# Properties

 Label 144.2.k.b Level $144$ Weight $2$ Character orbit 144.k Analytic conductor $1.150$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,2,Mod(37,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("144.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 144.k (of order $$4$$, 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: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.18939904.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ x^8 - 4*x^7 + 14*x^6 - 28*x^5 + 43*x^4 - 44*x^3 + 30*x^2 - 12*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} + (\beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{4} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_1 + 1) q^{8}+O(q^{10})$$ q - b6 * q^2 + (b7 - b5 + b4 - b3 - b1 - 1) * q^4 + (b6 + b4 + b3 + b2 - b1) * q^5 + (-b6 + 2*b5 + b4 + b3 + b2) * q^7 + (-b7 + b6 + b4 - b1 + 1) * q^8 $$q - \beta_{6} q^{2} + (\beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{4} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_1 + 1) q^{8} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{10} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + \beta_1) q^{11} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 1) q^{13} + (\beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_1 - 1) q^{14} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{16} + ( - \beta_{7} - 2 \beta_{4} + 2 \beta_{2} + \beta_1 + 2) q^{17} + (2 \beta_{6} - 2 \beta_{2} - 2) q^{19} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{20} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{22} - 2 \beta_{7} q^{23} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{25} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{26} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{28} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 4) q^{31} + (2 \beta_{7} - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{32} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{34} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_1 - 4) q^{35} + ( - 2 \beta_{7} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{37} + ( - 2 \beta_{7} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_1 + 2) q^{38} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 4 \beta_{2} + 4) q^{40} + (3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - \beta_1) q^{41} + (2 \beta_{7} - 4 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{43} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{44} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{46} - 2 \beta_1 q^{47} + ( - 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 1) q^{49} + ( - 4 \beta_{7} + 2 \beta_{6} - \beta_{4} + 4 \beta_{3} + 5) q^{50} + (2 \beta_{7} + \beta_{6} + 5 \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 3) q^{52} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{53} + ( - 4 \beta_{7} - 4 \beta_{5}) q^{55} + ( - 2 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} + 4 \beta_{2} + 