# Properties

 Label 35.2.e.a Level $35$ Weight $2$ Character orbit 35.e Analytic conductor $0.279$ 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] = [35,2,Mod(11,35)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(35, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("35.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 35.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: $$0.279476407074$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

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

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 $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−1.20711 2.09077i 0.207107 0.358719i −1.91421 + 3.31552i −0.500000 0.866025i −1.00000 2.62132 + 0.358719i 4.41421 1.41421 + 2.44949i −1.20711 + 2.09077i
11.2 0.207107 + 0.358719i −1.20711 + 2.09077i 0.914214 1.58346i −0.500000 0.866025i −1.00000 −1.62132 2.09077i 1.58579 −1.41421 2.44949i 0.207107 0.358719i
16.1 −1.20711 + 2.09077i 0.207107 + 0.358719i −1.91421 3.31552i −0.500000 + 0.866025i −1.00000 2.62132 0.358719i 4.41421 1.41421 2.44949i −1.20711 2.09077i
16.2 0.207107 0.358719i −1.20711 2.09077i 0.914214 + 1.58346i −0.500000 + 0.866025i −1.00000 −1.62132 + 2.09077i 1.58579 −1.41421 + 2.44949i 0.207107 + 0.358719i
 $$n$$: 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 35.2.e.a 4
3.b odd 2 1 315.2.j.e 4
4.b odd 2 1 560.2.q.k 4
5.b even 2 1 175.2.e.c 4
5.c odd 4 2 175.2.k.a 8
7.b odd 2 1 245.2.e.e 4
7.c even 3 1 inner 35.2.e.a 4
7.c even 3 1 245.2.a.h 2
7.d odd 6 1 245.2.a.g 2
7.d odd 6 1 245.2.e.e 4
21.g even 6 1 2205.2.a.q 2
21.h odd 6 1 315.2.j.e 4
21.h odd 6 1 2205.2.a.n 2
28.f even 6 1 3920.2.a.bv 2
28.g odd 6 1 560.2.q.k 4
28.g odd 6 1 3920.2.a.bq 2
35.i odd 6 1 1225.2.a.m 2
35.j even 6 1 175.2.e.c 4
35.j even 6 1 1225.2.a.k 2
35.k even 12 2 1225.2.b.h 4
35.l odd 12 2 175.2.k.a 8
35.l odd 12 2 1225.2.b.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 1.a even 1 1 trivial
35.2.e.a 4 7.c even 3 1 inner
175.2.e.c 4 5.b even 2 1
175.2.e.c 4 35.j even 6 1
175.2.k.a 8 5.c odd 4 2
175.2.k.a 8 35.l odd 12 2
245.2.a.g 2 7.d odd 6 1
245.2.a.h 2 7.c even 3 1
245.2.e.e 4 7.b odd 2 1
245.2.e.e 4 7.d odd 6 1
315.2.j.e 4 3.b odd 2 1
315.2.j.e 4 21.h odd 6 1
560.2.q.k 4 4.b odd 2 1
560.2.q.k 4 28.g odd 6 1
1225.2.a.k 2 35.j even 6 1
1225.2.a.m 2 35.i odd 6 1
1225.2.b.g 4 35.l odd 12 2
1225.2.b.h 4 35.k even 12 2
2205.2.a.n 2 21.h odd 6 1
2205.2.a.q 2 21.g even 6 1
3920.2.a.bq 2 28.g odd 6 1
3920.2.a.bv 2 28.f even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$3$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49$$
$11$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$13$ $$(T^{2} + 4 T - 4)^{2}$$
$17$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$19$ $$T^{4} + 8T^{2} + 64$$
$23$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T^{2} - 6 T + 36)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 10 T + 17)^{2}$$
$43$ $$(T^{2} - 10 T + 23)^{2}$$
$47$ $$(T^{2} + 2 T + 4)^{2}$$
$53$ $$T^{4} - 8 T^{3} + 56 T^{2} - 64 T + 64$$
$59$ $$T^{4} - 8 T^{3} + 120 T^{2} + \cdots + 3136$$
$61$ $$T^{4} - 6 T^{3} + 99 T^{2} + \cdots + 3969$$
$67$ $$T^{4} + 22 T^{3} + 365 T^{2} + \cdots + 14161$$
$71$ $$(T^{2} + 8 T - 56)^{2}$$
$73$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$79$ $$T^{4} + 24 T^{3} + 440 T^{2} + \cdots + 18496$$
$83$ $$(T^{2} - 2 T - 161)^{2}$$
$89$ $$T^{4} - 6 T^{3} + 59 T^{2} + 138 T + 529$$
$97$ $$(T^{2} - 12 T + 4)^{2}$$