# Properties

 Label 637.2.e.l Level $637$ Weight $2$ Character orbit 637.e Analytic conductor $5.086$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(79,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("637.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.4406832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1$$ x^6 - x^5 + 6*x^4 + 7*x^3 + 24*x^2 + 5*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 6\nu^{4} - 36\nu^{3} + 24\nu^{2} + 5\nu - 30 ) / 149$$ (-v^5 + 6*v^4 - 36*v^3 + 24*v^2 + 5*v - 30) / 149 $$\beta_{3}$$ $$=$$ $$( -6\nu^{5} + 36\nu^{4} - 67\nu^{3} + 144\nu^{2} + 30\nu + 416 ) / 149$$ (-6*v^5 + 36*v^4 - 67*v^3 + 144*v^2 + 30*v + 416) / 149 $$\beta_{4}$$ $$=$$ $$( 30\nu^{5} - 31\nu^{4} + 186\nu^{3} + 174\nu^{2} + 744\nu + 155 ) / 149$$ (30*v^5 - 31*v^4 + 186*v^3 + 174*v^2 + 744*v + 155) / 149 $$\beta_{5}$$ $$=$$ $$( 89\nu^{5} - 87\nu^{4} + 522\nu^{3} + 695\nu^{2} + 2088\nu + 435 ) / 149$$ (89*v^5 - 87*v^4 + 522*v^3 + 695*v^2 + 2088*v + 435) / 149
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3\beta_{4} - \beta_{2} + \beta_1$$ b5 - 3*b4 - b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 6\beta_{2} - 4$$ b3 - 6*b2 - 4 $$\nu^{4}$$ $$=$$ $$-6\beta_{5} + 19\beta_{4} + 6\beta_{3} - 12\beta _1 - 19$$ -6*b5 + 19*b4 + 6*b3 - 12*b1 - 19 $$\nu^{5}$$ $$=$$ $$-12\beta_{5} + 42\beta_{4} + 43\beta_{2} - 43\beta_1$$ -12*b5 + 42*b4 + 43*b2 - 43*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 1.43310 + 2.48220i −0.105378 − 0.182520i −0.827721 − 1.43366i 1.43310 − 2.48220i −0.105378 + 0.182520i −0.827721 + 1.43366i
−0.933099 + 1.61618i 1.67445 + 2.90023i −0.741348 1.28405i −0.433099 + 0.750150i −6.24970 0 −0.965392 −4.10755 + 7.11448i −0.808249 1.39993i
79.2 0.605378 1.04855i −0.872413 1.51106i 0.267035 + 0.462518i 1.10538 1.91457i −2.11256 0 3.06814 −0.0222090 + 0.0384672i −1.33834 2.31808i
79.3 1.32772 2.29968i 1.19797 + 2.07494i −2.52569 4.37462i 1.82772 3.16571i 6.36226 0 −8.10275 −1.37024 + 2.37333i −4.85341 8.40635i
508.1 −0.933099 1.61618i 1.67445 2.90023i −0.741348 + 1.28405i −0.433099 0.750150i −6.24970 0 −0.965392 −4.10755 7.11448i −0.808249 + 1.39993i
508.2 0.605378 + 1.04855i −0.872413 + 1.51106i 0.267035 0.462518i 1.10538 + 1.91457i −2.11256 0 3.06814 −0.0222090 0.0384672i −1.33834 + 2.31808i
508.3 1.32772 + 2.29968i 1.19797 2.07494i −2.52569 + 4.37462i 1.82772 + 3.16571i 6.36226 0 −8.10275 −1.37024 2.37333i −4.85341 + 8.40635i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.e.l 6
