Properties

 Label 325.2.k.b Level $325$ Weight $2$ Character orbit 325.k Analytic conductor $2.595$ Analytic rank $0$ Dimension $8$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(57,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 3]))

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

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

chi := DirichletCharacter("325.57");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.k (of order $$4$$, 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: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.619810816.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$$ x^8 - 2*x^5 + 14*x^4 - 8*x^3 + 2*x^2 + 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + ( - \beta_{3} + \beta_{2} + 1) q^{3} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10})$$ q - b5 * q^2 + (-b3 + b2 + 1) * q^3 + (-b7 - b5 + b4 + 1) * q^4 + (b6 - b5 - b3 - b2 - 1) * q^6 + (2*b6 - 2*b5 + 2*b4 - 2*b3 + b1) * q^7 + (-b7 - b5 - b3) * q^8 + (-b5 + b4 - b3 + b2 - b1) * q^9 $$q - \beta_{5} q^{2} + ( - \beta_{3} + \beta_{2} + 1) q^{3} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{99}+O(q^{100})$$ q - b5 * q^2 + (-b3 + b2 + 1) * q^3 + (-b7 - b5 + b4 + 1) * q^4 + (b6 - b5 - b3 - b2 - 1) * q^6 + (2*b6 - 2*b5 + 2*b4 - 2*b3 + b1) * q^7 + (-b7 - b5 - b3) * q^8 + (-b5 + b4 - b3 + b2 - b1) * q^9 + (b6 + b4 - b2 + 1) * q^11 + (-b7 - b6 + b4 + b3 + b1) * q^12 + (b4 + 2*b3 - 2*b2 - b1) * q^13 + (2*b6 - 3*b5 + 3*b4 - 3*b3 + 2*b2 + 3*b1) * q^14 + (-2*b4 - 2*b3 - 1) * q^16 + (b7 - b6 + 2*b5 - b4 + 2*b3 + 2*b2 - b1 + 2) * q^17 + (-b6 - 2*b2 - b1) * q^18 + (b5 - b4 - 2*b2 + 2) * q^19 + (-b7 + b6 + b4 + b2 - b1 - 1) * q^21 + (b7 - b6 - b5 + 2*b2 + b1 - 2) * q^22 + (b7 - b6 + b5 - 2*b4 + 2*b2 + b1 - 2) * q^23 + b3 * q^24 + (b7 - 4*b6 + b5 - b4 + 3*b3 + b2 - b1) * q^26 + (-2*b7 + b6 - b5 + 2*b4 + b2 - 2*b1 - 1) * q^27 + (4*b6 - 2*b5 + 2*b4 - 2*b3 + 3*b1) * q^28 + (-2*b6 + b5 - b4 + b3 + 2*b2 - b1) * q^29 + (-b6 + b5 - b3 + b2 + 1) * q^31 + (3*b5 + 2) * q^32 + (-b7 + b5 + b4 + 2) * q^33 + (2*b7 + 2*b5 - 2*b4 + 4*b3 - b2 - 2*b1 - 1) * q^34 + (-4*b6 + 3*b5 - 3*b4 + 3*b3 - 5*b2) * q^36 + (b5 - b4 + b3 - 6*b2 - b1) * q^37 + (-2*b6 - b5 - b4 + b2 - 1) * q^38 + (-2*b7 + 2*b4 + b3 - 4*b2 + b1) * q^39 + (-2*b7 + b6 - 3*b5 + 2*b4 - 2*b3 + 2*b2 + 2*b1 + 2) * q^41 + (-b6 + b5 - 2*b4 + 3*b2 - 3) * q^42 + (b7 + 3*b6 + b5 + 2*b4 + b1) * q^43 + (2*b6 + b5 + b4 - 2*b2 + 2) * q^44 + (4*b6 + 3*b5 + b4 - 2*b2 + 2) * q^46 + (2*b2 + 3*b1) * q^47 + (2*b7 + 2*b6 - 2*b4 - b3 + b2 - 2*b1 + 1) * q^48 + (4*b5 - 2*b4 - 2*b3 - 1) * q^49 + (-3*b5 + 3*b4 - 3*b3 + 4*b2) * q^51 + (b7 - b6 + 3*b5 - 3*b4 + b3 - 5*b2 - 3*b1 + 3) * q^52 + (b7 - 2*b6 + 3*b5 - b4 + 4*b3 + 3*b2 - b1 + 3) * q^53 + (-b7 - 3*b6 - 3*b4 + 3*b2 - b1 - 3) * q^54 + (2*b6 + b5 - b4 + b3 + 4*b2 + b1) * q^56 + (b7 + 3*b5 - 3*b4 - 2*b3 + 6) * q^57 + (2*b5 - 2*b4 + 2*b3 - 4*b2 - 