# Properties

 Label 126.2.e.c Level $126$ Weight $2$ Character orbit 126.e Analytic conductor $1.006$ 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] = [126,2,Mod(25,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 4]))

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

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

chi := DirichletCharacter("126.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 126.e (of order $$3$$, degree $$2$$, 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: $$1.00611506547$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3$$ (v^5 - v^4 + 5*v^3 + v^2 + 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3$$ (-v^5 + v^4 - 5*v^3 + 2*v^2 - 3*v) / 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3$$ (-2*v^5 + 5*v^4 - 16*v^3 + 19*v^2 - 21*v + 6) / 3 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3$$ (2*v^5 - 5*v^4 + 19*v^3 - 22*v^2 + 33*v - 9) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 - 2$$ b3 + b2 + b1 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1$$ b5 + b4 + b3 + b2 - 3*b1 - 1 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6$$ 2*b5 + 3*b4 - 5*b3 - 3*b2 - 6*b1 + 6 $$\nu^{5}$$ $$=$$ $$-3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7$$ -3*b5 - 2*b4 - 11*b3 - 6*b2 + 8*b1 + 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$-1 - \beta_{4}$$ $$-1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i 0.5 + 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i
−1.00000 −1.64400 0.545231i 1.00000 −0.794182 + 1.37556i 1.64400 + 0.545231i 1.23855 + 2.33795i −1.00000 2.40545 + 1.79272i 0.794182 1.37556i
25.2 −1.00000 0.933463 1.45899i 1.00000 −0.296790 + 0.514055i −0.933463 + 1.45899i 2.32383 1.26483i −1.00000 −1.25729 2.72382i 0.296790 0.514055i
25.3 −1.00000 1.71053 + 0.272169i 1.00000 1.59097 2.75564i −1.71053 0.272169i −2.56238 + 0.658939i −1.00000 2.85185 + 0.931107i −1.59097 + 2.75564i
121.1 −1.00000 −1.64400 + 0.545231i 1.00000 −0.794182 1.37556i 1.64400 0.545231i 1.23855 2.33795i −1.00000 2.40545 1.79272i 0.794182 + 1.37556i
121.2 −1.00000 0.933463 + 1.45899i 1.00000 −0.296790 0.514055i −0.933463 1.45899i 2.32383 + 1.26483i −1.00000 −1.25729 + 2.72382i 0.296790 + 0.514055i
121.3 −1.00000 1.71053 0.272169i 1.00000 1.59097 + 2.75564i −1.71053 + 0.272169i −2.56238 0.658939i −1.00000 2.85185 0.931107i −1.59097 2.75564i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.e.c 6
3.b odd 2 1 378.2.e.d 6
4.b odd 2 1 1008.2.q.g 6
7.b odd 2 1 882.2.e.o 6
7.c even 3 1 126.2.h.d yes 6
7.c even 3 1 882.2.f.n 6
7.d odd 6 1 882.2.f.o 6
7.d odd 6 1 882.2.h.p 6
9.c even 3 1 126.2.h.d yes 6
9.c even 3 1 1134.2.g.m 6
9.d odd 6 1 378.2.h.c 6
9.d odd 6 1 1134.2.g.l 6
12.b even 2 1 3024.2.q.g 6
21.c even 2 1 2646.2.e.p 6
21.g even 6 1 2646.2.f.m 6
21.g even 6 1 2646.2.h.o 6
21.h odd 6 1 378.2.h.c 6
21.h odd 6 1 2646.2.f.l 6
28.g odd 6 1 1008.2.t.h 6
36.f odd 6 1 1008.2.t.h 6
36.h even 6 1 3024.2.t.h 6
63.g even 3 1 882.2.f.n 6
63.g even 3 1 1134.2.g.m 6
