Properties

Label 288.2.r
Level 288
Weight 2
Character orbit r
Rep. character \(\chi_{288}(49,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 20
Newform subspaces 2
Sturm bound 96
Trace bound 1

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Defining parameters

Level: \( N \) = \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 288.r (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 72 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 112 28 84
Cusp forms 80 20 60
Eisenstein series 32 8 24

Trace form

\( 20q + 2q^{7} - 4q^{9} + O(q^{10}) \) \( 20q + 2q^{7} - 4q^{9} + 10q^{15} - 8q^{17} + 14q^{23} + 2q^{31} - 18q^{33} - 2q^{39} + 2q^{41} - 18q^{47} + 28q^{55} + 4q^{57} - 50q^{63} - 22q^{65} - 48q^{71} - 8q^{73} + 2q^{79} - 8q^{81} - 42q^{87} + 8q^{89} - 40q^{95} - 2q^{97} + O(q^{100}) \)

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.r.a \(4\) \(2.300\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(8\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{5}+4\zeta_{12}^{2}q^{7}+\cdots\)
288.2.r.b \(16\) \(2.300\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(-6\) \(q-\beta _{12}q^{3}+(-\beta _{4}-\beta _{7}+\beta _{8}-\beta _{11}+\cdots)q^{5}+\cdots\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 - T^{2} - 2 T^{4} + 9 T^{6} + 18 T^{8} + 81 T^{10} - 162 T^{12} - 729 T^{14} + 6561 T^{16} \))
$5$ (\( ( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} ) \))(\( 1 + 19 T^{2} + 176 T^{4} + 1031 T^{6} + 3893 T^{8} + 4928 T^{10} - 61926 T^{12} - 631122 T^{14} - 3717344 T^{16} - 15778050 T^{18} - 38703750 T^{20} + 77000000 T^{22} + 1520703125 T^{24} + 10068359375 T^{26} + 42968750000 T^{28} + 115966796875 T^{30} + 152587890625 T^{32} \))
$7$ (\( ( 1 - 5 T + 7 T^{2} )^{2}( 1 + T + 7 T^{2} )^{2} \))(\( ( 1 + 3 T - 14 T^{2} - 39 T^{3} + 139 T^{4} + 252 T^{5} - 1208 T^{6} - 666 T^{7} + 9424 T^{8} - 4662 T^{9} - 59192 T^{10} + 86436 T^{11} + 333739 T^{12} - 655473 T^{13} - 1647086 T^{14} + 2470629 T^{15} + 5764801 T^{16} )^{2} \))
$11$ (\( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} \))(\( 1 + 48 T^{2} + 1090 T^{4} + 17592 T^{6} + 242041 T^{8} + 2632140 T^{10} + 21031138 T^{12} + 158095260 T^{14} + 1558598596 T^{16} + 19129526460 T^{18} + 307916891458 T^{20} + 4662996570540 T^{22} + 51883637916121 T^{24} + 456291173580792 T^{26} + 3420886930625890 T^{28} + 18227992011995568 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( ( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} ) \))(\( 1 + 51 T^{2} + 1288 T^{4} + 16911 T^{6} + 73645 T^{8} - 1059264 T^{10} - 5456606 T^{12} + 411982806 T^{14} + 9084740848 T^{16} + 69625094214 T^{18} - 155846123966 T^{20} - 5112865008576 T^{22} + 60074488948045 T^{24} + 2331324955658439 T^{26} + 30007933637755528 T^{28} + 200806195670663739 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( ( 1 - 5 T + 17 T^{2} )^{4} \))(\( ( 1 + 7 T + 66 T^{2} + 309 T^{3} + 1702 T^{4} + 5253 T^{5} + 19074 T^{6} + 34391 T^{7} + 83521 T^{8} )^{4} \))
