Properties

Label 1200.4.f
Level $1200$
Weight $4$
Character orbit 1200.f
Rep. character $\chi_{1200}(49,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $24$
Sturm bound $960$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(960\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1200, [\chi])\).

Total New Old
Modular forms 756 54 702
Cusp forms 684 54 630
Eisenstein series 72 0 72

Trace form

\( 54 q - 486 q^{9} + O(q^{10}) \) \( 54 q - 486 q^{9} + 252 q^{19} + 284 q^{29} + 636 q^{31} - 312 q^{39} - 356 q^{41} - 2030 q^{49} + 132 q^{51} + 1376 q^{59} + 756 q^{61} - 528 q^{69} - 224 q^{71} - 2240 q^{79} + 4374 q^{81} - 1932 q^{89} - 812 q^{91} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1200, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1200.4.f.a \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+4iq^{7}-9q^{9}-72q^{11}+\cdots\)
1200.4.f.b \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+24iq^{7}-9q^{9}-52q^{11}+\cdots\)
1200.4.f.c \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+iq^{7}-9q^{9}-42q^{11}+67iq^{13}+\cdots\)
1200.4.f.d \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+8iq^{7}-9q^{9}-6^{2}q^{11}+\cdots\)
1200.4.f.e \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+2^{5}iq^{7}-9q^{9}-6^{2}q^{11}+\cdots\)
1200.4.f.f \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+19iq^{7}-9q^{9}-22q^{11}+\cdots\)
1200.4.f.g \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+8iq^{7}-9q^{9}-20q^{11}+\cdots\)
1200.4.f.h \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+20iq^{7}-9q^{9}-2^{4}q^{11}+\cdots\)
1200.4.f.i \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+5iq^{7}-9q^{9}-14q^{11}+\cdots\)
1200.4.f.j \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+2^{4}iq^{7}-9q^{9}-12q^{11}+\cdots\)
1200.4.f.k \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+13iq^{7}-9q^{9}-6q^{11}+\cdots\)
1200.4.f.l \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}-9q^{9}-4q^{11}+54iq^{13}+\cdots\)
1200.4.f.m \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+20iq^{7}-9q^{9}+24q^{11}+\cdots\)
1200.4.f.n \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+28iq^{7}-9q^{9}+24q^{11}+\cdots\)
1200.4.f.o \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+2^{4}iq^{7}-9q^{9}+28q^{11}+\cdots\)
1200.4.f.p \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+24iq^{7}-9q^{9}+28q^{11}+\cdots\)
1200.4.f.q \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+23iq^{7}-9q^{9}+30q^{11}+\cdots\)
1200.4.f.r \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+4iq^{7}-9q^{9}+48q^{11}+\cdots\)
1200.4.f.s \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+7iq^{7}-9q^{9}+54q^{11}+\cdots\)
1200.4.f.t \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+20iq^{7}-9q^{9}+56q^{11}+\cdots\)
1200.4.f.u \(2\) \(70.802\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+2^{5}iq^{7}-9q^{9}+60q^{11}+\cdots\)
1200.4.f.v \(4\) \(70.802\) \(\Q(i, \sqrt{19})\) None \(0\) \(0\) \(0\) \(0\) \(q+3\beta _{1}q^{3}+(-13\beta _{1}+\beta _{3})q^{7}-9q^{9}+\cdots\)
1200.4.f.w \(4\) \(70.802\) \(\Q(i, \sqrt{181})\) None \(0\) \(0\) \(0\) \(0\) \(q+3\beta _{1}q^{3}+(-3\beta _{1}+\beta _{2})q^{7}-9q^{9}+\cdots\)
1200.4.f.x \(4\) \(70.802\) \(\Q(i, \sqrt{109})\) None \(0\) \(0\) \(0\) \(0\) \(q+3\beta _{1}q^{3}+(\beta _{1}-\beta _{2})q^{7}-9q^{9}+(8+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1200, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)