# Properties

 Label 154.2.e.e Level $154$ Weight $2$ Character orbit 154.e Analytic conductor $1.230$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [154,2,Mod(23,154)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("154.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 154.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: $$1.22969619113$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{5} + (\beta_{3} - 1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{7} + q^{8} + (2 \beta_{3} + 2 \beta_1) q^{9}+O(q^{10})$$ q + b2 * q^2 + (b2 + b1 + 1) * q^3 + (-b2 - 1) * q^4 + (b3 - 2*b2 + b1) * q^5 + (b3 - 1) * q^6 + (-b3 + b2 + b1) * q^7 + q^8 + (2*b3 + 2*b1) * q^9 $$q + \beta_{2} q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{5} + (\beta_{3} - 1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{7} + q^{8} + (2 \beta_{3} + 2 \beta_1) q^{9} + (2 \beta_{2} - \beta_1 + 2) q^{10} + ( - \beta_{2} - 1) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{12} + (2 \beta_{3} - 1) q^{13} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{14} - \beta_{3} q^{15} + \beta_{2} q^{16} + (2 \beta_{2} - 4 \beta_1 + 2) q^{17} - 2 \beta_1 q^{18} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{19} + ( - \beta_{3} - 2) q^{20} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 1) q^{21} + q^{22} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{23} + (\beta_{2} + \beta_1 + 1) q^{24} + ( - \beta_{2} + 4 \beta_1 - 1) q^{25} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{26} + ( - \beta_{3} - 1) q^{27} + ( - \beta_{3} - 2 \beta_1 + 1) q^{28} + ( - 4 \beta_{3} - 3) q^{29} + (\beta_{3} + \beta_1) q^{30} + (4 \beta_{2} + 4) q^{31} + ( - \beta_{2} - 1) q^{32} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{33} + ( - 4 \beta_{3} - 2) q^{34} + ( - 4 \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{35} - 2 \beta_{3} q^{36} + ( - \beta_{3} - 8 \beta_{2} - \beta_1) q^{37} + (2 \beta_{2} - \beta_1 + 2) q^{38} + ( - 5 \beta_{2} - 3 \beta_1 - 5) q^{39} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{40} + (\beta_{3} - 4) q^{41} + ( - 3 \beta_{2} - 2 \beta_1 - 4) q^{42} + 4 \beta_{3} q^{43} + \beta_{2} q^{44} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{45} + (2 \beta_{2} + 3 \beta_1 + 2) q^{46} + ( - 6 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{47} + (\beta_{3} - 1) q^{48} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 5) q^{49} + (4 \beta_{3} + 1) q^{50} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{51} + (\beta_{2} + 2 \beta_1 + 1) q^{52} + (2 \beta_{2} - 7 \beta_1 + 2) q^{53} + (\beta_{3} - \beta_{2} + \beta_1) q^{54} + ( - \beta_{3} - 2) q^{55} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{56} - \beta_{3} q^{57} + (4 \beta_{3} - 3 \beta_{2} + 4 \beta_1) q^{58} + (7 \beta_{2} - \beta_1 + 7) q^{59} - \beta_1 q^{60} + (2 \beta_{3} + 9 \beta_{2} + 2 \beta_1) q^{61} - 4 q^{62} + (4 \beta_{2} - 2 \beta_1 - 