Properties

Label 189.2.f
Level 189
Weight 2
Character orbit f
Rep. character \(\chi_{189}(64,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 12
Newform subspaces 2
Sturm bound 48
Trace bound 2

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Defining parameters

Level: \( N \) = \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 189.f (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(48\)
Trace bound: \(2\)
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 12 48
Cusp forms 36 12 24
Eisenstein series 24 0 24

Trace form

\( 12q + 2q^{2} - 6q^{4} - 2q^{5} + O(q^{10}) \) \( 12q + 2q^{2} - 6q^{4} - 2q^{5} + 4q^{11} + 4q^{14} - 6q^{16} + 12q^{17} - 12q^{19} - 22q^{20} + 6q^{22} + 12q^{23} + 8q^{26} + 10q^{29} + 6q^{31} - 8q^{32} - 6q^{34} - 16q^{35} - 12q^{37} + 14q^{38} - 12q^{40} - 22q^{41} + 6q^{43} - 76q^{44} + 24q^{46} - 6q^{47} - 6q^{49} + 4q^{50} + 18q^{52} + 24q^{53} - 12q^{55} + 12q^{56} + 18q^{58} - 12q^{59} + 96q^{62} + 10q^{65} + 12q^{67} + 12q^{68} - 36q^{71} - 36q^{73} - 24q^{74} + 6q^{76} + 8q^{77} + 6q^{79} + 8q^{80} - 30q^{83} - 18q^{85} + 40q^{86} - 6q^{88} - 20q^{89} - 12q^{91} + 18q^{92} - 6q^{94} + 4q^{95} - 4q^{98} + O(q^{100}) \)

