Properties

Label 210.2.g
Level 210
Weight 2
Character orbit g
Rep. character \(\chi_{210}(169,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 2
Sturm bound 96
Trace bound 5

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Defining parameters

Level: \( N \) = \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 210.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(210, [\chi])\).

Total New Old
Modular forms 56 4 52
Cusp forms 40 4 36
Eisenstein series 16 0 16

Trace form

\( 4q - 4q^{4} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{9} - 8q^{15} + 4q^{16} + 16q^{19} - 4q^{21} - 12q^{25} + 16q^{26} + 4q^{30} + 8q^{31} - 24q^{34} - 8q^{35} + 4q^{36} - 8q^{39} + 16q^{41} - 16q^{46} - 4q^{49} + 16q^{50} - 8q^{51} + 8q^{55} + 32q^{59} + 8q^{60} - 4q^{64} - 16q^{65} + 8q^{66} + 16q^{69} + 4q^{70} - 40q^{71} - 24q^{74} - 16q^{76} + 16q^{79} + 4q^{81} + 4q^{84} - 16q^{85} + 24q^{86} - 8q^{91} - 24q^{94} + 8q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(210, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
210.2.g.a \(2\) \(1.677\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{2}+iq^{3}-q^{4}+(-1+2i)q^{5}+\cdots\)
210.2.g.b \(2\) \(1.677\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q+iq^{2}-iq^{3}-q^{4}+(1-2i)q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))
$3$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))
$7$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))
$11$ (\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))
$13$ (\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))
$17$ (\( ( 1 - 2 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} ) \))(\( 1 - 18 T^{2} + 289 T^{4} \))
$19$ (\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} )^{2} \))
$23$ (\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 18 T^{2} + 529 T^{4} \))
$29$ (\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))
$31$ (\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))
$37$ (\( 1 - 10 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))
$41$ (\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))
$43$ (\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))
$47$ (\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))
$53$ (\( 1 - 102 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))
$59$ (\( ( 1 - 8 T + 59 T^{2} )^{2} \))(\( ( 1 - 8 T + 59 T^{2} )^{2} \))
$61$ (\( ( 1 - 10 T + 61 T^{2} )^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))
$67$ (\( 1 + 10 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))
$71$ (\( ( 1 + 14 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))
$73$ (\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 + 50 T^{2} + 5329 T^{4} \))
$79$ (\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 12 T + 79 T^{2} )^{2} \))
$83$ (\( 1 + 90 T^{2} + 6889 T^{4} \))(\( 1 - 102 T^{2} + 6889 T^{4} \))
$89$ (\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))
$97$ (\( 1 - 94 T^{2} + 9409 T^{4} \))(\( 1 - 94 T^{2} + 9409 T^{4} \))
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