Properties

Label 450.2.e
Level $450$
Weight $2$
Character orbit 450.e
Rep. character $\chi_{450}(151,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $38$
Newform subspaces $14$
Sturm bound $180$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 14 \)
Sturm bound: \(180\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(450, [\chi])\).

Total New Old
Modular forms 204 38 166
Cusp forms 156 38 118
Eisenstein series 48 0 48

Trace form

\( 38q - q^{2} + q^{3} - 19q^{4} - 5q^{6} - 2q^{7} + 2q^{8} - 5q^{9} + O(q^{10}) \) \( 38q - q^{2} + q^{3} - 19q^{4} - 5q^{6} - 2q^{7} + 2q^{8} - 5q^{9} - q^{11} + 4q^{12} - 2q^{13} + 2q^{14} - 19q^{16} + 18q^{17} + 2q^{18} - 2q^{19} - 2q^{21} + 3q^{22} + 6q^{23} + q^{24} - 20q^{26} + 16q^{27} + 4q^{28} + 18q^{29} - 8q^{31} - q^{32} + 15q^{33} + 3q^{34} - 5q^{36} + 16q^{37} - 11q^{38} - 24q^{39} - 13q^{41} - 20q^{42} + q^{43} + 2q^{44} + 12q^{46} - 18q^{47} - 5q^{48} - 27q^{49} + 37q^{51} - 2q^{52} - 48q^{53} - 17q^{54} + 2q^{56} + 23q^{57} + 6q^{58} + 43q^{59} - 20q^{61} + 16q^{62} - 28q^{63} + 38q^{64} + 26q^{66} - 17q^{67} - 9q^{68} - 8q^{69} - 24q^{71} - 13q^{72} + 22q^{73} + 8q^{74} + q^{76} - 42q^{77} + 10q^{78} - 8q^{79} - 41q^{81} - 30q^{82} + 24q^{83} - 14q^{84} + 35q^{86} - 54q^{87} + 3q^{88} + 92q^{89} + 56q^{91} + 6q^{92} + 20q^{93} - 6q^{94} + 4q^{96} + 7q^{97} + 42q^{98} - 88q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
450.2.e.a \(2\) \(3.593\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-3\) \(0\) \(-4\) \(q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
450.2.e.b \(2\) \(3.593\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-3\) \(0\) \(1\) \(q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
450.2.e.c \(2\) \(3.593\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-2\) \(q+(-1+\zeta_{6})q^{2}+(-1+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
450.2.e.d \(2\) \(3.593\) \(\Q(\sqrt{-3}) \) None \(-1\) \(3\) \(0\) \(-4\) \(q+(-1+\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
450.2.e.e \(2\) \(3.593\) \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(0\) \(-1\) \(q+(1-\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
450.2.e.f \(2\) \(3.593\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(2\) \(q+(1-\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
450.2.e.g \(2\) \(3.593\) \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(0\) \(-1\) \(q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
450.2.e.h \(2\) \(3.593\) \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(0\) \(4\) \(q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
450.2.e.i \(2\) \(3.593\) \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(0\) \(2\) \(q+(1-\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
450.2.e.j \(4\) \(3.593\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(-2\) \(-2\) \(0\) \(1\) \(q+(-1+\beta _{2})q^{2}+(-\beta _{1}+\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots\)
450.2.e.k \(4\) \(3.593\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(-2\) \(2\) \(0\) \(4\) \(q-\beta _{2}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots\)
450.2.e.l \(4\) \(3.593\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(-2\) \(4\) \(0\) \(4\) \(q+(-1+\beta _{2})q^{2}+(1-\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots\)
450.2.e.m \(4\) \(3.593\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(2\) \(-4\) \(0\) \(-4\) \(q+(1-\beta _{2})q^{2}+(-1+\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots\)
450.2.e.n \(4\) \(3.593\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(2\) \(-2\) \(0\) \(-4\) \(q+\beta _{2}q^{2}+(-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)