Properties

Label 69.4.a
Level $69$
Weight $4$
Character orbit 69.a
Rep. character $\chi_{69}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $4$
Sturm bound $32$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(32\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(69))\).

Total New Old
Modular forms 26 12 14
Cusp forms 22 12 10
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
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(2\)
\(-\)\(+\)$-$\(2\)
\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(8\)
Minus space\(-\)\(4\)

Trace form

\( 12 q - 4 q^{2} + 56 q^{4} - 16 q^{5} + 12 q^{6} - 20 q^{7} + 36 q^{8} + 108 q^{9} + O(q^{10}) \) \( 12 q - 4 q^{2} + 56 q^{4} - 16 q^{5} + 12 q^{6} - 20 q^{7} + 36 q^{8} + 108 q^{9} + 20 q^{10} - 64 q^{13} + 156 q^{14} - 84 q^{15} + 328 q^{16} + 96 q^{17} - 36 q^{18} - 56 q^{19} - 636 q^{20} - 84 q^{21} - 244 q^{22} + 324 q^{24} + 420 q^{25} - 296 q^{26} - 824 q^{28} - 464 q^{29} - 408 q^{30} + 200 q^{31} + 844 q^{32} + 60 q^{33} - 584 q^{34} + 904 q^{35} + 504 q^{36} + 60 q^{37} - 196 q^{38} - 24 q^{39} + 124 q^{40} - 736 q^{41} - 876 q^{42} + 488 q^{43} - 800 q^{44} - 144 q^{45} + 184 q^{46} + 368 q^{47} - 96 q^{48} + 44 q^{49} - 1092 q^{50} - 120 q^{51} - 3064 q^{52} + 1848 q^{53} + 108 q^{54} + 1200 q^{55} + 1300 q^{56} + 468 q^{57} + 2616 q^{58} + 2056 q^{59} + 216 q^{60} - 1860 q^{61} - 864 q^{62} - 180 q^{63} + 1240 q^{64} + 176 q^{65} + 96 q^{66} - 1368 q^{67} + 2524 q^{68} + 276 q^{69} + 2984 q^{70} + 1208 q^{71} + 324 q^{72} - 696 q^{73} + 696 q^{74} - 1152 q^{75} + 2444 q^{76} - 1104 q^{77} + 192 q^{78} + 2660 q^{79} - 9044 q^{80} + 972 q^{81} + 976 q^{82} + 2008 q^{83} - 552 q^{84} - 2048 q^{85} - 1444 q^{86} - 48 q^{87} - 4820 q^{88} + 176 q^{89} + 180 q^{90} - 904 q^{91} - 1056 q^{93} - 5000 q^{94} - 2592 q^{95} + 3084 q^{96} - 1760 q^{97} - 2068 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(69))\) 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
69.4.a.a 69.a 1.a $2$ $4.071$ \(\Q(\sqrt{5}) \) None \(-4\) \(6\) \(-26\) \(-10\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{2}+3q^{3}+(1+4\beta )q^{4}+\cdots\)
69.4.a.b 69.a 1.a $2$ $4.071$ \(\Q(\sqrt{2}) \) None \(-2\) \(-6\) \(8\) \(-28\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-3q^{3}+(1-2\beta )q^{4}+\cdots\)
69.4.a.c 69.a 1.a $4$ $4.071$ 4.4.1140200.1 None \(-2\) \(-12\) \(-2\) \(32\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(6-\beta _{1}-2\beta _{2})q^{4}+\cdots\)
69.4.a.d 69.a 1.a $4$ $4.071$ 4.4.2009704.1 None \(4\) \(12\) \(4\) \(-14\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(7-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(69))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(69)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)