Properties

Label 189.2.p
Level 189
Weight 2
Character orbit p
Rep. character \(\chi_{189}(26,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 22
Newform subspaces 4
Sturm bound 48
Trace bound 4

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Defining parameters

Level: \( N \) = \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 189.p (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(48\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(189, [\chi])\).

Total New Old
Modular forms 60 22 38
Cusp forms 36 22 14
Eisenstein series 24 0 24

Trace form

\( 22q + 12q^{4} - 5q^{7} + O(q^{10}) \) \( 22q + 12q^{4} - 5q^{7} + 6q^{10} + 2q^{16} - 3q^{19} - 32q^{22} - 11q^{25} - 12q^{28} - 39q^{31} - 4q^{37} - 12q^{40} - 10q^{43} - 26q^{46} + 37q^{49} + 36q^{52} + 4q^{58} + 15q^{61} + 64q^{64} + 50q^{67} - 84q^{70} + 51q^{73} - 46q^{79} + 18q^{82} + 12q^{85} - 22q^{88} - 54q^{91} - 138q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(189, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
189.2.p.a \(2\) \(1.509\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1\) \(q+(-2+2\zeta_{6})q^{4}+(-1+3\zeta_{6})q^{7}+\cdots\)
189.2.p.b \(4\) \(1.509\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(10\) \(q+\beta _{1}q^{2}+(\beta _{1}-2\beta _{3})q^{5}+(2+\beta _{2})q^{7}+\cdots\)
189.2.p.c \(4\) \(1.509\) \(\Q(\sqrt{-3}, \sqrt{-5})\) None \(0\) \(0\) \(0\) \(-8\) \(q+\beta _{1}q^{2}+3\beta _{2}q^{4}+(-3+2\beta _{2})q^{7}+\cdots\)
189.2.p.d \(12\) \(1.509\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-8\) \(q-\beta _{1}q^{2}+(1-\beta _{3}-\beta _{7}+\beta _{9})q^{4}+(\beta _{5}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(189, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(189, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} + 4 T^{4} \))(\( ( 1 + 2 T^{2} )^{2}( 1 - 2 T^{2} + 4 T^{4} ) \))(\( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \))(\( 1 + 2 T^{2} - T^{4} - 12 T^{6} - 17 T^{8} + 7 T^{10} + 85 T^{12} + 28 T^{14} - 272 T^{16} - 768 T^{18} - 256 T^{20} + 2048 T^{22} + 4096 T^{24} \))
$3$ 1
$5$ (\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( 1 - 3 T^{2} - 24 T^{4} + 275 T^{6} - 297 T^{8} - 3096 T^{10} + 33369 T^{12} - 77400 T^{14} - 185625 T^{16} + 4296875 T^{18} - 9375000 T^{20} - 29296875 T^{22} + 244140625 T^{24} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( ( 1 - 5 T + 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 2 T^{2} - 11 T^{3} + 14 T^{4} + 196 T^{5} + 343 T^{6} )^{2} \))
$11$ (\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \))(\( 1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8} \))(\( 1 + 29 T^{2} + 440 T^{4} + 3147 T^{6} - 10709 T^{8} - 688988 T^{10} - 10240799 T^{12} - 83367548 T^{14} - 156790469 T^{16} + 5575102467 T^{18} + 94317907640 T^{20} + 752185313429 T^{22} + 3138428376721 T^{24} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 36 T^{2} + 720 T^{4} - 10271 T^{6} + 121680 T^{8} - 1028196 T^{10} + 4826809 T^{12} )^{2} \))
$17$ (\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 28 T^{2} + 495 T^{4} - 8092 T^{6} + 83521 T^{8} \))(\( 1 + 26 T^{2} + 387 T^{4} + 7514 T^{6} + 83521 T^{8} \))(\( 1 - 72 T^{2} + 2664 T^{4} - 74146 T^{6} + 1759536 T^{8} - 36331632 T^{10} + 658037283 T^{12} - 10499841648 T^{14} + 146958206256 T^{16} - 1789704191074 T^{18} + 18583417822824 T^{20} - 145151560832328 T^{22} + 582622237229761 T^{24} \))
$19$ (\( ( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( ( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 3 T + 36 T^{2} + 99 T^{3} + 549 T^{4} + 1788 T^{5} + 8413 T^{6} + 33972 T^{7} + 198189 T^{8} + 679041 T^{9} + 4691556 T^{10} + 7428297 T^{11} + 47045881 T^{12} )^{2} \))
$23$ (\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 4 T^{2} - 513 T^{4} - 2116 T^{6} + 279841 T^{8} \))(\( 1 + 26 T^{2} + 147 T^{4} + 13754 T^{6} + 279841 T^{8} \))(\( 1 + 44 T^{2} + 2132 T^{4} + 55302 T^{6} + 1548472 T^{8} + 27701944 T^{10} + 755610907 T^{12} + 14654328376 T^{14} + 433325952952 T^{16} + 8186680733478 T^{18} + 166959020619092 T^{20} + 1822766493400556 T^{22} + 21914624432020321 T^{24} \))
$29$ (\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 137 T^{2} + 8645 T^{4} - 319673 T^{6} + 7270445 T^{8} - 96897497 T^{10} + 594823321 T^{12} )^{2} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 + 9 T + 58 T^{2} + 279 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 42 T^{2} + 180 T^{3} - 168 T^{4} - 7836 T^{5} - 41627 T^{6} - 242916 T^{7} - 161448 T^{8} + 5362380 T^{9} + 38787882 T^{10} + 171774906 T^{11} + 887503681 T^{12} )^{2} \))
