# Properties

 Label 2738.2.a.w Level $2738$ Weight $2$ Character orbit 2738.a Self dual yes Analytic conductor $21.863$ 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] = [2738,2,Mod(1,2738)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2738, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2738.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2738 = 2 \cdot 37^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2738.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: $$21.8630400734$$ 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: $$x^{18} - 8 x^{17} - 8 x^{16} + 209 x^{15} - 253 x^{14} - 1996 x^{13} + 4196 x^{12} + 8667 x^{11} - 24522 x^{10} - 18284 x^{9} + 68697 x^{8} + 20865 x^{7} - 97011 x^{6} + \cdots + 113$$ x^18 - 8*x^17 - 8*x^16 + 209*x^15 - 253*x^14 - 1996*x^13 + 4196*x^12 + 8667*x^11 - 24522*x^10 - 18284*x^9 + 68697*x^8 + 20865*x^7 - 97011*x^6 - 20844*x^5 + 64568*x^4 + 17499*x^3 - 13387*x^2 - 3713*x + 113 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 8 x^{17} - 8 x^{16} + 209 x^{15} - 253 x^{14} - 1996 x^{13} + 4196 x^{12} + 8667 x^{11} - 24522 x^{10} - 18284 x^{9} + 68697 x^{8} + 20865 x^{7} - 97011 x^{6} + \cdots + 113$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 288951249349 \nu^{17} - 3591379024255 \nu^{16} + 5588056150692 \nu^{15} + 82055854684103 \nu^{14} - 283321663509840 \nu^{13} + \cdots + 481706958614797 ) / 299494704712751$$ (288951249349*v^17 - 3591379024255*v^16 + 5588056150692*v^15 + 82055854684103*v^14 - 283321663509840*v^13 - 579563545148231*v^12 + 3276687521980490*v^11 + 729251554315928*v^10 - 16401579703827456*v^9 + 6492197006019851*v^8 + 39940706355592698*v^7 - 24152766405752142*v^6 - 46921105903120778*v^5 + 26630279476534825*v^4 + 23808509570669154*v^3 - 8372343883337681*v^2 - 2524479297014846*v + 481706958614797) / 299494704712751 $$\beta_{3}$$ $$=$$ $$( - 339789756367 \nu^{17} - 4967748483248 \nu^{16} + 52736315932460 \nu^{15} + 52935633791109 \nu^{14} + \cdots + 569976441271657 ) / 299494704712751$$ (-339789756367*v^17 - 4967748483248*v^16 + 52736315932460*v^15 + 52935633791109*v^14 - 1254095928514007*v^13 + 809254783252041*v^12 + 11944428062161155*v^11 - 14766180970093030*v^10 - 54517157299618611*v^9 + 79541926491710231*v^8 + 127728873797066740*v^7 - 181746568578206627*v^6 - 157307843914450426*v^5 + 166945881220103345*v^4 + 99184277521293640*v^3 - 43139881097991377*v^2 - 17629045314485923*v + 569976441271657) / 299494704712751 $$\beta_{4}$$ $$=$$ $$( - 1076022812465 \nu^{17} + 6107715698223 \nu^{16} + 21640413863764 \nu^{15} - 164551539293128 \nu^{14} + \cdots - 13\!\cdots\!79 ) / 299494704712751$$ (-1076022812465*v^17 + 6107715698223*v^16 + 21640413863764*v^15 - 164551539293128*v^14 - 108463250649558*v^13 + 1656868039149756*v^12 - 190326075878805*v^11 - 7905291384952879*v^10 + 1753905090128988*v^9 + 19346973337731650*v^8 + 2464460227122344*v^7 - 23985800018328609*v^6 - 23211667202007462*v^5 + 11221047640701347*v^4 + 31050743105231780*v^3 + 4345425016131810*v^2 - 8731071258878247*v - 1331166494124979) / 299494704712751 $$\beta_{5}$$ $$=$$ $$( 1542780927592 \nu^{17} - 9170574598705 \nu^{16} - 28764108899129 \nu^{15} + 243809841597377 \nu^{14} + \cdots + 12\!