# Properties

 Label 153.4.l.a Level $153$ Weight $4$ Character orbit 153.l Analytic conductor $9.027$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,4,Mod(19,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 7]))

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

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

chi := DirichletCharacter("153.19");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$153 = 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 153.l (of order $$8$$, degree $$4$$, 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: $$9.02729223088$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 54x^{10} + 1085x^{8} + 9836x^{6} + 38276x^{4} + 49664x^{2} + 16384$$ x^12 + 54*x^10 + 1085*x^8 + 9836*x^6 + 38276*x^4 + 49664*x^2 + 16384 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 17) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{11} - \beta_{2}) q^{2} + (\beta_{9} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{5} + ( - 2 \beta_{11} + \beta_{10} + 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + \cdots - 2) q^{7}+ \cdots + (3 \beta_{10} + 2 \beta_{9} + \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots - 2) q^{8}+O(q^{10})$$ q + (-b11 - b2) * q^2 + (b9 + b6 - b5 - b3 - b2 + b1) * q^4 + (-b10 + b9 - b8 + b6 - b5 + b4 - 2*b3 - b2 + 2) * q^5 + (-2*b11 + b10 + 2*b7 + 2*b5 - 2*b4 - 2*b2 - b1 - 2) * q^7 + (3*b10 + 2*b9 + b8 - 3*b7 + 3*b6 - b5 - 2*b4 - 3*b3 + 2*b1 - 2) * q^8 $$q + ( - \beta_{11} - \beta_{2}) q^{2} + (\beta_{9} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{5} + ( - 2 \beta_{11} + \beta_{10} + 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + \cdots - 2) q^{7}+ \cdots + ( - 60 \beta_{11} - 60 \beta_{10} + 323 \beta_{9} + 38 \beta_{8} + \cdots + 147 \beta_1) q^{98}+O(q^{100})$$ q + (-b11 - b2) * q^2 + (b9 + b6 - b5 - b3 - b2 + b1) * q^4 + (-b10 + b9 - b8 + b6 - b5 + b4 - 2*b3 - b2 + 2) * q^5 + (-2*b11 + b10 + 2*b7 + 2*b5 - 2*b4 - 2*b2 - b1 - 2) * q^7 + (3*b10 + 2*b9 + b8 - 3*b7 + 3*b6 - b5 - 2*b4 - 3*b3 + 2*b1 - 2) * q^8 + (-4*b11 - 4*b10 + 3*b9 - 3*b7 + 3*b6 - 3*b5 - 4*b4 - 3*b3 - 10*b2 + 4*b1 - 10) * q^10 + (b11 - 7*b10 - 8*b9 - 5*b8 - b7 + b5 - b4 - 8*b3 + 2*b2 - 7*b1 - 2) * q^11 + (6*b11 + 6*b10 - 2*b9 - 4*b8 - 4*b7 + 3*b6 - 3*b5 + 11*b3 + 11*b2 - 6*b1) * q^13 + (-4*b11 + 2*b10 + 11*b9 - b8 + 6*b7 - b6 + 2*b4 + 7*b3 + 11*b2 - 4*b1 + 7) * q^14 + (8*b11 - 8*b10 - 2*b8 + 2*b7 + 5*b6 + 5*b5 - 5*b4 - 9*b3 + 9*b2 + 21) * q^16 + (-15*b11 + 8*b10 - 31*b9 - 2*b8 + b7 + 8*b6 - 4*b5 - b4 - 6*b3 - 10*b2 - 4*b1 - 12) * q^17 + (-14*b11 - 14*b9 - 3*b8 + 3*b7 + 3*b6 - 3*b5 - 19*b4 + 16*b2 - 19*b1 - 14) * q^19 + (8*b11 - 12*b10 + 29*b9 + 6*b8 - 9*b7 + 6*b6 - 12*b4 - 38*b3 + 29*b2 + 8*b1 - 38) * q^20 + (-7*b11 + 16*b10 - 13*b9 + 4*b8 - 4*b6 - b5 - 16*b4 + 65*b3 + 13*b2 + 7*b1 - 65) * q^22 + (24*b11 + 9*b10 + 24*b9 - 24*b4 + 24*b3 - 20*b2 + 9*b1 + 20) * q^23 + (-22*b10 + 27*b9 + 3*b8 - 10*b7 + 10*b6 - 3*b5 - 7*b4 + 47*b3 + 7*b1 - 27) * q^25 + (-2*b10 - 57*b9 - 3*b8 + 2*b7 - 2*b6 + 3*b5 - 4*b4 + 64*b3 + 4*b1 + 57) * q^26 + (-12*b11 + 2*b10 + 31*b9 - 12*b8 - 3*b7 + 3*b5 + 12*b4 + 31*b3 - 35*b2 + 2*b1 + 35) * q^28 + (-10*b11 - 3*b10 + 88*b9 + 14*b8 - 14*b6 + 9*b5 + 3*b4 + 51*b3 - 88*b2 + 10*b1 - 51) * q^29 + (-3*b11 - 4*b10 - 74*b9 - 6*b8 + 14*b7 - 6*b6 - 4*b4 + 6*b3 - 74*b2 - 3*b1 + 6) * q^31 + (-15*b11 - 72*b9 - 10*b8 + 2*b7 + 2*b6 - 10*b5 - 6*b4 - 3*b2 - 6*b1 - 72) * q^32 + (2*b11 + 8*b10 + 122*b9 - 2*b8 + b7 + 25*b6 - 21*b5 - 18*b4 + 45*b3 - 10*b2 + 13*b1 + 56) * q^34 + (-11*b11 + 11*b10 - 2*b8 + 2*b7 - 6*b6 - 6*b5 + 12*b4 + 48*b3 - 48*b2 + 44) * q^35 + (26*b11 + 15*b10 - 73*b9 + 12*b8 - 9*b7 + 12*b6 + 15*b4 + 98*b3 - 73*b2 + 26*b1 + 98) * q^37 + (2*b11 + 2*b10 + 58*b9 + 16*b8 + 16*b7 + 4*b6 - 4*b5 + 146*b3 + 146*b2 - 10*b1) * q^38 + (30*b11 - 6*b10 - 20*b9 + 21*b8 - 11*b7 + 11*b5 - 30*b4 - 20*b3 + 139*b2 - 6*b1 - 139) * q^40 + (7*b11 - 21*b10 - 44*b9 + 23*b7 + 12*b6 + 23*b5 + 7*b4 + 44*b3 + b2 + 21*b1 + 1) * q^41 + (18*b10 + 136*b9 + b8 + 28*b7 - 28*b6 - b5 + 6*b4 - 116*b3 - 6*b1 - 136) * q^43 + (22*b11 + 7*b10 + 51*b9 + 4*b7 - 33*b6 + 4*b5 + 22*b4 - 51*b3 + 115*b2 - 7*b1 + 115) * q^44 + (-20*b11 - 24*b10 - 187*b9 + 15*b8 - 15*b6 + 48*b5 + 24*b4 - 63*b3 + 187*b2 + 20*b1 + 63) * q^46 + (52*b11 + 52*b10 - 112*b9 - 22*b8 - 22*b7 - 6*b6 + 6*b5 - 30*b1) * q^47 + (-34*b11 + 58*b9 - 5*b8 - b7 - b6 - 5*b5 - 43*b4 - 125*b2 - 43*b1 + 58) * q^49 + (46*b11 - 46*b10 + 52*b8 - 52*b7 + 17*b6 + 17*b5 + 13*b4 - 111*b3 + 111*b2 - 157) * q^50 + (-20*b11 + 20*b10 - 25*b8 + 25*b7 - 30*b6 - 30*b5 + 18*b4 - 59*b3 + 59*b2 + 22) * q^52 + (-58*b11 - 29*b9 - 40*b8 - 13*b7 - 13*b6 - 40*b5 - 39*b4 - 236*b2 - 39*b1 - 29) * q^53 + (-15*b11 - 15*b10 - 132*b9 + 6*b8 + 6*b7 - 22*b6 + 22*b5 - 122*b3 - 122*b2 - 66*b1) * q^55 + (-40*b11 + 22*b10 + 101*b9 - 15*b8 + 15*b6 + 18*b5 - 22*b4 - 127*b3 - 101*b2 + 40*b1 + 127) * q^56 + (84*b11 - 56*b10 + 172*b9 + 10*b7 - 17*b6 + 10*b5 + 84*b4 - 172*b3 + 39*b2 + 56*b1 + 39) * q^58 + (-110*b10 - 94*b9 + 15*b8 + 14*b7 - 14*b6 - 15*b5 + 54*b4 - 356*b3 - 54*b1 + 94) * q^59 + (85*b11 + 68*b10 - 51*b9 - 29*b7 - b6 - 29*b5 + 85*b4 + 51*b3 - 54*b2 - 68*b1 - 54) * q^61 + (-40*b11 + 78*b10 + 89*b9 - 18*b8 + 19*b7 - 19*b5 + 40*b4 + 89*b3 + 13*b2 + 78*b1 - 13) * q^62 + (-8*b11 - 8*b10 - 41*b9 - 44*b8 - 44*b7 + 15*b6 - 15*b5 + 29*b3 + 29*b2 + 79*b1) * q^64 + (-12*b11 + 36*b10 + 23*b9 - 23*b8 + 48*b7 - 23*b6 + 36*b4 - 69*b3 + 23*b2 - 12*b1 - 69) * q^65 + (-35*b11 + 35*b10 + 66*b8 - 66*b7 + 25*b6 + 25*b5 + 50*b4 - 128*b3 + 128*b2 + 144) * q^67 + (-2*b11 - 93*b10 + 82*b9 - 32*b8 - 18*b7 + 43*b6 + 4*b5 - 16*b4 - 181*b3 + 27*b2 + 38*b1 - 175) * q^68 + (-40*b11 + 160*b9 - 33*b8 + 13*b7 + 13*b6 - 33*b5 + 68*b4 - 154*b2 + 68*b1 + 160) * q^70 + (-43*b11 + 44*b10 + 50*b9 + 42*b8 - 14*b7 + 42*b6 + 44*b4 + 16*b3 + 50*b2 - 43*b1 + 16) * q^71 + (-5*b11 - 73*b10 - 42*b9 + 41*b8 - 41*b6 + 70*b5 + 73*b4 - 139*b3 + 42*b2 + 5*b1 + 139) * q^73 + (-68*b11 + 46*b10 - 109*b9 + 3*b8 - 32*b7 + 32*b5 + 68*b4 - 109*b3 - 198*b2 + 46*b1 + 198) * q^74 + (6*b10 - 92*b9 - 22*b8 + 44*b7 - 44*b6 + 22*b5 + 38*b4 - 142*b3 - 38*b1 + 92) * q^76 + (32*b10 - 88*b9 + 67*b8 - 9*b7 + 9*b6 - 67*b5 - 35*b4 + 14*b3 + 35*b1 + 88) * q^77 + (-6*b11 - 27*b10 - 6*b9 + 24*b8 + 64*b7 - 64*b5 + 6*b4 - 6*b3 + 42*b2 - 27*b1 - 42) * q^79 + (96*b11 - 32*b10 - 132*b9 + 20*b8 - 20*b6 + 31*b5 + 32*b4 - 139*b3 + 132*b2 - 96*b1 + 139) * q^80 + (-82*b11 + 43*b10 - 87*b9 + 3*b8 - 8*b7 + 3*b6 + 43*b4 - 150*b3 - 87*b2 - 82*b1 - 150) * q^82 + (94*b11 + 118*b9 + 12*b8 + 19*b7 + 19*b6 + 12*b5 - 16*b4 + 272*b2 - 16*b1 + 118) * q^83 + (-98*b11 - 69*b10 + 108*b9 - 21*b8 - 15*b7 - b6 - 42*b5 - 53*b4 - 165*b3 - 122*b2 + 60*b1 - 279) * q^85 + (111*b11 - 111*b10 - 79*b8 + 79*b7 - 34*b6 - 34*b5 - 2*b4 - 72*b3 + 72*b2 + 98) * q^86 + (34*b11 + 75*b10 + 269*b9 - 34*b8 + 63*b7 - 34*b6 + 75*b4 - 37*b3 + 269*b2 + 34*b1 - 37) * q^88 + (7*b11 + 7*b10 + 194*b9 - 53*b8 - 53*b7 - 21*b6 + 21*b5 - 249*b3 - 249*b2 + 96*b1) * q^89 + (26*b11 + 34*b10 - 140*b9 + 102*b8 - 22*b7 + 22*b5 - 26*b4 - 140*b3 - 242*b2 + 34*b1 + 