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Properties

Label 58.4.a
Level $58$
Weight $4$
Character orbit 58.a
Rep. character $\chi_{58}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $4$
Sturm bound $30$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 58.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(30\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(58))\).

Total New Old
Modular forms 25 7 18
Cusp forms 21 7 14
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.

\(2\)\(29\)FrickeDim
\(+\)\(+\)$+$\(1\)
\(+\)\(-\)$-$\(2\)
\(-\)\(+\)$-$\(1\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(4\)
Minus space\(-\)\(3\)

Trace form

\( 7 q + 2 q^{2} + 28 q^{4} - 20 q^{6} - 12 q^{7} + 8 q^{8} + 9 q^{9} + O(q^{10}) \) \( 7 q + 2 q^{2} + 28 q^{4} - 20 q^{6} - 12 q^{7} + 8 q^{8} + 9 q^{9} + 20 q^{10} - 16 q^{11} - 84 q^{13} + 48 q^{14} - 52 q^{15} + 112 q^{16} - 130 q^{17} + 90 q^{18} + 8 q^{19} + 136 q^{21} + 180 q^{22} - 412 q^{23} - 80 q^{24} + 311 q^{25} - 76 q^{26} - 72 q^{27} - 48 q^{28} + 87 q^{29} + 4 q^{30} - 336 q^{31} + 32 q^{32} + 66 q^{33} - 180 q^{34} - 212 q^{35} + 36 q^{36} - 482 q^{37} - 64 q^{38} + 148 q^{39} + 80 q^{40} + 938 q^{41} - 120 q^{42} + 608 q^{43} - 64 q^{44} + 546 q^{45} - 600 q^{46} - 320 q^{47} + 391 q^{49} + 94 q^{50} + 884 q^{51} - 336 q^{52} - 736 q^{53} - 284 q^{54} + 156 q^{55} + 192 q^{56} - 912 q^{57} + 58 q^{58} - 280 q^{59} - 208 q^{60} + 90 q^{61} - 364 q^{62} - 1096 q^{63} + 448 q^{64} + 3310 q^{65} - 1408 q^{66} - 660 q^{67} - 520 q^{68} + 2516 q^{69} + 1600 q^{70} + 2176 q^{71} + 360 q^{72} + 370 q^{73} + 316 q^{74} - 3960 q^{75} + 32 q^{76} + 1320 q^{77} - 276 q^{78} - 1136 q^{79} - 2721 q^{81} - 884 q^{82} - 1096 q^{83} + 544 q^{84} - 552 q^{85} + 212 q^{86} + 720 q^{88} - 614 q^{89} - 724 q^{90} - 2380 q^{91} - 1648 q^{92} - 1226 q^{93} + 1828 q^{94} - 5448 q^{95} - 320 q^{96} - 134 q^{97} + 1298 q^{98} + 2832 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(58))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 29
58.4.a.a 58.a 1.a $1$ $3.422$ \(\Q\) None \(-2\) \(7\) \(5\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+7q^{3}+4q^{4}+5q^{5}-14q^{6}+\cdots\)
58.4.a.b 58.a 1.a $1$ $3.422$ \(\Q\) None \(2\) \(-7\) \(-15\) \(-18\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-7q^{3}+4q^{4}-15q^{5}-14q^{6}+\cdots\)
58.4.a.c 58.a 1.a $2$ $3.422$ \(\Q(\sqrt{6}) \) None \(-4\) \(-2\) \(-10\) \(-16\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+\beta )q^{3}+4q^{4}+(-5+\cdots)q^{5}+\cdots\)
58.4.a.d 58.a 1.a $3$ $3.422$ 3.3.19816.1 None \(6\) \(2\) \(20\) \(24\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(6+2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(58))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(58)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 2}\)