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Properties

Label 57.6.a
Level $57$
Weight $6$
Character orbit 57.a
Rep. character $\chi_{57}(1,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $6$
Sturm bound $40$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 57.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(40\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(57))\).

Total New Old
Modular forms 36 14 22
Cusp forms 32 14 18
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.

\(3\)\(19\)FrickeDim
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(3\)
\(-\)\(+\)$-$\(5\)
\(-\)\(-\)$+$\(2\)
Plus space\(+\)\(6\)
Minus space\(-\)\(8\)

Trace form

\( 14 q + 12 q^{2} + 236 q^{4} + 10 q^{5} - 72 q^{6} - 2 q^{7} + 156 q^{8} + 1134 q^{9} + O(q^{10}) \) \( 14 q + 12 q^{2} + 236 q^{4} + 10 q^{5} - 72 q^{6} - 2 q^{7} + 156 q^{8} + 1134 q^{9} - 696 q^{10} + 1010 q^{11} + 792 q^{12} + 436 q^{13} - 4636 q^{14} + 396 q^{15} + 6692 q^{16} - 2594 q^{17} + 972 q^{18} - 1444 q^{19} + 6512 q^{20} - 8292 q^{22} + 2684 q^{23} - 3564 q^{24} + 11000 q^{25} + 2368 q^{26} + 5728 q^{28} - 24080 q^{29} - 7704 q^{30} - 16556 q^{31} - 9156 q^{32} + 13032 q^{33} + 50364 q^{34} - 2382 q^{35} + 19116 q^{36} + 1120 q^{37} - 1008 q^{39} - 37644 q^{40} + 42412 q^{41} - 16128 q^{42} - 56546 q^{43} - 66992 q^{44} + 810 q^{45} + 6852 q^{46} - 3426 q^{47} + 70128 q^{48} + 39540 q^{49} - 99656 q^{50} + 15372 q^{51} + 11524 q^{52} - 23632 q^{53} - 5832 q^{54} + 42954 q^{55} - 109968 q^{56} - 6498 q^{57} - 5256 q^{58} + 30992 q^{59} + 107388 q^{60} + 10042 q^{61} - 24400 q^{62} - 162 q^{63} + 253772 q^{64} - 106260 q^{65} - 85032 q^{66} - 29816 q^{67} - 284440 q^{68} + 31644 q^{69} + 36276 q^{70} + 24984 q^{71} + 12636 q^{72} - 84482 q^{73} + 181928 q^{74} + 86040 q^{75} - 57760 q^{76} + 47774 q^{77} + 6300 q^{78} + 315136 q^{79} + 604016 q^{80} + 91854 q^{81} - 108864 q^{82} - 112244 q^{83} - 132552 q^{84} - 95094 q^{85} - 232836 q^{86} - 236196 q^{87} - 313200 q^{88} + 378064 q^{89} - 56376 q^{90} + 599060 q^{91} + 112360 q^{92} + 67680 q^{93} - 307920 q^{94} - 72922 q^{95} - 33300 q^{96} - 201020 q^{97} + 361712 q^{98} + 81810 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(57))\) 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 19
57.6.a.a 57.a 1.a $1$ $9.142$ \(\Q\) None \(-2\) \(9\) \(-98\) \(240\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+9q^{3}-28q^{4}-98q^{5}-18q^{6}+\cdots\)
57.6.a.b 57.a 1.a $1$ $9.142$ \(\Q\) None \(11\) \(9\) \(6\) \(-176\) $-$ $+$ $\mathrm{SU}(2)$ \(q+11q^{2}+9q^{3}+89q^{4}+6q^{5}+99q^{6}+\cdots\)
57.6.a.c 57.a 1.a $2$ $9.142$ \(\Q(\sqrt{17}) \) None \(-3\) \(18\) \(-87\) \(-251\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+9q^{3}+(7+3\beta )q^{4}+\cdots\)
57.6.a.d 57.a 1.a $3$ $9.142$ 3.3.616092.1 None \(-4\) \(27\) \(206\) \(186\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+9q^{3}+(26+4\beta _{1}+\cdots)q^{4}+\cdots\)
57.6.a.e 57.a 1.a $3$ $9.142$ 3.3.9153.1 None \(9\) \(-27\) \(-9\) \(141\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{2}-9q^{3}+(7-7\beta _{1}+2\beta _{2})q^{4}+\cdots\)
57.6.a.f 57.a 1.a $4$ $9.142$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(1\) \(-36\) \(-8\) \(-142\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-9q^{3}+(14-\beta _{1}-\beta _{2}+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(57))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(57)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)