242) * q^91 + (6*b11 + 106*b10 - 191*b9 + 19*b7 - 38*b6 + 19*b5 + 6*b4 + 191*b3 - 77*b2 - 106*b1 - 77) * q^92 + (180*b10 - 402*b9 - 58*b8 + 72*b7 - 72*b6 + 58*b5 - 60*b4 + 512*b3 + 60*b1 + 402) * q^94 + (-27*b11 + 55*b10 - 64*b9 + 28*b7 - 80*b6 + 28*b5 - 27*b4 + 64*b3 - 244*b2 - 55*b1 - 244) * q^95 + (104*b11 - 108*b10 + 331*b9 - 75*b8 + 75*b6 - 28*b5 + 108*b4 - 22*b3 - 331*b2 - 104*b1 + 22) * q^97 + (-60*b11 - 60*b10 + 323*b9 + 38*b8 + 38*b7 + 254*b3 + 254*b2 + 147*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{2} + 20 q^{5} - 4 q^{7} - 28 q^{8}+O(q^{10})$$ 12 * q + 4 * q^2 + 20 * q^5 - 4 * q^7 - 28 * q^8 $$12 q + 4 q^{2} + 20 q^{5} - 4 q^{7} - 28 q^{8} - 116 q^{10} - 40 q^{11} + 132 q^{14} + 184 q^{16} - 52 q^{17} - 12 q^{19} - 572 q^{20} - 620 q^{22} + 276 q^{23} - 464 q^{25} + 708 q^{26} + 452 q^{28} - 632 q^{29} + 188 q^{31} - 700 q^{32} + 764 q^{34} + 632 q^{35} + 940 q^{37} - 1864 q^{40} - 176 q^{41} - 1360 q^{43} + 1364 q^{44} + 452 q^{46} + 1044 q^{49} - 2856 q^{50} + 792 q^{52} + 360 q^{53} + 1788 q^{56} - 360 q^{58} + 584 q^{59} - 1052 q^{61} + 380 q^{62} - 404 q^{65} + 1080 q^{67} - 2532 q^{68} + 2072 q^{70} - 28 q^{71} + 824 q^{73} + 2292 q^{74} + 1328 q^{76} + 1252 q^{77} - 196 q^{79} + 904 q^{80} - 1528 q^{82} + 1008 q^{83} - 2824 q^{85} + 1200 q^{86} - 56 q^{88} + 2456 q^{91} - 396 q^{92} + 6360 q^{94} - 2172 q^{95} - 904 q^{97}+O(q^{100})$$ 12 * q + 4 * q^2 + 20 * q^5 - 4 * q^7 - 28 * q^8 - 116 * q^10 - 40 * q^11 + 132 * q^14 + 184 * q^16 - 52 * q^17 - 12 * q^19 - 572 * q^20 - 620 * q^22 + 276 * q^23 - 464 * q^25 + 708 * q^26 + 452 * q^28 - 632 * q^29 + 188 * q^31 - 700 * q^32 + 764 * q^34 + 632 * q^35 + 940 * q^37 - 1864 * q^40 - 176 * q^41 - 1360 * q^43 + 1364 * q^44 + 452 * q^46 + 1044 * q^49 - 2856 * q^50 + 792 * q^52 + 360 * q^53 + 1788 * q^56 - 360 * q^58 + 584 * q^59 - 1052 * q^61 + 380 * q^62 - 404 * q^65 + 1080 * q^67 - 2532 * q^68 + 2072 * q^70 - 28 * q^71 + 824 * q^73 + 2292 * q^74 + 1328 * q^76 + 1252 * q^77 - 196 * q^79 + 904 * q^80 - 1528 * q^82 + 1008 * q^83 - 2824 * q^85 + 1200 * q^86 - 56 * q^88 + 2456 * q^91 - 396 * q^92 + 6360 * q^94 - 2172 * q^95 - 904 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 54x^{10} + 1085x^{8} + 9836x^{6} + 38276x^{4} + 49664x^{2} + 16384$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - \nu^{11} - 408 \nu^{10} + 10 \nu^{9} - 