# Properties

 Label 4205.2.a.t Level $4205$ Weight $2$ Character orbit 4205.a Self dual yes Analytic conductor $33.577$ Analytic rank $0$ Dimension $18$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4205,2,Mod(1,4205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4205, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4205 = 5 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4205.a (trivial)

## 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: yes Analytic conductor: $$33.5770940499$$ Analytic rank: $$0$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} - x^{17} - 29 x^{16} + 26 x^{15} + 347 x^{14} - 277 x^{13} - 2215 x^{12} + 1567 x^{11} + \cdots + 41$$ x^18 - x^17 - 29*x^16 + 26*x^15 + 347*x^14 - 277*x^13 - 2215*x^12 + 1567*x^11 + 8142*x^10 - 5117*x^9 - 17296*x^8 + 9833*x^7 + 19923*x^6 - 10660*x^5 - 10213*x^4 + 5422*x^3 + 915*x^2 - 513*x + 41 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - x^{17} - 29 x^{16} + 26 x^{15} + 347 x^{14} - 277 x^{13} - 2215 x^{12} + 1567 x^{11} + \cdots + 41$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$( 29840 \nu^{17} - 79467 \nu^{16} - 733103 \nu^{15} + 1992300 \nu^{14} + 7037561 \nu^{13} + \cdots + 416901 ) / 65884$$ (29840*v^17 - 79467*v^16 - 733103*v^15 + 1992300*v^14 + 7037561*v^13 - 19889911*v^12 - 32964016*v^11 + 100651480*v^10 + 75176487*v^9 - 272167068*v^8 - 62098004*v^7 + 378846088*v^6 - 37251919*v^5 - 226687187*v^4 + 70620761*v^3 + 23772272*v^2 - 8015291*v + 416901) / 65884 $$\beta_{4}$$ $$=$$ $$( 238502 \nu^{17} - 646673 \nu^{16} - 5843139 \nu^{15} + 16273726 \nu^{14} + 55740803 \nu^{13} + \cdots + 8557209 ) / 461188$$ (238502*v^17 - 646673*v^16 - 5843139*v^15 + 16273726*v^14 + 55740803*v^13 - 163235383*v^12 - 257080060*v^11 + 830783190*v^10 + 559598327*v^9 - 2260235456*v^8 - 353016160*v^7 + 3156473562*v^6 - 552061125*v^5 - 1862906845*v^4 + 755583821*v^3 + 149415030*v^2 - 88406033*v + 8557209) / 461188 $$\beta_{5}$$ $$=$$ $$( 11948 \nu^{17} - 29039 \nu^{16} - 302635 \nu^{15} + 738738 \nu^{14} + 3028665 \nu^{13} + \cdots + 197727 ) / 17738$$ (11948*v^17 - 29039*v^16 - 302635*v^15 + 738738*v^14 + 3028665*v^13 - 7519167*v^12 - 15077870*v^11 + 39040172*v^10 + 38101221*v^9 - 109158096*v^8 - 41160140*v^7 + 158300438*v^6 - 1419761*v^5 - 99175679*v^4 + 26296021*v^3 + 10518250*v^2 - 2923369*v + 197727) / 17738 $$\beta_{6}$$ $$=$$ $$( 437793 \nu^{17} - 1163823 \nu^{16} - 10784092 \nu^{15} + 29347017 \nu^{14} + 103648205 \nu^{13} + \cdots + 14033920 ) / 461188$$ (437793*v^17 - 1163823*v^16 - 10784092*v^15 + 29347017*v^14 + 103648205*v^13 - 295207876*v^12 - 483382504*v^11 + 1508507787*v^10 + 1073183640*v^9 - 4127180412*v^8 - 729613528*v^7 + 