Properties

Label 13.12.a
Level 13
Weight 12
Character orbit a
Rep. character \(\chi_{13}(1,\cdot)\)
Character field \(\Q\)
Dimension 11
Newforms 2
Sturm bound 14
Trace bound 1

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 13.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(14\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(13))\).

Total New Old
Modular forms 15 11 4
Cusp forms 13 11 2
Eisenstein series 2 0 2

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

\(13\)Dim.
\(+\)\(6\)
\(-\)\(5\)

Trace form

\(11q \) \(\mathstrut +\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 18302q^{4} \) \(\mathstrut +\mathstrut 770q^{5} \) \(\mathstrut -\mathstrut 22356q^{6} \) \(\mathstrut -\mathstrut 40472q^{7} \) \(\mathstrut -\mathstrut 41544q^{8} \) \(\mathstrut +\mathstrut 1048863q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut +\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 18302q^{4} \) \(\mathstrut +\mathstrut 770q^{5} \) \(\mathstrut -\mathstrut 22356q^{6} \) \(\mathstrut -\mathstrut 40472q^{7} \) \(\mathstrut -\mathstrut 41544q^{8} \) \(\mathstrut +\mathstrut 1048863q^{9} \) \(\mathstrut -\mathstrut 6558q^{10} \) \(\mathstrut -\mathstrut 1443852q^{11} \) \(\mathstrut +\mathstrut 278982q^{12} \) \(\mathstrut -\mathstrut 371293q^{13} \) \(\mathstrut +\mathstrut 3197926q^{14} \) \(\mathstrut -\mathstrut 9404984q^{15} \) \(\mathstrut +\mathstrut 15961490q^{16} \) \(\mathstrut +\mathstrut 12448086q^{17} \) \(\mathstrut -\mathstrut 24241138q^{18} \) \(\mathstrut -\mathstrut 4602356q^{19} \) \(\mathstrut +\mathstrut 75833300q^{20} \) \(\mathstrut -\mathstrut 37638800q^{21} \) \(\mathstrut -\mathstrut 16924608q^{22} \) \(\mathstrut -\mathstrut 1353800q^{23} \) \(\mathstrut -\mathstrut 154650708q^{24} \) \(\mathstrut +\mathstrut 144966677q^{25} \) \(\mathstrut -\mathstrut 35644128q^{26} \) \(\mathstrut +\mathstrut 189892144q^{27} \) \(\mathstrut -\mathstrut 283857056q^{28} \) \(\mathstrut -\mathstrut 224809782q^{29} \) \(\mathstrut -\mathstrut 274325902q^{30} \) \(\mathstrut -\mathstrut 172350848q^{31} \) \(\mathstrut +\mathstrut 1139629712q^{32} \) \(\mathstrut -\mathstrut 818480720q^{33} \) \(\mathstrut -\mathstrut 755306916q^{34} \) \(\mathstrut +\mathstrut 622506648q^{35} \) \(\mathstrut +\mathstrut 3169855192q^{36} \) \(\mathstrut -\mathstrut 369863390q^{37} \) \(\mathstrut +\mathstrut 1769119492q^{38} \) \(\mathstrut -\mathstrut 360896796q^{39} \) \(\mathstrut -\mathstrut 873537294q^{40} \) \(\mathstrut -\mathstrut 2578116626q^{41} \) \(\mathstrut -\mathstrut 7013954238q^{42} \) \(\mathstrut -\mathstrut 794814044q^{43} \) \(\mathstrut +\mathstrut 830321236q^{44} \) \(\mathstrut +\mathstrut 8689697546q^{45} \) \(\mathstrut +\mathstrut 1817442660q^{46} \) \(\mathstrut -\mathstrut 8015746656q^{47} \) \(\mathstrut +\mathstrut 6283092766q^{48} \) \(\mathstrut +\mathstrut 9258535155q^{49} \) \(\mathstrut -\mathstrut 