Properties

Label 286.2.b
Level 286
Weight 2
Character orbit b
Rep. character \(\chi_{286}(155,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 4
Sturm bound 84
Trace bound 10

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Defining parameters

Level: \( N \) = \( 286 = 2 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 286.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(84\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(286, [\chi])\).

Total New Old
Modular forms 46 10 36
Cusp forms 38 10 28
Eisenstein series 8 0 8

Trace form

\( 10q + 4q^{3} - 10q^{4} + 6q^{9} + O(q^{10}) \) \( 10q + 4q^{3} - 10q^{4} + 6q^{9} - 8q^{10} - 4q^{12} + 4q^{13} + 8q^{14} + 10q^{16} + 2q^{22} + 20q^{23} + 18q^{25} + 2q^{26} + 4q^{27} - 16q^{29} - 4q^{30} - 20q^{35} - 6q^{36} - 8q^{38} - 32q^{39} + 8q^{40} - 28q^{42} - 8q^{43} + 4q^{48} - 34q^{49} + 4q^{51} - 4q^{52} + 20q^{53} - 8q^{56} + 40q^{61} + 36q^{62} - 10q^{64} - 24q^{65} + 8q^{66} - 64q^{69} + 16q^{74} + 16q^{75} - 8q^{77} - 4q^{78} + 48q^{79} + 26q^{81} + 16q^{82} + 8q^{87} - 2q^{88} - 12q^{90} + 12q^{91} - 20q^{92} + 24q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(286, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
286.2.b.a \(2\) \(2.284\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-q^{4}+3iq^{5}+iq^{7}-iq^{8}+\cdots\)
286.2.b.b \(2\) \(2.284\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-q^{4}-2iq^{5}-4iq^{7}-iq^{8}+\cdots\)
286.2.b.c \(2\) \(2.284\) \(\Q(\sqrt{-1}) \) None \(0\) \(6\) \(0\) \(0\) \(q+iq^{2}+3q^{3}-q^{4}+iq^{5}+3iq^{6}+\cdots\)
286.2.b.d \(4\) \(2.284\) \(\Q(i, \sqrt{17})\) None \(0\) \(-2\) \(0\) \(0\) \(q+\beta _{2}q^{2}+(-1+\beta _{3})q^{3}-q^{4}+\beta _{2}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(286, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(286, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(143, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))
$3$ (\( ( 1 + 3 T^{2} )^{2} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( ( 1 - 3 T + 3 T^{2} )^{2} \))(\( ( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} )^{2} \))
$5$ (\( 1 - T^{2} + 25 T^{4} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( 1 - 9 T^{2} + 25 T^{4} \))(\( ( 1 - 9 T^{2} + 25 T^{4} )^{2} \))
$7$ (\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( ( 1 + 3 T^{2} + 49 T^{4} )^{2} \))
$11$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 6 T + 18 T^{2} - 78 T^{3} + 169 T^{4} \))
$17$ (\( ( 1 + 8 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 + T + 17 T^{2} )^{2} \))(\( ( 1 - 7 T + 42 T^{2} - 119 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( 1 - 40 T^{2} + 1054 T^{4} - 14440 T^{6} + 130321 T^{8} \))
$23$ (\( ( 1 - 5 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 6 T + 23 T^{2} )^{2} \))(\( ( 1 - 11 T + 72 T^{2} - 253 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( ( 1 + 15 T + 110 T^{2} + 435 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 2 T^{2} + 961 T^{4} \))(\( 1 - 58 T^{2} + 961 T^{4} \))(\( 1 + 2 T^{2} + 961 T^{4} \))(\( ( 1 + 2 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 - 73 T^{2} + 1369 T^{4} \))(\( 1 - 79 T^{2} + 3784 T^{4} - 108151 T^{6} + 1874161 T^{8} \))
$41$ (\( 1 - T^{2} + 1681 T^{4} \))(\( 1 - 46 T^{2} + 1681 T^{4} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 75 T^{2} + 3812 T^{4} - 126075 T^{6} + 2825761 T^{8} \))
$43$ (\( ( 1 - 9 T + 43 T^{2} )^{2} \))(\( ( 1 + 6 T + 43 T^{2} )^{2} \))(\( ( 1 + 9 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} - 86 T^{3} + 1849 T^{4} )^{2} \))
$47$ (\( 1 + 6 T^{2} + 2209 T^{4} \))(\( 1 + 6 T^{2} + 2209 T^{4} \))(\( 1 + 27 T^{2} + 2209 T^{4} \))(\( 1 - 175 T^{2} + 12036 T^{4} - 386575 T^{6} + 4879681 T^{8} \))
$53$ (\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 8 T + 54 T^{2} + 424 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 - 117 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 114 T^{2} + 3481 T^{4} \))(\( 1 - 119 T^{2} + 7404 T^{4} - 414239 T^{6} + 12117361 T^{8} \))
$61$ (\( ( 1 + T + 61 T^{2} )^{2} \))(\( ( 1 - 4 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 25 T + 274 T^{2} - 1525 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 85 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 219 T^{2} + 20624 T^{4} - 983091 T^{6} + 20151121 T^{8} \))
$71$ (\( 1 + 54 T^{2} + 5041 T^{4} \))(\( 1 - 106 T^{2} + 5041 T^{4} \))(\( 1 - 61 T^{2} + 5041 T^{4} \))(\( 1 - 215 T^{2} + 21124 T^{4} - 1083815 T^{6} + 25411681 T^{8} \))
$73$ (\( 1 - 121 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 + 50 T^{2} + 5329 T^{4} \))(\( 1 - 191 T^{2} + 17904 T^{4} - 1017839 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 - 16 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 14 T + 79 T^{2} )^{2} \))(\( ( 1 + 2 T + 142 T^{2} + 158 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 162 T^{2} + 6889 T^{4} \))(\( ( 1 - 98 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( 1 - 114 T^{2} + 7921 T^{4} \))(\( 1 - 114 T^{2} + 7921 T^{4} \))(\( 1 + 18 T^{2} + 7921 T^{4} \))(\( 1 - 320 T^{2} + 41374 T^{4} - 2534720 T^{6} + 62742241 T^{8} \))
$97$ (\( 1 - 190 T^{2} + 9409 T^{4} \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( 1 + 62 T^{2} + 9409 T^{4} \))(\( 1 - 64 T^{2} + 14334 T^{4} - 602176 T^{6} + 88529281 T^{8} \))
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