Properties

Label 2.38.a
Level 2
Weight 38
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 2
Sturm bound 9
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(9\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_0(2))\).

Total New Old
Modular forms 10 4 6
Cusp forms 8 4 4
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.
\(+\)\(2\)
\(-\)\(2\)

Trace form

\( 4q - 78615600q^{3} + 274877906944q^{4} - 9324929696040q^{5} - 242419765346304q^{6} - 406504794575200q^{7} + 1870022118371332212q^{9} + O(q^{10}) \) \( 4q - 78615600q^{3} + 274877906944q^{4} - 9324929696040q^{5} - 242419765346304q^{6} - 406504794575200q^{7} + 1870022118371332212q^{9} + 4637361809664245760q^{10} + 29662824384101405808q^{11} - 5402422895286681600q^{12} - 568098141485386712200q^{13} - 1735092632420276502528q^{14} - 5680830860419105827360q^{15} + 18889465931478580854784q^{16} + 166092345674546990416200q^{17} - 32178957657687510220800q^{18} - 1187331645010775299663600q^{19} - 640804289311856331325440q^{20} + 1037957907379026036306048q^{21} + 5677270187643854074675200q^{22} + 28437628681165689800978400q^{23} - 16658959425061916711583744q^{24} - 157609474541449481232920900q^{25} + 70080651700984140986843136q^{26} - 35249328737993312469727200q^{27} - 27934796773882915402547200q^{28} + 1019898325065576960137855160q^{29} + 3272502404582066951647395840q^{30} - 5044931989506182998210198912q^{31} - 39542886088641287642965550400q^{33} + 43055203916586795813549637632q^{34} + 27257313622154294863345452480q^{35} + 128506941459224202151861420032q^{36} - 244342842739971685438525445800q^{37} + 476648489279064256600552243200q^{38} - 1269989421300575130722498191776q^{39} + 318677076995636996475306639360q^{40} - 630459916390438110955595460312q^{41} + 4785834047596026963647948390400q^{42} - 2142445644790311747845082902800q^{43} + 2038413770187310084687751282688q^{44} - 9768698992687879661455893987720q^{45} + 4845217992050117772649898704896q^{46} - 9607700580967414160668681819200q^{47} - 371251674470686880261839257600q^{48} + 30869913774611510149083682078308q^{49} - 24091460094454960115060991590400q^{50} + 67617102766005770789821523069088q^{51} - 39039407017569868652991427379200q^{52} + 90754574359927436497098574939800q^{53} - 420782302071524546329248226344960q^{54} + 215538537845952176283065263036320q^{55} - 119234657788410190490160341188608q^{56} + 369450367774050210270542664532800q^{57} - 348303278025997360128562141593600q^{58} + 1331013620448365739742226010780720q^{59} - 390383724153721606113187552296960q^{60} - 353989065654942037244215332857032q^{61} + 1576616316526411494882082711142400q^{62} - 8305081588797882869072135626668000q^{63} + 1298074214633706907132624082305024q^{64} - 339385835151877028329216289910960q^{65} + 13133254314358366172318608121659392q^{66} + 3665711708374496818987067079688400q^{67} + 11413779084609702135244870857523200q^{68} - 28442084615712409794997576539217536q^{69} + 1412299692836850246455802074234880q^{70} - 22391766667452507232398137896774752q^{71} - 2211321132146185910176027823308800q^{72} - 4871224337120534176819477926362200q^{73} - 3248013182587779255808754322505728q^{74} + 127042227529892469153602963464004400q^{75} - 81592809357234583674556188826009600q^{76} + 257576323431634470706604404682275200q^{77} - 248518528715476089149986488660787200q^{78} - 126995215811604848308995511956468160q^{79} - 44035735451695124609492882124963840q^{80} - 36659914971008034704581161546282396q^{81} - 416879092969140023747974699312742400q^{82} + 823463145952413972731717377383018000q^{83} + 71327924269080222436271756912099328q^{84} - 178347067147542690901468739681050320q^{85} + 1111797560771553491927563570539134976q^{86} - 1454165853712763693375774312151818400q^{87} + 390139036583778184738021213156147200q^{88} - 75158905971090583448683889815094040q^{89} + 2348071989038475617723982685069639680q^{90} - 6110718130363439317665023326715892032q^{91} + 1954218962582371981639727628838502400q^{92} - 5503333341909228136224718980327513600q^{93} + 3360802553789552011928088899802365952q^{94} + 13806544056226194570817284809773826400q^{95} - 1144794974656510320821248717523779584q^{96} - 613632711409297820260857919695739000q^{97} + 120119620461738472236394033761484800q^{98} - 11995052585300668254380468403393277776q^{99} + O(q^{100}) \)

