# Properties

 Label 201.2.p Level 201 Weight 2 Character orbit p Rep. character $$\chi_{201}(2,\cdot)$$ Character field $$\Q(\zeta_{66})$$ Dimension 420 Newform subspaces 2 Sturm bound 45 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$201 = 3 \cdot 67$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 201.p (of order $$66$$ and degree $$20$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$201$$ Character field: $$\Q(\zeta_{66})$$ Newform subspaces: $$2$$ Sturm bound: $$45$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(201, [\chi])$$.

Total New Old
Modular forms 500 500 0
Cusp forms 420 420 0
Eisenstein series 80 80 0

## Trace form

 $$420q - 22q^{3} - 22q^{4} - 20q^{6} - 44q^{7} - 24q^{9} + O(q^{10})$$ $$420q - 22q^{3} - 22q^{4} - 20q^{6} - 44q^{7} - 24q^{9} - 38q^{10} - 4q^{12} - 41q^{13} + 4q^{15} - 4q^{16} - 28q^{18} - 68q^{19} - 100q^{21} - 8q^{22} - 62q^{24} - 74q^{25} - 22q^{27} + 104q^{28} - 90q^{30} - 37q^{31} - 19q^{33} - 32q^{34} - 18q^{36} - 24q^{37} + 30q^{39} - 4q^{40} - 22q^{42} - 22q^{43} - 132q^{45} - 162q^{46} - 66q^{48} - 43q^{49} - 10q^{51} - 198q^{52} + 101q^{54} + 126q^{55} + 2q^{57} - 80q^{60} + 161q^{61} - 28q^{63} + 188q^{64} + q^{67} - q^{69} + 264q^{70} + 88q^{72} + 54q^{73} - 11q^{75} + 92q^{76} + 20q^{78} - 112q^{79} - 60q^{81} - 16q^{82} - 49q^{84} + 34q^{85} - 100q^{87} + 150q^{88} + 88q^{90} + 108q^{91} + 42q^{93} - 88q^{94} + 196q^{96} + 33q^{97} - 54q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(201, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
201.2.p.a $$20$$ $$1.605$$ $$\Q(\zeta_{33})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-6$$ $$q+(2\zeta_{33}+\zeta_{33}^{12})q^{3}-2\zeta_{33}^{4}q^{4}+\cdots$$
201.2.p.b $$400$$ $$1.605$$ None $$0$$ $$-22$$ $$0$$ $$-38$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T^{2} - 8 T^{6} - 16 T^{8} + 64 T^{12} + 128 T^{14} - 512 T^{18} - 1024 T^{20} - 2048 T^{22} + 8192 T^{26} + 16384 T^{28} - 65536 T^{32} - 131072 T^{34} + 524288 T^{38} + 1048576 T^{40}$$)
$3$ ($$1 - 3 T^{2} + 9 T^{4} - 27 T^{6} + 81 T^{8} - 243 T^{10} + 729 T^{12} - 2187 T^{14} + 6561 T^{16} - 19683 T^{18} + 59049 T^{20}$$)
$5$ ($$( 1 - 5 T^{2} + 25 T^{4} - 125 T^{6} + 625 T^{8} - 3125 T^{10} + 15625 T^{12} - 78125 T^{14} + 390625 T^{16} - 1953125 T^{18} + 9765625 T^{20} )^{2}$$)
