# Properties

 Label 189.4.e.g Level $189$ Weight $4$ Character orbit 189.e Analytic conductor $11.151$ Analytic rank $0$ Dimension $14$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,4,Mod(109,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("189.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 189.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: $$11.1513609911$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{14} - x^{13} + 44 x^{12} + 23 x^{11} + 1346 x^{10} + 854 x^{9} + 20545 x^{8} + 27750 x^{7} + \cdots + 254016$$ x^14 - x^13 + 44*x^12 + 23*x^11 + 1346*x^10 + 854*x^9 + 20545*x^8 + 27750*x^7 + 221349*x^6 + 172746*x^5 + 551772*x^4 + 275616*x^3 + 1006128*x^2 + 471744*x + 254016 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{9} - 4 \beta_{5} + \cdots + \beta_1) q^{4}+ \cdots + (\beta_{4} - \beta_{3} + 4 \beta_{2} - 11) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b9 - 4*b5 + b2 + b1) * q^4 + b11 * q^5 + (b11 - b8 + 2*b5 - b2 - b1) * q^7 + (b4 - b3 + 4*b2 - 11) * q^8 $$q + \beta_1 q^{2} + (\beta_{9} - 4 \beta_{5} + \cdots + \beta_1) q^{4}+ \cdots + ( - 6 \beta_{13} - 3 \beta_{12} + \cdots - 807) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b9 - 4*b5 + b2 + b1) * q^4 + b11 * q^5 + (b11 - b8 + 2*b5 - b2 - b1) * q^7 + (b4 - b3 + 4*b2 - 11) * q^8 + (2*b13 + 2*b12 + 4*b11 - 2*b8 - 2*b7 + 3*b6 - 5*b5 + 3*b3 - 2*b2 - 2*b1 + 2) * q^10 + (-b13 - b12 - b11 + b8 - 2*b6 + 15*b5 - 2*b3 - 3*b2 - 3*b1 - 1) * q^11 + (2*b13 + 2*b11 + 2*b10 - 2*b8 + 3*b7 + 2*b6 - 2*b5 + 2*b2 - 15) * q^13 + (3*b12 + 3*b11 - b10 + 2*b9 - b8 - 4*b7 + 2*b6 - 6*b5 - b4 + 2*b3 - 5*b2 - 3*b1 + 12) * q^14 + (b13 - b12 + 2*b11 + b9 + b7 + 18*b5 - b4 - 12*b1 - 18) * q^16 + (b13 + 4*b12 + 4*b11 + 3*b10 - 4*b9 - 4*b8 + 3*b6 - b5 - 3*b2 - 3*b1 + 4) * q^17 + (-3*b11 - 3*b10 - 2*b9 - 3*b8 + 3*b6 + 23*b5 + 2*b4 - 11*b1 - 26) * q^19 + (5*b13 + 8*b12 + 5*b11 + 5*b10 + 3*b8 - 9*b7 + 5*b6 - 5*b5 - 3*b4 + 7*b3 - 3*b2 + 24) * q^20 + (-2*b12 - 2*b8 + 8*b7 - b4 - 18*b2 + 36) * q^22 + (-3*b13 + 3*b12 - b11 + 5*b10 + 6*b9 + 5*b8 - 3*b7 + b6 + 4*b5 - 6*b4 - 3*b1 + 1) * q^23 + (3*b13 + 4*b12 + 2*b11 + b10 + 2*b9 - 4*b8 + 2*b7 + b6 - 55*b5 + 27*b2 + 27*b1 + 4) * q^25 + (-6*b13 + 6*b12 + 2*b11 - 2*b10 - 6*b9 - 2*b8 - 6*b7 + 5*b6 + 29*b5 + 6*b4 - 25*b1 - 31) * q^26 + (b13 + b12 + 4*b11 + 2*b10 + 3*b9 + 6*b8 - 13*b7 - 4*b6 - 6*b5 - 5*b4 + 3*b3 + 14*b2 + 18*b1 + 35) * q^28 + (2*b13 + 5*b12 + 2*b11 + 2*b10 + 3*b8 - 3*b7 + 2*b6 - 2*b5 + 14*b4 + 2*b3 - 3*b2 - 37) * q^29 + (-8*b13 - 6*b12 - 3*b11 + 2*b10 + 2*b9 + 6*b8 - 3*b7 - 4*b6 + 3*b5 - 6*b3 - 12*b2 - 12*b1 - 6) * q^31 + (4*b13 + 7*b12 + 11*b11 + 3*b10 - 6*b9 - 7*b8 - 4*b7 + 2*b6 + 48*b5 - b3 + 3*b2 + 3*b1 + 7) * q^32 + (-7*b13 + 2*b12 - 7*b11 - 7*b10 + 9*b8 - 11*b7 - 7*b6 + 7*b5 + 7*b4 - 31*b2 + 7) * q^34 + (b13 + b12 + 10*b11 + 4*b10 + 20*b9 - 3*b8 + b7 + 2*b6 - 