# Properties

 Label 140.2.c.b Level $140$ Weight $2$ Character orbit 140.c Analytic conductor $1.118$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,2,Mod(139,140)]

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, 1, 1]))

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

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

chi := DirichletCharacter("140.139");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.c (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: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 6x^{14} + 28x^{12} + 16x^{10} - 40x^{8} + 610x^{6} + 1625x^{4} - 524x^{2} + 1444$$ x^16 - 6*x^14 + 28*x^12 + 16*x^10 - 40*x^8 + 610*x^6 + 1625*x^4 - 524*x^2 + 1444 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - \beta_{9} q^{3} + (\beta_{6} - 1) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{15} + \beta_{11} - \beta_{5} - \beta_1) q^{6} + ( - \beta_{14} - \beta_{9} + \beta_{4} + \beta_1) q^{7} + ( - \beta_{12} - \beta_{3}) q^{8} + ( - \beta_{10} + \beta_{6}) q^{9}+O(q^{10})$$ q + b3 * q^2 - b9 * q^3 + (b6 - 1) * q^4 + (-b2 - b1) * q^5 + (-b15 + b11 - b5 - b1) * q^6 + (-b14 - b9 + b4 + b1) * q^7 + (-b12 - b3) * q^8 + (-b10 + b6) * q^9 $$q + \beta_{3} q^{2} - \beta_{9} q^{3} + (\beta_{6} - 1) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{15} + \beta_{11} - \beta_{5} - \beta_1) q^{6} + ( - \beta_{14} - \beta_{9} + \beta_{4} + \beta_1) q^{7} + ( - \beta_{12} - \beta_{3}) q^{8} + ( - \beta_{10} + \beta_{6}) q^{9} + (\beta_{15} + \beta_{14} - \beta_{13} - \beta_1) q^{10} + ( - \beta_{8} - \beta_{6}) q^{11} + (\beta_{14} + 2 \beta_{9} - \beta_{2} - \beta_1) q^{12} + ( - \beta_{15} + \beta_{9} + \beta_{5} + \beta_{2}) q^{13} + (\beta_{14} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{6}) q^{14} + ( - \beta_{10} + \beta_{8} + \beta_{4}) q^{15} + (\beta_{10} + \beta_{8} - \beta_{6}) q^{16} + (\beta_{15} - \beta_{14} - \beta_{9} - \beta_{5} + \beta_{2} + \beta_1) q^{17} + ( - \beta_{12} - \beta_{7}) q^{18} + (\beta_{15} + \beta_{13} + \beta_{5} + \beta_1) q^{19} + ( - 2 \beta_{14} - \beta_{11} - \beta_{5} + \beta_{2} + 2 \beta_1) q^{20} + ( - \beta_{15} + \beta_{13} - \beta_{10} + \beta_{6} - \beta_{5} + \beta_1 - 1) q^{21} + (\beta_{12} - 2 \beta_{4}) q^{22} + (\beta_{7} - 2 \beta_{4} - \beta_{3}) q^{23} + (2 \beta_{15} - \beta_{13} - 2 \beta_{11} + 2 \beta_{5} + \beta_1) q^{24} + (2 \beta_{12} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 1) q^{25} + ( - \beta_{14} + \beta_{13} + \beta_{11} - \beta_1) q^{26} + (\beta_{14} - \beta_{9} - \beta_1) q^{27} + ( - \beta_{15} + \beta_{12} + 2 \beta_{9} + \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{28} + (\beta_{10} - \beta_{6} - 1) q^{29} + (\beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{4}) q^{30} + (\beta_{15} - \beta_{14} - \beta_{13} - 