$7$ ($$( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} + 120 T^{5} - 83 T^{6} - 923 T^{7} - 342 T^{8} + 6119 T^{9} + 8513 T^{10} + 42833 T^{11} - 16758 T^{12} - 316589 T^{13} - 199283 T^{14} + 2016840 T^{15} + 3411821 T^{16} - 10706059 T^{17} - 34588806 T^{18} + 40353607 T^{19} + 282475249 T^{20} )( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 360 T^{5} + 757 T^{6} + 1265 T^{7} + 1026 T^{8} - 3725 T^{9} - 25807 T^{10} - 26075 T^{11} + 50274 T^{12} + 433895 T^{13} + 1817557 T^{14} + 6050520 T^{15} + 17529701 T^{16} + 45294865 T^{17} + 103766418 T^{18} + 201768035 T^{19} + 282475249 T^{20} )$$)
$11$ ($$( 1 - 11 T + 66 T^{2} - 363 T^{3} + 1815 T^{4} - 7986 T^{5} + 33275 T^{6} - 131769 T^{7} + 497794 T^{8} - 1771561 T^{9} + 5958887 T^{10} - 19487171 T^{11} + 60233074 T^{12} - 175384539 T^{13} + 487179275 T^{14} - 1286153286 T^{15} + 3215383215 T^{16} - 7073843073 T^{17} + 14147686146 T^{18} - 25937424601 T^{19} + 25937424601 T^{20} )( 1 + 11 T + 66 T^{2} + 363 T^{3} + 1815 T^{4} + 7986 T^{5} + 33275 T^{6} + 131769 T^{7} + 497794 T^{8} + 1771561 T^{9} + 5958887 T^{10} + 19487171 T^{11} + 60233074 T^{12} + 175384539 T^{13} + 487179275 T^{14} + 1286153286 T^{15} + 3215383215 T^{16} + 7073843073 T^{17} + 14147686146 T^{18} + 25937424601 T^{19} + 25937424601 T^{20} )$$)
$13$ ($$( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} - 630 T^{5} + 883 T^{6} + 6424 T^{7} - 24327 T^{8} - 34858 T^{9} + 385967 T^{10} - 453154 T^{11} - 4111263 T^{12} + 14113528 T^{13} + 25219363 T^{14} - 233914590 T^{15} + 139977461 T^{16} + 2760934748 T^{17} - 7341576489 T^{18} - 21208998746 T^{19} + 137858491849 T^{20} )( 1 + 5 T + 12 T^{2} - 5 T^{3} - 181 T^{4} - 840 T^{5} - 1847 T^{6} + 1685 T^{7} + 32436 T^{8} + 140275 T^{9} + 279707 T^{10} + 1823575 T^{11} + 5481684 T^{12} + 3701945 T^{13} - 52752167 T^{14} - 311886120 T^{15} - 873652429 T^{16} - 313742585 T^{17} + 9788768652 T^{18} + 53022496865 T^{19} + 137858491849 T^{20} )$$)
$17$ ($$1 - 17 T^{2} + 4913 T^{6} - 83521 T^{8} + 24137569 T^{12} - 410338673 T^{14} + 118587876497 T^{18} - 2015993900449 T^{20} + 34271896307633 T^{22} - 9904578032905937 T^{26} + 168377826559400929 T^{28} - 48661191875666868481 T^{32} +$$$$82\!\cdots\!77$$$$T^{34} -$$$$23\!\cdots\!53$$$$T^{38} +$$$$40\!\cdots\!01$$$$T^{40}$$)
$19$ ($$( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} + 1008 T^{5} - 4787 T^{6} - 23939 T^{7} + 67014 T^{8} + 521855 T^{9} - 751411 T^{10} + 9915245 T^{11} + 24192054 T^{12} - 164197601 T^{13} - 623846627 T^{14} + 2495907792 T^{15} + 14348993705 T^{16} - 33073254343 T^{17} - 305704134738 T^{18} + 322687697779 T^{19} + 6131066257801 T^{20} )( 1 + 7 T + 30 T^{2} + 77 T^{3} - 31 T^{4} - 1680 T^{5} - 11171 T^{6} - 46277 T^{7} - 111690 T^{8} + 97433 T^{9} + 2804141 T^{10} + 1851227 T^{11} - 40320090 T^{12} - 317413943 T^{13} - 1455815891 T^{14} - 4159846320 T^{15} - 1458422311 T^{16} + 68828123903 T^{17} + 509506891230 T^{18} + 2258813884453 T^{19} + 6131066257801 T^{20} )$$)