5808287417*v^6 - 970822167*v^5 - 3467501283*v^4 + 1404981308*v^3 + 290105677*v^2 - 163207731*v + 14033920) / 461188 $$\beta_{7}$$ $$=$$ $$( 470453 \nu^{17} - 1206984 \nu^{16} - 11756001 \nu^{15} + 30629767 \nu^{14} + 115378314 \nu^{13} + \cdots + 10737833 ) / 461188$$ (470453*v^17 - 1206984*v^16 - 11756001*v^15 + 30629767*v^14 + 115378314*v^13 - 310745163*v^12 - 556716820*v^11 + 1606362599*v^10 + 1323847705*v^9 - 4465099472*v^8 - 1179710126*v^7 + 6423837685*v^6 - 610649242*v^5 - 3974815708*v^4 + 1334218699*v^3 + 400245107*v^2 - 155267514*v + 10737833) / 461188 $$\beta_{8}$$ $$=$$ $$( 493901 \nu^{17} - 1243565 \nu^{16} - 12402410 \nu^{15} + 31606001 \nu^{14} + 122553663 \nu^{13} + \cdots + 11959042 ) / 461188$$ (493901*v^17 - 1243565*v^16 - 12402410*v^15 + 31606001*v^14 + 122553663*v^13 - 321333170*v^12 - 597418088*v^11 + 1665947459*v^10 + 1446388810*v^9 - 4648257800*v^8 - 1357946324*v^7 + 6716054173*v^6 - 535131821*v^5 - 4168189401*v^4 + 1378725650*v^3 + 408249105*v^2 - 160278389*v + 11959042) / 461188 $$\beta_{9}$$ $$=$$ $$( - 736531 \nu^{17} + 1887136 \nu^{16} + 18397989 \nu^{15} - 47868877 \nu^{14} - 180429682 \nu^{13} + \cdots - 19288573 ) / 461188$$ (-736531*v^17 + 1887136*v^16 + 18397989*v^15 - 47868877*v^14 - 180429682*v^13 + 485336575*v^12 + 869162748*v^11 - 2506580621*v^10 - 2057791471*v^9 + 6957244962*v^8 + 1797873336*v^7 - 9983484361*v^6 + 1040213272*v^5 + 6138955188*v^4 - 2150551059*v^3 - 585732009*v^2 + 252080222*v - 19288573) / 461188 $$\beta_{10}$$ $$=$$ $$( 749664 \nu^{17} - 1920719 \nu^{16} - 18764575 \nu^{15} + 48829984 \nu^{14} + 184522593 \nu^{13} + \cdots + 20249941 ) / 461188$$ (749664*v^17 - 1920719*v^16 - 18764575*v^15 + 48829984*v^14 + 184522593*v^13 - 496572627*v^12 - 892004592*v^11 + 2574953132*v^10 + 2120966383*v^9 - 7184565372*v^8 - 1859525640*v^7 + 10375121444*v^6 - 1096795259*v^5 - 6422936563*v^4 + 2269682117*v^3 + 612197804*v^2 - 265330255*v + 20249941) / 461188 $$\beta_{11}$$ $$=$$ $$( - 836399 \nu^{17} + 2186039 \nu^{16} + 20763412 \nu^{15} - 55350085 \nu^{14} - 201797451 \nu^{13} + \cdots - 26019218 ) / 461188$$ (-836399*v^17 + 2186039*v^16 + 20763412*v^15 - 55350085*v^14 - 201797451*v^13 + 559818274*v^12 + 957968356*v^11 - 2881751217*v^10 - 2202343386*v^9 + 7963328598*v^8 + 1721627096*v^7 - 11359063869*v^6 + 1554542395*v^5 + 6915591903*v^4 - 2620569968*v^3 - 621451757*v^2 + 305805797*v - 26019218) / 461188 $$\beta_{12}$$ $$=$$ $$( - 926803 \nu^{17} + 2424063 \nu^{16} + 23032284 \nu^{15} - 61478669 \nu^{14} - 224076039 \nu^{13} + \cdots - 31489058 ) / 461188$$ (-926803*v^17 + 2424063*v^16 + 23032284*v^15 - 61478669*v^14 - 224076039*v^13 + 623105062*v^12 + 1064105460*v^11 - 