↑

Properties

Label 504.4.a
Level $504$
Weight $4$
Character orbit 504.a
Rep. character $\chi_{504}(1,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $15$
Sturm bound $384$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(504))\).

Total New Old
Modular forms 304 22 282
Cusp forms 272 22 250
Eisenstein series 32 0 32

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

\(2\)\(3\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(2\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(3\)
\(+\)\(-\)\(-\)$+$\(4\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(+\)$+$\(4\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(12\)
Minus space\(-\)\(10\)

Trace form

\( 22 q + 14 q^{5} + O(q^{10}) \) \( 22 q + 14 q^{5} - 116 q^{11} - 66 q^{13} + 200 q^{17} - 90 q^{19} - 128 q^{23} + 542 q^{25} - 104 q^{29} + 236 q^{31} - 210 q^{35} + 24 q^{37} - 192 q^{41} + 756 q^{43} + 1404 q^{47} + 1078 q^{49} - 308 q^{53} - 1448 q^{55} + 42 q^{59} + 658 q^{61} - 796 q^{65} + 848 q^{67} + 2000 q^{71} + 1892 q^{73} + 308 q^{77} - 1288 q^{79} + 754 q^{83} - 3884 q^{85} + 1988 q^{89} + 126 q^{91} + 2040 q^{95} + 216 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(504))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
504.4.a.a 504.a 1.a $1$ $29.737$ \(\Q\) None \(0\) \(0\) \(-8\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{5}-7q^{7}-56q^{11}-28q^{13}+\cdots\)
504.4.a.b 504.a 1.a $1$ $29.737$ \(\Q\) None \(0\) \(0\) \(-4\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{5}-7q^{7}+26q^{11}+2q^{13}+6^{2}q^{17}+\cdots\)
504.4.a.c 504.a 1.a $1$ $29.737$ \(\Q\) None \(0\) \(0\) \(2\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}-7q^{7}-52q^{11}+86q^{13}+\cdots\)
504.4.a.d 504.a 1.a $1$ $29.737$ \(\Q\) None \(0\) \(0\) \(2\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}+7q^{7}-12q^{11}-66q^{13}+\cdots\)
504.4.a.e 504.a 1.a $1$ $29.737$ \(\Q\) None \(0\) \(0\) \(10\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+10q^{5}-7q^{7}+52q^{11}-10q^{13}+\cdots\)
504.4.a.f 504.a 1.a $1$ $29.737$ \(\Q\) None \(0\) \(0\) \(10\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+10q^{5}+7q^{7}+12q^{11}+30q^{13}+\cdots\)
504.4.a.g 504.a 1.a $1$ $29.737$ \(\Q\) None \(0\) \(0\) \(16\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{5}-7q^{7}-24q^{11}-68q^{13}+\cdots\)
504.4.a.h 504.a 1.a $1$ $29.737$ \(\Q\) None \(0\) \(0\) \(16\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{5}+7q^{7}+18q^{11}-54q^{13}+\cdots\)
504.4.a.i 504.a 1.a $2$ $29.737$ \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(-22\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-11-\beta )q^{5}+7q^{7}+(-18-6\beta )q^{11}+\cdots\)
504.4.a.j 504.a 1.a $2$ $29.737$ \(\Q(\sqrt{177}) \) None \(0\) \(0\) \(-14\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-7-\beta )q^{5}+7q^{7}+(-9+3\beta )q^{11}+\cdots\)
504.4.a.k 504.a 1.a $2$ $29.737$ \(\Q(\sqrt{22}) \) None \(0\) \(0\) \(-12\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-6+\beta )q^{5}-7q^{7}+(-2-\beta )q^{11}+\cdots\)
504.4.a.l 504.a 1.a $2$ $29.737$ \(\Q(\sqrt{30}) \) None \(0\) \(0\) \(-4\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{5}+7q^{7}+(-18-3\beta )q^{11}+\cdots\)
504.4.a.m 504.a 1.a $2$ $29.737$ \(\Q(\sqrt{30}) \) None \(0\) \(0\) \(4\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{5}+7q^{7}+(18-3\beta )q^{11}+\cdots\)
504.4.a.n 504.a 1.a $2$ $29.737$ \(\Q(\sqrt{337}) \) None \(0\) \(0\) \(6\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{5}-7q^{7}+(-13-\beta )q^{11}+\cdots\)
504.4.a.o 504.a 1.a $2$ $29.737$ \(\Q(\sqrt{22}) \) None \(0\) \(0\) \(12\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(6+\beta )q^{5}-7q^{7}+(2-\beta )q^{11}+(-30+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(504))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(504)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)