Properties

Label 507.4.b
Level $507$
Weight $4$
Character orbit 507.b
Rep. character $\chi_{507}(337,\cdot)$
Character field $\Q$
Dimension $76$
Newform subspaces $11$
Sturm bound $242$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(242\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(507, [\chi])\).

Total New Old
Modular forms 196 76 120
Cusp forms 168 76 92
Eisenstein series 28 0 28

Trace form

\( 76 q - 296 q^{4} + 684 q^{9} + O(q^{10}) \) \( 76 q - 296 q^{4} + 684 q^{9} + 32 q^{10} - 48 q^{12} + 304 q^{14} + 936 q^{16} - 316 q^{17} - 408 q^{22} + 80 q^{23} - 2100 q^{25} + 308 q^{29} + 528 q^{30} - 1040 q^{35} - 2664 q^{36} + 224 q^{38} + 280 q^{40} - 1296 q^{42} + 160 q^{43} + 1104 q^{48} - 3024 q^{49} + 192 q^{51} + 196 q^{53} - 208 q^{55} - 2208 q^{56} + 2340 q^{61} - 2788 q^{62} - 6012 q^{64} - 2184 q^{66} + 6348 q^{68} + 1272 q^{69} - 540 q^{74} + 480 q^{75} + 392 q^{77} - 5060 q^{79} + 6156 q^{81} + 3668 q^{82} + 1212 q^{87} - 2240 q^{88} + 288 q^{90} + 8620 q^{92} - 5520 q^{94} - 2000 q^{95} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(507, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
507.4.b.a 507.b 13.b $2$ $29.914$ \(\Q(\sqrt{-3}) \) None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-3q^{3}+8q^{4}+3\zeta_{6}q^{5}-6\zeta_{6}q^{7}+\cdots\)
507.4.b.b 507.b 13.b $2$ $29.914$ \(\Q(\sqrt{-1}) \) None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-3q^{3}+8q^{4}-6iq^{5}-iq^{7}+9q^{9}+\cdots\)
507.4.b.c 507.b 13.b $2$ $29.914$ \(\Q(\sqrt{-1}) \) None \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{2}+3q^{3}-q^{4}-9iq^{5}+9iq^{6}+\cdots\)
507.4.b.d 507.b 13.b $2$ $29.914$ \(\Q(\sqrt{-1}) \) None \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+3q^{3}+7q^{4}-7iq^{5}+3iq^{6}+\cdots\)
507.4.b.e 507.b 13.b $4$ $29.914$ \(\Q(\sqrt{-3}, \sqrt{-17})\) None \(0\) \(-12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-3q^{3}-9q^{4}+(-2\beta _{1}+3\beta _{2}+\cdots)q^{5}+\cdots\)
507.4.b.f 507.b 13.b $4$ $29.914$ \(\Q(i, \sqrt{14})\) None \(0\) \(-12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}-3q^{3}+(-7-\beta _{3})q^{4}+(7\beta _{1}+\cdots)q^{5}+\cdots\)
507.4.b.g 507.b 13.b $6$ $29.914$ 6.0.158155776.1 None \(0\) \(18\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{2}+3q^{3}+(-3+\beta _{1})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
507.4.b.h 507.b 13.b $8$ $29.914$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(-24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}-3q^{3}+(-6+\beta _{1})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
507.4.b.i 507.b 13.b $10$ $29.914$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(30\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+3q^{3}+(-6+\beta _{4})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
507.4.b.j 507.b 13.b $18$ $29.914$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None \(0\) \(-54\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{10}q^{2}-3q^{3}+(-4+\beta _{2})q^{4}+(2\beta _{10}+\cdots)q^{5}+\cdots\)
507.4.b.k 507.b 13.b $18$ $29.914$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None \(0\) \(54\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{9}q^{2}+3q^{3}+(-5+\beta _{2})q^{4}-\beta _{15}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(507, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(507, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)