3*b1) * q^58 + (4*b6 - 4*b5 - 5*b3 + 2*b2 + 2) * q^59 + (3*b7 - 2*b5 - 2*b4 + b3) * q^61 + (b7 + b6 - b4 - b3 - 4*b2 - b1 - 4) * q^62 + (-5*b7 - 5*b5 + 3*b4 - 2*b3 - 2) * q^63 + (3*b7 + b5 + b4 + 4*b3 - 7) * q^64 + (b7 - 2*b5 - 2*b4 - b3 - 6) * q^66 + (b7 - 2*b5 + b3) * q^67 + (-3*b6 + 3*b5 + 4*b3) * q^68 + (3*b7 - 4*b4 - b3 - 2) * q^69 + (b7 - 2*b6 + 3*b5 - b4 + 7*b3 - 2*b2 - b1 - 2) * q^71 + (-3*b6 + 3*b5 - 3*b4 + 3*b3 - 2*b2 - 2*b1) * q^72 + (3*b7 + 2*b5 - 2*b4 + b3) * q^73 + (-8*b6 + 2*b5 - 2*b4 + 2*b3 + 2*b2 - b1) * q^74 + (-2*b7 + b6 - 2*b5 + 3*b4 - b2 - 2*b1 + 1) * q^76 + (b6 - b5 - 2*b3 - 3*b2 - 3) * q^77 + (-2*b6 - 3*b5 - b4 - 2*b3 + b2 + b1 - 5) * q^78 + (-6*b6 + 5*b5 - 5*b4 + 5*b3 + 2*b2 - 2*b1) * q^79 + (-b7 + 2*b5 + 4*b4 + 3*b3 - 1) * q^81 + (-3*b7 + 6*b6 - 9*b5 + 3*b4 - 6*b3 + b2 + 3*b1 + 1) * q^82 + (-4*b6 - 4*b2 + 3*b1) * q^83 + (b7 + b6 + 4*b5 - 3*b4 - 3*b2 + b1 + 3) * q^84 + (4*b7 + 2*b6 + b5 + b4 + 6*b2 + 4*b1 - 6) * q^86 + (-b6 - 3*b5 + 2*b4 + 3*b2 - 3) * q^87 + (b5 - b4 + b2 - 1) * q^88 + (-b7 + 2*b6 - 3*b5 + b4 + 2*b3 - 5*b2 + b1 - 5) * q^89 + (-2*b7 + b6 - 8*b5 + 3*b4 - 5*b3 - 5*b2 + 3*b1 + 1) * q^91 + (2*b7 - b5 + b4 + 7*b2 + 2*b1 - 7) * q^92 + (-2*b6 + 6*b2 - b1) * q^93 + (8*b6 - 3*b5 + 3*b4 - 3*b3 + 3*b1) * q^94 + (-3*b6 + 3*b5 + b3 + 5*b2 + 5) * q^96 + (-2*b7 - 3*b5 - 3*b4 - 5*b3 - 2) * q^97 + (2*b7 + 5*b5 - 4*b4 - 2*b3 - 10) * q^98 + (-b7 - b5 + b4 - b3 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 6 q^{6}+O(q^{10})$$ 8 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 6 * q^6 $$8 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 6 q^{6} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 8 q^{16} + 16 q^{17} + 14 q^{19} - 12 q^{21} - 10 q^{22} - 14 q^{23} + 2 q^{24} + 6 q^{26} - 12 q^{27} + 2 q^{31} + 4 q^{32} + 8 q^{33} - 2 q^{38} - 6 q^{39} + 16 q^{41} - 24 q^{42} - 6 q^{43} + 10 q^{44} + 2 q^{46} + 14 q^{48} - 24 q^{49} + 22 q^{52} + 24 q^{53} - 20 q^{54} + 40 q^{57} + 22 q^{59} + 20 q^{61} - 30 q^{62} - 16 q^{63} - 48 q^{64} - 36 q^{66} + 12 q^{67} - 4 q^{68} - 4 q^{69} - 10 q^{71} + 4 q^{73} + 6 q^{76} - 24 q^{77} - 30 q^{78} - 20 q^{81} + 20 q^{82} + 16 q^{84} - 46 q^{86} - 16 q^{87} - 10 q^{88} - 28 q^{89} + 20 q^{91} - 50 q^{92} + 30 q^{96} - 12 q^{97} - 92 q^{98} - 2 q^{99}+O(q^{100})$$ 8 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 6 * q^6 + 6 * q^11 - 2 * q^12 + 2 * q^13 - 8 * q^16 + 16 * q^17 + 14 * q^19 - 12 * q^21 - 10 * q^22 - 14 * q^23 + 2 * q^24 + 6 * q^26 - 12 * q^27 + 2 * q^31 + 4 * q^32 + 8 * q^33 - 2 * q^38 - 6 * q^39 + 16 * q^41 - 24 * q^42 - 6 * q^43 + 10 * q^44 + 2 * q^46 + 14 * q^48 - 24 * q^49 + 22 * q^52 + 24 * q^53 - 20 * q^54 + 40 * q^57 + 22 * q^59 + 20 * q^61 - 30 * q^62 - 16 * q^63 - 48 * q^64 - 36 * q^66 + 12 * q^67 - 4 * q^68 - 4 * q^69 - 10 * q^71 + 4 * q^73 + 6 * q^76 - 24 * q^77 - 30 * q^78 - 20 * q^81 + 20 * q^82 + 16 * q^84 - 46 * q^86 - 16 * q^87 - 10 * q^88 - 28 * q^89 + 20 * q^91 - 50 * q^92 + 30 * q^96 - 12 * q^97 - 92 * q^98 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319$$ (64*v^7 + 16*v^6 + 4*v^5 - 127*v^4 + 944*v^3 - 276*v^2 + 378*v + 63) / 319 $$\beta_{2}$$ $$=$$ $$( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319$$ (-63*v^7 + 64*v^6 + 16*v^5 + 130*v^4 - 1009*v^3 + 1448*v^2 - 402*v - 67) / 319 $$\beta_{3}$$ $$=$$ $$( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319$$ (-67*v^7 + 63*v^6 - 64*v^5 + 118*v^4 - 1068*v^3 + 1545*v^2 - 1263*v + 268) / 319 $$\beta_{4}$$ $$=$$ $$( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319$$ (83*v^7 - 59*v^6 + 65*v^5 - 70*v^4 + 1304*v^3 - 1614*v^2 + 1198*v + 306) / 319 $$\beta_{5}$$ $$=$$ $$( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319$$ (-172*v^7 - 43*v^6 + 69*v^5 + 441*v^4 - 2218*v^3 + 662*v^2 + 619*v - 269) / 319 $$\beta_{6}$$ $$=$$ $$( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319$$ (-196*v^7 - 49*v^6 - 92*v^5 + 369*v^4 - 2572*v^3 + 1244*v^2 - 1038*v - 173) / 319 $$\beta_{7}$$ $$=$$ $$\nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2$$ v^7 - 2*v^4 + 14*v^3 - 8*v^2 + v + 2
 $$\nu$$ $$=$$ $$( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2$$ (-b7 - b5 + b4 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2}$$ b6 - b5 + b4 - b3 + 2*b2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2$$ (3*b7 + 5*b5 - 5*b4 - 2*b2 + 3*b1 + 2) / 2 $$\nu^{4}$$ $$=$$ $$-\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7$$ -b7 - b5 + 5*b4 + 4*b3 - 7 $$\nu^{5}$$ $$=$$ $$( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2$$ (11*b7 + 2*b6 + 9*b5 - 11*b4 - 12*b3 + 12*b2 - 11*b1 + 12) / 2 $$\nu^{6}$$ $$=$$ $$-15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1$$ -15*b6 + 16*b5 - 16*b4 + 16*b3 - 28*b2 + 7*b1 $$\nu^{7}$$ $$=$$ $$( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2$$ (-43*b7 + 16*b6 - 89*b5 + 105*b4 + 60*b2 - 43*b1 - 60) / 2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$-\beta_{2}$$ $$\beta_{2}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 0.561103 − 0.561103i −1.49094 + 1.49094i −0.252709 + 0.252709i 1.18254 − 1.18254i 0.561103 + 0.561103i −1.49094 − 1.49094i −0.252709 − 0.252709i 1.18254 + 1.18254i
−2.03032 1.33000 1.33000i 2.12221 0 −2.70032 + 2.70032i 1.61845i −0.248119 0.537789i 0
57.2 0.134632 2.15558 2.15558i −1.98187 0 0.290209 0.290209i 1.90970i −0.536087 6.29303i 0
57.3 1.57942 −0.725850 + 0.725850i 0.494582 0 −1.14643 + 1.14643i 4.24997i −2.37769 1.94628i 0
57.4 2.31627 0.240275 0.240275i 3.36509 0 0.556540 0.556540i 3.95872i 3.16190 2.88454i 0
268.1 −2.03032 1.33000 + 1.33000i 2.12221 0 −2.70032 2.70032i 1.61845i −0.248119 0.537789i 0
268.2 0.134632 2.15558 + 2.15558i −1.98187 0 0.290209 + 0.290209i 1.90970i −0.536087 6.29303i 0
268.3 1.57942 −0.725850 0.725850i 0.494582 0 −1.14643 1.14643i 4.24997i −2.37769 1.94628i 0
