# Properties

 Label 18.6.c.a Level $18$ Weight $6$ Character orbit 18.c Analytic conductor $2.887$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 18.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.88690875663$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \beta_1 q^{2} + (\beta_{3} + 6 \beta_1 - 3) q^{3} + (16 \beta_1 - 16) q^{4} + (2 \beta_{3} - 2 \beta_{2} + 27 \beta_1 - 27) q^{5} + (4 \beta_{2} + 12 \beta_1 - 24) q^{6} + (\beta_{2} + 37 \beta_1) q^{7} - 64 q^{8} + ( - 6 \beta_{3} + 12 \beta_{2} + 189) q^{9}+O(q^{10})$$ q + 4*b1 * q^2 + (b3 + 6*b1 - 3) * q^3 + (16*b1 - 16) * q^4 + (2*b3 - 2*b2 + 27*b1 - 27) * q^5 + (4*b2 + 12*b1 - 24) * q^6 + (b2 + 37*b1) * q^7 - 64 * q^8 + (-6*b3 + 12*b2 + 189) * q^9 $$q + 4 \beta_1 q^{2} + (\beta_{3} + 6 \beta_1 - 3) q^{3} + (16 \beta_1 - 16) q^{4} + (2 \beta_{3} - 2 \beta_{2} + 27 \beta_1 - 27) q^{5} + (4 \beta_{2} + 12 \beta_1 - 24) q^{6} + (\beta_{2} + 37 \beta_1) q^{7} - 64 q^{8} + ( - 6 \beta_{3} + 12 \beta_{2} + 189) q^{9} + (8 \beta_{3} - 108) q^{10} + ( - 43 \beta_{2} - 39 \beta_1) q^{11} + ( - 16 \beta_{3} + 16 \beta_{2} - 48 \beta_1 - 48) q^{12} + (20 \beta_{3} - 20 \beta_{2} - 553 \beta_1 + 553) q^{13} + ( - 4 \beta_{3} + 4 \beta_{2} + 148 \beta_1 - 148) q^{14} + ( - 21 \beta_{3} + 33 \beta_{2} - 513 \beta_1 + 351) q^{15} - 256 \beta_1 q^{16} + ( - 94 \beta_{3} + 246) q^{17} + ( - 48 \beta_{3} + 24 \beta_{2} + 756 \beta_1) q^{18} + (124 \beta_{3} - 820) q^{19} + (32 \beta_{2} - 432 \beta_1) q^{20} + ( - 6 \beta_{3} + 40 \beta_{2} + 327 \beta_1 - 222) q^{21} + (172 \beta_{3} - 172 \beta_{2} - 156 \beta_1 + 156) q^{22} + (25 \beta_{3} - 25 \beta_{2} + 2769 \beta_1 - 2769) q^{23} + ( - 64 \beta_{3} - 384 \beta_1 + 192) q^{24} + (108 \beta_{2} + 1532 \beta_1) q^{25} + (80 \beta_{3} + 2212) q^{26} + (135 \beta_{3} + 3726 \beta_1 - 1863) q^{27} + ( - 16 \beta_{3} - 592) q^{28} + ( - 124 \beta_{2} - 1947 \beta_1) q^{29} + ( - 132 \beta_{3} + 48 \beta_{2} - 648 \beta_1 + 2052) q^{30} + ( - 435 \beta_{3} + 435 \beta_{2} - 2359 \beta_1 + 2359) q^{31} + ( - 1024 \beta_1 + 1024) q^{32} + (258 \beta_{3} - 168 \beta_{2} - 9405 \beta_1 + 234) q^{33} + ( - 376 \beta_{2} + 984 \beta_1) q^{34} + (47 \beta_{3} - 567) q^{35} + ( - 96 \beta_{3} - 96 \beta_{2} + 3024 \beta_1 - 3024) q^{36} + ( - 826 \beta_{3} - 2398) q^{37} + (496 \beta_{2} - 3280 \beta_1) q^{38} + (613 \beta_{3} - 493 \beta_{2} - 2661 \beta_1 + 5979) q^{39} + ( - 128 \beta_{3} + 128 \beta_{2} - 1728 \beta_1 + 