65*b5 - 12*b4 - 6*b3 + 27*b2 + 23*b1 - 36) * q^35 + (-5*b13 + 5*b12 + 6*b11 - b10 - 6*b9 - b8 - 5*b7 - b6 - 23*b5 + 6*b4 - 35*b1 + 22) * q^37 + (-6*b12 + 6*b11 - 6*b10 - 17*b9 + 6*b8 - 12*b7 - 13*b6 + 139*b5 - 7*b3 - 39*b2 - 39*b1 - 6) * q^38 + (2*b13 - 2*b12 - 26*b11 + 12*b10 - 2*b9 + 12*b8 + 2*b7 - 8*b5 + 2*b4 + 47*b1 + 20) * q^40 + (-4*b13 - 7*b12 - 4*b11 - 4*b10 - 3*b8 + 8*b7 - 4*b6 + 4*b5 - 6*b4 - 14*b3 - 23*b2 + 63) * q^41 + (-5*b13 + 3*b12 - 5*b11 - 5*b10 + 8*b8 + 6*b7 - 5*b6 + 5*b5 - 8*b4 + 5*b2 - 2) * q^43 + (-6*b13 + 6*b12 + 16*b11 - 8*b10 + 22*b9 - 8*b8 - 6*b7 + b6 - 129*b5 - 22*b4 + 17*b1 + 121) * q^44 + (-12*b13 - 17*b12 - 37*b11 - 5*b10 - 5*b9 + 17*b8 + 20*b7 - 26*b6 + 78*b5 - 21*b3 + 50*b2 + 50*b1 - 17) * q^46 + (3*b13 - 3*b12 - b11 + 10*b10 - 18*b9 + 10*b8 + 3*b7 - 4*b6 - 40*b5 + 18*b4 + 22*b1 + 50) * q^47 + (-6*b13 - 8*b12 - 4*b11 - 6*b10 - 2*b9 - 7*b8 - b7 + 6*b6 - 47*b5 - 2*b4 - 6*b3 + 67*b2 + 53*b1 + 50) * q^49 + (-b13 + 2*b12 - b11 - b10 + 3*b8 - 5*b7 - b6 + b5 + 41*b4 - 4*b3 + 89*b2 - 343) * q^50 + (8*b13 - b12 + 13*b11 - 9*b10 - 26*b9 + b8 - 14*b7 - 12*b6 + 134*b5 - 3*b3 - 64*b2 - 64*b1 - 1) * q^52 + (3*b12 + 2*b11 + 3*b10 - 3*b8 + b7 + 15*b6 + 186*b5 + 12*b3 - 22*b2 - 22*b1 + 3) * q^53 + (-2*b13 - 7*b12 - 2*b11 - 2*b10 - 5*b8 + 25*b7 - 2*b6 + 2*b5 - 16*b4 - 12*b3 - 91*b2 + 2) * q^55 + (3*b13 - 27*b11 + 7*b10 + 21*b9 + 13*b8 + 7*b7 - 7*b6 + 35*b5 - 7*b4 + 7*b3 + 13*b2 + 44*b1 - 152) * q^56 + (7*b13 - 7*b12 - 14*b11 + 10*b10 - 5*b9 + 10*b8 + 7*b7 + 10*b6 - 28*b5 + 5*b4 - 122*b1 + 38) * q^58 + (7*b13 + 10*b12 + 5*b11 + 3*b10 - 38*b9 - 10*b8 + 5*b7 + 31*b6 - 5*b5 + 28*b3 - 27*b2 - 27*b1 + 10) * q^59 + (-9*b13 + 9*b12 - 6*b11 - 6*b10 + 10*b9 - 6*b8 - 9*b7 - 12*b6 + 17*b5 - 10*b4 + 28*b1 - 23) * q^61 + (-10*b13 - 10*b12 - 10*b11 - 10*b10 + 22*b7 - 10*b6 + 10*b5 - 22*b4 - 7*b3 + 15*b2 + 155) * q^62 + (16*b13 + 16*b12 + 16*b11 + 16*b10 + 2*b7 + 16*b6 - 16*b5 + 20*b4 + 21*b3 - 14*b2 - 213) * q^64 + (9*b13 - 9*b12 - 35*b11 + b10 + 14*b9 + b8 + 9*b7 + 3*b6 - 230*b5 - 14*b4 + 31*b1 + 231) * q^65 + (3*b13 + 12*b12 + 27*b11 + 9*b10 - 10*b9 - 12*b8 - 15*b7 + 51*b6 + 124*b5 + 42*b3 + 35*b2 + 35*b1 + 12) * q^67 + (3*b13 - 3*b12 - 8*b11 + 9*b9 + 3*b7 - 21*b6 - 369*b5 - 9*b4 - 2*b1 + 369) * q^68 + (6*b13 + 14*b12 + 28*b11 - 4*b10 - 20*b9 - 8*b8 - 14*b7 + 2*b6 + 54*b5 + 49*b4 - 3*b3 + 214*b2 + 33*b1 - 337) * q^70 + (b13 - 11*b12 + b11 + b10 - 12*b8 - 19*b7 + b6 - b5 + 8*b4 + 18*b3 - 59*b2 - 133) * q^71 + (5*b13 + 10*b12 + 20*b11 + 5*b10 + 58*b9 - 10*b8 - 10*b7 + 5*b6 - 214*b5 - 47*b2 - 47*b1 + 10) * q^73 + (14*b13 - b12 + 13*b11 - 15*b10 - 21*b9 + b8 - 14*b7 + b6 + 369*b5 + 16*b3 - 59*b2 - 59*b1 - 1) * q^74 + (25*b13 + 25*b11 + 25*b10 - 25*b8 - 9*b7 + 25*b6 - 25*b5 - 14*b4 + 24*b3 - 253*b2 + 259) * q^76 + (-14*b13 - 29*b12 - 24*b11 - 11*b10 + 22*b9 + 3*b8 + 20*b7 - 9*b6 + 69*b5 - 30*b4 - 8*b3 - 90*b2 - 50*b1 - 66) * q^77 + (-2*b13 + 2*b12 + 33*b11 - 19*b10 + 64*b9 - 19*b8 - 2*b7 - 19*b6 - 194*b5 - 64*b4 - 163*b1 + 175) * q^79 + (-12*b13 - 18*b12 - 70*b11 - 6*b10 + 7*b9 + 18*b8 + 52*b7 - 34*b6 - 314*b5 - 28*b3 + 77*b2 + 77*b1 - 18) * q^80 + (5*b13 - 5*b12 + 52*b11 - 24*b10 + 3*b9 - 24*b8 + 5*b7 - 9*b6 - 287*b5 - 3*b4 + 130*b1 + 263) * q^82 + (-17*b13 - 32*b12 - 17*b11 - 17*b10 - 15*b8 + 10*b7 - 17*b6 + 17*b5 + 10*b4 - 14*b3 + 112*b2 + 145) * q^83 + (-b13 - 33*b12 - b11 - b10 - 32*b8 + 9*b7 - b6 + b5 + 20*b4 - 42*b3 + 31*b2 + 19) * q^85 + (24*b13 - 24*b12 + 22*b11 - 22*b10 - b9 - 22*b8 + 24*b7 - 10*b6 + 18*b5 + b4 + 48*b1 - 40) * q^86 + (32*b13 + 21*b12 + 39*b11 - 11*b10 + 26*b9 - 21*b8 - 18*b7 + 10*b6 - 8*b5 + 21*b3 + 130*b2 + 130*b1 + 21) * q^88 + (33*b13 - 33*b12 - 40*b11 - 6*b10 - 40*b9 - 6*b8 + 33*b7 + 22*b6 - 175*b5 + 40*b4 - 60*b1 + 169) * q^89 + (14*b13 + 14*b12 - 20*b11 + 17*b10 + 22*b9 - 2*b8 + 28*b7 + 23*b6 + 137*b5 - 56*b4 + 18*b3 + 157*b2 + 110*b1 + 205) * q^91 + (-12*b13 - 24*b12 - 12*b11 - 12*b10 - 12*b8 + 110*b7 - 12*b6 + 12*b5 + 7*b4 - 16*b3 + 2*b2 - 560) * q^92 + (-22*b13 + b12 - 49*b11 + 23*b10 + 18*b9 - b8 + 50*b7 + 38*b6 - 230*b5 + 15*b3 - 66*b2 - 66*b1 + 1) * q^94 + (-40*b13 - 52*b12 - 91*b11 - 12*b10 + 48*b9 + 52*b8 + 39*b7 - 44*b6 + 579*b5 - 32*b3 + 234*b2 + 234*b1 - 52) * q^95 + (18*b12 + 18*b8 + 12*b7 + 68*b4 + 12*b3 - 70*b2 - 508) * q^97 + (-6*b13 - 3*b12 + 37*b11 - 27*b10 - 30*b9 + 3*b8 - 38*b7 - 30*b6 + 174*b5 + 57*b4 - 9*b3 + 112*b2 + 69*b1 - 807) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q + q^{2} - 31 q^{4} - q^{5} + 20 q^{7} - 168 q^{8}+O(q^{10})$$ 14 * q + q^2 - 31 * q^4 - q^5 + 20 * q^7 - 168 * q^8 $$14 q + q^{2} - 31 q^{4} - q^{5} + 20 q^{7} - 168 q^{8} - 12 q^{10} + 98 q^{11} - 248 q^{13} + 134 q^{14} - 139 q^{16} + 30 q^{17} - 182 q^{19} + 220 q^{20} + 552 q^{22} - 6 q^{23} - 388 q^{25} - 245 q^{26} + 425 q^{28} - 646 q^{29} - 26 q^{31} + 398 q^{32} + 228 q^{34} - 1025 q^{35} + 112 q^{37} + 1015 q^{38} + 147 q^{40} + 1048 q^{41} + 16 q^{43} + 937 q^{44} + 339 q^{46} + 288 q^{47} + 446 q^{49} - 5152 q^{50} + 1075 q^{52} + 1353 q^{53} + 312 q^{55} - 1980 q^{56} + 81 q^{58} + 165 q^{59} - 56 q^{61} + 2430 q^{62} - 3412 q^{64} + 1694 q^{65} + 988 q^{67} + 2625 q^{68} - 4941 q^{70} - 1584 q^{71} - 1487 q^{73} + 2736 q^{74} + 3904 q^{76} + 34 q^{77} + 1273 q^{79} - 2501 q^{80} + 2049 q^{82} + 2340 q^{83} + 432 q^{85} - 160 q^{86} - 9 q^{88} + 1058 q^{89} + 3538 q^{91} - 7668 q^{92} - 1653 q^{94} + 3260 q^{95} - 7460 q^{97} - 10160 q^{98}+O(q^{100})$$ 14 * q + q^2 - 31 * q^4 - q^5 + 20 * q^7 - 168 * q^8 - 12 * q^10 + 98 * q^11 - 248 * q^13 + 134 * q^14 - 139 * q^16 + 30 * q^17 - 182 * q^19 + 220 * q^20 + 552 * q^22 - 6 * q^23 - 388 * q^25 - 245 * q^26 + 425 * q^28 - 646 * q^29 - 26 * q^31 + 398 * q^32 + 228 * q^34 - 1025 * q^35 + 112 * q^37 + 