# Properties

 Label 209.2.e.b Level $209$ Weight $2$ Character orbit 209.e Analytic conductor $1.669$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [209,2,Mod(45,209)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(209, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("209.45");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$209 = 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 209.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.66887340224$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$9$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} - x^{17} + 14 x^{16} - 11 x^{15} + 130 x^{14} - 92 x^{13} + 629 x^{12} - 276 x^{11} + 2060 x^{10} - 681 x^{9} + 3955 x^{8} - 115 x^{7} + 4735 x^{6} + 105 x^{5} + 2904 x^{4} + \cdots + 9$$ x^18 - x^17 + 14*x^16 - 11*x^15 + 130*x^14 - 92*x^13 + 629*x^12 - 276*x^11 + 2060*x^10 - 681*x^9 + 3955*x^8 - 115*x^7 + 4735*x^6 + 105*x^5 + 2904*x^4 + 1230*x^3 + 639*x^2 + 81*x + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{12} + \beta_{4}) q^{3} + (\beta_{13} - \beta_{8} + \beta_{3} - 1) q^{4} + ( - \beta_{16} - \beta_{9} - \beta_1) q^{5} + ( - \beta_{14} - \beta_{10} - \beta_{7} - \beta_{2} - \beta_1) q^{6} + (\beta_{5} + \beta_{4}) q^{7} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2}) q^{8} + (\beta_{16} - \beta_{15} + \beta_{12} - \beta_{10} - \beta_{8} - 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b12 + b4) * q^3 + (b13 - b8 + b3 - 1) * q^4 + (-b16 - b9 - b1) * q^5 + (-b14 - b10 - b7 - b2 - b1) * q^6 + (b5 + b4) * q^7 + (b11 - b9 - b7 + b6 - b5 - b3 + b2) * q^8 + (b16 - b15 + b12 - b10 - b8 - 1) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{12} + \beta_{4}) q^{3} + (\beta_{13} - \beta_{8} + \beta_{3} - 1) q^{4} + ( - \beta_{16} - \beta_{9} - \beta_1) q^{5} + ( - \beta_{14} - \beta_{10} - \beta_{7} - \beta_{2} - \beta_1) q^{6} + (\beta_{5} + \beta_{4}) q^{7} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2}) q^{8} + (\beta_{16} - \beta_{15} + \beta_{12} - \beta_{10} - \beta_{8} - 1) q^{9} + (\beta_{15} - \beta_{13} - \beta_{12} + 2 \beta_{8} - \beta_{3} + 2) q^{10} - q^{11} + ( - \beta_{7} - \beta_{5} - \beta_{4} + 1) q^{12} + ( - \beta_{16} + \beta_{15} - \beta_{13} + \beta_{8} - \beta_{3} + 1) q^{13} + ( - \beta_{16} + \beta_{15} - \beta_{14} - \beta_{10} - \beta_{9} - \beta_{6} + \beta_{5} - \beta_1) q^{14} + ( - \beta_{16} - \beta_{15} + \beta_{14} + 2 \beta_{10} + \beta_{8} + \beta_{7} + 2 \beta_{2} + 2 \beta_1 + 1) q^{15} + (\beta_{16} + \beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - 2 \beta_{4} + \cdots + \beta_1) q^{16}+ \cdots + ( - \beta_{16} + \beta_{15} - \beta_{12} + \beta_{10} + \beta_{8} + 1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b12 + b4) * q^3 + (b13 - b8 + b3 - 1) * q^4 + (-b16 - b9 - b1) * q^5 + (-b14 - b10 - b7 - b2 - b1) * q^6 + (b5 + b4) * q^7 + (b11 - b9 - b7 + b6 - b5 - b3 + b2) * q^8 + (b16 - b15 + b12 - b10 - b8 - 1) * q^9 + (b15 - b13 - b12 + 2*b8 - b3 + 2) * q^10 - q^11 + (-b7 - b5 - b4 + 1) * q^12 + (-b16 + b15 - b13 + b8 - b3 + 1) * q^13 + (-b16 + b15 - b14 - b10 - b9 - b6 + b5 - b1) * q^14 + (-b16 - b15 + b14 + 2*b10 + b8 + b7 + 2*b2 + 2*b1 + 1) * q^15 + (b16 + b14 - b13 + 2*b12 - b10 + b9 + b8 - b6 - 2*b4 + b1) * q^16 + (-b16 + b14 + b13 - b12 + b10 - b9 - b8 + b6 + b4) * q^17 + (-b11 + b7 - 2*b6 + 2*b5 - b4 + b3 + b2) * q^18 + (b16 - b14 + b9 + b8 - b7 + b6 + 1) * q^19 + (-b11 + b3 - 3*b2) * q^20 + (b17 + 2*b16 + 2*b12 - b11 - b10 + 2*b9 - 3*b8 - b6 - 2*b4) * q^21 - b1 * q^22 + (-b17 + b13 + b3) * q^23 + (-b17 - 2*b15 + b12 + b11 + b8 - 2*b5 - b4 - b1) * q^24 + (b17 - b15 + b14 + b12 - b8 + b7 - 1) * q^25 + (-b11 + 2*b9 + b7 - b6 - b5 - b4 + b3 - 3*b2 - 1) * q^26 + (-2*b11 - b6 + b5 - b4 + b2 + 1) * q^27 + (-b16 + b15 - b14 - 2*b12 - b7) * q^28 + (b17 + b15 - b13 - 2*b12 + b8 - b3 + 1) * q^29 + (b11 - 2*b9 + b7 + b6 + b5 + 4*b4 - 4) * q^30 + (b11 + b7 + b6 - 2*b5 + b3 - 3) * q^31 + (-b17 - b16 - b15 + 2*b14 + b13 + 3*b10 - 2*b8 + 2*b7 + b3 - b2 - b1 - 