# Properties

 Label 209.2.e.a 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: $$x^{18} - x^{17} + 14 x^{16} - 15 x^{15} + 132 x^{14} - 132 x^{13} + 671 x^{12} - 520 x^{11} + 2312 x^{10} - 1413 x^{9} + 4605 x^{8} - 985 x^{7} + 5289 x^{6} - 427 x^{5} + 4276 x^{4} + \cdots + 81$$ x^18 - x^17 + 14*x^16 - 15*x^15 + 132*x^14 - 132*x^13 + 671*x^12 - 520*x^11 + 2312*x^10 - 1413*x^9 + 4605*x^8 - 985*x^7 + 5289*x^6 - 427*x^5 + 4276*x^4 + 426*x^3 + 1539*x^2 - 297*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{13} - \beta_{9} - \beta_{4} - 1) q^{4} + ( - \beta_{17} + \beta_{10}) q^{5} + (\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{6} + (\beta_{8} + \beta_{7} - \beta_{6} - 1) q^{7} + ( - \beta_{16} - \beta_{15} + \beta_{12} - \beta_{11} + \beta_{6} - \beta_{4} + \beta_{3}) q^{8} + ( - \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{8} + 2 \beta_{6} - \beta_{4} - \beta_{2} + \cdots + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 - b7 * q^3 + (b13 - b9 - b4 - 1) * q^4 + (-b17 + b10) * q^5 + (b15 + b14 - b13 - b12 - b6 - b5 + b4 - b3 + b1) * q^6 + (b8 + b7 - b6 - 1) * q^7 + (-b16 - b15 + b12 - b11 + b6 - b4 + b3) * q^8 + (-b16 - b15 - b14 + b13 + b12 - b8 + 2*b6 - b4 - b2 + 1) * q^9 $$q - \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{13} - \beta_{9} - \beta_{4} - 1) q^{4} + ( - \beta_{17} + \beta_{10}) q^{5} + (\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{6} + (\beta_{8} + \beta_{7} - \beta_{6} - 1) q^{7} + ( - \beta_{16} - \beta_{15} + \beta_{12} - \beta_{11} + \beta_{6} - \beta_{4} + \beta_{3}) q^{8} + ( - \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{8} + 2 \beta_{6} - \beta_{4} - \beta_{2} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{8} + 2 \beta_{6} - \beta_{4} - \beta_{2} + \cdots + 1) q^{99}+O(q^{100})$$ q - b1 * q^2 - b7 * q^3 + (b13 - b9 - b4 - 1) * q^4 + (-b17 + b10) * q^5 + (b15 + b14 - b13 - b12 - b6 - b5 + b4 - b3 + b1) * q^6 + (b8 + b7 - b6 - 1) * q^7 + (-b16 - b15 + b12 - b11 + b6 - b4 + b3) * q^8 + (-b16 - b15 - b14 + b13 + b12 - b8 + 2*b6 - b4 - b2 + 1) * q^9 + (b14 + b13 - b9 - b8 - b6 - b4 - 1) * q^10 + q^11 + (b11 - b10 + b8 + b7 + b6 - b5 + 2*b4 - b3 + 1) * q^12 + (b17 - b14 + b13 + b6 - b4 + b3 - b1) * q^13 + (b17 + b16 + b13 + b12 - b10 + b9 + b5) * q^14 + (-b16 + b6 - b5 - 2*b3 - b2 + 2*b1 + 1) * q^15 + (-b17 - b16 + b14 - 2*b13 - b12 + b10 - b9 - 2*b7 - b5 + 2*b1) * q^16 + (b17 + b16 - b14 + b12 - b10 + b9 + b7 + b5 - b1) * q^17 + (b12 - 2*b11 - b8 - b7 - b6 + b5 + b3) * q^18 + (-b15 - b14 - b11 + b9 + b6 + b4 - b2 + 1) * q^19 + (-2*b16 - 2*b15 + b12 - b11 + b10 + 3*b6 - 2*b4 + 4*b3 - 1) * q^20 + (-b15 + 3*b9 + b6 - b5 + 1) * q^21 - b1 * q^22 + (-b17 + b15 + 2*b14 - 2*b13 - 2*b6 + 2*b4 - b3 + b2 + b1) * q^23 + (b17 + 2*b16 + b15 - 2*b14 + 3*b13 + b12 + b11 - b10 - b9 + b7 - b6 + 2*b5 + b2 - 4*b1 - 1) * q^24 + (2*b17 + b16 - b15 - 2*b14 + b13 - b8 + b6 + b5 - b4 + 2*b3 - 2*b1 - 1) * q^25 + (-b16 - b15 + b12 - b11 + b10 + b8 + b7 + b6 - b4 + 2*b3 - 1) * q^26 + (b16 + b15 - 2*b12 + b11 - b10 + b8 + b7 - 2*b6 + b5 - b4 - 2*b3 - 1) * q^27 + (b16 + b9 - b6 + b5 + b2) * q^28 + (-b17 + b15 + b14 - 2*b13 - 2*b12 - 2*b9 - b6 - 2*b5 + 2*b4 - b3 - b2 + b1 - 2) * q^29 + (b16 + b15 + b12 - b11 - b10 + 2*b8 + 2*b7 - b6 + 4*b4 - b3 + 4) * q^30 + (b16 + b15 - b12 + b11 - b10 - 2*b6 + b4 - b3) * q^31 + (-b17 - b16 + 2*b15 + 3*b14 - 3*b13 - b12 + b9 + 2*b8 - 2*b6 - 2*b5 + 3*b4 - 2*b3 + 2*b1 + 2) * q^32 - b7 * q^33 + (b17 + b16 - b15 - 2*b14 + b13 + 2*b9 - 2*b8 + b6 + b5 - b4 - b3 + b1 + 1) * q^34 + (b16 + b15 + b14 - b12 + 2*b11 - 2*b9 - b7 - b6 + 2*b2 + b1 - 1) * q^35 + (-b17 - 2*b16 + b15 + 2*b14 - 4*b13 - 2*b12 + b10 + b9 + b7 - b6 - b5 + 2*b1 - 1) * q^36 + (b16 + b15 + b11 - b10 - b8 - b7 - b6 - b5 + b4 - 2*b3 + 2) * q^37 + (-b17 + b16 + 2*b15 + b14 - b13 - b12 + b11 + b10 - 3*b9 - b7 - 3*b6 + b5 + 2*b4 + 2*b2 - 2*b1 - 2) * q^38 + (b16 + b15 - b12 + b11 - b10 - b6 + 2*b4 - 1) * q^39 + (-3*b17 - 2*b16 + 3*b14 - 5*b13 - b12 - b11 + 3*b10 + 3*b9 - 2*b7 - b5 - b2 + 4*b1) * q^40 + (-b15 + b14 - b13 + 2*b9 + b6 - b5 + b1 + 1) * q^41 + (-b15 - b14 + b12 + b6 + b5 - b3 + b1) * q^42 + (-b17 - b14 + b12 - b11 + b10 - b9 + b5 - b2 - b1) * q^43 + (b13 - b9 - b4 - 1) * q^44 + (-b16 - b15 - b12 + 2*b10 + 2*b8 + 2*b7 + b5 - 3*b4 + 3*b3 - 3) * q^45 + (b16 + b15 - 2*b12 + 3*b11 - b10 - b5 - 4*b3 + 1) * q^46 + (-b17 + b16 - b14 + b9 - 2*b8 + b5 + b2) * q^47 + (b17 - b16 - 2*b15 - 2*b14 + 5*b13 + 3*b12 - 6*b9 - 2*b8 + 3*b6 + 2*b5 - 5*b4 + 4*b3 - 4*b1 - 5) * q^48 + (b16 + b15 - 3*b12 + 3*b11 + b10 - 2*b8 - 2*b7 - 2*b4 + 2*b3 + 1) * q^49 + (-2*b16 - 2*b15 - b11 + 2*b10 - b8 - b7 + 2*b6 + b5 - 2*b4 + 4*b3 - 1) * q^50 + (b17 + 2*b16 + b13 - b12 + 3*b9 + b8 - 2*b6 + b5 - b4 - b3 + b2 + b1 + 1) * q^51 + (b17 - b14 - b13 - b10 + 3*b9 - b7) * q^52 + (b17 + b15 + 2*b13 - 3*b9 + b8 - 2*b4 - b3 + b2 + b1 - 3) * q^53 + (-b16 - b15 - 2*b14 + 3*b13 + b12 - 2*b11 - 5*b9 - 2*b7 + b6 - 2*b2 + b1 + 1) * q^54 + (-b17 + b10) * q^55 + (b16 + b15 - b12 + b11 - b10 - 2*b8 - 2*b7 + b6) * q^56 + (-b17 - 3*b16 - 2*b15 - b14 - b13 + b12 + 2*b10 - b9 - b8 - 2*b7 + 5*b6 - b5 - 2*b4 + b3 - b2 + 1) * q^57 + (b16 + b15 + b11 - 2*b10 + 4*b8 + 4*b7 - b6 - b5 + 4*b4 - 2) * q^58 + (-b16 + b15 - b13 - b12 + 2*b9 - 2*b7 - b6 - 2*b1 - 1) * q^59 + (b17 + 4*b16 + 3*b15 - 2*b14 + 4*b13 + b12 + 3*b11 - b10 - b9 + 3*b7 - 3*b6 + 4*b5 + 3*b2 - 8*b1 - 3) * q^60 + (b17 + b16 - 2*b15 - 3*b14 + b12 + b8 + 2*b6 + 2*b5 + 2*b3 - 2*b1 - 1) * q^61 + (2*b17 + b16 - b14 + 3*b13 + b12 - 2*b10 + b9 + b7 + b5 - 2*b1) * q^62 + (b17 - 2*b16 + 2*b14 - b13 + 2*b12 - 3*b9 - 2*b8 + b4 + b3 - b1 - 1) * q^63 + (2*b16 + 2*b15 - b12 + 3*b11 - b10 + b8 + b7 - 2*b5 + 2*b4 - 5*b3 + 2) * q^64 + (-2*b16 - 2*b15 + 3*b12 - 2*b11 - b10 + b8 + b7 + 2*b6 - b5 - 2*b4 - b3 + 4) * q^65 + (b15 + b14 - b13 - b12 - b6 - b5 + b4 - b3 + b1) * q^66 + (b16 + b15 - 2*b14 - 2*b13 - b12 - 2*b9 + b6 + 2*b4 + 3*b3 + b2 - 3*b1 - 3) * q^67 + (-b16 - b15 - b11 + b5 - b4 + b3 + 5) * q^68 + (b16 + b15 + b12 - 3*b11 + b10 - b8 - b7 - 4*b6 + 2*b5 - b4 + b3) * q^69 + (b17 - b16 - 2*b15 - 2*b14 + 2*b13 + b12 + 5*b9 - b8 + 3*b6 - 2*b4 + 2*b3 - 2*b2 - 2*b1 + 6) * q^70 + (b15 - 2*b13 - 2*b12 + 2*b11 - b9 - b7 - b6 - b5 + 2*b2 - 1) * q^71 + (-2*b17 + 2*b15 + 3*b14 - 4*b13 - 2*b12 + b9 + 3*b8 - 3*b6 - 2*b5 + 4*b4 - 6*b3 + 6*b1 + 1) * q^72 + (b17 + b16 + b14 + b11 - b10 - b9 + 2*b7 + b2 + 2*b1) * q^73 + (3*b17 + b16 - b15 - b14 + 3*b13 + b12 - 3*b10 + 2*b9 + 4*b7 + b6 - 2*b1 + 1) * q^74 + (b10 + b6 + 2*b4 + 3) * q^75 + (b17 - b15 + 3*b13 + b11 - 5*b9 - b8 + b6 - b5 - 3*b4 - 3*b1 - 4) * q^76 + (b8 + b7 - b6 - 1) * q^77 + (b17 + 2*b16 + b15 - 2*b14 + 3*b13 + b12 + b11 - b10 + 2*b9 + b7 - b6 + 2*b5 + b2 - 3*b1 - 1) * q^78 + (-2*b17 - 2*b16 + 3*b14 - b13 - b12 - b11 + 2*b10 + 2*b7 - b5 - b2 + 4*b1) * q^79 + (-2*b17 - b16 + 2*b15 + b14 - 6*b13 - 3*b12 + b9 + 2*b8 - 4*b5 + 6*b4 - 7*b3 - 2*b2 + 7*b1 + 2) * q^80 + (-b17 - b16 - 2*b15 + 4*b14 - 2*b12 + b11 + b10 + 4*b9 - b7 + 2*b6 - 4*b5 + b2 + 2) * q^81 + (-b17 - b15 - 2*b14 - 2*b13 + 3*b9 + 2*b6 + 2*b4 - 2*b3 - b2 + 2*b1 + 3) * q^82 + (2*b16 + 2*b15 + b12 - 2*b11 - 2*b10 + b8 + b7 - b6 + b5 + 2*b4 + 3*b3 - 4) * q^83 + (-b16 - b15 + b12 + b10 - 3*b8 - 3*b7 + 2*b6 - b5 + b4 - b3 + 3) * q^84 + (-2*b17 + 2*b14 + b13 - b12 + 4*b9 - b8 - 2*b6 - b5 - b4 - 3*b3 - b2 + 3*b1 + 4) * q^85 + (b16 + 2*b15 + 3*b14 - 2*b12 - 3*b9 - 2*b8 - 4*b6 - b5 - b3 + b2 + b1 - 4) * q^86 + (-b16 - b15 + 2*b12 - 2*b11 + b10 + b8 + b7 + 2*b6 - 2*b4 + 5*b3 + 1) * q^87 + (-b16 - b15 + b12 - b11 + b6 - b4 + b3) * q^88 + (b17 - b16 - 2*b15 + 2*b13 + 3*b12 + b8 + b6 + 2*b5 - 2*b4 + 2*b3 - 2*b1 + 1) * q^89 + (-2*b17 - 2*b16 - b15 + b14 - 2*b13 + b12 - 3*b11 + 2*b10 + 2*b9 - 4*b7 + b6 - 3*b2 + 5*b1 + 1) * q^90 + (b16 + 2*b14 - b13 + b12 - 4*b9 + b8 - 3*b6 + 2*b5 + b4 - b3 + 2*b2 + b1 - 5) * q^91 + (b16 - b14 + 5*b13 + 2*b12 - b11 - 6*b9 + 2*b7 + 2*b5 - b2) * q^92 + (-b17 - b16 - 2*b15 + b14 - 2*b13 - b11 + b10 + b9 - b7 + 2*b6 - 2*b5 - b2 + 2*b1 + 2) * q^93 + (-3*b16 - 3*b15 - b12 + b11 + 2*b10 - 3*b8 - 3*b7 + b6 - 5*b4 + 2*b3 + 2) * q^94 + (b17 + b16 - b14 + b13 + 2*b12 + b11 - 4*b10 - b9 - b7 - b6 + b5 + 2*b4 - 3*b3 - 4*b1) * q^95 + (b12 - 3*b11 - 3*b8 - 3*b7 - 3*b6 + 2*b5 - 4*b4 + 6*b3 - 7) * q^96 + (3*b17 + 2*b16 - 4*b14 + 3*b13 + 2*b12 - 3*b10 + 7*b9 + b7 + 2*b5 - 5*b1) * q^97 + (-4*b16 - 3*b15 - b14 + b13 + b12 - 5*b11 + 7*b9 + b7 + 3*b6 - 2*b5 - 5*b2 + 3*b1 + 3) * q^98 + (-b16 - b15 - b14 + b13 + b12 - b8 + 2*b6 - b4 - b2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q - q^{2} - 4 q^{3} - 9 q^{4} + 7 q^{6} - 6 q^{7} - 12 q^{8} - 11 q^{9}+O(q^{10})$$ 18 * q - q^2 - 4 * q^3 - 9 * q^4 + 7 * q^6 - 6 * q^7 - 12 * q^8 - 11 * q^9 $$18 q - q^{2} - 4 q^{3} - 9 q^{4} + 7 q^{6} - 6 q^{7} - 12 q^{8} - 11 q^{9} - 11 q^{10} + 18 q^{11} + 8 q^{12} - q^{13} - 3 q^{14} - q^{15} - 5 q^{16} + 2 q^{17} + 12 q^{18} + 6 q^{19} - 44 q^{20} - 24 q^{21} - q^{22} + 11 q^{23} + 17 q^{24} - 15 q^{25} - 20 q^{26} + 14 q^{27} + 8 q^{28} - 13 q^{29} + 104 q^{30} + 16 q^{31} + 36 q^{32} - 4 q^{33} - 4 q^{34} + 6 q^{35} - 22 q^{36} + 28 q^{37} + 20 q^{38} - 4 q^{39} - 44 q^{40} - 16 q^{41} - 9 q^{42} + 15 q^{43} - 9 q^{44} - 34 q^{45} + 6 q^{46} - 2 q^{47} - 71 q^{48} + 12 q^{49} - 38 q^{50} + 26 q^{51} - 29 q^{52} - 16 q^{53} + 49 q^{54} - 10 q^{56} - 43 q^{57} - 37 q^{59} + 17 q^{60} - 15 q^{61} + q^{62} - 24 q^{63} + 32 q^{64} + 40 q^{65} + 7 q^{66} - 14 q^{67} + 92 q^{68} + 48 q^{69} + 26 q^{70} - 8 q^{71} + 33 q^{72} + 18 q^{73} + 7 q^{74} + 50 q^{75} - 56 q^{76} - 6 q^{77} - 9 q^{78} - q^{79} + 23 q^{80} - 53 q^{81} + 13 q^{82} - 16 q^{83} - 2 q^{84} + 31 q^{85} - 19 q^{86} + 20 q^{87} - 12 q^{88} - 3 q^{89} - 25 q^{90} - 28 q^{91} + 75 q^{92} - 8 q^{93} - 26 q^{94} - 5 q^{95} - 100 q^{96} - 44 q^{97} - 44 q^{98} - 11 q^{99}+O(q^{100})$$ 18 * q - q^2 - 4 * q^3 - 9 * q^4 + 7 * q^6 - 6 * q^7 - 12 * q^8 - 11 * q^9 - 11 * q^10 + 18 * q^11 + 8 * q^12 - q^13 - 3 * q^14 - q^15 - 5 * q^16 + 2 * q^17 + 12 * q^18 + 6 * q^19 - 44 * q^20 - 24 * q^21 - q^22 + 11 * q^23 + 17 * q^24 - 15 * q^25 - 20 * q^26 + 14 * q^27 + 8 * q^28 - 13 * q^29 + 104 * q^30 + 16 * q^31 + 36 * q^32 - 4 * q^33 - 4 * q^34 + 6 * q^35 - 22 * q^36 + 28 * q^37 + 20 * q^38 - 4 * q^39 - 44 * q^40 - 16 * q^41 - 9 * q^42 + 15 * q^43 - 9 * q^44 - 34 * q^45 + 6 * q^46 - 2 * q^47 - 71 * q^48 + 12 * q^49 - 38 * q^50 + 26 * q^51 - 29 * q^52 - 16 * q^53 + 49 * q^54 - 10 * q^56 - 43 * q^57 - 37 * q^59 + 17 * q^60 - 15 * q^61 + q^62 - 24 * q^63 + 32 * q^64 + 40 * q^65 + 7 * q^66 - 14 * q^67 + 92 * q^68 + 48 * q^69 + 26 * q^70 - 8 * q^71 + 33 * q^72 + 18 * q^73 + 7 * q^74 + 50 * q^75 - 56 * q^76 - 6 * q^77 - 9 * q^78 - q^79 + 23 * q^80 - 53 * q^81 + 13 * q^82 - 16 * q^83 - 2 * q^84 + 31 * q^85 - 19 * q^86 + 20 * q^87 - 12 * q^88 - 3 * q^89 - 25 * q^90 - 28 * q^91 + 75 * q^92 - 8 * q^93 - 26 * q^94 - 5 * q^95 - 100 * q^96 - 44 * q^97 - 44 * q^98 - 11 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - x^{17} + 14 x^{16} - 15 x^{15} + 132 x^{14} - 132 x^{13} + 671 x^{12} - 520 x^{11} + 2312 x^{10} - 1413 x^{9} + 4605 x^{8} - 985 x^{7} + 5289 x^{6} - 427 x^{5} + 4276 x^{4} + \cdots + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 10\!\cdots\!36 \nu^{17} + \cdots - 22\!\cdots\!24 ) / 18\!\cdots\!18$$ (1012720386363599036*v^17 - 94739171273474206175*v^16 + 57812798323091281903*v^15 - 1113127532414004579270*v^14 + 610414574657299721166*v^13 - 9629405855418767863809*v^12 + 3224944590594444024922*v^11 - 37220328656580677900777*v^10 - 9791376836995907689577*v^9 - 100582647222980916443262*v^8 - 81578794340984662732866*v^7 - 73350285228185102426042*v^6 - 400109977871415021730635*v^5 + 4177455110319818058139*v^4 - 356552067538101245885623*v^3 - 93718041862117461040791*v^2 - 288244945968983955330417*v - 2224623642468442977024) / 18473493573585888818418 $$\beta_{3}$$ $$=$$ $$( - 21\!\cdots\!69 \nu^{17} + \cdots + 15\!\cdots\!64 ) / 61\!