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Properties

Label 69.8.a
Level $69$
Weight $8$
Character orbit 69.a
Rep. character $\chi_{69}(1,\cdot)$
Character field $\Q$
Dimension $26$
Newform subspaces $4$
Sturm bound $64$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(64\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(69))\).

Total New Old
Modular forms 58 26 32
Cusp forms 54 26 28
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
\(+\)\(+\)$+$\(7\)
\(+\)\(-\)$-$\(5\)
\(-\)\(+\)$-$\(6\)
\(-\)\(-\)$+$\(8\)
Plus space\(+\)\(15\)
Minus space\(-\)\(11\)

Trace form

\( 26 q + 16 q^{2} + 54 q^{3} + 1664 q^{4} - 776 q^{5} + 432 q^{6} - 456 q^{7} + 5076 q^{8} + 18954 q^{9} + O(q^{10}) \) \( 26 q + 16 q^{2} + 54 q^{3} + 1664 q^{4} - 776 q^{5} + 432 q^{6} - 456 q^{7} + 5076 q^{8} + 18954 q^{9} - 15180 q^{10} - 4968 q^{12} + 14780 q^{13} + 36588 q^{14} + 21276 q^{15} + 43432 q^{16} - 91608 q^{17} + 11664 q^{18} + 17016 q^{19} - 41196 q^{20} - 40500 q^{21} + 281636 q^{22} - 16524 q^{24} + 431510 q^{25} + 70640 q^{26} + 39366 q^{27} - 108184 q^{28} - 70780 q^{29} + 338472 q^{30} + 249008 q^{31} + 1250924 q^{32} - 426168 q^{33} + 384752 q^{34} - 576616 q^{35} + 1213056 q^{36} - 43692 q^{37} + 1035916 q^{38} + 229932 q^{39} - 2037476 q^{40} + 738484 q^{41} + 432108 q^{42} - 1751776 q^{43} - 1736416 q^{44} - 565704 q^{45} + 389344 q^{46} - 1467512 q^{47} - 414720 q^{48} + 8044250 q^{49} - 4156872 q^{50} + 40716 q^{51} + 4543256 q^{52} - 2020056 q^{53} + 314928 q^{54} - 772400 q^{55} + 11274356 q^{56} - 3620916 q^{57} - 91824 q^{58} - 3516544 q^{59} + 5169096 q^{60} - 550988 q^{61} - 1696464 q^{62} - 332424 q^{63} - 15144 q^{64} + 3275416 q^{65} - 341280 q^{66} + 838296 q^{67} - 9512500 q^{68} + 1314036 q^{69} - 5169416 q^{70} - 9742064 q^{71} + 3700404 q^{72} - 7145036 q^{73} + 6737184 q^{74} + 8472978 q^{75} - 11316804 q^{76} - 3000672 q^{77} - 1157544 q^{78} - 661056 q^{79} - 27773044 q^{80} + 13817466 q^{81} - 25379608 q^{82} + 8019488 q^{83} - 27697464 q^{84} - 13052408 q^{85} - 15776420 q^{86} - 1099116 q^{87} + 16181804 q^{88} - 26507624 q^{89} - 11066220 q^{90} + 1221352 q^{91} + 4374648 q^{93} + 19106984 q^{94} + 35737608 q^{95} - 19873620 q^{96} + 18302620 q^{97} - 38455712 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(69))\) 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
69.8.a.a 69.a 1.a $5$ $21.555$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-135\) \(-266\) \(-496\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3^{3}q^{3}+(54+2\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
69.8.a.b 69.a 1.a $6$ $21.555$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-8\) \(162\) \(-372\) \(-1104\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+3^{3}q^{3}+(29+3\beta _{1}+\cdots)q^{4}+\cdots\)
69.8.a.c 69.a 1.a $7$ $21.555$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-189\) \(-516\) \(1018\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3^{3}q^{3}+(93+\beta _{1}+\beta _{2})q^{4}+\cdots\)
69.8.a.d 69.a 1.a $8$ $21.555$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(24\) \(216\) \(378\) \(126\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{2}+3^{3}q^{3}+(70-4\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(69))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(69)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)