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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

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

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 13
13.4.a.a 13.a 1.a $1$ $0.767$ \(\Q\) None \(-5\) \(-7\) \(-7\) \(-13\) $-$ $\mathrm{SU}(2)$ \(q-5q^{2}-7q^{3}+17q^{4}-7q^{5}+35q^{6}+\cdots\)
13.4.a.b 13.a 1.a $2$ $0.767$ \(\Q(\sqrt{17}) \) None \(1\) \(5\) \(-3\) \(-9\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(4-3\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)