Properties

Label 170.2.d
Level 170
Weight 2
Character orbit d
Rep. character \(\chi_{170}(169,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 3
Sturm bound 54
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 85 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(54\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(170, [\chi])\).

Total New Old
Modular forms 32 8 24
Cusp forms 24 8 16
Eisenstein series 8 0 8

Trace form

\( 8q - 8q^{4} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{4} + 12q^{9} + 16q^{15} + 8q^{16} - 12q^{19} - 40q^{21} - 4q^{25} + 20q^{26} - 20q^{30} - 4q^{34} + 4q^{35} - 12q^{36} + 32q^{49} + 4q^{50} + 36q^{51} - 68q^{59} - 16q^{60} - 8q^{64} - 32q^{69} + 44q^{70} + 12q^{76} + 8q^{81} + 40q^{84} + 16q^{85} - 48q^{86} + 68q^{89} + 28q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(170, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
170.2.d.a \(2\) \(1.357\) \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(-4\) \(-4\) \(q+iq^{2}-q^{3}-q^{4}+(-2+i)q^{5}-iq^{6}+\cdots\)
170.2.d.b \(2\) \(1.357\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(4\) \(4\) \(q+iq^{2}+q^{3}-q^{4}+(2-i)q^{5}+iq^{6}+\cdots\)
170.2.d.c \(4\) \(1.357\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}^{2}q^{2}+(-2\zeta_{8}+2\zeta_{8}^{3})q^{3}-q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(170, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(170, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))
$3$ (\( ( 1 + T + 3 T^{2} )^{2} \))(\( ( 1 - T + 3 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 9 T^{4} )^{2} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 8 T^{2} + 25 T^{4} \))
$7$ (\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T^{2} + 49 T^{4} )^{2} \))
$11$ (\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))
$13$ (\( 1 - 25 T^{2} + 169 T^{4} \))(\( 1 - 25 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )^{2}( 1 + 4 T + 13 T^{2} )^{2} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 2 T^{2} + 289 T^{4} \))
$19$ (\( ( 1 + 5 T + 19 T^{2} )^{2} \))(\( ( 1 + 5 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{4} \))
$23$ (\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 44 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( 1 + 23 T^{2} + 841 T^{4} \))(\( 1 + 23 T^{2} + 841 T^{4} \))(\( ( 1 - 40 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 - 37 T^{2} + 961 T^{4} \))(\( 1 - 37 T^{2} + 961 T^{4} \))(\( ( 1 - 44 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 56 T^{2} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 8 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} ) \))(\( ( 1 - 8 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} ) \))(\( ( 1 - 10 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 50 T^{2} + 1849 T^{4} \))(\( 1 - 50 T^{2} + 1849 T^{4} \))(\( ( 1 - 50 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( 1 - 45 T^{2} + 2209 T^{4} \))(\( 1 - 45 T^{2} + 2209 T^{4} \))(\( ( 1 - 47 T^{2} )^{4} \))
$53$ (\( 1 - 105 T^{2} + 2809 T^{4} \))(\( 1 - 105 T^{2} + 2809 T^{4} \))(\( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \))
$59$ (\( ( 1 + 5 T + 59 T^{2} )^{2} \))(\( ( 1 + 5 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{4} \))
$61$ (\( 1 - 97 T^{2} + 3721 T^{4} \))(\( 1 - 97 T^{2} + 3721 T^{4} \))(\( ( 1 - 104 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( ( 1 + 10 T^{2} + 4489 T^{4} )^{2} \))
$71$ (\( 1 - 117 T^{2} + 5041 T^{4} \))(\( 1 - 117 T^{2} + 5041 T^{4} \))(\( ( 1 - 124 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 + 11 T + 73 T^{2} )^{2} \))(\( ( 1 - 11 T + 73 T^{2} )^{2} \))(\( ( 1 + 74 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + 98 T^{2} + 6241 T^{4} \))(\( 1 + 98 T^{2} + 6241 T^{4} \))(\( ( 1 - 140 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( ( 1 - 130 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 - 5 T + 89 T^{2} )^{2} \))(\( ( 1 - 5 T + 89 T^{2} )^{2} \))(\( ( 1 - 12 T + 89 T^{2} )^{4} \))
$97$ (\( ( 1 + 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 + 97 T^{2} )^{4} \))
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