# Properties

 Label 2738.2.a.v Level $2738$ Weight $2$ Character orbit 2738.a Self dual yes Analytic conductor $21.863$ Analytic rank $1$ Dimension $9$ 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: $$1$$ Dimension: $$9$$ Coefficient field: $$\Q(\zeta_{38})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1$$ x^9 - x^8 - 8*x^7 + 7*x^6 + 21*x^5 - 15*x^4 - 20*x^3 + 10*x^2 + 5*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of $$\nu = \zeta_{38} + \zeta_{38}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 4\nu^{2} + 2$$ v^4 - 4*v^2 + 2 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 5\nu^{3} + 5\nu$$ v^5 - 5*v^3 + 5*v $$\beta_{6}$$ $$=$$ $$\nu^{6} - 6\nu^{4} + 9\nu^{2} - 2$$ v^6 - 6*v^4 + 9*v^2 - 2 $$\beta_{7}$$ $$=$$ $$\nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu$$ v^7 - 7*v^5 + 14*v^3 - 7*v $$\beta_{8}$$ $$=$$ $$\nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2$$ v^8 - 8*v^6 + 20*v^4 - 16*v^2 + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 4\beta_{2} + 6$$ b4 + 4*b2 + 6 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 5\beta_{3} + 10\beta_1$$ b5 + 5*b3 + 10*b1 $$\nu^{6}$$ $$=$$ $$\beta_{6} + 6\beta_{4} + 15\beta_{2} + 20$$ b6 + 6*b4 + 15*b2 + 20 $$\nu^{7}$$ $$=$$ $$\beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1$$ b7 + 7*b5 + 21*b3 + 35*b1 $$\nu^{8}$$ $$=$$ $$\beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70$$ b8 + 8*b6 + 28*b4 + 56*b2 + 70

## 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
 −1.89163 0.165159 −0.490971 −1.57828 −1.09390 1.97272 0.803391 1.75895 1.35456
1.00000 −3.38261 1.00000 −2.72648 −3.38261 −0.152624 1.00000 8.44202 −2.72648
1.2 1.00000 −2.72648 1.00000 −0.0314505 −2.72648 1.42593 1.00000 4.43367 −0.0314505
1.3 1.00000 −2.58487 1.00000 −3.38261 −2.58487 −3.95019 1.00000 3.68154 −3.38261
1.4 1.00000 −0.819334 1.00000 −0.605558 −0.819334 0.239042 1.00000 −2.32869 −0.605558
1.5 1.00000 −0.739333 1.00000 −2.58487 −0.739333 −0.650935 1.00000 −2.45339 −2.58487
1.6 1.00000 −0.605558 1.00000 2.32729 −0.605558 0.184773 1.00000 −2.63330 2.32729
1.7 1.00000 −0.0314505 1.00000 1.56234 −0.0314505 −4.80486 1.00000 −2.99901 1.56234
1.8 1.00000 1.56234 1.00000 −0.819334 1.56234 −1.93137 1.00000 −0.559099 −0.819334
1.9 1.00000 2.32729 1.00000 −0.739333 2.32729 −4.35976 1.00000 2.41626 −0.739333
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 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.v yes 9
37.b even 2 1 2738.2.a.u 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2738.2.a.u 9 37.b even 2 1
2738.2.a.v yes 9 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2738))$$:

