# Properties

 Label 140.2.g.c Level $140$ Weight $2$ Character orbit 140.g Analytic conductor $1.118$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,2,Mod(111,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("140.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.g (of order $$2$$, degree $$1$$, 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: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.342102016.5 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16$$ x^8 + x^6 + 4*x^4 + 4*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16$$ (-v^7 + v^5 + 2*v^3) / 16 $$\beta_{2}$$ $$=$$ $$( \nu^{7} + \nu^{5} + 4\nu^{3} + 4\nu ) / 8$$ (v^7 + v^5 + 4*v^3 + 4*v) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - \nu^{6} + 3\nu^{5} - 3\nu^{4} + 2\nu^{3} - 10\nu^{2} + 16\nu - 8 ) / 16$$ (v^7 - v^6 + 3*v^5 - 3*v^4 + 2*v^3 - 10*v^2 + 16*v - 8) / 16 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + \nu^{6} - 3\nu^{5} + 3\nu^{4} - 2\nu^{3} - 6\nu^{2} + 8 ) / 16$$ (-v^7 + v^6 - 3*v^5 + 3*v^4 - 2*v^3 - 6*v^2 + 8) / 16 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} + 10\nu^{2} + 16\nu + 8 ) / 16$$ (v^7 + v^6 + 3*v^5 + 3*v^4 + 2*v^3 + 10*v^2 + 16*v + 8) / 16 $$\beta_{6}$$ $$=$$ $$( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} - 6\nu^{2} + 8 ) / 16$$ (v^7 + v^6 + 3*v^5 + 3*v^4 + 2*v^3 - 6*v^2 + 8) / 16 $$\beta_{7}$$ $$=$$ $$( -\nu^{6} + \nu^{4} + 2\nu^{2} ) / 8$$ (-v^6 + v^4 + 2*v^2) / 8
 $$\nu$$ $$=$$ $$( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} ) / 2$$ (-b6 + b5 + b4 + b3) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} ) / 2$$ (-b6 + b5 - b4 - b3) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 4\beta_{2} + 4\beta_1 ) / 2$$ (-b6 - b5 + b4 - b3 + 4*b2 + 4*b1) / 2 $$\nu^{4}$$ $$=$$ $$( 4\beta_{7} + 3\beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{3} - 4 ) / 2$$ (4*b7 + 3*b6 + b5 + 3*b4 - b3 - 4) / 2 $$\nu^{5}$$ $$=$$ $$( 5\beta_{6} + \beta_{5} - 5\beta_{4} + \beta_{3} - 4\beta_{2} + 4\beta_1 ) / 2$$ (5*b6 + b5 - 5*b4 + b3 - 4*b2 + 4*b1) / 2 $$\nu^{6}$$ $$=$$ $$( -12\beta_{7} + \beta_{6} + 3\beta_{5} + \beta_{4} - 3\beta_{3} - 4 ) / 2$$ (-12*b7 + b6 + 3*b5 + b4 - 3*b3 - 4) / 2 $$\nu^{7}$$ $$=$$ $$( 3\beta_{6} - \beta_{5} - 3\beta_{4} - \beta_{3} + 4\beta_{2} - 20\beta_1 ) / 2$$ (3*b6 - b5 - 3*b4 - b3 + 4*b2 - 20*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$71$$ $$101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
111.1
 1.17915 + 0.780776i −1.17915 − 0.780776i 1.17915 − 0.780776i −1.17915 + 0.780776i −0.599676 + 1.28078i 0.599676 − 1.28078i −0.599676 − 1.28078i 0.599676 + 1.28078i
−0.780776 1.17915i −3.02045 −0.780776 + 1.84130i 1.00000i 2.35829 + 3.56155i 2.17238 1.51022i 2.78078 0.516994i 6.12311 1.17915 0.780776i
111.2 −0.780776 1.17915i 3.02045 −0.780776 + 1.84130i 1.00000i −2.35829 3.56155i −2.17238 1.51022i 2.78078 0.516994i 6.12311 −1.17915 + 0.780776i
111.3 −0.780776 + 1.17915i −3.02045 −0.780776 1.84130i 1.00000i 2.35829 3.56155i 2.17238 + 1.51022i 2.78078 + 0.516994i 6.12311 1.17915 + 0.780776i
111.4 −0.780776 + 1.17915i 3.02045 −0.780776 1.84130i 1.00000i −2.35829 + 3.56155i −2.17238 + 1.51022i 2.78078 + 0.516994i 6.12311 −1.17915 0.780776i
111.5 1.28078 0.599676i −0.936426 1.28078 1.53610i 1.00000i −1.19935 + 0.561553i 2.60399 + 0.468213i 0.719224 2.73546i −2.12311 −0.599676 1.28078i
111.6 1.28078 0.599676i 0.936426 1.28078 1.53610i 1.00000i 1.19935 0.561553i −2.60399 + 0.468213i 0.719224 2.73546i −2.12311 0.599676 + 1.28078i
111.7 1.28078 + 0.599676i −0.936426 1.28078 + 1.53610i 1.00000i −1.19935 0.561553i 2.60399 0.468213i 0.719224 + 2.73546i −2.12311 −0.599676 + 1.28078i
111.8 1.28078 + 0.599676i 0.936426 1.28078 + 1.53610i 1.00000i 1.19935 + 0.561553i −2.60399 0.468213i 0.719224 + 2.73546i −2.12311 0.599676 1.28078i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 111.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.g.c 8
3.b odd 2 1 1260.2.c.c 8
4.b odd 2 1 inner 140.2.g.c 8
5.b even 2 1 700.2.g.j 8
5.c odd 4 1 700.2.c.i 8
5.c odd 4 1 700.2.c.j 8
7.b odd 2 1 inner 140.2.g.c 8
7.c even 3 2 980.2.o.e 16
7.d odd 6 2 980.2.o.e 16
8.b even 2 1 2240.2.k.e 8
8.d odd 2 1 2240.2.k.e 8
12.b even 2 1 1260.2.c.c 8
20.d odd 2 1 700.2.g.j 8
20.e even 4 1 700.2.c.i 8
20.e even 4 1 700.2.c.j 8
21.c even 2 1 1260.2.c.c 8
28.d even 2 1 inner 140.2.g.c 8
28.f even 6 2 980.2.o.e 16
28.g odd 6 2 980.2.o.e 16
35.c odd 2 1 700.2.g.j 8
35.f even 4 1 700.2.c.i 8
35.f even 4 1 700.2.c.j 8
56.e even 2 1 2240.2.k.e 8
56.h odd 2 1 2240.2.k.e 8
84.h odd 2 1 1260.2.c.c 8
140.c even 2 1 700.2.g.j 8
140.j odd 4 1 700.2.c.i 8
140.j odd 4 1 700.2.c.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c 8 1.a even 1 1 trivial
140.2.g.c 8 4.b odd 2 1 inner
140.2.g.c 8 7.b odd 2 1 inner
140.2.g.c 8 28.d even 2 1 inner
700.2.c.i 8 5.c odd 4 1
700.2.c.i 8 20.e even 4 1
700.2.c.i 8 35.f even 4 1
700.2.c.i 8 140.j odd 4 1
700.2.c.j 8 5.c odd 4 1
700.2.c.j 8 20.e even 4 1
700.2.c.j 8 35.f even 4 1
700.2.c.j 8 140.j odd 4 1
700.2.g.j 8 5.b even 2 1
700.2.g.j 8 20.d odd 2 1
700.2.g.j 8 35.c odd 2 1
700.2.g.j 8 140.c even 2 1
980.2.o.e 16 7.c even 3 2
980.2.o.e 16 7.d odd 6 2
980.2.o.e 16 28.f even 6 2
980.2.o.e 16 28.g odd 6 2
1260.2.c.c 8 3.b odd 2 1
1260.2.c.c 8 12.b even 2 1
1260.2.c.c 8 21.c even 2 1
1260.2.c.c 8 84.h odd 2 1
2240.2.k.e 8 8.b even 2 1
2240.2.k.e 8 8.d odd 2 1
2240.2.k.e 8 56.e even 2 1
2240.2.k.e 8 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(140, [\chi])$$:

