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Properties

Label 539.4.a
Level $539$
Weight $4$
Character orbit 539.a
Rep. character $\chi_{539}(1,\cdot)$
Character field $\Q$
Dimension $102$
Newform subspaces $16$
Sturm bound $224$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 539.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(224\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(539))\).

Total New Old
Modular forms 176 102 74
Cusp forms 160 102 58
Eisenstein series 16 0 16

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

\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(28\)
\(+\)\(-\)\(-\)\(20\)
\(-\)\(+\)\(-\)\(24\)
\(-\)\(-\)\(+\)\(30\)
Plus space\(+\)\(58\)
Minus space\(-\)\(44\)

Trace form

\( 102 q + 2 q^{2} - 2 q^{3} + 392 q^{4} + 22 q^{5} - 10 q^{6} + 72 q^{8} + 888 q^{9} + O(q^{10}) \) \( 102 q + 2 q^{2} - 2 q^{3} + 392 q^{4} + 22 q^{5} - 10 q^{6} + 72 q^{8} + 888 q^{9} + 62 q^{10} - 22 q^{11} + 132 q^{12} + 88 q^{13} + 150 q^{15} + 1648 q^{16} - 28 q^{17} - 16 q^{18} - 160 q^{19} - 60 q^{20} - 22 q^{22} - 130 q^{23} - 56 q^{24} + 2764 q^{25} + 588 q^{26} + 106 q^{27} + 232 q^{29} + 798 q^{30} + 546 q^{31} + 648 q^{32} + 154 q^{33} + 372 q^{34} + 3988 q^{36} - 1738 q^{37} + 604 q^{38} - 1248 q^{39} + 876 q^{40} - 352 q^{41} - 580 q^{43} - 528 q^{44} - 1264 q^{45} + 1314 q^{46} + 708 q^{47} + 884 q^{48} - 3504 q^{50} + 1100 q^{51} + 372 q^{52} - 1108 q^{53} - 3354 q^{54} + 418 q^{55} + 1040 q^{57} - 428 q^{58} + 1626 q^{59} + 3376 q^{60} - 1000 q^{61} - 330 q^{62} + 6076 q^{64} + 1224 q^{65} - 286 q^{66} + 58 q^{67} - 2452 q^{68} - 1846 q^{69} - 2286 q^{71} + 2396 q^{72} + 760 q^{73} + 3518 q^{74} - 1544 q^{75} + 1080 q^{76} + 5112 q^{78} + 2748 q^{79} - 996 q^{80} + 9094 q^{81} - 3816 q^{82} - 2460 q^{83} + 3436 q^{85} - 2408 q^{86} - 1968 q^{87} - 660 q^{88} + 4350 q^{89} + 10012 q^{90} - 1260 q^{92} + 1018 q^{93} - 572 q^{94} - 4912 q^{95} + 1524 q^{96} - 1366 q^{97} - 484 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(539))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7 11
539.4.a.a 539.a 1.a $1$ $31.802$ \(\Q\) None 77.4.a.a \(3\) \(-4\) \(-12\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}-4q^{3}+q^{4}-12q^{5}-12q^{6}+\cdots\)
539.4.a.b 539.a 1.a $1$ $31.802$ \(\Q\) None 77.4.e.a \(3\) \(-3\) \(-2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}-3q^{3}+q^{4}-2q^{5}-9q^{6}+\cdots\)
539.4.a.c 539.a 1.a $1$ $31.802$ \(\Q\) None 77.4.e.a \(3\) \(3\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}+2q^{5}+9q^{6}+\cdots\)
539.4.a.d 539.a 1.a $2$ $31.802$ \(\Q(\sqrt{2}) \) None 77.4.a.b \(-2\) \(4\) \(4\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+2q^{3}+(1-2\beta )q^{4}+\cdots\)
539.4.a.e 539.a 1.a $2$ $31.802$ \(\Q(\sqrt{3}) \) None 11.4.a.a \(2\) \(2\) \(-2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+4\beta )q^{3}+(-4+2\beta )q^{4}+\cdots\)
539.4.a.f 539.a 1.a $4$ $31.802$ 4.4.509800.1 None 77.4.a.c \(-4\) \(12\) \(18\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{2})q^{2}+(3-\beta _{3})q^{3}+(6-\beta _{1}+\cdots)q^{4}+\cdots\)
539.4.a.g 539.a 1.a $4$ $31.802$ 4.4.522072.1 None 77.4.a.d \(-2\) \(-14\) \(-10\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(-4-\beta _{1})q^{3}+(6-\beta _{2}-2\beta _{3})q^{4}+\cdots\)
539.4.a.h 539.a 1.a $5$ $31.802$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 77.4.a.e \(1\) \(-2\) \(24\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(9+\beta _{1}+\beta _{2})q^{4}+\cdots\)
539.4.a.i 539.a 1.a $6$ $31.802$ 6.6.1536100192.1 None 539.4.a.i \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{5}q^{2}+\beta _{1}q^{3}+(1-\beta _{4})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
539.4.a.j 539.a 1.a $9$ $31.802$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 77.4.e.b \(-3\) \(-15\) \(-32\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-2+\beta _{1}+\beta _{4})q^{3}+(3+\beta _{2}+\cdots)q^{4}+\cdots\)
539.4.a.k 539.a 1.a $9$ $31.802$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 77.4.e.b \(-3\) \(15\) \(32\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2-\beta _{1}-\beta _{4})q^{3}+(3+\beta _{2}+\cdots)q^{4}+\cdots\)
539.4.a.l 539.a 1.a $10$ $31.802$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 539.4.a.l \(-4\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{5}q^{2}+(-\beta _{3}+\beta _{6})q^{3}+(\beta _{1}+\beta _{5}+\cdots)q^{4}+\cdots\)
539.4.a.m 539.a 1.a $10$ $31.802$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 77.4.e.c \(0\) \(0\) \(-10\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{4}q^{3}+(5+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
539.4.a.n 539.a 1.a $10$ $31.802$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 77.4.e.c \(0\) \(0\) \(10\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(5+\beta _{2})q^{4}+(1+\beta _{5}+\cdots)q^{5}+\cdots\)
539.4.a.o 539.a 1.a $10$ $31.802$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 539.4.a.o \(4\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{4}q^{2}+\beta _{8}q^{3}+(4-\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
539.4.a.p 539.a 1.a $18$ $31.802$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None 539.4.a.p \(4\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(5-\beta _{7})q^{4}+\beta _{8}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(539))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(539)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)