1015 * q^38 + 147 * q^40 + 1048 * q^41 + 16 * q^43 + 937 * q^44 + 339 * q^46 + 288 * q^47 + 446 * q^49 - 5152 * q^50 + 1075 * q^52 + 1353 * q^53 + 312 * q^55 - 1980 * q^56 + 81 * q^58 + 165 * q^59 - 56 * q^61 + 2430 * q^62 - 3412 * q^64 + 1694 * q^65 + 988 * q^67 + 2625 * q^68 - 4941 * q^70 - 1584 * q^71 - 1487 * q^73 + 2736 * q^74 + 3904 * q^76 + 34 * q^77 + 1273 * q^79 - 2501 * q^80 + 2049 * q^82 + 2340 * q^83 + 432 * q^85 - 160 * q^86 - 9 * q^88 + 1058 * q^89 + 3538 * q^91 - 7668 * q^92 - 1653 * q^94 + 3260 * q^95 - 7460 * q^97 - 10160 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - x^{13} + 44 x^{12} + 23 x^{11} + 1346 x^{10} + 854 x^{9} + 20545 x^{8} + 27750 x^{7} + \cdots + 254016$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 34\!\cdots\!97 \nu^{13} + \cdots + 40\!\cdots\!44 ) / 72\!\cdots\!20$$ (-3455031298486397*v^13 + 48196448829795521*v^12 - 20110423072368640*v^11 + 1645450782090378029*v^10 + 3729870784114661690*v^9 + 58523797219352106602*v^8 + 187869888951786203011*v^7 + 858219601631945703798*v^6 + 3561675320409133674231*v^5 + 12257447465942139293226*v^4 + 40150463029371601358364*v^3 + 25907324744010685143240*v^2 + 12387854102644392198624*v + 40729840670024027895744) / 72672625881267800515920 $$\beta_{3}$$ $$=$$ $$( - 20\!\cdots\!19 \nu^{13} + \cdots + 84\!\cdots\!08 ) / 72\!\cdots\!20$$ (-20904177139932419*v^13 + 1047643481827147727*v^12 + 1342818463580561000*v^11 + 39643907771594534603*v^10 + 132341938846438061750*v^9 + 1370805654146879254814*v^8 + 4523624610248881077757*v^7 + 20632615474304767535946*v^6 + 80526121390404010239777*v^5 + 274948354411900684175142*v^4 + 717045758327166937231788*v^3 + 580775853850398732076920*v^2 + 277728958905140646535968*v + 847417231842040651155408) / 72672625881267800515920 $$\beta_{4}$$ $$=$$ $$( 48\!\cdots\!21 \nu^{13} + \cdots + 83\!\cdots\!48 ) / 72\!\cdots\!20$$ (48196448829795521*v^13 + 83714505231237307*v^12 + 1745026925027933800*v^11 + 6734892129786974023*v^10 + 57744523164144827950*v^9 + 200329709759837122774*v^8 + 766226831213157017537*v^7 + 3468223441665853459986*v^6 + 9292614982221336755157*v^5 + 29799405093057898310622*v^4 - 13290876378997289419572*v^3 + 62629358970185029212120*v^2 + 29971876852252802563488*v + 832219303135505898915648) / 72672625881267800515920 $$\beta_{5}$$ $$=$$ $$( 11\!\cdots\!13 \nu^{13} + \cdots + 44\!\cdots\!04 ) / 50\!\cdots\!40$$ (1122405221285935513*v^13 - 1098220002196530734*v^12 + 49048454594772593925*v^11 + 25956093051083097279*v^10 + 1499239272376236554295*v^9 + 932424963489386296272*v^8 + 22650148690784080368371*v^7 + 29831655668022207064673*v^6 + 242435736114996919940451*v^5 + 168959285113396280409081*v^4 + 533509641497788234826454*v^3 + 28299596264343192842460*v^2 + 947932047273900929820984*v + 442772949991801617254304) / 508708381168874603611440 $$\beta_{6}$$ $$=$$ $$( 20\!