2) * q^32 + (b12 - b4) * q^33 + (b17 + b16 + b15 - b14 - 2*b13 + 2*b12 - 2*b10 + b8 - b7 - 2*b3 + 2*b2 + 2*b1 + 1) * q^34 + (-b17 - 2*b15 + b14 + b13 + b12 + b11 + 2*b10 + 2*b8 + 2*b6 - 2*b5 - b4 + 3*b1) * q^35 + (-b17 - b16 + 2*b13 - 3*b12 + b11 + b10 - b9 - b8 + b6 + 3*b4 - b1) * q^36 + (2*b7 + b6 + b4 - b2 - 3) * q^37 + (b17 + b16 - b15 - b13 + 3*b12 - b10 + 2*b8 - b7 - b5 - 3*b4 + b1) * q^38 + (b11 + b7 - b6 + b5 + 2*b4 + b2 + 1) * q^39 + (b15 + 2*b13 - 2*b12 + b10 - 4*b8 + b6 + b5 + 2*b4 - b1) * q^40 + (-b17 + 2*b16 + b15 - b13 + b11 - b10 + 2*b9 - b8 - b6 + b5 - b1) * q^41 + (-2*b15 + b14 + b13 + 3*b10 + 2*b8 + b7 + b3 + 5*b2 + 5*b1 + 2) * q^42 + (2*b16 + b15 - 2*b14 - b13 - b10 + 2*b9 - b6 + b5 - 2*b1) * q^43 + (-b13 + b8 - b3 + 1) * q^44 + (b11 - 2*b7 + 3*b6 - 2*b5 + 2*b4 - b3 - 3*b2 + 1) * q^45 + (b6 - b5 - b3 + b2 - 1) * q^46 + (-b14 - 2*b12 - b8 - b7 + b2 + b1 - 1) * q^47 + (-b17 + b16 - b13 - 2*b12 + b10 + 4*b8 - b3 - b2 - b1 + 4) * q^48 + (-b11 + 2*b9 - b7 - b6 - 2*b4 - b2) * q^49 + (-b9 + b7 - 2*b6 + 2*b5 + b4 + 2*b2 + 2) * q^50 + (b17 + b15 - b14 - b13 + b12 - b10 - 2*b8 - b7 - b3 - 2*b2 - 2*b1 - 2) * q^51 + (-b16 - b15 + 3*b13 - b12 + 2*b10 - b9 - 6*b8 + 2*b6 - b5 + b4 - b1) * q^52 + (-b17 - 2*b16 + 2*b15 + b13 - b12 - b8 + b3 - 2*b2 - 2*b1 - 1) * q^53 + (b17 + b15 - b14 - 2*b12 - b11 + 2*b10 + 4*b8 + 2*b6 + b5 + 2*b4 + 3*b1) * q^54 + (b16 + b9 + b1) * q^55 + (b11 - 2*b9 - b7 + b6 - b5 - 2*b4 + b2 - 2) * q^56 + (b17 + 3*b15 + b12 - b11 - 2*b10 - b6 + 2*b5 + b3 - 3*b1) * q^57 + (b9 - 2*b7 + b6 + b3 - 4*b2 + 1) * q^58 + (-b17 + 2*b16 - b15 - b14 + 2*b12 + b11 + 2*b9 - 3*b8 - b5 - 2*b4 - 2*b1) * q^59 + (b17 - 2*b16 + 2*b15 - 2*b14 - b13 - b12 - b11 - 2*b10 - 2*b9 - b8 - 2*b6 + 2*b5 + b4 - 3*b1) * q^60 + (b17 + b16 + b15 + 2*b14 + b12 - b10 - b8 + 2*b7 + b2 + b1 - 1) * q^61 + (2*b16 - 2*b15 + b14 + b12 - b10 + 2*b9 - b8 - b6 - 2*b5 - b4 - 4*b1) * q^62 + (-b16 + 2*b15 - b13 - 4*b12 + 2*b10 + 5*b8 - b3 + 5) * q^63 + (b11 + b5 + 3*b4 - 3*b3 + 2*b2 + 4) * q^64 + (b11 - b9 + b7 + b5 + b4 - b3 + 4*b2 - 3) * q^65 + (b14 + b10 + b7 + b2 + b1) * q^66 + (b16 - 4*b15 + 2*b14 + b13 + b10 + 2*b7 + b3 - b2 - b1) * q^67 + (-2*b11 + b9 - 3*b6 + b5 - 2*b4 + 4*b3 - 3*b2 - 5) * q^68 + (-b11 - 2*b9 - b7 + b6 + 2*b5 + b4 + b3 + 1) * q^69 + (b17 + 3*b16 - 2*b15 + 2*b14 + 5*b12 - b10 - 7*b8 + 2*b7 + b2 + b1 - 7) * q^70 + (-b17 - 2*b16 + 2*b14 - b12 + b11 + b10 - 2*b9 + b8 + b6 + b4 + 2*b1) * q^71 + (2*b16 - b15 - 2*b14 - 2*b13 - b12 - 2*b10 + 2*b8 - 2*b7 - 2*b3 + 4*b2 + 4*b1 + 2) * q^72 + (-2*b16 + 2*b15 - b14 - 2*b9 + 3*b8 + 2*b5 + b1) * q^73 + (3*b17 + 3*b16 + 2*b15 - 3*b14 - 3*b11 - 4*b10 + 3*b9 - 4*b8 - 4*b6 + 2*b5 - b1) * q^74 + (3*b9 - 2*b6 - 3*b5 - 4*b4 - 2*b3 - b2 + 2) * q^75 + (-2*b17 - b16 - 3*b15 + 3*b14 + 2*b13 + b12 + b11 + 3*b10 + b9 + 3*b7 - 2*b4 + 3*b3 - b2 - 2*b1 - 2) * q^76 + (-b5 - b4) * q^77 + (-b17 - 2*b16 + 2*b15 - 2*b14 - 3*b12 + b11 - 2*b10 - 2*b9 - 2*b6 + 2*b5 + 3*b4 - 2*b1) * q^78 + (2*b17 - b16 - b15 - 2*b11 + b10 - b9 - 3*b8 + b6 - b5 + 2*b1) * q^79 + (b17 + 2*b16 + 3*b15 - 4*b14 - 2*b13 + 2*b12 - 5*b10 + 4*b8 - 4*b7 - 2*b3 + b2 + b1 + 4) * q^80 + (2*b17 + 3*b15 + b14 - 2*b13 - b12 - 2*b11 + 2*b10 + b8 + 2*b6 + 3*b5 + b4 + 3*b1) * q^81 + (-3*b17 - 4*b16 - 2*b15 + 2*b14 + b13 + 2*b10 + 3*b8 + 2*b7 + b3 - 3*b2 - 3*b1 + 3) * q^82 + (-b11 - 2*b9 - b7 + b4 + b3 - 3*b2 + 2) * q^83 + (b11 - b9 + b6 + 2*b5 + 3*b4 + b3 + 2*b2 - 5) * q^84 + (-b17 - 3*b16 - b15 + b14 + b13 - 3*b12 + 3*b10 - b8 + b7 + b3 - b2 - b1 - 1) * q^85 + (-b16 - b14 - 2*b12 + 5*b8 - b7 - b2 - b1 + 5) * q^86 + (2*b11 - b9 + b7 + b6 - b5 + 3*b4 - b3 - 8) * q^87 + (-b11 + b9 + b7 - b6 + b5 + b3 - b2) * q^88 + (-b17 - 3*b16 - 2*b15 - b12 + b10 + b2 + b1) * q^89 + (b17 + 3*b16 - 3*b15 - b13 + 8*b12 - b11 - 4*b10 + 3*b9 - 5*b8 - 4*b6 - 3*b5 - 8*b4 + b1) * q^90 + (-b17 - 2*b15 + 2*b14 + b13 + 3*b12 + b10 - 2*b8 + 2*b7 + b3 + 3*b2 + 3*b1 - 2) * q^91 + (3*b16 - b14 - b13 + 2*b12 - 2*b10 + 3*b9 + 4*b8 - 2*b6 - 2*b4 + 2*b1) * q^92 + (-2*b17 - 