\cdots\!06$$ (-2128033936314699569*v^17 + 5458724537424206135*v^16 - 33495985837147298500*v^15 + 76569132350781580704*v^14 - 334018057082111013441*v^13 + 696200622227704165728*v^12 - 1888749540495685152013*v^11 + 3117939895998411333107*v^10 - 6670612517171256711232*v^9 + 9666800622044874031620*v^8 - 14687414512710319282566*v^7 + 14013733395563664168479*v^6 - 14702546900437300869921*v^5 + 12708497008881398216219*v^4 - 15427251839425111911257*v^3 + 7437918435843526749783*v^2 - 1553634095587192056960*v + 1579731844635043426764) / 6157831191195296272806 $$\beta_{4}$$ $$=$$ $$( 11\!\cdots\!22 \nu^{17} + \cdots - 61\!\cdots\!43 ) / 20\!\cdots\!02$$ (1110230200369835522*v^17 - 1234503576247168178*v^16 + 14882874435353695723*v^15 - 17705859162856890111*v^14 + 138433380878054607540*v^13 - 153612923076173913738*v^12 + 670454083038255852409*v^11 - 583532685470557102568*v^10 + 2219962890010734513541*v^9 - 1629272745327042589107*v^8 + 3972539989431228364338*v^7 - 1149125137089618283160*v^6 + 3933275506025007166752*v^5 - 2109259575914485518071*v^4 + 2781486964237862922059*v^3 - 1478807019598055231025*v^2 + 315901921849859218257*v - 6100374274914799384443) / 2052610397065098757602 $$\beta_{5}$$ $$=$$ $$( 50\!\cdots\!84 \nu^{17} + \cdots + 16\!\cdots\!00 ) / 61\!\cdots\!06$$ (5068373911969217684*v^17 + 25233339789554109763*v^16 + 49280306637406513471*v^15 + 306439993760139348540*v^14 + 375318299945568250614*v^13 + 2776889650180869168423*v^12 + 1112087995038306662506*v^11 + 12672498513256197572467*v^10 + 6559897132513592793427*v^9 + 40537137026235352292874*v^8 + 18341522438405460238146*v^7 + 66547196783284661914372*v^6 + 82977699564882746448771*v^5 + 64955336167610074656997*v^4 + 80375261349745839642467*v^3 + 73000389367222430464527*v^2 + 78149318772099043402767*v + 16649229379850399655900) / 6157831191195296272806 $$\beta_{6}$$ $$=$$ $$( - 11\!\cdots\!57 \nu^{17} + \cdots - 49\!\cdots\!02 ) / 92\!\cdots\!09$$ (-11934128132917906457*v^17 + 139486096990018545398*v^16 - 383242562731579497748*v^15 + 2005280792384086048731*v^14 - 4630486997240937876810*v^13 + 18974648343231804609399*v^12 - 35009023164887533714579*v^11 + 95846940382123340553878*v^10 - 139085048378033397368329*v^9 + 313974091801135265682612*v^8 - 370419961074177267635403*v^7 + 565754389195550354175158*v^6 - 374781973370824505109177*v^5 + 417418142702248809793241*v^4 - 197724859646014971603152*v^3 + 226881274916720108966310*v^2 - 46887400807752084023862*v - 490574411939484088902) / 9236746786792944409209 $$\beta_{7}$$ $$=$$ $$( - 90\!\cdots\!82 \nu^{17} + \cdots + 48\!\cdots\!31 ) / 61\!\cdots\!06$$ (-9008870397315943682*v^17 + 60326349462019315394*v^16 - 164992997066507653363*v^15 + 809598790542962504265*v^14 - 1725925490025956143194*v^13 + 7299208307846027647416*v^12 - 10371574594061708006503*v^11 + 32842410151236090689312*v^10 - 32008402869805471891333*v^9 + 100047118847585137600125*v^8 - 55937905810371635979630*v^7 + 143609866236965693017052*v^6 + 41241737793574240012674*v^5 + 97748006706951911735795*v^4 + 80171684354509576455205*v^3 + 80181102451896433709127*v^2 + 77378099559627947333319*v + 4844890607039941658931) / 6157831191195296272806 $$\beta_{8}$$ $$=$$ $$( - 58\!\cdots\!25 \nu^{17} + \cdots + 63\!\cdots\!60 ) / 18\!\cdots\!18$$ (-58005188972387748725*v^17 + 68683211494044809915*v^16 - 949432911649499686342*v^15 + 1234988133497495747430*v^14 - 9636173884713188483859*v^13 + 12133783689626192523222*v^12 - 57660444283870184947459*v^11 + 64451922381189926578631*v^10 - 229279292026664522386114*v^9 + 219225700168649345584866*v^8 - 579382802894050315729626*v^7 + 423626731083328603668257*v^6 - 874921046622351153657219*v^5 + 395707601544543045157223*v^4 - 663368771703028538086157*v^3 + 155792302431472194248133*v^2 - 314714655761330937068892*v + 63808345967751794451060) / 18473493573585888818418 $$\beta_{9}$$ $$=$$ $$( - 58\!\cdots\!32 \nu^{17} + \cdots - 57\!\cdots\!94 ) / 18\!\cdots\!18$$ (-58508586838334941732*v^17 + 52124485029390843025*v^16 - 802744042124416565843*v^15 + 777140845063582230480*v^14 - 7493426065607867566512*v^13 + 6721079291413879268301*v^12 - 37170659901839633404988*v^11 + 24758216534447114244601*v^10 - 125918033082235151285063*v^9 + 62660795651053502533620*v^8 - 240431640524397784581000*v^7 + 13568714497628959758322*v^6 - 267410715601262514315111*v^5 - 19124474121342882490199*v^4 - 212057226294076016197375*v^3 - 71206413511406020911603*v^2 - 67730959836666895076199*v - 5757345569361987294894) / 18473493573585888818418 $$\beta_{10}$$ $$=$$ $$( - 55\!