\cdots\!09 \nu^{13} + \cdots - 15\!\cdots\!88 ) / 50\!\cdots\!40$$ (2045781899713165309*v^13 - 3727537766578014602*v^12 + 88433071241684333205*v^11 - 23939185674236505933*v^10 + 2613428869590578789175*v^9 - 635359506475557433584*v^8 + 37565552597257274634023*v^7 + 20202310279386702905969*v^6 + 392751365519539208620143*v^5 - 110231006285582285020407*v^4 + 878929167204372983099742*v^3 - 101339842447458744881940*v^2 + 10175271138507223039700232*v - 151634115198718359793488) / 508708381168874603611440 $$\beta_{7}$$ $$=$$ $$( - 30\!\cdots\!51 \nu^{13} + \cdots - 70\!\cdots\!68 ) / 33\!\cdots\!60$$ (-3099003646113881951*v^13 - 7203563520624871877*v^12 - 117014204368024372200*v^11 - 530558121161619016713*v^10 - 4007021393313158629170*v^9 - 16220387883236909423994*v^8 - 60392029960707240254047*v^7 - 289045164051386066644126*v^6 - 791213015285569381004667*v^5 - 2575547869801197526046082*v^4 - 1707438894569011663554708*v^3 - 5419707710667771462442920*v^2 - 2593178912566558800556128*v - 7021942626775286677920768) / 339138920779249735740960 $$\beta_{8}$$ $$=$$ $$( 15\!\cdots\!79 \nu^{13} + \cdots - 24\!\cdots\!68 ) / 10\!\cdots\!80$$ (15021170859861634279*v^13 - 72071306011219214357*v^12 + 755228980756258833810*v^11 - 2222328519871851609543*v^10 + 20654226309838163246220*v^9 - 63929830957245815548674*v^8 + 311817131195674837507523*v^7 - 736628143612148601563956*v^6 + 2552254164024520265451423*v^5 - 9077669353584797641983792*v^4 + 6757273589600703179975832*v^3 - 19014971419082515051803960*v^2 + 23990771951904849642477792*v - 24274294050052298238583968) / 1017416762337749207222880 $$\beta_{9}$$ $$=$$ $$( 26\!\cdots\!87 \nu^{13} + \cdots + 10\!\cdots\!88 ) / 10\!\cdots\!88$$ (2698609574904126187*v^13 - 2703203033633387491*v^12 + 117744445619755541516*v^11 + 59990992227672904229*v^10 + 3592952434605207203942*v^9 + 2155886596267434161810*v^8 + 54097339013349292199875*v^7 + 70394466160968572969898*v^6 + 576859421227419820713159*v^5 + 388341857819832077971278*v^4 + 1224212491353571521681780*v^3 + 133390452626583624343656*v^2 + 2155952241479885161770000*v + 1005633303042290242356288) / 101741676233774920722288 $$\beta_{10}$$ $$=$$ $$( 36\!\cdots\!97 \nu^{13} + \cdots + 19\!\cdots\!36 ) / 10\!\cdots\!80$$ (36584533505283765397*v^13 - 31994237870822240281*v^12 + 1573711104078775681940*v^11 + 1131692614222411043231*v^10 + 47869410249743648518670*v^9 + 39378029727686957520578*v^8 + 715719600491710368447109*v^7 + 1155309683335030998087042*v^6 + 7641383848964266958610309*v^5 + 7581244112619044046382134*v^4 + 16109797668629437980485496*v^3 + 15831829390482953048665080*v^2 + 21650874405385349538812016*v + 19152938004169722822812736) / 1017416762337749207222880 $$\beta_{11}$$ $$=$$ $$( 93\!\cdots\!41 \nu^{13} + \cdots - 74\!\cdots\!12 ) / 25\!