Properties

Label 210.2.d
Level 210
Weight 2
Character orbit d
Rep. character \(\chi_{210}(209,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 2
Sturm bound 96
Trace bound 2

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Defining parameters

Level: \( N \) = \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 210.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(23\)

Dimensions

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

Total New Old
Modular forms 56 16 40
Cusp forms 40 16 24
Eisenstein series 16 0 16

Trace form

\( 16q + 16q^{4} + O(q^{10}) \) \( 16q + 16q^{4} - 8q^{15} + 16q^{16} - 16q^{21} - 8q^{30} - 48q^{39} - 32q^{46} - 16q^{49} - 32q^{51} - 8q^{60} + 16q^{64} - 32q^{70} - 32q^{79} + 80q^{81} - 16q^{84} + 64q^{85} + 32q^{91} + 64q^{99} + O(q^{100}) \)

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.d.a \(8\) \(1.677\) 8.0.\(\cdots\).11 None \(-8\) \(0\) \(0\) \(0\) \(q-q^{2}-\beta _{7}q^{3}+q^{4}+(\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
210.2.d.b \(8\) \(1.677\) 8.0.\(\cdots\).11 None \(8\) \(0\) \(0\) \(0\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(-\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T )^{8} \))(\( ( 1 - T )^{8} \))
$3$ (\( 1 - 10 T^{4} + 81 T^{8} \))(\( 1 - 10 T^{4} + 81 T^{8} \))
$5$ (\( 1 + 22 T^{4} + 625 T^{8} \))(\( 1 + 22 T^{4} + 625 T^{8} \))
$7$ (\( 1 + 4 T^{2} - 10 T^{4} + 196 T^{6} + 2401 T^{8} \))(\( 1 + 4 T^{2} - 10 T^{4} + 196 T^{6} + 2401 T^{8} \))
$11$ (\( ( 1 - 6 T + 11 T^{2} )^{4}( 1 + 6 T + 11 T^{2} )^{4} \))(\( ( 1 - 6 T + 11 T^{2} )^{4}( 1 + 6 T + 11 T^{2} )^{4} \))
$13$ (\( ( 1 + 40 T^{2} + 710 T^{4} + 6760 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 40 T^{2} + 710 T^{4} + 6760 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 - 44 T^{2} + 950 T^{4} - 12716 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 44 T^{2} + 950 T^{4} - 12716 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 24 T^{2} + 838 T^{4} - 8664 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 24 T^{2} + 838 T^{4} - 8664 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 - 4 T + 22 T^{2} - 92 T^{3} + 529 T^{4} )^{4} \))(\( ( 1 + 4 T + 22 T^{2} + 92 T^{3} + 529 T^{4} )^{4} \))
$29$ (\( ( 1 - 52 T^{2} + 1350 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 52 T^{2} + 1350 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 84 T^{2} + 3574 T^{4} - 80724 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 84 T^{2} + 3574 T^{4} - 80724 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 - 20 T^{2} + 38 T^{4} - 27380 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} + 38 T^{4} - 27380 T^{6} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 + 84 T^{2} + 4678 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 84 T^{2} + 4678 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 44 T^{2} + 2390 T^{4} - 81356 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 44 T^{2} + 2390 T^{4} - 81356 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 - 108 T^{2} + 6886 T^{4} - 238572 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 108 T^{2} + 6886 T^{4} - 238572 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 - 4 T - 2 T^{2} - 212 T^{3} + 2809 T^{4} )^{4} \))(\( ( 1 + 4 T - 2 T^{2} + 212 T^{3} + 2809 T^{4} )^{4} \))
$59$ (\( ( 1 + 216 T^{2} + 18598 T^{4} + 751896 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 216 T^{2} + 18598 T^{4} + 751896 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 136 T^{2} + 9798 T^{4} - 506056 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 136 T^{2} + 9798 T^{4} - 506056 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 - 140 T^{2} + 12086 T^{4} - 628460 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 140 T^{2} + 12086 T^{4} - 628460 T^{6} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 - 28 T^{2} + 9270 T^{4} - 141148 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 28 T^{2} + 9270 T^{4} - 141148 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 252 T^{2} + 26422 T^{4} + 1342908 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 + 252 T^{2} + 26422 T^{4} + 1342908 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 + 4 T + 134 T^{2} + 316 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 4 T + 134 T^{2} + 316 T^{3} + 6241 T^{4} )^{4} \))
$83$ (\( ( 1 - 224 T^{2} + 26294 T^{4} - 1543136 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 224 T^{2} + 26294 T^{4} - 1543136 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 - 60 T^{2} + 14950 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 60 T^{2} + 14950 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 108 T^{2} + 16246 T^{4} + 1016172 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 108 T^{2} + 16246 T^{4} + 1016172 T^{6} + 88529281 T^{8} )^{2} \))
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