17680 \nu^{8} - 1725 \nu^{7} - 268600 \nu^{6} - 73516 \nu^{5} - 1728288 \nu^{4} - 729732 \nu^{3} - 4623456 \nu^{2} + \cdots - 2889728 ) / 1392640$$ (-v^11 - 408*v^10 + 10*v^9 - 17680*v^8 - 1725*v^7 - 268600*v^6 - 73516*v^5 - 1728288*v^4 - 729732*v^3 - 4623456*v^2 - 1741056*v - 2889728) / 1392640 $$\beta_{3}$$ $$=$$ $$( - \nu^{11} + 408 \nu^{10} + 10 \nu^{9} + 17680 \nu^{8} - 1725 \nu^{7} + 268600 \nu^{6} - 73516 \nu^{5} + 1728288 \nu^{4} - 729732 \nu^{3} + 4623456 \nu^{2} + \cdots + 2889728 ) / 1392640$$ (-v^11 + 408*v^10 + 10*v^9 + 17680*v^8 - 1725*v^7 + 268600*v^6 - 73516*v^5 + 1728288*v^4 - 729732*v^3 + 4623456*v^2 - 1741056*v + 2889728) / 1392640 $$\beta_{4}$$ $$=$$ $$( \nu^{10} + 41\nu^{8} + 569\nu^{6} + 3051\nu^{4} + 5498\nu^{2} + 2432 ) / 544$$ (v^10 + 41*v^8 + 569*v^6 + 3051*v^4 + 5498*v^2 + 2432) / 544 $$\beta_{5}$$ $$=$$ $$( 241 \nu^{11} - 280 \nu^{10} + 19350 \nu^{9} + 2800 \nu^{8} + 502765 \nu^{7} + 387400 \nu^{6} + 5292396 \nu^{5} + 5701600 \nu^{4} + 20629572 \nu^{3} + \cdots + 12462080 ) / 1392640$$ (241*v^11 - 280*v^10 + 19350*v^9 + 2800*v^8 + 502765*v^7 + 387400*v^6 + 5292396*v^5 + 5701600*v^4 + 20629572*v^3 + 23023520*v^2 + 17121536*v + 12462080) / 1392640 $$\beta_{6}$$ $$=$$ $$( - 241 \nu^{11} - 280 \nu^{10} - 19350 \nu^{9} + 2800 \nu^{8} - 502765 \nu^{7} + 387400 \nu^{6} - 5292396 \nu^{5} + 5701600 \nu^{4} - 20629572 \nu^{3} + \cdots + 12462080 ) / 1392640$$ (-241*v^11 - 280*v^10 - 19350*v^9 + 2800*v^8 - 502765*v^7 + 387400*v^6 - 5292396*v^5 + 5701600*v^4 - 20629572*v^3 + 23023520*v^2 - 17121536*v + 12462080) / 1392640 $$\beta_{7}$$ $$=$$ $$( - 325 \nu^{11} + 872 \nu^{10} - 18510 \nu^{9} + 34800 \nu^{8} - 386545 \nu^{7} + 459720 \nu^{6} - 3460060 \nu^{5} + 2176992 \nu^{4} - 10989460 \nu^{3} + \cdots - 5347328 ) / 1392640$$ (-325*v^11 + 872*v^10 - 18510*v^9 + 34800*v^8 - 386545*v^7 + 459720*v^6 - 3460060*v^5 + 2176992*v^4 - 10989460*v^3 + 1717664*v^2 - 1301760*v - 5347328) / 1392640 $$\beta_{8}$$ $$=$$ $$( - 325 \nu^{11} - 872 \nu^{10} - 18510 \nu^{9} - 34800 \nu^{8} - 386545 \nu^{7} - 459720 \nu^{6} - 3460060 \nu^{5} - 2176992 \nu^{4} - 10989460 \nu^{3} + \cdots + 5347328 ) / 1392640$$ (-325*v^11 - 872*v^10 - 18510*v^9 - 34800*v^8 - 386545*v^7 - 459720*v^6 - 3460060*v^5 - 2176992*v^4 - 10989460*v^3 - 1717664*v^2 - 1301760*v + 5347328) / 1392640 $$\beta_{9}$$ $$=$$ $$( 19\nu^{11} + 