4 \beta_{11} + \beta_{5} + 2 \beta_1) q^{31} + (\beta_{12} + \beta_{7} + 2 \beta_{4}) q^{32} + ( - \beta_{15} + \beta_{14} + \beta_{9} + \beta_{5} - \beta_{2} - \beta_1) q^{33} + ( - \beta_{15} + \beta_{13} - \beta_{11} - \beta_{5} + 2 \beta_1) q^{34} + ( - \beta_{15} + 2 \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_1) q^{35} + (\beta_{10} + \beta_{8} + \beta_{6} + 2) q^{36} + ( - \beta_{7} - 3 \beta_{3}) q^{37} + ( - \beta_{15} + \beta_{14} + \beta_{5} + 2 \beta_{2} - \beta_1) q^{38} + (\beta_{8} + \beta_{6}) q^{39} + (\beta_{15} + \beta_{13} - \beta_{11} - 2 \beta_{9} - 2 \beta_{2} + 3 \beta_1) q^{40} + (3 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} + 3 \beta_{5}) q^{41} + ( - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{7} + \beta_{5} - \beta_{3} - 2 \beta_{2} - \beta_1) q^{42} + ( - \beta_{7} + \beta_{3}) q^{43} + ( - 3 \beta_{10} + \beta_{8} + 2 \beta_{6} + 1) q^{44} + ( - 2 \beta_{14} + \beta_{13} - \beta_{9} - 2 \beta_{5}) q^{45} + ( - 2 \beta_{10} + 2 \beta_{8} - 2) q^{46} + (\beta_{14} + 5 \beta_{9} - \beta_1) q^{47} + (\beta_{15} - 3 \beta_{14} - 4 \beta_{9} - \beta_{5} + 2 \beta_{2} + 3 \beta_1) q^{48} + ( - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{10} - 2 \beta_{6} - 2 \beta_{5} + \cdots + 1) q^{49}+ \cdots + ( - 2 \beta_{10} - 2 \beta_{6}) q^{99}+O(q^{100})$$ q + b3 * q^2 - b9 * q^3 + (b6 - 1) * q^4 + (-b2 - b1) * q^5 + (-b15 + b11 - b5 - b1) * q^6 + (-b14 - b9 + b4 + b1) * q^7 + (-b12 - b3) * q^8 + (-b10 + b6) * q^9 + (b15 + b14 - b13 - b1) * q^10 + (-b8 - b6) * q^11 + (b14 + 2*b9 - b2 - b1) * q^12 + (-b15 + b9 + b5 + b2) * q^13 + (b14 + b11 + b10 - b8 - b6) * q^14 + (-b10 + b8 + b4) * q^15 + (b10 + b8 - b6) * q^16 + (b15 - b14 - b9 - b5 + b2 + b1) * q^17 + (-b12 - b7) * q^18 + (b15 + b13 + b5 + b1) * q^19 + (-2*b14 - b11 - b5 + b2 + 2*b1) * q^20 + (-b15 + b13 - b10 + b6 - b5 + b1 - 1) * q^21 + (b12 - 2*b4) * q^22 + (b7 - 2*b4 - b3) * q^23 + (2*b15 - b13 - 2*b11 + 2*b5 + b1) * q^24 + (2*b12 + b10 + b7 - b6 - b4 + b3 + 1) * q^25 + (-b14 + b13 + b11 - b1) * q^26 + (b14 - b9 - b1) * q^27 + (-b15 + b12 + 2*b9 + b7 + b5 - 2*b4 + b2) * q^28 + (b10 - b6 - 1) * q^29 + (b10 - b8 - b7 - b6 + 2*b4) * q^30 + (b15 - b14 - b13 - 4*b11 + b5 + 2*b1) * q^31 + (b12 + b7 + 2*b4) * q^32 + (-b15 + b14 + b9 + b5 - b2 - b1) * q^33 + (-b15 + b13 - b11 - b5 + 2*b1) * q^34 + (-b15 + 2*b11 + b9 + b8 - b7 + b6 - b5 + b3 - b1) * q^35 + (b10 + b8 + b6 + 2) * q^36 + (-b7 - 3*b3) * q^37 + (-b15 + b14 + b5 + 2*b2 - b1) * q^38 + (b8 + b6) * q^39 + (b15 + b13 - b11 - 2*b9 - 2*b2 + 3*b1) * q^40 + (3*b15 + 3*b14 - 3*b13 + 3*b5) * q^41 + (-b15 + b14 - b12 - b7 + b5 - b3 - 2*b2 - b1) * q^42 + (-b7 + b3) * q^43 + (-3*b10 + b8 + 2*b6 + 1) * q^44 + (-2*b14 + b13 - b9 - 2*b5) * q^45 + (-2*b10 + 2*b8 - 2) * q^46 + (b14 + 5*b9 - b1) * q^47 + (b15 - 