1728) q^{40} + (14 \beta_{3} - 14 \beta_{2} - 7677 \beta_1 + 7677) q^{41} + ( - 160 \beta_{3} + 136 \beta_{2} + 420 \beta_1 - 1308) q^{42} + (267 \beta_{2} + 16429 \beta_1) q^{43} + (688 \beta_{3} + 624) q^{44} + (216 \beta_{3} - 540 \beta_{2} + 7695 \beta_1 - 2511) q^{45} + (100 \beta_{3} - 11076) q^{46} + (249 \beta_{2} + 12477 \beta_1) q^{47} + ( - 256 \beta_{2} - 768 \beta_1 + 1536) q^{48} + ( - 74 \beta_{3} + 74 \beta_{2} - 15222 \beta_1 + 15222) q^{49} + ( - 432 \beta_{3} + 432 \beta_{2} + 6128 \beta_1 - 6128) q^{50} + (528 \beta_{3} - 564 \beta_{2} + 1476 \beta_1 - 21042) q^{51} + (320 \beta_{2} + 8848 \beta_1) q^{52} + (178 \beta_{3} - 8166) q^{53} + (540 \beta_{2} + 7452 \beta_1 - 14904) q^{54} + (1083 \beta_{3} - 17523) q^{55} + ( - 64 \beta_{2} - 2368 \beta_1) q^{56} + ( - 1192 \beta_{3} + 744 \beta_{2} - 4920 \beta_1 + 29244) q^{57} + (496 \beta_{3} - 496 \beta_{2} - 7788 \beta_1 + 7788) q^{58} + ( - 311 \beta_{3} + 311 \beta_{2} - 10983 \beta_1 + 10983) q^{59} + ( - 192 \beta_{3} - 336 \beta_{2} + 5616 \beta_1 + 2592) q^{60} + ( - 708 \beta_{2} + 1525 \beta_1) q^{61} + ( - 1740 \beta_{3} + 9436) q^{62} + ( - 444 \beta_{3} + 411 \beta_{2} + 8289 \beta_1 - 2592) q^{63} + 4096 q^{64} + ( - 566 \beta_{2} + 6291 \beta_1) q^{65} + (672 \beta_{3} + 360 \beta_{2} - 36684 \beta_1 + 37620) q^{66} + (1671 \beta_{3} - 1671 \beta_{2} - 18379 \beta_1 + 18379) q^{67} + (1504 \beta_{3} - 1504 \beta_{2} + 3936 \beta_1 - 3936) q^{68} + ( - 2694 \beta_{3} + 2844 \beta_{2} - 13707 \beta_1 - 2907) q^{69} + (188 \beta_{2} - 2268 \beta_1) q^{70} + ( - 1798 \beta_{3} - 36924) q^{71} + (384 \beta_{3} - 768 \beta_{2} - 12096) q^{72} + (870 \beta_{3} - 25594) q^{73} + ( - 3304 \beta_{2} - 9592 \beta_1) q^{74} + ( - 648 \beta_{3} + 1856 \beta_{2} + 27924 \beta_1 - 9192) q^{75} + ( - 1984 \beta_{3} + 1984 \beta_{2} - 13120 \beta_1 + 13120) q^{76} + (1630 \beta_{3} - 1630 \beta_{2} - 10731 \beta_1 + 10731) q^{77} + (1972 \beta_{3} + 480 \beta_{2} + 13272 \beta_1 + 10644) q^{78} + (707 \beta_{2} - 7463 \beta_1) q^{79} + ( - 512 \beta_{3} + 6912) q^{80} + ( - 2268 \beta_{3} + 4536 \beta_{2} + 12393) q^{81} + (56 \beta_{3} + 30708) q^{82} + (147 \beta_{2} + 45381 \beta_1) q^{83} + ( - 544 \beta_{3} - 96 \beta_{2} - 3552 \beta_1 - 1680) q^{84} + (3030 \beta_{3} - 3030 \beta_{2} + 47250 \beta_1 - 47250) q^{85} + ( - 1068 \beta_{3} + 1068 \beta_{2} + 65716 \beta_1 - 65716) q^{86} + (744 \beta_{3} - 2319 \beta_{2} - 32625 \beta_1 + 11682) q^{87} + (2752 \beta_{2} + 