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

\( 68 q + O(q^{10}) \) \( 68 q + 38 q^{11} + 46 q^{19} - 188 q^{29} + 492 q^{31} - 234 q^{41} - 2676 q^{49} - 1144 q^{59} + 104 q^{61} + 984 q^{71} + 692 q^{79} + 4374 q^{89} - 40 q^{91} + O(q^{100}) \)

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

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

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

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