898\nu^{9} + 15367\nu^{7} + 114052\nu^{5} + 336716\nu^{3} + 239872\nu ) / 69632$$ (19*v^11 + 898*v^9 + 15367*v^7 + 114052*v^5 + 336716*v^3 + 239872*v) / 69632 $$\beta_{10}$$ $$=$$ $$( - 51 \nu^{11} + 8 \nu^{10} - 2210 \nu^{9} - 80 \nu^{8} - 33575 \nu^{7} - 7960 \nu^{6} - 216036 \nu^{5} - 86432 \nu^{4} - 577932 \nu^{3} - 211424 \nu^{2} - 361216 \nu + 2048 ) / 174080$$ (-51*v^11 + 8*v^10 - 2210*v^9 - 80*v^8 - 33575*v^7 - 7960*v^6 - 216036*v^5 - 86432*v^4 - 577932*v^3 - 211424*v^2 - 361216*v + 2048) / 174080 $$\beta_{11}$$ $$=$$ $$( - 51 \nu^{11} - 8 \nu^{10} - 2210 \nu^{9} + 80 \nu^{8} - 33575 \nu^{7} + 7960 \nu^{6} - 216036 \nu^{5} + 86432 \nu^{4} - 577932 \nu^{3} + 211424 \nu^{2} - 361216 \nu - 2048 ) / 174080$$ (-51*v^11 - 8*v^10 - 2210*v^9 + 80*v^8 - 33575*v^7 + 7960*v^6 - 216036*v^5 + 86432*v^4 - 577932*v^3 + 211424*v^2 - 361216*v - 2048) / 174080
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} - 8$$ b8 - b7 + b4 - b3 + b2 - 8 $$\nu^{3}$$ $$=$$ $$2\beta_{11} + 2\beta_{10} + 6\beta_{9} + \beta_{8} + \beta_{7} - \beta_{3} - \beta_{2} - 13\beta_1$$ 2*b11 + 2*b10 + 6*b9 + b8 + b7 - b3 - b2 - 13*b1 $$\nu^{4}$$ $$=$$ $$-16\beta_{8} + 16\beta_{7} - \beta_{6} - \beta_{5} - 21\beta_{4} + 31\beta_{3} - 31\beta_{2} + 106$$ -16*b8 + 16*b7 - b6 - b5 - 21*b4 + 31*b3 - 31*b2 + 106 $$\nu^{5}$$ $$=$$ $$- 50 \beta_{11} - 50 \beta_{10} - 138 \beta_{9} - 15 \beta_{8} - 15 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 19 \beta_{3} + 19 \beta_{2} + 189 \beta_1$$ -50*b11 - 50*b10 - 138*b9 - 15*b8 - 15*b7 - 4*b6 + 4*b5 + 19*b3 + 19*b2 + 189*b1 $$\nu^{6}$$ $$=$$ $$- 28 \beta_{11} + 28 \beta_{10} + 250 \beta_{8} - 250 \beta_{7} + 31 \beta_{6} + 31 \beta_{5} + 373 \beta_{4} - 619 \beta_{3} + 619 \beta_{2} - 1574$$ -28*b11 + 28*b10 + 250*b8 - 250*b7 + 31*b6 + 31*b5 + 373*b4 - 619*b3 + 619*b2 - 1574 $$\nu^{7}$$ $$=$$ $$990 \beta_{11} + 990 \beta_{10} + 2602 \beta_{9} + 191 \beta_{8} + 191 \beta_{7} + 120 \beta_{6} - 120 \beta_{5} - 535 \beta_{3} - 535 \beta_{2} - 2885 \beta_1$$ 990*b11 + 990*b10 + 2602*b9 + 191*b8 + 191*b7 + 120*b6 - 120*b5 - 535*b3 - 535*b2 - 2885*b1 $$\nu^{8}$$ $$=$$ $$1072 \beta_{11} - 1072 \beta_{10} - 3946 \beta_{8} + 3946 \beta_{7} - 679 \beta_{6} - 679 \beta_{5} - 6349 \beta_{4} + 11187 \beta_{3} - 11187 \beta_{2} + 24438$$ 1072*b11 - 1072*b10 - 3946*b8 + 3946*b7 - 679*b6 - 679*b5 - 6349*b4 + 