$19$ (\( ( 1 - 37 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 69 T^{2} + 2306 T^{4} - 53763 T^{6} + 1069146 T^{8} - 19408443 T^{10} + 300520226 T^{12} - 3246165789 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 5 T - 32 T^{2} - 45 T^{3} + 1385 T^{4} + 3040 T^{5} - 11142 T^{6} - 48640 T^{7} - 123716 T^{8} - 1118720 T^{9} - 5894118 T^{10} + 36987680 T^{11} + 387579785 T^{12} - 289635435 T^{13} - 4737148448 T^{14} - 17024127235 T^{15} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 + 29 T^{2} + 841 T^{4} )^{2} \))(\( 1 + 123 T^{2} + 7912 T^{4} + 340647 T^{6} + 10067965 T^{8} + 144056448 T^{10} - 3705324014 T^{12} - 328820871018 T^{14} - 12137186779472 T^{16} - 276538352526138 T^{18} - 2620705273945934 T^{20} + 85688134810823808 T^{22} + 5036463377066894365 T^{24} + \)\(14\!\cdots\!47\)\( T^{26} + \)\(27\!\cdots\!92\)\( T^{28} + \)\(36\!\cdots\!63\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T - 48 T^{2} + 155 T^{3} + 1121 T^{4} + 2040 T^{5} - 44678 T^{6} - 79040 T^{7} + 1805724 T^{8} - 2450240 T^{9} - 42935558 T^{10} + 60773640 T^{11} + 1035267041 T^{12} + 4437518405 T^{13} - 42600176688 T^{14} - 137563070555 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 144 T^{2} + 12668 T^{4} - 733824 T^{6} + 31784838 T^{8} - 1004605056 T^{10} + 23741871548 T^{12} - 369464602896 T^{14} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 4 T - 74 T^{2} + 288 T^{3} + 4517 T^{4} - 22796 T^{5} - 11538 T^{6} + 738968 T^{7} - 2462804 T^{8} + 30297688 T^{9} - 19395378 T^{10} - 1571123116 T^{11} + 12763962437 T^{12} + 33366585888 T^{13} - 351507713834 T^{14} + 779017095524 T^{15} + 7984925229121 T^{16} )^{2} \))
$43$ (\( 1 - 35 T^{2} - 624 T^{4} - 64715 T^{6} + 3418801 T^{8} \))(\( 1 + 324 T^{2} + 58258 T^{4} + 7305024 T^{6} + 705172105 T^{8} + 54954910764 T^{10} + 3558033934834 T^{12} + 194436327001344 T^{14} + 9048929543300068 T^{16} + 359512768625485056 T^{18} + 12164209974444414034 T^{20} + \)\(34\!\cdots\!36\)\( T^{22} + \)\(82\!\cdots\!05\)\( T^{24} + \)\(15\!\cdots\!76\)\( T^{26} + \)\(23\!\cdots\!58\)\( T^{28} + \)\(23\!\cdots\!76\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( ( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 3 T - 98 T^{2} - 219 T^{3} + 3523 T^{4} - 1116 T^{5} - 253856 T^{6} + 226218 T^{7} + 18909448 T^{8} + 10632246 T^{9} - 560767904 T^{10} - 115866468 T^{11} + 17191116163 T^{12} - 50226556533 T^{13} - 1056363102242 T^{14} + 1519869361389 T^{15} + 23811286661761 T^{16} )^{2} \))
