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Properties

Label 660.6.a
Level $660$
Weight $6$
Character orbit 660.a
Rep. character $\chi_{660}(1,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $9$
Sturm bound $864$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 660.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(864\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(660))\).

Total New Old
Modular forms 732 32 700
Cusp forms 708 32 676
Eisenstein series 24 0 24

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

\(2\)\(3\)\(5\)\(11\)FrickeDim
\(-\)\(+\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(+\)\(-\)$+$\(4\)
\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(-\)\(+\)\(-\)\(-\)$-$\(5\)
\(-\)\(-\)\(+\)\(+\)$+$\(4\)
\(-\)\(-\)\(+\)\(-\)$-$\(4\)
\(-\)\(-\)\(-\)\(+\)$-$\(5\)
\(-\)\(-\)\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(14\)
Minus space\(-\)\(18\)

Trace form

\( 32 q + 320 q^{7} + 2592 q^{9} + O(q^{10}) \) \( 32 q + 320 q^{7} + 2592 q^{9} + 560 q^{13} + 1912 q^{17} - 2360 q^{19} + 3888 q^{21} - 6720 q^{23} + 20000 q^{25} - 2128 q^{29} + 15296 q^{31} - 4356 q^{33} - 6200 q^{35} - 3760 q^{37} - 11160 q^{39} - 2592 q^{41} - 4800 q^{43} + 24464 q^{47} + 68912 q^{49} + 25776 q^{51} + 27360 q^{53} - 12888 q^{57} - 49840 q^{59} - 29280 q^{61} + 25920 q^{63} - 200 q^{65} - 28304 q^{67} + 58320 q^{69} - 203024 q^{71} + 192400 q^{73} - 59048 q^{77} - 171816 q^{79} + 209952 q^{81} - 34744 q^{83} + 3800 q^{85} - 47592 q^{87} + 34672 q^{89} + 282400 q^{91} - 155520 q^{93} + 192000 q^{95} - 294448 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(660))\) 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 5 11
660.6.a.a 660.a 1.a $1$ $105.853$ \(\Q\) None \(0\) \(9\) \(-25\) \(188\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}-5^{2}q^{5}+188q^{7}+3^{4}q^{9}+\cdots\)
660.6.a.b 660.a 1.a $3$ $105.853$ 3.3.67380.1 None \(0\) \(-27\) \(75\) \(16\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+5^{2}q^{5}+(7-3\beta _{1}+5\beta _{2})q^{7}+\cdots\)
660.6.a.c 660.a 1.a $3$ $105.853$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(27\) \(-75\) \(-26\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}-5^{2}q^{5}+(-9-\beta _{1}+\beta _{2})q^{7}+\cdots\)
660.6.a.d 660.a 1.a $3$ $105.853$ 3.3.132820.1 None \(0\) \(27\) \(75\) \(-96\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+5^{2}q^{5}+(-35-9\beta _{1}+2\beta _{2})q^{7}+\cdots\)
660.6.a.e 660.a 1.a $4$ $105.853$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-36\) \(-100\) \(-44\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}-5^{2}q^{5}+(-11+\beta _{3})q^{7}+3^{4}q^{9}+\cdots\)
660.6.a.f 660.a 1.a $4$ $105.853$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-36\) \(-100\) \(78\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}-5^{2}q^{5}+(20+\beta _{3})q^{7}+3^{4}q^{9}+\cdots\)
660.6.a.g 660.a 1.a $4$ $105.853$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(36\) \(-100\) \(88\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}-5^{2}q^{5}+(22-\beta _{2})q^{7}+3^{4}q^{9}+\cdots\)
660.6.a.h 660.a 1.a $5$ $105.853$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-45\) \(125\) \(-106\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}+5^{2}q^{5}+(-21-\beta _{1})q^{7}+3^{4}q^{9}+\cdots\)
660.6.a.i 660.a 1.a $5$ $105.853$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(45\) \(125\) \(222\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}+5^{2}q^{5}+(44+\beta _{1})q^{7}+3^{4}q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(660))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(660)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 2}\)