2) q^{56} + ( - 3 \beta_{6} + 5 \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{58} + ( - 4 \beta_{5} - 4) q^{59} + ( - 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 1) q^{61} + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{4} + 1) q^{62} + ( - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{64} + (\beta_{7} + 2 \beta_{4} - 2 \beta_{2} - 5 \beta_1) q^{65} + (4 \beta_{7} + 4 \beta_{4} - 4 \beta_{3} - 4) q^{67} + (4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 6) q^{68} + ( - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 6) q^{70} + ( - 2 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2}) q^{71} + ( - 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{73} + ( - 2 \beta_{7} + 3 \beta_{6} + \beta_{4} + 4 \beta_{2} - 2 \beta_1 - 5) q^{74} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 4) q^{76} + ( - 4 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{77} + ( - 4 \beta_{7} + 5 \beta_{6} - 3 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{79} + ( - 2 \beta_{7} - 2 \beta_{5} - 4 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{80} + ( - 2 \beta_{7} + 4 \beta_{5} + 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{82} + ( - \beta_{7} + 4 \beta_{6} - 6 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{83}+ \cdots + (4 \beta_{7} + \beta_{6} - 8 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 2) q^{98}+O(q^{100})$$ q - b6 * q^2 + (b7 - b5 + b4 - b3 - b1 - 1) * q^4 + (b6 + b4 + b3 + b2 - b1) * q^5 + (-b6 + 2*b5 + b4 + b3 + b2) * q^7 + (-b7 + b6 + b4 - b1 + 1) * q^8 + (-2*b7 + b6 - b5 - b4 + b3 - b2 - 1) * q^10 + (b7 - 2*b6 + 2*b5 - 2*b2 + b1) * q^11 + (b5 - 2*b4 + 2*b3 + 2*b1 + 1) * q^13 + (b6 - 2*b5 - b4 - 2*b1 - 1) * q^14 + (-2*b7 - b6 - b5 - b4 - b3 + b2 + 1) * q^16 + (-b7 - 2*b4 + 2*b2 + b1 + 2) * q^17 + (2*b6 - 2*b2 - 2) * q^19 + (-2*b3 + 2*b1 - 2) * q^20 + (2*b7 - 2*b6 + 2*b5 - 2*b3 - 2*b1) * q^22 - 2*b7 * q^23 + (2*b7 + 2*b6 - b5 + 2*b4 - 2*b3 + 2*b2 - 2*b1) * q^25 + (2*b7 - b6 + 2*b5 - b4 + 2*b3 - 2*b2 + 2*b1 - 3) * q^26 + (b6 - b5 + b4 - b3 - 3*b2 + 2*b1 - 1) * q^28 + (b7 + b6 - 2*b5 - b4 + b3 - b2 + 2*b1 + 2) * q^29 + (-b6 - b4 - b3 + b2 + 4) * q^31 + (2*b7 - 4*b5 + 2*b4 - 2*b3 + 2*b2) * q^32 + (2*b7 - 2*b5 + 2*b3 + 2*b1) * q^34 + (b7 + 2*b5 + 2*b4 - 2*b3 - b1 - 4) * q^35 + (-2*b7 - b5 + 2*b4 + 2*b3 + 2*b1 - 3) * q^37 + (-2*b7 + 4*b5 - 2*b4 + 2*b1 + 2) * q^38 + (2*b6 + 2*b5 + 2*b3 + 4*b2 + 4) * q^40 + (3*b7 + 2*b6 - 2*b5 - 2*b3 - b1) * q^41 + (2*b7 - 4*b6 - 2*b4 - 2*b3 - 4*b2 + 2*b1 - 2) * q^43 + (2*b7 - 2*b5 - 2*b3 - 2*b2 - 2*b1 + 4) * q^44 + (-2*b5 - 2*b3 + 2*b2) * q^46 - 2*b1 * q^47 + (-2*b6 - 2*b4 - 2*b3 + 2*b2 - 4*b1 + 1) * q^49 + (-4*b7 + 2*b6 - b4 + 4*b3 + 5) * q^50 + (2*b7 + b6 + 5*b5 - b4 + b3 - 3*b2 - 3) * q^52 + (-2*b7 - b6 - 2*b5 - b4 - b3 - b2 + 3*b1 - 2) * q^53 + (-4*b7 - 4*b5) * q^55 + (-2*b7 - 2*b6 + 6*b5 + 4*b2 + 2) * q^56 + (-3*b6 + 5*b5 + b4 + 3*b3 - b2 + 2*b1 - 3) * q^58 + (-4*b5 - 4) * q^59 + (-2*b7 + 