2*b16 - 4*b15 + b14 + 2*b13 + 3*b12 + 2*b11 - 2*b10 - 2*b9 - b8 - 2*b6 - 4*b5 - 3*b4 - b1) * q^93 + (b11 - b9 - 3*b7 + 3*b6 - b5 - b4 + b3 - 4) * q^94 + (-2*b17 - b16 - b14 + 2*b13 - 3*b12 + b11 + b10 - 3*b9 + 5*b8 + b6 + 3*b4 + b3 - b1 + 3) * q^95 + (b11 + b9 + 2*b6 - 2*b5 + b4 - b3 - 6*b2 + 2) * q^96 + (-3*b16 + 4*b14 - b13 + b12 + 4*b10 - 3*b9 - 3*b8 + 4*b6 - b4 + 2*b1) * q^97 + (-b17 - b16 - 3*b15 + 2*b14 + 2*b13 + b12 + b11 + 4*b10 - b9 + 4*b6 - 3*b5 - b4 + b1) * q^98 + (-b16 + b15 - b12 + b10 + b8 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q + q^{2} - 9 q^{4} - 3 q^{6} - 6 q^{7} - 11 q^{9}+O(q^{10})$$ 18 * q + q^2 - 9 * q^4 - 3 * q^6 - 6 * q^7 - 11 * q^9 $$18 q + q^{2} - 9 q^{4} - 3 q^{6} - 6 q^{7} - 11 q^{9} + 21 q^{10} - 18 q^{11} + 16 q^{12} + 11 q^{13} + q^{14} + 7 q^{15} - 13 q^{16} + 6 q^{17} - 8 q^{18} + 4 q^{19} + 4 q^{20} + 24 q^{21} - q^{22} - q^{23} - 3 q^{24} - 7 q^{25} - 4 q^{26} + 6 q^{27} - 2 q^{28} + 13 q^{29} - 64 q^{30} - 32 q^{31} - 14 q^{32} + 8 q^{34} - 12 q^{35} + 10 q^{36} - 36 q^{37} - 20 q^{38} + 20 q^{39} + 32 q^{40} + 4 q^{41} + 11 q^{42} + q^{43} + 9 q^{44} + 22 q^{45} - 14 q^{46} - 14 q^{47} + 37 q^{48} - 12 q^{49} + 30 q^{50} - 16 q^{51} + 57 q^{52} - 4 q^{53} - 33 q^{54} - 34 q^{56} - 7 q^{57} + 8 q^{58} + 31 q^{59} + 9 q^{60} + 3 q^{61} + 5 q^{62} + 50 q^{63} + 64 q^{64} - 56 q^{65} + 3 q^{66} - 2 q^{67} - 96 q^{68} - 58 q^{70} - 12 q^{71} + 5 q^{72} - 26 q^{73} + 35 q^{74} + 50 q^{75} - 36 q^{76} + 6 q^{77} + 3 q^{78} + 31 q^{79} + 31 q^{80} - 21 q^{81} + 25 q^{82} + 36 q^{83} - 102 q^{84} - 11 q^{85} + 41 q^{86} - 124 q^{87} - 11 q^{89} + 51 q^{90} - 20 q^{91} - 33 q^{92} + 20 q^{93} - 86 q^{94} + 17 q^{95} + 60 q^{96} + 16 q^{97} + 4 q^{98} + 11 q^{99}+O(q^{100})$$ 18 * q + q^2 - 9 * q^4 - 3 * q^6 - 6 * q^7 - 11 * q^9 + 21 * q^10 - 18 * q^11 + 16 * q^12 + 11 * q^13 + q^14 + 7 * q^15 - 13 * q^16 + 6 * q^17 - 8 * q^18 + 4 * q^19 + 4 * q^20 + 24 * q^21 - q^22 - q^23 - 3 * q^24 - 7 * q^25 - 4 * q^26 + 6 * q^27 - 2 * q^28 + 13 * q^29 - 64 * q^30 - 32 * q^31 - 14 * q^32 + 8 * q^34 - 12 * q^35 + 10 * q^36 - 36 * q^37 - 20 * q^38 + 20 * q^39 + 32 * q^40 + 4 * q^41 + 11 * q^42 + q^43 + 9 * q^44 + 22 * q^45 - 14 * q^46 - 14 * q^47 + 37 * q^48 - 12 * q^49 + 30 * q^50 - 16 * q^51 + 57 * q^52 - 4 * q^53 - 33 * q^54 - 34 * q^56 - 7 * q^57 + 8 * q^58 + 31 * q^59 + 9 * q^60 + 3 * q^61 + 5 * q^62 + 50 * q^63 + 64 * q^64 - 56 * q^65 + 3 * q^66 - 2 * q^67 - 96 * q^68 - 58 * q^70 - 12 * q^71 + 5 * q^72 - 26 * q^73 + 35 * q^74 + 50 * q^75 - 36 * q^76 + 6 * q^77 + 3 * q^78 + 31 * q^79 + 31 * q^80 - 21 * q^81 + 25 * q^82 + 36 * q^83 - 102 * q^84 - 11 * q^85 + 41 * q^86 - 124 * q^87 - 11 * q^89 + 51 * q^90 - 20 * q^91 - 33 * q^92 + 20 * q^93 - 86 * q^94 + 17 * q^95 + 60 * q^96 + 16 * q^97 + 4 * q^98 + 11 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - x^{17} + 14 x^{16} - 11 x^{15} + 130 x^{14} - 92 x^{13} + 629 x^{12} - 276 x^{11} + 2060 x^{10} - 681 x^{9} + 3955 x^{8} - 115 x^{7} + 4735 x^{6} + 105 x^{5} + 2904 x^{4} + \cdots + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 22\!\cdots\!19 \nu^{17} + \cdots + 65\!\cdots\!64 ) / 57\!\cdots\!02$$ (226390128672241019*v^17 - 658057456885613475*v^16 + 3771790359980738214*v^15 - 8495923446162102652*v^14 + 36341339174506414461*v^13 - 76127991009000161206*v^12 + 202112376378388540717*v^11 - 325507259171426993713*v^10 + 674964459490601087478*v^9 - 985451188344307393226*v^8 + 1504030925982783540634*v^7 - 1533730151792902572603*v^6 + 1708065285692301846865*v^5 - 1554335558829178175111*v^4 + 1688505772053611711371*v^3 - 492992759252098269147*v^2 - 63672080562628618110*v + 65226978328787314464) / 570032680884481371702 $$\beta_{3}$$ $$=$$ $$( - 43\!\cdots\!56 \nu^{17} + \cdots + 17\!\cdots\!35 ) / 57\!