Properties

Label 288.2.i
Level 288
Weight 2
Character orbit i
Rep. character \(\chi_{288}(97,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 24
Newform subspaces 6
Sturm bound 96
Trace bound 3

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Defining parameters

Level: \( N \) = \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 288.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 6 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(288, [\chi])\).

Total New Old
Modular forms 112 24 88
Cusp forms 80 24 56
Eisenstein series 32 0 32

Trace form

\( 24q + 4q^{9} + O(q^{10}) \) \( 24q + 4q^{9} + 8q^{17} + 8q^{21} - 12q^{25} - 8q^{29} + 28q^{33} + 12q^{41} - 40q^{45} - 12q^{49} - 48q^{53} - 4q^{57} - 16q^{65} - 48q^{69} + 24q^{73} - 16q^{77} - 68q^{81} - 64q^{89} + 64q^{93} - 12q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(288, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
288.2.i.a \(2\) \(2.300\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-4\) \(-2\) \(q+(-1-\zeta_{6})q^{3}-4\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+\cdots\)
288.2.i.b \(2\) \(2.300\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-4\) \(2\) \(q+(1+\zeta_{6})q^{3}-4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+\cdots\)
288.2.i.c \(4\) \(2.300\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(-4\) \(2\) \(2\) \(q+(-1+\beta _{3})q^{3}+\beta _{2}q^{5}+(1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
288.2.i.d \(4\) \(2.300\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(2\) \(0\) \(q-\zeta_{12}^{3}q^{3}+(1-\zeta_{12})q^{5}+\zeta_{12}^{2}q^{7}+\cdots\)
288.2.i.e \(4\) \(2.300\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(4\) \(2\) \(-2\) \(q+(1-\beta _{3})q^{3}+(1-\beta _{2})q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
288.2.i.f \(8\) \(2.300\) 8.0.170772624.1 None \(0\) \(0\) \(2\) \(0\) \(q+(-\beta _{1}+\beta _{5})q^{3}+(1-\beta _{4}+\beta _{7})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(288, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(288, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 + 3 T^{2} + 12 T^{4} + 27 T^{6} + 81 T^{8} \))
$5$ (\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( ( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 40 T^{5} - 25 T^{6} - 125 T^{7} + 625 T^{8} )^{2} \))
$7$ (\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 70 T^{5} - 245 T^{6} - 686 T^{7} + 2401 T^{8} \))(\( ( 1 - 13 T^{2} + 49 T^{4} )( 1 + 2 T^{2} + 49 T^{4} ) \))(\( 1 + 2 T - 5 T^{2} - 10 T^{3} + 4 T^{4} - 70 T^{5} - 245 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 - T^{2} - 23 T^{4} + 74 T^{6} - 1874 T^{8} + 3626 T^{10} - 55223 T^{12} - 117649 T^{14} + 5764801 T^{16} \))
$11$ (\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 110 T^{5} - 1573 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 - 19 T^{2} + 240 T^{4} - 2299 T^{6} + 14641 T^{8} \))(\( 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 110 T^{5} - 1573 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 - 8 T^{2} + 103 T^{4} + 2248 T^{6} - 20864 T^{8} + 272008 T^{10} + 1508023 T^{12} - 14172488 T^{14} + 214358881 T^{16} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 + 2 T + T^{2} - 46 T^{3} - 212 T^{4} - 598 T^{5} + 169 T^{6} + 4394 T^{7} + 28561 T^{8} \))(\( ( 1 - 3 T - 4 T^{2} - 39 T^{3} + 169 T^{4} )^{2} \))(\( 1 + 2 T + T^{2} - 46 T^{3} - 212 T^{4} - 598 T^{5} + 169 T^{6} + 4394 T^{7} + 28561 T^{8} \))(\( ( 1 + 3 T - 11 T^{2} - 18 T^{3} + 114 T^{4} - 234 T^{5} - 1859 T^{6} + 6591 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 + 3 T + 17 T^{2} )^{2} \))(\( ( 1 + 3 T + 17 T^{2} )^{2} \))(\( ( 1 + 10 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} )^{4} \))(\( ( 1 + 10 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - T + 26 T^{2} - 17 T^{3} + 289 T^{4} )^{4} \))
