Properties

Label 1200.2.f
Level $1200$
Weight $2$
Character orbit 1200.f
Rep. character $\chi_{1200}(49,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $9$
Sturm bound $480$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(480\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1200, [\chi])\).

Total New Old
Modular forms 276 18 258
Cusp forms 204 18 186
Eisenstein series 72 0 72

Trace form

\( 18 q - 18 q^{9} + O(q^{10}) \) \( 18 q - 18 q^{9} - 20 q^{19} + 4 q^{29} - 4 q^{31} + 8 q^{39} + 20 q^{41} - 26 q^{49} + 20 q^{51} - 32 q^{59} - 20 q^{61} + 16 q^{69} + 32 q^{71} + 18 q^{81} - 36 q^{89} + 68 q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1200, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1200.2.f.a 1200.f 5.b $2$ $9.582$ \(\Q(\sqrt{-1}) \) None 300.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+iq^{7}-q^{9}-6q^{11}-5iq^{13}+\cdots\)
1200.2.f.b 1200.f 5.b $2$ $9.582$ \(\Q(\sqrt{-1}) \) None 24.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}-q^{9}-4q^{11}+2iq^{13}+2iq^{17}+\cdots\)
1200.2.f.c 1200.f 5.b $2$ $9.582$ \(\Q(\sqrt{-1}) \) None 600.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+3iq^{7}-q^{9}-2q^{11}-3iq^{13}+\cdots\)
1200.2.f.d 1200.f 5.b $2$ $9.582$ \(\Q(\sqrt{-1}) \) None 75.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+3iq^{7}-q^{9}-2q^{11}-iq^{13}+\cdots\)
1200.2.f.e 1200.f 5.b $2$ $9.582$ \(\Q(\sqrt{-1}) \) None 30.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+4iq^{7}-q^{9}-2iq^{13}+6iq^{17}+\cdots\)
1200.2.f.f 1200.f 5.b $2$ $9.582$ \(\Q(\sqrt{-1}) \) None 120.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+4iq^{7}-q^{9}-6iq^{13}+2iq^{17}+\cdots\)
1200.2.f.g 1200.f 5.b $2$ $9.582$ \(\Q(\sqrt{-1}) \) None 120.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-q^{9}+4q^{11}-6iq^{13}-6iq^{17}+\cdots\)
1200.2.f.h 1200.f 5.b $2$ $9.582$ \(\Q(\sqrt{-1}) \) None 15.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-q^{9}+4q^{11}-2iq^{13}-2iq^{17}+\cdots\)
1200.2.f.i 1200.f 5.b $2$ $9.582$ \(\Q(\sqrt{-1}) \) None 600.2.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+5iq^{7}-q^{9}+6q^{11}+3iq^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

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