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Properties

Label 13.4.e
Level $13$
Weight $4$
Character orbit 13.e
Rep. character $\chi_{13}(4,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $4$
Newform subspaces $2$
Sturm bound $4$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.e (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 8 8 0
Cusp forms 4 4 0
Eisenstein series 4 4 0

Trace form

\( 4 q - 3 q^{2} + 5 q^{3} - q^{4} - 48 q^{6} + 15 q^{7} + q^{9} + O(q^{10}) \) \( 4 q - 3 q^{2} + 5 q^{3} - q^{4} - 48 q^{6} + 15 q^{7} + q^{9} + 51 q^{10} - 15 q^{11} + 76 q^{12} + 65 q^{13} - 204 q^{14} + 174 q^{15} + 79 q^{16} - 144 q^{17} - 351 q^{19} - 111 q^{20} + 102 q^{22} + 21 q^{23} + 246 q^{24} + 110 q^{25} + 351 q^{26} - 130 q^{27} + 276 q^{28} + 210 q^{29} - 330 q^{30} - 603 q^{32} - 321 q^{33} + 336 q^{35} + 203 q^{36} - 318 q^{37} + 216 q^{38} + 325 q^{39} - 462 q^{40} - 210 q^{41} - 498 q^{42} - 19 q^{43} + 597 q^{45} + 576 q^{46} - 562 q^{48} + 13 q^{49} + 768 q^{50} + 90 q^{51} - 494 q^{52} + 1038 q^{53} - 510 q^{54} + 336 q^{55} + 624 q^{56} + 9 q^{58} - 525 q^{59} - 128 q^{61} - 522 q^{62} - 1410 q^{63} - 742 q^{64} - 1209 q^{65} + 996 q^{66} - 1077 q^{67} - 477 q^{68} + 555 q^{69} + 2841 q^{71} + 1425 q^{72} - 111 q^{74} - 713 q^{75} + 378 q^{76} - 1398 q^{77} - 858 q^{78} + 64 q^{79} + 1923 q^{80} + 418 q^{81} + 1833 q^{82} + 852 q^{84} + 297 q^{85} - 201 q^{87} + 624 q^{88} - 1161 q^{89} - 1974 q^{90} - 741 q^{91} - 1236 q^{92} - 1422 q^{93} - 1710 q^{94} - 1026 q^{95} + 2487 q^{97} - 1437 q^{98} + O(q^{100}) \)

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.e.a 13.e 13.e $2$ $0.767$ \(\Q(\sqrt{-3}) \) None \(-6\) \(7\) \(0\) \(39\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2-2\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}+4\zeta_{6}q^{4}+\cdots\)
13.4.e.b 13.e 13.e $2$ $0.767$ \(\Q(\sqrt{-3}) \) None \(3\) \(-2\) \(0\) \(-24\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{4}+\cdots\)