3*b14 - 4*b9 - b5 + 2*b2 + 3*b1) * q^48 + (-2*b15 - 2*b14 + 2*b13 + 2*b10 - 2*b6 - 2*b5 + 1) * q^49 + (b12 - 3*b10 - b8 + b7 + b6 + b3 - 2) * q^50 + (2*b10 - 3*b8 - b6) * q^51 + (3*b14 + 2*b9 + b2 - 3*b1) * q^52 + (-4*b12 - 2*b7 + 2*b4 - 2*b3) * q^53 + (-2*b15 - b14 + b11 - 2*b5 - 2*b1) * q^54 + (b15 + 2*b14 + b13 + 3*b9 + b5 - b1) * q^55 + (b15 - b14 + b13 - 3*b10 + b8 + b6 + b5 - 2) * q^56 + (b7 + 3*b3) * q^57 + (b12 + b7 - b3) * q^58 + (-4*b15 + b14 - 2*b13 + 4*b11 - 4*b5 - 5*b1) * q^59 + (b12 + 2*b10 - 2*b8 + b7 - b6 - 2*b4 + 3) * q^60 + (b15 + 2*b14 - b13 + b5 + b1) * q^61 + (2*b15 - 4*b14 - 8*b9 - 2*b5 - 2*b2 + 4*b1) * q^62 + (-2*b14 + b7 - b4 - b3 + 2*b1) * q^63 + (b10 - 3*b8 - 3*b6 - 2) * q^64 + (-2*b12 + 2*b10 - 2*b6 + b4 + 2*b3 - 1) * q^65 + (b15 - b13 + b11 + b5 - 2*b1) * q^66 - 2*b4 * q^67 + (-b15 - 2*b9 + b5 - 3*b2) * q^68 + (3*b15 + b14 - 3*b13 + 3*b5 - 2*b1) * q^69 + (b15 + 2*b14 - b12 - b11 + 4*b9 + 2*b6 + 2*b5 + 2*b4 + 2) * q^70 + (4*b10 - 2*b8 + 2*b6) * q^71 + (-b12 + b7 + 2*b4 + 2*b3) * q^72 + (2*b15 - 2*b14 - 2*b9 - 2*b5 + 2*b2 + 2*b1) * q^73 + (-2*b6 + 6) * q^74 + (-2*b15 + 4*b11 - 3*b9 - 2*b5 - 2*b1) * q^75 + (-3*b15 - 3*b14 + 2*b13 + 2*b11 - 3*b5 - 3*b1) * q^76 + (b15 + b14 - b9 + b7 - b5 + 3*b3 - 3*b2 - b1) * q^77 + (-b12 + 2*b4) * q^78 + (-2*b10 + 3*b8 + b6) * q^79 + (-b15 + 2*b14 - 2*b13 + b11 - 2*b9 + 2*b5) * q^80 + (-4*b10 + 4*b6 - 5) * q^81 + (-6*b14 + 6*b2 + 6*b1) * q^82 + (5*b14 + 4*b9 - 5*b1) * q^83 + (b15 + b14 - 2*b13 + 2*b11 + b10 + b8 + b5 - 3*b1 + 3) * q^84 + (-3*b10 - b7 + 3*b6 - 3*b3 - 5) * q^85 + (2*b6 + 2) * q^86 + (-b14 - b9 + b1) * q^87 + (-2*b12 - 3*b7 + 2*b4 + b3) * q^88 + (-b15 - b14 + b13 - b5) * q^89 + (3*b14 - b11 - b5 - 2*b2 - b1) * q^90 + (-b15 - b13 - 2*b10 - b8 - 3*b6 - b5 - b1) * q^91 + (-2*b7 + 4*b4 - 2*b3) * q^92 + (4*b12 + 3*b7 - 2*b4 + 5*b3) * q^93 + (4*b15 - b14 - 5*b11 + 4*b5 + 4*b1) * q^94 + (b10 + 2*b8 + 2*b7 + 3*b6 - b4 - 2*b3) * q^95 + (-3*b15 + b14 + 2*b13 + 2*b11 - 3*b5 + b1) * q^96 + (-3*b15 + 3*b14 + 3*b9 + 3*b5 - 3*b2 - 3*b1) * q^97 + (4*b14 + 2*b12 + 2*b7 + b3 - 4*b2 - 4*b1) * q^98 + (-2*b10 - 2*b6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 12 q^{4} + 8 q^{9}+O(q^{10})$$ 16 * q - 12 * q^4 + 8 * q^9 $$16 q - 12 q^{4} + 8 q^{9} - 4 q^{14} - 12 q^{16} - 8 q^{21} + 8 q^{25} - 24 q^{29} - 4 q^{30} + 28 q^{36} + 32 q^{44} - 32 q^{46} - 12 q^{50} - 20 q^{56} + 44 q^{60} - 36 q^{64} - 32 q^{65} + 40 q^{70} + 88 q^{74} - 48 q^{81} + 40 q^{84} - 56 q^{85} + 40 q^{86}+O(q^{100})$$ 16 * q - 12 * q^4 + 8 * q^9 - 4 * q^14 - 12 * q^16 - 8 * q^21 + 8 * q^25 - 24 * q^29 - 4 * q^30 + 28 * q^36 + 32 * q^44 - 32 * q^46 - 12 * q^50 - 20 * q^56 + 44 * q^60 - 36 * q^64 - 32 * q^65 + 40 * q^70 + 88 * q^74 - 48 * q^81 + 40 * q^84 - 56 * q^85 + 40 * q^86