\cdots\!17 ) / 299494704712751$$ (1542780927592*v^17 - 9170574598705*v^16 - 28764108899129*v^15 + 243809841597377*v^14 + 96860268977836*v^13 - 2406621355973184*v^12 + 831812628338835*v^11 + 11156400754017960*v^10 - 4996435146501512*v^9 - 26453712008516361*v^8 + 2319993442458564*v^7 + 32750663628425992*v^6 + 24182404023606216*v^5 - 17893328470347055*v^4 - 33806492434112437*v^3 - 3315287526285751*v^2 + 7950745821081112*v + 1246679051402017) / 299494704712751 $$\beta_{6}$$ $$=$$ $$( - 1549894163847 \nu^{17} + 9333877080572 \nu^{16} + 29760010628922 \nu^{15} - 257398256579752 \nu^{14} + \cdots - 32933534425465 ) / 299494704712751$$ (-1549894163847*v^17 + 9333877080572*v^16 + 29760010628922*v^15 - 257398256579752*v^14 - 102610229841921*v^13 + 2690570464927454*v^12 - 1072377519402898*v^11 - 13649856978748989*v^10 + 8413483465613432*v^9 + 36796479602462196*v^8 - 18981264555772733*v^7 - 53409677809081450*v^6 + 9813625659342009*v^5 + 37542995882463788*v^4 + 8547745486866654*v^3 - 7110151562898516*v^2 - 3919784371093248*v - 32933534425465) / 299494704712751 $$\beta_{7}$$ $$=$$ $$( - 1997441834796 \nu^{17} + 11105151266670 \nu^{16} + 43686727585558 \nu^{15} - 315319923493599 \nu^{14} + \cdots + 429365903773692 ) / 299494704712751$$ (-1997441834796*v^17 + 11105151266670*v^16 + 43686727585558*v^15 - 315319923493599*v^14 - 270297533590283*v^13 + 3433457428546648*v^12 - 76024212846822*v^11 - 18383660814328288*v^10 + 5185389892600022*v^9 + 52499170576578220*v^8 - 12173132348872244*v^7 - 79434492073485422*v^6 - 2127175623686319*v^5 + 56031816868229648*v^4 + 22445631803143005*v^3 - 11121766496857281*v^2 - 8499063512109693*v + 429365903773692) / 299494704712751 $$\beta_{8}$$ $$=$$ $$( 2063781192451 \nu^{17} - 6051607290075 \nu^{16} - 81391785790979 \nu^{15} + 243596039548688 \nu^{14} + \cdots + 16075387819939 ) / 299494704712751$$ (2063781192451*v^17 - 6051607290075*v^16 - 81391785790979*v^15 + 243596039548688*v^14 + 1243361989681261*v^13 - 3781146012030868*v^12 - 9413525109133113*v^11 + 28742439801745577*v^10 + 38142728831692986*v^9 - 112480197845647430*v^8 - 86944891015356249*v^7 + 219779591171164104*v^6 + 116337067753103535*v^5 - 184973121914018463*v^4 - 85375491115456581*v^3 + 42174963743733590*v^2 + 15344131678380956*v + 16075387819939) / 299494704712751 $$\beta_{9}$$ $$=$$ $$( - 2977590344659 \nu^{17} + 17346377645528 \nu^{16} + 60178134110193 \nu^{15} - 481941385516172 \nu^{14} + \cdots - 321722815924100 ) / 299494704712751$$ (-2977590344659*v^17 + 17346377645528*v^16 + 60178134110193*v^15 - 481941385516172*v^14 - 278311256418702*v^13 + 5095011359397208*v^12 - 1241436602272699*v^11 - 26292206622322774*v^10 + 12276183663629322*v^9 + 72552540343060246*v^8 - 27195613105987253*v^7 - 108410123017997259*v^6 + 8085179896366297*v^5 + 78806942043966016*v^4 + 21300171149500287*v^3 - 15466325549684644*v^2 - 6974629750014819*v - 321722815924100) / 299494704712751 $$\beta_{10}$$ $$=$$ $$( - 5193825773890 \nu^{17} + 33161215679707 \nu^{16} + 86575370956769 \nu^{15} - 889001092131404 \nu^{14} + \cdots - 239751347303384 ) / 299494704712751$$ (-5193825773890*v^17 + 33161215679707*v^16 + 86575370956769*v^15 - 889001092131404*v^14 + 7134024413648*v^13 + 8879466003265165*v^12 - 7083041332797669*v^11 - 41812515764383033*v^10 + 44608613915547729*v^9 + 100675461032148408*v^8 - 103786691124385914*v^7 - 126352855145461890*v^6 + 86819412960715737*v^5 + 78173037643163086*v^4 - 10937142790704759*v^3 - 11919606620348286*v^2 - 2826900346170230*v - 239751347303384) / 299494704712751 $$\beta_{11}$$ $$=$$ $$( 6655598780904 \nu^{17} - 31157708048006 \nu^{16} - 185835614582087 \nu^{15} + 966004789070378 \nu^{14} + \cdots - 502765674810731 ) / 299494704712751$$ (6655598780904*v^17 - 31157708048006*v^16 - 185835614582087*v^15 + 966004789070378*v^14 + 1976056536782940*v^13 - 11822197119176959*v^12 - 10376055628046042*v^11 + 73399464203553831*v^10 + 31381258688046644*v^9 - 247510754593798846*v^8 - 68197298412169831*v^7 + 441373961727610603*v^6 + 120461052189341059*v^5 - 359057095152289879*v^4 - 119611631599895782*v^3 + 83425726149281352*v^2 + 24557124380356191*v - 502765674810731) / 299494704712751 $$\beta_{12}$$ $$=$$ $$( 6762901520895 \nu^{17} - 42145279389393 \nu^{16} - 120457688091949 \nu^{15} + \cdots - 671196427105821 ) / 299494704712751$$ (6762901520895*v^17 - 42145279389393*v^16 - 120457688091949*v^15 + 1147710449313395*v^14 + 187272621181785*v^13 - 11763661406594596*v^12 + 7433021077926929*v^11 + 57896428790527368*v^10 - 51048414425539692*v^9 - 149895593278800860*v^8 + 123654238282731551*v^7 + 209932349664738792*v^6 - 108539451252913884*v^5 - 149960779325544858*v^4 + 17661271262456289*v^3 + 33951543632064083*v^2 + 1941840097012444*v - 671196427105821) / 299494704712751 $$\beta_{13}$$ $$=$$ $$( 11661402651090 \nu^{17} - 79767685631079 \nu^{16} - 160932521842093 \nu^{15} + \cdots + 554147111971995 ) / 299494704712751$$ (11661402651090*v^17 - 79767685631079*v^16 - 160932521842093*v^15 + 2085899832022951*v^14 - 910211148671071*v^13 - 19966579885172080*v^12 + 24787297440159062*v^11 + 87029789607869533*v^10 - 141566514722023110*v^9 - 182300613084555561*v^8 + 330800007067610189*v^7 + 181024310455048509*v^6 - 313299020291160429*v^5 - 85833255803045477*v^4 + 93944686193658989*v^3 + 7890988424335866*v^2 - 3215086269990481*v + 554147111971995) / 299494704712751 $$\beta_{14}$$ $$=$$ $$( 11957932777767 \nu^{17} - 66354475924789 \nu^{16} - 265735968553660 \nu^{15} + \cdots - 10\!