2496 \beta_1) q^{88} + (6086 \beta_{3} - 4650) q^{89} + (2160 \beta_{3} - 1296 \beta_{2} + 20736 \beta_1 - 30780) q^{90} + (1293 \beta_{3} + 24781) q^{91} + (400 \beta_{2} - 44304 \beta_1) q^{92} + (1054 \beta_{3} - 3664 \beta_{2} + 101037 \beta_1 - 86883) q^{93} + ( - 996 \beta_{3} + 996 \beta_{2} + 49908 \beta_1 - 49908) q^{94} + ( - 4988 \beta_{3} + 4988 \beta_{2} - 75708 \beta_1 + 75708) q^{95} + (1024 \beta_{3} - 1024 \beta_{2} + 3072 \beta_1 + 3072) q^{96} + ( - 6574 \beta_{2} - 15131 \beta_1) q^{97} + ( - 296 \beta_{3} + 60888) q^{98} + (468 \beta_{3} - 8361 \beta_{2} - 63099 \beta_1 + 111456) q^{99}+O(q^{100})$$ q + 4*b1 * q^2 + (b3 + 6*b1 - 3) * q^3 + (16*b1 - 16) * q^4 + (2*b3 - 2*b2 + 27*b1 - 27) * q^5 + (4*b2 + 12*b1 - 24) * q^6 + (b2 + 37*b1) * q^7 - 64 * q^8 + (-6*b3 + 12*b2 + 189) * q^9 + (8*b3 - 108) * q^10 + (-43*b2 - 39*b1) * q^11 + (-16*b3 + 16*b2 - 48*b1 - 48) * q^12 + (20*b3 - 20*b2 - 553*b1 + 553) * q^13 + (-4*b3 + 4*b2 + 148*b1 - 148) * q^14 + (-21*b3 + 33*b2 - 513*b1 + 351) * q^15 - 256*b1 * q^16 + (-94*b3 + 246) * q^17 + (-48*b3 + 24*b2 + 756*b1) * q^18 + (124*b3 - 820) * q^19 + (32*b2 - 432*b1) * q^20 + (-6*b3 + 40*b2 + 327*b1 - 222) * q^21 + (172*b3 - 172*b2 - 156*b1 + 156) * q^22 + (25*b3 - 25*b2 + 2769*b1 - 2769) * q^23 + (-64*b3 - 384*b1 + 192) * q^24 + (108*b2 + 1532*b1) * q^25 + (80*b3 + 2212) * q^26 + (135*b3 + 3726*b1 - 1863) * q^27 + (-16*b3 - 592) * q^28 + (-124*b2 - 1947*b1) * q^29 + (-132*b3 + 48*b2 - 648*b1 + 2052) * q^30 + (-435*b3 + 435*b2 - 2359*b1 + 2359) * q^31 + (-1024*b1 + 1024) * q^32 + (258*b3 - 168*b2 - 9405*b1 + 234) * q^33 + (-376*b2 + 984*b1) * q^34 + (47*b3 - 567) * q^35 + (-96*b3 - 96*b2 + 3024*b1 - 3024) * q^36 + (-826*b3 - 2398) * q^37 + (496*b2 - 3280*b1) * q^38 + (613*b3 - 493*b2 - 2661*b1 + 5979) * q^39 + (-128*b3 + 128*b2 - 1728*b1 + 1728) * q^40 + (14*b3 - 14*b2 - 7677*b1 + 7677) * q^41 + (-160*b3 + 136*b2 + 420*b1 - 1308) * q^42 + (267*b2 + 16429*b1) * q^43 + (688*b3 + 624) * q^44 + (216*b3 - 540*b2 + 7695*b1 - 2511) * q^45 + (100*b3 - 11076) * q^46 + (249*b2 + 12477*b1) * q^47 + (-256*b2 - 768*b1 + 1536) * q^48 + (-74*b3 + 74*b2 - 15222*b1 + 15222) * q^49 + (-432*b3 + 432*b2 + 6128*b1 - 6128) * q^50 + (528*b3 - 564*b2 + 1476*b1 - 21042) * q^51 + (320*b2 + 8848*b1) * q^52 + (178*b3 - 8166) * q^53 + (540*b2 + 7452*b1 - 14904) * q^54 + (1083*b3 - 17523) * q^55 + (-64*b2 - 2368*b1) * q^56 + (-1192*b3 + 744*b2 - 4920*b1 + 29244) * q^57 + (496*b3 - 496*b2 - 7788*b1 + 7788) * q^58 + (-311*b3 + 311*b2 - 10983*b1 + 10983) * q^59 + (-192*b3 - 336*b2 + 5616*b1 + 2592) * q^60 + (-708*b2 + 1525*b1) * q^61 + (-1740*b3 + 9436) * q^62 + (-444*b3 + 411*b2 + 8289*b1 - 2592) * q^63 + 4096 * q^64 + (-566*b2 + 6291*b1) * q^65 + (672*b3 + 360*b2 - 36684*b1 + 37620) * q^66 + (1671*b3 - 1671*b2 - 18379*b1 + 18379) * q^67 + (1504*b3 - 1504*b2 + 3936*b1 - 3936) * q^68 + (-2694*b3 + 2844*b2 - 13707*b1 - 2907) * q^69 + (188*b2 - 2268*b1) * q^70 + (-1798*b3 - 36924) * q^71 + (384*b3 - 768*b2 - 12096) * q^72 + (870*b3 - 25594) * q^73 + (-3304*b2 - 9592*b1) * q^74 + (-648*b3 + 1856*b2 + 27924*b1 - 9192) * q^75 + (-1984*b3 + 1984*b2 - 13120*b1 + 13120) * q^76 + (1630*b3 - 1630*b2 - 10731*b1 + 10731) * q^77 + (1972*b3 + 480*b2 + 13272*b1 + 10644) * q^78 + (707*b2 - 7463*b1) * q^79 + (-512*b3 + 6912) * q^80 + (-2268*b3 + 4536*b2 + 12393) * q^81 + (56*b3 + 30708) * q^82 + (147*b2 + 45381*b1) * q^83 + (-544*b3 - 96*b2 - 3552*b1 - 1680) * q^84 + (3030*b3 - 3030*b2 + 47250*b1 - 47250) * q^85 + (-1068*b3 + 1068*b2 + 65716*b1 - 65716) * q^86 + (744*b3 - 2319*b2 - 32625*b1 + 11682) * q^87 + (2752*b2 + 2496*b1) * q^88 + (6086*b3 - 4650) * q^89 + (2160*b3 - 1296*b2 + 20736*b1 - 30780) * q^90 + (1293*b3 + 24781) * q^91 + (400*b2 - 44304*b1) * q^92 + (1054*b3 - 3664*b2 + 101037*b1 - 86883) * q^93 + (-996*b3 + 996*b2 + 49908*b1 - 49908) * q^94 + (-4988*b3 + 4988*b2 - 75708*b1 + 75708) * q^95 + (1024*b3 - 1024*b2 + 3072*b1 + 3072) * q^96 + (-6574*b2 - 15131*b1) * q^97 + (-296*b3 + 60888) * q^98 + (468*b3 - 8361*b2 - 63099*b1 + 111456) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{2} - 32 q^{4} - 54 q^{5} - 72 q^{6} + 74 q^{7} - 256 q^{8} + 756 q^{9}+O(q^{10})$$ 4 * q + 8 * q^2 - 32 * q^4 - 54 * q^5 - 72 * q^6 + 74 * q^7 - 256 * q^8 + 756 * q^9 $$4 q + 8 q^{2} - 32 q^{4} - 54 q^{5} - 72 q^{6} + 74 q^{7} - 256 q^{8} + 756 q^{9} - 432 q^{10} - 78 q^{11} - 288 q^{12} + 1106 q^{13} - 296 q^{14} + 378 q^{15} - 512 q^{16} + 984 q^{17} + 1512 q^{18} - 3280 q^{19} - 864 q^{20} - 234 q^{21} + 312 q^{22} - 5538 q^{23} + 3064 q^{25} + 8848 q^{26} - 2368 q^{28} - 3894 q^{29} + 6912 q^{30} + 4718 q^{31} + 2048 q^{32} - 17874 q^{33} + 1968 q^{34} - 2268 q^{35} - 6048 q^{36} - 9592 q^{37} - 6560 q^{38} + 18594 q^{39} + 3456 q^{40} + 15354 q^{41} - 4392 q^{42} + 32858 q^{43} + 2496 q^{44} + 5346 q^{45} - 44304 q^{46} + 24954 q^{47} + 4608 q^{48} + 30444 q^{49} - 12256 q^{50} - 81216 q^{51} + 17696 q^{52} - 32664 q^{53} - 44712 q^{54} - 70092 q^{55} - 