4) q^{63} + q^{64} + (3 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{65} + (\beta_{2} + \beta_1 + 1) q^{66} + ( - 7 \beta_{2} + 3 \beta_1 - 7) q^{67} + (4 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{68} + ( - 5 \beta_{3} + 8) q^{69} + (\beta_{3} - 4 \beta_{2} + 4 \beta_1 - 4) q^{70} + (5 \beta_{3} - 4) q^{71} + (2 \beta_{3} + 2 \beta_1) q^{72} + (8 \beta_{2} + \beta_1 + 8) q^{73} + (8 \beta_{2} + \beta_1 + 8) q^{74} + (3 \beta_{3} + 7 \beta_{2} + 3 \beta_1) q^{75} + ( - \beta_{3} - 2) q^{76} + ( - \beta_{3} - 2 \beta_1 + 1) q^{77} + ( - 3 \beta_{3} + 5) q^{78} + ( - 3 \beta_{3} - 9 \beta_{2} - 3 \beta_1) q^{79} + (2 \beta_{2} - \beta_1 + 2) q^{80} + (\beta_{2} + 6 \beta_1 + 1) q^{81} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{82} + (10 \beta_{3} + 2) q^{83} + ( - 2 \beta_{3} - \beta_{2} + 3) q^{84} + (10 \beta_{3} + 12) q^{85} + ( - 4 \beta_{3} - 4 \beta_1) q^{86} + (5 \beta_{2} + \beta_1 + 5) q^{87} + ( - \beta_{2} - 1) q^{88} + (6 \beta_{3} + 4 \beta_{2} + 6 \beta_1) q^{89} + (4 \beta_{3} + 4) q^{90} + ( - \beta_{3} - 5 \beta_{2} - 3 \beta_1 - 8) q^{91} + (3 \beta_{3} - 2) q^{92} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{93} + (2 \beta_{2} + 6 \beta_1 + 2) q^{94} + ( - 6 \beta_{2} + 4 \beta_1 - 6) q^{95} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{96} + (2 \beta_{3} - 1) q^{97} + ( - 2 \beta_{3} - 4 \beta_1 - 5) q^{98} - 2 \beta_{3} q^{99}+O(q^{100})$$ q + b2 * q^2 + (b2 + b1 + 1) * q^3 + (-b2 - 1) * q^4 + (b3 - 2*b2 + b1) * q^5 + (b3 - 1) * q^6 + (-b3 + b2 + b1) * q^7 + q^8 + (2*b3 + 2*b1) * q^9 + (2*b2 - b1 + 2) * q^10 + (-b2 - 1) * q^11 + (-b3 - b2 - b1) * q^12 + (2*b3 - 1) * q^13 + (2*b3 - b2 + b1 - 1) * q^14 - b3 * q^15 + b2 * q^16 + (2*b2 - 4*b1 + 2) * q^17 - 2*b1 * q^18 + (b3 - 2*b2 + b1) * q^19 + (-b3 - 2) * q^20 + (2*b3 + 4*b2 + 2*b1 + 1) * q^21 + q^22 + (-3*b3 - 2*b2 - 3*b1) * q^23 + (b2 + b1 + 1) * q^24 + (-b2 + 4*b1 - 1) * q^25 + (-2*b3 - b2 - 2*b1) * q^26 + (-b3 - 1) * q^27 + (-b3 - 2*b1 + 1) * q^28 + (-4*b3 - 3) * q^29 + (b3 + b1) * q^30 + (4*b2 + 4) * q^31 + (-b2 - 1) * q^32 + (-b3 - b2 - b1) * q^33 + (-4*b3 - 2) * q^34 + (-4*b3 + 4*b2 - 3*b1) * q^35 - 2*b3 * q^36 + (-b3 - 8*b2 - b1) * q^37 + (2*b2 - b1 + 2) * q^38 + (-5*b2 - 3*b1 - 5) * q^39 + (b3 - 2*b2 + b1) * q^40 + (b3 - 4) * q^41 + (-3*b2 - 2*b1 - 4) * q^42 + 4*b3 * q^43 + b2 * q^44 + (-4*b2 + 4*b1 - 4) * q^45 + (2*b2 + 3*b1 + 2) * q^46 + (-6*b3 - 2*b2 - 6*b1) * q^47 + (b3 - 1) * q^48 + (4*b3 + 5*b2 + 2*b1 + 5) * q^49 + (4*b3 + 1) * q^50 + (-2*b3 - 6*b2 - 2*b1) * q^51 + (b2 + 2*b1 + 1) * q^52 + (2*b2 - 7*b1 + 2) * q^53 + (b3 - b2 + b1) * q^54 + (-b3 - 2) * q^55 + (-b3 + b2 + b1) * q^56 - b3 * q^57 + (4*b3 - 3*b2 + 4*b1) * q^58 + (7*b2 - b1 + 7) * q^59 - b1 * q^60 + (2*b3 + 9*b2 + 2*b1) * q^61 - 4 * q^62 + (4*b2 - 2*b1 - 4) * q^63 + q^64 + (3*b3 - 2*b2 + 3*b1) * q^65 + (b2 + b1 + 1) * q^66 + (-7*b2 + 3*b1 - 7) * q^67 + (4*b3 - 2*b2 + 4*b1) * q^68 + (-5*b3 + 8) * q^69 + (b3 - 4*b2 + 4*b1 - 4) * q^70 + (5*b3 - 4) * q^71 + (2*b3 + 2*b1) * q^72 + (8*b2 + b1 + 8) * q^73 + (8*b2 + b1 + 8) * q^74 + (3*b3 + 7*b2 + 3*b1) * q^75 + (-b3 - 2) * q^76 + (-b3 - 2*b1 + 1) * q^77 + (-3*b3 + 5) * q^78 + (-3*b3 - 9*b2 - 3*b1) * q^79 + (2*b2 - b1 + 2) * q^80 + (b2 + 6*b1 + 1) * q^81 + (-b3 - 4*b2 - b1) * q^82 + (10*b3 + 2) * q^83 + (-2*b3 - b2 + 3) * q^84 + (10*b3 + 12) * q^85 + (-4*b3 - 4*b1) * q^86 + (5*b2 + b1 + 5) * q^87 + (-b2 - 1) * q^88 + (6*b3 + 4*b2 + 6*b1) * q^89 + (4*b3 + 4) * q^90 + (-b3 - 5*b2 - 3*b1 - 8) * q^91 + (3*b3 - 2) * q^92 + (4*b3 + 4*b2 + 4*b1) * q^93 + (2*b2 + 6*b1 + 2) * q^94 + (-6*b2 + 4*b1 - 6) * q^95 + (-b3 - b2 - b1) * q^96 + (2*b3 - 1) * q^97 + (-2*b3 - 4*b1 - 5) * q^98 - 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} + 4 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^5 - 4 * q^6 - 2 * q^7 + 4 * q^8 $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} + 4 q^{8} + 4 q^{10} - 2 q^{11} + 2 q^{12} - 4 q^{13} - 2 q^{14} - 2 q^{16} + 4 q^{17} + 4 q^{19} - 8 q^{20} - 4 q^{21} + 4 q^{22} + 4 q^{23} + 2 q^{24} - 2 q^{25} + 2 q^{26} - 4 q^{27} + 4 q^{28} - 12 q^{29} + 8 q^{31} - 2 q^{32} + 2 q^{33} - 8 q^{34} - 8 q^{35} + 16 q^{37} + 4 q^{38} - 10 q^{39} + 4 q^{40} - 16 q^{41} - 10 q^{42} - 2 q^{44} - 8 q^{45} + 4 q^{46} + 4 q^{47} - 4 q^{48} + 10 q^{49} + 4 q^{50} + 12 q^{51} + 2 q^{52} + 4 q^{53} + 2 q^{54} - 8 q^{55} - 2 q^{56} + 6 q^{58} + 14 q^{59} - 18 q^{61} - 16 q^{62} - 24 q^{63} + 4 q^{64} + 4 q^{65} + 2 q^{66} - 14 q^{67} + 4 q^{68} + 32 q^{69} - 8 q^{70} - 16 q^{71} + 16 q^{73} + 16 q^{74} - 14 q^{75} - 8 q^{76} + 4 q^{77} + 20 q^{78} + 18 q^{79} + 4 q^{80} + 2 q^{81} + 8 q^{82} + 8 q^{83} + 14 q^{84} + 48 q^{85} + 10 q^{87} - 2 q^{88} - 8 q^{89} + 16 q^{90} - 22 q^{91} - 8 q^{92} - 8 q^{93} + 4 q^{94} - 12 q^{95} + 2 q^{96} - 4 q^{97} - 20 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^5 - 4 * q^6 - 2 * q^7 + 4 * q^8 + 4 * q^10 - 2 * q^11 + 2 * q^12 - 4 * q^13 - 2 * q^14 - 2 * q^16 + 4 * q^17 + 4 * q^19 - 8 * q^20 - 4 * q^21 + 4 * q^22 + 4 * q^23 + 2 * q^24 - 2 * q^25 + 2 * q^26 - 4 * q^27 + 4 * q^28 - 12 * q^29 + 8 * q^31 - 2 * q^32 + 2 * q^33 - 8 * q^34 - 8 * q^35 + 16 * q^37 + 4 * q^38 - 10 * q^39 + 4 * q^40 - 16 * q^41 - 10 * q^42 - 2 * q^44 - 8 * q^45 + 4 * q^46 + 4 * q^47 - 4 * q^48 + 10 * q^49 + 4 * q^50 + 12 * q^51 + 2 * q^52 + 4 * q^53 + 2 * q^54 - 8 * q^55 - 2 * q^56 + 6 * q^58 + 14 * q^59 - 18 * q^61 - 16 * q^62 - 24 * q^63 + 4 * q^64 + 4 * q^65 + 2 * q^66 - 14 * q^67 + 4 * q^68 + 32 * q^69 - 8 * q^70 - 16 * q^71 + 16 * q^73 + 16 * q^74 - 14 * q^75 - 8 * q^76 + 4 * q^77 + 20 * q^78 + 18 * q^79 + 4 * q^80 + 2 * q^81 + 8 * q^82 + 8 * q^83 + 14 * q^84 + 48 * q^85 + 10 * q^87 - 2 * q^88 - 8 * q^89 + 16 * q^90 - 22 * q^91 - 8 * q^92 - 8 * q^93 + 4 * q^94 - 12 * q^95 + 2 * q^96 - 4 * q^97 - 20 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/154\mathbb{Z}\right)^\times$$.

