# Properties

 Label 63.2.f Level 63 Weight 2 Character orbit f Rep. character $$\chi_{63}(22,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 12 Newform subspaces 2 Sturm bound 16 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ = $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 63.f (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$2$$ Sturm bound: $$16$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(63, [\chi])$$.

Total New Old
Modular forms 20 12 8
Cusp forms 12 12 0
Eisenstein series 8 0 8

## Trace form

 $$12q - 2q^{2} - 4q^{3} - 6q^{4} + 2q^{5} - 8q^{6} - 4q^{9} + O(q^{10})$$ $$12q - 2q^{2} - 4q^{3} - 6q^{4} + 2q^{5} - 8q^{6} - 4q^{9} - 4q^{11} + 16q^{12} - 4q^{14} + 2q^{15} - 6q^{16} - 12q^{17} + 22q^{18} - 12q^{19} + 22q^{20} - 2q^{21} + 6q^{22} - 12q^{23} - 12q^{24} - 8q^{26} + 20q^{27} - 10q^{29} - 8q^{30} + 6q^{31} + 8q^{32} + 8q^{33} - 6q^{34} + 16q^{35} - 38q^{36} - 12q^{37} - 14q^{38} - 16q^{39} - 12q^{40} + 22q^{41} + 20q^{42} + 6q^{43} + 76q^{44} + 14q^{45} + 24q^{46} + 6q^{47} - 14q^{48} - 6q^{49} - 4q^{50} + 18q^{51} + 18q^{52} - 24q^{53} - 44q^{54} - 12q^{55} - 12q^{56} + 2q^{57} + 18q^{58} + 12q^{59} - 20q^{60} - 96q^{62} - 8q^{63} - 10q^{65} + 34q^{66} + 12q^{67} - 12q^{68} - 48q^{69} + 36q^{71} - 60q^{72} - 36q^{73} + 24q^{74} + 22q^{75} + 6q^{76} - 8q^{77} + 64q^{78} + 6q^{79} - 8q^{80} + 32q^{81} + 30q^{83} + 2q^{84} - 18q^{85} - 40q^{86} + 8q^{87} - 6q^{88} + 20q^{89} + 100q^{90} - 12q^{91} - 18q^{92} + 6q^{93} - 6q^{94} - 4q^{95} + 32q^{96} + 4q^{98} - 28q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(63, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
63.2.f.a $$6$$ $$0.503$$ $$\Q(\zeta_{18})$$ None $$-3$$ $$0$$ $$-3$$ $$-3$$ $$q+(-1+\zeta_{18}+\zeta_{18}^{5})q^{2}+(-\zeta_{18}^{3}+\cdots)q^{3}+\cdots$$
63.2.f.b $$6$$ $$0.503$$ 6.0.309123.1 None $$1$$ $$-4$$ $$5$$ $$3$$ $$q+(-\beta _{1}-\beta _{2}-\beta _{3}+\beta _{5})q^{2}+(\beta _{2}+\beta _{3}+\cdots)q^{3}+\cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T + 3 T^{2} - 3 T^{4} - 6 T^{5} - 11 T^{6} - 12 T^{7} - 12 T^{8} + 48 T^{10} + 96 T^{11} + 64 T^{12}$$)($$1 - T - T^{2} + 4 T^{3} - 3 T^{4} - 2 T^{5} + 13 T^{6} - 4 T^{7} - 12 T^{8} + 32 T^{9} - 16 T^{10} - 32 T^{11} + 64 T^{12}$$)
$3$ ($$1 - 9 T^{3} + 27 T^{6}$$)($$1 + 4 T + 10 T^{2} + 21 T^{3} + 30 T^{4} + 36 T^{5} + 27 T^{6}$$)
$5$ ($$1 + 3 T - 6 T^{2} - 9 T^{3} + 69 T^{4} + 30 T^{5} - 371 T^{6} + 150 T^{7} + 1725 T^{8} - 1125 T^{9} - 3750 T^{10} + 9375 T^{11} + 15625 T^{12}$$)($$1 - 5 T + 8 T^{2} - 7 T^{3} + 9 T^{4} + 62 T^{5} - 299 T^{6} + 310 T^{7} + 225 T^{8} - 875 T^{9} + 5000 T^{10} - 15625 T^{11} + 15625 T^{12}$$)
$7$ ($$( 1 + T + T^{2} )^{3}$$)($$( 1 - T + T^{2} )^{3}$$)
$11$ ($$1 + 6 T - 6 T^{2} - 18 T^{3} + 492 T^{4} + 852 T^{5} - 2873 T^{6} + 9372 T^{7} + 59532 T^{8} - 23958 T^{9} - 87846 T^{10} + 966306 T^{11} + 1771561 T^{12}$$)($$1 - 2 T - 10 T^{2} - 34 T^{3} + 48 T^{4} + 416 T^{5} + 31 T^{6} + 4576 T^{7} + 5808 T^{8} - 45254 T^{9} - 146410 T^{10} - 322102 T^{11} + 1771561 T^{12}$$)
