Properties

Label 650.6.b
Level $650$
Weight $6$
Character orbit 650.b
Rep. character $\chi_{650}(599,\cdot)$
Character field $\Q$
Dimension $90$
Newform subspaces $15$
Sturm bound $630$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 650.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(630\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(650, [\chi])\).

Total New Old
Modular forms 538 90 448
Cusp forms 514 90 424
Eisenstein series 24 0 24

Trace form

\( 90 q - 1440 q^{4} - 304 q^{6} - 6282 q^{9} + O(q^{10}) \) \( 90 q - 1440 q^{4} - 304 q^{6} - 6282 q^{9} + 148 q^{11} + 160 q^{14} + 23040 q^{16} + 1444 q^{19} + 14368 q^{21} + 4864 q^{24} - 4056 q^{26} - 9792 q^{29} + 7728 q^{31} + 9248 q^{34} + 100512 q^{36} - 20376 q^{41} - 2368 q^{44} + 47936 q^{46} - 256722 q^{49} + 29208 q^{51} + 15856 q^{54} - 2560 q^{56} + 122096 q^{59} - 237040 q^{61} - 368640 q^{64} + 282128 q^{66} + 35248 q^{69} - 184880 q^{71} - 15360 q^{74} - 23104 q^{76} - 39608 q^{79} + 394314 q^{81} - 229888 q^{84} + 212320 q^{86} - 110112 q^{89} - 181844 q^{91} + 192800 q^{94} - 77824 q^{96} - 1134344 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(650, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
650.6.b.a 650.b 5.b $2$ $104.249$ \(\Q(\sqrt{-1}) \) None 26.6.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{2}-2^{4}q^{4}+170iq^{7}-2^{6}iq^{8}+\cdots\)
650.6.b.b 650.b 5.b $4$ $104.249$ \(\Q(i, \sqrt{235})\) None 130.6.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}+(-13\beta _{1}+\beta _{3})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.c 650.b 5.b $4$ $104.249$ \(\Q(i, \sqrt{14})\) None 130.6.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{1}q^{2}+(8\beta _{1}+3\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.d 650.b 5.b $4$ $104.249$ \(\Q(i, \sqrt{2785})\) None 26.6.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{2}q^{2}+(\beta _{1}-4\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.e 650.b 5.b $4$ $104.249$ \(\Q(i, \sqrt{19})\) None 130.6.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}+(-3\beta _{1}+3\beta _{3})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.f 650.b 5.b $4$ $104.249$ \(\Q(i, \sqrt{145})\) None 130.6.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}+(-3\beta _{1}+\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.g 650.b 5.b $4$ $104.249$ \(\Q(i, \sqrt{10})\) None 130.6.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}+(2\beta _{1}+\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.h 650.b 5.b $4$ $104.249$ \(\Q(i, \sqrt{849})\) None 26.6.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{2}q^{2}+(\beta _{1}-4\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.i 650.b 5.b $6$ $104.249$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 130.6.a.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{2}q^{2}+(\beta _{1}+3\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.j 650.b 5.b $6$ $104.249$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 130.6.a.f \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}+(7\beta _{1}+\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.k 650.b 5.b $8$ $104.249$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 650.6.a.l \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{2}q^{2}+(\beta _{1}+5\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.l 650.b 5.b $8$ $104.249$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 130.6.a.h \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{1}q^{2}+(-5\beta _{1}-\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.m 650.b 5.b $10$ $104.249$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None 650.6.a.o \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{5}q^{2}+(\beta _{1}-2\beta _{5})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.n 650.b 5.b $10$ $104.249$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None 650.6.a.n \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{6}q^{2}+\beta _{1}q^{3}-2^{4}q^{4}-4\beta _{2}q^{6}+\cdots\)
650.6.b.o 650.b 5.b $12$ $104.249$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 650.6.a.s \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{7}q^{2}+(\beta _{1}+2\beta _{7})q^{3}-2^{4}q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(650, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(650, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 2}\)