Properties

Label 1200.3.l
Level $1200$
Weight $3$
Character orbit 1200.l
Rep. character $\chi_{1200}(401,\cdot)$
Character field $\Q$
Dimension $73$
Newform subspaces $25$
Sturm bound $720$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 25 \)
Sturm bound: \(720\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1200, [\chi])\).

Total New Old
Modular forms 516 79 437
Cusp forms 444 73 371
Eisenstein series 72 6 66

Trace form

\( 73q - q^{3} + 6q^{7} - 3q^{9} + O(q^{10}) \) \( 73q - q^{3} + 6q^{7} - 3q^{9} + 2q^{13} - 2q^{19} + 14q^{21} - 25q^{27} + 30q^{31} + 16q^{33} - 14q^{37} + 26q^{39} - 82q^{43} + 419q^{49} - 44q^{51} - 22q^{57} + 90q^{61} - 58q^{63} + 62q^{67} + 68q^{69} + 10q^{73} + 294q^{79} - 63q^{81} + 144q^{87} - 124q^{91} + 114q^{93} - 166q^{97} + 352q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1200, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1200.3.l.a \(1\) \(32.698\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(-13\) \(q-3q^{3}-13q^{7}+9q^{9}-23q^{13}-11q^{19}+\cdots\)
1200.3.l.b \(1\) \(32.698\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(2\) \(q-3q^{3}+2q^{7}+9q^{9}+22q^{13}-26q^{19}+\cdots\)
1200.3.l.c \(1\) \(32.698\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(11\) \(q-3q^{3}+11q^{7}+9q^{9}+q^{13}+37q^{19}+\cdots\)
1200.3.l.d \(1\) \(32.698\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(-11\) \(q+3q^{3}-11q^{7}+9q^{9}-q^{13}+37q^{19}+\cdots\)
1200.3.l.e \(1\) \(32.698\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(13\) \(q+3q^{3}+13q^{7}+9q^{9}+23q^{13}-11q^{19}+\cdots\)
1200.3.l.f \(2\) \(32.698\) \(\Q(\sqrt{-11}) \) None \(0\) \(-5\) \(0\) \(0\) \(q+(-3+\beta )q^{3}+(6-5\beta )q^{9}+(5-10\beta )q^{11}+\cdots\)
1200.3.l.g \(2\) \(32.698\) \(\Q(\sqrt{-5}) \) None \(0\) \(-4\) \(0\) \(-12\) \(q+(-2+\beta )q^{3}-6q^{7}+(-1-4\beta )q^{9}+\cdots\)
1200.3.l.h \(2\) \(32.698\) \(\Q(\sqrt{-5}) \) None \(0\) \(-4\) \(0\) \(16\) \(q+(-2+\beta )q^{3}+8q^{7}+(-1-4\beta )q^{9}+\cdots\)
1200.3.l.i \(2\) \(32.698\) \(\Q(\sqrt{-2}) \) None \(0\) \(-2\) \(0\) \(-14\) \(q+(-1+\beta )q^{3}-7q^{7}+(-7-2\beta )q^{9}+\cdots\)
1200.3.l.j \(2\) \(32.698\) \(\Q(\sqrt{-2}) \) None \(0\) \(-2\) \(0\) \(2\) \(q+(-1-\beta )q^{3}+q^{7}+(-7+2\beta )q^{9}+\cdots\)
1200.3.l.k \(2\) \(32.698\) \(\Q(\sqrt{-35}) \) None \(0\) \(-1\) \(0\) \(-16\) \(q-\beta q^{3}-8q^{7}+(-9+\beta )q^{9}+(3-6\beta )q^{11}+\cdots\)
1200.3.l.l \(2\) \(32.698\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}-9q^{9}+14iq^{17}-22q^{19}+\cdots\)
1200.3.l.m \(2\) \(32.698\) \(\Q(\sqrt{-35}) \) None \(0\) \(1\) \(0\) \(16\) \(q+\beta q^{3}+8q^{7}+(-9+\beta )q^{9}+(3-6\beta )q^{11}+\cdots\)
1200.3.l.n \(2\) \(32.698\) \(\Q(\sqrt{-2}) \) None \(0\) \(2\) \(0\) \(-12\) \(q+(1+\beta )q^{3}-6q^{7}+(-7+2\beta )q^{9}+\cdots\)
1200.3.l.o \(2\) \(32.698\) \(\Q(\sqrt{-2}) \) None \(0\) \(2\) \(0\) \(-2\) \(q+(1-\beta )q^{3}-q^{7}+(-7-2\beta )q^{9}+3\beta q^{11}+\cdots\)
1200.3.l.p \(2\) \(32.698\) \(\Q(\sqrt{-2}) \) None \(0\) \(2\) \(0\) \(14\) \(q+(1-\beta )q^{3}+7q^{7}+(-7-2\beta )q^{9}+\cdots\)
1200.3.l.q \(2\) \(32.698\) \(\Q(\sqrt{-5}) \) None \(0\) \(4\) \(0\) \(-16\) \(q+(2-\beta )q^{3}-8q^{7}+(-1-4\beta )q^{9}+\cdots\)
1200.3.l.r \(2\) \(32.698\) \(\Q(\sqrt{-5}) \) None \(0\) \(4\) \(0\) \(4\) \(q+(2+\beta )q^{3}+2q^{7}+(-1+4\beta )q^{9}+\cdots\)
1200.3.l.s \(2\) \(32.698\) \(\Q(\sqrt{-11}) \) None \(0\) \(5\) \(0\) \(0\) \(q+(3-\beta )q^{3}+(6-5\beta )q^{9}+(5-10\beta )q^{11}+\cdots\)
1200.3.l.t \(4\) \(32.698\) \(\Q(\sqrt{-2}, \sqrt{-17})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}+\beta _{2})q^{3}+(-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
1200.3.l.u \(4\) \(32.698\) \(\Q(\sqrt{-2}, \sqrt{-5})\) None \(0\) \(4\) \(0\) \(8\) \(q+(1+\beta _{3})q^{3}+(2+\beta _{1}-2\beta _{2}-2\beta _{3})q^{7}+\cdots\)
1200.3.l.v \(6\) \(32.698\) 6.0.574198272.1 None \(0\) \(-4\) \(0\) \(10\) \(q+(-1-\beta _{2})q^{3}+(2+\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
1200.3.l.w \(6\) \(32.698\) 6.0.574198272.1 None \(0\) \(4\) \(0\) \(-10\) \(q+(1+\beta _{2})q^{3}+(-2+\beta _{1})q^{7}+(3+\beta _{1}+\cdots)q^{9}+\cdots\)
1200.3.l.x \(8\) \(32.698\) 8.0.\(\cdots\).5 None \(0\) \(-4\) \(0\) \(16\) \(q+\beta _{2}q^{3}+(2+\beta _{3}-\beta _{4}-\beta _{7})q^{7}+(2+\cdots)q^{9}+\cdots\)
1200.3.l.y \(12\) \(32.698\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{3}-\beta _{5}q^{7}+(-1-\beta _{9})q^{9}+(\beta _{7}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1200, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)