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6x^{14} + 28x^{12} + 16x^{10} - 40x^{8} + 610x^{6} + 1625x^{4} - 524x^{2} + 1444$$ :

 $$\beta_{1}$$ $$=$$ $$( - 160577 \nu^{15} - 19792868 \nu^{13} + 118635540 \nu^{11} - 640577256 \nu^{9} + 5860864 \nu^{7} + 1292054998 \nu^{5} - 13534107197 \nu^{3} + \cdots - 3536177966 \nu ) / 39980301296$$ (-160577*v^15 - 19792868*v^13 + 118635540*v^11 - 640577256*v^9 + 5860864*v^7 + 1292054998*v^5 - 13534107197*v^3 - 3536177966*v) / 39980301296 $$\beta_{2}$$ $$=$$ $$( - 701627 \nu^{15} - 5034292 \nu^{13} + 18194892 \nu^{11} - 122627780 \nu^{9} - 828065124 \nu^{7} + 442514098 \nu^{5} - 6218975035 \nu^{3} + \cdots - 19599822798 \nu ) / 9995075324$$ (-701627*v^15 - 5034292*v^13 + 18194892*v^11 - 122627780*v^9 - 828065124*v^7 + 442514098*v^5 - 6218975035*v^3 - 19599822798*v) / 9995075324 $$\beta_{3}$$ $$=$$ $$( - 4622027 \nu^{14} + 18920224 \nu^{12} - 55700874 \nu^{10} - 428065102 \nu^{8} + 435253284 \nu^{6} - 2539415580 \nu^{4} - 9983737673 \nu^{2} + \cdots - 9488500630 ) / 9995075324$$ (-4622027*v^14 + 18920224*v^12 - 55700874*v^10 - 428065102*v^8 + 435253284*v^6 - 2539415580*v^4 - 9983737673*v^2 - 9488500630) / 9995075324 $$\beta_{4}$$ $$=$$ $$( - 778741 \nu^{14} + 5214580 \nu^{12} - 27045620 \nu^{10} + 25125688 \nu^{8} - 6532248 \nu^{6} - 613266210 \nu^{4} + 311886279 \nu^{2} + \cdots + 1386720738 ) / 1537703896$$ (-778741*v^14 + 5214580*v^12 - 27045620*v^10 + 25125688*v^8 - 6532248*v^6 - 613266210*v^4 + 311886279*v^2 + 1386720738) / 1537703896 $$\beta_{5}$$ $$=$$ $$( 10105411 \nu^{15} - 36502060 \nu^{13} + 157774044 \nu^{11} + 518633832 \nu^{9} + 1113094440 \nu^{7} + 506135110 \nu^{5} + \cdots - 1520006582 \nu ) / 39980301296$$ (10105411*v^15 - 36502060*v^13 + 157774044*v^11 + 518633832*v^9 + 1113094440*v^7 + 506135110*v^5 + 35184267703*v^3 - 1520006582*v) / 39980301296 $$\beta_{6}$$ $$=$$ $$( 2006632 \nu^{14} - 7648781 \nu^{12} + 31976398 \nu^{10} + 145329823 \nu^{8} - 58607182 \nu^{6} + 1241411587 \nu^{4} + 5422581996 \nu^{2} + \cdots + 2576062655 ) / 2498768831$$ (2006632*v^14 - 7648781*v^12 + 31976398*v^10 + 145329823*v^8 - 58607182*v^6 + 1241411587*v^4 + 5422581996*v^2 + 2576062655) / 2498768831 $$\beta_{7}$$ $$=$$ $$( - 4095595 \nu^{14} + 28568136 \nu^{12} - 115147725 \nu^{10} - 60918915 \nu^{8} + 765952940 \nu^{6} - 2161769415 \nu^{4} + \cdots + 16119618620 ) / 4997537662$$ (-4095595*v^14 + 28568136*v^12 - 115147725*v^10 - 60918915*v^8 + 765952940*v^6 - 2161769415*v^4 - 4185433928*v^2 + 16119618620) / 4997537662 $$\beta_{8}$$ $$=$$ $$( 2918186 \nu^{14} - 20701507 \nu^{12} + 98570518 \nu^{10} - 52671282 \nu^{8} + 2447580 \nu^{6} + 929333314 \nu^{4} + 2868437416 \nu^{2} + \cdots - 3350947293 ) / 2498768831$$ (2918186*v^14 - 20701507*v^12 + 98570518*v^10 - 52671282*v^8 + 2447580*v^6 + 929333314*v^4 + 2868437416*v^2 - 3350947293) / 2498768831 $$\beta_{9}$$ $$=$$ $$( 5290343 \nu^{15} - 30566954 \nu^{13} + 139846998 \nu^{11} + 73573152 \nu^{9} + 8711350 \nu^{7} + 2721018508 \nu^{5} + 5600946403 \nu^{3} + \cdots + 4522635990 \nu ) / 9995075324$$ (5290343*v^15 - 30566954*v^13 + 139846998*v^11 + 73573152*v^9 + 8711350*v^7 + 2721018508*v^5 + 5600946403*v^3 + 4522635990*v) / 9995075324 $$\beta_{10}$$ $$=$$ $$( 3003030 \nu^{14} - 18626943 \nu^{12} + 87707502 \nu^{10} + 56300402 \nu^{8} - 257526656 \nu^{6} + 2149887220 \nu^{4} + 4823331344 \nu^{2} + \cdots - 3210930232 ) / 2498768831$$ (3003030*v^14 - 18626943*v^12 + 87707502*v^10 + 56300402*v^8 - 257526656*v^6 + 2149887220*v^4 + 4823331344*v^2 - 3210930232) / 2498768831 $$\beta_{11}$$ $$=$$ $$( - 26047779 \nu^{15} + 212152652 \nu^{13} - 1049471708 \nu^{11} + 905836904 \nu^{9} + 2069617144 \nu^{7} - 15041745238 \nu^{5} + \cdots + 94111892774 \nu ) / 39980301296$$ (-26047779*v^15 + 212152652*v^13 - 1049471708*v^11 + 905836904*v^9 + 2069617144*v^7 - 15041745238*v^5 - 25596257271*v^3 + 94111892774*v) / 39980301296 $$\beta_{12}$$ $$=$$ $$( 16464459 \nu^{14} - 97577124 \nu^{12} + 428256914 \nu^{10} + 435323342 \nu^{8} - 945411436 \nu^{6} + 8697856648 \nu^{4} + 35362350517 \nu^{2} + \cdots - 8632792082 ) / 9995075324$$ (16464459*v^14 - 97577124*v^12 + 428256914*v^10 + 435323342*v^8 - 945411436*v^6 + 8697856648*v^4 + 35362350517*v^2 - 8632792082) / 9995075324 $$\beta_{13}$$ $$=$$ $$( - 50893351 \nu^{15} + 310852820 \nu^{13} - 1355898420 \nu^{11} - 1154887384 \nu^{9} + 4332540464 \nu^{7} - 31332033382 \nu^{5} + \cdots + 95495899390 \nu ) / 39980301296$$ (-50893351*v^15 + 310852820*v^13 - 1355898420*v^11 - 1154887384*v^9 + 4332540464*v^7 - 31332033382*v^5 - 74199704683*v^3 + 95495899390*v) / 39980301296 $$\beta_{14}$$ $$=$$ $$( - 54021013 \nu^{15} + 251129916 \nu^{13} - 1045847772 \nu^{11} - 2926553016 \nu^{9} + 1032001696 \nu^{7} - 26977866994 \nu^{5} + \cdots - 69936350326 \nu ) / 39980301296$$ (-54021013*v^15 + 251129916*v^13 - 1045847772*v^11 - 2926553016*v^9 + 1032001696*v^7 - 26977866994*v^5 - 126143819217*v^3 - 69936350326*v) / 39980301296 $$\beta_{15}$$ $$=$$ $$( - 57832737 \nu^{15} + 374118884 \nu^{13} - 1798365156 \nu^{11} - 138285752 \nu^{9} + 2995610328 \nu^{7} - 37488440626 \nu^{5} + \cdots + 99947547522 \nu ) / 39980301296$$ (-57832737*v^15 + 374118884*v^13 - 1798365156*v^11 - 138285752*v^9 + 2995610328*v^7 - 37488440626*v^5 - 85289354813*v^3 + 99947547522*v) / 39980301296