\cdots\!54 \nu^{17} + \cdots + 15\!\cdots\!19 ) / 92\!\cdots\!09$$ (-55737307064698246154*v^17 + 274360838412596100947*v^16 - 1152727506775656562102*v^15 + 4022977683597256962663*v^14 - 12599936245760377722033*v^13 + 37865066887331757825339*v^12 - 83819153896533292383874*v^11 + 188527222537537733051651*v^10 - 319922442535558052793244*v^9 + 612004505416492011847446*v^8 - 792551334707145401943783*v^7 + 1076903496677217295875833*v^6 - 799798676846045781451014*v^5 + 813961563542444007630746*v^4 - 408376366516504229135192*v^3 + 453921680033165549846025*v^2 - 94160168097166918290780*v + 1585426082536734927219) / 9236746786792944409209 $$\beta_{11}$$ $$=$$ $$( 39\!\cdots\!59 \nu^{17} + \cdots - 24\!\cdots\!51 ) / 61\!\cdots\!06$$ (39659972089298595059*v^17 - 77667707491299205832*v^16 + 624901425950809312318*v^15 - 1165539142089018396435*v^14 + 6264399467628424671789*v^13 - 10766797531781194060851*v^12 + 35930500265503146391786*v^11 - 50572236921127499239718*v^10 + 133256204851981614539416*v^9 - 159119677983574907506071*v^8 + 306157102355249817616056*v^7 - 248717649412476889935965*v^6 + 371920686152382093942516*v^5 - 198855291233771670768443*v^4 + 261296963431200305587433*v^3 - 78855845607335530175853*v^2 + 109794442250517075423882*v - 24128925972200890525251) / 6157831191195296272806 $$\beta_{12}$$ $$=$$ $$( 44\!\cdots\!05 \nu^{17} + \cdots - 32\!\cdots\!82 ) / 61\!\cdots\!06$$ (44612766994005039605*v^17 - 53715850295638962308*v^16 + 637248602995505334133*v^15 - 811986024706142835198*v^14 + 6108105447533198828379*v^13 - 7305279192892675539435*v^12 + 32055757055636454585961*v^11 - 31320231464286462127508*v^10 + 114183865537111489172197*v^9 - 93697244579686159348884*v^8 + 242742887090915537688276*v^7 - 115395382278934096392983*v^6 + 307084411803020138604984*v^5 - 111243084716931088787300*v^4 + 256680635286468998557994*v^3 - 33459296039767573247946*v^2 + 100482043685214965958477*v - 32013993452995533447882) / 6157831191195296272806 $$\beta_{13}$$ $$=$$ $$( - 55\!\cdots\!66 \nu^{17} + \cdots - 55\!\cdots\!05 ) / 61\!\cdots\!06$$ (-55177896237225435166*v^17 + 48420974300649338491*v^16 - 758095418818355478674*v^15 + 724023267575011560147*v^14 - 7078125922973703743892*v^13 + 6260240522185357527087*v^12 - 35159297652724865847761*v^11 + 23007618478035442936897*v^10 - 119258144412202947744440*v^9 + 57772977415072374766299*v^8 - 228514020556104099487986*v^7 + 10121339086360104908842*v^6 - 255610889083187492814855*v^5 - 25452252849086339044412*v^4 - 203712765401362427431198*v^3 - 69485003379004890331872*v^2 - 66783254071117317421428*v - 5584974820520496629805) / 6157831191195296272806 $$\beta_{14}$$ $$=$$ $$( - 15\!\cdots\!87 \nu^{17} + \cdots - 62\!\cdots\!10 ) / 92\!\cdots\!09$$ (-151597674815253257387*v^17 + 235346464740573556631*v^16 - 2203657474861696855219*v^15 + 3330749048575039248327*v^14 - 21048378268096033285638*v^13 + 29440787256780631086936*v^12 - 109832844444634600972669*v^11 + 119661853925291964368183*v^10 - 369306886152178336220716*v^9 + 330481289105750461121787*v^8 - 715540179930974812503219*v^7 + 281191609044682540532828*v^6 - 602313364292029234477272*v^5 + 52037773660384728298514*v^4 - 398382764419485734342327*v^3 - 43266902840444008061910*v^2 - 34789194035420748225417*v - 6283268336700719129610) / 9236746786792944409209 $$\beta_{15}$$ $$=$$ $$( 66\!\cdots\!60 \nu^{17} + \cdots + 17\!\cdots\!10 ) / 30\!\cdots\!03$$ (66735515158792444160*v^17 - 36837512138528793011*v^16 + 869182278837324785305*v^15 - 536118077619596692854*v^14 + 7855638943712920926492*v^13 - 4207579064449408525611*v^12 + 36074390336401310419099*v^11 - 8914040030287415809376*v^10 + 114476963614579837857523*v^9 - 3144683557080471387120*v^8 + 185390814646177039565841*v^7 + 128666217911549999011591*v^6 + 176768830930485446615517*v^5 + 154788400603113053782486*v^4 + 163681118129756955484469*v^3 + 126635982857813794336482*v^2 + 48646361125673754237513*v + 1761315113868612978510) / 3078915595597648136403 $$\beta_{16}$$ $$=$$ $$( - 45\!