268.4 2.31627 0.240275 + 0.240275i 3.36509 0 0.556540 + 0.556540i 3.95872i 3.16190 2.88454i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 57.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.k.b 8
5.b even 2 1 65.2.k.b yes 8
5.c odd 4 1 65.2.f.b 8
5.c odd 4 1 325.2.f.b 8
13.d odd 4 1 325.2.f.b 8
15.d odd 2 1 585.2.w.e 8
15.e even 4 1 585.2.n.e 8
20.d odd 2 1 1040.2.bg.n 8
20.e even 4 1 1040.2.cd.n 8
65.d even 2 1 845.2.k.b 8
65.f even 4 1 65.2.k.b yes 8
65.g odd 4 1 65.2.f.b 8
65.g odd 4 1 845.2.f.b 8
65.h odd 4 1 845.2.f.b 8
65.k even 4 1 inner 325.2.k.b 8
65.k even 4 1 845.2.k.b 8
65.l even 6 2 845.2.o.c 16
65.n even 6 2 845.2.o.d 16
65.o even 12 2 845.2.o.c 16
65.q odd 12 2 845.2.t.c 16
65.r odd 12 2 845.2.t.d 16
65.s odd 12 2 845.2.t.c 16
65.s odd 12 2 845.2.t.d 16
65.t even 12 2 845.2.o.d 16
195.n even 4 1 585.2.n.e 8
195.u odd 4 1 585.2.w.e 8
260.l odd 4 1 1040.2.bg.n 8
260.u even 4 1 1040.2.cd.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.b 8 5.c odd 4 1
65.2.f.b 8 65.g odd 4 1
65.2.k.b yes 8 5.b even 2 1
65.2.k.b yes 8 65.f even 4 1
325.2.f.b 8 5.c odd 4 1
325.2.f.b 8 13.d odd 4 1
325.2.k.b 8 1.a even 1 1 trivial
325.2.k.b 8 65.k even 4 1 inner
585.2.n.e 8 15.e even 4 1
585.2.n.e 8 195.n even 4 1
585.2.w.e 8 15.d odd 2 1
585.2.w.e 8 195.u odd 4 1
845.2.f.b 8 65.g odd 4 1
845.2.f.b 8 65.h odd 4 1
845.2.k.b 8 65.d even 2 1
845.2.k.b 8 65.k even 4 1
845.2.o.c 16 65.l even 6 2
845.2.o.c 16 65.o even 12 2
845.2.o.d 16 65.n even 6 2
845.2.o.d 16 65.t even 12 2
845.2.t.c 16 65.q odd 12 2
845.2.t.c 16 65.s odd 12 2
845.2.t.d 16 65.r odd 12 2
845.2.t.d 16 65.s odd 12 2
1040.2.bg.n 8 20.d odd 2 1
1040.2.bg.n 8 260.l odd 4 1
1040.2.cd.n 8 20.e even 4 1
1040.2.cd.n 8 260.u even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 8T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 2 T^{3} - 4 T^{2} + \cdots - 1)^{2}$$
$3$ $$T^{8} - 6 T^{7} + \cdots + 4$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 40 T^{6} + \cdots + 2704$$
$11$ $$T^{8} - 6 T^{7} + \cdots + 4$$
$13$ $$T^{8} - 2 T^{7} + \cdots + 28561$$
$17$ $$T^{8} - 16 T^{7} + \cdots + 13456$$
$19$ $$T^{8} - 14 T^{7} + \cdots + 100$$
$23$ $$T^{8} + 14 T^{7} + \cdots + 40804$$
$29$ $$T^{8} + 44 T^{6} + \cdots + 10000$$
$31$ $$T^{8} - 2 T^{7} + \cdots + 16900$$
$37$ $$T^{8} + 140 T^{6} + \cdots + 336400$$
$41$ $$T^{8} - 16 T^{7} + \cdots + 13456$$
$43$ $$T^{8} + 6 T^{7} + \cdots + 8836$$
$47$ $$T^{8} + 160 T^{6} + \cdots + 26896$$
$53$ $$T^{8} - 24 T^{7} + \cdots + 19600$$
$59$ $$T^{8} - 22 T^{7} + \cdots + 119716$$
$61$ $$(T^{4} - 10 T^{3} + \cdots - 3628)^{2}$$
$67$ $$(T^{4} - 6 T^{3} + \cdots - 148)^{2}$$
$71$ $$T^{8} + 10 T^{7} + \cdots + 1223236$$
$73$ $$(T^{4} - 2 T^{3} + \cdots + 740)^{2}$$
$79$ $$T^{8} + 292 T^{6} + \cdots + 13719616$$
$83$ $$T^{8} + 416 T^{6} + \cdots + 54346384$$
$89$ $$T^{8} + 28 T^{7} + \cdots + 1795600$$
$97$ $$(T^{4} + 6 T^{3} + \cdots + 3704)^{2}$$