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Properties

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

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 13.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
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 and the Fricke involution.

\(13\)Dim
\(+\)\(4\)
\(-\)\(3\)

Trace form

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\) 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
13.8.a.a 13.a 1.a $1$ $4.061$ \(\Q\) None \(10\) \(-73\) \(-295\) \(1373\) $-$ $\mathrm{SU}(2)$ \(q+10q^{2}-73q^{3}-28q^{4}-295q^{5}+\cdots\)
13.8.a.b 13.a 1.a $2$ $4.061$ \(\Q(\sqrt{337}) \) None \(-19\) \(45\) \(-353\) \(-2009\) $-$ $\mathrm{SU}(2)$ \(q+(-9-\beta )q^{2}+(21+3\beta )q^{3}+(37+19\beta )q^{4}+\cdots\)
13.8.a.c 13.a 1.a $4$ $4.061$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(15\) \(80\) \(258\) \(1692\) $+$ $\mathrm{SU}(2)$ \(q+(4-\beta _{1})q^{2}+(21-2\beta _{1}+\beta _{2})q^{3}+\cdots\)