\cdots\!02$$ (-431667328213372456*v^17 + 602328558569363948*v^16 - 6005632030767451443*v^15 + 6910622447115081991*v^14 - 55300099171153987458*v^13 + 59712985443548939766*v^12 - 263023583657888472469*v^11 + 208600794425784588338*v^10 - 831279510718511259287*v^9 + 608657967084070310489*v^8 - 1507695286995594855418*v^7 + 636108026429240621900*v^6 - 1578106522339763482106*v^5 + 1031068838389423792195*v^4 - 771452617518954722517*v^3 + 361697308100290742451*v^2 + 46889377906335791925*v + 1708060531495393945935) / 570032680884481371702 $$\beta_{4}$$ $$=$$ $$( - 27\!\cdots\!03 \nu^{17} + \cdots + 74\!\cdots\!51 ) / 41\!\cdots\!46$$ (-274473638250788355303*v^17 + 631719459543240136615*v^16 - 4312833572901823805463*v^15 + 8073731119061939687983*v^14 - 41144532590505202399095*v^13 + 72121191189589981771928*v^12 - 219585457619857811602414*v^11 + 303217395190094729723931*v^10 - 729658336164243654212791*v^9 + 917788304492156818152155*v^8 - 1539829797129157416047272*v^7 + 1400512001712863388119557*v^6 - 1719888033692942981489631*v^5 + 1470190200689493728631918*v^4 - 1192191288818493316849856*v^3 + 474481841026114092932190*v^2 + 61322742973395495646803*v + 74527304546508805072251) / 41612385704567140134246 $$\beta_{5}$$ $$=$$ $$( 18\!\cdots\!99 \nu^{17} + \cdots - 69\!\cdots\!97 ) / 20\!\cdots\!23$$ (187570417945165715599*v^17 - 416013672308888433700*v^16 + 2901961996383768516732*v^15 - 5242879284403089735699*v^14 + 27541464355364997082560*v^13 - 46635451117474592865991*v^12 + 145010939279950690814397*v^11 - 192315161121043520470808*v^10 + 478143807040636644649081*v^9 - 579460835534623703840850*v^8 + 996412545203605959464601*v^7 - 861344887294654229645188*v^6 + 1106689118417274785464703*v^5 - 929451129399577917463999*v^4 + 776349688681406684634604*v^3 - 301459065702000623531970*v^2 - 38968336584455222772084*v - 69535401065617040360997) / 20806192852283570067123 $$\beta_{6}$$ $$=$$ $$( 38\!\cdots\!49 \nu^{17} + \cdots - 23\!\cdots\!00 ) / 41\!\cdots\!46$$ (380619599716653489049*v^17 - 901417389096639205639*v^16 + 6027181323742451209950*v^15 - 11548106746507254037410*v^14 + 57556312694832756364041*v^13 - 103212926849207787857026*v^12 + 308875197153238283468235*v^11 - 435590573901565732324061*v^10 + 1026493035499480684285834*v^9 - 1318970600788019113650834*v^8 + 2183951896869063352241304*v^7 - 2029707111244853324288875*v^6 + 2445095243316857285777039*v^5 - 2108625223966543267227745*v^4 + 1768737702686687177504083*v^3 - 678833635443743721047487*v^2 - 87725064452392184202498*v - 23884276491265463923800) / 41612385704567140134246 $$\beta_{7}$$ $$=$$ $$( 49\!\cdots\!66 \nu^{17} + \cdots - 40\!\cdots\!41 ) / 41\!\cdots\!46$$ (495457466808399139866*v^17 - 1050522969380778211922*v^16 + 7656629143091434924989*v^15 - 13384115312802678314267*v^14 + 72759739156195972776684*v^13 - 119406956887950159139360*v^12 + 383189931502362798888707*v^11 - 499272461100073878395556*v^10 + 1268525950924436998542893*v^9 - 1511655381029197062477013*v^8 + 2632531801100182185025346*v^7 - 2306156297016631958153630*v^6 + 2922557560485655445337912*v^5 - 2437724430123642537987501*v^4 + 1915545307705675220651503*v^3 - 792297161603070672756033*v^2 - 102425365657304464825089*v - 40865792530456501535541) / 41612385704567140134246 $$\beta_{8}$$ $$=$$ $$( 21\!\cdots\!88 \nu^{17} + \cdots + 24\!\cdots\!52 ) / 17\!\cdots\!06$$ (21742326109595771488*v^17 - 22421496495612494545*v^16 + 306366737904997641257*v^15 - 250480958285495701010*v^14 + 2851990164585936601396*v^13 - 2109318019606330220279*v^12 + 13904307095962740749570*v^11 - 6607219135383598552839*v^10 + 45765713563281570246419*v^9 - 16831417459106523645762*v^8 + 88947253328484198414718*v^7 - 7012460280551864343022*v^6 + 107551104584314685713489*v^5 - 2841251615569349534355*v^4 + 67802721698753654926485*v^3 + 21677543798641963796127*v^2 + 15372324661787992788273*v + 242046613911699229752) / 1710098042653444115106 $$\beta_{9}$$ $$=$$ $$( 11\!\cdots\!49 \nu^{17} + \cdots - 33\!\cdots\!94 ) / 88\!