$19$ (\( ( 1 + T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{4} \))(\( ( 1 - 10 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{4} \))(\( ( 1 + 31 T^{2} + 888 T^{4} + 11191 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 6 T - 13 T^{2} + 18 T^{3} + 1044 T^{4} + 414 T^{5} - 6877 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( 1 + 29 T^{2} + 312 T^{4} + 15341 T^{6} + 279841 T^{8} \))(\( 1 - 6 T - 13 T^{2} - 18 T^{3} + 1044 T^{4} - 414 T^{5} - 6877 T^{6} - 73002 T^{7} + 279841 T^{8} \))(\( 1 - 65 T^{2} + 2185 T^{4} - 63830 T^{6} + 1646734 T^{8} - 33766070 T^{10} + 611452585 T^{12} - 9622332785 T^{14} + 78310985281 T^{16} \))
$29$ (\( 1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4} \))(\( 1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4} \))(\( 1 + 10 T + 41 T^{2} + 10 T^{3} - 260 T^{4} + 290 T^{5} + 34481 T^{6} + 243890 T^{7} + 707281 T^{8} \))(\( ( 1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4} )^{2} \))(\( 1 + 10 T + 41 T^{2} + 10 T^{3} - 260 T^{4} + 290 T^{5} + 34481 T^{6} + 243890 T^{7} + 707281 T^{8} \))(\( ( 1 - 5 T - 31 T^{2} + 10 T^{3} + 1570 T^{4} + 290 T^{5} - 26071 T^{6} - 121945 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 - 10 T + 19 T^{2} - 190 T^{3} + 2500 T^{4} - 5890 T^{5} + 18259 T^{6} - 297910 T^{7} + 923521 T^{8} \))(\( 1 - 35 T^{2} + 264 T^{4} - 33635 T^{6} + 923521 T^{8} \))(\( 1 + 10 T + 19 T^{2} + 190 T^{3} + 2500 T^{4} + 5890 T^{5} + 18259 T^{6} + 297910 T^{7} + 923521 T^{8} \))(\( 1 + 11 T^{2} - 1163 T^{4} - 7018 T^{6} + 608854 T^{8} - 6744298 T^{10} - 1074054923 T^{12} + 9762540491 T^{14} + 852891037441 T^{16} \))
$37$ (\( ( 1 + 8 T + 37 T^{2} )^{2} \))(\( ( 1 + 8 T + 37 T^{2} )^{2} \))(\( ( 1 - 8 T + 66 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 8 T + 37 T^{2} )^{4} \))(\( ( 1 - 8 T + 66 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{8} \))
$41$ (\( 1 + T - 40 T^{2} + 41 T^{3} + 1681 T^{4} \))(\( 1 + T - 40 T^{2} + 41 T^{3} + 1681 T^{4} \))(\( 1 - 14 T + 89 T^{2} - 350 T^{3} + 1732 T^{4} - 14350 T^{5} + 149609 T^{6} - 964894 T^{7} + 2825761 T^{8} \))(\( ( 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} )^{2} \))(\( 1 - 14 T + 89 T^{2} - 350 T^{3} + 1732 T^{4} - 14350 T^{5} + 149609 T^{6} - 964894 T^{7} + 2825761 T^{8} \))(\( ( 1 + T - 40 T^{2} + 41 T^{3} + 1681 T^{4} )^{4} \))
$43$ (\( 1 + 7 T + 6 T^{2} + 301 T^{3} + 1849 T^{4} \))(\( 1 - 7 T + 6 T^{2} - 301 T^{3} + 1849 T^{4} \))(\( ( 1 - 10 T + 43 T^{2} )^{2}( 1 + 10 T + 57 T^{2} + 430 T^{3} + 1849 T^{4} ) \))(\( 1 - 11 T^{2} - 1728 T^{4} - 20339 T^{6} + 3418801 T^{8} \))(\( ( 1 + 10 T + 43 T^{2} )^{2}( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} ) \))(\( 1 - 136 T^{2} + 10471 T^{4} - 588472 T^{6} + 26782720 T^{8} - 1088084728 T^{10} + 35798265271 T^{12} - 859705374664 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \))(\( 1 - 2 T - 37 T^{2} + 106 T^{3} - 716 T^{4} + 4982 T^{5} - 81733 T^{6} - 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 53 T^{2} + 600 T^{4} + 117077 T^{6} + 4879681 T^{8} \))(\( 1 + 2 T - 37 T^{2} - 106 T^{3} - 716 T^{4} - 4982 T^{5} - 81733 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 - 161 T^{2} + 15097 T^{4} - 1031366 T^{6} + 55019806 T^{8} - 2278287494 T^{10} + 73668544057 T^{12} - 1735453667969 T^{14} + 23811286661761 T^{16} \))
$53$ (\( ( 1 + 4 T + 53 T^{2} )^{2} \))(\( ( 1 + 4 T + 53 T^{2} )^{2} \))(\( ( 1 - 8 T + 98 T^{2} - 424 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 8 T + 53 T^{2} )^{4} \))(\( ( 1 - 8 T + 98 T^{2} - 424 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 4 T + 53 T^{2} )^{8} \))
