Properties

Label 1800.2.a
Level $1800$
Weight $2$
Character orbit 1800.a
Rep. character $\chi_{1800}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $24$
Sturm bound $720$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(720\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1800))\).

Total New Old
Modular forms 408 24 384
Cusp forms 313 24 289
Eisenstein series 95 0 95

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

\(2\)\(3\)\(5\)FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(+\)\(+\)\(+\)\(48\)\(2\)\(46\)\(37\)\(2\)\(35\)\(11\)\(0\)\(11\)
\(+\)\(+\)\(-\)\(-\)\(54\)\(3\)\(51\)\(42\)\(3\)\(39\)\(12\)\(0\)\(12\)
\(+\)\(-\)\(+\)\(-\)\(51\)\(4\)\(47\)\(39\)\(4\)\(35\)\(12\)\(0\)\(12\)
\(+\)\(-\)\(-\)\(+\)\(51\)\(3\)\(48\)\(39\)\(3\)\(36\)\(12\)\(0\)\(12\)
\(-\)\(+\)\(+\)\(-\)\(54\)\(2\)\(52\)\(42\)\(2\)\(40\)\(12\)\(0\)\(12\)
\(-\)\(+\)\(-\)\(+\)\(48\)\(3\)\(45\)\(36\)\(3\)\(33\)\(12\)\(0\)\(12\)
\(-\)\(-\)\(+\)\(+\)\(51\)\(3\)\(48\)\(39\)\(3\)\(36\)\(12\)\(0\)\(12\)
\(-\)\(-\)\(-\)\(-\)\(51\)\(4\)\(47\)\(39\)\(4\)\(35\)\(12\)\(0\)\(12\)
Plus space\(+\)\(198\)\(11\)\(187\)\(151\)\(11\)\(140\)\(47\)\(0\)\(47\)
Minus space\(-\)\(210\)\(13\)\(197\)\(162\)\(13\)\(149\)\(48\)\(0\)\(48\)

