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Properties

Label 300.6.a
Level $300$
Weight $6$
Character orbit 300.a
Rep. character $\chi_{300}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $10$
Sturm bound $360$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 300.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(360\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(300))\).

Total New Old
Modular forms 318 16 302
Cusp forms 282 16 266
Eisenstein series 36 0 36

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

\(2\)\(3\)\(5\)FrickeDim
\(-\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(-\)$+$\(4\)
\(-\)\(-\)\(+\)$+$\(3\)
\(-\)\(-\)\(-\)$-$\(5\)
Plus space\(+\)\(7\)
Minus space\(-\)\(9\)

Trace form

\( 16 q + 160 q^{7} + 1296 q^{9} + O(q^{10}) \) \( 16 q + 160 q^{7} + 1296 q^{9} + 356 q^{11} - 800 q^{13} + 1680 q^{17} + 650 q^{19} + 1674 q^{21} - 6240 q^{23} + 6864 q^{29} + 7454 q^{31} - 3240 q^{33} - 4400 q^{37} - 13950 q^{39} + 54140 q^{41} - 18080 q^{43} + 26400 q^{47} + 42294 q^{49} - 36 q^{51} + 12000 q^{53} - 19440 q^{57} - 6116 q^{59} + 11822 q^{61} + 12960 q^{63} + 4480 q^{67} + 46404 q^{69} + 4800 q^{71} + 119080 q^{73} - 62400 q^{77} + 6728 q^{79} + 104976 q^{81} + 36960 q^{83} - 25920 q^{87} - 121060 q^{89} + 209350 q^{91} + 58320 q^{93} - 182600 q^{97} + 28836 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(300))\) 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
300.6.a.a 300.a 1.a $1$ $48.115$ \(\Q\) None \(0\) \(-9\) \(0\) \(-91\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}-91q^{7}+3^{4}q^{9}-174q^{11}+\cdots\)
300.6.a.b 300.a 1.a $1$ $48.115$ \(\Q\) None \(0\) \(-9\) \(0\) \(-56\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}-56q^{7}+3^{4}q^{9}+156q^{11}+\cdots\)
300.6.a.c 300.a 1.a $1$ $48.115$ \(\Q\) None \(0\) \(-9\) \(0\) \(244\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+244q^{7}+3^{4}q^{9}-12^{2}q^{11}+\cdots\)
300.6.a.d 300.a 1.a $1$ $48.115$ \(\Q\) None \(0\) \(9\) \(0\) \(-44\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}-44q^{7}+3^{4}q^{9}+6^{3}q^{11}+\cdots\)
300.6.a.e 300.a 1.a $1$ $48.115$ \(\Q\) None \(0\) \(9\) \(0\) \(16\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}+2^{4}q^{7}+3^{4}q^{9}-564q^{11}+\cdots\)
300.6.a.f 300.a 1.a $1$ $48.115$ \(\Q\) None \(0\) \(9\) \(0\) \(91\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}+91q^{7}+3^{4}q^{9}-174q^{11}+\cdots\)
300.6.a.g 300.a 1.a $2$ $48.115$ \(\Q(\sqrt{7}) \) None \(0\) \(-18\) \(0\) \(-22\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+(-11+\beta )q^{7}+3^{4}q^{9}+(186+\cdots)q^{11}+\cdots\)
300.6.a.h 300.a 1.a $2$ $48.115$ \(\Q(\sqrt{7}) \) None \(0\) \(18\) \(0\) \(22\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+(11+\beta )q^{7}+3^{4}q^{9}+(186+\cdots)q^{11}+\cdots\)
300.6.a.i 300.a 1.a $3$ $48.115$ 3.3.535753.1 None \(0\) \(-27\) \(0\) \(-88\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}+(-29+\beta _{2})q^{7}+3^{4}q^{9}+(48+\cdots)q^{11}+\cdots\)
300.6.a.j 300.a 1.a $3$ $48.115$ 3.3.535753.1 None \(0\) \(27\) \(0\) \(88\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+(29-\beta _{2})q^{7}+3^{4}q^{9}+(48+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(300))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(300)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)