11187*b3 - 11187*b2 + 24438 $$\nu^{9}$$ $$=$$ $$- 18242 \beta_{11} - 18242 \beta_{10} - 46402 \beta_{9} - 2195 \beta_{8} - 2195 \beta_{7} - 2796 \beta_{6} + 2796 \beta_{5} + 13567 \beta_{3} + 13567 \beta_{2} + 45213 \beta_1$$ -18242*b11 - 18242*b10 - 46402*b9 - 2195*b8 - 2195*b7 - 2796*b6 + 2796*b5 + 13567*b3 + 13567*b2 + 45213*b1 $$\nu^{10}$$ $$=$$ $$- 28020 \beta_{11} + 28020 \beta_{10} + 62854 \beta_{8} - 62854 \beta_{7} + 13251 \beta_{6} + 13251 \beta_{5} + 107189 \beta_{4} - 195539 \beta_{3} + 195539 \beta_{2} - 388206$$ -28020*b11 + 28020*b10 + 62854*b8 - 62854*b7 + 13251*b6 + 13251*b5 + 107189*b4 - 195539*b3 + 195539*b2 - 388206 $$\nu^{11}$$ $$=$$ $$326166 \beta_{11} + 326166 \beta_{10} + 814346 \beta_{9} + 21583 \beta_{8} + 21583 \beta_{7} + 59104 \beta_{6} - 59104 \beta_{5} - 304847 \beta_{3} - 304847 \beta_{2} - 720309 \beta_1$$ 326166*b11 + 326166*b10 + 814346*b9 + 21583*b8 + 21583*b7 + 59104*b6 - 59104*b5 - 304847*b3 - 304847*b2 - 720309*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/153\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$137$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 4.15292i 0.705468i 3.86166i 2.49971i − 1.22788i − 3.68604i − 2.49971i 1.22788i 3.68604i 4.15292i − 0.705468i − 3.86166i
−2.22945 2.22945i 0 1.94089i 1.91633 + 4.62643i 0 1.06584 2.57316i −13.5085 + 13.5085i 0 6.04203 14.5867i
19.2 1.20595 + 1.20595i 0 5.09138i −2.60601 6.29147i 0 −5.31013 + 12.8198i 15.7875 15.7875i 0 4.44447 10.7299i
19.3 3.43772 + 3.43772i 0 15.6358i 7.10390 + 17.1503i 0 5.36561 12.9537i −26.2496 + 26.2496i 0 −34.5367 + 83.3791i
100.1 −2.47467 2.47467i 0 4.24796i 8.05561 3.33674i 0 −6.33320 2.62330i −9.28506 + 9.28506i 0 −28.1923 11.6776i
100.2 0.161134 + 0.161134i 0 7.94807i −2.54200 + 1.05293i 0 −19.8837 8.23610i 2.56978 2.56978i 0 −0.579266 0.239940i
100.3 1.89932 + 1.89932i 0 0.785167i −1.92782 + 0.798529i 0 23.0956 + 9.56650i 16.6858 16.6858i 0 −5.17821 2.14488i
127.1 −2.47467 + 2.47467i 0 4.24796i 8.05561 + 3.33674i 0 −6.33320 + 2.62330i −9.28506 9.28506i 0 −28.1923 + 11.6776i
127.2 0.161134 0.161134i 0 7.94807i −2.54200 1.05293i 0 −19.8837 + 8.23610i 2.56978 + 2.56978i 0 −0.579266 + 0.239940i
127.3 1.89932 1.89932i 0 0.785167i −1.92782 0.798529i 0 23.0956 9.56650i 16.6858 + 16.6858i 0 −5.17821 + 2.14488i
145.1 −2.22945 + 2.22945i 0 1.94089i 1.91633 4.62643i 0 1.06584 + 2.57316i −13.5085 13.5085i 0 6.04203 + 14.5867i
