Properties

Label 2.42.a
Level 2
Weight 42
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 10
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(10\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{42}(\Gamma_0(2))\).

Total New Old
Modular forms 11 3 8
Cusp forms 9 3 6
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim.
\(+\)\(1\)
\(-\)\(2\)

Trace form

\( 3q + 1048576q^{2} + 13906864044q^{3} + 3298534883328q^{4} + 49094989194930q^{5} + 4005383123238912q^{6} + 97714356676438488q^{7} + 1152921504606846976q^{8} + 30936992086866501639q^{9} + O(q^{10}) \) \( 3q + 1048576q^{2} + 13906864044q^{3} + 3298534883328q^{4} + 49094989194930q^{5} + 4005383123238912q^{6} + 97714356676438488q^{7} + 1152921504606846976q^{8} + 30936992086866501639q^{9} + 153200497216695828480q^{10} + 3264403333766748924036q^{11} + 15290758722277966086144q^{12} + 161882807152022753465994q^{13} + 352845035543974457114624q^{14} - 3539918996489522338057080q^{15} + 3626777458843887524118528q^{16} - 37011726883142944282175562q^{17} + 55583825877405288820113408q^{18} + 274230100811445606374810460q^{19} + 53980511485362616066375680q^{20} + 2665691359852998167875660896q^{21} - 3191041552072698539506925568q^{22} + 1420347666919737243157853064q^{23} + 4403965317698934946463219712q^{24} + 54412888097875513082262174525q^{25} + 193675559200226950852527521792q^{26} + 293544196044657269619833966520q^{27} + 107438071366395535287216242688q^{28} - 1238427538159622404128372563910q^{29} - 3198844223373791442948618977280q^{30} - 565073962679875259305481759904q^{31} + 1267650600228229401496703205376q^{32} + 2233551046463051372662468702608q^{33} + 17234179508571566375244140642304q^{34} - 97248489612709189284759665330160q^{35} + 34015582527923818408303461924864q^{36} - 101592630862331863544766032309742q^{37} + 144996736400908851639068697559040q^{38} + 640160633638918340067446036904168q^{39} + 168445728070821787776313699860480q^{40} + 768192138006768072331689412248126q^{41} + 4057996358704299679930133010972672q^{42} - 9163911628959853913209536056311836q^{43} + 3589249423227279134970318931623936q^{44} - 30457369032504081176750813707625910q^{45} + 29811534820383674194267711022825472q^{46} - 12260960588935821767780288588908272q^{47} + 16812367012661916406114713279135744q^{48} - 25596406892660163029921184065783829q^{49} + 147489600972651489691044136380006400q^{50} - 182678347564418655807945350847940584q^{51} + 177992028800668832354384608801849344q^{52} - 625375082980902027370788377449928766q^{53} + 810306378590012747923651603590021120q^{54} + 550426134204771570148170929066219160q^{55} + 387957219383635932970666138454196224q^{56} - 442953519980269300084782666393665040q^{57} - 1585298523526682507321181181449338880q^{58} + 177563983295764361517342239221260180q^{59} - 3892182098025379135645853384001454080q^{60} - 5463201135704660666208213208341875814q^{61} - 7012792278966542721658673238982524928q^{62} + 26310155382295499484506243647288915704q^{63} + 3987683987354747618711421180841033728q^{64} + 12473259455640304370441775300271028220q^{65} - 31015853645488527189477610831170502656q^{66} + 71683283838926171050853516149998220908q^{67} - 40694824072085237602584124037227610112q^{68} - 11480840010959325121257435993672114912q^{69} - 114117111792639301686346513623086530560q^{70} + 68110973123647230466528954560382296216q^{71} + 61115062868483640529874307679002820608q^{72} - 294908420905755625150102017496226933346q^{73} + 358933845813325666940616726553416433664q^{74} - 149075468778282886564381918940741001900q^{75} + 301519184528369137116851211238071336960q^{76} - 796259039077942095596584187532209589984q^{77} + 791944053109205052642588176234877812736q^{78} - 1467129308592083502686260888480869532560q^{79} + 59352200051452113588758453977538887680q^{80} + 1825606948109165162789233850315793931563q^{81} + 1858246690007324705570778546446744420352q^{82} + 665984279782956439537682608316709433884q^{83} + 2930958646220388991634913623732750647296q^{84} + 907701952915589923696197022820240971140q^{85} - 3069134989805620595871083685892449107968q^{86} - 17068192400013102750887440771225829791320q^{87} - 3508587291220306237862197575696953376768q^{88} + 14607841393604595000567679910075985192270q^{89} - 33059449148344087288488905373318654197760q^{90} + 18031837917819409202012303003569888867536q^{91} + 1561688775262764164166701740916069105664q^{92} + 55858944591888014581432169202537944899968q^{93} - 40445670686694062392590229249526392160256q^{94} + 90543084377116752054454370649641547666600q^{95} + 4842211075132204945687306119655049920512q^{96} - 95565478890036662057547911249774079434202q^{97} + 36731367279780021226536126103678601920512q^{98} - 158014553036426914776149850133844786493132q^{99} + O(q^{100}) \)