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.f.a \(6\) \(1.509\) 6.0.309123.1 None \(-1\) \(0\) \(-5\) \(3\) \(q+(\beta _{1}-\beta _{5})q^{2}+(-1+\beta _{2}+\beta _{4}+\beta _{5})q^{4}+\cdots\)
189.2.f.b \(6\) \(1.509\) \(\Q(\zeta_{18})\) None \(3\) \(0\) \(3\) \(-3\) \(q+(\zeta_{18}-\zeta_{18}^{3}-\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - T^{2} - 4 T^{3} - 3 T^{4} + 2 T^{5} + 13 T^{6} + 4 T^{7} - 12 T^{8} - 32 T^{9} - 16 T^{10} + 32 T^{11} + 64 T^{12} \))(\( 1 - 3 T + 3 T^{2} - 3 T^{4} + 6 T^{5} - 11 T^{6} + 12 T^{7} - 12 T^{8} + 48 T^{10} - 96 T^{11} + 64 T^{12} \))
$3$ 1
$5$ (\( 1 + 5 T + 8 T^{2} + 7 T^{3} + 9 T^{4} - 62 T^{5} - 299 T^{6} - 310 T^{7} + 225 T^{8} + 875 T^{9} + 5000 T^{10} + 15625 T^{11} + 15625 T^{12} \))(\( 1 - 3 T - 6 T^{2} + 9 T^{3} + 69 T^{4} - 30 T^{5} - 371 T^{6} - 150 T^{7} + 1725 T^{8} + 1125 T^{9} - 3750 T^{10} - 9375 T^{11} + 15625 T^{12} \))
$7$ (\( ( 1 - T + T^{2} )^{3} \))(\( ( 1 + T + T^{2} )^{3} \))
$11$ (\( 1 + 2 T - 10 T^{2} + 34 T^{3} + 48 T^{4} - 416 T^{5} + 31 T^{6} - 4576 T^{7} + 5808 T^{8} + 45254 T^{9} - 146410 T^{10} + 322102 T^{11} + 1771561 T^{12} \))(\( 1 - 6 T - 6 T^{2} + 18 T^{3} + 492 T^{4} - 852 T^{5} - 2873 T^{6} - 9372 T^{7} + 59532 T^{8} + 23958 T^{9} - 87846 T^{10} - 966306 T^{11} + 1771561 T^{12} \))
$13$ (\( ( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} )^{3} \))(\( 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 1755 T^{7} + 7605 T^{8} - 166972 T^{9} + 85683 T^{10} - 1113879 T^{11} + 4826809 T^{12} \))
$17$ (\( ( 1 - 12 T + 90 T^{2} - 435 T^{3} + 1530 T^{4} - 3468 T^{5} + 4913 T^{6} )^{2} \))(\( ( 1 + 6 T + 60 T^{2} + 207 T^{3} + 1020 T^{4} + 1734 T^{5} + 4913 T^{6} )^{2} \))
$19$ (\( ( 1 + 3 T + 51 T^{2} + 107 T^{3} + 969 T^{4} + 1083 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 + 3 T + 51 T^{2} + 97 T^{3} + 969 T^{4} + 1083 T^{5} + 6859 T^{6} )^{2} \))
$23$ (\( 1 - 36 T^{2} + 18 T^{3} + 468 T^{4} - 324 T^{5} - 5393 T^{6} - 7452 T^{7} + 247572 T^{8} + 219006 T^{9} - 10074276 T^{10} + 148035889 T^{12} \))(\( 1 - 12 T + 48 T^{2} - 54 T^{3} + 420 T^{4} - 6060 T^{5} + 37591 T^{6} - 139380 T^{7} + 222180 T^{8} - 657018 T^{9} + 13432368 T^{10} - 77236116 T^{11} + 148035889 T^{12} \))
$29$ (\( 1 - T - 82 T^{2} + 31 T^{3} + 4425 T^{4} - 758 T^{5} - 148595 T^{6} - 21982 T^{7} + 3721425 T^{8} + 756059 T^{9} - 57997042 T^{10} - 20511149 T^{11} + 594823321 T^{12} \))(\( 1 - 9 T + 30 T^{2} - 81 T^{3} - 579 T^{4} + 9414 T^{5} - 59051 T^{6} + 273006 T^{7} - 486939 T^{8} - 1975509 T^{9} + 21218430 T^{10} - 184600341 T^{11} + 594823321 T^{12} \))
$31$ (\( 1 - 3 T - 60 T^{2} + 219 T^{3} + 1983 T^{4} - 4746 T^{5} - 51289 T^{6} - 147126 T^{7} + 1905663 T^{8} + 6524229 T^{9} - 55411260 T^{10} - 85887453 T^{11} + 887503681 T^{12} \))(\( 1 - 3 T - 6 T^{2} - 319 T^{3} + 171 T^{4} + 1962 T^{5} + 62727 T^{6} + 60822 T^{7} + 164331 T^{8} - 9503329 T^{9} - 5541126 T^{10} - 85887453 T^{11} + 887503681 T^{12} \))
$37$ (\( ( 1 + 3 T + 57 T^{2} + 303 T^{3} + 2109 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 + 3 T + 33 T^{2} - 101 T^{3} + 1221 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2} \))
$41$ (\( 1 + 22 T + 206 T^{2} + 1802 T^{3} + 18432 T^{4} + 135116 T^{5} + 808243 T^{6} + 5539756 T^{7} + 30984192 T^{8} + 124195642 T^{9} + 582106766 T^{10} + 2548836422 T^{11} + 4750104241 T^{12} \))(\( 1 - 114 T^{2} + 18 T^{3} + 8322 T^{4} - 1026 T^{5} - 394913 T^{6} - 42066 T^{7} + 13989282 T^{8} + 1240578 T^{9} - 322136754 T^{10} + 4750104241 T^{12} \))
$43$ (\( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 566310 T^{7} + 227427 T^{8} + 45239483 T^{9} - 184615254 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \))(\( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 243810 T^{7} + 16757487 T^{8} + 11846543 T^{9} - 389743314 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \))
