# Properties

 Label 140.2.w.a Level $140$ Weight $2$ Character orbit 140.w Analytic conductor $1.118$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,2,Mod(23,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 9, 4]))

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

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

chi := DirichletCharacter("140.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.w (of order $$12$$, degree $$4$$, 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: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: 8.0.12745506816.9 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 49x^{4} + 2401$$ x^8 - 49*x^4 + 2401 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} + \beta_{2} - 1) q^{2} + \beta_1 q^{3} + (2 \beta_{6} - 2 \beta_{2}) q^{4} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{5} + (\beta_{5} + \beta_{3} - \beta_1) q^{6} + \beta_{7} q^{7} + ( - 2 \beta_{6} - 2) q^{8} + 4 \beta_{2} q^{9}+O(q^{10})$$ q + (b4 + b2 - 1) * q^2 + b1 * q^3 + (2*b6 - 2*b2) * q^4 + (-b4 - 2*b2 + 1) * q^5 + (b5 + b3 - b1) * q^6 + b7 * q^7 + (-2*b6 - 2) * q^8 + 4*b2 * q^9 $$q + (\beta_{4} + \beta_{2} - 1) q^{2} + \beta_1 q^{3} + (2 \beta_{6} - 2 \beta_{2}) q^{4} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{5} + (\beta_{5} + \beta_{3} - \beta_1) q^{6} + \beta_{7} q^{7} + ( - 2 \beta_{6} - 2) q^{8} + 4 \beta_{2} q^{9} + ( - 3 \beta_{6} - \beta_{4} + 3 \beta_{2}) q^{10} + ( - \beta_{7} + \beta_1) q^{11} + (2 \beta_{7} - 2 \beta_{3}) q^{12} + ( - 2 \beta_{6} - 2) q^{13} + (\beta_{5} - \beta_{3} - \beta_1) q^{14} + ( - \beta_{5} - 2 \beta_{3} + \beta_1) q^{15} + ( - 4 \beta_{4} + 4) q^{16} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{2}) q^{17} + (4 \beta_{6} + 4 \beta_{4} - 4 \beta_{2}) q^{18} + ( - \beta_{7} - \beta_{5} + \beta_{3}) q^{19} + (2 \beta_{6} + 4) q^{20} + (7 \beta_{4} - 7) q^{21} + 2 \beta_{3} q^{22} + \beta_{5} q^{23} + ( - 2 \beta_{7} - 2 \beta_1) q^{24} + (4 \beta_{6} + 3 \beta_{4} - 4 \beta_{2}) q^{25} + ( - 4 \beta_{4} + 4) q^{26} + \beta_{3} q^{27} - 2 \beta_{5} q^{28} - 3 \beta_{6} q^{29} + ( - 3 \beta_{7} - \beta_{5} + 3 \beta_{3}) q^{30} + (2 \beta_{7} - 2 \beta_1) q^{31} + ( - 4 \beta_{6} + 4 \beta_{4} + 4 \beta_{2}) q^{32} + ( - 7 \beta_{4} + 7 \beta_{2} + 7) q^{33} - 4 \beta_{6} q^{34} + ( - 2 \beta_{5} + \beta_{3} + 2 \beta_1) q^{35} - 8 q^{36} + 2 \beta_1 q^{38} + ( - 2 \beta_{7} - 2 \beta_1) q^{39} + (6 \beta_{4} + 2 \beta_{2} - 6) q^{40} + 3 q^{41} + (7 \beta_{6} - 7 \beta_{4} - 7 \beta_{2}) q^{42} + (3 \beta_{5} - 3 \beta_1) q^{43} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3}) q^{44} + ( - 