Properties

Label 168.2.i
Level 168
Weight 2
Character orbit i
Rep. character \(\chi_{168}(125,\cdot)\)
Character field \(\Q\)
Dimension 28
Newform subspaces 5
Sturm bound 64
Trace bound 4

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Defining parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.i (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 168 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(64\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

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

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

Trace form

\( 28q - 4q^{4} - 4q^{7} - 4q^{9} + O(q^{10}) \) \( 28q - 4q^{4} - 4q^{7} - 4q^{9} + 8q^{15} + 4q^{16} + 12q^{18} - 32q^{22} - 20q^{25} - 12q^{28} - 4q^{30} - 40q^{36} - 16q^{39} + 12q^{42} + 32q^{46} + 4q^{49} - 16q^{57} - 16q^{58} + 44q^{60} - 36q^{63} - 52q^{64} + 48q^{70} + 12q^{72} + 68q^{78} + 24q^{79} + 12q^{81} + 36q^{84} + 8q^{88} + O(q^{100}) \)

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.i.a \(4\) \(1.341\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(-4\) \(0\) \(-8\) \(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{3}+(1+\beta _{3})q^{4}+\cdots\)
168.2.i.b \(4\) \(1.341\) \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(4\) \(q+\beta _{1}q^{2}-\beta _{2}q^{3}+2q^{4}+2\beta _{2}q^{5}+\cdots\)
168.2.i.c \(4\) \(1.341\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(4\) \(0\) \(-8\) \(q+\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(1+\beta _{3})q^{4}+\cdots\)
168.2.i.d \(8\) \(1.341\) 8.0.\(\cdots\).11 \(\Q(\sqrt{-14}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}-\beta _{1}q^{3}-2q^{4}+(\beta _{6}-\beta _{7})q^{5}+\cdots\)
168.2.i.e \(8\) \(1.341\) 8.0.3317760000.1 None \(0\) \(0\) \(0\) \(8\) \(q+\beta _{5}q^{2}+(-\beta _{4}-\beta _{6})q^{3}+(-1-\beta _{2}+\cdots)q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T^{2} + 4 T^{4} \))(\( ( 1 - 2 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( ( 1 + 2 T^{2} )^{4} \))(\( ( 1 + T^{2} + 4 T^{4} )^{2} \))
$3$ (\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 - 10 T^{4} + 81 T^{8} \))(\( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \))
$5$ (\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 22 T^{4} + 625 T^{8} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{4} \))
$7$ (\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{4} \))(\( ( 1 - 2 T + 7 T^{2} )^{4} \))
$11$ (\( ( 1 + 16 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} )^{8} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{4} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 2 T + 13 T^{2} )^{4} \))(\( ( 1 + 310 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 + 16 T^{2} + 169 T^{4} )^{4} \))
$17$ (\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} )^{8} \))(\( ( 1 + 17 T^{2} )^{8} \))
$19$ (\( ( 1 + 4 T + 19 T^{2} )^{4} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 - 4 T + 19 T^{2} )^{4} \))(\( ( 1 - 650 T^{4} + 130321 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} + 361 T^{4} )^{4} \))
$23$ (\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 23 T^{2} )^{4} \))(\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 6 T + 23 T^{2} )^{4}( 1 + 6 T + 23 T^{2} )^{4} \))(\( ( 1 - 26 T^{2} + 529 T^{4} )^{4} \))
$29$ (\( ( 1 + 34 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 29 T^{2} )^{8} \))(\( ( 1 + 10 T^{2} + 841 T^{4} )^{4} \))
$31$ (\( ( 1 - 14 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 10 T + 31 T^{2} )^{2}( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{8} \))(\( ( 1 - 10 T + 31 T^{2} )^{4}( 1 + 10 T + 31 T^{2} )^{4} \))
$37$ (\( ( 1 + 34 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} )^{4} \))(\( ( 1 + 34 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} )^{8} \))(\( ( 1 - 37 T^{2} )^{8} \))
$41$ (\( ( 1 + 76 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 76 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{8} \))(\( ( 1 - 38 T^{2} + 1681 T^{4} )^{4} \))
$43$ (\( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 43 T^{2} )^{8} \))(\( ( 1 - 26 T^{2} + 1849 T^{4} )^{4} \))
$47$ (\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{8} \))(\( ( 1 - 26 T^{2} + 2209 T^{4} )^{4} \))
$53$ (\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 - 94 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 53 T^{2} )^{8} \))(\( ( 1 + 53 T^{2} )^{8} \))
$59$ (\( ( 1 - 86 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 86 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 1130 T^{4} + 12117361 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} + 3481 T^{4} )^{4} \))
$61$ (\( ( 1 + 10 T + 61 T^{2} )^{4} \))(\( ( 1 + 61 T^{2} )^{4} \))(\( ( 1 - 10 T + 61 T^{2} )^{4} \))(\( ( 1 - 7370 T^{4} + 13845841 T^{8} )^{2} \))(\( ( 1 + 112 T^{2} + 3721 T^{4} )^{4} \))
$67$ (\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{8} \))(\( ( 1 - 74 T^{2} + 4489 T^{4} )^{4} \))
$71$ (\( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 110 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 62 T^{2} + 5041 T^{4} )^{4} \))
$73$ (\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 73 T^{2} )^{8} \))(\( ( 1 + 70 T^{2} + 5329 T^{4} )^{4} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )^{4} \))(\( ( 1 - 10 T + 79 T^{2} )^{4} \))(\( ( 1 - 8 T + 79 T^{2} )^{4} \))(\( ( 1 + 130 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 + 10 T + 79 T^{2} )^{8} \))
$83$ (\( ( 1 - 38 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 13130 T^{4} + 47458321 T^{8} )^{2} \))(\( ( 1 - 116 T^{2} + 6889 T^{4} )^{4} \))
$89$ (\( ( 1 + 124 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 124 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{8} \))(\( ( 1 + 89 T^{2} )^{8} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2}( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 97 T^{2} )^{8} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{4} \))
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