$13$ ($$1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 1755 T^{7} + 7605 T^{8} - 166972 T^{9} + 85683 T^{10} - 1113879 T^{11} + 4826809 T^{12}$$)($$( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} )^{3}$$)
$17$ ($$( 1 - 6 T + 60 T^{2} - 207 T^{3} + 1020 T^{4} - 1734 T^{5} + 4913 T^{6} )^{2}$$)($$( 1 + 12 T + 90 T^{2} + 435 T^{3} + 1530 T^{4} + 3468 T^{5} + 4913 T^{6} )^{2}$$)
$19$ ($$( 1 + 3 T + 51 T^{2} + 97 T^{3} + 969 T^{4} + 1083 T^{5} + 6859 T^{6} )^{2}$$)($$( 1 + 3 T + 51 T^{2} + 107 T^{3} + 969 T^{4} + 1083 T^{5} + 6859 T^{6} )^{2}$$)
$23$ ($$1 + 12 T + 48 T^{2} + 54 T^{3} + 420 T^{4} + 6060 T^{5} + 37591 T^{6} + 139380 T^{7} + 222180 T^{8} + 657018 T^{9} + 13432368 T^{10} + 77236116 T^{11} + 148035889 T^{12}$$)($$1 - 36 T^{2} - 18 T^{3} + 468 T^{4} + 324 T^{5} - 5393 T^{6} + 7452 T^{7} + 247572 T^{8} - 219006 T^{9} - 10074276 T^{10} + 148035889 T^{12}$$)
$29$ ($$1 + 9 T + 30 T^{2} + 81 T^{3} - 579 T^{4} - 9414 T^{5} - 59051 T^{6} - 273006 T^{7} - 486939 T^{8} + 1975509 T^{9} + 21218430 T^{10} + 184600341 T^{11} + 594823321 T^{12}$$)($$1 + T - 82 T^{2} - 31 T^{3} + 4425 T^{4} + 758 T^{5} - 148595 T^{6} + 21982 T^{7} + 3721425 T^{8} - 756059 T^{9} - 57997042 T^{10} + 20511149 T^{11} + 594823321 T^{12}$$)
$31$ ($$1 - 3 T - 6 T^{2} - 319 T^{3} + 171 T^{4} + 1962 T^{5} + 62727 T^{6} + 60822 T^{7} + 164331 T^{8} - 9503329 T^{9} - 5541126 T^{10} - 85887453 T^{11} + 887503681 T^{12}$$)($$1 - 3 T - 60 T^{2} + 219 T^{3} + 1983 T^{4} - 4746 T^{5} - 51289 T^{6} - 147126 T^{7} + 1905663 T^{8} + 6524229 T^{9} - 55411260 T^{10} - 85887453 T^{11} + 887503681 T^{12}$$)
$37$ ($$( 1 + 3 T + 33 T^{2} - 101 T^{3} + 1221 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2}$$)($$( 1 + 3 T + 57 T^{2} + 303 T^{3} + 2109 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2}$$)
$41$ ($$1 - 114 T^{2} - 18 T^{3} + 8322 T^{4} + 1026 T^{5} - 394913 T^{6} + 42066 T^{7} + 13989282 T^{8} - 1240578 T^{9} - 322136754 T^{10} + 4750104241 T^{12}$$)($$1 - 22 T + 206 T^{2} - 1802 T^{3} + 18432 T^{4} - 135116 T^{5} + 808243 T^{6} - 5539756 T^{7} + 30984192 T^{8} - 124195642 T^{9} + 582106766 T^{10} - 2548836422 T^{11} + 4750104241 T^{12}$$)
$43$ ($$1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 243810 T^{7} + 16757487 T^{8} + 11846543 T^{9} - 389743314 T^{10} - 441025329 T^{11} + 6321363049 T^{12}$$)($$1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 566310 T^{7} + 227427 T^{8} + 45239483 T^{9} - 184615254 T^{10} - 441025329 T^{11} + 6321363049 T^{12}$$)
$47$ ($$1 + 3 T - 78 T^{2} - 405 T^{3} + 2481 T^{4} + 11064 T^{5} - 57089 T^{6} + 520008 T^{7} + 5480529 T^{8} - 42048315 T^{9} - 380615118 T^{10} + 688035021 T^{11} + 10779215329 T^{12}$$)($$1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 161586 T^{7} - 5374497 T^{8} + 55130013 T^{9} - 29278086 T^{10} - 2064105063 T^{11} + 10779215329 T^{12}$$)
