Properties

Label 1800.4.a
Level $1800$
Weight $4$
Character orbit 1800.a
Rep. character $\chi_{1800}(1,\cdot)$
Character field $\Q$
Dimension $71$
Newform subspaces $49$
Sturm bound $1440$
Trace bound $17$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 49 \)
Sturm bound: \(1440\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1800))\).

Total New Old
Modular forms 1128 71 1057
Cusp forms 1032 71 961
Eisenstein series 96 0 96

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.
\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(7\)
\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(12\)
\(-\)\(+\)\(+\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(+\)\(7\)
\(-\)\(-\)\(+\)\(+\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(11\)
Plus space\(+\)\(37\)
Minus space\(-\)\(34\)

Trace form

\( 71 q - 12 q^{7} + O(q^{10}) \) \( 71 q - 12 q^{7} - 22 q^{11} - 2 q^{13} - 46 q^{17} + 46 q^{19} + 28 q^{23} + 50 q^{29} - 100 q^{31} - 178 q^{37} - 16 q^{41} - 400 q^{43} + 52 q^{47} + 3527 q^{49} - 594 q^{53} - 252 q^{59} - 1446 q^{61} - 304 q^{67} - 136 q^{71} + 634 q^{73} + 32 q^{77} + 1172 q^{79} + 1040 q^{83} - 2772 q^{89} + 2768 q^{91} - 862 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1800))\) 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
1800.4.a.a 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-34\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-34q^{7}-18q^{11}-12q^{13}-106q^{17}+\cdots\)
1800.4.a.b 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-34\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-34q^{7}+18q^{11}-12q^{13}+106q^{17}+\cdots\)
1800.4.a.c 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-26\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-26q^{7}+59q^{11}-28q^{13}+5q^{17}+\cdots\)
1800.4.a.d 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-24\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-24q^{7}+44q^{11}-22q^{13}+50q^{17}+\cdots\)
1800.4.a.e 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-20\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-20q^{7}-2^{4}q^{11}-58q^{13}+38q^{17}+\cdots\)
1800.4.a.f 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-20\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-20q^{7}+56q^{11}+86q^{13}-106q^{17}+\cdots\)
1800.4.a.g 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-19\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-19q^{7}-22q^{11}+q^{13}+58q^{17}+\cdots\)
1800.4.a.h 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-16\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{7}-6^{2}q^{11}+42q^{13}-110q^{17}+\cdots\)
1800.4.a.i 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-10\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-10q^{7}+14q^{11}+82q^{13}+18q^{17}+\cdots\)
1800.4.a.j 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-10\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-10q^{7}+46q^{11}+34q^{13}+66q^{17}+\cdots\)
1800.4.a.k 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{7}-20q^{11}-22q^{13}-14q^{17}+\cdots\)
1800.4.a.l 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-6q^{7}+19q^{11}+12q^{13}+75q^{17}+\cdots\)
1800.4.a.m 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{7}-14q^{11}-q^{13}+46q^{17}+\cdots\)
1800.4.a.n 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{7}-72q^{11}+6q^{13}+38q^{17}+\cdots\)
1800.4.a.o 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{7}+28q^{11}+2^{4}q^{13}+108q^{17}+\cdots\)
1800.4.a.p 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{7}-39q^{11}-84q^{13}-61q^{17}+\cdots\)
1800.4.a.q 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}-34q^{11}+68q^{13}+38q^{17}+\cdots\)
1800.4.a.r 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}+34q^{11}+68q^{13}-38q^{17}+\cdots\)
1800.4.a.s 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{11}-54q^{13}+114q^{17}+44q^{19}+\cdots\)
1800.4.a.t 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{7}-39q^{11}+84q^{13}+61q^{17}+\cdots\)
1800.4.a.u 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{7}+28q^{11}-2^{4}q^{13}-108q^{17}+\cdots\)
1800.4.a.v 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{7}-14q^{11}+q^{13}-46q^{17}+\cdots\)
1800.4.a.w 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+6q^{7}+19q^{11}-12q^{13}-75q^{17}+\cdots\)
1800.4.a.x 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(10\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+10q^{7}+14q^{11}-82q^{13}-18q^{17}+\cdots\)