4 \beta_{6} - 8 \beta_{4} + 4 \beta_{2}) q^{45} + (\beta_{7} - \beta_1) q^{46} + ( - 4 \beta_{7} + 4 \beta_{3}) q^{47} + ( - 4 \beta_{5} + 4 \beta_1) q^{48} - 7 \beta_{2} q^{49} + ( - \beta_{6} - 7) q^{50} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3}) q^{51} + ( - 4 \beta_{6} + 4 \beta_{4} + 4 \beta_{2}) q^{52} + (5 \beta_{6} + 5 \beta_{4} - 5 \beta_{2}) q^{53} + (\beta_{7} + \beta_{5} - \beta_{3}) q^{54} + (\beta_{5} - 3 \beta_{3} - \beta_1) q^{55} + ( - 2 \beta_{7} + 2 \beta_1) q^{56} + ( - 7 \beta_{6} + 7) q^{57} + ( - 3 \beta_{4} + 3 \beta_{2} + 3) q^{58} + (\beta_{7} + \beta_1) q^{59} + (2 \beta_{7} + 4 \beta_1) q^{60} + ( - 3 \beta_{4} + 3) q^{61} - 4 \beta_{3} q^{62} + (4 \beta_{5} - 4 \beta_1) q^{63} + 8 \beta_{6} q^{64} + (6 \beta_{4} + 2 \beta_{2} - 6) q^{65} + 14 \beta_{4} q^{66} + 5 \beta_{7} q^{67} + ( - 4 \beta_{4} + 4 \beta_{2} + 4) q^{68} + 7 \beta_{6} q^{69} + ( - \beta_{7} + 3 \beta_{5} + \beta_{3}) q^{70} + (\beta_{5} + \beta_{3} - \beta_1) q^{71} + ( - 8 \beta_{4} - 8 \beta_{2} + 8) q^{72} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{2}) q^{73} + (4 \beta_{7} + 3 \beta_{5} - 4 \beta_{3}) q^{75} + (2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{76} + (7 \beta_{4} + 7 \beta_{2} - 7) q^{77} + ( - 4 \beta_{5} + 4 \beta_1) q^{78} + (\beta_{7} + \beta_{5} - \beta_{3}) q^{79} + (8 \beta_{6} - 4 \beta_{4} - 8 \beta_{2}) q^{80} - 5 \beta_{4} q^{81} + (3 \beta_{4} + 3 \beta_{2} - 3) q^{82} + ( - \beta_{5} + \beta_1) q^{83} - 14 \beta_{6} q^{84} + (6 \beta_{6} + 2) q^{85} + (3 \beta_{7} - 3 \beta_{5} - 3 \beta_{3}) q^{86} - 3 \beta_{7} q^{87} - 4 \beta_1 q^{88} - 3 \beta_{2} q^{89} + ( - 4 \beta_{6} + 12) q^{90} + ( - 2 \beta_{7} + 2 \beta_1) q^{91} - 2 \beta_{3} q^{92} + (14 \beta_{4} - 14 \beta_{2} - 14) q^{93} + (4 \beta_{7} + 4 \beta_1) q^{94} + (\beta_{7} - 3 \beta_1) q^{95} + ( - 4 \beta_{7} + 4 \beta_{5} + 4 \beta_{3}) q^{96} + (9 \beta_{6} - 9) q^{97} + ( - 7 \beta_{6} - 7 \beta_{4} + 7 \beta_{2}) q^{98} + ( - 4 \beta_{5} + 4 \beta_{3} + 4 \beta_1) q^{99}+O(q^{100})$$ q + (b4 + b2 - 1) * q^2 + b1 * q^3 + (2*b6 - 2*b2) * q^4 + (-b4 - 2*b2 + 1) * q^5 + (b5 + b3 - b1) * q^6 + b7 * q^7 + (-2*b6 - 2) * q^8 + 4*b2 * q^9 + (-3*b6 - b4 + 3*b2) * q^10 + (-b7 + b1) * q^11 + (2*b7 - 2*b3) * q^12 + (-2*b6 - 2) * q^13 + (b5 - b3 - b1) * q^14 + (-b5 - 2*b3 + b1) * q^15 + (-4*b4 + 4) * q^16 + (2*b6 - 2*b4 - 2*b2) * q^17 + (4*b6 + 4*b4 - 4*b2) * q^18 + (-b7 - b5 + b3) * q^19 + (2*b6 + 4) * q^20 + (7*b4 - 7) * q^21 + 2*b3 * q^22 + b5 * q^23 + (-2*b7 - 2*b1) * q^24 + (4*b6 + 3*b4 - 4*b2) * q^25 + (-4*b4 + 4) * q^26 + b3 * q^27 - 2*b5 * q^28 - 3*b6 * q^29 + (-3*b7 - b5 + 3*b3) * q^30 + (2*b7 - 2*b1) * q^31 + (-4*b6 + 4*b4 + 4*b2) * q^32 + (-7*b4 + 7*b2 + 7) * q^33 - 4*b6 * q^34 + (-2*b5 + b3 + 2*b1) * q^35 - 8 * q^36 + 2*b1 * q^38 + (-2*b7 - 2*b1) * q^39 + (6*b4 + 2*b2 - 6) * q^40 + 3 * q^41 + (7*b6 - 7*b4 - 7*b2) * q^42 + (3*b5 - 3*b1) * q^43 + (2*b7 + 2*b5 - 2*b3) * q^44 + (-4*b6 - 8*b4 + 4*b2) * q^45 + (b7 - b1) * q^46 + (-4*b7 + 4*b3) * q^47 + (-4*b5 + 4*b1) * q^48 - 7*b2 * q^49 + (-b6 - 7) * q^50 + (2*b7 - 2*b5 - 2*b3) * q^51 + (-4*b6 + 4*b4 + 4*b2) * q^52 + (5*b6 + 5*b4 - 5*b2) * q^53 + (b7 + b5 - b3) * q^54 + (b5 - 3*b3 - b1) * q^55 + (-2*b7 + 2*b1) * q^56 + (-7*b6 + 7) * q^57 + (-3*b4 + 3*b2 + 3) * q^58 + (b7 + b1) * q^59 + (2*b7 + 4*b1) * q^60 + (-3*b4 + 3) * q^61 - 4*b3 * q^62 + (4*b5 - 4*b1) * q^63 + 8*b6 * q^64 + (6*b4 + 2*b2 - 6) * q^65 + 14*b4 * q^66 + 5*b7 * q^67 + (-4*b4 + 4*b2 + 4) * q^68 + 7*b6 * q^69 + (-b7 + 3*b5 + b3) * q^70 + (b5 + b3 - b1) * q^71 + (-8*b4 - 8*b2 + 8) * q^72 + (-2*b6 - 2*b4 + 2*b2) * q^73 + (4*b7 + 3*b5 - 4*b3) * q^75 + (2*b5 + 2*b3 - 2*b1) * q^76 + (7*b4 + 7*b2 - 7) * q^77 + (-4*b5 + 4*b1) * q^78 + (b7 + b5 - b3) * q^79 + (8*b6 - 4*b4 - 8*b2) * q^80 - 5*b4 * q^81 + (3*b4 + 3*b2 - 3) * q^82 + (-b5 + b1) * q^83 - 14*b6 * q^84 + (6*b6 + 2) * q^85 + (3*b7 - 3*b5 - 3*b3) * q^86 - 3*b7 * q^87 - 4*b1 * q^88 - 3*b2 * q^89 + (-4*b6 + 12) * q^90 + (-2*b7 + 2*b1) * q^91 - 2*b3 * q^92 + (14*b4 - 14*b2 - 14) * q^93 + (4*b7 + 4*b1) * q^94 + (b7 - 3*b1) * q^95 + (-4*b7 + 4*b5 + 4*b3) * q^96 + (9*b6 - 9) * q^97 + (-7*b6 - 7*b4 + 7*b2) * q^98 + (-4*b5 + 4*b3 + 4*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} + 4 q^{5} - 16 q^{8}+O(q^{10})$$ 8 * q - 4 * q^2 + 4 * q^5 - 16 * q^8 $$8 q - 4 q^{2} + 4 q^{5} - 16 q^{8} - 4 q^{10} - 16 q^{13} + 16 q^{16} - 8 q^{17} + 16 q^{18} + 32 q^{20} - 28 q^{21} + 12 q^{25} + 16 q^{26} + 16 q^{32} + 28 q^{33} - 64 q^{36} - 24 q^{40} + 24 q^{41} - 28 q^{42} - 32 q^{45} - 56 q^{50} + 16 q^{52} + 20 q^{53} + 56 q^{57} + 12 q^{58} + 12 q^{61} - 24 q^{65} + 56 q^{66} + 16 q^{68} + 32 q^{72} - 8 q^{73} - 28 q^{77} - 16 q^{80} - 20 q^{81} - 12 q^{82} + 16 q^{85} + 96 q^{90} - 56 q^{93} - 72 q^{97} - 28 q^{98}+O(q^{100})$$ 8 * q - 4 * q^2 + 4 * q^5 - 16 * q^8 - 4 * q^10 - 16 * q^13 + 16 * q^16 - 8 * q^17 + 16 * q^18 + 32 * q^20 - 28 * q^21 + 12 * q^25 + 16 * q^26 + 16 * q^32 + 28 * q^33 - 64 * q^36 - 24 * q^40 + 24 * q^41 - 28 * q^42 - 32 * q^45 - 56 * q^50 + 16 * q^52 + 20 * q^53 + 56 * q^57 + 12 * q^58 + 12 * q^61 - 24 * q^65 + 56 * q^66 + 16 * q^68 + 32 * q^72 - 8 * q^73 - 28 * q^77 - 16 * q^80 - 20 * q^81 - 12 * q^82 + 16 * q^85 + 96 * q^90 - 56 * q^93 - 72 * q^97 - 28 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 49x^{4} + 2401$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7 $$\beta_{4}$$ $$=$$ $$( \nu^{4} ) / 49$$ (v^4) / 49 $$\beta_{5}$$ $$=$$ $$( \nu^{5} ) / 49$$ (v^5) / 49 $$\beta_{6}$$ $$=$$ $$( \nu^{6} ) / 343$$ (v^6) / 343 $$\beta_{7}$$ $$=$$ $$( \nu^{7} ) / 343$$ (v^7) / 343
