Properties

Label 168.2.j
Level 168
Weight 2
Character orbit j
Rep. character \(\chi_{168}(155,\cdot)\)
Character field \(\Q\)
Dimension 24
Newform subspaces 4
Sturm bound 64
Trace bound 2

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Defining parameters

Level: \( N \) = \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 168.j (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 24 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(64\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 36 24 12
Cusp forms 28 24 4
Eisenstein series 8 0 8

Trace form

\( 24q + 4q^{4} - 6q^{6} + O(q^{10}) \) \( 24q + 4q^{4} - 6q^{6} - 12q^{10} - 6q^{12} - 12q^{16} + 8q^{18} - 16q^{19} + 30q^{24} + 24q^{25} - 24q^{27} - 8q^{28} - 12q^{30} - 8q^{33} - 24q^{34} + 12q^{36} - 20q^{40} - 10q^{42} + 32q^{43} - 24q^{46} + 6q^{48} - 24q^{49} + 44q^{52} - 10q^{54} - 8q^{57} - 24q^{58} + 20q^{60} + 52q^{64} + 40q^{66} - 32q^{67} - 12q^{70} - 56q^{75} - 28q^{76} + 44q^{78} + 8q^{81} + 24q^{82} + 14q^{84} - 32q^{90} + 48q^{94} + 2q^{96} + 64q^{99} + O(q^{100}) \)