4736 q^{56} + 107136 q^{57} + 15576 q^{58} + 21966 q^{59} + 21600 q^{60} + 3050 q^{61} + 37744 q^{62} + 6210 q^{63} + 16384 q^{64} + 12582 q^{65} + 77112 q^{66} + 36758 q^{67} - 7872 q^{68} - 39042 q^{69} - 4536 q^{70} - 147696 q^{71} - 48384 q^{72} - 102376 q^{73} - 19184 q^{74} + 19080 q^{75} + 26240 q^{76} + 21462 q^{77} + 69120 q^{78} - 14926 q^{79} + 27648 q^{80} + 49572 q^{81} + 122832 q^{82} + 90762 q^{83} - 13824 q^{84} - 94500 q^{85} - 131432 q^{86} - 18522 q^{87} + 4992 q^{88} - 18600 q^{89} - 81648 q^{90} + 99124 q^{91} - 88608 q^{92} - 145458 q^{93} - 99816 q^{94} + 151416 q^{95} + 18432 q^{96} - 30262 q^{97} + 243552 q^{98} + 319626 q^{99}+O(q^{100})$$ 4 * q + 8 * q^2 - 32 * q^4 - 54 * q^5 - 72 * q^6 + 74 * q^7 - 256 * q^8 + 756 * q^9 - 432 * q^10 - 78 * q^11 - 288 * q^12 + 1106 * q^13 - 296 * q^14 + 378 * q^15 - 512 * q^16 + 984 * q^17 + 1512 * q^18 - 3280 * q^19 - 864 * q^20 - 234 * q^21 + 312 * q^22 - 5538 * q^23 + 3064 * q^25 + 8848 * q^26 - 2368 * q^28 - 3894 * q^29 + 6912 * q^30 + 4718 * q^31 + 2048 * q^32 - 17874 * q^33 + 1968 * q^34 - 2268 * q^35 - 6048 * q^36 - 9592 * q^37 - 6560 * q^38 + 18594 * q^39 + 3456 * q^40 + 15354 * q^41 - 4392 * q^42 + 32858 * q^43 + 2496 * q^44 + 5346 * q^45 - 44304 * q^46 + 24954 * q^47 + 4608 * q^48 + 30444 * q^49 - 12256 * q^50 - 81216 * q^51 + 17696 * q^52 - 32664 * q^53 - 44712 * q^54 - 70092 * q^55 - 4736 * q^56 + 107136 * q^57 + 15576 * q^58 + 21966 * q^59 + 21600 * q^60 + 3050 * q^61 + 37744 * q^62 + 6210 * q^63 + 16384 * q^64 + 12582 * q^65 + 77112 * q^66 + 36758 * q^67 - 7872 * q^68 - 39042 * q^69 - 4536 * q^70 - 147696 * q^71 - 48384 * q^72 - 102376 * q^73 - 19184 * q^74 + 19080 * q^75 + 26240 * q^76 + 21462 * q^77 + 69120 * q^78 - 14926 * q^79 + 27648 * q^80 + 49572 * q^81 + 122832 * q^82 + 90762 * q^83 - 13824 * q^84 - 94500 * q^85 - 131432 * q^86 - 18522 * q^87 + 4992 * q^88 - 18600 * q^89 - 81648 * q^90 + 99124 * q^91 - 88608 * q^92 - 145458 * q^93 - 99816 * q^94 + 151416 * q^95 + 18432 * q^96 - 30262 * q^97 + 243552 * q^98 + 319626 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$3\nu^{3} + 6\nu$$ 3*v^3 + 6*v $$\beta_{3}$$ $$=$$ $$-3\nu^{3} + 12\nu$$ -3*v^3 + 12*v
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 18$$ (b3 + b2) / 18 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 9$$ (-b3 + 2*b2) / 9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
2.00000 3.46410i −14.6969 5.19615i −8.00000 13.8564i −28.1969 48.8385i −47.3939 + 40.5194i 11.1515 19.3150i −64.0000 189.000 + 152.735i −225.576