\cdots\!18$$ (11681428364458569449*v^17 - 21867240932612159655*v^16 + 174589474719244157658*v^15 - 271844369962795687150*v^14 + 1641956322991459079937*v^13 - 2407288366239823166128*v^12 + 8391154407174482712127*v^11 - 9704123369814297737173*v^10 + 27376753003324908469242*v^9 - 29181938560611704855462*v^8 + 54927909738953059661632*v^7 - 42371058619499267007075*v^6 + 60108342007547596981927*v^5 - 47525192267321901299213*v^4 + 38492306448825703423261*v^3 - 15689910362638511166699*v^2 - 2029531143269283926406*v - 3392657704425493602294) / 885369908607811492218 $$\beta_{10}$$ $$=$$ $$( 33\!\cdots\!75 \nu^{17} + \cdots + 12\!\cdots\!04 ) / 15\!\cdots\!86$$ (33397098538601316475*v^17 - 24160528347341335165*v^16 + 452127140353618105178*v^15 - 232421462059278982826*v^14 + 4152167316143268330835*v^13 - 1809571680721452834218*v^12 + 19336674859005842487281*v^11 - 2891427597133754789631*v^10 + 62204533593714563346026*v^9 - 2198583002162166572298*v^8 + 112305451366520264187574*v^7 + 36154042108859877930491*v^6 + 129943471320397013388469*v^5 + 46786382087956761627591*v^4 + 64369018576615419843909*v^3 + 63937376058976528058283*v^2 + 10414322161832589202296*v + 1291033462766356179504) / 1504062133900017113286 $$\beta_{11}$$ $$=$$ $$( 92\!\cdots\!95 \nu^{17} + \cdots - 21\!\cdots\!58 ) / 41\!\cdots\!46$$ (924861724186685684895*v^17 - 1724732292195874890889*v^16 + 13823962504282002754980*v^15 - 21492965026864432590982*v^14 + 130156806314502704972103*v^13 - 190461720831299114379224*v^12 + 665749131061088908667293*v^11 - 770368000231214053006323*v^10 + 2177982488665979582752174*v^9 - 2319035848316366153521406*v^8 + 4363983708434730139033024*v^7 - 3384331324143808638088861*v^6 + 4777287022949591880341385*v^5 - 3779084997345306496005015*v^4 + 2877637123302724644049985*v^3 - 1247461183989001131169761*v^2 - 161361704114957112287166*v - 214626658569266929367658) / 41612385704567140134246 $$\beta_{12}$$ $$=$$ $$( 40\!\cdots\!23 \nu^{17} + \cdots + 27\!\cdots\!00 ) / 12\!\cdots\!38$$ (4073315540396521218623*v^17 - 3679222500942063483011*v^16 + 56434986241640930690848*v^15 - 39092105397479000590870*v^14 + 522162387317951985109631*v^13 - 321175997053173398108428*v^12 + 2496861354762213697362751*v^11 - 857028275675199219661509*v^10 + 8128698261862766760525748*v^9 - 1900011920106720368953404*v^8 + 15316094811297304254058934*v^7 + 1206691198337986153609327*v^6 + 18121914623219546405185583*v^5 + 2234151380814643311101469*v^4 + 10501211223456609786252615*v^3 + 5820383693824426193138799*v^2 + 2154492637559186682631266*v + 271893469514165709474600) / 124837157113701420402738 $$\beta_{13}$$ $$=$$ $$( 22\!\cdots\!44 \nu^{17} + \cdots + 24\!\cdots\!23 ) / 57\!\cdots\!02$$ (22173993437809143944*v^17 - 23023825054181858493*v^16 + 312372369935765092700*v^15 - 257391580732610783001*v^14 + 2907290263757090588854*v^13 - 2169031005049879160045*v^12 + 14167330679620629222039*v^11 - 6815819929809383141177*v^10 + 46596993074000081505706*v^9 - 17440075426190593956251*v^8 + 90454948615479793270136*v^7 - 7648568306981104964922*v^6 + 109129211106654449195595*v^5 - 3872320453958773326550*v^4 + 68574174316272609649002*v^3 + 21885879171426154425378*v^2 + 15325435283881656996348*v + 244084125069749398923) / 570032680884481371702 $$\beta_{14}$$ $$=$$ $$( - 49\!\cdots\!08 \nu^{17} + \cdots - 42\!\cdots\!13 ) / 12\!\cdots\!38$$ (-4985555441731225078508*v^17 + 6120183932291608929275*v^16 - 70771659915487459698574*v^15 + 70324180444150066315903*v^14 - 657907674735941899803584*v^13 + 600657847527839757161653*v^12 - 3213957617855831207829937*v^11 + 2039488136369657771377389*v^10 - 10435070665306860655150096*v^9 + 5502253865085385578924957*v^8 - 19980050753970440166071690*v^7 + 4349101908425570462025650*v^6 - 22617623647661003486390243*v^5 + 3627965488613830385743014*v^4 - 13085094898142892689421336*v^3 - 3980435500010768429751288*v^2 - 1016329710923097757141194*v - 42913736168582456720613) / 124837157113701420402738 $$\beta_{15}$$ $$=$$ $$( 22\!\cdots\!36 \nu^{17} + \cdots + 16\!\cdots\!02 ) / 41\!