 $$\nu$$ $$=$$ $$( -\beta_{14} + \beta_{13} + \beta_{2} + 2\beta_1 ) / 2$$ (-b14 + b13 + b2 + 2*b1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{12} - \beta_{10} - \beta_{8} + 2\beta_{3} + 1 ) / 2$$ (2*b12 - b10 - b8 + 2*b3 + 1) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{14} - 3\beta_{9} + \beta_{5} + 4\beta_1$$ -b14 - 3*b9 + b5 + 4*b1 $$\nu^{4}$$ $$=$$ $$( 8\beta_{12} - \beta_{10} - 7\beta_{8} + 4\beta_{7} - 8\beta_{6} - 8\beta_{4} - 4\beta_{3} - 5 ) / 2$$ (8*b12 - b10 - 7*b8 + 4*b7 - 8*b6 - 8*b4 - 4*b3 - 5) / 2 $$\nu^{5}$$ $$=$$ $$( -2\beta_{15} + 3\beta_{14} - 11\beta_{13} + 6\beta_{11} - 14\beta_{9} - 10\beta_{5} - 15\beta_{2} + 12\beta_1 ) / 2$$ (-2*b15 + 3*b14 - 11*b13 + 6*b11 - 14*b9 - 10*b5 - 15*b2 + 12*b1) / 2 $$\nu^{6}$$ $$=$$ $$9\beta_{12} - \beta_{8} + 15\beta_{7} - 17\beta_{6} - 6\beta_{4} - 20\beta_{3} - 38$$ 9*b12 - b8 + 15*b7 - 17*b6 - 6*b4 - 20*b3 - 38 $$\nu^{7}$$ $$=$$ $$( 34\beta_{15} + 35\beta_{14} - 69\beta_{13} - 36\beta_{11} - 16\beta_{9} - 58\beta_{5} - 115\beta_{2} - 60\beta_1 ) / 2$$ (34*b15 + 35*b14 - 69*b13 - 36*b11 - 16*b9 - 58*b5 - 115*b2 - 60*b1) / 2 $$\nu^{8}$$ $$=$$ $$( -96\beta_{12} + 155\beta_{10} + 67\beta_{8} + 88\beta_{7} - 78\beta_{6} + 112\beta_{4} - 184\beta_{3} - 273 ) / 2$$ (-96*b12 + 155*b10 + 67*b8 + 88*b7 - 78*b6 + 112*b4 - 184*b3 - 273) / 2 $$\nu^{9}$$ $$=$$ $$51\beta_{15} + 41\beta_{14} - 49\beta_{13} - 114\beta_{11} + 51\beta_{9} - 180\beta_{5} - 137\beta_{2} - 351\beta_1$$ 51*b15 + 41*b14 - 49*b13 - 114*b11 + 51*b9 - 180*b5 - 137*b2 - 351*b1 $$\nu^{10}$$ $$=$$ $$( -930\beta_{12} + 881\beta_{10} + 527\beta_{8} + 48\beta_{7} + 506\beta_{6} + 792\beta_{4} + 234\beta_{3} - 133 ) / 2$$ (-930*b12 + 881*b10 + 527*b8 + 48*b7 + 506*b6 + 792*b4 + 234*b3 - 133) / 2 $$\nu^{11}$$ $$=$$ $$( - 674 \beta_{15} + 117 \beta_{14} + 1233 \beta_{13} - 410 \beta_{11} + 1330 \beta_{9} - 774 \beta_{5} + 531 \beta_{2} - 3614 \beta_1 ) / 2$$ (-674*b15 + 117*b14 + 1233*b13 - 410*b11 + 1330*b9 - 774*b5 + 531*b2 - 3614*b1) / 2 $$\nu^{12}$$ $$=$$ $$- 2012 \beta_{12} + 566 \beta_{10} + 821 \beta_{8} - 1128 \beta_{7} + 3040 \beta_{6} + 920 \beta_{4} + 2648 \beta_{3} + 2279$$ -2012*b12 + 566*b10 + 821*b8 - 1128*b7 + 3040*b6 + 920*b4 + 2648*b3 + 2279 $$\nu^{13}$$ $$=$$ $$( - 7262 \beta_{15} - 169 \beta_{14} + 9417 \beta_{13} + 4980 \beta_{11} + 7652 \beta_{9} + 4530 \beta_{5} + 10397 \beta_{2} - 8374 \beta_1 ) / 2$$ (-7262*b15 - 169*b14 + 9417*b13 + 4980*b11 + 7652*b9 + 4530*b5 + 10397*b2 - 8374*b1) / 2 $$\nu^{14}$$ $$=$$ $$( - 3394 \beta_{12} - 17629 \beta_{10} - 17 \beta_{8} - 17336 \beta_{7} + 27208 \beta_{6} - 9120 \beta_{4} + 25686 \beta_{3} + 34869 ) / 2$$ (-3394*b12 - 17629*b10 - 17*b8 - 17336*b7 + 27208*b6 - 9120*b4 + 25686*b3 + 34869) / 2 $$\nu^{15}$$ $$=$$ $$- 12294 \beta_{15} - 1919 \beta_{14} + 14048 \beta_{13} + 19878 \beta_{11} + 12497 \beta_{9} + 27381 \beta_{5} + 28448 \beta_{2} + 20330 \beta_1$$ -12294*b15 - 1919*b14 + 14048*b13 + 19878*b11 + 12497*b9 + 27381*b5 + 28448*b2 + 20330*b1

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 0.744612 + 0.556573i −0.744612 − 0.556573i −0.744612 + 0.556573i 0.744612 − 0.556573i −0.328458 + 1.49331i 0.328458 − 1.49331i 0.328458 + 1.49331i −0.328458 − 1.49331i 1.61596 − 1.02509i −1.61596 + 1.02509i −1.61596 − 1.02509i 1.61596 + 1.02509i 2.05580 + 0.953651i −2.05580 − 0.953651i −2.05580 + 0.953651i 2.05580 − 0.953651i