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.38.a.a \(2\) \(17.343\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-524288\) \(423071208\) \(-1\!\cdots\!40\) \(31\!\cdots\!56\) \(+\) \(q-2^{18}q^{2}+(211535604-\beta )q^{3}+2^{36}q^{4}+\cdots\)
2.38.a.b \(2\) \(17.343\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(524288\) \(-501686808\) \(41\!\cdots\!00\) \(-3\!\cdots\!56\) \(-\) \(q+2^{18}q^{2}+(-250843404-\beta )q^{3}+\cdots\)

Decomposition of \(S_{38}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{38}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{38}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 262144 T )^{2} \))(\( ( 1 - 262144 T )^{2} \))
$3$ (\( 1 - 423071208 T + 41584754370593142 T^{2} - \)\(19\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!69\)\( T^{4} \))(\( 1 + 501686808 T + 139311448405639542 T^{2} + \)\(22\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!69\)\( T^{4} \))
$5$ (\( 1 + 13507530555540 T + \)\(18\!\cdots\!50\)\( T^{2} + \)\(98\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!25\)\( T^{4} \))(\( 1 - 4182600859500 T + \)\(14\!\cdots\!50\)\( T^{2} - \)\(30\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!25\)\( T^{4} \))
$7$ (\( 1 - 3106174162962256 T + \)\(15\!\cdots\!98\)\( T^{2} - \)\(57\!\cdots\!92\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \))(\( 1 + 3512678957537456 T + \)\(16\!\cdots\!98\)\( T^{2} + \)\(65\!\cdots\!92\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \))
$11$ (\( 1 - 4002878661541032504 T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(13\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - 25659945722560373304 T + \)\(80\!\cdots\!46\)\( T^{2} - \)\(87\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \))
$13$ (\( 1 + \)\(41\!\cdots\!72\)\( T + \)\(37\!\cdots\!62\)\( T^{2} + \)\(68\!\cdots\!76\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \))(\( 1 + \)\(15\!\cdots\!28\)\( T - \)\(64\!\cdots\!38\)\( T^{2} + \)\(24\!\cdots\!24\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \))
$17$ (\( 1 - \)\(92\!\cdots\!36\)\( T + \)\(18\!\cdots\!78\)\( T^{2} - \)\(31\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(16\!\cdots\!64\)\( T + \)\(13\!\cdots\!78\)\( T^{2} - \)\(55\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))
$19$ (\( 1 + \)\(15\!\cdots\!00\)\( T + \)\(97\!\cdots\!78\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(31\!\cdots\!00\)\( T + \)\(18\!\cdots\!78\)\( T^{2} - \)\(64\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} \))
$23$ (\( 1 - \)\(49\!\cdots\!08\)\( T + \)\(32\!\cdots\!22\)\( T^{2} - \)\(12\!\cdots\!24\)\( T^{3} + \)\(58\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(23\!\cdots\!92\)\( T + \)\(62\!\cdots\!22\)\( T^{2} - \)\(56\!\cdots\!76\)\( T^{3} + \)\(58\!\cdots\!09\)\( T^{4} \))
$29$ (\( 1 - \)\(11\!\cdots\!80\)\( T + \)\(22\!\cdots\!18\)\( T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \))(\( 1 + \)\(15\!\cdots\!20\)\( T + \)\(25\!\cdots\!18\)\( T^{2} + \)\(19\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \))
$31$ (\( 1 + \)\(55\!\cdots\!56\)\( T + \)\(37\!\cdots\!06\)\( T^{2} + \)\(83\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(48\!\cdots\!44\)\( T + \)\(20\!\cdots\!06\)\( T^{2} - \)\(73\!\cdots\!84\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} \))
$37$ (\( 1 + \)\(11\!\cdots\!44\)\( T + \)\(23\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))(\( 1 + \)\(12\!\cdots\!56\)\( T + \)\(21\!\cdots\!18\)\( T^{2} + \)\(13\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))