3215793453*v^10 - 2439196510*v^9 + 8912406558*v^8 + 1853235048*v^7 - 12745071613*v^6 + 1885514071*v^5 + 7755456419*v^4 - 3059409820*v^3 - 664325197*v^2 + 358939781*v - 31489058) / 461188 $$\beta_{13}$$ $$=$$ $$( 1151055 \nu^{17} - 3014842 \nu^{16} - 28643183 \nu^{15} + 76581673 \nu^{14} + 279158892 \nu^{13} + \cdots + 36483943 ) / 461188$$ (1151055*v^17 - 3014842*v^16 - 28643183*v^15 + 76581673*v^14 + 279158892*v^13 - 777744085*v^12 - 1329069908*v^11 + 4024550241*v^10 + 3059460805*v^9 - 11193612562*v^8 - 2354054752*v^7 + 16084945661*v^6 - 2322007618*v^5 - 9862574390*v^4 + 3825927789*v^3 + 880770685*v^2 - 446012956*v + 36483943) / 461188 $$\beta_{14}$$ $$=$$ $$( 207099 \nu^{17} - 534156 \nu^{16} - 5174933 \nu^{15} + 13579877 \nu^{14} + 50745382 \nu^{13} + \cdots + 6535141 ) / 65884$$ (207099*v^17 - 534156*v^16 - 5174933*v^15 + 13579877*v^14 + 50745382*v^13 - 138080879*v^12 - 244059696*v^11 + 715749525*v^10 + 573699267*v^9 - 1995554966*v^8 - 480037620*v^7 + 2877298169*v^6 - 350640116*v^5 - 1773834796*v^4 + 656569259*v^3 + 162345393*v^2 - 77532558*v + 6535141) / 65884 $$\beta_{15}$$ $$=$$ $$( 1498205 \nu^{17} - 3838619 \nu^{16} - 37449726 \nu^{15} + 97428583 \nu^{14} + 367638783 \nu^{13} + \cdots + 38645688 ) / 461188$$ (1498205*v^17 - 3838619*v^16 - 37449726*v^15 + 97428583*v^14 + 367638783*v^13 - 988585476*v^12 - 1773843996*v^11 + 5110881791*v^10 + 4212955536*v^9 - 14204648630*v^8 - 3721354028*v^7 + 20418275671*v^6 - 2041910311*v^5 - 12585729035*v^4 + 4338023390*v^3 + 1212528931*v^2 - 503620941*v + 38645688) / 461188 $$\beta_{16}$$ $$=$$ $$( - 960073 \nu^{17} + 2473038 \nu^{16} + 23990266 \nu^{15} - 62844453 \nu^{14} - 235297994 \nu^{13} + \cdots - 27550128 ) / 230594$$ (-960073*v^17 + 2473038*v^16 + 23990266*v^15 - 62844453*v^14 - 235297994*v^13 + 638655658*v^12 + 1132567346*v^11 - 3308304637*v^10 - 2669871287*v^9 + 9216992117*v^8 + 2269986733*v^7 - 13282462434*v^6 + 1532565027*v^5 + 8195829640*v^4 - 2964871162*v^3 - 768562023*v^2 + 347844740*v - 27550128) / 230594 $$\beta_{17}$$ $$=$$ $$( 4633386 \nu^{17} - 11901709 \nu^{16} - 115689811 \nu^{15} + 301959672 \nu^{14} + 1133854541 \nu^{13} + \cdots + 119961671 ) / 461188$$ (4633386*v^17 - 11901709*v^16 - 115689811*v^15 + 301959672*v^14 + 1133854541*v^13 - 3062401549*v^12 - 5455998304*v^11 + 15822491714*v^10 + 12886071967*v^9 - 43941672604*v^8 - 11147121846*v^7 + 63103767254*v^6 - 6785389499*v^5 - 38840578173*v^4 + 13690303229*v^3 + 3711312148*v^2 - 1597680943*v + 119961671) / 461188
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{16} + \beta_{14} + \beta_{7} + \beta_{2} + 5\beta_1$$ b16 + b14 + b7 + b2 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{15} - \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - 2\beta_{8} - \beta_{3} + 7\beta_{2} + 16$$ b15 - b13 - b11 + b10 + b9 - 2*b8 - b3 + 7*b2 + 16 $$\nu^{5}$$ $$=$$ $$9 \beta_{16} + 8 \beta_{14} - \beta_{11} + 2 \beta_{10} - 2 \beta_{8} + 9 \beta_{7} + \beta_{5} + \cdots + 2$$ 9*b16 + 8*b14 - b11 + 2*b10 - 2*b8 + 9*b7 + b5 - b3 + 9*b2 + 30*b1 + 2 $$\nu^{6}$$ $$=$$ $$\beta_{17} + 2 \beta_{16} + 9 \beta_{15} + \beta_{14} - 10 \beta_{13} - 9 \beta_{11} + 12 \beta_{10} + \cdots + 99$$ b17 + 2*b16 + 9*b15 + b14 - 10*b13 - 9*b11 + 12*b10 + 11*b9 - 23*b8 + b7 + b5 + b4 - 11*b3 + 48*b2 + 2*b1 + 99 $$\nu^{7}$$ $$=$$ $$\beta_{17} + 67 \beta_{16} - 2 \beta_{15} + 57 \beta_{14} - 3 \beta_{13} - \beta_{12} - 11 \beta_{11} + \cdots + 29$$ b17 + 67*b16 - 2*b15 + 57*b14 - 3*b13 - b12 - 11*b11 + 26*b10 + 3*b9 - 30*b8 + 70*b7 + b6 + 14*b5 + b4 - 13*b3 + 71*b2 + 192*b1 + 29 $$\nu^{8}$$ $$=$$ $$14 \beta_{17} + 30 \beta_{16} + 63 \beta_{15} + 19 \beta_{14} - 81 \beta_{13} - \beta_{12} - 65 \beta_{11} + \cdots + 655$$ 14*b17 + 30*b16 + 63*b15 + 19*b14 - 81*b13 - b12 - 65*b11 + 109*b10 + 99*b9 - 204*b8 + 20*b7 + 4*b6 + 19*b5 + 12*b4 - 92*b3 + 333*b2 + 34*b1 + 655 $$\nu^{9}$$ $$=$$ $$15 \beta_{17} + 475 \beta_{16} - 33 \beta_{15} + 400 \beta_{14} - 49 \beta_{13} - 19 \beta_{12} + \cdots + 317$$ 15*b17 + 475*b16 - 33*b15 + 400*b14 - 49*b13 - 19*b12 - 93*b11 + 252*b10 + 55*b9 - 317*b8 + 530*b7 + 22*b6 + 147*b5 + 9*b4 - 122*b3 + 545*b2 + 1271*b1 + 317 $$\nu^{10}$$ $$=$$ $$142 \beta_{17} + 321 \beta_{16} + 401 \beta_{15} + 235 \beta_{14} - 622 \beta_{13} - 27 \beta_{12} + \cdots + 4492$$ 142*b17 + 321*b16 + 401*b15 + 235*b14 - 622*b13 - 27*b12 - 440*b11 + 903*b10 + 838*b9 - 1669*b8 + 261*b7 + 73*b6 + 236*b5 + 94*b4 - 699*b3 + 2341*b2 + 405*b1 + 4492 $$\nu^{11}$$ $$=$$ $$165 \beta_{17} + 3337 \beta_{16} - 378 \beta_{15} + 2828 \beta_{14} - 557 \beta_{13} - 252 \beta_{12} + \cdots + 3099$$ 165*b17 + 3337*b16 - 378*b15 + 2828*b14 - 557*b13 - 252*b12 - 713*b11 + 2197*b10 + 689*b9 - 2935*b8 + 3998*b7 + 288*b6 + 1374*b5 + 33*b4 - 1009*b3 + 4158*b2 + 8595*b1 + 3099 $$\nu^{12}$$ $$=$$ $$1279 \beta_{17} + 3017 \beta_{16} + 2399 \beta_{15} + 2427 \beta_{14} - 4711 \beta_{13} - 441 \beta_{12} + \cdots + 31518$$ 1279*b17 + 3017*b16 + 2399*b15 + 2427*b14 - 4711*b13 - 441*b12 - 2902*b11 + 7201*b10 + 6903*b9 - 13229*b8 + 2840*b7 + 876*b6 + 2461*b5 + 584*b4 - 5087*b3 + 16654*b2 + 4136*b1 + 31518 $$\nu^{13}$$ $$=$$ $$1623 \beta_{17} + 23560 \beta_{16} - 3729 \beta_{15} + 20255 \beta_{14} - 5478 \beta_{13} - 2847 \beta_{12} + \cdots + 28438$$ 1623*b17 + 23560*b16 - 3729*b15 + 20255*b14 - 5478*b13 - 2847*b12 - 5202*b11 + 18228*b10 + 7365*b9 - 25486*b8 + 30225*b7 + 