\cdots\!69 \nu^{17} + \cdots + 43\!\cdots\!90 ) / 18\!\cdots\!18$$ (-451335543352520157469*v^17 + 664844238645778185061*v^16 - 6580922924828412380888*v^15 + 9595819120361814659856*v^14 - 63119861001179235512277*v^13 + 85404851730926743646274*v^12 - 332453947832562585313103*v^11 + 354955029925705471220359*v^10 - 1142287750155821659273802*v^9 + 1002749049195036097935768*v^8 - 2299491533413955456966724*v^7 + 980026399489690621985341*v^6 - 2260623438433432133473635*v^5 + 378553292652792385740919*v^4 - 1629354079200697153350403*v^3 - 44985658896703278976581*v^2 - 439737792795909635687730*v + 43395104194284271044690) / 18473493573585888818418 $$\beta_{17}$$ $$=$$ $$( 50\!\cdots\!80 \nu^{17} + \cdots + 23\!\cdots\!86 ) / 18\!\cdots\!18$$ (505333260053646242080*v^17 - 406425275426803369093*v^16 + 6557141719077132055829*v^15 - 5503047742410695250780*v^14 + 59188996969652460607872*v^13 - 43874966405710862755359*v^12 + 268480428504054308975054*v^11 - 110014167579409972157155*v^10 + 818909257814120245857749*v^9 - 130540965073751017884138*v^8 + 1194407876941309038329898*v^7 + 954530321415797155211480*v^6 + 697084630472215147839831*v^5 + 1249582998386145433968623*v^4 + 656000449102725300845629*v^3 + 1008542024036242284754395*v^2 - 59081009221657649667789*v + 23428063165369596254586) / 18473493573585888818418
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{13} - 3\beta_{9} - \beta_{4} - 3$$ b13 - 3*b9 - b4 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{16} + \beta_{15} - \beta_{12} + \beta_{11} - \beta_{6} + \beta_{4} - 5\beta_{3}$$ b16 + b15 - b12 + b11 - b6 + b4 - 5*b3 $$\nu^{4}$$ $$=$$ $$- \beta_{17} - \beta_{16} + \beta_{14} - 8 \beta_{13} - \beta_{12} + \beta_{10} + 13 \beta_{9} - 2 \beta_{7} - \beta_{5} + 2 \beta_1$$ -b17 - b16 + b14 - 8*b13 - b12 + b10 + 13*b9 - 2*b7 - b5 + 2*b1 $$\nu^{5}$$ $$=$$ $$\beta_{17} + \beta_{16} - 2 \beta_{15} - 3 \beta_{14} + 11 \beta_{13} + 9 \beta_{12} - 9 \beta_{9} - 2 \beta_{8} + 2 \beta_{6} + 10 \beta_{5} - 11 \beta_{4} + 30 \beta_{3} + 8 \beta_{2} - 30 \beta _1 - 10$$ b17 + b16 - 2*b15 - 3*b14 + 11*b13 + 9*b12 - 9*b9 - 2*b8 + 2*b6 + 10*b5 - 11*b4 + 30*b3 + 8*b2 - 30*b1 - 10 $$\nu^{6}$$ $$=$$ $$12 \beta_{16} + 12 \beta_{15} - \beta_{12} + 3 \beta_{11} - 11 \beta_{10} + 21 \beta_{8} + 21 \beta_{7} - 10 \beta_{6} - 2 \beta_{5} + 58 \beta_{4} - 25 \beta_{3} + 68$$ 12*b16 + 12*b15 - b12 + 3*b11 - 11*b10 + 21*b8 + 21*b7 - 10*b6 - 2*b5 + 58*b4 - 25*b3 + 68 $$\nu^{7}$$ $$=$$ $$- 14 \beta_{17} - 81 \beta_{16} - 44 \beta_{15} + 37 \beta_{14} - 99 \beta_{13} - 26 \beta_{12} - 55 \beta_{11} + 14 \beta_{10} + 69 \beta_{9} - 28 \beta_{7} + 44 \beta_{6} - 70 \beta_{5} - 55 \beta_{2} + 194 \beta _1 + 44$$ -14*b17 - 81*b16 - 44*b15 + 37*b14 - 99*b13 - 26*b12 - 55*b11 + 14*b10 + 69*b9 - 28*b7 + 44*b6 - 70*b5 - 55*b2 + 194*b1 + 44 $$\nu^{8}$$ $$=$$ $$92 \beta_{17} + 26 \beta_{16} - 94 \beta_{15} - 97 \beta_{14} + 417 \beta_{13} + 112 \beta_{12} - 387 \beta_{9} - 174 \beta_{8} + 71 \beta_{6} + 138 \beta_{5} - 417 \beta_{4} + 236 \beta_{3} + 44 \beta_{2} - 236 \beta _1 - 413$$ 92*b17 + 26*b16 - 94*b15 - 97*b14 + 417*b13 + 112*b12 - 387*b9 - 174*b8 + 71*b6 + 138*b5 - 417*b4 + 236*b3 + 44*b2 - 236*b1 - 413 $$\nu^{9}$$ $$=$$ $$523 \beta_{16} + 523 \beta_{15} - 279 \beta_{12} + 373 \beta_{11} - 141 \beta_{10} + 278 \beta_{8} + 278 \beta_{7} - 518 \beta_{6} - 94 \beta_{5} + 823 \beta_{4} - 1309 \beta_{3} + 337$$ 523*b16 + 523*b15 - 279*b12 + 373*b11 - 141*b10 + 278*b8 + 278*b7 - 518*b6 - 94*b5 + 823*b4 - 1309*b3 + 337 $$\nu^{10}$$ $$=$$ $$- 706 \beta_{17} - 1195 \beta_{16} - 206 \beta_{15} + 797 \beta_{14} - 3015 \beta_{13} - 745 \beta_{12} - 450 \beta_{11} + 706 \beta_{10} + 2425 \beta_{9} - 1337 \beta_{7} + 206 \beta_{6} - 951 \beta_{5} - 450 \beta_{2} + \cdots + 206$$ -706*b17 - 1195*b16 - 206*b15 + 797*b14 - 3015*b13 - 745*b12 - 450*b11 + 706*b10 + 2425*b9 - 1337*b7 + 206*b6 - 