145.2 1.20595 1.20595i 0 5.09138i −2.60601 + 6.29147i 0 −5.31013 12.8198i 15.7875 + 15.7875i 0 4.44447 + 10.7299i
145.3 3.43772 3.43772i 0 15.6358i 7.10390 17.1503i 0 5.36561 + 12.9537i −26.2496 26.2496i 0 −34.5367 83.3791i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.4.l.a 12
3.b odd 2 1 17.4.d.a 12
17.d even 8 1 inner 153.4.l.a 12
51.g odd 8 1 17.4.d.a 12
51.i even 16 2 289.4.a.g 12
51.i even 16 2 289.4.b.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.d.a 12 3.b odd 2 1
17.4.d.a 12 51.g odd 8 1
153.4.l.a 12 1.a even 1 1 trivial
153.4.l.a 12 17.d even 8 1 inner
289.4.a.g 12 51.i even 16 2
289.4.b.e 12 51.i even 16 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 4 T_{2}^{11} + 8 T_{2}^{10} + 20 T_{2}^{9} + 322 T_{2}^{8} - 924 T_{2}^{7} + 1320 T_{2}^{6} + 468 T_{2}^{5} + 18817 T_{2}^{4} - 54040 T_{2}^{3} + 76832 T_{2}^{2} - 21952 T_{2} + 3136$$ acting on $$S_{4}^{\mathrm{new}}(153, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 4 T^{11} + 8 T^{10} + 20 T^{9} + \cdots + 3136$$
$3$ $$T^{12}$$
$5$ $$T^{12} - 20 T^{11} + \cdots + 1004236928$$
$7$ $$T^{12} + 4 T^{11} + \cdots + 3993906708992$$
$11$ $$T^{12} + 40 T^{11} + \cdots + 44\!\cdots\!12$$
$13$ $$T^{12} + 11916 T^{10} + \cdots + 34\!\cdots\!44$$
$17$ $$T^{12} + 52 T^{11} + \cdots + 14\!\cdots\!09$$
$19$ $$T^{12} + 12 T^{11} + \cdots + 13\!\cdots\!36$$
$23$ $$T^{12} - 276 T^{11} + \cdots + 78\!\cdots\!08$$
$29$ $$T^{12} + 632 T^{11} + \cdots + 73\!\cdots\!48$$
$31$ $$T^{12} - 188 T^{11} + \cdots + 18\!\cdots\!52$$
$37$ $$T^{12} - 940 T^{11} + \cdots + 16\!\cdots\!68$$
$41$ $$T^{12} + 176 T^{11} + \cdots + 36\!\cdots\!72$$
$43$ $$T^{12} + 1360 T^{11} + \cdots + 65\!\cdots\!16$$
$47$ $$T^{12} + 665464 T^{10} + \cdots + 18\!\cdots\!04$$
$53$ $$T^{12} - 360 T^{11} + \cdots + 95\!\cdots\!04$$
$59$ $$T^{12} - 584 T^{11} + \cdots + 71\!\cdots\!76$$
$61$ $$T^{12} + 1052 T^{11} + \cdots + 23\!\cdots\!32$$
$67$ $$(T^{6} - 540 T^{5} + \cdots + 61\!\cdots\!36)^{2}$$
$71$ $$T^{12} + 28 T^{11} + \cdots + 32\!\cdots\!48$$
$73$ $$T^{12} - 824 T^{11} + \cdots + 99\!\cdots\!28$$
$79$ $$T^{12} + 196 T^{11} + \cdots + 93\!\cdots\!12$$
$83$ $$T^{12} - 1008 T^{11} + \cdots + 15\!\cdots\!96$$
$89$ $$T^{12} + 2740280 T^{10} + \cdots + 22\!\cdots\!56$$
$97$ $$T^{12} + 904 T^{11} + \cdots + 16\!\cdots\!52$$