# Properties

 Label 169.2.e.b Level $169$ Weight $2$ Character orbit 169.e Analytic conductor $1.349$ Analytic rank $0$ Dimension $12$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(23,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

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

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

chi := DirichletCharacter("169.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.e (of order $$6$$, degree $$2$$, not 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: $$1.34947179416$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.17213603549184.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1$$ x^12 - 5*x^10 + 19*x^8 - 28*x^6 + 31*x^4 - 6*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{10} + \beta_{8} - \beta_1) q^{2} + (\beta_{9} + \beta_{4} - \beta_{3} + 1) q^{3} + ( - \beta_{9} + \beta_{4}) q^{4} + ( - \beta_{11} - 2 \beta_{10} + \cdots - \beta_{2}) q^{5}+ \cdots + (\beta_{9} - 2 \beta_{7} + 2 \beta_{4} + 2) q^{9}+O(q^{10})$$ q + (b10 + b8 - b1) * q^2 + (b9 + b4 - b3 + 1) * q^3 + (-b9 + b4) * q^4 + (-b11 - 2*b10 - b8 + 2*b6 - b2) * q^5 - b1 * q^6 + (b6 - b2 + b1) * q^7 + (-2*b11 - b8 - 2*b2) * q^8 + (b9 - 2*b7 + 2*b4 + 2) * q^9 $$q + (\beta_{10} + \beta_{8} - \beta_1) q^{2} + (\beta_{9} + \beta_{4} - \beta_{3} + 1) q^{3} + ( - \beta_{9} + \beta_{4}) q^{4} + ( - \beta_{11} - 2 \beta_{10} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - 2 \beta_{11} - 3 \beta_{10} + \cdots - 2 \beta_{2}) q^{99}+O(q^{100})$$ q + (b10 + b8 - b1) * q^2 + (b9 + b4 - b3 + 1) * q^3 + (-b9 + b4) * q^4 + (-b11 - 2*b10 - b8 + 2*b6 - b2) * q^5 - b1 * q^6 + (b6 - b2 + b1) * q^7 + (-2*b11 - b8 - 2*b2) * q^8 + (b9 - 2*b7 + 2*b4 + 2) * q^9 + (2*b7 - b5 - b4) * q^10 + (b11 - 2*b10 + b8 - b1) * q^11 + (-b5 + b3) * q^12 + (2*b5 + 2*b3 - 3) * q^14 + (b11 + b10) * q^15 + (b9 - b7 - b5 - b3 + 1) * q^16 + (b9 + b7 - 2*b4 - 1) * q^17 + (-b11 + 3*b10 + b8 - 3*b6 - b2) * q^18 + (-b6 - 2*b2 + b1) * q^19 + (-b6 + 3*b2) * q^20 + (2*b11 + b10 + 3*b8 - b6 + 2*b2) * q^21 + (2*b7 - 3*b4 - 2) * q^22 + (2*b9 - 3*b7 + 2*b4 - 2*b3 + 2) * q^23 + (-b11 + 2*b10 - 2*b8 + 2*b1) * q^24 + (-b5 - 3*b3 + 3) * q^25 + (-2*b5 - 3*b3 + 2) * q^27 + (-5*b10 - b8 + b1) * q^28 + (-5*b9 + 3*b7 + 2*b5 - 3*b4 + 5*b3 - 5) * q^29 + b9 * q^30 + (5*b11 + b10 + 3*b8 - b6 + 5*b2) * q^31 + (5*b2 + 2*b1) * q^32 + (-b6 - 3*b2 - 4*b1) * q^33 + (3*b11 - 4*b10 - 2*b8 + 4*b6 + 3*b2) * q^34 + (-4*b9 + 3*b7 - b4 - 3) * q^35 + (-4*b9 - b7 + 4*b5 + 4*b3 - 4) * q^36 + (-2*b11 + 3*b10 - b8 + b1) * q^37 + (6*b5 + 3*b3 - 7) * q^38 + (-3*b5 - 3*b3 + 1) * q^40 + (-4*b11 - b10 - 6*b8 + 6*b1) * q^41 + (-b9 - 2*b7 - b4 + b3 - 1) * q^42 + (2*b9 - 6*b7 + 3*b4 + 6) * q^43 + (b11 - b10 - 4*b8 + b6 + b2) * q^44 + (5*b6 - 2*b2 - 2*b1) * q^45 + (-3*b6 + b1) * q^46 + (b11 + 6*b10 - b8 - 6*b6 + b2) * q^47 + (-b9 + b7 - b4 - 1) * q^48 + (-b9 - b7 + b5 + b3 - 1) * q^49 + (-b11 + b10 + 3*b8 - 3*b1) * q^50 + (2*b5 - b3) * q^51 + (-3*b5 + 4*b3) * q^53 + (-2*b11 + b10 + 2*b8 - 2*b1) * q^54 + (b9 - 3*b7 + b5 + 2*b4 - b3 + 1) * q^55 + (-3*b9 + 4*b7 - 5*b4 - 4) * q^56 + (2*b10 + b8 - 2*b6) * q^57 + (b6 - 2*b2 + 2*b1) * q^58 + (5*b6 + 4*b2) * q^59 + (-b11 - 3*b10 - b8 + 3*b6 - b2) * q^60 + (6*b9 - b7 + b4 + 1) * q^61 + (2*b9 + b7 - 6*b5 - 4*b4 - 2*b3 + 2) * q^62 + (5*b11 + b10 + 2*b8 - 2*b1) * q^63 + (-6*b5 - b3 + 6) * q^64 + (3*b5 - b3 + 1) * q^66 + (6*b11 + 4*b10 + 5*b8 - 5*b1) * q^67 + (3*b9 + 7*b7 - 6*b5 - 3*b4 - 3*b3 + 3) * q^68 + (-b9 + 2*b7 + b4 - 2) * q^69 + (-3*b11 + b8 - 3*b2) * q^70 + (-10*b6 + 3*b2) * q^71 + (b6 - 2*b2 + 3*b1) * q^72 + (-6*b11 - 4*b10 + 3*b8 + 4*b6 - 6*b2) * q^73 + (-b9 - 4*b7 + 5*b4 + 4) * q^74 + (2*b9 + 3*b7 + 2*b5 + 4*b4 - 2*b3 + 2) * q^75 + (2*b11 - 8*b10 - 5*b8 + 5*b1) * q^76 + (-5*b5 - 1) * q^77 + (9*b5 + b3 - 5) * q^79 + (3*b11 + 3*b10 + b8 - b1) * q^80 + (3*b9 - 3*b7 + 4*b5 + 7*b4 - 3*b3 + 3) * q^81 + (2*b9 + 3*b7 + 3*b4 - 3) * q^82 + (-9*b11 - 6*b10 - 2*b8 + 6*b6 - 9*b2) * q^83 + (-4*b2 - 3*b1) * q^84 + (-b6 - 3*b2 + b1) * q^85 + (-b11 + 7*b10 + 4*b8 - 7*b6 - b2) * q^86 + (-5*b7 - 3*b4 + 5) * q^87 + (5*b9 + 2*b7 - b5 + 4*b4 - 5*b3 + 5) * q^88 + (-7*b11 - 6*b10) * q^89 + (-3*b5 + 5) * q^90 + (4*b5 + 5*b3 - 3) * q^92 + (2*b11 - 5*b10 + 5*b8 - 5*b1) * q^93 + (2*b9 - 4*b7 + 3*b5 + 5*b4 - 2*b3 + 2) * q^94 + (-4*b9 + b4) * q^95 + (2*b11 - 5*b10 + 4*b8 + 5*b6 + 2*b2) * q^96 + (5*b6 - b2 - 7*b1) * q^97 + (-2*b6 - b2 + 2*b1) * q^98 + (-2*b11 - 3*b10 - 6*b8 + 3*b6 - 2*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{3} + 6 q^{9}+O(q^{10})$$ 12 * q + 4 * q^3 + 6 * q^9 $$12 q + 4 q^{3} + 6 q^{9} + 10 q^{10} - 20 q^{14} - 4 q^{16} - 4 q^{17} - 6 q^{22} - 10 q^{23} + 20 q^{25} + 4 q^{27} + 2 q^{29} - 2 q^{30} - 8 q^{35} - 14 q^{36} - 48 q^{38} - 12 q^{40} - 16 q^{42} + 26 q^{43} - 2 q^{48} - 8 q^{49} + 4 q^{51} + 4 q^{53} - 12 q^{55} - 8 q^{56} - 8 q^{61} + 2 q^{62} + 44 q^{64} + 20 q^{66} + 42 q^{68} - 12 q^{69} + 16 q^{74} + 30 q^{75} - 32 q^{77} - 20 q^{79} + 2 q^{81} - 28 q^{82} + 36 q^{87} + 30 q^{88} + 48 q^{90} - 10 q^{94} + 6 q^{95}+O(q^{100})$$ 12 * q + 4 * q^3 + 6 * q^9 + 10 * q^10 - 20 * q^14 - 4 * q^16 - 4 * q^17 - 6 * q^22 - 10 * q^23 + 20 * q^25 + 4 * q^27 + 2 * q^29 - 2 * q^30 - 8 * q^35 - 14 * q^36 - 48 * q^38 - 12 * q^40 - 16 * q^42 + 26 * q^43 - 2 * q^48 - 8 * q^49 + 4 * q^51 + 4 * q^53 - 12 * q^55 - 8 * q^56 - 8 * q^61 + 2 * q^62 + 44 * q^64 + 20 * q^66 + 42 * q^68 - 12 * q^69 + 16 * q^74 + 30 * q^75 - 32 * q^77 - 20 * q^79 + 2 * q^81 - 28 * q^82 + 36 * q^87 + 30 * q^88 + 48 * q^90 - 10 * q^94 + 6 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559$$ (-25*v^11 + 95*v^9 - 361*v^7 + 155*v^5 - 30*v^3 - 1563*v) / 559 $$\beta_{3}$$ $$=$$ $$( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559$$ (25*v^10 - 95*v^8 + 361*v^6 - 155*v^4 + 30*v^2 + 1004) / 559 $$\beta_{4}$$ $$=$$ $$( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43$$ (3*v^10 - 20*v^8 + 76*v^6 - 139*v^4 + 124*v^2 - 24) / 43 $$\beta_{5}$$ $$=$$ $$( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559$$ (45*v^10 - 171*v^8 + 538*v^6 - 279*v^4 + 54*v^2 + 242) / 559 $$\beta_{6}$$ $$=$$ $$( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559$$ (70*v^11 - 266*v^9 + 899*v^7 - 434*v^5 + 84*v^3 + 1246*v) / 559 $$\beta_{7}$$ $$=$$ $$( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559$$ (114*v^10 - 545*v^8 + 2071*v^6 - 2831*v^4 + 3379*v^2 - 95) / 559 $$\beta_{8}$$ $$=$$ $$( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559$$ (114*v^11 - 545*v^9 + 2071*v^7 - 2831*v^5 + 3379*v^3 - 95*v) / 559 $$\beta_{9}$$ $$=$$ $$( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559$$ (-128*v^10 + 710*v^8 - 2698*v^6 + 4483*v^4 - 4402*v^2 + 852) / 559 $$\beta_{10}$$ $$=$$ $$( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559$$ (-242*v^11 + 1255*v^9 - 4769*v^7 + 7314*v^5 - 7781*v^3 + 1506*v) / 559 $$\beta_{11}$$ $$=$$ $$( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559$$ (-317*v^11 + 1540*v^9 - 5852*v^7 + 8338*v^5 - 9548*v^3 + 1848*v) / 559
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1$$ b9 + b7 + b4 - b3 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{11} + 3\beta_{8} + \beta_{2}$$ b11 + 3*b8 + b2 $$\nu^{4}$$ $$=$$ $$3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2$$ 3*b9 + 2*b7 + 4*b4 - 2 $$\nu^{5}$$ $$=$$ $$4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1$$ 4*b11 - b10 + 9*b8 - 9*b1 $$\nu^{6}$$ $$=$$ $$-5\beta_{5} + 9\beta_{3} - 14$$ -5*b5 + 9*b3 - 14 $$\nu^{7}$$ $$=$$ $$-5\beta_{6} - 14\beta_{2} - 28\beta_1$$ -5*b6 - 14*b2 - 28*b1 $$\nu^{8}$$ $$=$$ $$-28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28$$ -28*b9 - 14*b7 - 19*b5 - 47*b4 + 28*b3 - 28 $$\nu^{9}$$ $$=$$ $$-47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2}$$ -47*b11 + 19*b10 - 89*b8 - 19*b6 - 47*b2 $$\nu^{10}$$ $$=$$ $$-89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42$$ -89*b9 - 42*b7 - 155*b4 + 42 $$\nu^{11}$$ $$=$$ $$-155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1$$ -155*b11 + 66*b10 - 286*b8 + 286*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$1 - \beta_{7}$$