\cdots\!86 ) / 299494704712751$$ (11957932777767*v^17 - 66354475924789*v^16 - 265735968553660*v^15 + 1898286705968220*v^14 + 1735090029111824*v^13 - 20944113703748491*v^12 - 717638691069597*v^11 + 114791456669847498*v^10 - 26242701870756680*v^9 - 340936807498192264*v^8 + 68824409431264617*v^7 + 547536388190630961*v^6 - 8994860024009478*v^5 - 419005754138692071*v^4 - 84392470082077522*v^3 + 92476802757233809*v^2 + 24439456919977314*v - 1063702576573886) / 299494704712751 $$\beta_{15}$$ $$=$$ $$( - 16593280437543 \nu^{17} + 100278150348575 \nu^{16} + 315461018420552 \nu^{15} + \cdots - 11\!\cdots\!58 ) / 299494704712751$$ (-16593280437543*v^17 + 100278150348575*v^16 + 315461018420552*v^15 - 2760889895744933*v^14 - 999797485644219*v^13 + 28780404182123421*v^12 - 12724468402938301*v^11 - 145331220565157578*v^10 + 98263745533067174*v^9 + 389082547345593234*v^8 - 233813841994024799*v^7 - 561017860397639906*v^6 + 168921804333386413*v^5 + 395534400058961939*v^4 + 25747897955902161*v^3 - 75676296797743638*v^2 - 18218719442563657*v - 1151909443176258) / 299494704712751 $$\beta_{16}$$ $$=$$ $$( 19304209555584 \nu^{17} - 121340222289561 \nu^{16} - 336951345493012 \nu^{15} + \cdots - 532122377771700 ) / 299494704712751$$ (19304209555584*v^17 - 121340222289561*v^16 - 336951345493012*v^15 + 3291285076634187*v^14 + 349959671054308*v^13 - 33505044800373537*v^12 + 23048012522751799*v^11 + 162874757405004653*v^10 - 154089069756671294*v^9 - 412466291696692572*v^8 + 371210138539940529*v^7 + 556061007581793797*v^6 - 326166007476849501*v^5 - 373183695428986822*v^4 + 51985707590541362*v^3 + 71514732442213384*v^2 + 7502074529610405*v - 532122377771700) / 299494704712751 $$\beta_{17}$$ $$=$$ $$( 23081286448329 \nu^{17} - 134279987934766 \nu^{16} - 472778247498487 \nu^{15} + \cdots - 260913772800456 ) / 299494704712751$$ (23081286448329*v^17 - 134279987934766*v^16 - 472778247498487*v^15 + 3762080277451093*v^14 + 2279884841882579*v^13 - 40263374240388962*v^12 + 9177604707166099*v^11 + 211416288480430388*v^10 - 99465154053716808*v^9 - 595747890083786009*v^8 + 246922138483313209*v^7 + 907821126074097072*v^6 - 155505773574389718*v^5 - 669656338916165128*v^4 - 78334027427529390*v^3 + 142037698973146694*v^2 + 33960501965489954*v - 260913772800456) / 299494704712751
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{14} - 2\beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{8} - \beta_{3} - \beta_{2} + \beta _1 + 5$$ b14 - 2*b11 - b10 + 2*b9 + b8 - b3 - b2 + b1 + 5 $$\nu^{3}$$ $$=$$ $$- \beta_{17} + \beta_{16} + \beta_{15} - \beta_{14} + 2 \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{5} - 2 \beta_{4} + 6 \beta _1 + 2$$ -b17 + b16 + b15 - b14 + 2*b12 + b11 - 2*b10 + b9 + 2*b8 - b5 - 2*b4 + 6*b1 + 2 $$\nu^{4}$$ $$=$$ $$- \beta_{17} - 2 \beta_{16} - 2 \beta_{15} + 9 \beta_{14} + 3 \beta_{12} - 16 \beta_{11} - 10 \beta_{10} + 20 \beta_{9} + 8 \beta_{8} - \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 8 \beta_{3} - 7 \beta_{2} + 8 \beta _1 + 36$$ -b17 - 2*b16 - 2*b15 + 9*b14 + 3*b12 - 16*b11 - 10*b10 + 20*b9 + 8*b8 - b7 + 3*b6 - 2*b5 - 3*b4 - 8*b3 - 7*b2 + 8*b1 + 36 $$\nu^{5}$$ $$=$$ $$- 12 \beta_{17} + 11 \beta_{16} + 11 \beta_{15} - 13 \beta_{14} - \beta_{13} + 28 \beta_{12} + 15 \beta_{11} - 24 \beta_{10} + 19 \beta_{9} + 20 \beta_{8} - 2 \beta_{7} - 13 \beta_{5} - 29 \beta_{4} + 2 \beta_{2} + 44 \beta _1 + 20$$ -12*b17 + 11*b16 + 11*b15 - 13*b14 - b13 + 28*b12 + 15*b11 - 24*b10 + 19*b9 + 20*b8 - 2*b7 - 13*b5 - 29*b4 + 2*b2 + 44*b1 + 20 $$\nu^{6}$$ $$=$$ $$- 20 \beta_{17} - 24 \beta_{16} - 29 \beta_{15} + 75 \beta_{14} - 2 \beta_{13} + 50 \beta_{12} - 122 \beta_{11} - 100 \beta_{10} + 198 \beta_{9} + 62 \beta_{8} - 21 \beta_{7} + 44 \beta_{6} - 32 \beta_{5} - 50 \beta_{4} - 68 \beta_{3} + \cdots + 294$$ -20*b17 - 24*b16 - 29*b15 + 75*b14 - 2*b13 + 50*b12 - 122*b11 - 100*b10 + 198*b9 + 62*b8 - 21*b7 + 44*b6 - 32*b5 - 50*b4 - 68*b3 - 52*b2 + 64*b1 + 294 $$\nu^{7}$$ $$=$$ $$- 134 \beta_{17} + 109 \beta_{16} + 98 \beta_{15} - 142 \beta_{14} - 17 \beta_{13} + 317 \beta_{12} + 188 \beta_{11} - 248 \beta_{10} + 244 \beta_{9} + 172 \beta_{8} - 44 \beta_{7} + 24 \beta_{6} - 142 \beta_{5} - 332 \beta_{4} + \cdots + 183$$ -134*b17 + 109*b16 + 98*b15 - 142*b14 - 17*b13 + 317*b12 + 188*b11 - 248*b10 + 244*b9 + 172*b8 - 44*b7 + 24*b6 - 142*b5 - 332*b4 + 37*b2 + 358*b1 + 183 $$\nu^{8}$$ $$=$$ $$- 273 \beta_{17} - 216 \beta_{16} - 317 \beta_{15} + 614 \beta_{14} - 32 \beta_{13} + 616 \beta_{12} - 914 \beta_{11} - 969 \beta_{10} + 1953 \beta_{9} + 487 \beta_{8} - 290 \beta_{7} + 538 \beta_{6} - 376 \beta_{5} + \cdots + 2549$$ -273*b17 - 216*b16 - 317*b15 + 614*b14 - 32*b13 + 616*b12 - 914*b11 - 969*b10 + 1953*b9 + 487*b8 - 290*b7 + 538*b6 - 376*b5 - 634*b4 - 601*b3 - 399*b2 + 541*b1 + 2549 $$\nu^{9}$$ $$=$$ $$- 1451 \beta_{17} + 1080 \beta_{16} + 843 \beta_{15} - 1478 \beta_{14} - 192 \beta_{13} + 3338 \beta_{12} + 2177 \beta_{11} - 2446 \beta_{10} + 2795 \beta_{9} + 1438 \beta_{8} - 663 \beta_{7} + 578 \beta_{6} + \cdots + 1717$$ -1451*b17 + 1080*b16 + 843*b15 - 1478*b14 - 192*b13 + 3338*b12 + 2177*b11 - 2446*b10 + 2795*b9 + 1438*b8 - 663*b7 + 578*b6 - 1459*b5 - 3535*b4 - 17*b3 + 495*b2 + 3090*b1 + 1717 $$\nu^{10}$$ $$=$$ $$- 3282 \beta_{17} - 1707 \beta_{16} - 3156 \beta_{15} + 4942 \beta_{14} - 345 \beta_{13} + 6902 \beta_{12} - 6633 \beta_{11} - 9231 \beta_{10} + 19244 \beta_{9} + 3913 \beta_{8} - 3513 \beta_{7} + 6210 \beta_{6} + \cdots + 22832$$ -3282*b17 - 1707*b16 - 3156*b15 + 4942*b14 - 345*b13 + 6902*b12 - 6633*b11 - 9231*b10 + 19244*b9 + 3913*b8 - 3513*b7 + 6210*b6 - 4012*b5 - 7310*b4 - 5429*b3 - 3112*b2 + 4807*b1 + 22832 $$\nu^{11}$$ $$=$$ $$- 15402 \beta_{17} + 10783 \beta_{16} + 7257 \beta_{15} - 15083 \beta_{14} - 1805 \beta_{13} + 34131 \beta_{12} + 24100 \beta_{11} - 23685 \beta_{10} + 30688 \beta_{9} + 12070 \beta_{8} - 8599 \beta_{7} + \cdots + 16778$$ -15402*b17 + 10783*b16 + 7257*b15 - 15083*b14 - 1805*b13 + 34131*b12 + 24100*b11 - 23685*b10 + 30688*b9 + 12070*b8 - 8599*b7 + 9309*b6 - 14603*b5 - 36578*b4 - 423*b3 + 5791*b2 + 27639*b1 + 16778 $$\nu^{12}$$ $$=$$ $$- 37278 \beta_{17} - 12128 \beta_{16} - 30258 \beta_{15} + 38910 \beta_{14} - 3016 \beta_{13} + 74391 \beta_{12} - 45235 \beta_{11} - 87350 \beta_{10} + 189912 \beta_{9} + 32242 \beta_{8} + \cdots + 208575$$ -37278*b17 - 12128*b16 - 30258*b15 + 38910*b14 - 3016*b13 + 74391*b12 - 45235*b11 - 87350*b10 + 189912*b9 + 32242*b8 - 40185*b7 + 69434*b6 - 41262*b5 - 80665*b4 - 49568*b3 - 24412*b2 + 44417*b1 + 208575 $$\nu^{13}$$ $$=$$ $$- 161516 \beta_{17} + 107998 \beta_{16} + 62841 \beta_{15} - 152320 \beta_{14} - 14923 \beta_{13} + 344904 \beta_{12} + 259562 \beta_{11} - 227821 \beta_{10} + 330721 \beta_{9} + 102930 \beta_{8} + \cdots + 169512$$ -161516*b17 + 107998*b16 + 62841*b15 - 152320*b14 - 14923*b13 + 344904*b12 + 259562*b11 - 227821*b10 + 330721*b9 + 102930*b8 - 103415*b7 + 126411*b6 - 144621*b5 - 373815*b4 - 6827*b3 + 63229*b2 + 252992*b1 + 169512 $$\nu^{14}$$ $$=$$ $$- 410845 \beta_{17} - 74224 \beta_{16} - 285391 \beta_{15} + 297077 \beta_{14} - 21476 \beta_{13} + 787408 \beta_{12} - 271036 \beta_{11} - 825518 \beta_{10} + 1879525 \beta_{9} + \cdots + 1931025$$ -410845*b17 - 74224*b16 - 285391*b15 + 297077*b14 - 21476*b13 + 787408*b12 - 271036*b11 - 825518*b10 + 1879525*b9 + 272623*b8 - 445662*b7 + 759365*b6 - 418057*b5 - 869378*b4 - 454556*b3 - 190780*b2 + 422504*b1 + 1931025 $$\nu^{15}$$ $$=$$ $$- 1681533 \beta_{17} + 1081255 \beta_{16} + 546312 \beta_{15} - 1528015 \beta_{14} - 107233 \beta_{13} + 3471341 \beta_{12} + 2747354 \beta_{11} - 2188379 \beta_{10} + 3528559 \beta_{9} + \cdots + 1750513$$ -1681533*b17 + 1081255*b16 + 546312*b15 - 1528015*b14 - 107233*b13 + 3471341*b12 + 2747354*b11 - 2188379*b10 + 3528559*b9 + 895697*b8 - 1190432*b7 + 1567113*b6 - 1427691*b5 - 3799173*b4 - 90557*b3 + 664312*b2 + 2353529*b1 + 1750513 $$\nu^{16}$$ $$=$$ $$- 4446713 \beta_{17} - 316333 \beta_{16} - 2672210 \beta_{15} + 2165884 \beta_{14} - 102250 \beta_{13} + 8256849 \beta_{12} - 1126223 \beta_{11} - 7812646 \beta_{10} + 18661330 \beta_{9} + \cdots + 18060389$$ -4446713*b17 - 316333*b16 - 2672210*b15 + 2165884*b14 - 102250*b13 + 8256849*b12 - 1126223*b11 - 7812646*b10 + 18661330*b9 + 2364632*b8 - 4848222*b7 + 8171384*b6 - 4209887*b5 - 9238590*b4 - 4171808*b3 - 1469480*b2 + 4104042*b1 + 18060389 $$\nu^{17}$$ $$=$$ $$- 17428676 \beta_{17} + 10805515 \beta_{16} + 4755694 \beta_{15} - 15256403 \beta_{14} - 599792 \beta_{13} + 34916469 \beta_{12} + 28755474 \beta_{11} - 21041972 \beta_{10} + \cdots + 18301860$$ -17428676*b17 + 10805515*b16 + 4755694*b15 - 15256403*b14 - 599792*b13 + 34916469*b12 + 28755474*b11 - 21041972*b10 + 37395882*b9 + 7958973*b8 - 13325220*b7 + 18396516*b6 - 14096836*b5 - 38520916*b4 - 1069928*b3 + 6826470*b2 + 22164431*b1 + 18301860