$41$ (\( 1 - \)\(47\!\cdots\!44\)\( T + \)\(89\!\cdots\!46\)\( T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \))(\( 1 + \)\(11\!\cdots\!56\)\( T + \)\(10\!\cdots\!46\)\( T^{2} + \)\(52\!\cdots\!36\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \))
$43$ (\( 1 + \)\(31\!\cdots\!52\)\( T + \)\(63\!\cdots\!62\)\( T^{2} + \)\(87\!\cdots\!36\)\( T^{3} + \)\(75\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(10\!\cdots\!52\)\( T + \)\(41\!\cdots\!62\)\( T^{2} - \)\(28\!\cdots\!36\)\( T^{3} + \)\(75\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 + \)\(11\!\cdots\!04\)\( T + \)\(12\!\cdots\!78\)\( T^{2} + \)\(82\!\cdots\!48\)\( T^{3} + \)\(54\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(16\!\cdots\!04\)\( T + \)\(13\!\cdots\!78\)\( T^{2} - \)\(11\!\cdots\!48\)\( T^{3} + \)\(54\!\cdots\!69\)\( T^{4} \))
$53$ (\( 1 - \)\(53\!\cdots\!28\)\( T + \)\(13\!\cdots\!22\)\( T^{2} - \)\(33\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(37\!\cdots\!72\)\( T + \)\(21\!\cdots\!22\)\( T^{2} - \)\(23\!\cdots\!36\)\( T^{3} + \)\(39\!\cdots\!69\)\( T^{4} \))
$59$ (\( 1 - \)\(58\!\cdots\!60\)\( T + \)\(64\!\cdots\!38\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(12\!\cdots\!60\)\( T + \)\(10\!\cdots\!38\)\( T^{2} - \)\(42\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \))
$61$ (\( 1 - \)\(94\!\cdots\!84\)\( T + \)\(19\!\cdots\!06\)\( T^{2} - \)\(10\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!41\)\( T^{4} \))(\( 1 + \)\(13\!\cdots\!16\)\( T + \)\(23\!\cdots\!06\)\( T^{2} + \)\(14\!\cdots\!36\)\( T^{3} + \)\(13\!\cdots\!41\)\( T^{4} \))
$67$ (\( 1 - \)\(32\!\cdots\!96\)\( T + \)\(76\!\cdots\!58\)\( T^{2} - \)\(12\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(38\!\cdots\!04\)\( T + \)\(31\!\cdots\!58\)\( T^{2} - \)\(14\!\cdots\!08\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \))
$71$ (\( 1 + \)\(36\!\cdots\!76\)\( T + \)\(84\!\cdots\!26\)\( T^{2} + \)\(11\!\cdots\!16\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(14\!\cdots\!24\)\( T + \)\(67\!\cdots\!26\)\( T^{2} - \)\(44\!\cdots\!84\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} \))
$73$ (\( 1 + \)\(10\!\cdots\!72\)\( T + \)\(44\!\cdots\!02\)\( T^{2} + \)\(94\!\cdots\!16\)\( T^{3} + \)\(76\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(58\!\cdots\!72\)\( T + \)\(13\!\cdots\!02\)\( T^{2} - \)\(51\!\cdots\!16\)\( T^{3} + \)\(76\!\cdots\!09\)\( T^{4} \))
$79$ (\( 1 - \)\(41\!\cdots\!20\)\( T + \)\(22\!\cdots\!18\)\( T^{2} - \)\(68\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \))(\( 1 + \)\(16\!\cdots\!80\)\( T + \)\(39\!\cdots\!18\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \))
$83$ (\( 1 - \)\(17\!\cdots\!88\)\( T + \)\(18\!\cdots\!82\)\( T^{2} - \)\(17\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(65\!\cdots\!12\)\( T + \)\(22\!\cdots\!82\)\( T^{2} - \)\(65\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \))
$89$ (\( 1 + \)\(77\!\cdots\!20\)\( T + \)\(24\!\cdots\!58\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(69\!\cdots\!80\)\( T + \)\(91\!\cdots\!58\)\( T^{2} - \)\(93\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!41\)\( T^{4} \))
$97$ (\( 1 + \)\(65\!\cdots\!44\)\( T + \)\(45\!\cdots\!58\)\( T^{2} + \)\(21\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(58\!\cdots\!44\)\( T + \)\(69\!\cdots\!58\)\( T^{2} - \)\(19\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \))
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