Properties

Label 13.10.a
Level 13
Weight 10
Character orbit a
Rep. character \(\chi_{13}(1,\cdot)\)
Character field \(\Q\)
Dimension 9
Newforms 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: \(\Q\)
Newforms: \( 2 \)
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.

\(13\)Dim.
\(+\)\(4\)
\(-\)\(5\)

Trace form

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

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(13))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 13
13.10.a.a \(4\) \(6.695\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-33\) \(-163\) \(471\) \(-11241\) \(+\) \(q+(-8-\beta _{1})q^{2}+(-41+2\beta _{1}-\beta _{3})q^{3}+\cdots\)
13.10.a.b \(5\) \(6.695\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(15\) \(161\) \(1803\) \(10099\) \(-\) \(q+(3+\beta _{1})q^{2}+(2^{5}+2\beta _{1}-\beta _{2})q^{3}+\cdots\)