Properties

Label 507.4.a
Level $507$
Weight $4$
Character orbit 507.a
Rep. character $\chi_{507}(1,\cdot)$
Character field $\Q$
Dimension $78$
Newform subspaces $18$
Sturm bound $242$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(242\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(507))\).

Total New Old
Modular forms 196 78 118
Cusp forms 168 78 90
Eisenstein series 28 0 28

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(20\)
\(+\)\(-\)\(-\)\(19\)
\(-\)\(+\)\(-\)\(16\)
\(-\)\(-\)\(+\)\(23\)
Plus space\(+\)\(43\)
Minus space\(-\)\(35\)

Trace form

\( 78 q - 4 q^{2} + 320 q^{4} - 16 q^{5} - 32 q^{7} - 48 q^{8} + 702 q^{9} + O(q^{10}) \) \( 78 q - 4 q^{2} + 320 q^{4} - 16 q^{5} - 32 q^{7} - 48 q^{8} + 702 q^{9} + 72 q^{10} + 96 q^{11} + 272 q^{14} + 24 q^{15} + 1384 q^{16} + 64 q^{17} - 36 q^{18} - 216 q^{19} + 92 q^{20} - 84 q^{21} + 272 q^{22} + 224 q^{23} + 180 q^{24} + 1850 q^{25} + 48 q^{28} - 544 q^{29} - 288 q^{30} - 128 q^{31} - 28 q^{32} - 192 q^{33} - 440 q^{34} - 224 q^{35} + 2880 q^{36} + 12 q^{37} - 664 q^{38} - 112 q^{40} - 296 q^{41} + 360 q^{42} + 416 q^{43} - 20 q^{44} - 144 q^{45} - 248 q^{46} + 48 q^{47} + 528 q^{48} + 4394 q^{49} + 380 q^{50} + 864 q^{51} + 600 q^{53} + 1024 q^{55} + 440 q^{56} + 84 q^{57} + 896 q^{58} + 1072 q^{59} + 1188 q^{60} - 584 q^{61} - 1068 q^{62} - 288 q^{63} + 7204 q^{64} - 1608 q^{66} + 360 q^{67} - 972 q^{68} - 168 q^{69} - 656 q^{70} + 976 q^{71} - 432 q^{72} + 1804 q^{73} - 2108 q^{74} - 480 q^{75} - 2248 q^{76} - 1704 q^{77} - 700 q^{79} + 700 q^{80} + 6318 q^{81} - 3948 q^{82} + 3072 q^{83} - 624 q^{84} - 2432 q^{85} - 2640 q^{86} + 1476 q^{87} - 1272 q^{88} - 1376 q^{89} + 648 q^{90} - 1156 q^{92} - 1428 q^{93} + 824 q^{94} + 4944 q^{95} - 840 q^{96} - 2820 q^{97} + 5644 q^{98} + 864 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(507))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 13
507.4.a.a 507.a 1.a $1$ $29.914$ \(\Q\) None \(-3\) \(3\) \(9\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+3q^{3}+q^{4}+9q^{5}-9q^{6}+\cdots\)
507.4.a.b 507.a 1.a $1$ $29.914$ \(\Q\) None \(-1\) \(3\) \(7\) \(-10\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}-7q^{4}+7q^{5}-3q^{6}+\cdots\)
507.4.a.c 507.a 1.a $1$ $29.914$ \(\Q\) None \(0\) \(-3\) \(12\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{4}+12q^{5}-2q^{7}+9q^{9}+\cdots\)
507.4.a.d 507.a 1.a $1$ $29.914$ \(\Q\) None \(1\) \(3\) \(-7\) \(10\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}-7q^{4}-7q^{5}+3q^{6}+\cdots\)
507.4.a.e 507.a 1.a $1$ $29.914$ \(\Q\) None \(3\) \(3\) \(-9\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}-9q^{5}+9q^{6}+\cdots\)
507.4.a.f 507.a 1.a $2$ $29.914$ \(\Q(\sqrt{14}) \) None \(-2\) \(-6\) \(-24\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-3q^{3}+(7-2\beta )q^{4}+\cdots\)
507.4.a.g 507.a 1.a $2$ $29.914$ \(\Q(\sqrt{3}) \) None \(0\) \(-6\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{4}+3\beta q^{5}+6\beta q^{7}+9q^{9}+\cdots\)
507.4.a.h 507.a 1.a $3$ $29.914$ 3.3.3144.1 None \(-2\) \(9\) \(-4\) \(-30\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+3q^{3}+(3+\beta _{2})q^{4}+\cdots\)
507.4.a.i 507.a 1.a $4$ $29.914$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-2\) \(-12\) \(6\) \(14\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
507.4.a.j 507.a 1.a $4$ $29.914$ 4.4.5054412.1 None \(0\) \(-12\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(7+\beta _{3})q^{4}-\beta _{2}q^{5}+\cdots\)
507.4.a.k 507.a 1.a $4$ $29.914$ \(\Q(\sqrt{3}, \sqrt{17})\) None \(0\) \(-12\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}-3q^{3}+9q^{4}+(3\beta _{1}-2\beta _{3})q^{5}+\cdots\)
507.4.a.l 507.a 1.a $4$ $29.914$ 4.4.1362828.1 None \(0\) \(12\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{3})q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)
507.4.a.m 507.a 1.a $4$ $29.914$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(2\) \(-12\) \(-6\) \(-14\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
507.4.a.n 507.a 1.a $9$ $29.914$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-8\) \(-27\) \(-41\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(3-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
507.4.a.o 507.a 1.a $9$ $29.914$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-6\) \(27\) \(-33\) \(-83\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{3})q^{2}+3q^{3}+(5+\beta _{3}+\beta _{5}+\cdots)q^{4}+\cdots\)
507.4.a.p 507.a 1.a $9$ $29.914$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(6\) \(27\) \(33\) \(83\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{3})q^{2}+3q^{3}+(5+\beta _{3}+\beta _{5}+\cdots)q^{4}+\cdots\)
507.4.a.q 507.a 1.a $9$ $29.914$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(8\) \(-27\) \(41\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-3q^{3}+(3-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
507.4.a.r 507.a 1.a $10$ $29.914$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(30\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{4})q^{4}+(\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(507))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(507)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)