Properties

Label 210.2.b
Level 210
Weight 2
Character orbit b
Rep. character \(\chi_{210}(41,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 2
Sturm bound 96
Trace bound 3

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Defining parameters

Level: \( N \) = \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 210.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(17\)

Dimensions

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

Total New Old
Modular forms 56 8 48
Cusp forms 40 8 32
Eisenstein series 16 0 16

Trace form

\( 8q - 8q^{4} + 12q^{7} + O(q^{10}) \) \( 8q - 8q^{4} + 12q^{7} - 4q^{15} + 8q^{16} - 16q^{18} - 16q^{21} + 8q^{25} - 12q^{28} - 4q^{30} + 24q^{37} - 24q^{39} - 4q^{42} + 32q^{43} + 32q^{46} - 32q^{51} + 24q^{57} + 8q^{58} + 4q^{60} + 20q^{63} - 8q^{64} - 96q^{67} - 12q^{70} + 16q^{72} - 16q^{78} + 8q^{81} + 16q^{84} + 24q^{85} - 32q^{91} + 40q^{93} + O(q^{100}) \)

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.b.a \(4\) \(1.677\) \(\Q(i, \sqrt{5})\) None \(0\) \(-2\) \(4\) \(6\) \(q+\beta _{2}q^{2}+(-1-\beta _{1})q^{3}-q^{4}+q^{5}+\cdots\)
210.2.b.b \(4\) \(1.677\) \(\Q(i, \sqrt{5})\) None \(0\) \(2\) \(-4\) \(6\) \(q-\beta _{2}q^{2}+(\beta _{2}+\beta _{3})q^{3}-q^{4}-q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))
$3$ (\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))
$5$ (\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))
$7$ (\( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4} \))(\( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4} \))
$11$ (\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( 1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8} \))(\( 1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8} \))
$17$ (\( ( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( 1 - 4 T^{2} - 554 T^{4} - 1444 T^{6} + 130321 T^{8} \))(\( 1 - 4 T^{2} - 554 T^{4} - 1444 T^{6} + 130321 T^{8} \))
$23$ (\( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( 1 - 24 T^{2} + 1646 T^{4} - 20184 T^{6} + 707281 T^{8} \))(\( 1 - 24 T^{2} + 1646 T^{4} - 20184 T^{6} + 707281 T^{8} \))
$31$ (\( 1 - 64 T^{2} + 2446 T^{4} - 61504 T^{6} + 923521 T^{8} \))(\( 1 - 64 T^{2} + 2446 T^{4} - 61504 T^{6} + 923521 T^{8} \))
$37$ (\( ( 1 - 6 T + 78 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 6 T + 78 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 8 T + 22 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 8 T + 22 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))
$47$ (\( ( 1 + 4 T + 78 T^{2} + 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T + 78 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \))
$53$ (\( 1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8} \))(\( 1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 + 98 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( 1 - 184 T^{2} + 15406 T^{4} - 684664 T^{6} + 13845841 T^{8} \))(\( 1 - 184 T^{2} + 15406 T^{4} - 684664 T^{6} + 13845841 T^{8} \))
$67$ (\( ( 1 + 12 T + 67 T^{2} )^{4} \))(\( ( 1 + 12 T + 67 T^{2} )^{4} \))
$71$ (\( 1 - 224 T^{2} + 22126 T^{4} - 1129184 T^{6} + 25411681 T^{8} \))(\( 1 - 224 T^{2} + 22126 T^{4} - 1129184 T^{6} + 25411681 T^{8} \))
$73$ (\( 1 - 120 T^{2} + 12638 T^{4} - 639480 T^{6} + 28398241 T^{8} \))(\( 1 - 120 T^{2} + 12638 T^{4} - 639480 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( ( 1 + 2 T - 78 T^{2} + 166 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 2 T - 78 T^{2} - 166 T^{3} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 + 20 T + 258 T^{2} + 1780 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 20 T + 258 T^{2} - 1780 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 360 T^{2} + 51038 T^{4} - 3387240 T^{6} + 88529281 T^{8} \))(\( 1 - 360 T^{2} + 51038 T^{4} - 3387240 T^{6} + 88529281 T^{8} \))
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