Properties

Label 900.4.a
Level $900$
Weight $4$
Character orbit 900.a
Rep. character $\chi_{900}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $20$
Sturm bound $720$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(720\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(900))\).

Total New Old
Modular forms 576 24 552
Cusp forms 504 24 480
Eisenstein series 72 0 72

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.
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(7\)
Plus space\(+\)\(13\)
Minus space\(-\)\(11\)

Trace form

\( 24 q + O(q^{10}) \) \( 24 q + 6 q^{11} + 12 q^{13} + 60 q^{17} - 190 q^{19} - 144 q^{23} - 384 q^{29} - 44 q^{31} - 180 q^{37} + 390 q^{41} + 264 q^{43} + 936 q^{47} + 2304 q^{49} - 612 q^{53} - 1104 q^{59} + 1848 q^{61} - 408 q^{67} - 360 q^{71} + 180 q^{73} + 3072 q^{77} + 1628 q^{79} - 1584 q^{83} - 2070 q^{89} + 160 q^{91} - 1164 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(900))\) 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
900.4.a.a 900.a 1.a $1$ $53.102$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-37\) $-$ $+$ $-$ $N(\mathrm{U}(1))$ \(q-37q^{7}+89q^{13}-163q^{19}-17^{2}q^{31}+\cdots\)
900.4.a.b 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(-32\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{7}-6^{2}q^{11}+10q^{13}-78q^{17}+\cdots\)
900.4.a.c 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(-26\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-26q^{7}-45q^{11}-44q^{13}+117q^{17}+\cdots\)
900.4.a.d 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(-22\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-22q^{7}+14q^{11}+30q^{13}+62q^{17}+\cdots\)
900.4.a.e 900.a 1.a $1$ $53.102$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-17\) $-$ $+$ $+$ $N(\mathrm{U}(1))$ \(q-17q^{7}+19q^{13}+107q^{19}-19q^{31}+\cdots\)
900.4.a.f 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(-13\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-13q^{7}-6q^{11}+5q^{13}+78q^{17}+\cdots\)
900.4.a.g 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(-8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{7}-6^{2}q^{11}+10q^{13}+18q^{17}+\cdots\)
900.4.a.h 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(-7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-7q^{7}+54q^{11}-55q^{13}-18q^{17}+\cdots\)
900.4.a.i 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}-30q^{11}+4q^{13}+90q^{17}+\cdots\)
900.4.a.j 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}+30q^{11}+4q^{13}-90q^{17}+\cdots\)
900.4.a.k 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+7q^{7}+54q^{11}+55q^{13}+18q^{17}+\cdots\)
900.4.a.l 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(13\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+13q^{7}-6q^{11}-5q^{13}-78q^{17}+\cdots\)
900.4.a.m 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(16\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{7}+60q^{11}-86q^{13}+18q^{17}+\cdots\)
900.4.a.n 900.a 1.a $1$ $53.102$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(17\) $-$ $+$ $-$ $N(\mathrm{U}(1))$ \(q+17q^{7}-19q^{13}+107q^{19}-19q^{31}+\cdots\)
900.4.a.o 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(22\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+22q^{7}+14q^{11}-30q^{13}-62q^{17}+\cdots\)
900.4.a.p 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(26\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+26q^{7}-45q^{11}+44q^{13}-117q^{17}+\cdots\)
900.4.a.q 900.a 1.a $1$ $53.102$ \(\Q\) None \(0\) \(0\) \(0\) \(28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+28q^{7}+24q^{11}+70q^{13}+102q^{17}+\cdots\)
900.4.a.r 900.a 1.a $1$ $53.102$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(37\) $-$ $+$ $+$ $N(\mathrm{U}(1))$ \(q+37q^{7}-89q^{13}-163q^{19}-17^{2}q^{31}+\cdots\)
900.4.a.s 900.a 1.a $2$ $53.102$ \(\Q(\sqrt{19}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{7}-20q^{11}-6\beta q^{13}+8\beta q^{17}+\cdots\)
900.4.a.t 900.a 1.a $4$ $53.102$ \(\Q(\sqrt{10}, \sqrt{34})\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{7}+\beta _{2}q^{11}+3\beta _{1}q^{13}-\beta _{3}q^{17}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(900))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(900)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)