# Properties

 Label 192.2.j.a Level $192$ Weight $2$ Character orbit 192.j Analytic conductor $1.533$ 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] = [192,2,Mod(49,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, 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("192.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 192.j (of order $$4$$, degree $$2$$, not 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: $$1.53312771881$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + (\beta_{5} - \beta_{3}) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4}) q^{7} + \beta_{4} q^{9}+O(q^{10})$$ q + b2 * q^3 + (b5 - b3) * q^5 + (b7 + b6 + b5 + b4) * q^7 + b4 * q^9 $$q + \beta_{2} q^{3} + (\beta_{5} - \beta_{3}) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4}) q^{7} + \beta_{4} q^{9} + ( - \beta_{7} - \beta_{4} - \beta_1 + 1) q^{11} + ( - \beta_{7} + 2 \beta_{2} + \beta_1) q^{13} + (\beta_1 + 1) q^{15} + (\beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_1 + 1) q^{19} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{23} + (2 \beta_{7} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2}) q^{25} + \beta_{3} q^{27} + ( - \beta_{6} - 2 \beta_{4} - 3 \beta_{2} - 2) q^{29} + (\beta_{6} - \beta_{5} - \beta_1 - 3) q^{31} + (\beta_{6} - \beta_{5} - \beta_{3} + \beta_{2}) q^{33} + (\beta_{7} - 3 \beta_{4} - \beta_1 - 3) q^{35} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - \beta_1 - 2) q^{37} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4}) q^{39} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{3} - 3 \beta_{2}) q^{41} + (\beta_{7} - 2 \beta_{5} - \beta_{4} + \beta_1 + 1) q^{43} + ( - \beta_{6} + \beta_{2}) q^{45} + (2 \beta_{3} - 2 \beta_{2}) q^{47} + ( - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 1) q^{49} + (\beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_1 - 1) q^{51} + ( - \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 2) q^{53} + ( - 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2}) q^{55} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2}) q^{57} + (4 \beta_{4} - 4) q^{59} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 4 \beta_{2} - \beta_1 + 2) q^{61} + ( - \beta_{6} + \beta_{5} + \beta_1 - 1) q^{63} + ( - \beta_{6} + \beta_{5} - 5 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 2) q^{65} + ( - 2 \beta_{7} + 2 \beta_{4} - 4 \beta_{2} + 2 \beta_1 + 2) q^{67} + ( - 2 \beta_{4} + 2) q^{69} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4}) q^{71} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2}) q^{73} + (2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 2) q^{75} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 6 \beta_{2} - 2 \beta_1 + 