2*b6 - 3*b5 - 2*b2 - 2*b1 + 1) * q^61 + (2*b7 - 3*b6 - 2*b5 + b4 + 1) * q^62 + (-2*b7 + 2*b6 + 4*b4 + 4*b3 + 2*b2 - 2*b1) * q^64 + (b7 + 2*b4 - 2*b2 - 5*b1) * q^65 + (4*b7 + 4*b4 - 4*b3 - 4) * q^67 + (4*b5 + 2*b4 + 4*b3 - 2*b2 - 6) * q^68 + (-2*b7 + 4*b6 - 2*b4 + 2*b2 - 2*b1 + 6) * q^70 + (-2*b6 + 4*b5 + 2*b4 + 2*b3 + 2*b2) * q^71 + (-4*b7 + 2*b6 - 2*b4 - 2*b3 - 2*b2) * q^73 + (-2*b7 + 3*b6 + b4 + 4*b2 - 2*b1 - 5) * q^74 + (2*b7 - 2*b6 - 4*b4 + 2*b2 + 2*b1 + 4) * q^76 + (-4*b7 - 2*b6 + 4*b5 - 2*b4 + 2*b3 + 2*b2 - 2*b1) * q^77 + (-4*b7 + 5*b6 - 3*b4 + 5*b3 + 3*b2 - 4*b1) * q^79 + (-2*b7 - 2*b5 - 4*b4 + 6*b3 - 2*b2 + 2*b1) * q^80 + (-2*b7 + 4*b5 + 4*b3 - 2*b2 + 2*b1 + 4) * q^82 + (-b7 + 4*b6 - 6*b5 - 2*b4 + 2*b3 - 4*b2 + b1 + 4) * q^83 + (4*b7 - 2*b6 + 2*b4 + 2*b3 - 2*b2 - 2*b1 - 4) * q^85 + (6*b7 - 2*b6 + 4*b5 + 4*b4 - 4*b3 - 2*b1) * q^86 + (-6*b6 + 2*b5 + 2*b4 - 2*b3 - 2*b2 + 2) * q^88 + (-6*b7 + 2*b5 + 4*b4 + 4*b2 - 2*b1) * q^89 + (2*b7 + 2*b4 + 2*b3 - 2*b1 - 2) * q^91 + (2*b6 - 2*b5 + 2*b4 + 2*b3 + 2*b2 + 2) * q^92 + (-2*b5 - 2*b3 - 2*b2) * q^94 + (-2*b6 - 2*b4 - 2*b3 + 2*b2 + 2*b1 + 8) * q^95 + (2*b7 - 4*b6 - 4*b3 + 6*b1) * q^97 + (4*b7 + b6 - 8*b5 + 2*b4 - 4*b3 - 4*b2 + 2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{4} + 12 q^{8}+O(q^{10})$$ 8 * q - 4 * q^4 + 12 * q^8 $$8 q - 4 q^{4} + 12 q^{8} - 8 q^{10} + 8 q^{11} - 12 q^{14} - 8 q^{19} - 16 q^{20} - 20 q^{26} + 8 q^{28} + 16 q^{29} + 24 q^{31} - 24 q^{35} - 16 q^{37} + 8 q^{38} + 16 q^{40} - 8 q^{43} + 40 q^{44} - 8 q^{46} - 8 q^{49} + 36 q^{50} - 16 q^{52} - 16 q^{53} - 16 q^{58} - 32 q^{59} + 16 q^{61} + 12 q^{62} + 8 q^{64} + 16 q^{65} - 16 q^{67} - 32 q^{68} + 32 q^{70} - 52 q^{74} + 8 q^{76} - 16 q^{77} - 24 q^{79} - 8 q^{80} + 40 q^{82} + 40 q^{83} - 16 q^{85} + 16 q^{86} + 32 q^{88} - 8 q^{91} + 16 q^{92} + 8 q^{94} + 48 q^{95} + 40 q^{98}+O(q^{100})$$ 8 * q - 4 * q^4 + 12 * q^8 - 8 * q^10 + 8 * q^11 - 12 * q^14 - 8 * q^19 - 16 * q^20 - 20 * q^26 + 8 * q^28 + 16 * q^29 + 24 * q^31 - 24 * q^35 - 16 * q^37 + 8 * q^38 + 16 * q^40 - 8 * q^43 + 40 * q^44 - 8 * q^46 - 8 * q^49 + 36 * q^50 - 16 * q^52 - 16 * q^53 - 16 * q^58 - 32 * q^59 + 16 * q^61 + 12 * q^62 + 8 * q^64 + 16 * q^65 - 16 * q^67 - 32 * q^68 + 32 * q^70 - 52 * q^74 + 8 * q^76 - 16 * q^77 - 24 * q^79 - 8 * q^80 + 40 * q^82 + 40 * q^83 - 16 * q^85 + 16 * q^86 + 32 * q^88 - 8 * q^91 + 16 * q^92 + 8 * q^94 + 48 * q^95 + 40 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{6} - 3\nu^{5} + 10\nu^{4} - 15\nu^{3} + 19\nu^{2} - 12\nu + 4$$ v^6 - 3*v^5 + 10*v^4 - 15*v^3 + 19*v^2 - 12*v + 4 $$\beta_{2}$$ $$=$$ $$-2\nu^{7} + 7\nu^{6} - 24\nu^{5} + 42\nu^{4} - 59\nu^{3} + 48\nu^{2} - 24\nu + 4$$ -2*v^7 + 7*v^6 - 24*v^5 + 42*v^4 - 59*v^3 + 48*v^2 - 24*v + 4 $$\beta_{3}$$ $$=$$ $$-3\nu^{7} + 10\nu^{6} - 35\nu^{5} + 60\nu^{4} - 87\nu^{3} + 73\nu^{2} - 42\nu + 11$$ -3*v^7 + 10*v^6 - 35*v^5 + 60*v^4 - 87*v^3 + 73*v^2 - 42*v + 11 $$\beta_{4}$$ $$=$$ $$-7\nu^{7} + 25\nu^{6} - 87\nu^{5} + 