Properties

Label 600.4.a
Level $600$
Weight $4$
Character orbit 600.a
Rep. character $\chi_{600}(1,\cdot)$
Character field $\Q$
Dimension $29$
Newform subspaces $23$
Sturm bound $480$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 600.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 23 \)
Sturm bound: \(480\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(600))\).

Total New Old
Modular forms 384 29 355
Cusp forms 336 29 307
Eisenstein series 48 0 48

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.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(16\)
Minus space\(-\)\(13\)

Trace form

\( 29 q + 3 q^{3} - 12 q^{7} + 261 q^{9} + O(q^{10}) \) \( 29 q + 3 q^{3} - 12 q^{7} + 261 q^{9} + 28 q^{11} + 58 q^{13} - 90 q^{17} - 272 q^{19} + 12 q^{21} - 160 q^{23} + 27 q^{27} + 158 q^{29} + 20 q^{31} - 48 q^{33} - 598 q^{37} - 198 q^{39} - 22 q^{41} + 116 q^{43} + 928 q^{47} + 2297 q^{49} - 366 q^{51} + 1394 q^{53} - 420 q^{57} - 1156 q^{59} + 1578 q^{61} - 108 q^{63} + 500 q^{67} + 576 q^{69} + 1464 q^{71} + 802 q^{73} - 768 q^{77} + 392 q^{79} + 2349 q^{81} + 836 q^{83} + 6 q^{87} + 5226 q^{89} - 1396 q^{91} - 1128 q^{93} + 50 q^{97} + 252 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(600))\) 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
600.4.a.a 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(-3\) \(0\) \(-20\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-20q^{7}+9q^{9}+2^{4}q^{11}-58q^{13}+\cdots\)
600.4.a.b 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(-3\) \(0\) \(-10\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-10q^{7}+9q^{9}-46q^{11}+34q^{13}+\cdots\)
600.4.a.c 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(-3\) \(0\) \(-10\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-10q^{7}+9q^{9}-14q^{11}+82q^{13}+\cdots\)
600.4.a.d 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(-3\) \(0\) \(-8\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{7}+9q^{9}+20q^{11}-22q^{13}+\cdots\)
600.4.a.e 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(-3\) \(0\) \(-5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{7}+9q^{9}+14q^{11}-q^{13}+\cdots\)
600.4.a.f 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(-3\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+4q^{7}+9q^{9}-28q^{11}-2^{4}q^{13}+\cdots\)
600.4.a.g 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(-3\) \(0\) \(19\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+19q^{7}+9q^{9}+22q^{11}-q^{13}+\cdots\)
600.4.a.h 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(-3\) \(0\) \(24\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+24q^{7}+9q^{9}-28q^{11}+74q^{13}+\cdots\)
600.4.a.i 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(3\) \(0\) \(-20\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-20q^{7}+9q^{9}-56q^{11}+86q^{13}+\cdots\)
600.4.a.j 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(3\) \(0\) \(-19\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-19q^{7}+9q^{9}+22q^{11}+q^{13}+\cdots\)
600.4.a.k 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(3\) \(0\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-4q^{7}+9q^{9}-28q^{11}+2^{4}q^{13}+\cdots\)
600.4.a.l 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(3\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-4q^{7}+9q^{9}+72q^{11}+6q^{13}+\cdots\)
600.4.a.m 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(3\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+9q^{9}+4q^{11}-54q^{13}-114q^{17}+\cdots\)
600.4.a.n 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(3\) \(0\) \(5\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{7}+9q^{9}+14q^{11}+q^{13}+\cdots\)
600.4.a.o 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(3\) \(0\) \(10\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+10q^{7}+9q^{9}-46q^{11}-34q^{13}+\cdots\)
600.4.a.p 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(3\) \(0\) \(10\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+10q^{7}+9q^{9}-14q^{11}-82q^{13}+\cdots\)
600.4.a.q 600.a 1.a $1$ $35.401$ \(\Q\) None \(0\) \(3\) \(0\) \(16\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+2^{4}q^{7}+9q^{9}-28q^{11}+26q^{13}+\cdots\)
600.4.a.r 600.a 1.a $2$ $35.401$ \(\Q(\sqrt{181}) \) None \(0\) \(-6\) \(0\) \(-6\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-3-\beta )q^{7}+9q^{9}+(4+\beta )q^{11}+\cdots\)
600.4.a.s 600.a 1.a $2$ $35.401$ \(\Q(\sqrt{109}) \) None \(0\) \(-6\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(1+\beta )q^{7}+9q^{9}+(-8+3\beta )q^{11}+\cdots\)
600.4.a.t 600.a 1.a $2$ $35.401$ \(\Q(\sqrt{129}) \) None \(0\) \(-6\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(1+3\beta )q^{7}+9q^{9}+(37-\beta )q^{11}+\cdots\)
600.4.a.u 600.a 1.a $2$ $35.401$ \(\Q(\sqrt{109}) \) None \(0\) \(6\) \(0\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-1-\beta )q^{7}+9q^{9}+(-8+\cdots)q^{11}+\cdots\)
600.4.a.v 600.a 1.a $2$ $35.401$ \(\Q(\sqrt{129}) \) None \(0\) \(6\) \(0\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-1-3\beta )q^{7}+9q^{9}+(37+\cdots)q^{11}+\cdots\)
600.4.a.w 600.a 1.a $2$ $35.401$ \(\Q(\sqrt{181}) \) None \(0\) \(6\) \(0\) \(6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(3+\beta )q^{7}+9q^{9}+(4+\beta )q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(600))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(600)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)