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Properties

Label 13.10.e
Level $13$
Weight $10$
Character orbit 13.e
Rep. character $\chi_{13}(4,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $20$
Newform subspaces $1$
Sturm bound $11$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 13.e (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(11\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(13, [\chi])\).

Total New Old
Modular forms 24 24 0
Cusp forms 20 20 0
Eisenstein series 4 4 0

Trace form

\( 20 q - 3 q^{2} - 163 q^{3} + 2559 q^{4} - 3288 q^{6} + 1023 q^{7} - 64763 q^{9} + O(q^{10}) \) \( 20 q - 3 q^{2} - 163 q^{3} + 2559 q^{4} - 3288 q^{6} + 1023 q^{7} - 64763 q^{9} + 44195 q^{10} - 94599 q^{11} - 427652 q^{12} + 187013 q^{13} + 473556 q^{14} + 754950 q^{15} - 594809 q^{16} - 49656 q^{17} - 3879 q^{19} - 154815 q^{20} + 2536766 q^{22} + 3126189 q^{23} - 6472626 q^{24} - 1529274 q^{25} - 13931889 q^{26} + 18052718 q^{27} + 4918980 q^{28} - 2712414 q^{29} + 15022758 q^{30} - 14390595 q^{32} - 34050309 q^{33} + 7549080 q^{35} + 27039443 q^{36} - 36102006 q^{37} - 39021096 q^{38} - 32365163 q^{39} + 134360674 q^{40} + 30265110 q^{41} - 60800034 q^{42} + 26621029 q^{43} + 31870857 q^{45} + 121485864 q^{46} - 41385538 q^{48} + 7027649 q^{49} - 58518456 q^{50} - 10208934 q^{51} - 109817238 q^{52} - 36429786 q^{53} - 393493974 q^{54} + 65727080 q^{55} + 63636336 q^{56} + 470295633 q^{58} + 3987867 q^{59} - 268212896 q^{61} - 445379898 q^{62} + 506780166 q^{63} + 343337066 q^{64} + 292006563 q^{65} - 445758060 q^{66} - 944780397 q^{67} - 168297045 q^{68} + 541942791 q^{69} + 1318764849 q^{71} + 1409393649 q^{72} - 541934631 q^{74} - 609413441 q^{75} - 594798654 q^{76} - 1470187374 q^{77} - 2076950874 q^{78} + 1556703616 q^{79} - 968911845 q^{80} - 1069563758 q^{81} + 1859661377 q^{82} + 7275171300 q^{84} - 318942363 q^{85} - 644123073 q^{87} - 2054580464 q^{88} - 2287100109 q^{89} - 6489878934 q^{90} - 4972612749 q^{91} + 10148843820 q^{92} - 1470324870 q^{93} - 2447356414 q^{94} + 4625601270 q^{95} + 7273214547 q^{97} - 1557570597 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{10}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.10.e.a 13.e 13.e $20$ $6.695$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None 13.10.e.a \(-3\) \(-163\) \(0\) \(1023\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}-\beta _{2})q^{2}+(2^{4}\beta _{5}-\beta _{6})q^{3}+(2^{8}+\cdots)q^{4}+\cdots\)