Decomposition of \(S_{42}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.42.a.a \(1\) \(21.294\) \(\Q\) None \(-1048576\) \(5043516516\) \(-4\!\cdots\!50\) \(-1\!\cdots\!68\) \(+\) \(q-2^{20}q^{2}+5043516516q^{3}+2^{40}q^{4}+\cdots\)
2.42.a.b \(2\) \(21.294\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(2097152\) \(8863347528\) \(97\!\cdots\!80\) \(21\!\cdots\!56\) \(-\) \(q+2^{20}q^{2}+(4431673764-\beta )q^{3}+\cdots\)

Decomposition of \(S_{42}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{42}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{42}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 1048576 T \))(\( ( 1 - 1048576 T )^{2} \))
$3$ (\( 1 - 5043516516 T + 36472996377170786403 T^{2} \))(\( 1 - 8863347528 T + 54765996269786386902 T^{2} - \)\(32\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!09\)\( T^{4} \))
$5$ (\( 1 + 48504195130650 T + \)\(45\!\cdots\!25\)\( T^{2} \))(\( 1 - 97599184325580 T + \)\(14\!\cdots\!50\)\( T^{2} - \)\(44\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!25\)\( T^{4} \))
$7$ (\( 1 + 119392445696650168 T + \)\(44\!\cdots\!07\)\( T^{2} \))(\( 1 - 217106802373088656 T + \)\(65\!\cdots\!98\)\( T^{2} - \)\(96\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!49\)\( T^{4} \))
$11$ (\( 1 - \)\(31\!\cdots\!52\)\( T + \)\(49\!\cdots\!11\)\( T^{2} \))(\( 1 - \)\(11\!\cdots\!84\)\( T + \)\(86\!\cdots\!86\)\( T^{2} - \)\(55\!\cdots\!24\)\( T^{3} + \)\(24\!\cdots\!21\)\( T^{4} \))
$13$ (\( 1 + \)\(11\!\cdots\!74\)\( T + \)\(46\!\cdots\!13\)\( T^{2} \))(\( 1 - \)\(17\!\cdots\!68\)\( T + \)\(16\!\cdots\!82\)\( T^{2} - \)\(81\!\cdots\!84\)\( T^{3} + \)\(22\!\cdots\!69\)\( T^{4} \))
$17$ (\( 1 + \)\(26\!\cdots\!58\)\( T + \)\(28\!\cdots\!17\)\( T^{2} \))(\( 1 + \)\(10\!\cdots\!04\)\( T + \)\(58\!\cdots\!38\)\( T^{2} + \)\(28\!\cdots\!68\)\( T^{3} + \)\(78\!\cdots\!89\)\( T^{4} \))
$19$ (\( 1 - \)\(67\!\cdots\!60\)\( T + \)\(26\!\cdots\!19\)\( T^{2} \))(\( 1 - \)\(20\!\cdots\!00\)\( T + \)\(45\!\cdots\!38\)\( T^{2} - \)\(55\!\cdots\!00\)\( T^{3} + \)\(72\!\cdots\!61\)\( T^{4} \))
$23$ (\( 1 + \)\(13\!\cdots\!04\)\( T + \)\(67\!\cdots\!23\)\( T^{2} \))(\( 1 - \)\(14\!\cdots\!68\)\( T + \)\(19\!\cdots\!02\)\( T^{2} - \)\(10\!\cdots\!64\)\( T^{3} + \)\(45\!\cdots\!29\)\( T^{4} \))
$29$ (\( 1 - \)\(13\!\cdots\!10\)\( T + \)\(90\!\cdots\!29\)\( T^{2} \))(\( 1 + \)\(13\!\cdots\!20\)\( T + \)\(13\!\cdots\!58\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(82\!\cdots\!41\)\( T^{4} \))
$31$ (\( 1 - \)\(30\!\cdots\!12\)\( T + \)\(13\!\cdots\!31\)\( T^{2} \))(\( 1 + \)\(36\!\cdots\!16\)\( T + \)\(10\!\cdots\!26\)\( T^{2} + \)\(50\!\cdots\!96\)\( T^{3} + \)\(19\!\cdots\!61\)\( T^{4} \))
$37$ (\( 1 + \)\(22\!\cdots\!78\)\( T + \)\(19\!\cdots\!37\)\( T^{2} \))(\( 1 - \)\(12\!\cdots\!36\)\( T + \)\(42\!\cdots\!98\)\( T^{2} - \)\(23\!\cdots\!32\)\( T^{3} + \)\(39\!\cdots\!69\)\( T^{4} \))
