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Properties

Label 13.4.c
Level $13$
Weight $4$
Character orbit 13.c
Rep. character $\chi_{13}(3,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $6$
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.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(13, [\chi])\).

Total New Old
Modular forms 10 10 0
Cusp forms 6 6 0
Eisenstein series 4 4 0

Trace form

\( 6 q + q^{2} - 7 q^{3} - 13 q^{4} + 4 q^{5} + 30 q^{6} - 35 q^{7} - 30 q^{8} - 12 q^{9} + O(q^{10}) \) \( 6 q + q^{2} - 7 q^{3} - 13 q^{4} + 4 q^{5} + 30 q^{6} - 35 q^{7} - 30 q^{8} - 12 q^{9} - 63 q^{10} + 15 q^{11} + 312 q^{12} + 34 q^{13} + 68 q^{14} - 124 q^{15} + 7 q^{16} - 57 q^{17} - 614 q^{18} + 111 q^{19} - 311 q^{20} + 206 q^{21} + 298 q^{22} - 223 q^{23} + 216 q^{24} + 478 q^{25} + 237 q^{26} + 470 q^{27} - 240 q^{28} - 231 q^{29} + 4 q^{30} - 428 q^{31} + 151 q^{32} - 361 q^{33} - 182 q^{34} - 270 q^{35} - 541 q^{36} + 417 q^{37} + 860 q^{38} - 51 q^{39} - 370 q^{40} - 373 q^{41} + 210 q^{42} - 299 q^{43} + 848 q^{44} + 16 q^{45} - 230 q^{46} - 204 q^{47} + 368 q^{48} + 508 q^{49} - 1106 q^{50} - 518 q^{51} - 102 q^{52} + 1276 q^{53} + 1314 q^{54} + 34 q^{55} + 172 q^{56} - 330 q^{57} - 193 q^{58} + 1673 q^{59} + 144 q^{60} - 647 q^{61} + 796 q^{62} + 70 q^{63} - 3566 q^{64} - 2102 q^{65} - 3708 q^{66} - 387 q^{67} + 609 q^{68} + 323 q^{69} + 2560 q^{70} - 781 q^{71} + 1155 q^{72} + 1600 q^{73} + 59 q^{74} + 1397 q^{75} - 100 q^{76} - 770 q^{77} + 3278 q^{78} + 328 q^{79} + 2153 q^{80} - 543 q^{81} + 2175 q^{82} + 1776 q^{83} - 1540 q^{84} + 1426 q^{85} - 5172 q^{86} - 2009 q^{87} + 1020 q^{88} - 655 q^{89} - 4578 q^{90} - 767 q^{91} - 832 q^{92} - 2052 q^{93} - 392 q^{94} - 1780 q^{95} + 2816 q^{96} + 177 q^{97} - 1513 q^{98} + 5892 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.4.c.a 13.c 13.c $2$ $0.767$ \(\Q(\sqrt{-3}) \) None \(-4\) \(-2\) \(34\) \(-20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4+4\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}+\cdots\)
13.4.c.b 13.c 13.c $4$ $0.767$ \(\Q(\sqrt{-3}, \sqrt{17})\) None \(5\) \(-5\) \(-30\) \(-15\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-\beta _{1}-2\beta _{2}-\beta _{3})q^{2}+(-1+3\beta _{1}+\cdots)q^{3}+\cdots\)