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Properties

Label 51.4.a
Level $51$
Weight $4$
Character orbit 51.a
Rep. character $\chi_{51}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $5$
Sturm bound $24$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 51.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(24\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(51))\).

Total New Old
Modular forms 20 8 12
Cusp forms 16 8 8
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.

\(3\)\(17\)FrickeDim
\(+\)\(+\)$+$\(3\)
\(+\)\(-\)$-$\(1\)
\(-\)\(+\)$-$\(1\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(2\)

Trace form

\( 8 q + 4 q^{2} + 12 q^{4} + 12 q^{6} + 8 q^{7} + 48 q^{8} + 72 q^{9} + O(q^{10}) \) \( 8 q + 4 q^{2} + 12 q^{4} + 12 q^{6} + 8 q^{7} + 48 q^{8} + 72 q^{9} + 100 q^{10} + 16 q^{11} + 52 q^{13} - 288 q^{14} - 72 q^{15} - 44 q^{16} + 36 q^{18} - 204 q^{19} + 32 q^{20} - 84 q^{21} - 284 q^{22} + 144 q^{24} + 532 q^{25} - 204 q^{26} - 720 q^{28} + 432 q^{29} - 408 q^{30} - 48 q^{31} - 312 q^{32} - 132 q^{33} + 136 q^{34} + 480 q^{35} + 108 q^{36} + 192 q^{37} - 756 q^{38} - 24 q^{39} + 176 q^{40} + 1440 q^{41} + 252 q^{42} - 700 q^{43} + 1056 q^{44} + 820 q^{46} + 104 q^{47} - 96 q^{48} + 496 q^{49} + 48 q^{50} + 204 q^{51} + 1072 q^{52} - 632 q^{53} + 108 q^{54} + 124 q^{55} - 1056 q^{56} + 312 q^{57} + 400 q^{58} - 72 q^{59} - 12 q^{60} + 64 q^{61} - 248 q^{62} + 72 q^{63} - 1580 q^{64} - 656 q^{65} - 348 q^{66} - 1688 q^{67} - 72 q^{69} - 40 q^{70} - 2128 q^{71} + 432 q^{72} - 1048 q^{73} - 96 q^{74} - 1104 q^{75} - 400 q^{76} - 1272 q^{77} + 1620 q^{78} + 1768 q^{79} - 1968 q^{80} + 648 q^{81} - 1076 q^{82} + 2512 q^{83} - 588 q^{84} - 68 q^{85} + 780 q^{86} - 972 q^{87} - 184 q^{88} - 120 q^{89} + 900 q^{90} - 600 q^{91} - 352 q^{92} - 1836 q^{93} + 4632 q^{94} - 3024 q^{95} - 936 q^{96} + 3160 q^{97} + 4508 q^{98} + 144 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(51))\) 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 17
51.4.a.a 51.a 1.a $1$ $3.009$ \(\Q\) None \(-1\) \(-3\) \(16\) \(34\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}-7q^{4}+2^{4}q^{5}+3q^{6}+\cdots\)
51.4.a.b 51.a 1.a $1$ $3.009$ \(\Q\) None \(-1\) \(3\) \(-20\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}-7q^{4}-20q^{5}-3q^{6}+\cdots\)
51.4.a.c 51.a 1.a $1$ $3.009$ \(\Q\) None \(1\) \(-3\) \(-10\) \(-8\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}-7q^{4}-10q^{5}-3q^{6}+\cdots\)
51.4.a.d 51.a 1.a $2$ $3.009$ \(\Q(\sqrt{2}) \) None \(0\) \(-6\) \(6\) \(-8\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-3q^{3}+10q^{4}+(3+4\beta )q^{5}+\cdots\)
51.4.a.e 51.a 1.a $3$ $3.009$ 3.3.5912.1 None \(5\) \(9\) \(8\) \(-8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+3q^{3}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(51))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(51)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)