 $$T_{3}^{9} + 7T_{3}^{8} + 7T_{3}^{7} - 45T_{3}^{6} - 100T_{3}^{5} + 13T_{3}^{4} + 170T_{3}^{3} + 139T_{3}^{2} + 36T_{3} + 1$$ T3^9 + 7*T3^8 + 7*T3^7 - 45*T3^6 - 100*T3^5 + 13*T3^4 + 170*T3^3 + 139*T3^2 + 36*T3 + 1 $$T_{5}^{9} + 7T_{5}^{8} + 7T_{5}^{7} - 45T_{5}^{6} - 100T_{5}^{5} + 13T_{5}^{4} + 170T_{5}^{3} + 139T_{5}^{2} + 36T_{5} + 1$$ T5^9 + 7*T5^8 + 7*T5^7 - 45*T5^6 - 100*T5^5 + 13*T5^4 + 170*T5^3 + 139*T5^2 + 36*T5 + 1 $$T_{7}^{9} + 14T_{7}^{8} + 66T_{7}^{7} + 96T_{7}^{6} - 99T_{7}^{5} - 287T_{7}^{4} - 64T_{7}^{3} + 46T_{7}^{2} + T_{7} - 1$$ T7^9 + 14*T7^8 + 66*T7^7 + 96*T7^6 - 99*T7^5 - 287*T7^4 - 64*T7^3 + 46*T7^2 + T7 - 1 $$T_{13}^{9} + 3 T_{13}^{8} - 53 T_{13}^{7} - 170 T_{13}^{6} + 732 T_{13}^{5} + 2334 T_{13}^{4} - 2040 T_{13}^{3} - 5321 T_{13}^{2} - 46 T_{13} + 379$$ T13^9 + 3*T13^8 - 53*T13^7 - 170*T13^6 + 732*T13^5 + 2334*T13^4 - 2040*T13^3 - 5321*T13^2 - 46*T13 + 379 $$T_{17}^{9} + 10 T_{17}^{8} - 40 T_{17}^{7} - 540 T_{17}^{6} + 202 T_{17}^{5} + 8595 T_{17}^{4} + 5594 T_{17}^{3} - 37295 T_{17}^{2} - 52545 T_{17} - 12769$$ T17^9 + 10*T17^8 - 40*T17^7 - 540*T17^6 + 202*T17^5 + 8595*T17^4 + 5594*T17^3 - 37295*T17^2 - 52545*T17 - 12769

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{9}$$
$3$ $$T^{9} + 7 T^{8} + 7 T^{7} - 45 T^{6} + \cdots + 1$$
$5$ $$T^{9} + 7 T^{8} + 7 T^{7} - 45 T^{6} + \cdots + 1$$
$7$ $$T^{9} + 14 T^{8} + 66 T^{7} + 96 T^{6} + \cdots - 1$$
$11$ $$T^{9} + T^{8} - 46 T^{7} - 83 T^{6} + \cdots + 1901$$
$13$ $$T^{9} + 3 T^{8} - 53 T^{7} - 170 T^{6} + \cdots + 379$$
$17$ $$T^{9} + 10 T^{8} - 40 T^{7} + \cdots - 12769$$
$19$ $$T^{9} + 9 T^{8} - 59 T^{7} + \cdots - 1481$$
$23$ $$T^{9} - 3 T^{8} - 129 T^{7} + \cdots - 379$$
$29$ $$T^{9} + 12 T^{8} - 69 T^{7} + \cdots + 154697$$
$31$ $$T^{9} + 2 T^{8} - 89 T^{7} + \cdots + 1747$$
$37$ $$T^{9}$$
$41$ $$T^{9} + 16 T^{8} - 15 T^{7} + \cdots + 60041$$
$43$ $$T^{9} - 6 T^{8} - 193 T^{7} + \cdots + 289103$$
$47$ $$T^{9} + 15 T^{8} - 185 T^{7} + \cdots + 9187069$$
$53$ $$T^{9} + 7 T^{8} - 297 T^{7} + \cdots + 47107043$$
$59$ $$T^{9} + T^{8} - 312 T^{7} + \cdots - 565249$$
$61$ $$T^{9} + 7 T^{8} - 297 T^{7} + \cdots - 47151197$$
$67$ $$T^{9} + 66 T^{8} + 1727 T^{7} + \cdots - 30468247$$
$71$ $$T^{9} + 15 T^{8} - 128 T^{7} + \cdots - 453683$$
$73$ $$T^{9} + 50 T^{8} + \cdots - 393016331$$
$79$ $$T^{9} - 29 T^{8} + 36 T^{7} + \cdots + 11315299$$
$83$ $$T^{9} + 43 T^{8} + 427 T^{7} + \cdots - 99198581$$
$89$ $$T^{9} + 6 T^{8} - 345 T^{7} + \cdots + 11018443$$
$97$ $$T^{9} + 50 T^{8} + 881 T^{7} + \cdots - 18092827$$