Properties

Label 13.10.a
Level $13$
Weight $10$
Character orbit 13.a
Rep. character $\chi_{13}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $2$
Sturm bound $11$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 13.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(11\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(13))\).

Total New Old
Modular forms 11 9 2
Cusp forms 9 9 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
\(+\)\(4\)
\(-\)\(5\)

Trace form

\( 9 q - 18 q^{2} - 2 q^{3} + 1790 q^{4} + 2274 q^{5} + 1164 q^{6} - 1142 q^{7} - 22392 q^{8} + 31107 q^{9} + O(q^{10}) \) \( 9 q - 18 q^{2} - 2 q^{3} + 1790 q^{4} + 2274 q^{5} + 1164 q^{6} - 1142 q^{7} - 22392 q^{8} + 31107 q^{9} + 16674 q^{10} + 81606 q^{11} - 42090 q^{12} + 28561 q^{13} - 269178 q^{14} + 189280 q^{15} + 404146 q^{16} - 416952 q^{17} + 1034798 q^{18} - 631074 q^{19} - 724764 q^{20} - 461036 q^{21} - 659504 q^{22} - 4849596 q^{23} + 1265916 q^{24} - 1229795 q^{25} + 1370928 q^{26} + 4770082 q^{27} + 6623136 q^{28} + 9030246 q^{29} + 4747058 q^{30} + 1519294 q^{31} - 26176032 q^{32} + 2865076 q^{33} + 16323580 q^{34} - 5409066 q^{35} - 32071016 q^{36} + 11808714 q^{37} - 396348 q^{38} + 9253764 q^{39} - 31822654 q^{40} + 23153550 q^{41} + 6044466 q^{42} - 20536106 q^{43} + 25010148 q^{44} + 8711270 q^{45} + 12049636 q^{46} + 39411210 q^{47} - 87495458 q^{48} + 48544411 q^{49} - 20488518 q^{50} - 2758570 q^{51} - 30503148 q^{52} - 77787546 q^{53} - 5963196 q^{54} - 21711692 q^{55} - 81953742 q^{56} + 42344352 q^{57} + 230985704 q^{58} + 183106794 q^{59} + 127806048 q^{60} - 55182422 q^{61} + 62256984 q^{62} - 572650678 q^{63} + 859981570 q^{64} + 38043252 q^{65} - 366129504 q^{66} - 410562606 q^{67} - 479443350 q^{68} - 33344204 q^{69} + 264198964 q^{70} + 464611446 q^{71} + 146804976 q^{72} + 341685262 q^{73} - 764952582 q^{74} - 219139040 q^{75} - 1280673160 q^{76} + 1011616572 q^{77} + 291950542 q^{78} - 1131272744 q^{79} + 391542492 q^{80} + 120090585 q^{81} - 705588852 q^{82} + 1617316434 q^{83} + 1356883180 q^{84} - 225900296 q^{85} + 2701696572 q^{86} - 1702056976 q^{87} - 2547791592 q^{88} - 1906094658 q^{89} + 3173815920 q^{90} + 609491740 q^{91} - 4199605332 q^{92} + 725481888 q^{93} + 2131980718 q^{94} + 168927792 q^{95} + 4102066876 q^{96} + 1052922690 q^{97} + 1991147070 q^{98} + 5148380898 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(13))\) 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
13.10.a.a 13.a 1.a $4$ $6.695$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-33\) \(-163\) \(471\) \(-11241\) $+$ $\mathrm{SU}(2)$ \(q+(-8-\beta _{1})q^{2}+(-41+2\beta _{1}-\beta _{3})q^{3}+\cdots\)
13.10.a.b 13.a 1.a $5$ $6.695$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(15\) \(161\) \(1803\) \(10099\) $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{2}+(2^{5}+2\beta _{1}-\beta _{2})q^{3}+\cdots\)