# Properties

 Label 1183.2.a.k Level $1183$ Weight $2$ Character orbit 1183.a Self dual yes Analytic conductor $9.446$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(1,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1183.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.a (trivial)

## 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: yes Analytic conductor: $$9.44630255912$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.27004.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 3x + 1$$ x^4 - x^3 - 6*x^2 + 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6x^{2} + 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 6\nu + 2$$ v^3 - v^2 - 6*v + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 7\beta _1 + 1$$ b3 + b2 + 7*b1 + 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.74108 0.710287 −0.231361 −2.22001
−2.74108 1.36482 5.51353 0.741082 −3.74108 −1.00000 −9.63087 −1.13727 −2.03137
1.2 −0.710287 2.40788 −1.49549 −1.28971 −1.71029 −1.00000 2.48280 2.79790 0.916066
1.3 0.231361 −3.32225 −1.94647 −2.23136 −0.768639 −1.00000 −0.913059 8.03736 −0.516249
1.4 2.22001 0.549551 2.92843 −4.22001 1.22001 −1.00000 2.06113 −2.69799 −9.36845
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.a.k 4
7.b odd 2 1 8281.2.a.bp 4
13.b even 2 1 1183.2.a.l 4
13.c even 3 2 91.2.f.c 8
13.d odd 4 2 1183.2.c.g 8
39.i odd 6 2 819.2.o.h 8
52.j odd 6 2 1456.2.s.q 8
91.b odd 2 1 8281.2.a.bt 4
91.g even 3 2 637.2.g.k 8
91.h even 3 2 637.2.h.h 8
91.m odd 6 2 637.2.g.j 8
91.n odd 6 2 637.2.f.i 8
91.v odd 6 2 637.2.h.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 13.c even 3 2
637.2.f.i 8 91.n odd 6 2
637.2.g.j 8 91.m odd 6 2
637.2.g.k 8 91.g even 3 2
637.2.h.h 8 91.h even 3 2
637.2.h.i 8 91.v odd 6 2
819.2.o.h 8 39.i odd 6 2
1183.2.a.k 4 1.a even 1 1 trivial
1183.2.a.l 4 13.b even 2 1
1183.2.c.g 8 13.d odd 4 2
1456.2.s.q 8 52.j odd 6 2
8281.2.a.bp 4 7.b odd 2 1
8281.2.a.bt 4 91.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1183))$$:

 $$T_{2}^{4} + T_{2}^{3} - 6T_{2}^{2} - 3T_{2} + 1$$ T2^4 + T2^3 - 6*T2^2 - 3*T2 + 1 $$T_{11}^{4} + T_{11}^{3} - 9T_{11}^{2} - 16T_{11} - 6$$ T11^4 + T11^3 - 9*T11^2 - 16*T11 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} - 6 T^{2} - 3 T + 1$$
$3$ $$T^{4} - T^{3} - 9 T^{2} + 16 T - 6$$
$5$ $$T^{4} + 7 T^{3} + 12 T^{2} - T - 9$$
$7$ $$(T + 1)^{4}$$
$11$ $$T^{4} + T^{3} - 9 T^{2} - 16 T - 6$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 4 T^{3} - 12 T^{2} - 60 T - 53$$
$19$ $$T^{4} - T^{3} - 55 T^{2} + 500$$
$23$ $$T^{4} + 2 T^{3} - 38 T^{2} - 56 T + 72$$
$29$ $$T^{4} - T^{3} - 22 T^{2} - 21 T - 5$$
$31$ $$T^{4} + 4 T^{3} - 13 T^{2} - 26 T + 54$$
$37$ $$T^{4} + 10 T^{3} + 17 T^{2} - 44 T - 16$$
$41$ $$T^{4} + 22 T^{3} + 149 T^{2} + \cdots - 564$$
$43$ $$T^{4} - 3 T^{3} - 3 T^{2} + 8 T - 2$$
$47$ $$T^{4} - 2 T^{3} - 51 T^{2} - 12 T + 100$$
$53$ $$T^{4} - 2 T^{3} - 140 T^{2} + \cdots - 1389$$
$59$ $$T^{4} + 8 T^{3} - 77 T^{2} - 550 T - 706$$
$61$ $$T^{4} - 8 T^{3} - 15 T^{2} + 176 T - 100$$
$67$ $$T^{4} + 6 T^{3} - 211 T^{2} + \cdots + 11010$$
$71$ $$T^{4} + 14 T^{3} - 16 T^{2} + \cdots - 2032$$
$73$ $$T^{4} + 8 T^{3} - 119 T^{2} + \cdots + 1772$$
$79$ $$T^{4} + 26 T^{3} + 134 T^{2} + \cdots - 7680$$
$83$ $$T^{4} - 97 T^{2} - 442 T - 426$$
$89$ $$T^{4} + T^{3} - 71 T^{2} - 184 T - 108$$
$97$ $$T^{4} - 3 T^{3} - 305 T^{2} + \cdots + 15692$$