\cdots\!46$$ (2264901108529346509536*v^17 - 2091014811290125176631*v^16 + 31512882540195856497119*v^15 - 22512130874785873228434*v^14 + 292146217289369828415028*v^13 - 186281883514831296285393*v^12 + 1404751146286636451422110*v^11 - 520750327376800330790761*v^10 + 4594232807431268856938645*v^9 - 1211977877833487564237804*v^8 + 8745054442954062724689866*v^7 + 328323039842707036570862*v^6 + 10455556768359298632610811*v^5 + 849528453915768870722443*v^4 + 6202491041404248737557531*v^3 + 3042267389223090898353339*v^2 + 1312764152710243641822591*v + 166005699226131168275502) / 41612385704567140134246 $$\beta_{16}$$ $$=$$ $$( 25\!\cdots\!33 \nu^{17} + \cdots + 18\!\cdots\!08 ) / 41\!\cdots\!46$$ (2547901320017570612833*v^17 - 2447234122378027536938*v^16 + 35470310595171451848877*v^15 - 26535824100327563513162*v^14 + 328815144152539939884875*v^13 - 220468525321391704614611*v^12 + 1581480219245692711511903*v^11 - 632784343384805919115528*v^10 + 5167263792184203151218295*v^9 - 1508491521014935952253902*v^8 + 9835816354092401184224970*v^7 + 159986641476033882406135*v^6 + 11742134977406751780593436*v^5 + 761019841743451503787240*v^4 + 7010393375221362953370952*v^3 + 3494426712165931906203942*v^2 + 1499672653113340467468537*v + 189766076725908269742108) / 41612385704567140134246 $$\beta_{17}$$ $$=$$ $$( - 35\!\cdots\!91 \nu^{17} + \cdots - 25\!\cdots\!96 ) / 41\!\cdots\!46$$ (-3563111243528487285091*v^17 + 3460014847170294660404*v^16 - 49578044602733796950131*v^15 + 37431192424478046445688*v^14 - 459156828396083537536091*v^13 + 310684272142067953922405*v^12 - 2204666526253152780522125*v^11 + 886347849111010276466074*v^10 - 7178772932338066158792145*v^9 + 2098500308970985606687772*v^8 - 13599872259842720615078082*v^7 - 330607799793726613837279*v^6 - 16070029311198980313308610*v^5 - 1219800526605753753260506*v^4 - 9568418077658791679641870*v^3 - 5106412495677884343226632*v^2 - 2030554791382026982399755*v - 256808447330875662135396) / 41612385704567140134246
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{13} - 3\beta_{8} + \beta_{3} - 3$$ b13 - 3*b8 + b3 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 5\beta_{2}$$ b11 - b9 - b7 + b6 - b5 - b3 + 5*b2 $$\nu^{4}$$ $$=$$ $$\beta_{16} + \beta_{14} - 7\beta_{13} + 2\beta_{12} - \beta_{10} + \beta_{9} + 15\beta_{8} - \beta_{6} - 2\beta_{4} + \beta_1$$ b16 + b14 - 7*b13 + 2*b12 - b10 + b9 + 15*b8 - b6 - 2*b4 + b1 $$\nu^{5}$$ $$=$$ $$- 9 \beta_{17} - 9 \beta_{16} - 9 \beta_{15} + 10 \beta_{14} + 9 \beta_{13} + 11 \beta_{10} - 2 \beta_{8} + 10 \beta_{7} + 9 \beta_{3} - 29 \beta_{2} - 29 \beta _1 - 2$$ -9*b17 - 9*b16 - 9*b15 + 10*b14 + 9*b13 + 11*b10 - 2*b8 + 10*b7 + 9*b3 - 29*b2 - 29*b1 - 2 $$\nu^{6}$$ $$=$$ $$\beta_{11} - 10\beta_{9} + 10\beta_{7} + 10\beta_{6} + \beta_{5} + 23\beta_{4} - 49\beta_{3} + 12\beta_{2} + 90$$ b11 - 10*b9 + 10*b7 + 10*b6 + b5 + 23*b4 - 49*b3 + 12*b2 + 90 $$\nu^{7}$$ $$=$$ $$68 \beta_{17} + 67 \beta_{16} + 70 \beta_{15} - 81 \beta_{14} - 71 \beta_{13} - 68 \beta_{11} - 92 \beta_{10} + 67 \beta_{9} + 25 \beta_{8} - 92 \beta_{6} + 70 \beta_{5} + 184 \beta_1$$ 68*b17 + 67*b16 + 70*b15 - 81*b14 - 71*b13 - 68*b11 - 92*b10 + 67*b9 + 25*b8 - 92*b6 + 70*b5 + 184*b1 $$\nu^{8}$$ $$=$$ $$- 14 \beta_{17} - 84 \beta_{16} + 13 \beta_{15} - 78 \beta_{14} + 347 \beta_{13} - 198 \beta_{12} + 82 \beta_{10} - 590 \beta_{8} - 78 \beta_{7} + 347 \beta_{3} - 110 \beta_{2} - 110 \beta _1 - 590$$ -14*b17 - 84*b16 + 13*b15 - 78*b14 + 347*b13 - 198*b12 + 82*b10 - 590*b8 - 78*b7 + 347*b3 - 110*b2 - 110*b1 - 590 $$\nu^{9}$$ $$=$$ $$493 \beta_{11} - 480 \beta_{9} - 609 \beta_{7} + 705 \beta_{6} - 522 \beta_{5} + 2 \beta_{4} - 539 \beta_{3} + 1232 \beta_{2} + 236$$ 493*b11 - 480*b9 - 609*b7 + 705*b6 - 522*b5 + 2*b4 - 539*b3 + 1232*b2 + 236 $$\nu^{10}$$ $$=$$ $$142 \beta_{17} + 664 \beta_{16} - 112 \beta_{15} + 561 \beta_{14} - 2476 \beta_{13} + 1539 \beta_{12} - 142 \beta_{11} - 637 \beta_{10} + 664 \beta_{9} + 4037 \beta_{8} - 637 \beta_{6} - 112 \beta_{5} - 1539 \beta_{4} + \cdots + 915 \beta_1$$ 142*b17 + 664*b16 - 112*b15 + 561*b14 - 2476*b13 + 1539*b12 - 142*b11 - 637*b10 + 664*b9 + 4037*b8 - 637*b6 - 112*b5 - 1539*b4 + 915*b1 $$\nu^{11}$$ $$=$$ $$- 3532 \beta_{17} - 3420 \beta_{16} - 3813 \beta_{15} + 