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3 $$\nu^{4}$$ $$=$$ $$49\beta_{4}$$ 49*b4 $$\nu^{5}$$ $$=$$ $$49\beta_{5}$$ 49*b5 $$\nu^{6}$$ $$=$$ $$343\beta_{6}$$ 343*b6 $$\nu^{7}$$ $$=$$ $$343\beta_{7}$$ 343*b7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$71$$ $$101$$ $$\chi(n)$$ $$-\beta_{6}$$ $$-1$$ $$-1 + \beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −2.55560 − 0.684771i 2.55560 + 0.684771i −2.55560 + 0.684771i 2.55560 − 0.684771i −0.684771 + 2.55560i 0.684771 − 2.55560i −0.684771 − 2.55560i 0.684771 + 2.55560i
0.366025 + 1.36603i −2.55560 0.684771i −1.73205 + 1.00000i −1.23205 1.86603i 3.74166i 0.684771 2.55560i −2.00000 2.00000i 3.46410 + 2.00000i 2.09808 2.36603i
23.2 0.366025 + 1.36603i 2.55560 + 0.684771i −1.73205 + 1.00000i −1.23205 1.86603i 3.74166i −0.684771 + 2.55560i −2.00000 2.00000i 3.46410 + 2.00000i 2.09808 2.36603i
67.1 0.366025 1.36603i −2.55560 + 0.684771i −1.73205 1.00000i −1.23205 + 1.86603i 3.74166i 0.684771 + 2.55560i −2.00000 + 2.00000i 3.46410 2.00000i 2.09808 + 2.36603i
67.2 0.366025 1.36603i 2.55560 0.684771i −1.73205 1.00000i −1.23205 + 1.86603i 3.74166i −0.684771 2.55560i −2.00000 + 2.00000i 3.46410 2.00000i 2.09808 + 2.36603i
107.1 −1.36603 + 0.366025i −0.684771 + 2.55560i 1.73205 1.00000i 2.23205 + 0.133975i 3.74166i 2.55560 + 0.684771i −2.00000 + 2.00000i −3.46410 2.00000i −3.09808 + 0.633975i
107.2 −1.36603 + 0.366025i 0.684771 2.55560i 1.73205 1.00000i 2.23205 + 0.133975i 3.74166i −2.55560 0.684771i −2.00000 + 2.00000i −3.46410 2.00000i −3.09808 + 0.633975i
123.1 −1.36603 0.366025i −0.684771 2.55560i 1.73205 + 1.00000i 2.23205 0.133975i 3.74166i 2.55560 0.684771i −2.00000 2.00000i −3.46410 + 2.00000i −3.09808 0.633975i
123.2 −1.36603 0.366025i 0.684771 + 2.55560i 1.73205 + 1.00000i 2.23205 0.133975i 3.74166i −2.55560 + 0.684771i −2.00000 2.00000i −3.46410 + 2.00000i −3.09808 0.633975i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
20.e even 4 1 inner
28.g odd 6 1 inner
35.l odd 12 1 inner
140.w even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.w.a 8
4.b odd 2 1 inner 140.2.w.a 8
5.b even 2 1 700.2.be.c 8
5.c odd 4 1 inner 140.2.w.a 8
5.c odd 4 1 700.2.be.c 8
7.b odd 2 1 980.2.x.f 8
7.c even 3 1 inner 140.2.w.a 8
7.c even 3 1 980.2.k.e 4
7.d odd 6 1 980.2.k.g 4