$59$ (\( 1 - 5 T - 34 T^{2} - 295 T^{3} + 3481 T^{4} \))(\( 1 + 5 T - 34 T^{2} + 295 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 83 T^{2} + 70 T^{3} - 2276 T^{4} + 4130 T^{5} + 288923 T^{6} - 2875306 T^{7} + 12117361 T^{8} \))(\( 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} \))(\( 1 + 14 T + 83 T^{2} - 70 T^{3} - 2276 T^{4} - 4130 T^{5} + 288923 T^{6} + 2875306 T^{7} + 12117361 T^{8} \))(\( 1 - 56 T^{2} - 4313 T^{4} - 27272 T^{6} + 32453824 T^{8} - 94933832 T^{10} - 52262177993 T^{12} - 2362109883896 T^{14} + 146830437604321 T^{16} \))
$61$ (\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 - 6 T - 71 T^{2} + 90 T^{3} + 5532 T^{4} + 5490 T^{5} - 264191 T^{6} - 1361886 T^{7} + 13845841 T^{8} \))(\( ( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} )^{2} \))(\( 1 - 6 T - 71 T^{2} + 90 T^{3} + 5532 T^{4} + 5490 T^{5} - 264191 T^{6} - 1361886 T^{7} + 13845841 T^{8} \))(\( ( 1 + 13 T + 79 T^{2} - 416 T^{3} - 5930 T^{4} - 25376 T^{5} + 293959 T^{6} + 2950753 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 - 10 T - 5 T^{2} + 290 T^{3} - 164 T^{4} + 19430 T^{5} - 22445 T^{6} - 3007630 T^{7} + 20151121 T^{8} \))(\( 1 - 59 T^{2} - 1008 T^{4} - 264851 T^{6} + 20151121 T^{8} \))(\( 1 + 10 T - 5 T^{2} - 290 T^{3} - 164 T^{4} - 19430 T^{5} - 22445 T^{6} + 3007630 T^{7} + 20151121 T^{8} \))(\( 1 - 88 T^{2} - 2873 T^{4} - 144232 T^{6} + 57806752 T^{8} - 647457448 T^{10} - 57894170633 T^{12} - 7960337630872 T^{14} + 406067677556641 T^{16} \))
$71$ (\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 50 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 130 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 4 T + 50 T^{2} + 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 176 T^{2} + 16638 T^{4} + 887216 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 - 3 T + 73 T^{2} )^{2} \))(\( ( 1 - 3 T + 73 T^{2} )^{2} \))(\( ( 1 + 122 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 12 T + 73 T^{2} )^{4} \))(\( ( 1 + 122 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 15 T + 194 T^{2} - 1095 T^{3} + 5329 T^{4} )^{4} \))
$79$ (\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 22 T + 211 T^{2} - 2530 T^{3} + 30052 T^{4} - 199870 T^{5} + 1316851 T^{6} - 10846858 T^{7} + 38950081 T^{8} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )( 1 + 11 T^{2} + 6241 T^{4} ) \))(\( 1 + 22 T + 211 T^{2} + 2530 T^{3} + 30052 T^{4} + 199870 T^{5} + 1316851 T^{6} + 10846858 T^{7} + 38950081 T^{8} \))(\( 1 - 181 T^{2} + 12757 T^{4} - 1361482 T^{6} + 156748534 T^{8} - 8497009162 T^{10} + 496886183317 T^{12} - 43998829449301 T^{14} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))(\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 1494 T^{5} - 916237 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 91 T^{2} + 1392 T^{4} - 626899 T^{6} + 47458321 T^{8} \))(\( 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 1494 T^{5} - 916237 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 125 T^{2} + 6925 T^{4} + 634750 T^{6} - 79408946 T^{8} + 4372792750 T^{10} + 328648872925 T^{12} - 40867546671125 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 16 T + 218 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 4 T + 89 T^{2} )^{4} \))(\( ( 1 + 16 T + 218 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 14 T + 194 T^{2} - 1246 T^{3} + 7921 T^{4} )^{4} \))
$97$ (\( 1 - 11 T + 24 T^{2} - 1067 T^{3} + 9409 T^{4} \))(\( 1 - 11 T + 24 T^{2} - 1067 T^{3} + 9409 T^{4} \))(\( 1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 4462 T^{5} - 1571303 T^{6} + 1825346 T^{7} + 88529281 T^{8} \))(\( ( 1 - 3 T - 88 T^{2} - 291 T^{3} + 9409 T^{4} )^{2} \))(\( 1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 4462 T^{5} - 1571303 T^{6} + 1825346 T^{7} + 88529281 T^{8} \))(\( ( 1 + 9 T - 16 T^{2} + 873 T^{3} + 9409 T^{4} )^{4} \))
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