951*b5 - 450*b2 + 2017*b1 + 206 $$\nu^{11}$$ $$=$$ $$1247 \beta_{17} + 745 \beta_{16} - 2031 \beta_{15} - 2685 \beta_{14} + 6563 \beta_{13} + 3851 \beta_{12} - 3978 \beta_{9} - 2429 \beta_{8} + 1940 \beta_{6} + 4596 \beta_{5} - 6563 \beta_{4} + 9085 \beta_{3} + \cdots - 4723$$ 1247*b17 + 745*b16 - 2031*b15 - 2685*b14 + 6563*b13 + 3851*b12 - 3978*b9 - 2429*b8 + 1940*b6 + 4596*b5 - 6563*b4 + 9085*b3 + 2565*b2 - 9085*b1 - 4723 $$\nu^{12}$$ $$=$$ $$7706 \beta_{16} + 7706 \beta_{15} - 1967 \beta_{12} + 3998 \beta_{11} - 5250 \beta_{10} + 9976 \beta_{8} + 9976 \beta_{7} - 6268 \beta_{6} - 2031 \beta_{5} + 21960 \beta_{4} - 16427 \beta_{3} + 16162$$ 7706*b16 + 7706*b15 - 1967*b12 + 3998*b11 - 5250*b10 + 9976*b8 + 9976*b7 - 6268*b6 - 2031*b5 + 21960*b4 - 16427*b3 + 16162 $$\nu^{13}$$ $$=$$ $$- 10330 \beta_{17} - 33967 \beta_{16} - 12223 \beta_{15} + 20662 \beta_{14} - 51109 \beta_{13} - 16005 \beta_{12} - 17962 \beta_{11} + 10330 \beta_{10} + 30448 \beta_{9} - 19963 \beta_{7} + \cdots + 12223$$ -10330*b17 - 33967*b16 - 12223*b15 + 20662*b14 - 51109*b13 - 16005*b12 - 17962*b11 + 10330*b10 + 30448*b9 - 19963*b7 + 12223*b6 - 28228*b5 - 17962*b2 + 64319*b1 + 12223 $$\nu^{14}$$ $$=$$ $$38624 \beta_{17} + 16005 \beta_{16} - 43664 \beta_{15} - 49337 \beta_{14} + 160985 \beta_{13} + 60806 \beta_{12} - 111173 \beta_{9} - 73609 \beta_{8} + 33332 \beta_{6} + 76811 \beta_{5} + \cdots - 127178$$ 38624*b17 + 16005*b16 - 43664*b15 - 49337*b14 + 160985*b13 + 60806*b12 - 111173*b9 - 73609*b8 + 33332*b6 + 76811*b5 - 160985*b4 + 130140*b3 + 33147*b2 - 130140*b1 - 127178 $$\nu^{15}$$ $$=$$ $$206935 \beta_{16} + 206935 \beta_{15} - 84174 \beta_{12} + 127838 \beta_{11} - 82484 \beta_{10} + 158662 \beta_{8} + 158662 \beta_{7} - 196222 \beta_{6} - 43664 \beta_{5} + 392404 \beta_{4} + \cdots + 192501$$ 206935*b16 + 206935*b15 - 84174*b12 + 127838*b11 - 82484*b10 + 158662*b8 + 158662*b7 - 196222*b6 - 43664*b5 + 392404*b4 - 461951*b3 + 192501 $$\nu^{16}$$ $$=$$ $$- 283550 \beta_{17} - 594730 \beta_{16} - 141805 \beta_{15} + 379697 \beta_{14} - 1186209 \beta_{13} - 330164 \beta_{12} - 264566 \beta_{11} + 283550 \beta_{10} + 788264 \beta_{9} + \cdots + 141805$$ -283550*b17 - 594730*b16 - 141805*b15 + 379697*b14 - 1186209*b13 - 330164*b12 - 264566*b11 + 283550*b10 + 788264*b9 - 541475*b7 + 141805*b6 - 471969*b5 - 264566*b2 + 1013210*b1 + 141805 $$\nu^{17}$$ $$=$$ $$644263 \beta_{17} + 330164 \beta_{16} - 928802 \beta_{15} - 1162819 \beta_{14} + 2986476 \beta_{13} + 1520281 \beta_{12} - 1778402 \beta_{9} - 1236074 \beta_{8} + 832655 \beta_{6} + \cdots - 2108566$$ 644263*b17 + 330164*b16 - 928802*b15 - 1162819*b14 + 2986476*b13 + 1520281*b12 - 1778402*b9 - 1236074*b8 + 832655*b6 + 1850445*b5 - 2986476*b4 + 3352949*b3 + 921643*b2 - 3352949*b1 - 2108566

## Character values

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

 $$n$$ $$78$$ $$134$$ $$\chi(n)$$ $$-1 - \beta_{9}$$ $$1$$

## 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
 1.17624 + 2.03730i 1.01915 + 1.76522i 0.831273 + 1.43981i 0.605399 + 1.04858i 0.111589 + 0.193278i −0.424456 − 0.735179i −0.448336 − 0.776541i −1.00350 − 1.73811i −1.36735 − 2.36833i 1.17624 − 2.03730i 1.01915 − 1.76522i 0.831273 − 1.43981i 0.605399 − 1.04858i 0.111589 − 0.193278i −0.424456 + 0.735179i −0.448336 + 0.776541i −1.00350 + 1.73811i −1.36735 + 2.36833i
−1.17624 2.03730i −0.948876 1.64350i −1.76706 + 3.06064i −1.45451 2.51928i −2.23220 + 3.86629i −0.513749 3.60897 −0.300733 + 0.520885i −3.42169 + 5.92654i
45.2 −1.01915 1.76522i 1.62111 + 2.80784i −1.07732 + 1.86598i 0.953652 + 1.65177i 3.30430 5.72321i −1.44214 0.315214 −3.75599 + 6.50557i 1.94382 3.36680i
45.3 −0.831273 1.43981i −0.205287 0.355568i −0.382031 + 0.661697i 1.09220 + 1.89174i −0.341299 + 0.591148i 4.21687 −2.05481 1.41571 2.45209i 1.81583 3.14511i
45.4 −0.605399 1.04858i 0.391191 + 0.677562i 0.266984 0.462430i −0.857641 1.48548i 0.473653 0.820391i −5.17559 −3.06812 1.19394 2.06796i −1.03843 + 1.79861i
45.5 −0.111589 0.193278i −1.70486 2.95291i 0.975096 1.68892i −0.909261 1.57489i −0.380488 + 0.659025i 3.46341 −0.881596 −4.31312 + 7.47054i −0.202927 + 0.351480i
