Properties

Label 13.8.a
Level 13
Weight 8
Character orbit a
Rep. character \(\chi_{13}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newforms 3
Sturm bound 9
Trace bound 1

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 13.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(9\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(13))\).

Total New Old
Modular forms 9 7 2
Cusp forms 7 7 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.
\(+\)\(4\)
\(-\)\(3\)

Trace form

\(7q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut +\mathstrut 52q^{3} \) \(\mathstrut +\mathstrut 318q^{4} \) \(\mathstrut -\mathstrut 390q^{5} \) \(\mathstrut -\mathstrut 84q^{6} \) \(\mathstrut +\mathstrut 1056q^{7} \) \(\mathstrut -\mathstrut 1320q^{8} \) \(\mathstrut +\mathstrut 4791q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut +\mathstrut 52q^{3} \) \(\mathstrut +\mathstrut 318q^{4} \) \(\mathstrut -\mathstrut 390q^{5} \) \(\mathstrut -\mathstrut 84q^{6} \) \(\mathstrut +\mathstrut 1056q^{7} \) \(\mathstrut -\mathstrut 1320q^{8} \) \(\mathstrut +\mathstrut 4791q^{9} \) \(\mathstrut -\mathstrut 2238q^{10} \) \(\mathstrut -\mathstrut 7620q^{11} \) \(\mathstrut +\mathstrut 10086q^{12} \) \(\mathstrut -\mathstrut 2197q^{13} \) \(\mathstrut +\mathstrut 13014q^{14} \) \(\mathstrut -\mathstrut 21704q^{15} \) \(\mathstrut -\mathstrut 27694q^{16} \) \(\mathstrut -\mathstrut 17694q^{17} \) \(\mathstrut +\mathstrut 23462q^{18} \) \(\mathstrut +\mathstrut 46580q^{19} \) \(\mathstrut -\mathstrut 73740q^{20} \) \(\mathstrut -\mathstrut 70952q^{21} \) \(\mathstrut +\mathstrut 105472q^{22} \) \(\mathstrut +\mathstrut 128712q^{23} \) \(\mathstrut +\mathstrut 53196q^{24} \) \(\mathstrut +\mathstrut 154757q^{25} \) \(\mathstrut -\mathstrut 52728q^{26} \) \(\mathstrut +\mathstrut 33400q^{27} \) \(\mathstrut -\mathstrut 273952q^{28} \) \(\mathstrut +\mathstrut 34218q^{29} \) \(\mathstrut -\mathstrut 150142q^{30} \) \(\mathstrut -\mathstrut 316488q^{31} \) \(\mathstrut -\mathstrut 250704q^{32} \) \(\mathstrut +\mathstrut 169144q^{33} \) \(\mathstrut +\mathstrut 56716q^{34} \) \(\mathstrut +\mathstrut 255168q^{35} \) \(\mathstrut +\mathstrut 100504q^{36} \) \(\mathstrut +\mathstrut 684986q^{37} \) \(\mathstrut -\mathstrut 1099260q^{38} \) \(\mathstrut -\mathstrut 237276q^{39} \) \(\mathstrut +\mathstrut 1315346q^{40} \) \(\mathstrut +\mathstrut 843054q^{41} \) \(\mathstrut -\mathstrut 10158q^{42} \) \(\mathstrut +\mathstrut 5052q^{43} \) \(\mathstrut -\mathstrut 590604q^{44} \) \(\mathstrut -\mathstrut 2624614q^{45} \) \(\mathstrut -\mathstrut 496316q^{46} \) \(\mathstrut -\mathstrut 1610664q^{47} \) \(\mathstrut +\mathstrut 