# Properties

 Label 18.3.d.a Level $18$ Weight $3$ Character orbit 18.d Analytic conductor $0.490$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.490464475849$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ 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: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + (3 \beta_{2} - 6) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{6} + ( - 6 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{7} + 2 \beta_{3} q^{8} + (6 \beta_{3} + 3) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b3 - 2*b2 - 2*b1 + 1) * q^3 + 2*b2 * q^4 + (3*b2 - 6) * q^5 + (-2*b3 - 2*b2 + b1 - 2) * q^6 + (-6*b3 - b2 + 3*b1 + 1) * q^7 + 2*b3 * q^8 + (6*b3 + 3) * q^9 $$q + \beta_1 q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + (3 \beta_{2} - 6) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{6} + ( - 6 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{7} + 2 \beta_{3} q^{8} + (6 \beta_{3} + 3) q^{9} + (3 \beta_{3} - 6 \beta_1) q^{10} + (3 \beta_{2} - 3 \beta_1 + 3) q^{11} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{12} + (6 \beta_{3} - 5 \beta_{2} + 6 \beta_1) q^{13} + ( - \beta_{3} - 6 \beta_{2} + \beta_1 + 12) q^{14} + ( - 9 \beta_{3} + 9 \beta_{2} + 9 \beta_1) q^{15} + (4 \beta_{2} - 4) q^{16} + ( - 6 \beta_{3} - 12 \beta_{2} + 6) q^{17} + (12 \beta_{2} + 3 \beta_1 - 12) q^{18} + (6 \beta_{3} - 12 \beta_1 - 10) q^{19} + ( - 6 \beta_{2} - 6) q^{20} + (2 \beta_{3} + 17 \beta_{2} - 10 \beta_1 - 19) q^{21} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1) q^{22} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 6) q^{23} + ( - 2 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 4) q^{24} + ( - 2 \beta_{2} + 2) q^{25} + ( - 5 \beta_{3} + 24 \beta_{2} - 12) q^{26} + ( - 3 \beta_{3} - 30 \beta_{2} + 6 \beta_1 + 15) q^{27} + ( - 6 \beta_{3} + 12 \beta_1 + 2) q^{28} + (3 \beta_{2} + 6 \beta_1 + 3) q^{29} + (9 \beta_{3} + 18) q^{30} + ( - 9 \beta_{3} + 19 \beta_{2} - 9 \beta_1) q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + (6 \beta_{3} - 3 \beta_{2} - 12 \beta_1 + 15) q^{33} + ( - 12 \beta_{3} - 12 \beta_{2} + 6 \beta_1 + 12) q^{34} + (27 \beta_{3} + 6 \beta_{2} - 3) q^{35} + (12 \beta_{3} + 6 \beta_{2} - 12 \beta_1) q^{36} + ( - 6 \beta_{3} + 12 \beta_1 + 32) q^{37} + ( - 12 \beta_{2} - 10 \beta_1 - 12) q^{38} + ( - 13 \beta_{3} - 31 \beta_{2} + 23 \beta_1 - 10) q^{39} + ( - 6 \beta_{3} - 6 \beta_1) q^{40} + ( - 18 \beta_{3} + 21 \beta_{2} + 18 \beta_1 - 42) q^{41} + (17 \beta_{3} - 16 \beta_{2} - 19 \beta_1 - 4) q^{42} + (18 \beta_{3} + 23 \beta_{2} - 9 \beta_1 - 23) q^{43} + ( - 6 \beta_{3} + 12 \beta_{2} - 6) q^{44} + ( - 18 \beta_{3} + 9 \beta_{2} - 18 \beta_1 - 18) q^{45} + ( - 3 \beta_{3} + 6 \beta_1 - 6) q^{46} + (9 \beta_{2} + 21 \beta_1 + 9) q^{47} + ( - 8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 4) q^{48} + ( - 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{49} + ( - 2 \beta_{3} + 2 \beta_1) q^{50} + (24 \beta_{3} + 24 \beta_{2} - 12 \beta_1 - 30) q^{51} + (24 \beta_{3} - 10 \beta_{2} - 12 \beta_1 + 10) q^{52} + ( - 30 \beta_{3} - 60 \beta_{2} + 30) q^{53} + ( - 30 \beta_{3} + 6 \beta_{2} + 15 \beta_1 + 6) q^{54} + ( - 9 \beta_{3} + 18 \beta_1 - 27) q^{55} + (12 \beta_{2} + 2 \beta_1 + 12) q^{56} + (8 \beta_{3} + 20 \beta_{2} + 20 \beta_1 + 26) q^{57} + (3 \beta_{3} + 12 \beta_{2} + 3 \beta_1) q^{58} + (39 \beta_{3} - 21 \beta_{2} - 39 \beta_1 + 42) q^{59} + (18 \beta_{2} + 18 \beta_1 - 18) q^{60} + (36 \beta_{3} - 31 \beta_{2} - 18 \beta_1 + 31) q^{61} + (19 \beta_{3} - 36 \beta_{2} + 18) q^{62} + ( - 18 \beta_{3} + 33 \beta_{2} + 15 \beta_1 + 39) q^{63} - 8 q^{64} + (15 \beta_{2} - 54 \beta_1 + 15) q^{65} + ( - 3 \beta_{3} - 12 \beta_{2} + 15 \beta_1 - 12) q^{66} + ( - 9 \beta_{3} - 53 \beta_{2} - 9 \beta_1) q^{67} + ( - 12 \beta_{3} - 12 \beta_{2} + 12 \beta_1 + 24) q^{68} + (12 \beta_{3} - 15 \beta_{2} - 6 \beta_1 + 12) q^{69} + (6 \beta_{3} + 54 \beta_{2} - 3 \beta_1 - 54) q^{70} + ( - 24 \beta_{3} + 60 \beta_{2} - 30) q^{71} + (6 \beta_{3} - 24) q^{72} + ( - 18 \beta_{3} + 36 \beta_1 - 52) q^{73} + (12 \beta_{2} + 32 \beta_1 + 12) q^{74} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{75} + ( - 12 \beta_{3} - 20 \beta_{2} - 12 \beta_1) q^{76} + ( - 24 \beta_{3} + 15 \beta_{2} + 24 \beta_1 - 30) q^{77} + ( - 31 \beta_{3} + 20 \beta_{2} - 10 \beta_1 + 26) q^{78} + ( - 30 \beta_{3} - 7 \beta_{2} + 15 \beta_1 + 7) q^{79} + ( - 24 \beta_{2} + 12) q^{80} + (36 \beta_{3} - 63) q^{81} + (21 \beta_{3} - 42 \beta_1 + 36) q^{82} + ( - 63 \beta_{2} - 15 \beta_1 - 63) q^{83} + ( - 16 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 34) q^{84} + (18 \beta_{3} + 54 \beta_{2} + 18 \beta_1) q^{85} + (23 \beta_{3} + 18 \beta_{2} - 23 \beta_1 - 36) q^{86} + ( - 12 \beta_{3} - 21 \beta_{2} - 3 \beta_1 - 3) q^{87} + (12 \beta_{3} - 12 \beta_{2} - 6 \beta_1 + 12) q^{88} + (66 \beta_{3} + 60 \beta_{2} - 30) q^{89} + (9 \beta_{3} - 72 \beta_{2} - 18 \beta_1 + 36) q^{90} + (9 \beta_{3} - 18 \beta_1 + 103) q^{91} + (6 \beta_{2} - 6 \beta_1 + 6) q^{92} + (8 \beta_{3} + 35 \beta_{2} - 46 \beta_1 + 38) q^{93} + (9 \beta_{3} + 42 \beta_{2} + 9 \beta_1) q^{94} + ( - 54 \beta_{3} - 30 \beta_{2} + 54 \beta_1 + 60) q^{95} + (4 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 16) q^{96} + ( - 84 \beta_{3} - 7 \beta_{2} + 42 \beta_1 + 7) q^{97} + ( - 6 \beta_{3} - 24 \beta_{2} + 12) q^{98} + (36 \beta_{3} - 27 \beta_{2} - 27 \beta_1 + 45) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b3 - 2*b2 - 2*b1 + 1) * q^3 + 2*b2 * q^4 + (3*b2 - 6) * q^5 + (-2*b3 - 2*b2 + b1 - 2) * q^6 + (-6*b3 - b2 + 3*b1 + 1) * q^7 + 2*b3 * q^8 + (6*b3 + 3) * q^9 + (3*b3 - 6*b1) * q^10 + (3*b2 - 3*b1 + 3) * q^11 + (-2*b3 - 2*b2 - 2*b1 + 4) * q^12 + (6*b3 - 5*b2 + 6*b1) * q^13 + (-b3 - 6*b2 + b1 + 12) * q^14 + (-9*b3 + 9*b2 + 9*b1) * q^15 + (4*b2 - 4) * q^16 + (-6*b3 - 12*b2 + 6) * q^17 + (12*b2 + 3*b1 - 12) * q^18 + (6*b3 - 12*b1 - 10) * q^19 + (-6*b2 - 6) * q^20 + (2*b3 + 17*b2 - 10*b1 - 19) * q^21 + (3*b3 - 6*b2 + 3*b1) * q^22 + (3*b3 - 3*b2 - 3*b1 + 6) * q^23 + (-2*b3 - 8*b2 + 4*b1 + 4) * q^24 + (-2*b2 + 2) * q^25 + (-5*b3 + 24*b2 - 12) * q^26 + (-3*b3 - 30*b2 + 6*b1 + 15) * q^27 + (-6*b3 + 12*b1 + 2) * q^28 + (3*b2 + 6*b1 + 3) * q^29 + (9*b3 + 18) * q^30 + (-9*b3 + 19*b2 - 9*b1) * q^31 + (4*b3 - 4*b1) * q^32 + (6*b3 - 3*b2 - 12*b1 + 15) * q^33 + (-12*b3 - 12*b2 + 6*b1 + 12) * q^34 + (27*b3 + 6*b2 - 3) * q^35 + (12*b3 + 6*b2 - 12*b1) * q^36 + (-6*b3 + 12*b1 + 32) * q^37 + (-12*b2 - 10*b1 - 12) * q^38 + (-13*b3 - 31*b2 + 23*b1 - 10) * q^39 + (-6*b3 - 6*b1) * q^40 + (-18*b3 + 21*b2 + 18*b1 - 42) * q^41 + (17*b3 - 16*b2 - 19*b1 - 4) * q^42 + (18*b3 + 23*b2 - 9*b1 - 23) * q^43 + (-6*b3 + 12*b2 - 6) * q^44 + (-18*b3 + 9*b2 - 18*b1 - 18) * q^45 + (-3*b3 + 6*b1 - 6) * q^46 + (9*b2 + 21*b1 + 9) * q^47 + (-8*b3 + 4*b2 + 4*b1 + 4) * q^48 + (-6*b3 - 6*b2 - 6*b1) * q^49 + (-2*b3 + 2*b1) * q^50 + (24*b3 + 24*b2 - 12*b1 - 30) * q^51 + (24*b3 - 10*b2 - 12*b1 + 10) * q^52 + (-30*b3 - 60*b2 + 30) * q^53 + (-30*b3 + 6*b2 + 15*b1 + 6) * q^54 + (-9*b3 + 18*b1 - 27) * q^55 + (12*b2 + 2*b1 + 12) * q^56 + (8*b3 + 20*b2 + 20*b1 + 26) * q^57 + (3*b3 + 12*b2 + 3*b1) * q^58 + (39*b3 - 21*b2 - 39*b1 + 42) * q^59 + (18*b2 + 18*b1 - 18) * q^60 + (36*b3 - 31*b2 - 18*b1 + 31) * q^61 + (19*b3 - 36*b2 + 18) * q^62 + (-18*b3 + 33*b2 + 15*b1 + 39) * q^63 - 8 * q^64 + (15*b2 - 54*b1 + 15) * q^65 + (-3*b3 - 12*b2 + 15*b1 - 12) * q^66 + (-9*b3 - 53*b2 - 9*b1) * q^67 + (-12*b3 - 12*b2 + 12*b1 + 24) * q^68 + (12*b3 - 15*b2 - 6*b1 + 12) * q^69 + (6*b3 + 54*b2 - 3*b1 - 54) * q^70 + (-24*b3 + 60*b2 - 30) * q^71 + (6*b3 - 24) * q^72 + (-18*b3 + 36*b1 - 52) * q^73 + (12*b2 + 32*b1 + 12) * q^74 + (4*b3 - 2*b2 - 2*b1 - 2) * q^75 + (-12*b3 - 20*b2 - 12*b1) * q^76 + (-24*b3 + 15*b2 + 24*b1 - 30) * q^77 + (-31*b3 + 20*b2 - 10*b1 + 26) * q^78 + (-30*b3 - 7*b2 + 15*b1 + 7) * q^79 + (-24*b2 + 12) * q^80 + (36*b3 - 63) * q^81 + (21*b3 - 42*b1 + 36) * q^82 + (-63*b2 - 15*b1 - 63) * q^83 + (-16*b3 - 4*b2 - 4*b1 - 34) * q^84 + (18*b3 + 54*b2 + 18*b1) * q^85 + (23*b3 + 18*b2 - 23*b1 - 36) * q^86 + (-12*b3 - 21*b2 - 3*b1 - 3) * q^87 + (12*b3 - 12*b2 - 6*b1 + 12) * q^88 + (66*b3 + 60*b2 - 30) * q^89 + (9*b3 - 72*b2 - 18*b1 + 36) * q^90 + (9*b3 - 18*b1 + 103) * q^91 + (6*b2 - 6*b1 + 6) * q^92 + (8*b3 + 35*b2 - 46*b1 + 38) * q^93 + (9*b3 + 42*b2 + 9*b1) * q^94 + (-54*b3 - 30*b2 + 54*b1 + 60) * q^95 + (4*b3 - 8*b2 + 4*b1 + 16) * q^96 + (-84*b3 - 7*b2 + 42*b1 + 7) * q^97 + (-6*b3 - 24*b2 + 12) * q^98 + (36*b3 - 27*b2 - 27*b1 + 45) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} - 18 q^{5} - 12 q^{6} + 2 q^{7} + 12 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 - 18 * q^5 - 12 * q^6 + 2 * q^7 + 12 * q^9 $$4 q + 4 q^{4} - 18 q^{5} - 12 q^{6} + 2 q^{7} + 12 q^{9} + 18 q^{11} + 12 q^{12} - 10 q^{13} + 36 q^{14} + 18 q^{15} - 8 q^{16} - 24 q^{18} - 40 q^{19} - 36 q^{20} - 42 q^{21} - 12 q^{22} + 18 q^{23} + 4 q^{25} + 8 q^{28} + 18 q^{29} + 72 q^{30} + 38 q^{31} + 54 q^{33} + 24 q^{34} + 12 q^{36} + 128 q^{37} - 72 q^{38} - 102 q^{39} - 126 q^{41} - 48 q^{42} - 46 q^{43} - 54 q^{45} - 24 q^{46} + 54 q^{47} + 24 q^{48} - 12 q^{49} - 72 q^{51} + 20 q^{52} + 36 q^{54} - 108 q^{55} + 72 q^{56} + 144 q^{57} + 24 q^{58} + 126 q^{59} - 36 q^{60} + 62 q^{61} + 222 q^{63} - 32 q^{64} + 90 q^{65} - 72 q^{66} - 106 q^{67} + 72 q^{68} + 18 q^{69} - 108 q^{70} - 96 q^{72} - 208 q^{73} + 72 