## 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
 1.07992 + 0.623490i 1.56052 + 0.900969i −0.385418 − 0.222521i 0.385418 + 0.222521i −1.56052 − 0.900969i −1.07992 − 0.623490i 1.07992 − 0.623490i 1.56052 − 0.900969i −0.385418 + 0.222521i 0.385418 − 0.222521i −1.56052 + 0.900969i −1.07992 + 0.623490i
−1.94594 + 1.12349i 0.277479 + 0.480608i 1.52446 2.64044i 1.44504i −1.07992 0.623490i 1.77441 + 1.02446i 2.35690i 1.34601 2.33136i 1.62349 + 2.81197i
23.2 −0.694498 + 0.400969i 1.12349 + 1.94594i −0.678448 + 1.17511i 0.246980i −1.56052 0.900969i 2.04113 + 1.17845i 2.69202i −1.02446 + 1.77441i 0.0990311 + 0.171527i
23.3 −0.480608 + 0.277479i −0.400969 0.694498i −0.846011 + 1.46533i 2.80194i 0.385418 + 0.222521i −2.33136 1.34601i 2.04892i 1.17845 2.04113i 0.777479 + 1.34663i
23.4 0.480608 0.277479i −0.400969 0.694498i −0.846011 + 1.46533i 2.80194i −0.385418 0.222521i 2.33136 + 1.34601i 2.04892i 1.17845 2.04113i 0.777479 + 1.34663i
23.5 0.694498 0.400969i 1.12349 + 1.94594i −0.678448 + 1.17511i 0.246980i 1.56052 + 0.900969i −2.04113 1.17845i 2.69202i −1.02446 + 1.77441i 0.0990311 + 0.171527i
23.6 1.94594 1.12349i 0.277479 + 0.480608i 1.52446 2.64044i 1.44504i 1.07992 + 0.623490i −1.77441 1.02446i 2.35690i 1.34601 2.33136i 1.62349 + 2.81197i
147.1 −1.94594 1.12349i 0.277479 0.480608i 1.52446 + 2.64044i 1.44504i −1.07992 + 0.623490i 1.77441 1.02446i 2.35690i 1.34601 + 2.33136i 1.62349 2.81197i
147.2 −0.694498 0.400969i 1.12349 1.94594i −0.678448 1.17511i 0.246980i −1.56052 + 0.900969i 2.04113 1.17845i 2.69202i −1.02446 1.77441i 0.0990311 0.171527i
147.3 −0.480608 0.277479i −0.400969 + 0.694498i −0.846011 1.46533i 2.80194i 0.385418 0.222521i −2.33136 + 1.34601i 2.04892i 1.17845 + 2.04113i 0.777479 1.34663i
147.4 0.480608 + 0.277479i −0.400969 + 0.694498i −0.846011 1.46533i 2.80194i −0.385418 + 0.222521i 2.33136 1.34601i 2.04892i 1.17845 + 2.04113i 0.777479 1.34663i
147.5 0.694498 + 0.400969i 1.12349 1.94594i −0.678448 1.17511i 0.246980i 1.56052 0.900969i −2.04113 + 1.17845i 2.69202i −1.02446 1.77441i 0.0990311 0.171527i
147.6 1.94594 + 1.12349i 0.277479 0.480608i 1.52446 + 2.64044i 1.44504i 1.07992 0.623490i −1.77441 + 1.02446i 2.35690i 1.34601 + 2.33136i 1.62349 2.81197i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.6 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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.e.b 12