158\nu^{4} - 231\nu^{3} + 206\nu^{2} - 118\nu + 31$$ -7*v^7 + 25*v^6 - 87*v^5 + 158*v^4 - 231*v^3 + 206*v^2 - 118*v + 31 $$\beta_{5}$$ $$=$$ $$-8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31$$ -8*v^7 + 28*v^6 - 98*v^5 + 175*v^4 - 256*v^3 + 223*v^2 - 126*v + 31 $$\beta_{6}$$ $$=$$ $$8\nu^{7} - 28\nu^{6} + 98\nu^{5} - 175\nu^{4} + 257\nu^{3} - 224\nu^{2} + 130\nu - 32$$ 8*v^7 - 28*v^6 + 98*v^5 - 175*v^4 + 257*v^3 - 224*v^2 + 130*v - 32 $$\beta_{7}$$ $$=$$ $$10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 166\nu - 42$$ 10*v^7 - 35*v^6 + 123*v^5 - 220*v^4 + 325*v^3 - 285*v^2 + 166*v - 42
 $$\nu$$ $$=$$ $$( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2$$ (-b7 + b6 - b5 + b4 - b3 + b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -2\beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 ) / 2$$ (-2*b7 + 3*b6 - b5 + b4 + b3 + b2 - 3) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} + \beta_{6} + 5\beta_{5} - 3\beta_{4} + 5\beta_{3} - 3\beta_{2} + 4\beta _1 - 5 ) / 2$$ (2*b7 + b6 + 5*b5 - 3*b4 + 5*b3 - 3*b2 + 4*b1 - 5) / 2 $$\nu^{4}$$ $$=$$ $$( 12\beta_{7} - 11\beta_{6} + 11\beta_{5} - 5\beta_{4} - \beta_{3} - 9\beta_{2} + 2\beta _1 + 5 ) / 2$$ (12*b7 - 11*b6 + 11*b5 - 5*b4 - b3 - 9*b2 + 2*b1 + 5) / 2 $$\nu^{5}$$ $$=$$ $$( 6\beta_{7} - 17\beta_{6} - 13\beta_{5} + 13\beta_{4} - 23\beta_{3} + 3\beta_{2} - 18\beta _1 + 21 ) / 2$$ (6*b7 - 17*b6 - 13*b5 + 13*b4 - 23*b3 + 3*b2 - 18*b1 + 21) / 2 $$\nu^{6}$$ $$=$$ $$( -46\beta_{7} + 29\beta_{6} - 67\beta_{5} + 37\beta_{4} - 15\beta_{3} + 47\beta_{2} - 24\beta _1 - 1 ) / 2$$ (-46*b7 + 29*b6 - 67*b5 + 37*b4 - 15*b3 + 47*b2 - 24*b1 - 1) / 2 $$\nu^{7}$$ $$=$$ $$( -76\beta_{7} + 105\beta_{6} - 7\beta_{5} - 31\beta_{4} + 91\beta_{3} + 39\beta_{2} + 68\beta _1 - 83 ) / 2$$ (-76*b7 + 105*b6 - 7*b5 - 31*b4 + 91*b3 + 39*b2 + 68*b1 - 83) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 − 0.0297061i 0.5 + 0.691860i 0.5 + 1.44392i 0.5 − 2.10607i 0.5 + 0.0297061i 0.5 − 0.691860i 0.5 − 1.44392i 0.5 + 2.10607i
−0.874559 + 1.11137i 0 −0.470294 1.94392i 0.334904 + 0.334904i 0 4.55765i 2.57172 + 1.17740i 0 −0.665096 + 0.0793096i
37.2 −0.635665 1.26330i 0 −1.19186 + 1.60607i 2.68554 + 2.68554i 0 2.15894i 2.78658 + 0.484753i 0 1.68554 5.09976i
37.3 0.167452 1.40426i 0 −1.94392 0.470294i −1.74912 1.74912i 0 2.55765i −0.985930 + 2.65103i 0 −2.74912 + 2.16333i
37.4 1.34277 0.443806i 0 1.60607 1.19186i −1.27133 1.27133i 0 0.158942i 1.62764 2.31318i 0 −2.27133 1.14288i
109.1 −0.874559 1.11137i 0 −0.470294 + 1.94392i 0.334904 0.334904i 0 4.55765i 2.57172 1.17740i 0 −0.665096 0.0793096i
109.2 −0.635665 + 1.26330i 0 −1.19186 1.60607i 2.68554 2.68554i 0 2.15894i 2.78658 0.484753i 0 1.68554 + 5.09976i
109.3 0.167452 + 1.40426i 0 −1.94392 + 0.470294i −1.74912 + 1.74912i 0 2.55765i −0.985930 2.65103i 0 −2.74912 2.16333i
109.4 1.34277 + 0.443806i 0 1.60607 + 1.19186i −1.27133 + 1.27133i 0 0.158942i 1.62764 + 2.31318i 0 −2.27133 + 1.14288i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.k.b 8