7.2 2.00000 3.46410i 14.6969 5.19615i −8.00000 13.8564i 1.19694 + 2.07316i 11.3939 61.3040i 25.8485 44.7709i −64.0000 189.000 152.735i 9.57551
13.1 2.00000 + 3.46410i −14.6969 + 5.19615i −8.00000 + 13.8564i −28.1969 + 48.8385i −47.3939 40.5194i 11.1515 + 19.3150i −64.0000 189.000 152.735i −225.576
13.2 2.00000 + 3.46410i 14.6969 + 5.19615i −8.00000 + 13.8564i 1.19694 2.07316i 11.3939 + 61.3040i 25.8485 + 44.7709i −64.0000 189.000 + 152.735i 9.57551
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.6.c.a 4
3.b odd 2 1 54.6.c.a 4
4.b odd 2 1 144.6.i.a 4
9.c even 3 1 inner 18.6.c.a 4
9.c even 3 1 162.6.a.e 2
9.d odd 6 1 54.6.c.a 4
9.d odd 6 1 162.6.a.f 2
12.b even 2 1 432.6.i.a 4
36.f odd 6 1 144.6.i.a 4
36.h even 6 1 432.6.i.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.c.a 4 1.a even 1 1 trivial
18.6.c.a 4 9.c even 3 1 inner
54.6.c.a 4 3.b odd 2 1
54.6.c.a 4 9.d odd 6 1
144.6.i.a 4 4.b odd 2 1
144.6.i.a 4 36.f odd 6 1
162.6.a.e 2 9.c even 3 1
162.6.a.f 2 9.d odd 6 1
432.6.i.a 4 12.b even 2 1
432.6.i.a 4 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 54T_{5}^{3} + 3051T_{5}^{2} - 7290T_{5} + 18225$$ acting on $$S_{6}^{\mathrm{new}}(18, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 4 T + 16)^{2}$$
$3$ $$T^{4} - 378 T^{2} + 59049$$
$5$ $$T^{4} + 54 T^{3} + 3051 T^{2} + \cdots + 18225$$
$7$ $$T^{4} - 74 T^{3} + 4323 T^{2} + \cdots + 1329409$$
$11$ $$T^{4} + 78 T^{3} + \cdots + 158294966769$$
$13$ $$T^{4} - 1106 T^{3} + \cdots + 48140309281$$
$17$ $$(T^{2} - 492 T - 1848060)^{2}$$
$19$ $$(T^{2} + 1640 T - 2648816)^{2}$$
$23$ $$T^{4} + 5538 T^{3} + \cdots + 56736462234321$$
$29$ $$T^{4} + 3894 T^{3} + \cdots + 220517585649$$
$31$ $$T^{4} - 4718 T^{3} + \cdots + 12\!\cdots\!61$$
$37$ $$(T^{2} + 4796 T - 141621212)^{2}$$
$41$ $$T^{4} - 15354 T^{3} + \cdots + 34\!\cdots\!49$$
$43$ $$T^{4} - 32858 T^{3} + \cdots + 64\!\cdots\!89$$
$47$ $$T^{4} - 24954 T^{3} + \cdots + 20\!\cdots\!69$$
$53$ $$(T^{2} + 16332 T + 59839812)^{2}$$
$59$ $$T^{4} - 21966 T^{3} + \cdots + 99\!\cdots\!09$$
$61$ $$T^{4} - 3050 T^{3} + \cdots + 11\!\cdots\!01$$
$67$ $$T^{4} - 36758 T^{3} + \cdots + 70\!\cdots\!25$$
$71$ $$(T^{2} + 73848 T + 665096112)^{2}$$
$73$ $$(T^{2} + 51188 T + 491562436)^{2}$$
$79$ $$T^{4} + 14926 T^{3} + \cdots + 27\!\cdots\!25$$
$83$ $$T^{4} - 90762 T^{3} + \cdots + 42\!\cdots\!89$$
$89$ $$(T^{2} + 9300 T - 7978887036)^{2}$$
$97$ $$T^{4} + 30262 T^{3} + \cdots + 82\!\cdots\!25$$