\cdots\!20$$ (9368321844795477541*v^13 - 18654933007643345443*v^12 + 424682705356542091640*v^11 - 185618042321941263307*v^10 + 12511791838234115320460*v^9 - 4050916727390361265456*v^8 + 188337476463722852167477*v^7 + 82524177653675181815496*v^6 + 1866970694222456365863747*v^5 - 218889890364544164221328*v^4 + 4185077983227115304971698*v^3 - 552420063880752537792540*v^2 + 7746894686707004703008268*v - 742329174155107762832112) / 254354190584437301805720 $$\beta_{12}$$ $$=$$ $$( - 21\!\cdots\!27 \nu^{13} + \cdots - 20\!\cdots\!36 ) / 50\!\cdots\!40$$ (-21049398563957379327*v^13 + 2795109761782199936*v^12 - 894361152045150317555*v^11 - 1319011904096999698391*v^10 - 28178849414416216089215*v^9 - 42743750110246751820338*v^8 - 432557843858831477448749*v^7 - 963112878010510321900597*v^6 - 4955823262072307980764099*v^5 - 7472579831896103245722429*v^4 - 12962820287434431278685816*v^3 - 15773777237836645828731720*v^2 - 24091359366607891939574496*v - 20493335910010359841368336) / 508708381168874603611440 $$\beta_{13}$$ $$=$$ $$( - 45\!\cdots\!73 \nu^{13} + \cdots - 18\!\cdots\!64 ) / 10\!\cdots\!80$$ (-45831542345574856173*v^13 + 59286086789569118519*v^12 - 2055269535023719846250*v^11 - 494861196284468985959*v^10 - 62277453684521348826620*v^9 - 24364934481680548957622*v^8 - 957648442465118303442401*v^7 - 1083845759612645148733468*v^6 - 10127224453694779667736441*v^5 - 6503055305421576432934656*v^4 - 26654079036549678482573244*v^3 - 14052924908851408110806760*v^2 - 38332375731633038893222464*v - 18748417766656815858356064) / 1017416762337749207222880
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} - 12\beta_{5} + \beta_{2} + \beta_1$$ b9 - 12*b5 + b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + 20\beta_{2} - 11$$ b4 - b3 + 20*b2 - 11 $$\nu^{4}$$ $$=$$ $$\beta_{13} - \beta_{12} + 2\beta_{11} - 23\beta_{9} + \beta_{7} + 242\beta_{5} + 23\beta_{4} - 36\beta _1 - 242$$ b13 - b12 + 2*b11 - 23*b9 + b7 + 242*b5 + 23*b4 - 36*b1 - 242 $$\nu^{5}$$ $$=$$ $$4 \beta_{13} + 7 \beta_{12} + 11 \beta_{11} + 3 \beta_{10} - 38 \beta_{9} - 7 \beta_{8} - 4 \beta_{7} + \cdots + 7$$ 4*b13 + 7*b12 + 11*b11 + 3*b10 - 38*b9 - 7*b8 - 4*b7 + 34*b6 + 400*b5 + 31*b3 - 445*b2 - 445*b1 + 7 $$\nu^{6}$$ $$=$$ $$- 24 \beta_{13} + 16 \beta_{12} - 24 \beta_{11} - 24 \beta_{10} + 40 \beta_{8} - 118 \beta_{7} + \cdots + 5331$$ -24*b13 + 16*b12 - 24*b11 - 24*b10 + 40*b8 - 118*b7 - 24*b6 + 24*b5 - 516*b4 + 21*b3 - 1070*b2 + 5331 $$\nu^{7}$$ $$=$$ $$99 \beta_{13} - 99 \beta_{12} - 230 \beta_{11} + 188 \beta_{10} + 1149 \beta_{9} + 188 \beta_{8} + \cdots + 12373$$ 99*b13 - 99*b12 - 230*b11 + 188*b10 + 1149*b9 + 188*b8 + 99*b7 - 653*b6 - 12185*b5 - 1149*b4 + 10263*b1 + 12373 $$\nu^{8}$$ $$=$$ $$- 836 \beta_{13} + 302 \beta_{12} - 2216 \beta_{11} + 1138 \beta_{10} + 11836 \beta_{9} - 302 \beta_{8} + \cdots + 302$$ -836*b13 + 302*b12 - 2216*b11 + 1138*b10 + 11836*b9 - 302*b8 + 2518*b7 - 38*b6 - 122242*b5 - 1176*b3 + 29635*b2 + 29635*b1 + 302 $$\nu^{9}$$ $$=$$ $$- 8946 \beta_{13} - 6708 \beta_{12} - 8946 \beta_{11} - 8946 \beta_{10} + 2238 \beta_{8} + \cdots - 353276$$ -8946*b13 - 6708*b12 - 8946*b11 - 8946*b10 + 2238*b8 + 6822*b7 - 8946*b6 + 8946*b5 + 32095*b4 - 22162*b3 + 241895*b2 - 353276 $$\nu^{10}$$ $$=$$ $$28876 \beta_{13} - 28876 \beta_{12} + 75860 \beta_{11} - 31536 \beta_{10} - 277073 \beta_{9} + \cdots - 2878871$$ 28876*b13 - 28876*b12 + 75860*b11 - 31536*b10 - 277073*b9 - 31536*b8 + 28876*b7 + 14517*b6 + 2847335*b5 + 277073*b4 - 793626*b1 - 2878871 $$\nu^{11}$$ $$=$$ $$214792 \beta_{13} + 255367 \beta_{12} + 562265 \beta_{11} + 40575 \beta_{10} - 863393 \beta_{9} + \cdots + 255367$$ 214792*b13 + 255367*b12 + 562265*b11 + 40575*b10 - 863393*b9 - 255367*b8 - 306898*b7 + 619999*b6 + 8764867*b5 + 579424*b3 - 5792446*b2 - 5792446*b1 + 255367 $$\nu^{12}$$ $$=$$ $$342231 \beta_{13} + 1043380 \beta_{12} + 342231 \beta_{11} + 342231 \beta_{10} + 701149 \beta_{8} + \cdots + 68968512$$ 342231*b13 + 1043380*b12 + 342231*b11 + 342231*b10 + 701149*b8 - 2903377*b7 + 342231*b6 - 342231*b5 - 6590400*b4 + 1553097*b3 - 20874605*b2 + 68968512 $$\nu^{13}$$ $$=$$ $$550350 \beta_{13} - 550350 \beta_{12} - 9597410 \beta_{11} + 6491216 \beta_{10} + 22762674 \beta_{9} + \cdots + 242906239$$ 550350*b13 - 550350*b12 - 9597410*b11 + 6491216*b10 + 22762674*b9 + 6491216*b8 + 550350*b7 - 8641829*b6 - 236415023*b5 - 22762674*b4 + 140421006*b1 + 242906239

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{5}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −2.24639 − 3.89087i −1.81482 − 3.14336i −0.815927 − 1.41323i −0.270836 − 0.469102i 0.720983 + 1.24878i 2.38108 + 4.12414i 2.54592 + 4.40966i −2.24639 + 3.89087i −1.81482 + 3.14336i −0.815927 + 1.41323i −0.270836 + 0.469102i 0.720983 − 1.24878i 2.38108 − 4.12414i 2.54592 − 4.40966i
−2.24639 3.89087i 0 −6.09257 + 10.5526i −4.09907 7.09980i 0 −18.4138 1.98297i 18.8030 0 −18.4163 + 31.8979i
109.2 −1.81482 3.14336i 0 −2.58715 + 4.48108i 2.42081 + 4.19297i 0 16.3234 8.74903i −10.2563 0 8.78669 15.2190i
109.3 −0.815927 1.41323i 0 2.66853 4.62202i 6.18949 + 10.7205i 0 −6.28171 + 17.4224i −21.7641 0 10.1003 17.4943i
109.4 −0.270836 0.469102i 0 3.85330 6.67410i −8.36445 14.4876i 0 18.5069 0.702629i −8.50782 0 −4.53079 + 7.84756i
109.5 0.720983 + 1.24878i 0 2.96037 5.12751i 3.30182 + 5.71892i 0 −10.3191 15.3791i 20.0732 0 −4.76111 + 8.24648i
109.6 2.38108 + 4.12414i 0 −7.33904 + 12.7116i 9.35067 + 16.1958i 0 14.2072 + 11.8809i −31.8020 0 −44.5293 + 77.1270i
109.7 2.54592 + 4.40966i 0 −8.96343 + 15.5251i −9.29927 16.1068i 0 −4.02294 18.0781i −50.5460 0 47.3504 82.0134i
163.1 −2.24639 + 3.89087i 0 −6.09257 10.5526i −4.09907 + 7.09980i 0 −18.4138 + 1.98297i 18.8030 0 −18.4163 31.8979i
163.2 −1.81482 + 3.14336i 0 −2.58715 4.48108i 2.42081 4.19297i 0 16.3234 + 8.74903i −10.2563 0 8.78669 + 15.2190i
