Properties

Label 51.2.d
Level $51$
Weight $2$
Character orbit 51.d
Rep. character $\chi_{51}(16,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $2$
Sturm bound $12$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 51.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(51, [\chi])\).

Total New Old
Modular forms 8 4 4
Cusp forms 4 4 0
Eisenstein series 4 0 4

Trace form

\( 4q - 2q^{2} + 2q^{4} - 6q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 2q^{4} - 6q^{8} - 4q^{9} + 2q^{13} - 6q^{15} - 10q^{16} + 10q^{17} + 2q^{18} + 2q^{19} + 4q^{21} + 2q^{25} + 8q^{26} + 12q^{30} + 26q^{32} + 2q^{33} - 14q^{34} - 12q^{35} - 2q^{36} - 28q^{38} + 16q^{42} - 10q^{43} - 20q^{47} - 12q^{49} + 26q^{50} - 10q^{51} - 8q^{52} + 30q^{55} - 24q^{59} - 12q^{60} - 2q^{64} - 28q^{66} + 14q^{68} + 10q^{69} + 24q^{70} + 6q^{72} + 28q^{76} + 52q^{77} + 4q^{81} + 12q^{83} - 16q^{84} - 6q^{85} - 4q^{86} - 12q^{87} - 40q^{89} + 28q^{93} - 8q^{94} - 30q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(51, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
51.2.d.a \(2\) \(0.407\) \(\Q(\sqrt{-1}) \) None \(-4\) \(0\) \(0\) \(0\) \(q-2q^{2}+iq^{3}+2q^{4}+3iq^{5}-2iq^{6}+\cdots\)
51.2.d.b \(2\) \(0.407\) \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(0\) \(0\) \(q+q^{2}+iq^{3}-q^{4}+iq^{6}-4iq^{7}+\cdots\)