Properties

Label 600.2.a
Level 600
Weight 2
Character orbit a
Rep. character \(\chi_{600}(1,\cdot)\)
Character field \(\Q\)
Dimension 9
Newforms 9
Sturm bound 240
Trace bound 7

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 600.a (trivial)
Character field: \(\Q\)
Newforms: \( 9 \)
Sturm bound: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(600))\).

Total New Old
Modular forms 144 9 135
Cusp forms 97 9 88
Eisenstein series 47 0 47

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.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(6\)

Trace form

\(9q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 16q^{23} \) \(\mathstrut -\mathstrut q^{27} \) \(\mathstrut +\mathstrut 22q^{29} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut +\mathstrut 8q^{33} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 34q^{41} \) \(\mathstrut -\mathstrut 12q^{43} \) \(\mathstrut -\mathstrut 16q^{47} \) \(\mathstrut +\mathstrut 29q^{49} \) \(\mathstrut +\mathstrut 10q^{51} \) \(\mathstrut -\mathstrut 14q^{53} \) \(\mathstrut -\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 20q^{59} \) \(\mathstrut +\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut -\mathstrut 24q^{71} \) \(\mathstrut +\mathstrut 10q^{73} \) \(\mathstrut -\mathstrut 48q^{79} \) \(\mathstrut +\mathstrut 9q^{81} \) \(\mathstrut +\mathstrut 4q^{83} \) \(\mathstrut +\mathstrut 14q^{87} \) \(\mathstrut +\mathstrut 18q^{89} \) \(\mathstrut -\mathstrut 20q^{91} \) \(\mathstrut +\mathstrut 16q^{93} \) \(\mathstrut -\mathstrut 6q^{97} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(600))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
600.2.a.a \(1\) \(4.791\) \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) \(-\) \(+\) \(+\) \(q-q^{3}-4q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots\)
600.2.a.b \(1\) \(4.791\) \(\Q\) None \(0\) \(-1\) \(0\) \(-3\) \(+\) \(+\) \(+\) \(q-q^{3}-3q^{7}+q^{9}+2q^{11}+3q^{13}+\cdots\)
600.2.a.c \(1\) \(4.791\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{9}-4q^{11}-6q^{13}+6q^{17}+\cdots\)
600.2.a.d \(1\) \(4.791\) \(\Q\) None \(0\) \(-1\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q-q^{3}+2q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
600.2.a.e \(1\) \(4.791\) \(\Q\) None \(0\) \(-1\) \(0\) \(5\) \(-\) \(+\) \(+\) \(q-q^{3}+5q^{7}+q^{9}-6q^{11}+3q^{13}+\cdots\)
600.2.a.f \(1\) \(4.791\) \(\Q\) None \(0\) \(1\) \(0\) \(-5\) \(+\) \(-\) \(-\) \(q+q^{3}-5q^{7}+q^{9}-6q^{11}-3q^{13}+\cdots\)
600.2.a.g \(1\) \(4.791\) \(\Q\) None \(0\) \(1\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(q+q^{3}-2q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
600.2.a.h \(1\) \(4.791\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\)
600.2.a.i \(1\) \(4.791\) \(\Q\) None \(0\) \(1\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q+q^{3}+3q^{7}+q^{9}+2q^{11}-3q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(600)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)