−1.06789 0.927153i 0.662153i 0.280776 + 1.98019i 1.31119 + 1.81129i −0.613917 + 0.707107i 1.19935 + 2.35829i 1.53610 2.37495i 2.56155 0.279135 3.14993i
139.2 −1.06789 0.927153i 0.662153i 0.280776 + 1.98019i −1.31119 1.81129i 0.613917 0.707107i 1.19935 2.35829i 1.53610 2.37495i 2.56155 −0.279135 + 3.14993i
139.3 −1.06789 + 0.927153i 0.662153i 0.280776 1.98019i −1.31119 + 1.81129i 0.613917 + 0.707107i 1.19935 + 2.35829i 1.53610 + 2.37495i 2.56155 −0.279135 3.14993i
139.4 −1.06789 + 0.927153i 0.662153i 0.280776 1.98019i 1.31119 1.81129i −0.613917 0.707107i 1.19935 2.35829i 1.53610 + 2.37495i 2.56155 0.279135 + 3.14993i
139.5 −0.331077 1.37491i 2.13578i −1.78078 + 0.910404i 1.94442 1.10418i −2.93651 + 0.707107i −2.35829 1.19935i 1.84130 + 2.14700i −1.56155 −2.16191 2.30784i
139.6 −0.331077 1.37491i 2.13578i −1.78078 + 0.910404i −1.94442 + 1.10418i 2.93651 0.707107i −2.35829 + 1.19935i 1.84130 + 2.14700i −1.56155 2.16191 + 2.30784i
139.7 −0.331077 + 1.37491i 2.13578i −1.78078 0.910404i −1.94442 1.10418i 2.93651 + 0.707107i −2.35829 1.19935i 1.84130 2.14700i −1.56155 2.16191 2.30784i
139.8 −0.331077 + 1.37491i 2.13578i −1.78078 0.910404i 1.94442 + 1.10418i −2.93651 0.707107i −2.35829 + 1.19935i 1.84130 2.14700i −1.56155 −2.16191 + 2.30784i
139.9 0.331077 1.37491i 2.13578i −1.78078 0.910404i −1.94442 + 1.10418i −2.93651 0.707107i 2.35829 1.19935i −1.84130 + 2.14700i −1.56155 0.874406 + 3.03898i
139.10 0.331077 1.37491i 2.13578i −1.78078 0.910404i 1.94442 1.10418i 2.93651 + 0.707107i 2.35829 + 1.19935i −1.84130 + 2.14700i −1.56155 −0.874406 3.03898i
139.11 0.331077 + 1.37491i 2.13578i −1.78078 + 0.910404i 1.94442 + 1.10418i 2.93651 0.707107i 2.35829 1.19935i −1.84130 2.14700i −1.56155 −0.874406 + 3.03898i
139.12 0.331077 + 1.37491i 2.13578i −1.78078 + 0.910404i −1.94442 1.10418i −2.93651 + 0.707107i 2.35829 + 1.19935i −1.84130 2.14700i −1.56155 0.874406 3.03898i
139.13 1.06789 0.927153i 0.662153i 0.280776 1.98019i −1.31119 1.81129i −0.613917 0.707107i −1.19935 + 2.35829i −1.53610 2.37495i 2.56155 −3.07955 0.718585i
139.14 1.06789 0.927153i 0.662153i 0.280776 1.98019i 1.31119 + 1.81129i 0.613917 + 0.707107i −1.19935 2.35829i −1.53610 2.37495i 2.56155 3.07955 + 0.718585i
139.15 1.06789 + 0.927153i 0.662153i 0.280776 + 1.98019i 1.31119 1.81129i 0.613917 0.707107i −1.19935 + 2.35829i −1.53610 + 2.37495i 2.56155 3.07955 0.718585i
139.16 1.06789 + 0.927153i 0.662153i 0.280776 + 1.98019i −1.31119 + 1.81129i −0.613917 + 0.707107i −1.19935 2.35829i −1.53610 + 2.37495i 2.56155 −3.07955 + 0.718585i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 139.16 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