$37$ (\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 4 T - 76 T^{2} + 90 T^{3} + 4144 T^{4} + 688 T^{5} - 181325 T^{6} + 25456 T^{7} + 5673136 T^{8} + 4558770 T^{9} - 142436236 T^{10} - 277375828 T^{11} + 2565726409 T^{12} )^{2} \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 76 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 22 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 132 T^{2} + 8472 T^{4} + 388519 T^{6} + 14241432 T^{8} + 373000452 T^{10} + 4750104241 T^{12} )^{2} \))
$43$ (\( ( 1 - 13 T + 43 T^{2} )^{2} \))(\( ( 1 + 7 T + 43 T^{2} )^{4} \))(\( ( 1 + 7 T + 43 T^{2} )^{4} \))(\( ( 1 - 5 T + 113 T^{2} - 431 T^{3} + 4859 T^{4} - 9245 T^{5} + 79507 T^{6} )^{4} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8} \))(\( 1 - 34 T^{2} - 1053 T^{4} - 75106 T^{6} + 4879681 T^{8} \))(\( 1 - 147 T^{2} + 9984 T^{4} - 446341 T^{6} + 16664751 T^{8} - 505982232 T^{10} + 16085788593 T^{12} - 1117714750488 T^{14} + 81318668824431 T^{16} - 4811205749161189 T^{18} + 237731886031021824 T^{20} - 7732072438667017203 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 104 T^{2} + 8007 T^{4} + 292136 T^{6} + 7890481 T^{8} \))(\( 1 + 86 T^{2} + 4587 T^{4} + 241574 T^{6} + 7890481 T^{8} \))(\( 1 + 200 T^{2} + 21164 T^{4} + 1425678 T^{6} + 67732372 T^{8} + 2466420796 T^{10} + 102771938515 T^{12} + 6928176015964 T^{14} + 534440994350932 T^{16} + 31599242045670462 T^{18} + 1317664087866044204 T^{20} + 34977494073102609800 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 98 T^{2} + 6123 T^{4} + 341138 T^{6} + 12117361 T^{8} \))(\( 1 - 58 T^{2} - 117 T^{4} - 201898 T^{6} + 12117361 T^{8} \))(\( 1 - 291 T^{2} + 46596 T^{4} - 5329417 T^{6} + 480785571 T^{8} - 35925708996 T^{10} + 2284341412497 T^{12} - 125057393015076 T^{14} + 5825852327398131 T^{16} - 224797653055417297 T^{18} + 6841711070610941316 T^{20} - \)\(14\!\cdots\!91\)\( T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( ( 1 + T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 - T + 61 T^{2} )^{2} \))(\( ( 1 - 9 T + 159 T^{2} - 1188 T^{3} + 11505 T^{4} - 60363 T^{5} + 676942 T^{6} - 3682143 T^{7} + 42810105 T^{8} - 269653428 T^{9} + 2201488719 T^{10} - 7601366709 T^{11} + 51520374361 T^{12} )^{2} \))
$67$ (\( ( 1 - 11 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} ) \))(\( ( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - T - 66 T^{2} - 67 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 18 T + 108 T^{2} - 418 T^{3} + 954 T^{4} + 65430 T^{5} - 968637 T^{6} + 4383810 T^{7} + 4282506 T^{8} - 125718934 T^{9} + 2176321068 T^{10} - 24302251926 T^{11} + 90458382169 T^{12} )^{2} \))
$71$ (\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 62 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 341 T^{2} + 53765 T^{4} - 4892609 T^{6} + 271029365 T^{8} - 8665383221 T^{10} + 128100283921 T^{12} )^{2} \))
$73$ (\( ( 1 - 17 T + 73 T^{2} )( 1 - 10 T + 73 T^{2} ) \))(\( ( 1 - 21 T + 220 T^{2} - 1533 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 21 T + 342 T^{2} + 4095 T^{3} + 40149 T^{4} + 343806 T^{5} + 3034951 T^{6} + 25097838 T^{7} + 213954021 T^{8} + 1593024615 T^{9} + 9712198422 T^{10} + 43534503453 T^{11} + 151334226289 T^{12} )^{2} \))
$79$ (\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \))(\( ( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 18 T + 72 T^{2} - 1138 T^{3} - 7470 T^{4} + 110574 T^{5} + 1995747 T^{6} + 8735346 T^{7} - 46620270 T^{8} - 561078382 T^{9} + 2804405832 T^{10} + 55387015182 T^{11} + 243087455521 T^{12} )^{2} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 106 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 339 T^{2} + 56361 T^{4} + 5822683 T^{6} + 388270929 T^{8} + 16088370819 T^{10} + 326940373369 T^{12} )^{2} \))
$89$ (\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 172 T^{2} + 21663 T^{4} - 1362412 T^{6} + 62742241 T^{8} \))(\( 1 + 62 T^{2} - 4077 T^{4} + 491102 T^{6} + 62742241 T^{8} \))(\( 1 - 333 T^{2} + 53955 T^{4} - 6596860 T^{6} + 732206169 T^{8} - 72458005383 T^{10} + 6566534648358 T^{12} - 573939860638743 T^{14} + 45940255917084729 T^{16} - 3278515999088982460 T^{18} + \)\(21\!\cdots\!55\)\( T^{20} - \)\(10\!\cdots\!33\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( ( 1 - 119 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 191 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 378 T^{2} + 65727 T^{4} - 7461452 T^{6} + 618425343 T^{8} - 33464068218 T^{10} + 832972004929 T^{12} )^{2} \))
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