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Properties

Label 225.4.e
Level $225$
Weight $4$
Character orbit 225.e
Rep. character $\chi_{225}(76,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $108$
Newform subspaces $7$
Sturm bound $120$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 7 \)
Sturm bound: \(120\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(225, [\chi])\).

Total New Old
Modular forms 192 120 72
Cusp forms 168 108 60
Eisenstein series 24 12 12

Trace form

\( 108 q - q^{2} + 5 q^{3} - 203 q^{4} - 27 q^{6} - 5 q^{7} + 42 q^{8} - 75 q^{9} + O(q^{10}) \) \( 108 q - q^{2} + 5 q^{3} - 203 q^{4} - 27 q^{6} - 5 q^{7} + 42 q^{8} - 75 q^{9} + 12 q^{11} - 158 q^{12} + 13 q^{13} + 174 q^{14} - 715 q^{16} + 302 q^{17} - 256 q^{18} - 146 q^{19} - 69 q^{21} + 3 q^{22} - 159 q^{23} + 135 q^{24} - 648 q^{26} + 308 q^{27} + 52 q^{28} + 69 q^{29} - 81 q^{31} + 61 q^{32} + 1024 q^{33} + 85 q^{34} + 2103 q^{36} - 236 q^{37} - 649 q^{38} - 711 q^{39} + 678 q^{41} - 2184 q^{42} - 86 q^{43} + 186 q^{44} + 512 q^{46} - q^{47} - 431 q^{48} - 1841 q^{49} - 75 q^{51} + 532 q^{52} + 1916 q^{53} - 1743 q^{54} + 1974 q^{56} + 1475 q^{57} + 534 q^{58} + 738 q^{59} - 405 q^{61} + 1416 q^{62} - 2331 q^{63} + 4562 q^{64} + 1116 q^{66} - 374 q^{67} - 2039 q^{68} + 1917 q^{69} - 6672 q^{71} + 3069 q^{72} - 902 q^{73} + 1764 q^{74} + 251 q^{76} - 1221 q^{77} + 6508 q^{78} - 383 q^{79} + 3489 q^{81} - 138 q^{82} - 1695 q^{83} - 4524 q^{84} - 591 q^{86} - 6127 q^{87} + 1653 q^{88} - 4488 q^{89} - 1934 q^{91} - 4998 q^{92} - 1251 q^{93} + 814 q^{94} - 7308 q^{96} + 1210 q^{97} - 1850 q^{98} - 27 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(225, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
225.4.e.a 225.e 9.c $4$ $13.275$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(-1\) \(6\) \(0\) \(9\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{3})q^{2}+(1-\beta _{1}-\beta _{2}+2\beta _{3})q^{3}+\cdots\)
225.4.e.b 225.e 9.c $4$ $13.275$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(3\) \(3\) \(0\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{3})q^{2}+(1-\beta _{1}-\beta _{2}-\beta _{3})q^{3}+\cdots\)
225.4.e.c 225.e 9.c $6$ $13.275$ 6.0.15759792.1 None \(-1\) \(-9\) \(0\) \(-43\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{5})q^{2}+(-1-\beta _{2}+\beta _{5})q^{3}+\cdots\)
225.4.e.d 225.e 9.c $14$ $13.275$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(-2\) \(5\) \(0\) \(22\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{2})q^{2}+(-\beta _{7}+\beta _{10})q^{3}+(-5+\cdots)q^{4}+\cdots\)
225.4.e.e 225.e 9.c $24$ $13.275$ None \(-4\) \(-1\) \(0\) \(6\) $\mathrm{SU}(2)[C_{3}]$
225.4.e.f 225.e 9.c $24$ $13.275$ None \(4\) \(1\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{3}]$
225.4.e.g 225.e 9.c $32$ $13.275$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{4}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(225, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)