2) q^{77} + ( - \beta_{6} + \beta_{5} - 4 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 3) q^{79} - q^{81} + (\beta_{7} + 4 \beta_{6} + 5 \beta_{4} - \beta_1 + 5) q^{83} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} - 2 \beta_1 - 2) q^{85} + ( - \beta_{7} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{87} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 6 \beta_{2}) q^{89} + (\beta_{7} + 2 \beta_{5} - \beta_{4} - 4 \beta_{3} + \beta_1 + 1) q^{91} + (\beta_{7} + \beta_{6} - 3 \beta_{2} - \beta_1) q^{93} + ( - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{95} + ( - 2 \beta_{6} + 2 \beta_{5} - 6 \beta_{3} + 6 \beta_{2}) q^{97} + (\beta_{7} + \beta_{4} - \beta_1 + 1) q^{99}+O(q^{100})$$ q + b2 * q^3 + (b5 - b3) * q^5 + (b7 + b6 + b5 + b4) * q^7 + b4 * q^9 + (-b7 - b4 - b1 + 1) * q^11 + (-b7 + 2*b2 + b1) * q^13 + (b1 + 1) * q^15 + (b6 - b5 + b3 - b2 - 2*b1) * q^17 + (-b7 - 2*b6 + b4 + b1 + 1) * q^19 + (b7 + b5 + b3 + b1) * q^21 + (-2*b3 - 2*b2) * q^23 + (2*b7 - b4 + 2*b3 + 2*b2) * q^25 + b3 * q^27 + (-b6 - 2*b4 - 3*b2 - 2) * q^29 + (b6 - b5 - b1 - 3) * q^31 + (b6 - b5 - b3 + b2) * q^33 + (b7 - 3*b4 - b1 - 3) * q^35 + (-b7 - 2*b5 + 2*b4 - 4*b3 - b1 - 2) * q^37 + (-b6 - b5 + 2*b4) * q^39 + (-2*b7 - b6 - b5 - 3*b3 - 3*b2) * q^41 + (b7 - 2*b5 - b4 + b1 + 1) * q^43 + (-b6 + b2) * q^45 + (2*b3 - 2*b2) * q^47 + (-2*b6 + 2*b5 + 4*b3 - 4*b2 + 2*b1 - 1) * q^49 + (b7 + 2*b6 - b4 - b1 - 1) * q^51 + (-b5 - 2*b4 + 5*b3 + 2) * q^53 + (-4*b4 + 4*b3 + 4*b2) * q^55 + (-2*b7 - b6 - b5 + b3 + b2) * q^57 + (4*b4 - 4) * q^59 + (b7 + 2*b6 + 2*b4 - 4*b2 - b1 + 2) * q^61 + (-b6 + b5 + b1 - 1) * q^63 + (-b6 + b5 - 5*b3 + 5*b2 + 2*b1 - 2) * q^65 + (-2*b7 + 2*b4 - 4*b2 + 2*b1 + 2) * q^67 + (-2*b4 + 2) * q^69 + (-2*b7 - 2*b6 - 2*b5 - 2*b4) * q^71 + (2*b7 + 2*b6 + 2*b5 - 2*b4 - 4*b3 - 4*b2) * q^73 + (2*b5 + 2*b4 - b3 - 2) * q^75 + (2*b7 + 2*b6 + 2*b4 + 6*b2 - 2*b1 + 2) * q^77 + (-b6 + b5 - 4*b3 + 4*b2 - 3*b1 + 3) * q^79 - q^81 + (b7 + 4*b6 + 5*b4 - b1 + 5) * q^83 + (-2*b7 - 2*b5 + 2*b4 + 6*b3 - 2*b1 - 2) * q^85 + (-b7 - 3*b4 - 2*b3 - 2*b2) * q^87 + (2*b6 + 2*b5 + 2*b4 + 6*b3 + 6*b2) * q^89 + (b7 + 2*b5 - b4 - 4*b3 + b1 + 1) * q^91 + (b7 + b6 - 3*b2 - b1) * q^93 + (-2*b6 + 2*b5 - 2*b3 + 2*b2 + 2*b1 + 6) * q^95 + (-2*b6 + 2*b5 - 6*b3 + 6*b2) * q^97 + (b7 + b4 - b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 8 q^{11} + 8 q^{15} + 8 q^{19} - 16 q^{29} - 24 q^{31} - 24 q^{35} - 16 q^{37} + 8 q^{43} - 8 q^{49} - 8 q^{51} + 16 q^{53} - 32 q^{59} + 16 q^{61} - 8 q^{63} - 16 q^{65} + 16 q^{67} + 16 q^{69} - 16 q^{75} + 16 q^{77} + 24 q^{79} - 8 q^{81} + 40 q^{83} - 16 q^{85} + 8 q^{91} + 48 q^{95} + 8 q^{99}+O(q^{100})$$ 8 * q + 8 * q^11 + 8 * q^15 + 8 * q^19 - 16 * q^29 - 24 * q^31 - 24 * q^35 - 16 * q^37 + 8 * q^43 - 8 * q^49 - 8 * q^51 + 16 * q^53 - 32 * q^59 + 16 * q^61 - 8 * q^63 - 16 * q^65 + 16 * q^67 + 16 * q^69 - 16 * q^75 + 16 * q^77 + 24 * q^79 - 8 * q^81 + 40 * q^83 - 16 * q^85 + 8 * q^91 + 48 * q^95 + 8 * q^99