7.b odd 2 1 637.2.e.k 6
7.c even 3 1 637.2.a.h 3
7.c even 3 1 inner 637.2.e.l 6
7.d odd 6 1 637.2.a.i yes 3
7.d odd 6 1 637.2.e.k 6
21.g even 6 1 5733.2.a.bd 3
21.h odd 6 1 5733.2.a.be 3
91.r even 6 1 8281.2.a.bh 3
91.s odd 6 1 8281.2.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.h 3 7.c even 3 1
637.2.a.i yes 3 7.d odd 6 1
637.2.e.k 6 7.b odd 2 1
637.2.e.k 6 7.d odd 6 1
637.2.e.l 6 1.a even 1 1 trivial
637.2.e.l 6 7.c even 3 1 inner
5733.2.a.bd 3 21.g even 6 1
5733.2.a.be 3 21.h odd 6 1
8281.2.a.bh 3 91.r even 6 1
8281.2.a.bk 3 91.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{6} - 2T_{2}^{5} + 8T_{2}^{4} - 4T_{2}^{3} + 28T_{2}^{2} - 24T_{2} + 36$$ T2^6 - 2*T2^5 + 8*T2^4 - 4*T2^3 + 28*T2^2 - 24*T2 + 36 $$T_{3}^{6} - 4T_{3}^{5} + 18T_{3}^{4} - 20T_{3}^{3} + 60T_{3}^{2} - 28T_{3} + 196$$ T3^6 - 4*T3^5 + 18*T3^4 - 20*T3^3 + 60*T3^2 - 28*T3 + 196 $$T_{5}^{6} - 5T_{5}^{5} + 22T_{5}^{4} - 29T_{5}^{3} + 44T_{5}^{2} + 21T_{5} + 49$$ T5^6 - 5*T5^5 + 22*T5^4 - 29*T5^3 + 44*T5^2 + 21*T5 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 2 T^{5} + 8 T^{4} - 4 T^{3} + \cdots + 36$$
$3$ $$T^{6} - 4 T^{5} + 18 T^{4} - 20 T^{3} + \cdots + 196$$
$5$ $$T^{6} - 5 T^{5} + 22 T^{4} - 29 T^{3} + \cdots + 49$$
$7$ $$T^{6}$$
$11$ $$T^{6} - 4 T^{5} + 16 T^{4} - 4 T^{3} + \cdots + 4$$
$13$ $$(T - 1)^{6}$$
$17$ $$T^{6} - 4 T^{5} + 18 T^{4} - 20 T^{3} + \cdots + 196$$
$19$ $$T^{6} - 7 T^{5} + 52 T^{4} + \cdots + 3969$$
$23$ $$T^{6} + T^{5} + 42 T^{4} - 127 T^{3} + \cdots + 1849$$
$29$ $$(T^{3} + 7 T^{2} - 13 T - 3)^{2}$$
$31$ $$T^{6} + 3 T^{5} + 50 T^{4} + \cdots + 2401$$
$37$ $$T^{6} - 10 T^{5} + 92 T^{4} + \cdots + 6724$$
$41$ $$(T^{3} + 6 T^{2} - 116 T - 504)^{2}$$
$43$ $$(T^{3} - 9 T^{2} - 5 T + 101)^{2}$$
$47$ $$T^{6} - 17 T^{5} + 200 T^{4} + \cdots + 21609$$
$53$ $$T^{6} + 13 T^{5} + 130 T^{4} + \cdots + 81$$
$59$ $$T^{6} - 22 T^{5} + 340 T^{4} + \cdots + 63504$$
$61$ $$T^{6} + 24 T^{5} + 416 T^{4} + \cdots + 50176$$
$67$ $$T^{6} - 14 T^{5} + 232 T^{4} + \cdots + 419904$$
$71$ $$(T^{3} - 4 T^{2} - 44 T + 194)^{2}$$
$73$ $$T^{6} - 5 T^{5} + 244 T^{4} + \cdots + 2436721$$
$79$ $$T^{6} + T^{5} + 70 T^{4} - 267 T^{3} + \cdots + 9801$$
$83$ $$(T^{3} + 23 T^{2} + 127 T + 203)^{2}$$
$89$ $$T^{6} + 11 T^{5} + 176 T^{4} + \cdots + 441$$
$97$ $$(T^{3} - 3 T^{2} - 71 T + 7)^{2}$$