794435 T^{13} + 88578050 T^{14} + 77948684 T^{15} + 214358881 T^{16} )^{2} \))
$13$ (\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - 83 T^{2} + 3652 T^{4} - 114842 T^{6} + 2886577 T^{8} - 60801425 T^{10} + 1099853842 T^{12} - 17341481474 T^{14} + 240230833040 T^{16} - 2930710369106 T^{18} + 31412925581362 T^{20} - 293476865402825 T^{22} + 2354669537432017 T^{24} - 15831944920922858 T^{26} + 85084606867300612 T^{28} - 326802240013040987 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( ( 1 + 3 T + 17 T^{2} )^{2} \))(\( ( 1 - 7 T + 92 T^{2} - 453 T^{3} + 3596 T^{4} - 13372 T^{5} + 85371 T^{6} - 262757 T^{7} + 1557053 T^{8} - 4466869 T^{9} + 24672219 T^{10} - 65696636 T^{11} + 300341516 T^{12} - 643195221 T^{13} + 2220656348 T^{14} - 2872370711 T^{15} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 + T + 19 T^{2} )^{2} \))(\( ( 1 + T + 86 T^{2} + 52 T^{3} + 3849 T^{4} + 2105 T^{5} + 116145 T^{6} + 57198 T^{7} + 2545210 T^{8} + 1086762 T^{9} + 41928345 T^{10} + 14438195 T^{11} + 501605529 T^{12} + 128757148 T^{13} + 4045945766 T^{14} + 893871739 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 - 26 T^{2} + 529 T^{4} \))(\( 1 - 111 T^{2} + 7708 T^{4} - 359067 T^{6} + 12935139 T^{8} - 361013314 T^{10} + 8484083118 T^{12} - 175407125231 T^{14} + 3842668659170 T^{16} - 92790369247199 T^{18} + 2374194303824238 T^{20} - 53442926878826146 T^{22} + 1012963479836689059 T^{24} - 14874893101951305483 T^{26} + \)\(16\!\cdots\!68\)\( T^{28} - \)\(12\!\cdots\!99\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 38 T^{2} + 841 T^{4} \))(\( 1 - 191 T^{2} + 18847 T^{4} - 1240684 T^{6} + 59820260 T^{8} - 2223160735 T^{10} + 66767157151 T^{12} - 1767490893446 T^{14} + 48098134782170 T^{16} - 1486459841388086 T^{18} + 47223141676916431 T^{20} - 1322387851509500935 T^{22} + 29924870487394389860 T^{24} - \)\(52\!\cdots\!84\)\( T^{26} + \)\(66\!\cdots\!27\)\( T^{28} - \)\(56\!\cdots\!71\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( ( 1 - 31 T^{2} )^{2} \))(\( 1 - 240 T^{2} + 31020 T^{4} - 2784587 T^{6} + 192020923 T^{8} - 10703010697 T^{10} + 496384321394 T^{12} - 19480388722295 T^{14} + 652666853734742 T^{16} - 18720653562125495 T^{18} + 458421344878108274 T^{20} - 9498961391369875657 T^{22} + \)\(16\!\cdots\!43\)\( T^{24} - \)\(22\!\cdots\!87\)\( T^{26} + \)\(24\!\cdots\!20\)\( T^{28} - \)\(18\!\cdots\!40\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 17 T + 281 T^{2} + 3142 T^{3} + 33367 T^{4} + 285756 T^{5} + 2303952 T^{6} + 15952696 T^{7} + 104146760 T^{8} + 590249752 T^{9} + 3154110288 T^{10} + 14474398668 T^{11} + 62535130087 T^{12} + 217878712894 T^{13} + 720969120929 T^{14} + 1613841911261 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 12 T + 41 T^{2} )( 1 + 12 T + 41 T^{2} ) \))(\( 1 - 440 T^{2} + 94975 T^{4} - 13407451 T^{6} + 1389803177 T^{8} - 112379988628 T^{10} + 7339258780819 T^{12} - 395046576538985 T^{14} + 17715215154082664 T^{16} - 664073295162033785 T^{18} + 20738991231745878259 T^{20} - \)\(53\!\cdots\!48\)\( T^{22} + \)\(11\!\cdots\!17\)\( T^{24} - \)\(17\!\cdots\!51\)\( T^{26} + \)\(21\!\cdots\!75\)\( T^{28} - \)\(16\!\cdots\!40\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 19 T + 310 T^{2} - 3194 T^{3} + 29757 T^{4} - 206765 T^{5} + 1416853 T^{6} - 8006500 T^{7} + 54339474 T^{8} - 344279500 T^{9} + 2619761197 T^{10} - 16439264855 T^{11} + 101733261357 T^{12} - 469544966942 T^{13} + 1959622545190 T^{14} - 5164553611033 T^{15} + 11688200277601 T^{16} )^{2} \))
$47$ (\( 1 - 14 T^{2} + 2209 T^{4} \))(\( 1 - 429 T^{2} + 89248 T^{4} - 12073318 T^{6} + 1201206099 T^{8} - 94346541797 T^{10} + 6146317617691 T^{12} - 345063082939647 T^{14} + 17150099258501604 T^{16} - 762244350213680223 T^{18} + 29992069299012036571 T^{20} - \)\(10\!\cdots\!13\)\( T^{22} + \)\(28\!\cdots\!39\)\( T^{24} - \)\(63\!\cdots\!82\)\( T^{26} + \)\(10\!\cdots\!68\)\( T^{28} - \)\(11\!\cdots\!01\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 14 T + 262 T^{2} + 2641 T^{3} + 30531 T^{4} + 261945 T^{5} + 2383262 T^{6} + 17866475 T^{7} + 140328262 T^{8} + 946923175 T^{9} + 6694582958 T^{10} + 38997585765 T^{11} + 240904275411 T^{12} + 1104454297013 T^{13} + 5807062615798 T^{14} + 16445955957718 T^{15} + 62259690411361 T^{16} )^{2} \))
