# Properties

 Label 91.2.f.a Level $91$ Weight $2$ Character orbit 91.f Analytic conductor $0.727$ 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] = [91,2,Mod(22,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, 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("91.22");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.f (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: $$0.726638658394$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*x^2 + x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.809017 − 1.40126i −0.309017 + 0.535233i 0.809017 + 1.40126i −0.309017 − 0.535233i
−1.30902 + 2.26728i 1.30902 2.26728i −2.42705 4.20378i 2.61803 3.42705 + 5.93583i −0.500000 0.866025i 7.47214 −1.92705 3.33775i −3.42705 + 5.93583i
22.2 −0.190983 + 0.330792i 0.190983 0.330792i 0.927051 + 1.60570i 0.381966 0.0729490 + 0.126351i −0.500000 0.866025i −1.47214 1.42705 + 2.47172i −0.0729490 + 0.126351i
29.1 −1.30902 2.26728i 1.30902 + 2.26728i −2.42705 + 4.20378i 2.61803 3.42705 5.93583i −0.500000 + 0.866025i 7.47214 −1.92705 + 3.33775i −3.42705 5.93583i
29.2 −0.190983 0.330792i 0.190983 + 0.330792i 0.927051 1.60570i 0.381966 0.0729490 0.126351i −0.500000 + 0.866025i −1.47214 1.42705 2.47172i −0.0729490 0.126351i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.f.a 4
3.b odd 2 1 819.2.o.c 4
4.b odd 2 1 1456.2.s.h 4
7.b odd 2 1 637.2.f.c 4
7.c even 3 1 637.2.g.b 4
7.c even 3 1 637.2.h.g 4
7.d odd 6 1 637.2.g.c 4
7.d odd 6 1 637.2.h.f 4
13.c even 3 1 inner 91.2.f.a 4
13.c even 3 1 1183.2.a.g 2
13.e even 6 1 1183.2.a.c 2
13.f odd 12 2 1183.2.c.c 4
39.i odd 6 1 819.2.o.c 4
52.j odd 6 1 1456.2.s.h 4
91.g even 3 1 637.2.h.g 4
91.h even 3 1 637.2.g.b 4
91.m odd 6 1 637.2.h.f 4
91.n odd 6 1 637.2.f.c 4
91.n odd 6 1 8281.2.a.bb 2
91.t odd 6 1 8281.2.a.n 2
91.v odd 6 1 637.2.g.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 1.a even 1 1 trivial
91.2.f.a 4 13.c even 3 1 inner
637.2.f.c 4 7.b odd 2 1
637.2.f.c 4 91.n odd 6 1
637.2.g.b 4 7.c even 3 1
637.2.g.b 4 91.h even 3 1
637.2.g.c 4 7.d odd 6 1
637.2.g.c 4 91.v odd 6 1
637.2.h.f 4 7.d odd 6 1
637.2.h.f 4 91.m odd 6 1
637.2.h.g 4 7.c even 3 1
637.2.h.g 4 91.g even 3 1
819.2.o.c 4 3.b odd 2 1
819.2.o.c 4 39.i odd 6 1
1183.2.a.c 2 13.e even 6 1
1183.2.a.g 2 13.c even 3 1
1183.2.c.c 4 13.f odd 12 2
1456.2.s.h 4 4.b odd 2 1
1456.2.s.h 4 52.j odd 6 1
8281.2.a.n 2 91.t odd 6 1
8281.2.a.bb 2 91.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{3} + 8T_{2}^{2} + 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} + 8 T^{2} + 3 T + 1$$
$3$ $$T^{4} - 3 T^{3} + 8 T^{2} - 3 T + 1$$
$5$ $$(T^{2} - 3 T + 1)^{2}$$
$7$ $$(T^{2} + T + 1)^{2}$$
$11$ $$T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81$$
$13$ $$(T^{2} + 5 T + 13)^{2}$$
$17$ $$T^{4} + 6 T^{3} + 47 T^{2} - 66 T + 121$$
$19$ $$T^{4} + 3 T^{3} + 18 T^{2} - 27 T + 81$$
$23$ $$T^{4} + 20T^{2} + 400$$
$29$ $$T^{4} + 3 T^{3} + 38 T^{2} - 87 T + 841$$
$31$ $$(T^{2} - 4 T - 41)^{2}$$
$37$ $$(T^{2} + 4 T + 16)^{2}$$
$41$ $$T^{4} + 6 T^{3} + 32 T^{2} + 24 T + 16$$
$43$ $$T^{4} + 5 T^{3} + 120 T^{2} + \cdots + 9025$$
$47$ $$(T^{2} - 5)^{2}$$
$53$ $$(T^{2} - 12 T + 31)^{2}$$
$59$ $$T^{4} + 5T^{2} + 25$$
$61$ $$(T^{2} - 6 T + 36)^{2}$$
$67$ $$T^{4} - 12 T^{3} + 153 T^{2} + \cdots + 81$$
$71$ $$T^{4} - 6 T^{3} + 152 T^{2} + \cdots + 13456$$
$73$ $$(T + 2)^{4}$$
$79$ $$(T - 4)^{4}$$
$83$ $$(T^{2} - 45)^{2}$$
$89$ $$T^{4} + 21 T^{3} + 362 T^{2} + \cdots + 6241$$
$97$ $$T^{4} + 31 T^{3} + 732 T^{2} + \cdots + 52441$$