$23$ ($$1 - 23 T^{2} + 12167 T^{6} - 279841 T^{8} + 148035889 T^{12} - 3404825447 T^{14} + 1801152661463 T^{18} - 41426511213649 T^{20} + 952809757913927 T^{22} - 504036361936467383 T^{26} + 11592836324538749809 T^{28} -$$$$61\!\cdots\!61$$$$T^{32} +$$$$14\!\cdots\!03$$$$T^{34} -$$$$74\!\cdots\!87$$$$T^{38} +$$$$17\!\cdots\!01$$$$T^{40}$$)
$29$ ($$( 1 + 29 T^{2} + 841 T^{4} )^{10}$$)
$31$ ($$( 1 - 11 T + 90 T^{2} - 649 T^{3} + 4349 T^{4} - 27720 T^{5} + 170101 T^{6} - 1011791 T^{7} + 5856570 T^{8} - 33056749 T^{9} + 182070569 T^{10} - 1024759219 T^{11} + 5628163770 T^{12} - 30142265681 T^{13} + 157091845621 T^{14} - 793600065720 T^{15} + 3859753508669 T^{16} - 17855686558039 T^{17} + 76760193369690 T^{18} - 290835843767381 T^{19} + 819628286980801 T^{20} )( 1 - 4 T - 15 T^{2} + 184 T^{3} - 271 T^{4} - 4620 T^{5} + 26881 T^{6} + 35696 T^{7} - 976095 T^{8} + 2797804 T^{9} + 19067729 T^{10} + 86731924 T^{11} - 938027295 T^{12} + 1063419536 T^{13} + 24825168001 T^{14} - 132266677620 T^{15} - 240513497551 T^{16} + 5062320996424 T^{17} - 12793365561615 T^{18} - 105758488642684 T^{19} + 819628286980801 T^{20} )$$)
$37$ ($$( 1 - 11 T + 84 T^{2} - 517 T^{3} + 2579 T^{4} - 9240 T^{5} + 6217 T^{6} + 273493 T^{7} - 3238452 T^{8} + 25503731 T^{9} - 160718317 T^{10} + 943638047 T^{11} - 4433440788 T^{12} + 13853240929 T^{13} + 11651658937 T^{14} - 640738162680 T^{15} + 6617008408811 T^{16} - 49079780477761 T^{17} + 295048274129364 T^{18} - 1429579137745847 T^{19} + 4808584372417849 T^{20} )( 1 + T - 36 T^{2} - 73 T^{3} + 1259 T^{4} + 3960 T^{5} - 42623 T^{6} - 189143 T^{7} + 1387908 T^{8} + 8386199 T^{9} - 42966397 T^{10} + 310289363 T^{11} + 1900046052 T^{12} - 9580660379 T^{13} - 79882364303 T^{14} + 274602069720 T^{15} + 3230249548931 T^{16} - 6930027030709 T^{17} - 126449260341156 T^{18} + 129961739795077 T^{19} + 4808584372417849 T^{20} )$$)
$41$ ($$1 + 41 T^{2} - 68921 T^{6} - 2825761 T^{8} + 4750104241 T^{12} + 194754273881 T^{14} - 327381934393961 T^{18} - 13422659310152401 T^{20} - 550329031716248441 T^{22} +$$$$92\!\cdots\!21$$$$T^{26} +$$$$37\!\cdots\!61$$$$T^{28} -$$$$63\!\cdots\!41$$$$T^{32} -$$$$26\!\cdots\!81$$$$T^{34} +$$$$43\!