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Properties

Label 13.6.a
Level $13$
Weight $6$
Character orbit 13.a
Rep. character $\chi_{13}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $2$
Sturm bound $7$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 13.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(7\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(13))\).

Total New Old
Modular forms 7 5 2
Cusp forms 5 5 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
\(+\)\(2\)
\(-\)\(3\)

Trace form

\( 5 q + 2 q^{2} - 20 q^{3} + 78 q^{4} + 14 q^{5} - 180 q^{6} - 96 q^{7} + 552 q^{8} + 21 q^{9} + O(q^{10}) \) \( 5 q + 2 q^{2} - 20 q^{3} + 78 q^{4} + 14 q^{5} - 180 q^{6} - 96 q^{7} + 552 q^{8} + 21 q^{9} - 846 q^{10} + 180 q^{11} - 234 q^{12} + 169 q^{13} - 1298 q^{14} + 520 q^{15} + 3250 q^{16} - 1722 q^{17} + 1874 q^{18} - 164 q^{19} - 1564 q^{20} + 5464 q^{21} - 3184 q^{22} + 1000 q^{23} - 14436 q^{24} + 10255 q^{25} + 2028 q^{26} - 9008 q^{27} - 2912 q^{28} - 9570 q^{29} + 1658 q^{30} + 5112 q^{31} + 29696 q^{32} + 7720 q^{33} + 21476 q^{34} - 18696 q^{35} - 16376 q^{36} - 37490 q^{37} + 34804 q^{38} + 6084 q^{39} - 62174 q^{40} - 3158 q^{41} + 12666 q^{42} + 4452 q^{43} + 61444 q^{44} + 38750 q^{45} + 15188 q^{46} + 44232 q^{47} - 101858 q^{48} - 35887 q^{49} - 99898 q^{50} + 56816 q^{51} + 27716 q^{52} - 56290 q^{53} + 54420 q^{54} + 14648 q^{55} - 28942 q^{56} + 4656 q^{57} - 16688 q^{58} + 25012 q^{59} + 160128 q^{60} - 10770 q^{61} - 63448 q^{62} - 105424 q^{63} + 120610 q^{64} + 16562 q^{65} - 116640 q^{66} - 13796 q^{67} + 12810 q^{68} - 55592 q^{69} + 105284 q^{70} + 144976 q^{71} - 5904 q^{72} + 31386 q^{73} - 105014 q^{74} - 6020 q^{75} - 94120 q^{76} + 35696 q^{77} - 36842 q^{78} - 117168 q^{79} - 146948 q^{80} - 15459 q^{81} + 106068 q^{82} + 22380 q^{83} - 29108 q^{84} - 43516 q^{85} - 35844 q^{86} + 147992 q^{87} + 144984 q^{88} + 150202 q^{89} - 24360 q^{90} - 4056 q^{91} + 297100 q^{92} + 20160 q^{93} - 34282 q^{94} - 180488 q^{95} - 202148 q^{96} + 41578 q^{97} - 54926 q^{98} - 220140 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(13))\) 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
13.6.a.a 13.a 1.a $2$ $2.085$ \(\Q(\sqrt{17}) \) None \(-5\) \(-28\) \(-42\) \(-36\) $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{2}+(-17+6\beta )q^{3}+(-24+\cdots)q^{4}+\cdots\)
13.6.a.b 13.a 1.a $3$ $2.085$ 3.3.168897.1 None \(7\) \(8\) \(56\) \(-60\) $-$ $\mathrm{SU}(2)$ \(q+(2+\beta _{1})q^{2}+(3-\beta _{1}-\beta _{2})q^{3}+(40+\cdots)q^{4}+\cdots\)