3031*b6 + 12078*b5 - 230*b4 - 7844*b3 + 31698*b2 + 59038*b1 + 28438 $$\nu^{14}$$ $$=$$ $$10889 \beta_{17} + 26593 \beta_{16} + 13438 \beta_{15} + 22799 \beta_{14} - 35638 \beta_{13} + \cdots + 224873$$ 10889*b17 + 26593*b16 + 13438*b15 + 22799*b14 - 35638*b13 - 5673*b12 - 18892*b11 + 56435*b10 + 56039*b9 - 103417*b8 + 28060*b7 + 8798*b6 + 23422*b5 + 2823*b4 - 36239*b3 + 119735*b2 + 38753*b1 + 224873 $$\nu^{15}$$ $$=$$ $$15105 \beta_{17} + 168028 \beta_{16} - 34035 \beta_{15} + 147045 \beta_{14} - 50015 \beta_{13} + \cdots + 250318$$ 15105*b17 + 168028*b16 - 34035*b15 + 147045*b14 - 50015*b13 - 29267*b12 - 36838*b11 + 147278*b10 + 72260*b9 - 213576*b8 + 229415*b7 + 28479*b6 + 102380*b5 - 6194*b4 - 58978*b3 + 241800*b2 + 410735*b1 + 250318 $$\nu^{16}$$ $$=$$ $$89928 \beta_{17} + 225980 \beta_{16} + 68039 \beta_{15} + 202516 \beta_{14} - 270408 \beta_{13} + \cdots + 1626032$$ 89928*b17 + 225980*b16 + 68039*b15 + 202516*b14 - 270408*b13 - 63532*b12 - 121973*b11 + 438690*b10 + 450990*b9 - 803394*b8 + 261520*b7 + 80464*b6 + 211179*b5 + 6933*b4 - 255383*b3 + 869040*b2 + 344022*b1 + 1626032 $$\nu^{17}$$ $$=$$ $$135905 \beta_{17} + 1212365 \beta_{16} - 296893 \beta_{15} + 1080928 \beta_{14} - 437159 \beta_{13} + \cdots + 2140037$$ 135905*b17 + 1212365*b16 - 296893*b15 + 1080928*b14 - 437159*b13 - 282603*b12 - 255530*b11 + 1171831*b10 + 672136*b9 - 1751801*b8 + 1749086*b7 + 250660*b6 + 848171*b5 - 83129*b4 - 435216*b3 + 1846447*b2 + 2889639*b1 + 2140037

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.54760 −2.47806 −2.43831 −1.57163 −1.56310 −1.51612 −1.46716 −0.370427 0.117638 0.200159 0.589764 1.14331 1.33016 1.81057 2.03141 2.30810 2.62027 2.80103
−2.54760 0.347586 4.49025 1.00000 −0.885510 3.89213 −6.34414 −2.87918 −2.54760
1.2 −2.47806 −0.0240901 4.14079 1.00000 0.0596967 −2.90004 −5.30501 −2.99942 −2.47806
1.3 −2.43831 −1.39369 3.94538 1.00000 3.39825 −0.911770 −4.74344 −1.05763 −2.43831
1.4 −1.57163 3.15664 0.470008 1.00000 −4.96105 1.09217 2.40457 6.96437 −1.57163
1.5 −1.56310 0.512231 0.443278 1.00000 −0.800667 −3.87624 2.43331 −2.73762 −1.56310
1.6 −1.51612 −3.04030 0.298613 1.00000 4.60945 2.32872 2.57950 6.24343 −1.51612
1.7 −1.46716 2.83912 0.152571 1.00000 −4.16545 4.97656 2.71048 5.06058 −1.46716
1.8 −0.370427 −0.193908 −1.86278 1.00000 0.0718287 −0.400866 1.43088 −2.96240 −0.370427
1.9 0.117638 −1.90235 −1.98616 1.00000 −0.223789 2.81872 −0.468926 0.618929 0.117638
1.10 0.200159 −0.328234 −1.95994 1.00000 −0.0656991 4.42737 −0.792618 −2.89226 0.200159
1.11 0.589764 −2.80954 −1.65218 1.00000 −1.65696 0.145738 −2.15392 4.89351 0.589764
1.12 1.14331 0.291587 −0.692836 1.00000 0.333376 −2.48191 −3.07875 −2.91498 1.14331
