# Properties

 Label 2738.2.a.s Level $2738$ Weight $2$ Character orbit 2738.a Self dual yes Analytic conductor $21.863$ Analytic rank $1$ Dimension $6$ 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: $$6$$ Coefficient field: $$\Q(\zeta_{36})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 6x^{4} + 9x^{2} - 3$$ x^6 - 6*x^4 + 9*x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of $$\nu = \zeta_{36} + \zeta_{36}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5\nu^{2} + 4$$ v^4 - 5*v^2 + 4 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 5\nu^{3} + 4\nu$$ v^5 - 5*v^3 + 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5\beta_{2} + 6$$ b4 + 5*b2 + 6 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 5\beta_{3} + 11\beta_1$$ b5 + 5*b3 + 11*b1

## 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
 −1.28558 1.28558 −0.684040 0.684040 1.96962 −1.96962
1.00000 −1.87939 1.00000 −4.26414 −1.87939 3.61144 1.00000 0.532089 −4.26414
1.2 1.00000 −1.87939 1.00000 −0.800038 −1.87939 0.147334 1.00000 0.532089 −0.800038
1.3 1.00000 0.347296 1.00000 −0.852666 0.347296 1.38475 1.00000 −2.87939 −0.852666
1.4 1.00000 0.347296 1.00000 2.61144 0.347296 −2.07935 1.00000 −2.87939 2.61144
1.5 1.00000 1.53209 1.00000 −3.07935 1.53209 0.199962 1.00000 −0.652704 −3.07935
1.6 1.00000 1.53209 1.00000 0.384754 1.53209 −3.26414 1.00000 −0.652704 0.384754
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.s 6
37.b even 2 1 2738.2.a.r 6
37.i odd 36 2 74.2.h.a 12
111.q even 36 2 666.2.bj.c 12
148.q even 36 2 592.2.bq.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.h.a 12 37.i odd 36 2
592.2.bq.b 12 148.q even 36 2
666.2.bj.c 12 111.q even 36 2
2738.2.a.r 6 37.b even 2 1
2738.2.a.s 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2738))$$:

 $$T_{3}^{3} - 3T_{3} + 1$$ T3^3 - 3*T3 + 1 $$T_{5}^{6} + 6T_{5}^{5} - 42T_{5}^{3} - 45T_{5}^{2} + 9$$ T5^6 + 6*T5^5 - 42*T5^3 - 45*T5^2 + 9 $$T_{7}^{6} - 15T_{7}^{4} - 2T_{7}^{3} + 36T_{7}^{2} - 12T_{7} + 1$$ T7^6 - 15*T7^4 - 2*T7^3 + 36*T7^2 - 12*T7 + 1 $$T_{13}^{6} + 12T_{13}^{5} + 39T_{13}^{4} - 30T_{13}^{3} - 288T_{13}^{2} - 90T_{13} + 537$$ T13^6 + 12*T13^5 + 39*T13^4 - 30*T13^3 - 288*T13^2 - 90*T13 + 537 $$T_{17}^{6} + 12T_{17}^{5} + 27T_{17}^{4} - 120T_{17}^{3} - 477T_{17}^{2} - 324T_{17} + 9$$ T17^6 + 12*T17^5 + 27*T17^4 - 120*T17^3 - 477*T17^2 - 324*T17 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{6}$$
$3$ $$(T^{3} - 3 T + 1)^{2}$$
$5$ $$T^{6} + 6 T^{5} - 42 T^{3} - 45 T^{2} + \cdots + 9$$
$7$ $$T^{6} - 15 T^{4} - 2 T^{3} + 36 T^{2} + \cdots + 1$$
$11$ $$T^{6} + 6 T^{5} - 27 T^{4} - 150 T^{3} + \cdots - 639$$
$13$ $$T^{6} + 12 T^{5} + 39 T^{4} + \cdots + 537$$
$17$ $$T^{6} + 12 T^{5} + 27 T^{4} - 120 T^{3} + \cdots + 9$$
$19$ $$T^{6} + 12 T^{5} + 9 T^{4} + \cdots - 3231$$
$23$ $$T^{6} - 6 T^{5} - 45 T^{4} + 366 T^{3} + \cdots + 657$$
$29$ $$T^{6} - 6 T^{5} - 18 T^{4} + 114 T^{3} + \cdots + 333$$
$31$ $$T^{6} + 18 T^{5} + 57 T^{4} + \cdots + 17817$$
$37$ $$T^{6}$$
$41$ $$T^{6} + 12 T^{5} - 36 T^{4} + \cdots + 576$$
$43$ $$T^{6} - 78 T^{4} + 144 T^{3} + \cdots + 1509$$
$47$ $$T^{6} + 6 T^{5} - 117 T^{4} + \cdots - 45027$$
$53$ $$T^{6} + 24 T^{5} + 9 T^{4} + \cdots + 212229$$
$59$ $$T^{6} + 12 T^{5} - 135 T^{4} + \cdots + 132201$$
$61$ $$T^{6} + 12 T^{5} - 72 T^{4} + \cdots + 576$$
$67$ $$T^{6} - 18 T^{5} - 123 T^{4} + \cdots - 244331$$
$71$ $$T^{6} - 24 T^{5} - 144 T^{4} + \cdots + 2032704$$
$73$ $$T^{6} - 366 T^{4} - 322 T^{3} + \cdots + 94609$$
$79$ $$T^{6} + 12 T^{5} + 39 T^{4} + \cdots - 219$$
$83$ $$T^{6} + 6 T^{5} - 315 T^{4} + \cdots + 333$$
$89$ $$T^{6} - 243 T^{4} + 972 T^{3} + \cdots + 26217$$
$97$ $$T^{6} - 12 T^{5} - 240 T^{4} + \cdots + 165696$$