163.3 −0.815927 + 1.41323i 0 2.66853 + 4.62202i 6.18949 10.7205i 0 −6.28171 17.4224i −21.7641 0 10.1003 + 17.4943i
163.4 −0.270836 + 0.469102i 0 3.85330 + 6.67410i −8.36445 + 14.4876i 0 18.5069 + 0.702629i −8.50782 0 −4.53079 7.84756i
163.5 0.720983 1.24878i 0 2.96037 + 5.12751i 3.30182 5.71892i 0 −10.3191 + 15.3791i 20.0732 0 −4.76111 8.24648i
163.6 2.38108 4.12414i 0 −7.33904 12.7116i 9.35067 16.1958i 0 14.2072 11.8809i −31.8020 0 −44.5293 77.1270i
163.7 2.54592 4.40966i 0 −8.96343 15.5251i −9.29927 + 16.1068i 0 −4.02294 + 18.0781i −50.5460 0 47.3504 + 82.0134i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.7 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 189.4.e.g yes 14
3.b odd 2 1 189.4.e.f 14
7.c even 3 1 inner 189.4.e.g yes 14
7.c even 3 1 1323.4.a.bi 7
7.d odd 6 1 1323.4.a.bh 7
21.g even 6 1 1323.4.a.bk 7
21.h odd 6 1 189.4.e.f 14
21.h odd 6 1 1323.4.a.bj 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.f 14 3.b odd 2 1
189.4.e.f 14 21.h odd 6 1
189.4.e.g yes 14 1.a even 1 1 trivial
189.4.e.g yes 14 7.c even 3 1 inner
1323.4.a.bh 7 7.d odd 6 1
1323.4.a.bi 7 7.c even 3 1
1323.4.a.bj 7 21.h odd 6 1
1323.4.a.bk 7 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(189, [\chi])$$:

 $$T_{2}^{14} - T_{2}^{13} + 44 T_{2}^{12} + 23 T_{2}^{11} + 1346 T_{2}^{10} + 854 T_{2}^{9} + \cdots + 254016$$ T2^14 - T2^13 + 44*T2^12 + 23*T2^11 + 1346*T2^10 + 854*T2^9 + 20545*T2^8 + 27750*T2^7 + 221349*T2^6 + 172746*T2^5 + 551772*T2^4 + 275616*T2^3 + 1006128*T2^2 + 471744*T2 + 254016 $$T_{13}^{7} + 124 T_{13}^{6} - 354 T_{13}^{5} - 620994 T_{13}^{4} - 26760399 T_{13}^{3} + \cdots + 86896171316$$ T13^7 + 124*T13^6 - 354*T13^5 - 620994*T13^4 - 26760399*T13^3 - 167423118*T13^2 + 7969485833*T13 + 86896171316

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14} - T^{13} + \cdots + 254016$$
$3$ $$T^{14}$$
$5$ $$T^{14} + \cdots + 356440542214689$$
$7$ $$T^{14} + \cdots + 55\!\cdots\!07$$
$11$ $$T^{14} + \cdots + 87\!\cdots\!96$$
$13$ $$(T^{7} + 124 T^{6} + \cdots + 86896171316)^{2}$$
$17$ $$T^{14} + \cdots + 14\!\cdots\!24$$
$19$ $$T^{14} + \cdots + 86\!\cdots\!56$$
$23$ $$T^{14} + \cdots + 40\!\cdots\!76$$
$29$ $$(T^{7} + \cdots + 495493458583509)^{2}$$
$31$ $$T^{14} + \cdots + 14\!\cdots\!76$$
$37$ $$T^{14} + \cdots + 13\!\cdots\!56$$
$41$ $$(T^{7} + \cdots + 6874920774882)^{2}$$
$43$ $$(T^{7} + \cdots - 765727707363728)^{2}$$
$47$ $$T^{14} + \cdots + 48\!\cdots\!36$$
$53$ $$T^{14} + \cdots + 11\!\cdots\!29$$
$59$ $$T^{14} + \cdots + 32\!\cdots\!25$$
$61$ $$T^{14} + \cdots + 21\!\cdots\!96$$
$67$ $$T^{14} + \cdots + 91\!\cdots\!00$$
$71$ $$(T^{7} + \cdots - 23\!\cdots\!58)^{2}$$
$73$ $$T^{14} + \cdots + 15\!\cdots\!25$$
$79$ $$T^{14} + \cdots + 64\!\cdots\!61$$
$83$ $$(T^{7} + \cdots - 86\!\cdots\!12)^{2}$$
$89$ $$T^{14} + \cdots + 16\!\cdots\!96$$
$97$ $$(T^{7} + \cdots + 43\!\cdots\!00)^{2}$$