4434 \beta_{14} + 4028 \beta_{13} - 49 \beta_{12} + 5213 \beta_{10} - 2015 \beta_{8} + 4434 \beta_{7} + 4028 \beta_{3} - 8506 \beta_{2} - 8506 \beta _1 - 2015$$ -3532*b17 - 3420*b16 - 3813*b15 + 4434*b14 + 4028*b13 - 49*b12 + 5213*b10 - 2015*b8 + 4434*b7 + 4028*b3 - 8506*b2 - 8506*b1 - 2015 $$\nu^{12}$$ $$=$$ $$1275 \beta_{11} - 5088 \beta_{9} + 3889 \beta_{7} + 4856 \beta_{6} + 799 \beta_{5} + 11440 \beta_{4} - 17747 \beta_{3} + 7269 \beta_{2} + 28213$$ 1275*b11 - 5088*b9 + 3889*b7 + 4856*b6 + 799*b5 + 11440*b4 - 17747*b3 + 7269*b2 + 28213 $$\nu^{13}$$ $$=$$ $$25217 \beta_{17} + 24418 \beta_{16} + 27523 \beta_{15} - 31801 \beta_{14} - 29872 \beta_{13} + 735 \beta_{12} - 25217 \beta_{11} - 37932 \beta_{10} + 24418 \beta_{9} + 16411 \beta_{8} - 37932 \beta_{6} + \cdots + 59737 \beta_1$$ 25217*b17 + 24418*b16 + 27523*b15 - 31801*b14 - 29872*b13 + 735*b12 - 25217*b11 - 37932*b10 + 24418*b9 + 16411*b8 - 37932*b6 + 27523*b5 - 735*b4 + 59737*b1 $$\nu^{14}$$ $$=$$ $$- 10786 \beta_{17} - 38309 \beta_{16} + 5034 \beta_{15} - 26411 \beta_{14} + 127541 \beta_{13} - 83247 \beta_{12} + 36738 \beta_{10} - 199309 \beta_{8} - 26411 \beta_{7} + 127541 \beta_{3} + \cdots - 199309$$ -10786*b17 - 38309*b16 + 5034*b15 - 26411*b14 + 127541*b13 - 83247*b12 + 36738*b10 - 199309*b8 - 26411*b7 + 127541*b3 - 56334*b2 - 56334*b1 - 199309 $$\nu^{15}$$ $$=$$ $$179904 \beta_{11} - 174870 \beta_{9} - 226413 \beta_{7} + 273937 \beta_{6} - 197295 \beta_{5} + 8756 \beta_{4} - 220613 \beta_{3} + 423507 \beta_{2} + 130234$$ 179904*b11 - 174870*b9 - 226413*b7 + 273937*b6 - 197295*b5 + 8756*b4 - 220613*b3 + 423507*b2 + 130234 $$\nu^{16}$$ $$=$$ $$88233 \beta_{17} + 285528 \beta_{16} - 28225 \beta_{15} + 176948 \beta_{14} - 918057 \beta_{13} + 599417 \beta_{12} - 88233 \beta_{11} - 276893 \beta_{10} + 285528 \beta_{9} + 1416097 \beta_{8} + \cdots + 430324 \beta_1$$ 88233*b17 + 285528*b16 - 28225*b15 + 176948*b14 - 918057*b13 + 599417*b12 - 88233*b11 - 276893*b10 + 285528*b9 + 1416097*b8 - 276893*b6 - 28225*b5 - 599417*b4 + 430324*b1 $$\nu^{17}$$ $$=$$ $$- 1283665 \beta_{17} - 1255440 \beta_{16} - 1408758 \beta_{15} + 1606189 \beta_{14} + 1625274 \beta_{13} - 91310 \beta_{12} + 1971315 \beta_{10} - 1017156 \beta_{8} + 1606189 \beta_{7} + \cdots - 1017156$$ -1283665*b17 - 1255440*b16 - 1408758*b15 + 1606189*b14 + 1625274*b13 - 91310*b12 + 1971315*b10 - 1017156*b8 + 1606189*b7 + 1625274*b3 - 3018402*b2 - 3018402*b1 - 1017156

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/209\mathbb{Z}\right)^\times$$.

 $$n$$ $$78$$ $$134$$ $$\chi(n)$$ $$-1 - \beta_{8}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1
 −1.34782 − 2.33450i −0.918695 − 1.59123i −0.615930 − 1.06682i −0.198213 − 0.343315i −0.0694302 − 0.120257i 0.527190 + 0.913120i 0.797832 + 1.38189i 1.00688 + 1.74398i 1.31818 + 2.28316i −1.34782 + 2.33450i −0.918695 + 1.59123i −0.615930 + 1.06682i −0.198213 + 0.343315i −0.0694302 + 0.120257i 0.527190 − 0.913120i 0.797832 − 1.38189i 1.00688 − 1.74398i 1.31818 − 2.28316i
−1.34782 2.33450i 0.0468836 + 0.0812047i −2.63325 + 4.56093i 1.12231 + 1.94390i 0.126382 0.218899i 0.215810 8.80535 1.49560 2.59046i 3.02535 5.24006i
45.2 −0.918695 1.59123i −1.56355 2.70815i −0.688002 + 1.19165i 1.63491 + 2.83176i −2.87286 + 4.97594i −3.63195 −1.14653 −3.38940 + 5.87061i 3.00398 5.20304i
45.3 −0.615930 1.06682i 1.01385 + 1.75603i 0.241261 0.417876i −1.31548 2.27848i 1.24892 2.16319i 3.75846 −3.05812 −0.555768 + 0.962618i −1.62049 + 2.80677i
45.4 −0.198213 0.343315i −0.634005 1.09813i 0.921423 1.59595i −0.220960 0.382715i −0.251336 + 0.435327i −0.667657 −1.52341 0.696076 1.20564i −0.0875945 + 0.151718i
45.5 −0.0694302 0.120257i 0.748393 + 1.29625i 0.990359 1.71535i 1.75825 + 3.04538i 0.103922 0.179998i −1.46902 −0.552764 0.379817 0.657862i 0.244151 0.422882i
45.6 0.527190 + 0.913120i 1.63062 + 2.82432i 0.444141 0.769275i −0.650102 1.12601i −1.71929 + 2.97791i −1.25919 3.04535 −3.81785 + 6.61271i 0.685455 1.18724i
