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Properties

Label 300.4.a
Level $300$
Weight $4$
Character orbit 300.a
Rep. character $\chi_{300}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $9$
Sturm bound $240$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 300.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(300))\).

Total New Old
Modular forms 198 9 189
Cusp forms 162 9 153
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
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(5\)
Minus space\(-\)\(4\)

Trace form

\( 9 q + 3 q^{3} - 12 q^{7} + 81 q^{9} + O(q^{10}) \) \( 9 q + 3 q^{3} - 12 q^{7} + 81 q^{9} - 76 q^{11} + 90 q^{13} - 42 q^{17} - 100 q^{19} - 84 q^{21} + 192 q^{23} + 27 q^{27} + 294 q^{29} + 256 q^{31} - 72 q^{33} + 474 q^{37} + 30 q^{39} - 190 q^{41} - 108 q^{43} - 768 q^{47} + 189 q^{49} + 186 q^{51} + 114 q^{53} + 780 q^{57} + 1444 q^{59} - 702 q^{61} - 108 q^{63} + 372 q^{67} - 1056 q^{69} - 840 q^{71} + 186 q^{73} - 2112 q^{77} + 8 q^{79} + 729 q^{81} + 2532 q^{83} + 1638 q^{87} - 670 q^{89} + 880 q^{91} - 408 q^{93} + 1146 q^{97} - 684 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(300))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
300.4.a.a 300.a 1.a $1$ $17.701$ \(\Q\) None 300.4.a.a \(0\) \(-3\) \(0\) \(-13\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-13q^{7}+9q^{9}+6q^{11}+5q^{13}+\cdots\)
300.4.a.b 300.a 1.a $1$ $17.701$ \(\Q\) None 12.4.a.a \(0\) \(-3\) \(0\) \(-8\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{7}+9q^{9}+6^{2}q^{11}+10q^{13}+\cdots\)
300.4.a.c 300.a 1.a $1$ $17.701$ \(\Q\) None 300.4.a.c \(0\) \(-3\) \(0\) \(7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+7q^{7}+9q^{9}-54q^{11}+55q^{13}+\cdots\)
300.4.a.d 300.a 1.a $1$ $17.701$ \(\Q\) None 60.4.d.a \(0\) \(-3\) \(0\) \(22\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+22q^{7}+9q^{9}-14q^{11}-30q^{13}+\cdots\)
300.4.a.e 300.a 1.a $1$ $17.701$ \(\Q\) None 60.4.a.b \(0\) \(3\) \(0\) \(-32\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-2^{5}q^{7}+9q^{9}+6^{2}q^{11}+10q^{13}+\cdots\)
300.4.a.f 300.a 1.a $1$ $17.701$ \(\Q\) None 60.4.d.a \(0\) \(3\) \(0\) \(-22\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-22q^{7}+9q^{9}-14q^{11}+30q^{13}+\cdots\)
300.4.a.g 300.a 1.a $1$ $17.701$ \(\Q\) None 300.4.a.c \(0\) \(3\) \(0\) \(-7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-7q^{7}+9q^{9}-54q^{11}-55q^{13}+\cdots\)
300.4.a.h 300.a 1.a $1$ $17.701$ \(\Q\) None 300.4.a.a \(0\) \(3\) \(0\) \(13\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+13q^{7}+9q^{9}+6q^{11}-5q^{13}+\cdots\)
300.4.a.i 300.a 1.a $1$ $17.701$ \(\Q\) None 60.4.a.a \(0\) \(3\) \(0\) \(28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+28q^{7}+9q^{9}-24q^{11}+70q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(300))\) into lower level spaces

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