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Properties

Label 69.6.a
Level $69$
Weight $6$
Character orbit 69.a
Rep. character $\chi_{69}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $5$
Sturm bound $48$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(48\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(69))\).

Total New Old
Modular forms 42 18 24
Cusp forms 38 18 20
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(23\)FrickeDim
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(6\)
\(-\)\(+\)$-$\(5\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(11\)

Trace form

\( 18 q + 8 q^{2} - 18 q^{3} + 288 q^{4} + 32 q^{5} - 72 q^{6} + 40 q^{7} - 108 q^{8} + 1458 q^{9} + O(q^{10}) \) \( 18 q + 8 q^{2} - 18 q^{3} + 288 q^{4} + 32 q^{5} - 72 q^{6} + 40 q^{7} - 108 q^{8} + 1458 q^{9} + 372 q^{10} - 72 q^{12} - 1156 q^{13} - 84 q^{14} + 396 q^{15} + 8680 q^{16} - 1488 q^{17} + 648 q^{18} - 3560 q^{19} + 9588 q^{20} + 2484 q^{21} + 1108 q^{22} - 3564 q^{24} + 18222 q^{25} + 2368 q^{26} - 1458 q^{27} + 1160 q^{28} - 7868 q^{29} - 1368 q^{30} - 19072 q^{31} - 19988 q^{32} + 13032 q^{33} + 29920 q^{34} - 5576 q^{35} + 23328 q^{36} - 29564 q^{37} - 49492 q^{38} - 22644 q^{39} - 20740 q^{40} + 40388 q^{41} + 5148 q^{42} + 8480 q^{43} + 53632 q^{44} + 2592 q^{45} - 8464 q^{46} + 65768 q^{47} - 6912 q^{48} + 20530 q^{49} - 35328 q^{50} + 12924 q^{51} - 7624 q^{52} - 139440 q^{53} - 5832 q^{54} - 23344 q^{55} - 144332 q^{56} + 17892 q^{57} - 15744 q^{58} - 11936 q^{59} - 30456 q^{60} + 68772 q^{61} + 61776 q^{62} + 3240 q^{63} - 28552 q^{64} - 103144 q^{65} - 147168 q^{66} + 96856 q^{67} - 11636 q^{68} - 19044 q^{69} - 380296 q^{70} + 155120 q^{71} - 8748 q^{72} + 103044 q^{73} + 91152 q^{74} + 66186 q^{75} - 51908 q^{76} + 126720 q^{77} + 5544 q^{78} - 87152 q^{79} + 335788 q^{80} + 118098 q^{81} - 218600 q^{82} + 4096 q^{83} + 220104 q^{84} + 101384 q^{85} - 82756 q^{86} - 196380 q^{87} + 646444 q^{88} + 310784 q^{89} + 30132 q^{90} + 198664 q^{91} - 227016 q^{93} + 39304 q^{94} + 258216 q^{95} + 18540 q^{96} - 251844 q^{97} + 401336 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(69))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 23
69.6.a.a 69.a 1.a $2$ $11.066$ \(\Q(\sqrt{29}) \) None \(0\) \(-18\) \(94\) \(-118\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2\beta q^{2}-9q^{3}+84q^{4}+(47-\beta )q^{5}+\cdots\)
69.6.a.b 69.a 1.a $3$ $11.066$ 3.3.5333.1 None \(-8\) \(27\) \(-56\) \(-114\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{2})q^{2}+9q^{3}+(9-4\beta _{1}-9\beta _{2})q^{4}+\cdots\)
69.6.a.c 69.a 1.a $4$ $11.066$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(4\) \(-36\) \(-122\) \(62\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}-9q^{3}+(-12+4\beta _{1}+\cdots)q^{4}+\cdots\)
69.6.a.d 69.a 1.a $4$ $11.066$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(4\) \(-36\) \(22\) \(-62\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-9q^{3}+(7-\beta _{1}+\beta _{3})q^{4}+\cdots\)
69.6.a.e 69.a 1.a $5$ $11.066$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(8\) \(45\) \(94\) \(272\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{2})q^{2}+9q^{3}+(24-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(69))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(69)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)