Properties

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

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
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 and the Fricke involution.

\(13\)Dim.
\(+\)\(2\)
\(-\)\(1\)

Trace form

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

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\) into newform subspaces

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 \(1\) \(5\) \(-3\) \(-9\) \(+\) \(q+\beta q^{2}+(4-3\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)