Properties

Label 550.6.b
Level $550$
Weight $6$
Character orbit 550.b
Rep. character $\chi_{550}(199,\cdot)$
Character field $\Q$
Dimension $74$
Newform subspaces $16$
Sturm bound $540$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 550.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(540\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(550, [\chi])\).

Total New Old
Modular forms 462 74 388
Cusp forms 438 74 364
Eisenstein series 24 0 24

Trace form

\( 74 q - 1184 q^{4} - 160 q^{6} - 4786 q^{9} + O(q^{10}) \) \( 74 q - 1184 q^{4} - 160 q^{6} - 4786 q^{9} - 242 q^{11} + 1728 q^{14} + 18944 q^{16} + 2440 q^{19} + 18256 q^{21} + 2560 q^{24} - 17664 q^{26} - 14132 q^{29} + 21708 q^{31} + 6608 q^{34} + 76576 q^{36} - 58984 q^{39} + 49668 q^{41} + 3872 q^{44} + 37920 q^{46} - 124626 q^{49} + 8896 q^{51} - 37760 q^{54} - 27648 q^{56} + 168672 q^{59} - 5100 q^{61} - 303104 q^{64} - 34848 q^{66} + 176696 q^{69} - 268844 q^{71} - 152112 q^{74} - 39040 q^{76} + 226648 q^{79} + 203050 q^{81} - 292096 q^{84} + 2000 q^{86} + 39720 q^{89} + 104648 q^{91} - 107168 q^{94} - 40960 q^{96} - 171094 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(550, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
550.6.b.a 550.b 5.b $2$ $88.211$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{2}+29iq^{3}-2^{4}q^{4}-116q^{6}+\cdots\)
550.6.b.b 550.b 5.b $2$ $88.211$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+6iq^{3}-2^{4}q^{4}-48q^{6}+\cdots\)
550.6.b.c 550.b 5.b $2$ $88.211$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+6iq^{3}-2^{4}q^{4}-48q^{6}+\cdots\)
550.6.b.d 550.b 5.b $2$ $88.211$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+iq^{3}-2^{4}q^{4}-8q^{6}-102iq^{7}+\cdots\)
550.6.b.e 550.b 5.b $2$ $88.211$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+iq^{3}-2^{4}q^{4}-8q^{6}+20iq^{7}+\cdots\)
550.6.b.f 550.b 5.b $2$ $88.211$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{2}+iq^{3}-2^{4}q^{4}-4q^{6}+166iq^{7}+\cdots\)
550.6.b.g 550.b 5.b $2$ $88.211$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4iq^{2}+21iq^{3}-2^{4}q^{4}+84q^{6}+\cdots\)
550.6.b.h 550.b 5.b $4$ $88.211$ \(\Q(i, \sqrt{889})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\beta _{2}q^{2}+(\beta _{1}-5\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
550.6.b.i 550.b 5.b $4$ $88.211$ \(\Q(i, \sqrt{1321})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\beta _{2}q^{2}+(-\beta _{1}-\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
550.6.b.j 550.b 5.b $4$ $88.211$ \(\Q(i, \sqrt{793})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\beta _{2}q^{2}+(-\beta _{1}+7\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
550.6.b.k 550.b 5.b $6$ $88.211$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\beta _{2}q^{2}+(\beta _{1}+\beta _{2})q^{3}-2^{4}q^{4}+(10+\cdots)q^{6}+\cdots\)
550.6.b.l 550.b 5.b $6$ $88.211$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta _{1}q^{2}+(-4\beta _{1}-\beta _{5})q^{3}-2^{4}q^{4}+\cdots\)
550.6.b.m 550.b 5.b $8$ $88.211$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{3}q^{2}+(\beta _{1}-4\beta _{3})q^{3}-2^{4}q^{4}+\cdots\)
550.6.b.n 550.b 5.b $8$ $88.211$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}+(-4\beta _{1}+\beta _{6})q^{3}-2^{4}q^{4}+\cdots\)
550.6.b.o 550.b 5.b $10$ $88.211$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{6}q^{2}+(\beta _{1}-\beta _{6})q^{3}-2^{4}q^{4}+(4+\cdots)q^{6}+\cdots\)
550.6.b.p 550.b 5.b $10$ $88.211$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{6}q^{2}+(\beta _{1}+\beta _{6})q^{3}-2^{4}q^{4}+(4+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(550, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(550, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(110, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(275, [\chi])\)\(^{\oplus 2}\)