Properties

Label 13.4.a
Level 13
Weight 4
Character orbit a
Rep. character \(\chi_{13}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 2
Sturm bound 4
Trace bound 1

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 13.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(13))\).

Total New Old
Modular forms 5 3 2
Cusp forms 3 3 0
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.
\(+\)\(2\)
\(-\)\(1\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 12q^{6} \) \(\mathstrut -\mathstrut 22q^{7} \) \(\mathstrut -\mathstrut 48q^{8} \) \(\mathstrut +\mathstrut 57q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 12q^{6} \) \(\mathstrut -\mathstrut 22q^{7} \) \(\mathstrut -\mathstrut 48q^{8} \) \(\mathstrut +\mathstrut 57q^{9} \) \(\mathstrut +\mathstrut 42q^{10} \) \(\mathstrut +\mathstrut 54q^{11} \) \(\mathstrut -\mathstrut 162q^{12} \) \(\mathstrut -\mathstrut 13q^{13} \) \(\mathstrut +\mathstrut 154q^{14} \) \(\mathstrut +\mathstrut 16q^{15} \) \(\mathstrut +\mathstrut 50q^{16} \) \(\mathstrut +\mathstrut 96q^{17} \) \(\mathstrut -\mathstrut 220q^{18} \) \(\mathstrut -\mathstrut 210q^{19} \) \(\mathstrut -\mathstrut 100q^{20} \) \(\mathstrut -\mathstrut 212q^{21} \) \(\mathstrut +\mathstrut 272q^{22} \) \(\mathstrut +\mathstrut 100q^{23} \) \(\mathstrut +\mathstrut 588q^{24} \) \(\mathstrut -\mathstrut 313q^{25} \) \(\mathstrut -\mathstrut 78q^{26} \) \(\mathstrut +\mathstrut 370q^{27} \) \(\mathstrut -\mathstrut 96q^{28} \) \(\mathstrut -\mathstrut 126q^{29} \) \(\mathstrut -\mathstrut 202q^{30} \) \(\mathstrut +\mathstrut 110q^{31} \) \(\mathstrut -\mathstrut 208q^{32} \) \(\mathstrut +\mathstrut 76q^{33} \) \(\mathstrut -\mathstrut 520q^{34} \) \(\mathstrut +\mathstrut 198q^{35} \) \(\mathstrut +\mathstrut 124q^{36} \) \(\mathstrut +\mathstrut 78q^{37} \) \(\mathstrut +\mathstrut 316q^{38} \) \(\mathstrut -\mathstrut 156q^{39} \) \(\mathstrut +\mathstrut 226q^{40} \) \(\mathstrut +\mathstrut 106q^{41} \) \(\mathstrut -\mathstrut 258q^{42} \) \(\mathstrut +\mathstrut 86q^{43} \) \(\mathstrut -\mathstrut 620q^{44} \) \(\mathstrut -\mathstrut 334q^{45} \) \(\mathstrut +\mathstrut 476q^{46} \) \(\mathstrut +\mathstrut 330q^{47} \) \(\mathstrut -\mathstrut 338q^{48} \) \(\mathstrut +\mathstrut 209q^{49} \) \(\mathstrut +\mathstrut 236q^{50} \) \(\mathstrut -\mathstrut 58q^{51} \) \(\mathstrut +\mathstrut 312q^{52} \) \(\mathstrut -\mathstrut 550q^{53} \) \(\mathstrut -\mathstrut 84q^{54} \) \(\mathstrut +\mathstrut 164q^{55} \) \(\mathstrut -\mathstrut 430q^{56} \) \(\mathstrut +\mathstrut 1488q^{57} \) \(\mathstrut +\mathstrut 1204q^{58} \) \(\mathstrut -\mathstrut 662q^{59} \) \(\mathstrut +\mathstrut 1008q^{60} \) \(\mathstrut -\mathstrut 1114q^{61} \) \(\mathstrut -\mathstrut 1312q^{62} \) \(\mathstrut -\mathstrut 1846q^{63} \) \(\mathstrut +\mathstrut 482q^{64} \) \(\mathstrut -\mathstrut 52q^{65} \) \(\mathstrut -\mathstrut 1728q^{66} \) \(\mathstrut +\mathstrut 546q^{67} \) \(\mathstrut +\mathstrut 1098q^{68} \) \(\mathstrut +\mathstrut 1468q^{69} \) \(\mathstrut -\mathstrut 580q^{70} \) \(\mathstrut -\mathstrut 122q^{71} \) \(\mathstrut +\mathstrut 360q^{72} \) \(\mathstrut +\mathstrut 554q^{73} \) \(\mathstrut +\mathstrut 802q^{74} \) \(\mathstrut +\mathstrut 16q^{75} \) \(\mathstrut -\mathstrut 2120q^{76} \) \(\mathstrut +\mathstrut 1100q^{77} \) \(\mathstrut +\mathstrut 754q^{78} \) \(\mathstrut +\mathstrut 296q^{79} \) \(\mathstrut -\mathstrut 692q^{80} \) \(\mathstrut -\mathstrut 717q^{81} \) \(\mathstrut -\mathstrut 1608q^{82} \) \(\mathstrut +\mathstrut 1650q^{83} \) \(\mathstrut +\mathstrut 2956q^{84} \) \(\mathstrut -\mathstrut 712q^{85} \) \(\mathstrut +\mathstrut 2364q^{86} \) \(\mathstrut -\mathstrut 1984q^{87} \) \(\mathstrut -\mathstrut 72q^{88} \) \(\mathstrut -\mathstrut 1910q^{89} \) \(\mathstrut +\mathstrut 1020q^{90} \) \(\mathstrut -\mathstrut 52q^{91} \) \(\mathstrut -\mathstrut 2420q^{92} \) \(\mathstrut -\mathstrut 720q^{93} \) \(\mathstrut -\mathstrut 286q^{94} \) \(\mathstrut +\mathstrut 736q^{95} \) \(\mathstrut -\mathstrut 1268q^{96} \) \(\mathstrut -\mathstrut 858q^{97} \) \(\mathstrut +\mathstrut 220q^{98} \) \(\mathstrut -\mathstrut 702q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 13
13.4.a.a \(1\) \(0.767\) \(\Q\) None \(-5\) \(-7\) \(-7\) \(-13\) \(-\) \(q-5q^{2}-7q^{3}+17q^{4}-7q^{5}+35q^{6}+\cdots\)
13.4.a.b \(2\) \(0.767\) \(\Q(\sqrt{17}) \) None \(1\) \(5\) \(-3\) \(-9\) \(+\) \(q+\beta q^{2}+(4-3\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)