$53$ ($$( 1 - 6 T + 150 T^{2} - 639 T^{3} + 7950 T^{4} - 16854 T^{5} + 148877 T^{6} )^{2}$$)($$( 1 + 18 T + 234 T^{2} + 1917 T^{3} + 12402 T^{4} + 50562 T^{5} + 148877 T^{6} )^{2}$$)
$59$ ($$1 - 3 T - 96 T^{2} + 495 T^{3} + 3615 T^{4} - 15798 T^{5} - 107021 T^{6} - 932082 T^{7} + 12583815 T^{8} + 101662605 T^{9} - 1163266656 T^{10} - 2144772897 T^{11} + 42180533641 T^{12}$$)($$1 - 9 T - 90 T^{2} + 459 T^{3} + 10161 T^{4} - 20556 T^{5} - 598421 T^{6} - 1212804 T^{7} + 35370441 T^{8} + 94268961 T^{9} - 1090562490 T^{10} - 6434318691 T^{11} + 42180533641 T^{12}$$)
$61$ ($$1 + 6 T - 132 T^{2} - 418 T^{3} + 13698 T^{4} + 19134 T^{5} - 893289 T^{6} + 1167174 T^{7} + 50970258 T^{8} - 94878058 T^{9} - 1827651012 T^{10} + 5067577806 T^{11} + 51520374361 T^{12}$$)($$1 - 6 T - 126 T^{2} + 358 T^{3} + 12372 T^{4} - 11472 T^{5} - 838653 T^{6} - 699792 T^{7} + 46036212 T^{8} + 81259198 T^{9} - 1744575966 T^{10} - 5067577806 T^{11} + 51520374361 T^{12}$$)
$67$ ($$1 - 12 T - 78 T^{2} + 518 T^{3} + 15318 T^{4} - 50094 T^{5} - 815637 T^{6} - 3356298 T^{7} + 68762502 T^{8} + 155795234 T^{9} - 1571787438 T^{10} - 16201501284 T^{11} + 90458382169 T^{12}$$)($$1 + 6 T^{2} + 1366 T^{3} + 438 T^{4} + 4098 T^{5} + 1065603 T^{6} + 274566 T^{7} + 1966182 T^{8} + 410842258 T^{9} + 120906726 T^{10} + 90458382169 T^{12}$$)
$71$ ($$( 1 - 9 T + 159 T^{2} - 1305 T^{3} + 11289 T^{4} - 45369 T^{5} + 357911 T^{6} )^{2}$$)($$( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 14697 T^{4} - 45369 T^{5} + 357911 T^{6} )^{2}$$)
$73$ ($$( 1 + 21 T + 303 T^{2} + 2797 T^{3} + 22119 T^{4} + 111909 T^{5} + 389017 T^{6} )^{2}$$)($$( 1 - 3 T + 51 T^{2} - 681 T^{3} + 3723 T^{4} - 15987 T^{5} + 389017 T^{6} )^{2}$$)
$79$ ($$1 - 21 T + 84 T^{2} - 499 T^{3} + 25767 T^{4} - 195678 T^{5} + 408327 T^{6} - 15458562 T^{7} + 160811847 T^{8} - 246026461 T^{9} + 3271806804 T^{10} - 64618184379 T^{11} + 243087455521 T^{12}$$)($$1 + 15 T + 36 T^{2} - 367 T^{3} - 3225 T^{4} - 51726 T^{5} - 676905 T^{6} - 4086354 T^{7} - 20127225 T^{8} - 180945313 T^{9} + 1402202916 T^{10} + 46155845985 T^{11} + 243087455521 T^{12}$$)
$83$ ($$1 - 18 T + 30 T^{2} + 702 T^{3} + 8088 T^{4} - 126648 T^{5} + 719359 T^{6} - 10511784 T^{7} + 55718232 T^{8} + 401394474 T^{9} + 1423749630 T^{10} - 70902731574 T^{11} + 326940373369 T^{12}$$)($$1 - 12 T - 144 T^{2} + 582 T^{3} + 34812 T^{4} - 90444 T^{5} - 2656433 T^{6} - 7506852 T^{7} + 239819868 T^{8} + 332780034 T^{9} - 6833998224 T^{10} - 47268487716 T^{11} + 326940373369 T^{12}$$)
$89$ ($$( 1 - 12 T + 204 T^{2} - 1323 T^{3} + 18156 T^{4} - 95052 T^{5} + 704969 T^{6} )^{2}$$)($$( 1 + 2 T + 116 T^{2} + 735 T^{3} + 10324 T^{4} + 15842 T^{5} + 704969 T^{6} )^{2}$$)
$97$ ($$1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 1115694 T^{7} + 22779189 T^{8} + 135988277 T^{9} - 10092338034 T^{10} - 25762020771 T^{11} + 832972004929 T^{12}$$)($$1 + 3 T - 168 T^{2} + 573 T^{3} + 14223 T^{4} - 78504 T^{5} - 1297807 T^{6} - 7614888 T^{7} + 133824207 T^{8} + 522961629 T^{9} - 14872919208 T^{10} + 25762020771 T^{11} + 832972004929 T^{12}$$)