7.d odd 6 1 980.2.x.f 8
20.d odd 2 1 700.2.be.c 8
20.e even 4 1 inner 140.2.w.a 8
20.e even 4 1 700.2.be.c 8
28.d even 2 1 980.2.x.f 8
28.f even 6 1 980.2.k.g 4
28.f even 6 1 980.2.x.f 8
28.g odd 6 1 inner 140.2.w.a 8
28.g odd 6 1 980.2.k.e 4
35.f even 4 1 980.2.x.f 8
35.j even 6 1 700.2.be.c 8
35.k even 12 1 980.2.k.g 4
35.k even 12 1 980.2.x.f 8
35.l odd 12 1 inner 140.2.w.a 8
35.l odd 12 1 700.2.be.c 8
35.l odd 12 1 980.2.k.e 4
140.j odd 4 1 980.2.x.f 8
140.p odd 6 1 700.2.be.c 8
140.w even 12 1 inner 140.2.w.a 8
140.w even 12 1 700.2.be.c 8
140.w even 12 1 980.2.k.e 4
140.x odd 12 1 980.2.k.g 4
140.x odd 12 1 980.2.x.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.w.a 8 1.a even 1 1 trivial
140.2.w.a 8 4.b odd 2 1 inner
140.2.w.a 8 5.c odd 4 1 inner
140.2.w.a 8 7.c even 3 1 inner
140.2.w.a 8 20.e even 4 1 inner
140.2.w.a 8 28.g odd 6 1 inner
140.2.w.a 8 35.l odd 12 1 inner
140.2.w.a 8 140.w even 12 1 inner
700.2.be.c 8 5.b even 2 1
700.2.be.c 8 5.c odd 4 1
700.2.be.c 8 20.d odd 2 1
700.2.be.c 8 20.e even 4 1
700.2.be.c 8 35.j even 6 1
700.2.be.c 8 35.l odd 12 1
700.2.be.c 8 140.p odd 6 1
700.2.be.c 8 140.w even 12 1
980.2.k.e 4 7.c even 3 1
980.2.k.e 4 28.g odd 6 1
980.2.k.e 4 35.l odd 12 1
980.2.k.e 4 140.w even 12 1
980.2.k.g 4 7.d odd 6 1
980.2.k.g 4 28.f even 6 1
980.2.k.g 4 35.k even 12 1
980.2.k.g 4 140.x odd 12 1
980.2.x.f 8 7.b odd 2 1
980.2.x.f 8 7.d odd 6 1
980.2.x.f 8 28.d even 2 1
980.2.x.f 8 28.f even 6 1
980.2.x.f 8 35.f even 4 1
980.2.x.f 8 35.k even 12 1
980.2.x.f 8 140.j odd 4 1
980.2.x.f 8 140.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 49T_{3}^{4} + 2401$$ acting on $$S_{2}^{\mathrm{new}}(140, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4)^{2}$$
$3$ $$T^{8} - 49T^{4} + 2401$$
$5$ $$(T^{4} - 2 T^{3} - T^{2} - 10 T + 25)^{2}$$
$7$ $$T^{8} - 49T^{4} + 2401$$
$11$ $$(T^{4} - 14 T^{2} + 196)^{2}$$
$13$ $$(T^{2} + 4 T + 8)^{4}$$
$17$ $$(T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64)^{2}$$
$19$ $$(T^{4} + 14 T^{2} + 196)^{2}$$
$23$ $$T^{8} - 49T^{4} + 2401$$
$29$ $$(T^{2} + 9)^{4}$$
$31$ $$(T^{4} - 56 T^{2} + 3136)^{2}$$
$37$ $$T^{8}$$
$41$ $$(T - 3)^{8}$$
$43$ $$(T^{4} + 3969)^{2}$$
$47$ $$T^{8} - 12544 T^{4} + \cdots + 157351936$$
$53$ $$(T^{4} - 10 T^{3} + 50 T^{2} - 500 T + 2500)^{2}$$
$59$ $$(T^{4} + 14 T^{2} + 196)^{2}$$
$61$ $$(T^{2} - 3 T + 9)^{4}$$
$67$ $$T^{8} - 30625 T^{4} + \cdots + 937890625$$
$71$ $$(T^{2} + 14)^{4}$$
$73$ $$(T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64)^{2}$$
$79$ $$(T^{4} + 14 T^{2} + 196)^{2}$$
$83$ $$(T^{4} + 49)^{2}$$
$89$ $$(T^{4} - 9 T^{2} + 81)^{2}$$
$97$ $$(T^{2} + 18 T + 162)^{4}$$