## 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
 2.24542 3.18423 2.72365 1.39715 0.0277653 2.93806 2.26200 3.21182 −1.00049 2.01728 −1.75172 0.561002 −3.00437 −0.771034 −1.88881 −0.844166 −2.98762 −0.320182
−1.00000 −3.00437 1.00000 −4.06246 3.00437 0.951094 −1.00000 6.02625 4.06246
1.2 −1.00000 −2.98762 1.00000 2.67223 2.98762 2.38780 −1.00000 5.92589 −2.67223
1.3 −1.00000 −1.88881 1.00000 0.545033 1.88881 −1.80620 −1.00000 0.567610 −0.545033
1.4 −1.00000 −1.75172 1.00000 2.09131 1.75172 1.21116 −1.00000 0.0685155 −2.09131
1.5 −1.00000 −1.00049 1.00000 −2.36317 1.00049 3.55615 −1.00000 −1.99902 2.36317
1.6 −1.00000 −0.844166 1.00000 −2.97364 0.844166 4.24887 −1.00000 −2.28738 2.97364
1.7 −1.00000 −0.771034 1.00000 0.437041 0.771034 −4.61128 −1.00000 −2.40551 −0.437041
1.8 −1.00000 −0.320182 1.00000 −3.73221 0.320182 2.93372 −1.00000 −2.89748 3.73221
1.9 −1.00000 0.0277653 1.00000 3.81163 −0.0277653 −3.33489 −1.00000 −2.99923 −3.81163
1.10 −1.00000 0.561002 1.00000 −3.29795 −0.561002 −3.86349 −1.00000 −2.68528 3.29795
1.11 −1.00000 1.39715 1.00000 −3.69423 −1.39715 −0.398050 −1.00000 −1.04796 3.69423
1.12 −1.00000 2.01728 1.00000 2.91611 −2.01728 5.00881 −1.00000 1.06941 −2.91611
1.13 −1.00000 2.24542 1.00000 −0.488539 −2.24542 3.12577 −1.00000 2.04193 0.488539
1.14 −1.00000 2.26200 1.00000 −3.49383 −2.26200 4.18290 −1.00000 2.11666 3.49383
1.15 −1.00000 2.72365 1.00000 −2.50059 −2.72365 −0.328765 −1.00000 4.41829 2.50059
1.16 −1.00000 2.93806 1.00000 3.37430 −2.93806 2.74077 −1.00000 5.63221 −3.37430
1.17 −1.00000 3.18423 1.00000 −0.335004 −3.18423 3.81934 −1.00000 7.13933 0.335004
1.18 −1.00000 3.21182 1.00000 2.09398 −3.21182 −1.82370 −1.00000 7.31576 −2.09398
 $$n$$: 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
$$2$$ $$1$$
$$37$$ $$-1$$

## 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 2738.2.a.w 18
37.b even 2 1 2738.2.a.x yes 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2738.2.a.w 18 1.a even 1 1 trivial
2738.2.a.x yes 18 37.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2738))$$:

 $$T_{3}^{18} - 8 T_{3}^{17} - 8 T_{3}^{16} + 209 T_{3}^{15} - 253 T_{3}^{14} - 1996 T_{3}^{13} + 4196 T_{3}^{12} + 8667 T_{3}^{11} - 24522 T_{3}^{10} - 18284 T_{3}^{9} + 68697 T_{3}^{8} + 20865 T_{3}^{7} - 97011 T_{3}^{6} + \cdots + 113$$ T3^18 - 8*T3^17 - 8*T3^16 + 209*T3^15 - 253*T3^14 - 1996*T3^13 + 4196*T3^12 + 8667*T3^11 - 24522*T3^10 - 18284*T3^9 + 68697*T3^8 + 20865*T3^7 - 97011*T3^6 - 20844*T3^5 + 64568*T3^4 + 17499*T3^3 - 13387*T3^2 - 3713*T3 + 113 $$T_{5}^{18} + 9 T_{5}^{17} - 29 T_{5}^{16} - 455 T_{5}^{15} - 78 T_{5}^{14} + 9261 T_{5}^{13} + 12116 T_{5}^{12} - 97547 T_{5}^{11} - 183708 T_{5}^{10} + 567401 T_{5}^{9} + 1261800 T_{5}^{8} - 1790132 T_{5}^{7} + \cdots + 194048$$ T5^18 + 9*T5^17 - 29*T5^16 - 455*T5^15 - 78*T5^14 + 9261*T5^13 + 12116*T5^12 - 97547*T5^11 - 183708*T5^10 + 567401*T5^9 + 1261800*T5^8 - 1790132*T5^7 - 4296840*T5^6 + 2763248*T5^5 + 6371680*T5^4 - 1596544*T5^3 - 2128512*T5^2 + 235008*T5 + 194048 $$T_{7}^{18} - 18 T_{7}^{17} + 74 T_{7}^{16} + 524 T_{7}^{15} - 5105 T_{7}^{14} + 4893 T_{7}^{13} + 84052 T_{7}^{12} - 285632 T_{7}^{11} - 300591 T_{7}^{10} + 2942857 T_{7}^{9} - 2894898 T_{7}^{8} - 9557644 T_{7}^{7} + \cdots - 2140672$$ T7^18 - 18*T7^17 + 74*T7^16 + 524*T7^15 - 5105*T7^14 + 4893*T7^13 + 84052*T7^12 - 285632*T7^11 - 300591*T7^10 + 2942857*T7^9 - 2894898*T7^8 - 9557644*T7^7 + 20607624*T7^6 + 3522176*T7^5 - 37007488*T7^4 + 19908800*T7^3 + 11367040*T7^2 - 6192640*T7 - 2140672 $$T_{13}^{18} + 9 T_{13}^{17} - 105 T_{13}^{16} - 1082 T_{13}^{15} + 3874 T_{13}^{14} + 50966 T_{13}^{13} - 51990 T_{13}^{12} - 1202511 T_{13}^{11} - 153574 T_{13}^{10} + 15178743 T_{13}^{9} + 10405398 T_{13}^{8} + \cdots + 66558464$$ T13^18 + 9*T13^17 - 105*T13^16 - 1082*T13^15 + 3874*T13^14 + 50966*T13^13 - 51990*T13^12 - 1202511*T13^11 - 153574*T13^10 + 15178743*T13^9 + 10405398*T13^8 - 102059896*T13^7 - 94152856*T13^6 + 347985040*T13^5 + 340623328*T13^4 - 513256512*T13^3 - 458989440*T13^2 + 186667264*T13 + 66558464 $$T_{17}^{18} + 13 T_{17}^{17} - 78 T_{17}^{16} - 1565 T_{17}^{15} + 593 T_{17}^{14} + 73468 T_{17}^{13} + 98241 T_{17}^{12} - 1750289 T_{17}^{11} - 3531210 T_{17}^{10} + 23200272 T_{17}^{9} + 50807507 T_{17}^{8} + \cdots + 278237$$ T17^18 + 13*T17^17 - 78*T17^16 - 1565*T17^15 + 593*T17^14 + 73468*T17^13 + 98241*T17^12 - 1750289*T17^11 - 3531210*T17^10 + 23200272*T17^9 + 50807507*T17^8 - 176233700*T17^7 - 350656811*T17^6 + 741755987*T17^5 + 1099798988*T17^4 - 1477977255*T17^3 - 1137831084*T17^2 + 698947892*T17 + 278237

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{18}$$
$3$ $$T^{18} - 8 T^{17} - 8 T^{16} + 209 T^{15} + \cdots + 113$$
$5$ $$T^{18} + 9 T^{17} - 29 T^{16} + \cdots + 194048$$
$7$ $$T^{18} - 18 T^{17} + 74 T^{16} + \cdots - 2140672$$
$11$ $$T^{18} - 10 T^{17} - 67 T^{16} + \cdots - 1678687$$
$13$ $$T^{18} + 9 T^{17} - 105 T^{16} + \cdots + 66558464$$
$17$ $$T^{18} + 13 T^{17} - 78 T^{16} + \cdots + 278237$$
$19$ $$T^{18} + 2 T^{17} - 186 T^{16} + \cdots + 153521$$
$23$ $$T^{18} - 11 T^{17} + \cdots + 702069248$$
$29$ $$T^{18} + 30 T^{17} + \cdots - 755457536$$
$31$ $$T^{18} - 8 T^{17} + \cdots + 12374814208$$
$37$ $$T^{18}$$
$41$ $$T^{18} - 5 T^{17} + \cdots + 11940934483$$
$43$ $$T^{18} - 3 T^{17} + \cdots - 605849481581$$
$47$ $$T^{18} - 37 T^{17} + \cdots + 35338905088$$
$53$ $$T^{18} - 25 T^{17} + \cdots - 53823866757632$$
$59$ $$T^{18} + 26 T^{17} + \cdots + 5283151765979$$
$61$ $$T^{18} + 27 T^{17} + \cdots + 147836571136$$
$67$ $$T^{18} - 23 T^{17} + \cdots - 26881668076057$$
$71$ $$T^{18} + 25 T^{17} + \cdots + 42725861888$$
$73$ $$T^{18} - 77 T^{17} + \cdots + 13545811777$$
$79$ $$T^{18} - 13 T^{17} + \cdots - 862427513344$$
$83$ $$T^{18} + 10 T^{17} + \cdots - 40118279748961$$
$89$ $$T^{18} + 55 T^{17} + \cdots + 21059632979023$$
$97$ $$T^{18} + \cdots + 160408755241271$$