1.13 1.33016 2.65160 −0.230684 1.00000 3.52704 −1.42983 −2.96716 4.03098 1.33016
1.14 1.81057 −1.85470 1.27815 1.00000 −3.35806 −2.49392 −1.30696 0.439927 1.81057
1.15 2.03141 2.41065 2.12661 1.00000 4.89701 4.27304 0.257201 2.81123 2.03141
1.16 2.30810 2.53573 3.32733 1.00000 5.85273 3.10066 3.06362 3.42994 2.30810
1.17 2.62027 −2.90669 4.86584 1.00000 −7.61632 2.62123 7.50929 5.44882 2.62027
1.18 2.80103 0.708357 5.84576 1.00000 1.98413 1.81823 10.7721 −2.49823 2.80103
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$29$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4205.2.a.t 18
29.b even 2 1 4205.2.a.s 18
29.e even 14 2 145.2.k.b 36
145.l even 14 2 725.2.l.e 36
145.q odd 28 4 725.2.r.d 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.k.b 36 29.e even 14 2
725.2.l.e 36 145.l even 14 2
725.2.r.d 72 145.q odd 28 4
4205.2.a.s 18 29.b even 2 1
4205.2.a.t 18 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4205))$$:

 $$T_{2}^{18} - T_{2}^{17} - 29 T_{2}^{16} + 26 T_{2}^{15} + 347 T_{2}^{14} - 277 T_{2}^{13} - 2215 T_{2}^{12} + \cdots + 41$$ T2^18 - T2^17 - 29*T2^16 + 26*T2^15 + 347*T2^14 - 277*T2^13 - 2215*T2^12 + 1567*T2^11 + 8142*T2^10 - 5117*T2^9 - 17296*T2^8 + 9833*T2^7 + 19923*T2^6 - 10660*T2^5 - 10213*T2^4 + 5422*T2^3 + 915*T2^2 - 513*T2 + 41 $$T_{3}^{18} - T_{3}^{17} - 36 T_{3}^{16} + 32 T_{3}^{15} + 518 T_{3}^{14} - 394 T_{3}^{13} - 3781 T_{3}^{12} + \cdots - 1$$ T3^18 - T3^17 - 36*T3^16 + 32*T3^15 + 518*T3^14 - 394*T3^13 - 3781*T3^12 + 2331*T3^11 + 14583*T3^10 - 6861*T3^9 - 27587*T3^8 + 10269*T3^7 + 19309*T3^6 - 9296*T3^5 - 2178*T3^4 + 1273*T3^3 + 53*T3^2 - 41*T3 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18} - T^{17} + \cdots + 41$$
$3$ $$T^{18} - T^{17} + \cdots - 1$$
$5$ $$(T - 1)^{18}$$
$7$ $$T^{18} - 17 T^{17} + \cdots - 205729$$
$11$ $$T^{18} - 5 T^{17} + \cdots + 17681728$$
$13$ $$T^{18} - 17 T^{17} + \cdots + 59082688$$
$17$ $$T^{18} + 14 T^{17} + \cdots - 3879616$$
$19$ $$T^{18} + \cdots + 575326144$$
$23$ $$T^{18} - 28 T^{17} + \cdots - 63413$$
$29$ $$T^{18}$$
$31$ $$T^{18} + 7 T^{17} + \cdots - 80576$$
$37$ $$T^{18} + \cdots + 4756936947008$$
$41$ $$T^{18} + \cdots + 457112949881$$
$43$ $$T^{18} + \cdots - 28751525207$$
$47$ $$T^{18} + \cdots - 64893278323$$
$53$ $$T^{18} + \cdots + 1789774633472$$
$59$ $$T^{18} + \cdots + 600525977408$$
$61$ $$T^{18} + \cdots - 363101625152$$
$67$ $$T^{18} + \cdots + 388848546868928$$
$71$ $$T^{18} + \cdots + 806647744$$
$73$ $$T^{18} + \cdots - 46786424768$$
$79$ $$T^{18} + \cdots + 11470874176$$
$83$ $$T^{18} + \cdots - 772624375419197$$
$89$ $$T^{18} + \cdots + 28001490013$$
$97$ $$T^{18} + \cdots + 44674838248768$$