$59$ (\( 1 - 38 T^{2} + 3481 T^{4} \))(\( 1 - 603 T^{2} + 179192 T^{4} - 34906353 T^{6} + 5003565479 T^{8} - 561658458824 T^{10} + 51272316369020 T^{12} - 3896641819837207 T^{14} + 249793193575778362 T^{16} - 13564210174853317567 T^{18} + \)\(62\!\cdots\!20\)\( T^{20} - \)\(23\!\cdots\!84\)\( T^{22} + \)\(73\!\cdots\!59\)\( T^{24} - \)\(17\!\cdots\!53\)\( T^{26} + \)\(31\!\cdots\!52\)\( T^{28} - \)\(37\!\cdots\!83\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 - 5 T + 61 T^{2} )^{2} \))(\( ( 1 - 13 T + 375 T^{2} - 4329 T^{3} + 65315 T^{4} - 671683 T^{5} + 7003540 T^{6} - 62874787 T^{7} + 510878041 T^{8} - 3835362007 T^{9} + 26060172340 T^{10} - 152459279023 T^{11} + 904341104915 T^{12} - 3656257387029 T^{13} + 19320140385375 T^{14} - 40855656868273 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( 1 + 46 T^{2} + 4489 T^{4} \))(\( 1 - 527 T^{2} + 143352 T^{4} - 26726780 T^{6} + 3818427235 T^{8} - 442828927195 T^{10} + 43091101327723 T^{12} - 3587173226330813 T^{14} + 258101323783461752 T^{16} - 16102820612999019557 T^{18} + \)\(86\!\cdots\!83\)\( T^{20} - \)\(40\!\cdots\!55\)\( T^{22} + \)\(15\!\cdots\!35\)\( T^{24} - \)\(48\!\cdots\!20\)\( T^{26} + \)\(11\!\cdots\!72\)\( T^{28} - \)\(19\!\cdots\!83\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 + 15 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 368 T^{2} - 1574 T^{3} + 63707 T^{4} - 197279 T^{5} + 7139319 T^{6} - 17214035 T^{7} + 583961990 T^{8} - 1222196485 T^{9} + 35989307079 T^{10} - 70608324169 T^{11} + 1618901961467 T^{12} - 2839856998474 T^{13} + 47140904482928 T^{14} - 54570720950346 T^{15} + 645753531245761 T^{16} )^{2} \))
$73$ (\( 1 + 34 T^{2} + 5329 T^{4} \))(\( 1 - 540 T^{2} + 153292 T^{4} - 30340255 T^{6} + 4660528956 T^{8} - 585237069848 T^{10} + 61880730121118 T^{12} - 5607687633270269 T^{14} + 439551876124546322 T^{16} - 29883367397697263501 T^{18} + \)\(17\!\cdots\!38\)\( T^{20} - \)\(88\!\cdots\!72\)\( T^{22} + \)\(37\!\cdots\!36\)\( T^{24} - \)\(13\!\cdots\!95\)\( T^{26} + \)\(35\!\cdots\!32\)\( T^{28} - \)\(65\!\cdots\!60\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 + 22 T^{2} + 6241 T^{4} \))(\( 1 - 538 T^{2} + 131982 T^{4} - 20804869 T^{6} + 2625285951 T^{8} - 301355576425 T^{10} + 31081350208218 T^{12} - 2786901221599379 T^{14} + 226521248661222974 T^{16} - 17393050524001724339 T^{18} + \)\(12\!\cdots\!58\)\( T^{20} - \)\(73\!\cdots\!25\)\( T^{22} + \)\(39\!\cdots\!11\)\( T^{24} - \)\(19\!\cdots\!69\)\( T^{26} + \)\(77\!\cdots\!62\)\( T^{28} - \)\(19\!\cdots\!78\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 426 T^{2} - 1909 T^{3} + 86691 T^{4} - 414059 T^{5} + 11394137 T^{6} - 53249160 T^{7} + 1089259614 T^{8} - 4419680280 T^{9} + 78494209793 T^{10} - 236753553433 T^{11} + 4114209305811 T^{12} - 7519628587487 T^{13} + 139276599055194 T^{14} - 108544203958508 T^{15} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( ( 1 - 6 T + 89 T^{2} )( 1 + 6 T + 89 T^{2} ) \))(\( 1 - 764 T^{2} + 301867 T^{4} - 81214762 T^{6} + 16576635524 T^{8} - 2711045835319 T^{10} + 366347122484839 T^{12} - 41631440722564547 T^{14} + 4015766180229082034 T^{16} - \)\(32\!\cdots\!87\)\( T^{18} + \)\(22\!\cdots\!99\)\( T^{20} - \)\(13\!\cdots\!59\)\( T^{22} + \)\(65\!\cdots\!44\)\( T^{24} - \)\(25\!\cdots\!62\)\( T^{26} + \)\(74\!\cdots\!07\)\( T^{28} - \)\(14\!\cdots\!24\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( ( 1 + 7 T + 97 T^{2} )^{2} \))(\( ( 1 + 15 T + 681 T^{2} + 8689 T^{3} + 211459 T^{4} + 2267717 T^{5} + 38955080 T^{6} + 346855759 T^{7} + 4644048029 T^{8} + 33645008623 T^{9} + 366528347720 T^{10} + 2069684077541 T^{11} + 18720313230979 T^{12} + 74615399493073 T^{13} + 567253935356649 T^{14} + 1211974267171695 T^{15} + 7837433594376961 T^{16} )^{2} \))
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