\cdots\!61$$$$T^{38} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$43$ ($$( 1 - 5 T - 18 T^{2} + 305 T^{3} - 751 T^{4} - 9360 T^{5} + 79093 T^{6} + 7015 T^{7} - 3436074 T^{8} + 16878725 T^{9} + 63357557 T^{10} + 725785175 T^{11} - 6353300826 T^{12} + 557741605 T^{13} + 270403227493 T^{14} - 1375999026480 T^{15} - 4747343649799 T^{16} + 82904676387635 T^{17} - 210387604996818 T^{18} - 2512963059684215 T^{19} + 21611482313284249 T^{20} )( 1 + 5 T - 18 T^{2} - 305 T^{3} - 751 T^{4} + 9360 T^{5} + 79093 T^{6} - 7015 T^{7} - 3436074 T^{8} - 16878725 T^{9} + 63357557 T^{10} - 725785175 T^{11} - 6353300826 T^{12} - 557741605 T^{13} + 270403227493 T^{14} + 1375999026480 T^{15} - 4747343649799 T^{16} - 82904676387635 T^{17} - 210387604996818 T^{18} + 2512963059684215 T^{19} + 21611482313284249 T^{20} )$$)
$47$ ($$1 - 47 T^{2} + 103823 T^{6} - 4879681 T^{8} + 10779215329 T^{12} - 506623120463 T^{14} + 1119130473102767 T^{18} - 52599132235830049 T^{20} + 2472159215084012303 T^{22} -$$$$54\!\cdots\!27$$$$T^{26} +$$$$25\!\cdots\!69$$$$T^{28} -$$$$56\!\cdots\!21$$$$T^{32} +$$$$26\!\cdots\!87$$$$T^{34} -$$$$58\!\cdots\!83$$$$T^{38} +$$$$27\!\cdots\!01$$$$T^{40}$$)
$53$ ($$( 1 - 53 T^{2} + 2809 T^{4} - 148877 T^{6} + 7890481 T^{8} - 418195493 T^{10} + 22164361129 T^{12} - 1174711139837 T^{14} + 62259690411361 T^{16} - 3299763591802133 T^{18} + 174887470365513049 T^{20} )^{2}$$)
$59$ ($$( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} + 714924299 T^{10} + 42180533641 T^{12} + 2488651484819 T^{14} + 146830437604321 T^{16} + 8662995818654939 T^{18} + 511116753300641401 T^{20} )^{2}$$)
$61$ ($$( 1 - 14 T + 135 T^{2} - 1036 T^{3} + 6269 T^{4} - 24570 T^{5} - 38429 T^{6} + 2036776 T^{7} - 26170695 T^{8} + 242146394 T^{9} - 1793637121 T^{10} + 14770930034 T^{11} - 97381156095 T^{12} + 462309453256 T^{13} - 532081823789 T^{14} - 20751731115570 T^{15} + 322981226869109 T^{16} - 3255881578117756 T^{17} + 25880487254632935 T^{18} - 163718045299677974 T^{19} + 713342911662882601 T^{20} )( 1 - T - 60 T^{2} + 121 T^{3} + 3539 T^{4} - 10920 T^{5} - 204959 T^{6} + 871079 T^{7} + 11631420 T^{8} - 64767239 T^{9} - 644749381 T^{10} - 3950801579 T^{11} + 43280513820 T^{12} + 197718382499 T^{13} - 2837829725519 T^{14} - 9222991606920 T^{15} + 182330604863579 T^{16} + 380271883158541 T^{17} - 11502438779836860 T^{18} - 11694146092834141 T^{19} + 713342911662882601 T^{20} )$$)
$67$ ($$1 - 5 T - 42 T^{2} + 545 T^{3} + 89 T^{4} - 36960 T^{5} + 178837 T^{6} + 1582135 T^{7} - 19892754 T^{8} - 6539275 T^{9} + 1365510893 T^{10} - 438131425 T^{11} - 89298572706 T^{12} + 475847669005 T^{13} + 3603766026277 T^{14} - 49900623954720 T^{15} + 8050796013041 T^{16} + 3303087824901035 T^{17} - 17054842457378922 T^{18} - 136032671981474735 T^{19} + 1822837804551761449 T^{20}$$)