$47$ (\( 1 + 9 T - 6 T^{2} - 531 T^{3} - 2433 T^{4} + 3438 T^{5} + 104623 T^{6} + 161586 T^{7} - 5374497 T^{8} - 55130013 T^{9} - 29278086 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} \))(\( 1 - 3 T - 78 T^{2} + 405 T^{3} + 2481 T^{4} - 11064 T^{5} - 57089 T^{6} - 520008 T^{7} + 5480529 T^{8} + 42048315 T^{9} - 380615118 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \))
$53$ (\( ( 1 - 18 T + 234 T^{2} - 1917 T^{3} + 12402 T^{4} - 50562 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 + 6 T + 150 T^{2} + 639 T^{3} + 7950 T^{4} + 16854 T^{5} + 148877 T^{6} )^{2} \))
$59$ (\( 1 + 9 T - 90 T^{2} - 459 T^{3} + 10161 T^{4} + 20556 T^{5} - 598421 T^{6} + 1212804 T^{7} + 35370441 T^{8} - 94268961 T^{9} - 1090562490 T^{10} + 6434318691 T^{11} + 42180533641 T^{12} \))(\( 1 + 3 T - 96 T^{2} - 495 T^{3} + 3615 T^{4} + 15798 T^{5} - 107021 T^{6} + 932082 T^{7} + 12583815 T^{8} - 101662605 T^{9} - 1163266656 T^{10} + 2144772897 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 - 6 T - 126 T^{2} + 358 T^{3} + 12372 T^{4} - 11472 T^{5} - 838653 T^{6} - 699792 T^{7} + 46036212 T^{8} + 81259198 T^{9} - 1744575966 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} \))(\( 1 + 6 T - 132 T^{2} - 418 T^{3} + 13698 T^{4} + 19134 T^{5} - 893289 T^{6} + 1167174 T^{7} + 50970258 T^{8} - 94878058 T^{9} - 1827651012 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 + 6 T^{2} + 1366 T^{3} + 438 T^{4} + 4098 T^{5} + 1065603 T^{6} + 274566 T^{7} + 1966182 T^{8} + 410842258 T^{9} + 120906726 T^{10} + 90458382169 T^{12} \))(\( 1 - 12 T - 78 T^{2} + 518 T^{3} + 15318 T^{4} - 50094 T^{5} - 815637 T^{6} - 3356298 T^{7} + 68762502 T^{8} + 155795234 T^{9} - 1571787438 T^{10} - 16201501284 T^{11} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 14697 T^{4} + 45369 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 11289 T^{4} + 45369 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( ( 1 - 3 T + 51 T^{2} - 681 T^{3} + 3723 T^{4} - 15987 T^{5} + 389017 T^{6} )^{2} \))(\( ( 1 + 21 T + 303 T^{2} + 2797 T^{3} + 22119 T^{4} + 111909 T^{5} + 389017 T^{6} )^{2} \))
$79$ (\( 1 + 15 T + 36 T^{2} - 367 T^{3} - 3225 T^{4} - 51726 T^{5} - 676905 T^{6} - 4086354 T^{7} - 20127225 T^{8} - 180945313 T^{9} + 1402202916 T^{10} + 46155845985 T^{11} + 243087455521 T^{12} \))(\( 1 - 21 T + 84 T^{2} - 499 T^{3} + 25767 T^{4} - 195678 T^{5} + 408327 T^{6} - 15458562 T^{7} + 160811847 T^{8} - 246026461 T^{9} + 3271806804 T^{10} - 64618184379 T^{11} + 243087455521 T^{12} \))
$83$ (\( 1 + 12 T - 144 T^{2} - 582 T^{3} + 34812 T^{4} + 90444 T^{5} - 2656433 T^{6} + 7506852 T^{7} + 239819868 T^{8} - 332780034 T^{9} - 6833998224 T^{10} + 47268487716 T^{11} + 326940373369 T^{12} \))(\( 1 + 18 T + 30 T^{2} - 702 T^{3} + 8088 T^{4} + 126648 T^{5} + 719359 T^{6} + 10511784 T^{7} + 55718232 T^{8} - 401394474 T^{9} + 1423749630 T^{10} + 70902731574 T^{11} + 326940373369 T^{12} \))
$89$ (\( ( 1 - 2 T + 116 T^{2} - 735 T^{3} + 10324 T^{4} - 15842 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 + 12 T + 204 T^{2} + 1323 T^{3} + 18156 T^{4} + 95052 T^{5} + 704969 T^{6} )^{2} \))
$97$ (\( 1 + 3 T - 168 T^{2} + 573 T^{3} + 14223 T^{4} - 78504 T^{5} - 1297807 T^{6} - 7614888 T^{7} + 133824207 T^{8} + 522961629 T^{9} - 14872919208 T^{10} + 25762020771 T^{11} + 832972004929 T^{12} \))(\( 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 1115694 T^{7} + 22779189 T^{8} + 135988277 T^{9} - 10092338034 T^{10} - 25762020771 T^{11} + 832972004929 T^{12} \))
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