Properties

Label 210.3.f
Level $210$
Weight $3$
Character orbit 210.f
Rep. character $\chi_{210}(181,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $1$
Sturm bound $144$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 210.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(144\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(210, [\chi])\).

Total New Old
Modular forms 104 8 96
Cusp forms 88 8 80
Eisenstein series 16 0 16

Trace form

\( 8 q + 16 q^{4} - 24 q^{9} + O(q^{10}) \) \( 8 q + 16 q^{4} - 24 q^{9} - 16 q^{11} + 32 q^{14} + 32 q^{16} + 96 q^{22} - 40 q^{25} - 144 q^{29} - 80 q^{35} - 48 q^{36} - 48 q^{37} - 48 q^{39} - 48 q^{42} - 64 q^{43} - 32 q^{44} + 128 q^{46} - 24 q^{49} + 128 q^{53} + 64 q^{56} + 144 q^{57} + 224 q^{58} + 64 q^{64} + 80 q^{65} - 192 q^{67} + 176 q^{71} - 160 q^{74} + 192 q^{77} - 96 q^{78} - 288 q^{79} + 72 q^{81} - 240 q^{85} - 64 q^{86} + 192 q^{88} + 64 q^{91} + 336 q^{93} + 80 q^{95} + 48 q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(210, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
210.3.f.a 210.f 7.b $8$ $5.722$ 8.0.3317760000.3 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{5}q^{3}+2q^{4}+\beta _{6}q^{5}+\beta _{3}q^{6}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)