3.b odd 2 1 48.2.j.a 8
4.b odd 2 1 576.2.k.b 8
8.b even 2 1 1152.2.k.c 8
8.d odd 2 1 1152.2.k.f 8
12.b even 2 1 192.2.j.a 8
16.e even 4 1 inner 144.2.k.b 8
16.e even 4 1 1152.2.k.c 8
16.f odd 4 1 576.2.k.b 8
16.f odd 4 1 1152.2.k.f 8
24.f even 2 1 384.2.j.a 8
24.h odd 2 1 384.2.j.b 8
32.g even 8 1 9216.2.a.y 4
32.g even 8 1 9216.2.a.bo 4
32.h odd 8 1 9216.2.a.x 4
32.h odd 8 1 9216.2.a.bn 4
48.i odd 4 1 48.2.j.a 8
48.i odd 4 1 384.2.j.b 8
48.k even 4 1 192.2.j.a 8
48.k even 4 1 384.2.j.a 8
96.o even 8 1 3072.2.a.n 4
96.o even 8 1 3072.2.a.o 4
96.o even 8 2 3072.2.d.i 8
96.p odd 8 1 3072.2.a.i 4
96.p odd 8 1 3072.2.a.t 4
96.p odd 8 2 3072.2.d.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 3.b odd 2 1
48.2.j.a 8 48.i odd 4 1
144.2.k.b 8 1.a even 1 1 trivial
144.2.k.b 8 16.e even 4 1 inner
192.2.j.a 8 12.b even 2 1
192.2.j.a 8 48.k even 4 1
384.2.j.a 8 24.f even 2 1
384.2.j.a 8 48.k even 4 1
384.2.j.b 8 24.h odd 2 1
384.2.j.b 8 48.i odd 4 1
576.2.k.b 8 4.b odd 2 1
576.2.k.b 8 16.f odd 4 1
1152.2.k.c 8 8.b even 2 1
1152.2.k.c 8 16.e even 4 1
1152.2.k.f 8 8.d odd 2 1
1152.2.k.f 8 16.f odd 4 1
3072.2.a.i 4 96.p odd 8 1
3072.2.a.n 4 96.o even 8 1
3072.2.a.o 4 96.o even 8 1
3072.2.a.t 4 96.p odd 8 1
3072.2.d.f 8 96.p odd 8 2
3072.2.d.i 8 96.o even 8 2
9216.2.a.x 4 32.h odd 8 1
9216.2.a.y 4 32.g even 8 1
9216.2.a.bn 4 32.h odd 8 1
9216.2.a.bo 4 32.g even 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 16T_{5}^{5} + 128T_{5}^{4} + 192T_{5}^{3} + 128T_{5}^{2} - 128T_{5} + 64$$ acting on $$S_{2}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 2 T^{6} - 4 T^{5} + 2 T^{4} + \cdots + 16$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 16 T^{5} + 128 T^{4} + \cdots + 64$$
$7$ $$T^{8} + 32 T^{6} + 264 T^{4} + \cdots + 16$$
$11$ $$T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 1024$$
$13$ $$T^{8} - 64 T^{5} + 776 T^{4} + \cdots + 16$$
$17$ $$(T^{4} - 32 T^{2} - 64 T + 16)^{2}$$
$19$ $$T^{8} + 8 T^{7} + 32 T^{6} - 32 T^{5} + \cdots + 256$$
$23$ $$(T^{2} + 8)^{4}$$
$29$ $$T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 61504$$
$31$ $$(T^{4} - 12 T^{3} + 40 T^{2} - 24 T - 28)^{2}$$
$37$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 1106704$$
$41$ $$T^{8} + 128 T^{6} + 3872 T^{4} + \cdots + 12544$$
$43$ $$T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 12544$$
$47$ $$(T^{2} - 8)^{4}$$
$53$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 18496$$
$59$ $$(T^{2} + 8 T + 32)^{4}$$
$61$ $$T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 1106704$$
$67$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 65536$$
$71$ $$T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096$$
$73$ $$T^{8} + 256 T^{6} + 8320 T^{4} + \cdots + 4096$$
$79$ $$(T^{4} + 12 T^{3} - 168 T^{2} + \cdots - 10108)^{2}$$
$83$ $$T^{8} - 40 T^{7} + 800 T^{6} + \cdots + 1024$$
$89$ $$T^{8} + 464 T^{6} + 62304 T^{4} + \cdots + 3625216$$
$97$ $$(T^{4} - 224 T^{2} + 768 T + 512)^{2}$$