Trace form

\( 24 q - 4 q^{7} + 6 q^{11} - 4 q^{13} - 4 q^{17} - 6 q^{19} - 12 q^{23} - 8 q^{29} - 4 q^{31} - 4 q^{37} - 26 q^{41} + 12 q^{43} + 20 q^{47} + 32 q^{49} + 20 q^{53} + 32 q^{59} + 8 q^{61} + 4 q^{67} + 32 q^{71}+ \cdots + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1800))\) 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
1800.2.a.a 1800.a 1.a $1$ $14.373$ \(\Q\) None 600.2.a.e \(0\) \(0\) \(0\) \(-5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-5q^{7}+6q^{11}-3q^{13}+2q^{17}+q^{19}+\cdots\)
1800.2.a.b 1800.a 1.a $1$ $14.373$ \(\Q\) None 360.2.f.b \(0\) \(0\) \(0\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{7}-4q^{11}+4q^{13}+6q^{17}-4q^{19}+\cdots\)
1800.2.a.c 1800.a 1.a $1$ $14.373$ \(\Q\) None 120.2.a.a \(0\) \(0\) \(0\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{7}+6q^{13}-2q^{17}+4q^{19}-8q^{23}+\cdots\)
1800.2.a.d 1800.a 1.a $1$ $14.373$ \(\Q\) None 360.2.f.b \(0\) \(0\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{7}+4q^{11}+4q^{13}-6q^{17}-4q^{19}+\cdots\)
1800.2.a.e 1800.a 1.a $1$ $14.373$ \(\Q\) None 600.2.a.b \(0\) \(0\) \(0\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{7}-2q^{11}+3q^{13}+6q^{17}-7q^{19}+\cdots\)
1800.2.a.f 1800.a 1.a $1$ $14.373$ \(\Q\) None 360.2.a.c \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}-2q^{11}-4q^{13}-2q^{17}+4q^{19}+\cdots\)
1800.2.a.g 1800.a 1.a $1$ $14.373$ \(\Q\) None 120.2.f.a \(0\) \(0\) \(0\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{7}-2q^{11}+2q^{13}-6q^{17}+8q^{19}+\cdots\)
1800.2.a.h 1800.a 1.a $1$ $14.373$ \(\Q\) None 200.2.a.a \(0\) \(0\) \(0\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}-q^{11}-4q^{13}+5q^{17}+q^{19}+\cdots\)
1800.2.a.i 1800.a 1.a $1$ $14.373$ \(\Q\) None 360.2.a.c \(0\) \(0\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}+2q^{11}-4q^{13}+2q^{17}+4q^{19}+\cdots\)
1800.2.a.j 1800.a 1.a $1$ $14.373$ \(\Q\) None 40.2.c.a \(0\) \(0\) \(0\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{7}+4q^{11}-4q^{13}-4q^{19}-2q^{23}+\cdots\)
1800.2.a.k 1800.a 1.a $1$ $14.373$ \(\Q\) None 1800.2.a.k \(0\) \(0\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{7}-4q^{11}+q^{13}+4q^{17}+q^{19}+\cdots\)
1800.2.a.l 1800.a 1.a $1$ $14.373$ \(\Q\) None 1800.2.a.k \(0\) \(0\) \(0\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{7}+4q^{11}+q^{13}-4q^{17}+q^{19}+\cdots\)
1800.2.a.m 1800.a 1.a $1$ $14.373$ \(\Q\) None 24.2.a.a \(0\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{11}+2q^{13}+2q^{17}-4q^{19}+\cdots\)
1800.2.a.n 1800.a 1.a $1$ $14.373$ \(\Q\) None 120.2.a.b \(0\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{11}-6q^{13}-6q^{17}-4q^{19}+\cdots\)
1800.2.a.o 1800.a 1.a $1$ $14.373$ \(\Q\) None 1800.2.a.k \(0\) \(0\) \(0\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{7}-4q^{11}-q^{13}-4q^{17}+q^{19}+\cdots\)
1800.2.a.p 1800.a 1.a $1$ $14.373$ \(\Q\) None 1800.2.a.k \(0\) \(0\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{7}+4q^{11}-q^{13}+4q^{17}+q^{19}+\cdots\)
1800.2.a.q 1800.a 1.a $1$ $14.373$ \(\Q\) None 120.2.f.a \(0\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{7}-2q^{11}-2q^{13}+6q^{17}+8q^{19}+\cdots\)
1800.2.a.r 1800.a 1.a $1$ $14.373$ \(\Q\) None 200.2.a.a \(0\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{7}-q^{11}+4q^{13}-5q^{17}+q^{19}+\cdots\)
1800.2.a.s 1800.a 1.a $1$ $14.373$ \(\Q\) None 40.2.c.a \(0\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{7}+4q^{11}+4q^{13}-4q^{19}+2q^{23}+\cdots\)
1800.2.a.t 1800.a 1.a $1$ $14.373$ \(\Q\) None 600.2.a.b \(0\) \(0\) \(0\) \(3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{7}-2q^{11}-3q^{13}-6q^{17}-7q^{19}+\cdots\)
1800.2.a.u 1800.a 1.a $1$ $14.373$ \(\Q\) None 360.2.f.b \(0\) \(0\) \(0\) \(4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{7}-4q^{11}-4q^{13}-6q^{17}-4q^{19}+\cdots\)
1800.2.a.v 1800.a 1.a $1$ $14.373$ \(\Q\) None 40.2.a.a \(0\) \(0\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{7}-4q^{11}+2q^{13}+2q^{17}+4q^{19}+\cdots\)
1800.2.a.w 1800.a 1.a $1$ $14.373$ \(\Q\) None 360.2.f.b \(0\) \(0\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{7}+4q^{11}-4q^{13}+6q^{17}-4q^{19}+\cdots\)
1800.2.a.x 1800.a 1.a $1$ $14.373$ \(\Q\) None 600.2.a.e \(0\) \(0\) \(0\) \(5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+5q^{7}+6q^{11}+3q^{13}-2q^{17}+q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1800)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 2}\)