$71$ ($$1 - 71 T^{2} + 357911 T^{6} - 25411681 T^{8} + 128100283921 T^{12} - 9095120158391 T^{14} + 45848500718449031 T^{18} - 3255243551009881201 T^{20} +$$$$23\!\cdots\!71$$$$T^{22} -$$$$11\!\cdots\!11$$$$T^{26} +$$$$82\!\cdots\!81$$$$T^{28} -$$$$41\!\cdots\!21$$$$T^{32} +$$$$29\!\cdots\!91$$$$T^{34} -$$$$14\!\cdots\!31$$$$T^{38} +$$$$10\!\cdots\!01$$$$T^{40}$$)
$73$ ($$( 1 - 7 T + 73 T^{2} )^{10}( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 32130 T^{5} + 158383 T^{6} + 3929320 T^{7} + 27731241 T^{8} - 9527950 T^{9} - 2119660093 T^{10} - 695540350 T^{11} + 147779783289 T^{12} + 1528572278440 T^{13} + 4497798604303 T^{14} - 66607790283090 T^{15} - 994417200945019 T^{16} - 5081803318784620 T^{17} + 21774422481140187 T^{18} + 588715867082679130 T^{19} + 4297625829703557649 T^{20} )$$)
$79$ ($$( 1 - 13 T + 79 T^{2} )^{10}( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} + 4363376 T^{7} - 18170271 T^{8} - 272025620 T^{9} + 2523553889 T^{10} - 21490023980 T^{11} - 113400661311 T^{12} + 2151314539664 T^{13} + 353394084913 T^{14} - 171367424973108 T^{15} + 657551567184305 T^{16} + 10907820304138312 T^{17} - 95577855024113343 T^{18} - 479406383930473276 T^{19} + 9468276082626847201 T^{20} )$$)
$83$ ($$1 - 83 T^{2} + 571787 T^{6} - 47458321 T^{8} + 326940373369 T^{12} - 27136050989627 T^{14} + 186940255267540403 T^{18} - 15516041187205853449 T^{20} +$$$$12\!\cdots\!67$$$$T^{22} -$$$$88\!\cdots\!63$$$$T^{26} +$$$$73\!\cdots\!29$$$$T^{28} -$$$$50\!\cdots\!81$$$$T^{32} +$$$$42\!\cdots\!23$$$$T^{34} -$$$$29\!\cdots\!47$$$$T^{38} +$$$$24\!\cdots\!01$$$$T^{40}$$)
$89$ ($$( 1 + 89 T^{2} + 7921 T^{4} + 704969 T^{6} + 62742241 T^{8} + 5584059449 T^{10} + 496981290961 T^{12} + 44231334895529 T^{14} + 3936588805702081 T^{16} + 350356403707485209 T^{18} + 31181719929966183601 T^{20} )^{2}$$)
$97$ ($$( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 526792 T^{7} + 84766473 T^{8} - 1237829446 T^{9} + 9107264363 T^{10} - 120069456262 T^{11} + 797567744457 T^{12} + 480788835016 T^{13} - 84095114902553 T^{14} + 1130695091639190 T^{15} - 7672505137401019 T^{16} - 2262351965387164 T^{17} + 775905925843319139 T^{18} - 10643234821163913038 T^{19} + 73742412689492826049 T^{20} )( 1 + 5 T - 72 T^{2} - 845 T^{3} + 2759 T^{4} + 95760 T^{5} + 211177 T^{6} - 8232835 T^{7} - 61648344 T^{8} + 490343275 T^{9} + 8431605743 T^{10} + 47563297675 T^{11} - 580049268696 T^{12} - 7513886217955 T^{13} + 18695347973737 T^{14} + 822323703010320 T^{15} + 2298169761599111 T^{16} - 68274550384005485 T^{17} - 564295218795141192 T^{18} + 3801155293272826085 T^{19} + 73742412689492826049 T^{20} )$$)