45.6 0.424456 + 0.735179i 0.287689 + 0.498292i 0.639674 1.10795i −2.13253 3.69366i −0.244223 + 0.423006i 0.190562 2.78388 1.33447 2.31137i 1.81033 3.13559i
45.7 0.448336 + 0.776541i −0.820353 1.42089i 0.597989 1.03575i 1.31548 + 2.27848i 0.735588 1.27408i −1.27368 2.86575 0.154042 0.266809i −1.17956 + 2.04305i
45.8 1.00350 + 1.73811i 0.669895 + 1.16029i −1.01402 + 1.75633i 0.304819 + 0.527963i −1.34448 + 2.32870i −2.87509 −0.0562672 0.602481 1.04353i −0.611771 + 1.05962i
45.9 1.36735 + 2.36833i −1.29050 2.23522i −2.73931 + 4.74463i 1.68779 + 2.92334i 3.52915 6.11267i 0.409409 −9.51302 −1.83080 + 3.17105i −4.61562 + 7.99448i
144.1 −1.17624 + 2.03730i −0.948876 + 1.64350i −1.76706 3.06064i −1.45451 + 2.51928i −2.23220 3.86629i −0.513749 3.60897 −0.300733 0.520885i −3.42169 5.92654i
144.2 −1.01915 + 1.76522i 1.62111 2.80784i −1.07732 1.86598i 0.953652 1.65177i 3.30430 + 5.72321i −1.44214 0.315214 −3.75599 6.50557i 1.94382 + 3.36680i
144.3 −0.831273 + 1.43981i −0.205287 + 0.355568i −0.382031 0.661697i 1.09220 1.89174i −0.341299 0.591148i 4.21687 −2.05481 1.41571 + 2.45209i 1.81583 + 3.14511i
144.4 −0.605399 + 1.04858i 0.391191 0.677562i 0.266984 + 0.462430i −0.857641 + 1.48548i 0.473653 + 0.820391i −5.17559 −3.06812 1.19394 + 2.06796i −1.03843 1.79861i
144.5 −0.111589 + 0.193278i −1.70486 + 2.95291i 0.975096 + 1.68892i −0.909261 + 1.57489i −0.380488 0.659025i 3.46341 −0.881596 −4.31312 7.47054i −0.202927 0.351480i
144.6 0.424456 0.735179i 0.287689 0.498292i 0.639674 + 1.10795i −2.13253 + 3.69366i −0.244223 0.423006i 0.190562 2.78388 1.33447 + 2.31137i 1.81033 + 3.13559i
144.7 0.448336 0.776541i −0.820353 + 1.42089i 0.597989 + 1.03575i 1.31548 2.27848i 0.735588 + 1.27408i −1.27368 2.86575 0.154042 + 0.266809i −1.17956 2.04305i
144.8 1.00350 1.73811i 0.669895 1.16029i −1.01402 1.75633i 0.304819 0.527963i −1.34448 2.32870i −2.87509 −0.0562672 0.602481 + 1.04353i −0.611771 1.05962i
144.9 1.36735 2.36833i −1.29050 + 2.23522i −2.73931 4.74463i 1.68779 2.92334i 3.52915 + 6.11267i 0.409409 −9.51302 −1.83080 3.17105i −4.61562 7.99448i
 $$n$$: 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.a 18
19.c even 3 1 inner 209.2.e.a 18
19.c even 3 1 3971.2.a.m 9
19.d odd 6 1 3971.2.a.j 9

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{18} + T_{2}^{17} + 14 T_{2}^{16} + 15 T_{2}^{15} + 132 T_{2}^{14} + 132 T_{2}^{13} + 671 T_{2}^{12} + 520 T_{2}^{11} + 2312 T_{2}^{10} + 1413 T_{2}^{9} + 4605 T_{2}^{8} + 985 T_{2}^{7} + 5289 T_{2}^{6} + 427 T_{2}^{5} + 4276 T_{2}^{4} + \cdots + 81$$ acting on $$S_{2}^{\mathrm{new}}(209, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18} + T^{17} + 14 T^{16} + 15 T^{15} + \cdots + 81$$
$3$ $$T^{18} + 4 T^{17} + 27 T^{16} + 66 T^{15} + \cdots + 484$$
$5$ $$T^{18} + 30 T^{16} - 26 T^{15} + \cdots + 762129$$
$7$ $$(T^{9} + 3 T^{8} - 30 T^{7} - 82 T^{6} + \cdots + 16)^{2}$$
$11$ $$(T - 1)^{18}$$
$13$ $$T^{18} + T^{17} + 45 T^{16} + 150 T^{15} + \cdots + 4$$
$17$ $$T^{18} - 2 T^{17} + 70 T^{16} + \cdots + 18974736$$
$19$ $$T^{18} - 6 T^{17} + \cdots + 322687697779$$
$23$ $$T^{18} - 11 T^{17} + \cdots + 19115274564$$
$29$ $$T^{18} + 13 T^{17} + \cdots + 38699545284$$
$31$ $$(T^{9} - 8 T^{8} - 35 T^{7} + 485 T^{6} + \cdots + 3872)^{2}$$
$37$ $$(T^{9} - 14 T^{8} - 66 T^{7} + \cdots - 471948)^{2}$$
$41$ $$T^{18} + 16 T^{17} + \cdots + 77720518656$$
$43$ $$T^{18} - 15 T^{17} + \cdots + 37122499584$$
$47$ $$T^{18} + 2 T^{17} + \cdots + 361636255044$$
$53$ $$T^{18} + 16 T^{17} + \cdots + 49461315201$$
$59$ $$T^{18} + 37 T^{17} + \cdots + 1647254170116$$
$61$ $$T^{18} + 15 T^{17} + \cdots + 8564391936$$
$67$ $$T^{18} + 14 T^{17} + \cdots + 36309590753536$$
$71$ $$T^{18} + 8 T^{17} + \cdots + 343531965456$$
$73$ $$T^{18} - 18 T^{17} + \cdots + 981997757764$$
$79$ $$T^{18} + T^{17} + \cdots + 432502207201$$
$83$ $$(T^{9} + 8 T^{8} - 567 T^{7} + \cdots + 141072192)^{2}$$
$89$ $$T^{18} + \cdots + 111035618574336$$
$97$ $$T^{18} + 44 T^{17} + \cdots + 17\!\cdots\!49$$