1800.4.a.y 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(10\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+10q^{7}+46q^{11}-34q^{13}-66q^{17}+\cdots\)
1800.4.a.z 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(12\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+12q^{7}-2^{6}q^{11}-58q^{13}+2^{5}q^{17}+\cdots\)
1800.4.a.ba 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(12\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+12q^{7}+2^{6}q^{11}-58q^{13}-2^{5}q^{17}+\cdots\)
1800.4.a.bb 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(16\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{7}+28q^{11}+26q^{13}-62q^{17}+\cdots\)
1800.4.a.bc 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(18\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+18q^{7}-34q^{11}-12q^{13}-102q^{17}+\cdots\)
1800.4.a.bd 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(18\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+18q^{7}+2^{4}q^{11}+6q^{13}-6q^{17}+\cdots\)
1800.4.a.be 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(18\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+18q^{7}+34q^{11}-12q^{13}+102q^{17}+\cdots\)
1800.4.a.bf 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(19\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+19q^{7}-22q^{11}-q^{13}-58q^{17}+\cdots\)
1800.4.a.bg 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(24\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+24q^{7}+28q^{11}+74q^{13}+82q^{17}+\cdots\)
1800.4.a.bh 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(26\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+26q^{7}+59q^{11}+28q^{13}-5q^{17}+\cdots\)
1800.4.a.bi 1800.a 1.a $1$ $106.203$ \(\Q\) None \(0\) \(0\) \(0\) \(34\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+34q^{7}-2^{4}q^{11}-58q^{13}-70q^{17}+\cdots\)
1800.4.a.bj 1800.a 1.a $2$ $106.203$ \(\Q(\sqrt{181}) \) None \(0\) \(0\) \(0\) \(-6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{7}+(-4-\beta )q^{11}+(-7+\cdots)q^{13}+\cdots\)
1800.4.a.bk 1800.a 1.a $2$ $106.203$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2+3\beta )q^{7}+(-20-8\beta )q^{11}+\cdots\)
1800.4.a.bl 1800.a 1.a $2$ $106.203$ \(\Q(\sqrt{129}) \) None \(0\) \(0\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-3\beta )q^{7}+(-37+\beta )q^{11}+(7^{2}+\cdots)q^{13}+\cdots\)
1800.4.a.bm 1800.a 1.a $2$ $106.203$ \(\Q(\sqrt{109}) \) None \(0\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{7}+(8-3\beta )q^{11}+(-41+\cdots)q^{13}+\cdots\)
1800.4.a.bn 1800.a 1.a $2$ $106.203$ \(\Q(\sqrt{129}) \) None \(0\) \(0\) \(0\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+3\beta )q^{7}+(-37+\beta )q^{11}+(-7^{2}+\cdots)q^{13}+\cdots\)
1800.4.a.bo 1800.a 1.a $2$ $106.203$ \(\Q(\sqrt{109}) \) None \(0\) \(0\) \(0\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{7}+(8-3\beta )q^{11}+(41+2\beta )q^{13}+\cdots\)
1800.4.a.bp 1800.a 1.a $2$ $106.203$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(2+3\beta )q^{7}+(-20+8\beta )q^{11}+(52+\cdots)q^{13}+\cdots\)
1800.4.a.bq 1800.a 1.a $2$ $106.203$ \(\Q(\sqrt{181}) \) None \(0\) \(0\) \(0\) \(6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{7}+(-4-\beta )q^{11}+(7-2\beta )q^{13}+\cdots\)
1800.4.a.br 1800.a 1.a $3$ $106.203$ 3.3.121909.1 None \(0\) \(0\) \(0\) \(-9\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{7}+(-3+\beta _{1}+\beta _{2})q^{11}+\cdots\)
1800.4.a.bs 1800.a 1.a $3$ $106.203$ 3.3.121909.1 None \(0\) \(0\) \(0\) \(-9\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{7}+(3-\beta _{1}-\beta _{2})q^{11}+\cdots\)
1800.4.a.bt 1800.a 1.a $3$ $106.203$ 3.3.121909.1 None \(0\) \(0\) \(0\) \(9\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{7}+(-3+\beta _{1}+\beta _{2})q^{11}+\cdots\)
1800.4.a.bu 1800.a 1.a $3$ $106.203$ 3.3.121909.1 None \(0\) \(0\) \(0\) \(9\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{7}+(3-\beta _{1}-\beta _{2})q^{11}+(5+\cdots)q^{13}+\cdots\)
1800.4.a.bv 1800.a 1.a $4$ $106.203$ \(\Q(\sqrt{6}, \sqrt{46})\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}+\beta _{2})q^{7}+(\beta _{1}-2\beta _{2})q^{11}+\cdots\)
1800.4.a.bw 1800.a 1.a $4$ $106.203$ \(\Q(\sqrt{6}, \sqrt{46})\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}+\beta _{2})q^{7}+(-\beta _{1}+2\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1800))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1800)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 24}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 2}\)