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Properties

Label 177.6.a
Level $177$
Weight $6$
Character orbit 177.a
Rep. character $\chi_{177}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $4$
Sturm bound $120$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(177))\).

Total New Old
Modular forms 102 48 54
Cusp forms 98 48 50
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.

\(3\)\(59\)FrickeDim
\(+\)\(+\)$+$\(12\)
\(+\)\(-\)$-$\(13\)
\(-\)\(+\)$-$\(12\)
\(-\)\(-\)$+$\(11\)
Plus space\(+\)\(23\)
Minus space\(-\)\(25\)

Trace form

\( 48 q + 12 q^{2} - 18 q^{3} + 792 q^{4} - 12 q^{5} + 180 q^{6} + 4 q^{7} + 156 q^{8} + 3888 q^{9} + O(q^{10}) \) \( 48 q + 12 q^{2} - 18 q^{3} + 792 q^{4} - 12 q^{5} + 180 q^{6} + 4 q^{7} + 156 q^{8} + 3888 q^{9} - 524 q^{10} + 1524 q^{11} - 864 q^{12} - 1004 q^{13} - 2156 q^{14} - 504 q^{15} + 11000 q^{16} - 4420 q^{17} + 972 q^{18} - 2332 q^{19} - 9316 q^{20} + 720 q^{21} - 12904 q^{22} + 1280 q^{23} + 432 q^{24} + 41768 q^{25} + 5956 q^{26} - 1458 q^{27} + 16560 q^{28} - 18040 q^{29} + 8352 q^{30} + 15812 q^{31} - 2156 q^{32} + 360 q^{33} + 3160 q^{34} - 3712 q^{35} + 64152 q^{36} + 4844 q^{37} + 29668 q^{38} - 10476 q^{39} - 22092 q^{40} + 32608 q^{41} + 8352 q^{42} - 33344 q^{43} + 18204 q^{44} - 972 q^{45} - 36692 q^{46} - 8860 q^{47} - 32256 q^{48} + 80812 q^{49} - 85172 q^{50} - 18324 q^{51} + 31560 q^{52} - 9964 q^{53} + 14580 q^{54} + 56812 q^{55} - 60860 q^{56} + 24336 q^{57} - 122348 q^{58} + 30024 q^{60} - 79920 q^{61} + 232776 q^{62} + 324 q^{63} + 214204 q^{64} - 73796 q^{65} - 86760 q^{66} - 28336 q^{67} - 316328 q^{68} - 12924 q^{69} + 128556 q^{70} - 17556 q^{71} + 12636 q^{72} + 164832 q^{73} + 168032 q^{74} - 59886 q^{75} - 99872 q^{76} - 290508 q^{77} - 73044 q^{78} - 307920 q^{79} + 69856 q^{80} + 314928 q^{81} + 295140 q^{82} - 333972 q^{83} + 135432 q^{84} - 207884 q^{85} + 131048 q^{86} + 64008 q^{87} - 376260 q^{88} - 126816 q^{89} - 42444 q^{90} - 84968 q^{91} - 77420 q^{92} - 103716 q^{93} - 210576 q^{94} - 174036 q^{95} - 47916 q^{96} - 158084 q^{97} + 905776 q^{98} + 123444 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(177))\) 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 59
177.6.a.a 177.a 1.a $11$ $28.388$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-6\) \(99\) \(-192\) \(-371\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+9q^{3}+(14-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
177.6.a.b 177.a 1.a $12$ $28.388$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-4\) \(-108\) \(36\) \(-411\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-9q^{3}+(17+\beta _{2})q^{4}+(3+\beta _{1}+\cdots)q^{5}+\cdots\)
177.6.a.c 177.a 1.a $12$ $28.388$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(22\) \(108\) \(158\) \(413\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+9q^{3}+(17-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
177.6.a.d 177.a 1.a $13$ $28.388$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(0\) \(-117\) \(-14\) \(373\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-9q^{3}+(19+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(177))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(177)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 2}\)