Properties

Label 168.2.ba
Level 168
Weight 2
Character orbit ba
Rep. character \(\chi_{168}(5,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 56
Newform subspaces 3
Sturm bound 64
Trace bound 3

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Defining parameters

Level: \( N \) = \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 168.ba (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 168 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(64\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 72 72 0
Cusp forms 56 56 0
Eisenstein series 16 16 0

Trace form

\( 56q - 2q^{4} - 8q^{7} - 2q^{9} + O(q^{10}) \) \( 56q - 2q^{4} - 8q^{7} - 2q^{9} - 6q^{10} - 18q^{12} - 20q^{15} - 10q^{16} - 4q^{22} - 12q^{24} + 8q^{25} - 18q^{28} + 22q^{30} - 12q^{31} - 6q^{33} + 4q^{36} - 8q^{39} - 66q^{40} + 36q^{42} - 8q^{46} - 16q^{49} - 48q^{52} + 60q^{54} + 4q^{57} + 22q^{58} - 8q^{60} - 30q^{63} + 4q^{64} + 54q^{66} + 42q^{70} + 42q^{72} - 36q^{73} - 68q^{78} - 36q^{79} + 6q^{81} + 48q^{82} + 48q^{84} - 24q^{87} - 26q^{88} + 48q^{94} + 90q^{96} + O(q^{100}) \)

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.ba.a \(4\) \(1.341\) \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Q(\sqrt{-6}) \) \(0\) \(-6\) \(6\) \(-2\) \(q+\beta _{1}q^{2}+(-2-\beta _{2})q^{3}+2\beta _{2}q^{4}+\cdots\)
168.2.ba.b \(4\) \(1.341\) \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Q(\sqrt{-6}) \) \(0\) \(6\) \(-6\) \(-2\) \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{3}+2\beta _{2}q^{4}+(-1+\cdots)q^{5}+\cdots\)
168.2.ba.c \(48\) \(1.341\) None \(0\) \(0\) \(0\) \(-4\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))
$3$ (\( ( 1 + 3 T + 3 T^{2} )^{2} \))(\( ( 1 - 3 T + 3 T^{2} )^{2} \))
$5$ (\( ( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 2 T^{2} + 25 T^{4} ) \))(\( ( 1 + 2 T^{2} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} ) \))
$7$ (\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))
$11$ (\( ( 1 - 10 T^{2} + 121 T^{4} )( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} ) \))(\( ( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} )( 1 - 10 T^{2} + 121 T^{4} ) \))
$13$ (\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))
$17$ (\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 19 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 + 18 T + 137 T^{2} + 522 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 18 T + 137 T^{2} - 522 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 + 10 T + 31 T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} ) \))(\( ( 1 + 10 T + 31 T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} ) \))
$37$ (\( ( 1 + 37 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 37 T^{2} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))
$43$ (\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 43 T^{2} )^{4} \))
$47$ (\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( ( 1 - 94 T^{2} + 2809 T^{4} )( 1 + 6 T + 65 T^{2} + 318 T^{3} + 2809 T^{4} ) \))(\( ( 1 - 6 T + 65 T^{2} - 318 T^{3} + 2809 T^{4} )( 1 - 94 T^{2} + 2809 T^{4} ) \))
$59$ (\( ( 1 - 18 T + 167 T^{2} - 1062 T^{3} + 3481 T^{4} )( 1 - 10 T^{2} + 3481 T^{4} ) \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )( 1 + 18 T + 167 T^{2} + 1062 T^{3} + 3481 T^{4} ) \))
$61$ (\( ( 1 - 61 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( ( 1 + 67 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 67 T^{2} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 71 T^{2} )^{4} \))
$73$ (\( ( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} )( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} ) \))(\( ( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} )( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} ) \))
$79$ (\( ( 1 - 10 T + 79 T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} ) \))(\( ( 1 - 10 T + 79 T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} ) \))
$83$ (\( ( 1 - 30 T + 383 T^{2} - 2490 T^{3} + 6889 T^{4} )( 1 + 30 T + 383 T^{2} + 2490 T^{3} + 6889 T^{4} ) \))(\( ( 1 - 30 T + 383 T^{2} - 2490 T^{3} + 6889 T^{4} )( 1 + 30 T + 383 T^{2} + 2490 T^{3} + 6889 T^{4} ) \))
$89$ (\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} )( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} ) \))(\( ( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} )( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} ) \))
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