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Properties

Label 471.6.a
Level $471$
Weight $6$
Character orbit 471.a
Rep. character $\chi_{471}(1,\cdot)$
Character field $\Q$
Dimension $130$
Newform subspaces $4$
Sturm bound $316$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 471.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(316\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(471))\).

Total New Old
Modular forms 266 130 136
Cusp forms 262 130 132
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.

\(3\)\(157\)FrickeDim
\(+\)\(+\)$+$\(30\)
\(+\)\(-\)$-$\(35\)
\(-\)\(+\)$-$\(38\)
\(-\)\(-\)$+$\(27\)
Plus space\(+\)\(57\)
Minus space\(-\)\(73\)

Trace form

\( 130 q + 4 q^{2} + 2104 q^{4} + 108 q^{6} - 120 q^{7} - 228 q^{8} + 10530 q^{9} + O(q^{10}) \) \( 130 q + 4 q^{2} + 2104 q^{4} + 108 q^{6} - 120 q^{7} - 228 q^{8} + 10530 q^{9} + 596 q^{10} + 852 q^{11} + 404 q^{13} - 4636 q^{14} - 504 q^{15} + 33512 q^{16} + 1444 q^{17} + 324 q^{18} + 2360 q^{19} + 780 q^{20} - 1764 q^{21} - 14328 q^{22} - 5708 q^{23} - 3024 q^{24} + 79462 q^{25} + 15448 q^{26} + 7212 q^{28} - 2472 q^{29} + 8352 q^{30} + 5812 q^{31} + 17108 q^{32} + 13032 q^{33} - 1276 q^{34} - 14436 q^{35} + 170424 q^{36} + 32120 q^{37} + 13584 q^{38} + 14840 q^{40} - 2764 q^{41} - 9072 q^{42} - 1360 q^{43} - 56412 q^{44} - 1408 q^{46} + 42136 q^{47} - 17424 q^{48} + 321678 q^{49} + 154316 q^{50} - 7956 q^{51} + 104144 q^{52} + 24284 q^{53} + 8748 q^{54} - 87520 q^{55} - 157884 q^{56} - 9216 q^{57} - 33416 q^{58} + 75948 q^{59} - 25776 q^{60} - 52092 q^{61} - 89136 q^{62} - 9720 q^{63} + 600612 q^{64} - 3712 q^{65} - 16632 q^{66} - 131508 q^{67} + 158632 q^{68} + 27792 q^{69} - 178816 q^{70} - 22636 q^{71} - 18468 q^{72} - 43996 q^{73} - 43284 q^{74} + 37728 q^{75} + 254944 q^{76} + 123312 q^{77} + 95076 q^{78} - 29916 q^{79} + 65488 q^{80} + 852930 q^{81} - 30032 q^{82} - 30008 q^{83} + 78408 q^{84} + 150200 q^{85} + 43024 q^{86} - 196380 q^{87} - 102840 q^{88} - 157588 q^{89} + 48276 q^{90} + 251300 q^{91} - 936884 q^{92} + 61488 q^{93} - 297824 q^{94} + 25768 q^{95} - 176940 q^{96} + 53860 q^{97} - 31720 q^{98} + 69012 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(471))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 157
471.6.a.a 471.a 1.a $27$ $75.541$ None \(-2\) \(243\) \(-164\) \(-618\) $-$ $-$ $\mathrm{SU}(2)$
471.6.a.b 471.a 1.a $30$ $75.541$ None \(-8\) \(-270\) \(-136\) \(68\) $+$ $+$ $\mathrm{SU}(2)$
471.6.a.c 471.a 1.a $35$ $75.541$ None \(4\) \(-315\) \(164\) \(-30\) $+$ $-$ $\mathrm{SU}(2)$
471.6.a.d 471.a 1.a $38$ $75.541$ None \(10\) \(342\) \(136\) \(460\) $-$ $+$ $\mathrm{SU}(2)$

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(471))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(471)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(157))\)\(^{\oplus 2}\)