↑

Properties

Label 57.8.a
Level $57$
Weight $8$
Character orbit 57.a
Rep. character $\chi_{57}(1,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $4$
Sturm bound $53$
Trace bound $1$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 57.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(53\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(57))\).

Total New Old
Modular forms 48 22 26
Cusp forms 44 22 22
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
\(+\)\(+\)$+$\(5\)
\(+\)\(-\)$-$\(6\)
\(-\)\(+\)$-$\(4\)
\(-\)\(-\)$+$\(7\)
Plus space\(+\)\(12\)
Minus space\(-\)\(10\)

Trace form

\( 22 q - 12 q^{2} + 1900 q^{4} - 778 q^{5} + 432 q^{6} - 370 q^{7} + 636 q^{8} + 16038 q^{9} + O(q^{10}) \) \( 22 q - 12 q^{2} + 1900 q^{4} - 778 q^{5} + 432 q^{6} - 370 q^{7} + 636 q^{8} + 16038 q^{9} - 2232 q^{10} + 4090 q^{11} - 15336 q^{12} + 7124 q^{13} + 8356 q^{14} + 21276 q^{15} + 108772 q^{16} - 41254 q^{17} - 8748 q^{18} + 27436 q^{19} - 195920 q^{20} + 93036 q^{22} + 97660 q^{23} - 16524 q^{24} + 432628 q^{25} + 70640 q^{26} - 151360 q^{28} - 155224 q^{29} + 283608 q^{30} + 293444 q^{31} - 411684 q^{32} - 426168 q^{33} - 367188 q^{34} + 156714 q^{35} + 1385100 q^{36} - 1597960 q^{37} - 30672 q^{39} + 204276 q^{40} + 848972 q^{41} + 630720 q^{42} - 314410 q^{43} + 3801776 q^{44} - 567162 q^{45} - 1873884 q^{46} - 2184114 q^{47} - 1418256 q^{48} + 5942976 q^{49} + 1916240 q^{50} - 1025028 q^{51} + 32036 q^{52} - 3677528 q^{53} + 314928 q^{54} + 717834 q^{55} - 6518064 q^{56} + 370386 q^{57} - 3912072 q^{58} - 2321552 q^{59} + 2150172 q^{60} + 3590006 q^{61} - 3778640 q^{62} - 269730 q^{63} + 10039756 q^{64} + 322116 q^{65} + 2037096 q^{66} - 1187992 q^{67} + 3656392 q^{68} - 1631340 q^{69} - 6445836 q^{70} - 13737960 q^{71} + 463644 q^{72} + 3559322 q^{73} - 17052296 q^{74} + 1151928 q^{75} + 4389760 q^{76} + 7297210 q^{77} + 9923580 q^{78} - 2100976 q^{79} - 50960816 q^{80} + 11691702 q^{81} - 9896544 q^{82} - 1249156 q^{83} + 20902968 q^{84} - 714306 q^{85} + 24820476 q^{86} + 3196908 q^{87} - 7897200 q^{88} - 18199288 q^{89} - 1627128 q^{90} - 1193660 q^{91} + 30872456 q^{92} - 3868128 q^{93} + 8371248 q^{94} - 7668362 q^{95} + 26704620 q^{96} + 5234180 q^{97} - 25856264 q^{98} + 2981610 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(57))\) 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
57.8.a.a 57.a 1.a $4$ $17.806$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-7\) \(108\) \(32\) \(-2074\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}+3^{3}q^{3}+(22-8\beta _{1}+\cdots)q^{4}+\cdots\)
57.8.a.b 57.a 1.a $5$ $17.806$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(1\) \(-135\) \(138\) \(670\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3^{3}q^{3}+(111+\beta _{4})q^{4}+(29+\cdots)q^{5}+\cdots\)
57.8.a.c 57.a 1.a $6$ $17.806$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-15\) \(-162\) \(-921\) \(-855\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{2}-3^{3}q^{3}+(113+\beta _{1}+\cdots)q^{4}+\cdots\)
57.8.a.d 57.a 1.a $7$ $17.806$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(9\) \(189\) \(-27\) \(1889\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+3^{3}q^{3}+(83+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(57))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(57)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)