63.h even 3 1 inner 126.2.e.c 6
63.h even 3 1 7938.2.a.bv 3
63.i even 6 1 2646.2.e.p 6
63.i even 6 1 7938.2.a.bz 3
63.j odd 6 1 378.2.e.d 6
63.j odd 6 1 7938.2.a.ca 3
63.k odd 6 1 882.2.f.o 6
63.l odd 6 1 882.2.h.p 6
63.n odd 6 1 1134.2.g.l 6
63.n odd 6 1 2646.2.f.l 6
63.o even 6 1 2646.2.h.o 6
63.s even 6 1 2646.2.f.m 6
63.t odd 6 1 882.2.e.o 6
63.t odd 6 1 7938.2.a.bw 3
84.n even 6 1 3024.2.t.h 6
252.u odd 6 1 1008.2.q.g 6
252.bb even 6 1 3024.2.q.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 1.a even 1 1 trivial
126.2.e.c 6 63.h even 3 1 inner
126.2.h.d yes 6 7.c even 3 1
126.2.h.d yes 6 9.c even 3 1
378.2.e.d 6 3.b odd 2 1
378.2.e.d 6 63.j odd 6 1
378.2.h.c 6 9.d odd 6 1
378.2.h.c 6 21.h odd 6 1
882.2.e.o 6 7.b odd 2 1
882.2.e.o 6 63.t odd 6 1
882.2.f.n 6 7.c even 3 1
882.2.f.n 6 63.g even 3 1
882.2.f.o 6 7.d odd 6 1
882.2.f.o 6 63.k odd 6 1
882.2.h.p 6 7.d odd 6 1
882.2.h.p 6 63.l odd 6 1
1008.2.q.g 6 4.b odd 2 1
1008.2.q.g 6 252.u odd 6 1
1008.2.t.h 6 28.g odd 6 1
1008.2.t.h 6 36.f odd 6 1
1134.2.g.l 6 9.d odd 6 1
1134.2.g.l 6 63.n odd 6 1
1134.2.g.m 6 9.c even 3 1
1134.2.g.m 6 63.g even 3 1
2646.2.e.p 6 21.c even 2 1
2646.2.e.p 6 63.i even 6 1
2646.2.f.l 6 21.h odd 6 1
2646.2.f.l 6 63.n odd 6 1
2646.2.f.m 6 21.g even 6 1
2646.2.f.m 6 63.s even 6 1
2646.2.h.o 6 21.g even 6 1
2646.2.h.o 6 63.o even 6 1
3024.2.q.g 6 12.b even 2 1
3024.2.q.g 6 252.bb even 6 1
3024.2.t.h 6 36.h even 6 1
3024.2.t.h 6 84.n even 6 1
7938.2.a.bv 3 63.h even 3 1
7938.2.a.bw 3 63.t odd 6 1
7938.2.a.bz 3 63.i even 6 1
7938.2.a.ca 3 63.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - T_{5}^{5} + 7T_{5}^{4} + 12T_{5}^{3} + 33T_{5}^{2} + 18T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{6}$$
$3$ $$T^{6} - 2 T^{5} - 2 T^{4} + 9 T^{3} + \cdots + 27$$
$5$ $$T^{6} - T^{5} + 7 T^{4} + 12 T^{3} + \cdots + 9$$
$7$ $$T^{6} - 2 T^{5} - 4 T^{4} + 31 T^{3} + \cdots + 343$$
$11$ $$T^{6} + T^{5} + 7 T^{4} - 12 T^{3} + \cdots + 9$$
$13$ $$T^{6} - 8 T^{5} + 63 T^{4} + \cdots + 4761$$
$17$ $$T^{6} + 4 T^{5} + 28 T^{4} + 240 T^{2} + \cdots + 576$$
$19$ $$T^{6} + 3 T^{5} + 45 T^{4} + \cdots + 2401$$
$23$ $$T^{6} + 7 T^{5} + 37 T^{4} + 78 T^{3} + \cdots + 9$$
$29$ $$T^{6} + 5 T^{5} + 91 T^{4} + \cdots + 131769$$
$31$ $$(T^{3} + 20 T^{2} + 121 T + 201)^{2}$$
$37$ $$(T^{2} - T + 1)^{3}$$
$41$ $$T^{6} + 33 T^{4} + 18 T^{3} + 1089 T^{2} + \cdots + 81$$
$43$ $$T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 16129$$
$47$ $$(T^{3} - 9 T^{2} - 54 T + 189)^{2}$$
$53$ $$T^{6} - 15 T^{5} + 159 T^{4} + \cdots + 6561$$
$59$ $$(T^{3} - 14 T^{2} + 39 T + 63)^{2}$$
$61$ $$(T^{3} + 8 T^{2} - 5 T - 93)^{2}$$
$67$ $$(T^{3} + T^{2} - 112 T - 211)^{2}$$
$71$ $$(T^{3} - 7 T^{2} - 198 T + 1593)^{2}$$
$73$ $$T^{6} - 19 T^{5} + 353 T^{4} + \cdots + 398161$$
$79$ $$(T^{3} + 5 T^{2} - 74 T - 321)^{2}$$
$83$ $$T^{6} - 2 T^{5} + 67 T^{4} + \cdots + 21609$$
$89$ $$T^{6} + 9 T^{5} + 123 T^{4} - 396 T^{3} + \cdots + 81$$
$97$ $$T^{6} - 28 T^{5} + 572 T^{4} + \cdots + 61504$$