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c 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.c.b 16
4.b odd 2 1 inner 140.2.c.b 16
5.b even 2 1 inner 140.2.c.b 16
5.c odd 4 2 700.2.g.l 16
7.b odd 2 1 inner 140.2.c.b 16
7.c even 3 2 980.2.s.f 32
7.d odd 6 2 980.2.s.f 32
8.b even 2 1 2240.2.e.f 16
8.d odd 2 1 2240.2.e.f 16
20.d odd 2 1 inner 140.2.c.b 16
20.e even 4 2 700.2.g.l 16
28.d even 2 1 inner 140.2.c.b 16
28.f even 6 2 980.2.s.f 32
28.g odd 6 2 980.2.s.f 32
35.c odd 2 1 inner 140.2.c.b 16
35.f even 4 2 700.2.g.l 16
35.i odd 6 2 980.2.s.f 32
35.j even 6 2 980.2.s.f 32
40.e odd 2 1 2240.2.e.f 16
40.f even 2 1 2240.2.e.f 16
56.e even 2 1 2240.2.e.f 16
56.h odd 2 1 2240.2.e.f 16
140.c even 2 1 inner 140.2.c.b 16
140.j odd 4 2 700.2.g.l 16
140.p odd 6 2 980.2.s.f 32
140.s even 6 2 980.2.s.f 32
280.c odd 2 1 2240.2.e.f 16
280.n even 2 1 2240.2.e.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.b 16 1.a even 1 1 trivial
140.2.c.b 16 4.b odd 2 1 inner
140.2.c.b 16 5.b even 2 1 inner
140.2.c.b 16 7.b odd 2 1 inner
140.2.c.b 16 20.d odd 2 1 inner
140.2.c.b 16 28.d even 2 1 inner
140.2.c.b 16 35.c odd 2 1 inner
140.2.c.b 16 140.c even 2 1 inner
700.2.g.l 16 5.c odd 4 2
700.2.g.l 16 20.e even 4 2
700.2.g.l 16 35.f even 4 2
700.2.g.l 16 140.j odd 4 2
980.2.s.f 32 7.c even 3 2
980.2.s.f 32 7.d odd 6 2
980.2.s.f 32 28.f even 6 2
980.2.s.f 32 28.g odd 6 2
980.2.s.f 32 35.i odd 6 2
980.2.s.f 32 35.j even 6 2
980.2.s.f 32 140.p odd 6 2
980.2.s.f 32 140.s even 6 2
2240.2.e.f 16 8.b even 2 1
2240.2.e.f 16 8.d odd 2 1
2240.2.e.f 16 40.e odd 2 1
2240.2.e.f 16 40.f even 2 1
2240.2.e.f 16 56.e even 2 1
2240.2.e.f 16 56.h odd 2 1
2240.2.e.f 16 280.c odd 2 1
2240.2.e.f 16 280.n even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 5T_{3}^{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(140, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} + 3 T^{6} + 6 T^{4} + 12 T^{2} + \cdots + 16)^{2}$$
$3$ $$(T^{4} + 5 T^{2} + 2)^{4}$$
$5$ $$(T^{8} - 2 T^{6} + 34 T^{4} - 50 T^{2} + \cdots + 625)^{2}$$
$7$ $$(T^{8} + 30 T^{4} + 2401)^{2}$$
$11$ $$(T^{4} + 15 T^{2} + 52)^{4}$$
$13$ $$(T^{4} - 23 T^{2} + 26)^{4}$$
$17$ $$(T^{4} - 29 T^{2} + 104)^{4}$$
$19$ $$(T^{4} - 38 T^{2} + 208)^{4}$$
$23$ $$(T^{4} - 40 T^{2} + 128)^{4}$$
$29$ $$(T^{2} + 3 T - 2)^{8}$$
$31$ $$(T^{4} - 120 T^{2} + 3328)^{4}$$
$37$ $$(T^{4} + 44 T^{2} + 416)^{4}$$
$41$ $$(T^{2} + 72)^{8}$$
$43$ $$(T^{4} - 20 T^{2} + 32)^{4}$$
$47$ $$(T^{4} + 95 T^{2} + 8)^{4}$$
$53$ $$(T^{4} + 164 T^{2} + 6656)^{4}$$
$59$ $$(T^{4} - 270 T^{2} + 13312)^{4}$$
$61$ $$(T^{4} + 42 T^{2} + 16)^{4}$$
$67$ $$(T^{4} - 28 T^{2} + 128)^{4}$$
$71$ $$(T^{4} + 236 T^{2} + 13312)^{4}$$
$73$ $$(T^{4} - 116 T^{2} + 1664)^{4}$$
$79$ $$(T^{4} + 115 T^{2} + 208)^{4}$$
$83$ $$(T^{4} + 170 T^{2} + 2312)^{4}$$
$89$ $$(T^{2} + 8)^{8}$$
$97$ $$(T^{4} - 261 T^{2} + 8424)^{4}$$