1839262q^{48} \) \(\mathstrut +\mathstrut 678811q^{49} \) \(\mathstrut +\mathstrut 5871426q^{50} \) \(\mathstrut -\mathstrut 3096904q^{51} \) \(\mathstrut -\mathstrut 413036q^{52} \) \(\mathstrut +\mathstrut 2453442q^{53} \) \(\mathstrut +\mathstrut 1043412q^{54} \) \(\mathstrut -\mathstrut 42296q^{55} \) \(\mathstrut +\mathstrut 760050q^{56} \) \(\mathstrut -\mathstrut 2563680q^{57} \) \(\mathstrut -\mathstrut 4502440q^{58} \) \(\mathstrut +\mathstrut 399996q^{59} \) \(\mathstrut -\mathstrut 11818512q^{60} \) \(\mathstrut +\mathstrut 5212914q^{61} \) \(\mathstrut +\mathstrut 1682232q^{62} \) \(\mathstrut +\mathstrut 10585472q^{63} \) \(\mathstrut -\mathstrut 9723598q^{64} \) \(\mathstrut -\mathstrut 1990482q^{65} \) \(\mathstrut +\mathstrut 19895328q^{66} \) \(\mathstrut -\mathstrut 6391324q^{67} \) \(\mathstrut -\mathstrut 2959782q^{68} \) \(\mathstrut +\mathstrut 4028200q^{69} \) \(\mathstrut -\mathstrut 13184060q^{70} \) \(\mathstrut -\mathstrut 4184352q^{71} \) \(\mathstrut -\mathstrut 4021200q^{72} \) \(\mathstrut -\mathstrut 17699298q^{73} \) \(\mathstrut +\mathstrut 18282570q^{74} \) \(\mathstrut +\mathstrut 16262716q^{75} \) \(\mathstrut +\mathstrut 27841480q^{76} \) \(\mathstrut -\mathstrut 14950008q^{77} \) \(\mathstrut -\mathstrut 7122674q^{78} \) \(\mathstrut +\mathstrut 18741312q^{79} \) \(\mathstrut +\mathstrut 4873068q^{80} \) \(\mathstrut -\mathstrut 10908753q^{81} \) \(\mathstrut +\mathstrut 2920092q^{82} \) \(\mathstrut -\mathstrut 13691388q^{83} \) \(\mathstrut -\mathstrut 21686852q^{84} \) \(\mathstrut -\mathstrut 6734372q^{85} \) \(\mathstrut -\mathstrut 31170660q^{86} \) \(\mathstrut +\mathstrut 30644360q^{87} \) \(\mathstrut +\mathstrut 21699000q^{88} \) \(\mathstrut +\mathstrut 33292110q^{89} \) \(\mathstrut -\mathstrut 50053200q^{90} \) \(\mathstrut -\mathstrut 5114616q^{91} \) \(\mathstrut +\mathstrut 35518716q^{92} \) \(\mathstrut -\mathstrut 9696336q^{93} \) \(\mathstrut +\mathstrut 5824414q^{94} \) \(\mathstrut +\mathstrut 1053576q^{95} \) \(\mathstrut -\mathstrut 36116084q^{96} \) \(\mathstrut -\mathstrut 25966114q^{97} \) \(\mathstrut -\mathstrut 12157482q^{98} \) \(\mathstrut -\mathstrut 30302100q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 13
13.8.a.a \(1\) \(4.061\) \(\Q\) None \(10\) \(-73\) \(-295\) \(1373\) \(-\) \(q+10q^{2}-73q^{3}-28q^{4}-295q^{5}+\cdots\)
13.8.a.b \(2\) \(4.061\) \(\Q(\sqrt{337}) \) None \(-19\) \(45\) \(-353\) \(-2009\) \(-\) \(q+(-9-\beta )q^{2}+(21+3\beta )q^{3}+(37+19\beta )q^{4}+\cdots\)
13.8.a.c \(4\) \(4.061\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(15\) \(80\) \(258\) \(1692\) \(+\) \(q+(4-\beta _{1})q^{2}+(21-2\beta _{1}+\beta _{2})q^{3}+\cdots\)