$53$ (\( ( 1 - 53 T^{2} )^{4} \))(\( ( 1 - 264 T^{2} + 35516 T^{4} - 3123720 T^{6} + 194863110 T^{8} - 8774529480 T^{10} + 280238323196 T^{12} - 5851391338056 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( 1 + 117 T^{2} + 10208 T^{4} + 407277 T^{6} + 12117361 T^{8} \))(\( 1 + 364 T^{2} + 70130 T^{4} + 9549968 T^{6} + 1028964713 T^{8} + 92991430988 T^{10} + 7291973853618 T^{12} + 506327334867240 T^{14} + 31497451193778532 T^{16} + 1762525452672862440 T^{18} + 88359479586850462098 T^{20} + \)\(39\!\cdots\!08\)\( T^{22} + \)\(15\!\cdots\!73\)\( T^{24} + \)\(48\!\cdots\!68\)\( T^{26} + \)\(12\!\cdots\!30\)\( T^{28} + \)\(22\!\cdots\!04\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} )( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} ) \))(\( 1 + 323 T^{2} + 52152 T^{4} + 6035887 T^{6} + 584536925 T^{8} + 49803739968 T^{10} + 3801157073266 T^{12} + 264102902456582 T^{14} + 16823476283802768 T^{16} + 982726900040941622 T^{18} + 52630216452466386706 T^{20} + \)\(25\!\cdots\!48\)\( T^{22} + \)\(11\!\cdots\!25\)\( T^{24} + \)\(43\!\cdots\!87\)\( T^{26} + \)\(13\!\cdots\!92\)\( T^{28} + \)\(31\!\cdots\!43\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8} \))(\( 1 + 296 T^{2} + 39450 T^{4} + 3796936 T^{6} + 362561921 T^{8} + 33090030780 T^{10} + 2631719645962 T^{12} + 193036342550180 T^{14} + 13423375489686516 T^{16} + 866540141707758020 T^{18} + 53032101023857423402 T^{20} + \)\(29\!\cdots\!20\)\( T^{22} + \)\(14\!\cdots\!61\)\( T^{24} + \)\(69\!\cdots\!64\)\( T^{26} + \)\(32\!\cdots\!50\)\( T^{28} + \)\(10\!\cdots\!84\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 - 6 T + 71 T^{2} )^{4} \))(\( ( 1 + 18 T + 332 T^{2} + 3582 T^{3} + 36198 T^{4} + 254322 T^{5} + 1673612 T^{6} + 6442398 T^{7} + 25411681 T^{8} )^{4} \))
$73$ (\( ( 1 - 9 T + 73 T^{2} )^{4} \))(\( ( 1 + 11 T + 278 T^{2} + 2325 T^{3} + 29966 T^{4} + 169725 T^{5} + 1481462 T^{6} + 4279187 T^{7} + 28398241 T^{8} )^{4} \))
$79$ (\( ( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 15 T - 80 T^{2} + 1305 T^{3} + 13273 T^{4} - 59400 T^{5} - 1907150 T^{6} + 913560 T^{7} + 190569148 T^{8} + 72171240 T^{9} - 11902523150 T^{10} - 29286516600 T^{11} + 516984425113 T^{12} + 4015558600695 T^{13} - 19446996441680 T^{14} - 288058634792385 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( 1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8} \))(\( 1 + 559 T^{2} + 168260 T^{4} + 35967119 T^{6} + 6055752221 T^{8} + 843016361960 T^{10} + 99713806040442 T^{12} + 10182388433060610 T^{14} + 904920089638581976 T^{16} + 70146473915354542290 T^{18} + \)\(47\!\cdots\!82\)\( T^{20} + \)\(27\!\cdots\!40\)\( T^{22} + \)\(13\!\cdots\!61\)\( T^{24} + \)\(55\!\cdots\!31\)\( T^{26} + \)\(17\!\cdots\!60\)\( T^{28} + \)\(41\!\cdots\!11\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 + 14 T + 89 T^{2} )^{4} \))(\( ( 1 - 16 T + 408 T^{2} - 4116 T^{3} + 56278 T^{4} - 366324 T^{5} + 3231768 T^{6} - 11279504 T^{7} + 62742241 T^{8} )^{4} \))
$97$ (\( ( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 254 T^{2} + 1560 T^{3} + 35209 T^{4} - 273780 T^{5} - 2055806 T^{6} + 15575820 T^{7} + 104325124 T^{8} + 1510854540 T^{9} - 19343078654 T^{10} - 249871613940 T^{11} + 3117027454729 T^{12} + 13396250800920 T^{13} - 211574889251966 T^{14} + 7837433594376961 T^{16} )^{2} \))
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