Properties

Label 2.38.a
Level 2
Weight 38
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newforms 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: \(\Q\)
Newforms: \( 2 \)
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.

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

Trace form

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

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

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

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