q^{74} - 12 q^{75} - 40 q^{76} - 90 q^{77} + 144 q^{78} + 14 q^{79} - 252 q^{81} + 144 q^{82} - 378 q^{83} - 144 q^{84} + 108 q^{85} - 108 q^{86} - 54 q^{87} + 24 q^{88} + 412 q^{91} + 36 q^{92} + 222 q^{93} + 84 q^{94} + 180 q^{95} + 48 q^{96} + 14 q^{97} + 126 q^{99}+O(q^{100})$$ 4 * q + 4 * q^4 - 18 * q^5 - 12 * q^6 + 2 * q^7 + 12 * q^9 + 18 * q^11 + 12 * q^12 - 10 * q^13 + 36 * q^14 + 18 * q^15 - 8 * q^16 - 24 * q^18 - 40 * q^19 - 36 * q^20 - 42 * q^21 - 12 * q^22 + 18 * q^23 + 4 * q^25 + 8 * q^28 + 18 * q^29 + 72 * q^30 + 38 * q^31 + 54 * q^33 + 24 * q^34 + 12 * q^36 + 128 * q^37 - 72 * q^38 - 102 * q^39 - 126 * q^41 - 48 * q^42 - 46 * q^43 - 54 * q^45 - 24 * q^46 + 54 * q^47 + 24 * q^48 - 12 * q^49 - 72 * q^51 + 20 * q^52 + 36 * q^54 - 108 * q^55 + 72 * q^56 + 144 * q^57 + 24 * q^58 + 126 * q^59 - 36 * q^60 + 62 * q^61 + 222 * q^63 - 32 * q^64 + 90 * q^65 - 72 * q^66 - 106 * q^67 + 72 * q^68 + 18 * q^69 - 108 * q^70 - 96 * q^72 - 208 * q^73 + 72 * q^74 - 12 * q^75 - 40 * q^76 - 90 * q^77 + 144 * q^78 + 14 * q^79 - 252 * q^81 + 144 * q^82 - 378 * q^83 - 144 * q^84 + 108 * q^85 - 108 * q^86 - 54 * q^87 + 24 * q^88 + 412 * q^91 + 36 * q^92 + 222 * q^93 + 84 * q^94 + 180 * q^95 + 48 * q^96 + 14 * q^97 + 126 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$11$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i 2.44949 + 1.73205i 1.00000 1.73205i −4.50000 2.59808i −4.22474 0.389270i −3.17423 5.49794i 2.82843i 3.00000 + 8.48528i 7.34847
5.2 1.22474 0.707107i −2.44949 + 1.73205i 1.00000 1.73205i −4.50000 2.59808i −1.77526 + 3.85337i 4.17423 + 7.22999i 2.82843i 3.00000 8.48528i −7.34847
11.1 −1.22474 0.707107i 2.44949 1.73205i 1.00000 + 1.73205i −4.50000 + 2.59808i −4.22474 + 0.389270i −3.17423 + 5.49794i 2.82843i 3.00000 8.48528i 7.34847
11.2 1.22474 + 0.707107i −2.44949 1.73205i 1.00000 + 1.73205i −4.50000 + 2.59808i −1.77526 3.85337i 4.17423 7.22999i 2.82843i 3.00000 + 8.48528i −7.34847
 $$n$$: 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.3.d.a 4
3.b odd 2 1 54.3.d.a 4
4.b odd 2 1 144.3.q.c 4
5.b even 2 1 450.3.i.b 4
5.c odd 4 2 450.3.k.a 8
8.b even 2 1 576.3.q.f 4
8.d odd 2 1 576.3.q.e 4
9.c even 3 1 54.3.d.a 4
9.c even 3 1 162.3.b.a 4
9.d odd 6 1 inner 18.3.d.a 4
9.d odd 6 1 162.3.b.a 4
12.b even 2 1 432.3.q.d 4
15.d odd 2 1 1350.3.i.b 4
15.e even 4 2 1350.3.k.a 8
24.f even 2 1 1728.3.q.c 4
