Properties

Label 618.2.a
Level $618$
Weight $2$
Character orbit 618.a
Rep. character $\chi_{618}(1,\cdot)$
Character field $\Q$
Dimension $17$
Newform subspaces $11$
Sturm bound $208$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 618 = 2 \cdot 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 618.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(208\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(618))\).

Total New Old
Modular forms 108 17 91
Cusp forms 101 17 84
Eisenstein series 7 0 7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(103\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(11\)

Trace form

\( 17q + q^{2} - q^{3} + 17q^{4} + 6q^{5} - q^{6} + q^{8} + 17q^{9} + O(q^{10}) \) \( 17q + q^{2} - q^{3} + 17q^{4} + 6q^{5} - q^{6} + q^{8} + 17q^{9} - 2q^{10} - 4q^{11} - q^{12} - 6q^{13} - 6q^{15} + 17q^{16} + 2q^{17} + q^{18} - 16q^{19} + 6q^{20} - 8q^{21} - 4q^{22} - q^{24} + 19q^{25} - 2q^{26} - q^{27} + 6q^{29} - 2q^{30} - 24q^{31} + q^{32} - 8q^{33} + 18q^{34} + 24q^{35} + 17q^{36} - 2q^{37} + 4q^{38} - 6q^{39} - 2q^{40} + 2q^{41} + 4q^{43} - 4q^{44} + 6q^{45} - 8q^{46} + 8q^{47} - q^{48} + 9q^{49} + 31q^{50} - 2q^{51} - 6q^{52} - 2q^{53} - q^{54} + 24q^{55} - 4q^{57} + 2q^{58} - 20q^{59} - 6q^{60} + 34q^{61} + 24q^{62} + 17q^{64} + 20q^{65} + 4q^{66} - 20q^{67} + 2q^{68} + 8q^{69} + 8q^{70} - 32q^{71} + q^{72} - 54q^{73} - 2q^{74} + q^{75} - 16q^{76} - 16q^{77} - 14q^{78} - 16q^{79} + 6q^{80} + 17q^{81} + 10q^{82} - 28q^{83} - 8q^{84} - 28q^{85} - 4q^{86} + 2q^{87} - 4q^{88} - 30q^{89} - 2q^{90} + 24q^{91} - 28q^{93} - 16q^{94} - 32q^{95} - q^{96} + 10q^{97} + 9q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(618))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 103
618.2.a.a \(1\) \(4.935\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
618.2.a.b \(1\) \(4.935\) \(\Q\) None \(-1\) \(-1\) \(2\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-2q^{7}+\cdots\)
618.2.a.c \(1\) \(4.935\) \(\Q\) None \(-1\) \(1\) \(-3\) \(2\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}+2q^{7}+\cdots\)
618.2.a.d \(1\) \(4.935\) \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-4q^{7}-q^{8}+\cdots\)
618.2.a.e \(1\) \(4.935\) \(\Q\) None \(1\) \(-1\) \(-2\) \(-2\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}-2q^{7}+\cdots\)
618.2.a.f \(1\) \(4.935\) \(\Q\) None \(1\) \(1\) \(-4\) \(-4\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}-4q^{5}+q^{6}-4q^{7}+\cdots\)
618.2.a.g \(1\) \(4.935\) \(\Q\) None \(1\) \(1\) \(3\) \(-2\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}-2q^{7}+\cdots\)
618.2.a.h \(2\) \(4.935\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(6\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+(1+\beta )q^{5}+q^{6}+\cdots\)
618.2.a.i \(2\) \(4.935\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(4\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+(2+\beta )q^{5}-q^{6}+\cdots\)
618.2.a.j \(2\) \(4.935\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+\beta q^{5}+q^{6}+(2+\beta )q^{7}+\cdots\)
618.2.a.k \(4\) \(4.935\) 4.4.54332.1 None \(4\) \(-4\) \(5\) \(4\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+(1+\beta _{1})q^{5}-q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(618))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(618)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(103))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(206))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(309))\)\(^{\oplus 2}\)