45.7 0.797832 + 1.38189i −0.583506 1.01066i −0.273072 + 0.472974i 0.206639 + 0.357909i 0.931080 1.61268i 2.20560 2.31987 0.819041 1.41862i −0.329727 + 0.571103i
45.8 1.00688 + 1.74398i −1.29029 2.23484i −1.02763 + 1.77991i −1.77613 3.07635i 2.59834 4.50045i −4.09817 −0.111292 −1.82967 + 3.16909i 3.57672 6.19505i
45.9 1.31818 + 2.28316i 0.631607 + 1.09397i −2.47522 + 4.28721i −0.759439 1.31539i −1.66515 + 2.88412i 1.94612 −7.77846 0.702146 1.21615i 2.00216 3.46785i
144.1 −1.34782 + 2.33450i 0.0468836 0.0812047i −2.63325 4.56093i 1.12231 1.94390i 0.126382 + 0.218899i 0.215810 8.80535 1.49560 + 2.59046i 3.02535 + 5.24006i
144.2 −0.918695 + 1.59123i −1.56355 + 2.70815i −0.688002 1.19165i 1.63491 2.83176i −2.87286 4.97594i −3.63195 −1.14653 −3.38940 5.87061i 3.00398 + 5.20304i
144.3 −0.615930 + 1.06682i 1.01385 1.75603i 0.241261 + 0.417876i −1.31548 + 2.27848i 1.24892 + 2.16319i 3.75846 −3.05812 −0.555768 0.962618i −1.62049 2.80677i
144.4 −0.198213 + 0.343315i −0.634005 + 1.09813i 0.921423 + 1.59595i −0.220960 + 0.382715i −0.251336 0.435327i −0.667657 −1.52341 0.696076 + 1.20564i −0.0875945 0.151718i
144.5 −0.0694302 + 0.120257i 0.748393 1.29625i 0.990359 + 1.71535i 1.75825 3.04538i 0.103922 + 0.179998i −1.46902 −0.552764 0.379817 + 0.657862i 0.244151 + 0.422882i
144.6 0.527190 0.913120i 1.63062 2.82432i 0.444141 + 0.769275i −0.650102 + 1.12601i −1.71929 2.97791i −1.25919 3.04535 −3.81785 6.61271i 0.685455 + 1.18724i
144.7 0.797832 1.38189i −0.583506 + 1.01066i −0.273072 0.472974i 0.206639 0.357909i 0.931080 + 1.61268i 2.20560 2.31987 0.819041 + 1.41862i −0.329727 0.571103i
144.8 1.00688 1.74398i −1.29029 + 2.23484i −1.02763 1.77991i −1.77613 + 3.07635i 2.59834 + 4.50045i −4.09817 −0.111292 −1.82967 3.16909i 3.57672 + 6.19505i
144.9 1.31818 2.28316i 0.631607 1.09397i −2.47522 4.28721i −0.759439 + 1.31539i −1.66515 2.88412i 1.94612 −7.77846 0.702146 + 1.21615i 2.00216 + 3.46785i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 45.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.e.b 18
19.c even 3 1 inner 209.2.e.b 18
19.c even 3 1 3971.2.a.k 9
19.d odd 6 1 3971.2.a.l 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.e.b 18 1.a even 1 1 trivial
209.2.e.b 18 19.c even 3 1 inner
3971.2.a.k 9 19.c even 3 1
3971.2.a.l 9 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{18} - T_{2}^{17} + 14 T_{2}^{16} - 11 T_{2}^{15} + 130 T_{2}^{14} - 92 T_{2}^{13} + 629 T_{2}^{12} - 276 T_{2}^{11} + 2060 T_{2}^{10} - 681 T_{2}^{9} + 3955 T_{2}^{8} - 115 T_{2}^{7} + 4735 T_{2}^{6} + 105 T_{2}^{5} + 2904 T_{2}^{4} + \cdots + 9$$ acting on $$S_{2}^{\mathrm{new}}(209, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18} - T^{17} + 14 T^{16} - 11 T^{15} + \cdots + 9$$
$3$ $$T^{18} + 19 T^{16} - 2 T^{15} + 251 T^{14} + \cdots + 196$$
$5$ $$T^{18} + 26 T^{16} + 14 T^{15} + \cdots + 7569$$
$7$ $$(T^{9} + 3 T^{8} - 24 T^{7} - 64 T^{6} + \cdots + 64)^{2}$$
$11$ $$(T + 1)^{18}$$
$13$ $$T^{18} - 11 T^{17} + 119 T^{16} + \cdots + 23020804$$
$17$ $$T^{18} - 6 T^{17} + 78 T^{16} + \cdots + 13571856$$
$19$ $$T^{18} - 4 T^{17} + \cdots + 322687697779$$
$23$ $$T^{18} + T^{17} + 61 T^{16} + 168 T^{15} + \cdots + 72900$$
$29$ $$T^{18} - 13 T^{17} + \cdots + 463344321636$$
$31$ $$(T^{9} + 16 T^{8} - 75 T^{7} + \cdots - 331264)^{2}$$
$37$ $$(T^{9} + 18 T^{8} - 78 T^{7} + \cdots - 1086460)^{2}$$
$41$ $$T^{18} + \cdots + 127125805400064$$
$43$ $$T^{18} - T^{17} + 143 T^{16} + \cdots + 41783296$$
$47$ $$T^{18} + 14 T^{17} + 230 T^{16} + \cdots + 101124$$
$53$ $$T^{18} + 4 T^{17} + 231 T^{16} + \cdots + 267813225$$
$59$ $$T^{18} - 31 T^{17} + \cdots + 76585986800964$$
$61$ $$T^{18} - 3 T^{17} + \cdots + 16374647833600$$
$67$ $$T^{18} + 2 T^{17} + \cdots + 44\!\cdots\!00$$
$71$ $$T^{18} + 12 T^{17} + \cdots + 10122094962576$$
$73$ $$T^{18} + 26 T^{17} + \cdots + 120629793124$$
$79$ $$T^{18} - 31 T^{17} + \cdots + 51046624225$$
$83$ $$(T^{9} - 18 T^{8} - 155 T^{7} + \cdots - 3773520)^{2}$$
$89$ $$T^{18} + 11 T^{17} + \cdots + 3020601600$$
$97$ $$T^{18} - 16 T^{17} + \cdots + 15\!\cdots\!89$$