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

 $$\beta_{1}$$ $$=$$ $$-\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 24\nu^{2} + 16\nu - 5$$ -v^6 + 3*v^5 - 11*v^4 + 17*v^3 - 24*v^2 + 16*v - 5 $$\beta_{2}$$ $$=$$ $$5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19$$ 5*v^7 - 17*v^6 + 60*v^5 - 105*v^4 + 155*v^3 - 133*v^2 + 77*v - 19 $$\beta_{3}$$ $$=$$ $$5\nu^{7} - 18\nu^{6} + 63\nu^{5} - 115\nu^{4} + 170\nu^{3} - 152\nu^{2} + 89\nu - 23$$ 5*v^7 - 18*v^6 + 63*v^5 - 115*v^4 + 170*v^3 - 152*v^2 + 89*v - 23 $$\beta_{4}$$ $$=$$ $$8\nu^{7} - 28\nu^{6} + 98\nu^{5} - 175\nu^{4} + 256\nu^{3} - 223\nu^{2} + 126\nu - 31$$ 8*v^7 - 28*v^6 + 98*v^5 - 175*v^4 + 256*v^3 - 223*v^2 + 126*v - 31 $$\beta_{5}$$ $$=$$ $$9\nu^{7} - 31\nu^{6} + 108\nu^{5} - 190\nu^{4} + 275\nu^{3} - 236\nu^{2} + 131\nu - 33$$ 9*v^7 - 31*v^6 + 108*v^5 - 190*v^4 + 275*v^3 - 236*v^2 + 131*v - 33 $$\beta_{6}$$ $$=$$ $$9\nu^{7} - 32\nu^{6} + 111\nu^{5} - 200\nu^{4} + 290\nu^{3} - 253\nu^{2} + 141\nu - 33$$ 9*v^7 - 32*v^6 + 111*v^5 - 200*v^4 + 290*v^3 - 253*v^2 + 141*v - 33 $$\beta_{7}$$ $$=$$ $$10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 168\nu - 43$$ 10*v^7 - 35*v^6 + 123*v^5 - 220*v^4 + 325*v^3 - 285*v^2 + 168*v - 43
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{3} - \beta_{2} + 1 ) / 2$$ (b7 - b3 - b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{3} - 3 ) / 2$$ (b7 + b6 - b5 - 2*b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$( -2\beta_{7} + 2\beta_{6} - \beta_{5} - 3\beta_{4} + 2\beta_{3} + 5\beta_{2} - 5 ) / 2$$ (-2*b7 + 2*b6 - b5 - 3*b4 + 2*b3 + 5*b2 - 5) / 2 $$\nu^{4}$$ $$=$$ $$( -5\beta_{7} - \beta_{6} + 3\beta_{5} - 6\beta_{4} + 12\beta_{3} + 4\beta_{2} - 2\beta _1 + 7 ) / 2$$ (-5*b7 - b6 + 3*b5 - 6*b4 + 12*b3 + 4*b2 - 2*b1 + 7) / 2 $$\nu^{5}$$ $$=$$ $$( 3\beta_{7} - 10\beta_{6} + 5\beta_{5} + 10\beta_{4} + 6\beta_{3} - 19\beta_{2} - 5\beta _1 + 26 ) / 2$$ (3*b7 - 10*b6 + 5*b5 + 10*b4 + 6*b3 - 19*b2 - 5*b1 + 26) / 2 $$\nu^{6}$$ $$=$$ $$( 22\beta_{7} - 9\beta_{6} - 11\beta_{5} + 45\beta_{4} - 48\beta_{3} - 32\beta_{2} + 5\beta _1 - 6 ) / 2$$ (22*b7 - 9*b6 - 11*b5 + 45*b4 - 48*b3 - 32*b2 + 5*b1 - 6) / 2 $$\nu^{7}$$ $$=$$ $$( 7\beta_{7} + 33\beta_{6} - 30\beta_{5} - 83\beta_{3} + 64\beta_{2} + 35\beta _1 - 118 ) / 2$$ (7*b7 + 33*b6 - 30*b5 - 83*b3 + 64*b2 + 35*b1 - 118) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.5 + 0.691860i 0.5 − 2.10607i 0.5 − 0.0297061i 0.5 + 1.44392i 0.5 − 0.691860i 0.5 + 2.10607i 0.5 + 0.0297061i 0.5 − 1.44392i
