Properties

Label 144.2.s
Level 144
Weight 2
Character orbit s
Rep. character \(\chi_{144}(47,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 12
Newform subspaces 5
Sturm bound 48
Trace bound 7

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Defining parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.s (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 36 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 5 \)
Sturm bound: \(48\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(144, [\chi])\).

Total New Old
Modular forms 60 12 48
Cusp forms 36 12 24
Eisenstein series 24 0 24

Trace form

\( 12q + 6q^{9} + O(q^{10}) \) \( 12q + 6q^{9} - 12q^{21} + 6q^{25} - 36q^{29} - 18q^{33} - 18q^{41} - 36q^{45} + 6q^{49} + 30q^{57} + 72q^{65} + 72q^{69} - 36q^{73} + 72q^{77} + 54q^{81} - 18q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(144, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
144.2.s.a \(2\) \(1.150\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-6\) \(-6\) \(q+(-1-\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+(-2+\cdots)q^{7}+\cdots\)
144.2.s.b \(2\) \(1.150\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(-3\) \(q+(1-2\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(-1-\zeta_{6})q^{7}+\cdots\)
144.2.s.c \(2\) \(1.150\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(3\) \(q+(-1+2\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(1+\zeta_{6})q^{7}+\cdots\)
144.2.s.d \(2\) \(1.150\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-6\) \(6\) \(q+(1+\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+(2+2\zeta_{6})q^{7}+\cdots\)
144.2.s.e \(4\) \(1.150\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(6\) \(0\) \(q-\zeta_{12}^{3}q^{3}+(1+\zeta_{12})q^{5}+(-\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(144, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(144, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))
$5$ (\( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} \))(\( ( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( ( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( ( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 4 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 - T + 7 T^{2} ) \))(\( 1 + 5 T^{2} - 24 T^{4} + 245 T^{6} + 2401 T^{8} \))
$11$ (\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 5 T^{2} - 96 T^{4} + 605 T^{6} + 14641 T^{8} \))
$13$ (\( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} \))(\( ( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} )^{2} \))
$17$ (\( 1 - 31 T^{2} + 289 T^{4} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 - 31 T^{2} + 289 T^{4} \))(\( ( 1 - 22 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( 1 - 35 T^{2} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 35 T^{2} + 361 T^{4} \))(\( ( 1 - 2 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 19 T^{2} - 168 T^{4} - 10051 T^{6} + 279841 T^{8} \))
$29$ (\( 1 + 6 T + 41 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 41 T^{2} + 174 T^{3} + 841 T^{4} \))(\( ( 1 + 15 T + 104 T^{2} + 435 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 31 T^{2} + 961 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 279 T^{3} + 961 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} \))(\( 1 + 31 T^{2} + 961 T^{4} \))(\( 1 + 53 T^{2} + 1848 T^{4} + 50933 T^{6} + 923521 T^{8} \))
$37$ (\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{4} \))
$41$ (\( 1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( ( 1 - 9 T + 68 T^{2} - 369 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 9 T + 70 T^{2} - 387 T^{3} + 1849 T^{4} \))(\( 1 - 9 T + 70 T^{2} - 387 T^{3} + 1849 T^{4} \))(\( 1 + 9 T + 70 T^{2} + 387 T^{3} + 1849 T^{4} \))(\( 1 + 9 T + 70 T^{2} + 387 T^{3} + 1849 T^{4} \))(\( 1 + 77 T^{2} + 4080 T^{4} + 142373 T^{6} + 3418801 T^{8} \))
$47$ (\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 67 T^{2} + 2280 T^{4} - 148003 T^{6} + 4879681 T^{8} \))
$53$ (\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 + 2 T^{2} + 2809 T^{4} )^{2} \))
$59$ (\( 1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4} \))(\( 1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4} \))(\( 1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 + 11 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 11 T + 73 T^{2} )^{2} \))(\( ( 1 - 4 T + 73 T^{2} )^{4} \))
$79$ (\( 1 - 6 T + 91 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 + 15 T + 154 T^{2} + 1185 T^{3} + 6241 T^{4} \))(\( 1 - 15 T + 154 T^{2} - 1185 T^{3} + 6241 T^{4} \))(\( 1 + 6 T + 91 T^{2} + 474 T^{3} + 6241 T^{4} \))(\( 1 - 67 T^{2} - 1752 T^{4} - 418147 T^{6} + 38950081 T^{8} \))
$83$ (\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4} \))(\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 - 139 T^{2} + 12432 T^{4} - 957571 T^{6} + 47458321 T^{8} \))
$89$ (\( 1 + 14 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 + 14 T^{2} + 7921 T^{4} \))(\( ( 1 - 166 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( 1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( 1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4} \))(\( ( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2} \))
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