24.h odd 2 1 1728.3.q.d 4
36.f odd 6 1 432.3.q.d 4
36.f odd 6 1 1296.3.e.g 4
36.h even 6 1 144.3.q.c 4
36.h even 6 1 1296.3.e.g 4
45.h odd 6 1 450.3.i.b 4
45.j even 6 1 1350.3.i.b 4
45.k odd 12 2 1350.3.k.a 8
45.l even 12 2 450.3.k.a 8
72.j odd 6 1 576.3.q.f 4
72.l even 6 1 576.3.q.e 4
72.n even 6 1 1728.3.q.d 4
72.p odd 6 1 1728.3.q.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 1.a even 1 1 trivial
18.3.d.a 4 9.d odd 6 1 inner
54.3.d.a 4 3.b odd 2 1
54.3.d.a 4 9.c even 3 1
144.3.q.c 4 4.b odd 2 1
144.3.q.c 4 36.h even 6 1
162.3.b.a 4 9.c even 3 1
162.3.b.a 4 9.d odd 6 1
432.3.q.d 4 12.b even 2 1
432.3.q.d 4 36.f odd 6 1
450.3.i.b 4 5.b even 2 1
450.3.i.b 4 45.h odd 6 1
450.3.k.a 8 5.c odd 4 2
450.3.k.a 8 45.l even 12 2
576.3.q.e 4 8.d odd 2 1
576.3.q.e 4 72.l even 6 1
576.3.q.f 4 8.b even 2 1
576.3.q.f 4 72.j odd 6 1
1296.3.e.g 4 36.f odd 6 1
1296.3.e.g 4 36.h even 6 1
1350.3.i.b 4 15.d odd 2 1
1350.3.i.b 4 45.j even 6 1
1350.3.k.a 8 15.e even 4 2
1350.3.k.a 8 45.k odd 12 2
1728.3.q.c 4 24.f even 2 1
1728.3.q.c 4 72.p odd 6 1
1728.3.q.d 4 24.h odd 2 1
1728.3.q.d 4 72.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(18, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4} - 6T^{2} + 81$$
$5$ $$(T^{2} + 9 T + 27)^{2}$$
$7$ $$T^{4} - 2 T^{3} + 57 T^{2} + \cdots + 2809$$
$11$ $$T^{4} - 18 T^{3} + 117 T^{2} + \cdots + 81$$
$13$ $$T^{4} + 10 T^{3} + 291 T^{2} + \cdots + 36481$$
$17$ $$T^{4} + 360T^{2} + 1296$$
$19$ $$(T^{2} + 20 T - 116)^{2}$$
$23$ $$T^{4} - 18 T^{3} + 117 T^{2} + \cdots + 81$$
$29$ $$T^{4} - 18 T^{3} + 63 T^{2} + \cdots + 2025$$
$31$ $$T^{4} - 38 T^{3} + 1569 T^{2} + \cdots + 15625$$
$37$ $$(T^{2} - 64 T + 808)^{2}$$
$41$ $$T^{4} + 126 T^{3} + 5967 T^{2} + \cdots + 455625$$
$43$ $$T^{4} + 46 T^{3} + 2073 T^{2} + \cdots + 1849$$
$47$ $$T^{4} - 54 T^{3} + 333 T^{2} + \cdots + 408321$$
$53$ $$T^{4} + 9000 T^{2} + 810000$$
$59$ $$T^{4} - 126 T^{3} + 3573 T^{2} + \cdots + 2954961$$
$61$ $$T^{4} - 62 T^{3} + 4827 T^{2} + \cdots + 966289$$
$67$ $$T^{4} + 106 T^{3} + 8913 T^{2} + \cdots + 5396329$$
$71$ $$T^{4} + 7704 T^{2} + \cdots + 2396304$$
$73$ $$(T^{2} + 104 T + 760)^{2}$$
$79$ $$T^{4} - 14 T^{3} + 1497 T^{2} + \cdots + 1692601$$
$83$ $$T^{4} + 378 T^{3} + \cdots + 131262849$$
$89$ $$T^{4} + 22824 T^{2} + \cdots + 36144144$$
$97$ $$T^{4} - 14 T^{3} + \cdots + 110986225$$