0 −0.707107 + 0.707107i 0 −2.68554 2.68554i 0 2.15894i 0 1.00000i 0
49.2 0 −0.707107 + 0.707107i 0 1.27133 + 1.27133i 0 0.158942i 0 1.00000i 0
49.3 0 0.707107 0.707107i 0 −0.334904 0.334904i 0 4.55765i 0 1.00000i 0
49.4 0 0.707107 0.707107i 0 1.74912 + 1.74912i 0 2.55765i 0 1.00000i 0
145.1 0 −0.707107 0.707107i 0 −2.68554 + 2.68554i 0 2.15894i 0 1.00000i 0
145.2 0 −0.707107 0.707107i 0 1.27133 1.27133i 0 0.158942i 0 1.00000i 0
145.3 0 0.707107 + 0.707107i 0 −0.334904 + 0.334904i 0 4.55765i 0 1.00000i 0
145.4 0 0.707107 + 0.707107i 0 1.74912 1.74912i 0 2.55765i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.4 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 192.2.j.a 8
3.b odd 2 1 576.2.k.b 8
4.b odd 2 1 48.2.j.a 8
8.b even 2 1 384.2.j.a 8
8.d odd 2 1 384.2.j.b 8
12.b even 2 1 144.2.k.b 8
16.e even 4 1 inner 192.2.j.a 8
16.e even 4 1 384.2.j.a 8
16.f odd 4 1 48.2.j.a 8
16.f odd 4 1 384.2.j.b 8
24.f even 2 1 1152.2.k.c 8
24.h odd 2 1 1152.2.k.f 8
32.g even 8 1 3072.2.a.n 4
32.g even 8 1 3072.2.a.o 4
32.g even 8 2 3072.2.d.i 8
32.h odd 8 1 3072.2.a.i 4
32.h odd 8 1 3072.2.a.t 4
32.h odd 8 2 3072.2.d.f 8
48.i odd 4 1 576.2.k.b 8
48.i odd 4 1 1152.2.k.f 8
48.k even 4 1 144.2.k.b 8
48.k even 4 1 1152.2.k.c 8
96.o even 8 1 9216.2.a.y 4
96.o even 8 1 9216.2.a.bo 4
96.p odd 8 1 9216.2.a.x 4
96.p odd 8 1 9216.2.a.bn 4

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 1)^{2}$$
$5$ $$T^{8} - 16 T^{5} + 128 T^{4} + \cdots + 64$$
$7$ $$T^{8} + 32 T^{6} + 264 T^{4} + \cdots + 16$$
$11$ $$T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 1024$$
$13$ $$T^{8} - 64 T^{5} + 776 T^{4} + \cdots + 16$$
$17$ $$(T^{4} - 32 T^{2} + 64 T + 16)^{2}$$
$19$ $$T^{8} - 8 T^{7} + 32 T^{6} + 32 T^{5} + \cdots + 256$$
$23$ $$(T^{2} + 8)^{4}$$
$29$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 61504$$
$31$ $$(T^{4} + 12 T^{3} + 40 T^{2} + 24 T - 28)^{2}$$
$37$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 1106704$$
$41$ $$T^{8} + 128 T^{6} + 3872 T^{4} + \cdots + 12544$$
$43$ $$T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 12544$$
$47$ $$(T^{2} - 8)^{4}$$
$53$ $$T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 18496$$
$59$ $$(T^{2} + 8 T + 32)^{4}$$
$61$ $$T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 1106704$$
$67$ $$T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 65536$$
$71$ $$T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096$$
$73$ $$T^{8} + 256 T^{6} + 8320 T^{4} + \cdots + 4096$$
$79$ $$(T^{4} - 12 T^{3} - 168 T^{2} + \cdots - 10108)^{2}$$
$83$ $$T^{8} - 40 T^{7} + 800 T^{6} + \cdots + 1024$$
$89$ $$T^{8} + 464 T^{6} + 62304 T^{4} + \cdots + 3625216$$
$97$ $$(T^{4} - 224 T^{2} + 768 T + 512)^{2}$$