Properties

Label 1800.4.f
Level $1800$
Weight $4$
Character orbit 1800.f
Rep. character $\chi_{1800}(649,\cdot)$
Character field $\Q$
Dimension $68$
Newform subspaces $28$
Sturm bound $1440$
Trace bound $29$

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Defining parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 28 \)
Sturm bound: \(1440\)
Trace bound: \(29\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1800, [\chi])\).

Total New Old
Modular forms 1128 68 1060
Cusp forms 1032 68 964
Eisenstein series 96 0 96

Trace form

\( 68 q + 38 q^{11} + 46 q^{19} - 188 q^{29} + 492 q^{31} - 234 q^{41} - 2676 q^{49} - 1144 q^{59} + 104 q^{61} + 984 q^{71} + 692 q^{79} + 4374 q^{89} - 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{4}^{\mathrm{new}}(1800, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1800.4.f.a 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 120.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta q^{7}-72 q^{11}+3\beta q^{13}-19\beta q^{17}+\cdots\)
1800.4.f.b 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 72.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+6\beta q^{7}-64 q^{11}+29\beta q^{13}+16\beta q^{17}+\cdots\)
1800.4.f.c 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 200.4.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2 i q^{7}-39 q^{11}-84 i q^{13}+61 i q^{17}+\cdots\)
1800.4.f.d 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 40.4.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta q^{7}-36 q^{11}+21\beta q^{13}+55\beta q^{17}+\cdots\)
1800.4.f.e 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 360.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+9\beta q^{7}-34 q^{11}+6\beta q^{13}-51\beta q^{17}+\cdots\)
1800.4.f.f 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 360.4.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{7}-34 q^{11}+34\beta q^{13}-19\beta q^{17}+\cdots\)
1800.4.f.g 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 600.4.a.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+19 i q^{7}-22 q^{11}+i q^{13}-58 i q^{17}+\cdots\)
1800.4.f.h 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 120.4.a.f \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta q^{7}-20 q^{11}-11\beta q^{13}+7\beta q^{17}+\cdots\)
1800.4.f.i 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 360.4.a.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+17\beta q^{7}-18 q^{11}-6\beta q^{13}+53\beta q^{17}+\cdots\)
1800.4.f.j 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 40.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+17\beta q^{7}-16 q^{11}+29\beta q^{13}-35\beta q^{17}+\cdots\)
1800.4.f.k 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 120.4.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+10\beta q^{7}-16 q^{11}-29\beta q^{13}-19\beta q^{17}+\cdots\)
1800.4.f.l 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 600.4.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+5 i q^{7}-14 q^{11}-i q^{13}-46 i q^{17}+\cdots\)
1800.4.f.m 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 120.4.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4 q^{11}-27\beta q^{13}-57\beta q^{17}-44 q^{19}+\cdots\)
1800.4.f.n 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 40.4.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+9\beta q^{7}+16 q^{11}-3\beta q^{13}-3\beta q^{17}+\cdots\)
1800.4.f.o 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 360.4.a.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+17\beta q^{7}+18 q^{11}-6\beta q^{13}-53\beta q^{17}+\cdots\)
1800.4.f.p 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 200.4.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+6 i q^{7}+19 q^{11}+12 i q^{13}-75 i q^{17}+\cdots\)
1800.4.f.q 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 24.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+12\beta q^{7}+28 q^{11}-37\beta q^{13}+41\beta q^{17}+\cdots\)
1800.4.f.r 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 120.4.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta q^{7}+28 q^{11}-13\beta q^{13}-31\beta q^{17}+\cdots\)
1800.4.f.s 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 360.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+9\beta q^{7}+34 q^{11}+6\beta q^{13}+51\beta q^{17}+\cdots\)
1800.4.f.t 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 360.4.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{7}+34 q^{11}+34\beta q^{13}+19\beta q^{17}+\cdots\)
1800.4.f.u 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 8.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+12\beta q^{7}+44 q^{11}-11\beta q^{13}-25\beta q^{17}+\cdots\)
1800.4.f.v 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 120.4.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+10\beta q^{7}+56 q^{11}+43\beta q^{13}+53\beta q^{17}+\cdots\)
1800.4.f.w 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 200.4.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+26 i q^{7}+59 q^{11}-28 i q^{13}-5 i q^{17}+\cdots\)
1800.4.f.x 1800.f 5.b $2$ $106.203$ \(\Q(\sqrt{-1}) \) None 72.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+6\beta q^{7}+64 q^{11}+29\beta q^{13}-16\beta q^{17}+\cdots\)
1800.4.f.y 1800.f 5.b $4$ $106.203$ \(\Q(i, \sqrt{181})\) None 600.4.a.r \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-3\beta _{1}+\beta _{2})q^{7}+(-4+\beta _{3})q^{11}+\cdots\)
1800.4.f.z 1800.f 5.b $4$ $106.203$ \(\Q(i, \sqrt{109})\) None 600.4.a.s \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}-\beta _{2})q^{7}+(8+3\beta _{3})q^{11}+(-41\beta _{1}+\cdots)q^{13}+\cdots\)
1800.4.f.ba 1800.f 5.b $6$ $106.203$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 1800.4.a.br \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-3\beta _{1}-\beta _{4})q^{7}+(-3+\beta _{2}-\beta _{3}+\cdots)q^{11}+\cdots\)
1800.4.f.bb 1800.f 5.b $6$ $106.203$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 1800.4.a.br \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-3\beta _{1}-\beta _{4})q^{7}+(3-\beta _{2}+\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1800, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1800, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 2}\)