10817398534q^{50} \) \(\mathstrut +\mathstrut 13339519280q^{51} \) \(\mathstrut -\mathstrut 4572844588q^{52} \) \(\mathstrut +\mathstrut 4511046890q^{53} \) \(\mathstrut -\mathstrut 26339445228q^{54} \) \(\mathstrut -\mathstrut 4629796776q^{55} \) \(\mathstrut +\mathstrut 4056068978q^{56} \) \(\mathstrut -\mathstrut 14931881088q^{57} \) \(\mathstrut +\mathstrut 22177924776q^{58} \) \(\mathstrut +\mathstrut 3853933924q^{59} \) \(\mathstrut -\mathstrut 2270304432q^{60} \) \(\mathstrut +\mathstrut 2078626882q^{61} \) \(\mathstrut -\mathstrut 561722056q^{62} \) \(\mathstrut +\mathstrut 361546328q^{63} \) \(\mathstrut -\mathstrut 15176929966q^{64} \) \(\mathstrut -\mathstrut 2173549222q^{65} \) \(\mathstrut +\mathstrut 24816629664q^{66} \) \(\mathstrut +\mathstrut 38915233708q^{67} \) \(\mathstrut -\mathstrut 14369028870q^{68} \) \(\mathstrut +\mathstrut 387900928q^{69} \) \(\mathstrut +\mathstrut 25212520260q^{70} \) \(\mathstrut -\mathstrut 58066820168q^{71} \) \(\mathstrut -\mathstrut 16884990288q^{72} \) \(\mathstrut -\mathstrut 6608698322q^{73} \) \(\mathstrut -\mathstrut 19505266742q^{74} \) \(\mathstrut -\mathstrut 14978188484q^{75} \) \(\mathstrut +\mathstrut 146173699240q^{76} \) \(\mathstrut -\mathstrut 20579410408q^{77} \) \(\mathstrut +\mathstrut 34618616734q^{78} \) \(\mathstrut -\mathstrut 97023255680q^{79} \) \(\mathstrut +\mathstrut 69870018508q^{80} \) \(\mathstrut +\mathstrut 134641759827q^{81} \) \(\mathstrut -\mathstrut 48818256084q^{82} \) \(\mathstrut +\mathstrut 2601089100q^{83} \) \(\mathstrut -\mathstrut 303502689380q^{84} \) \(\mathstrut -\mathstrut 49120712172q^{85} \) \(\mathstrut +\mathstrut 109680301404q^{86} \) \(\mathstrut +\mathstrut 5242109864q^{87} \) \(\mathstrut -\mathstrut 163780604808q^{88} \) \(\mathstrut +\mathstrut 86246478958q^{89} \) \(\mathstrut -\mathstrut 476289754080q^{90} \) \(\mathstrut -\mathstrut 11925931160q^{91} \) \(\mathstrut +\mathstrut 355165655260q^{92} \) \(\mathstrut +\mathstrut 332006816880q^{93} \) \(\mathstrut +\mathstrut 245017459950q^{94} \) \(\mathstrut +\mathstrut 133983808456q^{95} \) \(\mathstrut -\mathstrut 184167544724q^{96} \) \(\mathstrut -\mathstrut 270561627674q^{97} \) \(\mathstrut +\mathstrut 317151283102q^{98} \) \(\mathstrut -\mathstrut 271122388524q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(13))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 13
13.12.a.a \(5\) \(9.988\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-41\) \(-496\) \(-2542\) \(-36296\) \(-\) \(q+(-8-\beta _{1})q^{2}+(-99-2\beta _{1}-\beta _{4})q^{3}+\cdots\)
13.12.a.b \(6\) \(9.988\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(55\) \(476\) \(3312\) \(-4176\) \(+\) \(q+(9+\beta _{1})q^{2}+(80-2\beta _{1}+\beta _{4})q^{3}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(13))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(13)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)