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.j.a \(4\) \(1.341\) \(\Q(i, \sqrt{5})\) None \(-4\) \(-2\) \(4\) \(0\) \(q+(-1-\beta _{2})q^{2}+(\beta _{1}+\beta _{2})q^{3}+2\beta _{2}q^{4}+\cdots\)
168.2.j.b \(4\) \(1.341\) \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(0\) \(q-\zeta_{8}^{3}q^{2}+(1-\zeta_{8}^{2})q^{3}+2q^{4}+2\zeta_{8}^{3}q^{5}+\cdots\)
168.2.j.c \(4\) \(1.341\) \(\Q(i, \sqrt{5})\) None \(4\) \(-2\) \(-4\) \(0\) \(q+(1+\beta _{2})q^{2}+(-\beta _{2}-\beta _{3})q^{3}+2\beta _{2}q^{4}+\cdots\)
168.2.j.d \(12\) \(1.341\) 12.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{2}-\beta _{6}q^{3}+(-\beta _{2}-\beta _{3})q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(168, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T + 2 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} )^{2} \))(\( 1 + 2 T^{2} + T^{4} - 4 T^{6} + 4 T^{8} + 32 T^{10} + 64 T^{12} \))
$3$ (\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( ( 1 - T^{2} - 4 T^{3} - 3 T^{4} + 27 T^{6} )^{2} \))
$5$ (\( ( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 107 T^{4} + 588 T^{6} + 2675 T^{8} + 8750 T^{10} + 15625 T^{12} )^{2} \))
$7$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{6} \))
$11$ (\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{12} \))
$13$ (\( 1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( 1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8} \))(\( ( 1 - 14 T^{2} + 331 T^{4} - 5132 T^{6} + 55939 T^{8} - 399854 T^{10} + 4826809 T^{12} )^{2} \))
$17$ (\( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \))(\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \))(\( ( 1 - 38 T^{2} + 1151 T^{4} - 20724 T^{6} + 332639 T^{8} - 3173798 T^{10} + 24137569 T^{12} )^{2} \))
$19$ (\( ( 1 - 10 T + 58 T^{2} - 190 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{4} \))(\( ( 1 - 10 T + 58 T^{2} - 190 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 12 T + 95 T^{2} + 476 T^{3} + 1805 T^{4} + 4332 T^{5} + 6859 T^{6} )^{4} \))
$23$ (\( ( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 8 T + 42 T^{2} + 184 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 66 T^{2} + 2399 T^{4} + 64348 T^{6} + 1269071 T^{8} + 18469506 T^{10} + 148035889 T^{12} )^{2} \))
$29$ (\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 26 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 46 T^{2} + 2695 T^{4} + 76516 T^{6} + 2266495 T^{8} + 32534926 T^{10} + 594823321 T^{12} )^{2} \))
$31$ (\( ( 1 + 18 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 + 18 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{12} \))
$37$ (\( 1 - 100 T^{2} + 4918 T^{4} - 136900 T^{6} + 1874161 T^{8} \))(\( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \))(\( 1 - 100 T^{2} + 4918 T^{4} - 136900 T^{6} + 1874161 T^{8} \))(\( ( 1 - 94 T^{2} + 3191 T^{4} - 75972 T^{6} + 4368479 T^{8} - 176171134 T^{10} + 2565726409 T^{12} )^{2} \))
$41$ (\( ( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 74 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 86 T^{2} + 5647 T^{4} - 231764 T^{6} + 9492607 T^{8} - 243015446 T^{10} + 4750104241 T^{12} )^{2} \))
$43$ (\( ( 1 - 8 T + 82 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{4} \))(\( ( 1 - 8 T + 82 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 10 T + 129 T^{2} - 844 T^{3} + 5547 T^{4} - 18490 T^{5} + 79507 T^{6} )^{4} \))
$47$ (\( ( 1 + 4 T + 78 T^{2} + 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 - 4 T + 78 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 4591 T^{4} + 251884 T^{6} + 10141519 T^{8} + 283021498 T^{10} + 10779215329 T^{12} )^{2} \))
$53$ (\( ( 1 + 12 T + 122 T^{2} + 636 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 74 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 12 T + 122 T^{2} - 636 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 6551 T^{4} + 242436 T^{6} + 18401759 T^{8} + 489209822 T^{10} + 22164361129 T^{12} )^{2} \))
$59$ (\( 1 - 224 T^{2} + 19486 T^{4} - 779744 T^{6} + 12117361 T^{8} \))(\( ( 1 - 46 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 224 T^{2} + 19486 T^{4} - 779744 T^{6} + 12117361 T^{8} \))(\( ( 1 - 226 T^{2} + 25867 T^{4} - 1888084 T^{6} + 90043027 T^{8} - 2738523586 T^{10} + 42180533641 T^{12} )^{2} \))
$61$ (\( 1 - 104 T^{2} + 5646 T^{4} - 386984 T^{6} + 13845841 T^{8} \))(\( ( 1 - 10 T + 61 T^{2} )^{2}( 1 + 10 T + 61 T^{2} )^{2} \))(\( 1 - 104 T^{2} + 5646 T^{4} - 386984 T^{6} + 13845841 T^{8} \))(\( ( 1 - 222 T^{2} + 25163 T^{4} - 1831724 T^{6} + 93631523 T^{8} - 3073776702 T^{10} + 51520374361 T^{12} )^{2} \))
$67$ (\( ( 1 + 4 T + 58 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 6 T + 67 T^{2} )^{4} \))(\( ( 1 + 4 T + 58 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 10 T + 185 T^{2} + 1116 T^{3} + 12395 T^{4} + 44890 T^{5} + 300763 T^{6} )^{4} \))
$71$ (\( ( 1 - 4 T + 66 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 4 T + 66 T^{2} + 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 362 T^{2} + 58607 T^{4} + 5382060 T^{6} + 295437887 T^{8} + 9199028522 T^{10} + 128100283921 T^{12} )^{2} \))
$73$ (\( ( 1 - 8 T + 142 T^{2} - 584 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{4} \))(\( ( 1 - 8 T + 142 T^{2} - 584 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{12} \))
$79$ (\( 1 - 124 T^{2} + 11206 T^{4} - 773884 T^{6} + 38950081 T^{8} \))(\( ( 1 - 94 T^{2} + 6241 T^{4} )^{2} \))(\( 1 - 124 T^{2} + 11206 T^{4} - 773884 T^{6} + 38950081 T^{8} \))(\( ( 1 - 154 T^{2} + 18095 T^{4} - 1857324 T^{6} + 112930895 T^{8} - 5998312474 T^{10} + 243087455521 T^{12} )^{2} \))
$83$ (\( 1 - 80 T^{2} + 12958 T^{4} - 551120 T^{6} + 47458321 T^{8} \))(\( ( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2} \))(\( 1 - 80 T^{2} + 12958 T^{4} - 551120 T^{6} + 47458321 T^{8} \))(\( ( 1 - 162 T^{2} + 19931 T^{4} - 2214676 T^{6} + 137304659 T^{8} - 7688248002 T^{10} + 326940373369 T^{12} )^{2} \))
$89$ (\( ( 1 - 142 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 170 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 278 T^{2} + 40879 T^{4} - 4202324 T^{6} + 323802559 T^{8} - 17442342998 T^{10} + 496981290961 T^{12} )^{2} \))
$97$ (\( ( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{4} \))(\( ( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 14 T + 159 T^{2} + 1124 T^{3} + 15423 T^{4} + 131726 T^{5} + 912673 T^{6} )^{4} \))
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