$41$ (\( 1 + \)\(50\!\cdots\!38\)\( T + \)\(13\!\cdots\!41\)\( T^{2} \))(\( 1 - \)\(12\!\cdots\!64\)\( T + \)\(75\!\cdots\!06\)\( T^{2} - \)\(16\!\cdots\!24\)\( T^{3} + \)\(17\!\cdots\!81\)\( T^{4} \))
$43$ (\( 1 + \)\(31\!\cdots\!84\)\( T + \)\(93\!\cdots\!43\)\( T^{2} \))(\( 1 + \)\(60\!\cdots\!52\)\( T + \)\(20\!\cdots\!62\)\( T^{2} + \)\(56\!\cdots\!36\)\( T^{3} + \)\(87\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 - \)\(13\!\cdots\!92\)\( T + \)\(35\!\cdots\!47\)\( T^{2} \))(\( 1 + \)\(25\!\cdots\!64\)\( T + \)\(84\!\cdots\!18\)\( T^{2} + \)\(91\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \))
$53$ (\( 1 + \)\(32\!\cdots\!14\)\( T + \)\(49\!\cdots\!53\)\( T^{2} \))(\( 1 + \)\(30\!\cdots\!52\)\( T + \)\(11\!\cdots\!82\)\( T^{2} + \)\(14\!\cdots\!56\)\( T^{3} + \)\(24\!\cdots\!09\)\( T^{4} \))
$59$ (\( 1 - \)\(34\!\cdots\!20\)\( T + \)\(40\!\cdots\!59\)\( T^{2} \))(\( 1 + \)\(32\!\cdots\!40\)\( T + \)\(78\!\cdots\!18\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \))
$61$ (\( 1 + \)\(97\!\cdots\!78\)\( T + \)\(15\!\cdots\!61\)\( T^{2} \))(\( 1 + \)\(44\!\cdots\!36\)\( T + \)\(17\!\cdots\!46\)\( T^{2} + \)\(70\!\cdots\!96\)\( T^{3} + \)\(24\!\cdots\!21\)\( T^{4} \))
$67$ (\( 1 - \)\(16\!\cdots\!52\)\( T + \)\(73\!\cdots\!67\)\( T^{2} \))(\( 1 - \)\(55\!\cdots\!56\)\( T + \)\(15\!\cdots\!18\)\( T^{2} - \)\(40\!\cdots\!52\)\( T^{3} + \)\(54\!\cdots\!89\)\( T^{4} \))
$71$ (\( 1 - \)\(11\!\cdots\!32\)\( T + \)\(79\!\cdots\!71\)\( T^{2} \))(\( 1 + \)\(48\!\cdots\!16\)\( T + \)\(10\!\cdots\!06\)\( T^{2} + \)\(38\!\cdots\!36\)\( T^{3} + \)\(63\!\cdots\!41\)\( T^{4} \))
$73$ (\( 1 - \)\(19\!\cdots\!66\)\( T + \)\(24\!\cdots\!73\)\( T^{2} \))(\( 1 + \)\(48\!\cdots\!12\)\( T + \)\(10\!\cdots\!82\)\( T^{2} + \)\(12\!\cdots\!76\)\( T^{3} + \)\(62\!\cdots\!29\)\( T^{4} \))
$79$ (\( 1 + \)\(56\!\cdots\!80\)\( T + \)\(63\!\cdots\!79\)\( T^{2} \))(\( 1 + \)\(90\!\cdots\!80\)\( T + \)\(14\!\cdots\!58\)\( T^{2} + \)\(57\!\cdots\!20\)\( T^{3} + \)\(40\!\cdots\!41\)\( T^{4} \))
$83$ (\( 1 + \)\(60\!\cdots\!84\)\( T + \)\(48\!\cdots\!83\)\( T^{2} \))(\( 1 - \)\(12\!\cdots\!68\)\( T + \)\(95\!\cdots\!22\)\( T^{2} - \)\(61\!\cdots\!44\)\( T^{3} + \)\(23\!\cdots\!89\)\( T^{4} \))
$89$ (\( 1 - \)\(11\!\cdots\!90\)\( T + \)\(84\!\cdots\!89\)\( T^{2} \))(\( 1 - \)\(26\!\cdots\!80\)\( T + \)\(16\!\cdots\!78\)\( T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(70\!\cdots\!21\)\( T^{4} \))
$97$ (\( 1 + \)\(63\!\cdots\!98\)\( T + \)\(28\!\cdots\!97\)\( T^{2} \))(\( 1 + \)\(31\!\cdots\!04\)\( T + \)\(59\!\cdots\!98\)\( T^{2} + \)\(91\!\cdots\!88\)\( T^{3} + \)\(82\!\cdots\!09\)\( T^{4} \))
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