 $$T_{3}^{4} - 10T_{3}^{2} + 8$$ T3^4 - 10*T3^2 + 8 $$T_{19}^{4} - 28T_{19}^{2} + 128$$ T19^4 - 28*T19^2 + 128

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{3} - 2 T + 4)^{2}$$
$3$ $$(T^{4} - 10 T^{2} + 8)^{2}$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$T^{8} - 18 T^{6} + 162 T^{4} + \cdots + 2401$$
$11$ $$(T^{4} + 28 T^{2} + 128)^{2}$$
$13$ $$(T^{2} + 4)^{4}$$
$17$ $$(T^{4} + 52 T^{2} + 64)^{2}$$
$19$ $$(T^{4} - 28 T^{2} + 128)^{2}$$
$23$ $$(T^{4} + 74 T^{2} + 1352)^{2}$$
$29$ $$(T + 2)^{8}$$
$31$ $$(T^{4} - 56 T^{2} + 512)^{2}$$
$37$ $$(T - 2)^{8}$$
$41$ $$(T^{4} + 52 T^{2} + 64)^{2}$$
$43$ $$(T^{4} + 58 T^{2} + 8)^{2}$$
$47$ $$(T^{4} - 126 T^{2} + 2592)^{2}$$
$53$ $$(T - 2)^{8}$$
$59$ $$(T^{4} - 124 T^{2} + 512)^{2}$$
$61$ $$(T^{2} + 4)^{4}$$
$67$ $$(T^{4} + 218 T^{2} + 2888)^{2}$$
$71$ $$(T^{4} + 92 T^{2} + 2048)^{2}$$
$73$ $$(T^{4} + 324 T^{2} + 20736)^{2}$$
$79$ $$(T^{4} + 20 T^{2} + 32)^{2}$$
$83$ $$(T^{4} - 10 T^{2} + 8)^{2}$$
$89$ $$(T^{2} + 144)^{4}$$
$97$ $$(T^{4} + 52 T^{2} + 64)^{2}$$