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.500000 + 0.866025i −0.207107 0.358719i −0.500000 0.866025i 1.70711 2.95680i 0.414214 −2.62132 0.358719i 1.00000 1.41421 2.44949i 1.70711 + 2.95680i
23.2 −0.500000 + 0.866025i 1.20711 + 2.09077i −0.500000 0.866025i 0.292893 0.507306i −2.41421 1.62132 + 2.09077i 1.00000 −1.41421 + 2.44949i 0.292893 + 0.507306i
67.1 −0.500000 0.866025i −0.207107 + 0.358719i −0.500000 + 0.866025i 1.70711 + 2.95680i 0.414214 −2.62132 + 0.358719i 1.00000 1.41421 + 2.44949i 1.70711 2.95680i
67.2 −0.500000 0.866025i 1.20711 2.09077i −0.500000 + 0.866025i 0.292893 + 0.507306i −2.41421 1.62132 2.09077i 1.00000 −1.41421 2.44949i 0.292893 0.507306i
 $$n$$: e.g. 2-40 or 990-1000 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 154.2.e.e 4
3.b odd 2 1 1386.2.k.t 4
4.b odd 2 1 1232.2.q.f 4
7.b odd 2 1 1078.2.e.m 4
7.c even 3 1 inner 154.2.e.e 4
7.c even 3 1 1078.2.a.t 2
7.d odd 6 1 1078.2.a.x 2
7.d odd 6 1 1078.2.e.m 4
21.g even 6 1 9702.2.a.ch 2
21.h odd 6 1 1386.2.k.t 4
21.h odd 6 1 9702.2.a.cx 2
28.f even 6 1 8624.2.a.bh 2
28.g odd 6 1 1232.2.q.f 4
28.g odd 6 1 8624.2.a.cc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 1.a even 1 1 trivial
154.2.e.e 4 7.c even 3 1 inner
1078.2.a.t 2 7.c even 3 1
1078.2.a.x 2 7.d odd 6 1
1078.2.e.m 4 7.b odd 2 1
1078.2.e.m 4 7.d odd 6 1
1232.2.q.f 4 4.b odd 2 1
1232.2.q.f 4 28.g odd 6 1
1386.2.k.t 4 3.b odd 2 1
1386.2.k.t 4 21.h odd 6 1
8624.2.a.bh 2 28.f even 6 1
8624.2.a.cc 2 28.g odd 6 1
9702.2.a.ch 2 21.g even 6 1
9702.2.a.cx 2 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(154, [\chi])$$:

 $$T_{3}^{4} - 2T_{3}^{3} + 5T_{3}^{2} + 2T_{3} + 1$$ T3^4 - 2*T3^3 + 5*T3^2 + 2*T3 + 1 $$T_{13}^{2} + 2T_{13} - 7$$ T13^2 + 2*T13 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$5$ $$T^{4} - 4 T^{3} + 14 T^{2} - 8 T + 4$$
$7$ $$T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49$$
$11$ $$(T^{2} + T + 1)^{2}$$
$13$ $$(T^{2} + 2 T - 7)^{2}$$
$17$ $$T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784$$
$19$ $$T^{4} - 4 T^{3} + 14 T^{2} - 8 T + 4$$
$23$ $$T^{4} - 4 T^{3} + 30 T^{2} + 56 T + 196$$
$29$ $$(T^{2} + 6 T - 23)^{2}$$
$31$ $$(T^{2} - 4 T + 16)^{2}$$
$37$ $$T^{4} - 16 T^{3} + 194 T^{2} + \cdots + 3844$$
$41$ $$(T^{2} + 8 T + 14)^{2}$$
$43$ $$(T^{2} - 32)^{2}$$
$47$ $$T^{4} - 4 T^{3} + 84 T^{2} + \cdots + 4624$$
$53$ $$T^{4} - 4 T^{3} + 110 T^{2} + \cdots + 8836$$
$59$ $$T^{4} - 14 T^{3} + 149 T^{2} + \cdots + 2209$$
$61$ $$T^{4} + 18 T^{3} + 251 T^{2} + \cdots + 5329$$
$67$ $$T^{4} + 14 T^{3} + 165 T^{2} + \cdots + 961$$
$71$ $$(T^{2} + 8 T - 34)^{2}$$
$73$ $$T^{4} - 16 T^{3} + 194 T^{2} + \cdots + 3844$$
$79$ $$T^{4} - 18 T^{3} + 261 T^{2} + \cdots + 3969$$
$83$ $$(T^{2} - 4 T - 196)^{2}$$
$89$ $$T^{4} + 8 T^{3} + 120 T^{2} + \cdots + 3136$$
$97$ $$(T^{2} + 2 T - 7)^{2}$$