Properties

Label 168.2.k
Level 168
Weight 2
Character orbit k
Rep. character \(\chi_{168}(41,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 1
Sturm bound 64
Trace bound 0

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Defining parameters

Level: \( N \) = \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 168.k (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(64\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(168, [\chi])\).

Total New Old
Modular forms 40 8 32
Cusp forms 24 8 16
Eisenstein series 16 0 16

Trace form

\( 8q - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 8q - 4q^{7} + 4q^{9} + 4q^{15} - 8q^{21} - 16q^{37} + 4q^{39} - 32q^{43} + 16q^{49} - 24q^{51} - 28q^{57} + 32q^{63} + 16q^{67} + 56q^{79} + 32q^{85} - 24q^{91} + 56q^{93} + 64q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(168, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
168.2.k.a \(8\) \(1.341\) 8.0.342102016.5 None \(0\) \(0\) \(0\) \(-4\) \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(-1-\beta _{5}-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(168, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(168, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \)
$5$ \( ( 1 + 10 T^{2} + 58 T^{4} + 250 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 8 T^{2} + 190 T^{4} - 968 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 48 T^{2} + 1086 T^{4} + 13872 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 2 T^{2} + 706 T^{4} - 722 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 42 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 32 T^{2} + 238 T^{4} - 26912 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 32 T^{2} + 2110 T^{4} - 30752 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{8} \)
$41$ \( ( 1 + 48 T^{2} + 606 T^{4} + 80688 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{8} \)
$47$ \( ( 1 + 4 T^{2} + 4150 T^{4} + 8836 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 128 T^{2} + 8014 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 110 T^{2} + 8610 T^{4} + 382910 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 118 T^{2} + 9546 T^{4} - 439078 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 4 T + 70 T^{2} - 268 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 88 T^{2} + 6510 T^{4} - 443608 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 132 T^{2} + 10662 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 14 T + 190 T^{2} - 1106 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 318 T^{2} + 39042 T^{4} + 2190702 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 24 T^{2} + 12654 T^{4} - 190104 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} )^{2} \)
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