13.b even 2 1 inner 169.2.e.b 12
13.c even 3 1 169.2.b.b 6
13.c even 3 1 inner 169.2.e.b 12
13.d odd 4 1 169.2.c.b 6
13.d odd 4 1 169.2.c.c 6
13.e even 6 1 169.2.b.b 6
13.e even 6 1 inner 169.2.e.b 12
13.f odd 12 1 169.2.a.b 3
13.f odd 12 1 169.2.a.c yes 3
13.f odd 12 1 169.2.c.b 6
13.f odd 12 1 169.2.c.c 6
39.h odd 6 1 1521.2.b.l 6
39.i odd 6 1 1521.2.b.l 6
39.k even 12 1 1521.2.a.o 3
39.k even 12 1 1521.2.a.r 3
52.i odd 6 1 2704.2.f.o 6
52.j odd 6 1 2704.2.f.o 6
52.l even 12 1 2704.2.a.z 3
52.l even 12 1 2704.2.a.ba 3
65.s odd 12 1 4225.2.a.bb 3
65.s odd 12 1 4225.2.a.bg 3
91.bc even 12 1 8281.2.a.bf 3
91.bc even 12 1 8281.2.a.bj 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b 3 13.f odd 12 1
169.2.a.c yes 3 13.f odd 12 1
169.2.b.b 6 13.c even 3 1
169.2.b.b 6 13.e even 6 1
169.2.c.b 6 13.d odd 4 1
169.2.c.b 6 13.f odd 12 1
169.2.c.c 6 13.d odd 4 1
169.2.c.c 6 13.f odd 12 1
169.2.e.b 12 1.a even 1 1 trivial
169.2.e.b 12 13.b even 2 1 inner
169.2.e.b 12 13.c even 3 1 inner
169.2.e.b 12 13.e even 6 1 inner
1521.2.a.o 3 39.k even 12 1
1521.2.a.r 3 39.k even 12 1
1521.2.b.l 6 39.h odd 6 1
1521.2.b.l 6 39.i odd 6 1
2704.2.a.z 3 52.l even 12 1
2704.2.a.ba 3 52.l even 12 1
2704.2.f.o 6 52.i odd 6 1
2704.2.f.o 6 52.j odd 6 1
4225.2.a.bb 3 65.s odd 12 1
4225.2.a.bg 3 65.s odd 12 1
8281.2.a.bf 3 91.bc even 12 1
8281.2.a.bj 3 91.bc even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 6T_{2}^{10} + 31T_{2}^{8} - 28T_{2}^{6} + 19T_{2}^{4} - 5T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 6 T^{10} + \cdots + 1$$
$3$ $$(T^{6} - 2 T^{5} + 5 T^{4} + \cdots + 1)^{2}$$
$5$ $$(T^{6} + 10 T^{4} + 17 T^{2} + 1)^{2}$$
$7$ $$T^{12} - 17 T^{10} + \cdots + 28561$$
$11$ $$T^{12} - 26 T^{10} + \cdots + 28561$$
$13$ $$T^{12}$$
$17$ $$(T^{6} + 2 T^{5} + \cdots + 169)^{2}$$
$19$ $$T^{12} - 38 T^{10} + \cdots + 1$$
$23$ $$(T^{6} + 5 T^{5} + \cdots + 169)^{2}$$
$29$ $$(T^{6} - T^{5} + 45 T^{4} + \cdots + 6889)^{2}$$
$31$ $$(T^{6} + 97 T^{4} + \cdots + 27889)^{2}$$
$37$ $$T^{12} - 62 T^{10} + \cdots + 707281$$
$41$ $$T^{12} - 147 T^{10} + \cdots + 5764801$$
$43$ $$(T^{6} - 13 T^{5} + \cdots + 169)^{2}$$
$47$ $$(T^{6} + 122 T^{4} + \cdots + 27889)^{2}$$
$53$ $$(T^{3} - T^{2} - 86 T + 337)^{4}$$
$59$ $$T^{12} - 195 T^{10} + \cdots + 1$$
$61$ $$(T^{6} + 4 T^{5} + \cdots + 57121)^{2}$$
$67$ $$T^{12} - 145 T^{10} + \cdots + 2825761$$
$71$ $$T^{12} + \cdots + 89526025681$$
$73$ $$(T^{6} + 321 T^{4} + \cdots + 829921)^{2}$$
$79$ $$(T^{3} + 5 T^{2} + \cdots + 127)^{4}$$
$83$ $$(T^{6} + 329 T^{4} + \cdots + 41209)^{2}$$
$89$ $$T^{12} + \cdots + 6234839521$$
$97$ $$T^{12} + \cdots + 8208541201$$