# Properties

 Label 378.4.f.a Level $378$ Weight $4$ Character orbit 378.f Analytic conductor $22.303$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(127,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("378.127");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 378.f (of order $$3$$, 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: $$22.3027219822$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 126) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{3} q^{2} + (4 \beta_{3} - 4) q^{4} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 3) q^{5} - 7 \beta_{3} q^{7} + 8 q^{8}+O(q^{10})$$ q - 2*b3 * q^2 + (4*b3 - 4) * q^4 + (-b5 + b4 - 3*b3 - 2*b1 + 3) * q^5 - 7*b3 * q^7 + 8 * q^8 $$q - 2 \beta_{3} q^{2} + (4 \beta_{3} - 4) q^{4} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 3) q^{5} - 7 \beta_{3} q^{7} + 8 q^{8} + ( - 2 \beta_{4} + 4 \beta_{2} + 4 \beta_1 - 6) q^{10} + ( - 5 \beta_{5} + 16 \beta_{3} - 6 \beta_{2}) q^{11} + ( - 19 \beta_{3} + 18 \beta_1 + 19) q^{13} + (14 \beta_{3} - 14) q^{14} - 16 \beta_{3} q^{16} + ( - 2 \beta_{4} - 33 \beta_{2} - 33 \beta_1 - 8) q^{17} + (4 \beta_{4} + 29 \beta_{2} + 29 \beta_1 - 47) q^{19} + (4 \beta_{5} + 12 \beta_{3} - 8 \beta_{2}) q^{20} + (10 \beta_{5} - 10 \beta_{4} - 32 \beta_{3} - 12 \beta_1 + 32) q^{22} + ( - 17 \beta_{5} + 17 \beta_{4} + 10 \beta_{3} - 3 \beta_1 - 10) q^{23} + ( - 5 \beta_{5} + 70 \beta_{3} - \beta_{2}) q^{25} + ( - 36 \beta_{2} - 36 \beta_1 - 38) q^{26} + 28 q^{28} + ( - 35 \beta_{5} + 39 \beta_{3} + 43 \beta_{2}) q^{29} + (9 \beta_{5} - 9 \beta_{4} - 55 \beta_{3} - 36 \beta_1 + 55) q^{31} + (32 \beta_{3} - 32) q^{32} + (4 \beta_{5} + 16 \beta_{3} + 66 \beta_{2}) q^{34} + ( - 7 \beta_{4} + 14 \beta_{2} + 14 \beta_1 - 21) q^{35} + ( - 27 \beta_{4} - 90 \beta_{2} - 90 \beta_1 - 133) q^{37} + ( - 8 \beta_{5} + 94 \beta_{3} - 58 \beta_{2}) q^{38} + ( - 8 \beta_{5} + 8 \beta_{4} - 24 \beta_{3} - 16 \beta_1 + 24) q^{40} + ( - 19 \beta_{5} + 19 \beta_{4} + 128 \beta_{3} + 12 \beta_1 - 128) q^{41} + (5 \beta_{5} + 257 \beta_{3} - 62 \beta_{2}) q^{43} + (20 \beta_{4} + 24 \beta_{2} + 24 \beta_1 - 64) q^{44} + ( - 34 \beta_{4} + 6 \beta_{2} + 6 \beta_1 + 20) q^{46} + ( - 108 \beta_{5} + 27 \beta_{3} - 9 \beta_{2}) q^{47} + (49 \beta_{3} - 49) q^{49} + (10 \beta_{5} - 10 \beta_{4} - 140 \beta_{3} - 2 \beta_1 + 140) q^{50} + (76 \beta_{3} + 72 \beta_{2}) q^{52} + ( - 85 \beta_{4} - 145 \beta_{2} - 145 \beta_1 - 174) q^{53} + ( - 26 \beta_{4} - 67 \beta_{2} - 67 \beta_1 + 26) q^{55} - 56 \beta_{3} q^{56} + (70 \beta_{5} - 70 \beta_{4} - 78 \beta_{3} + 86 \beta_1 + 78) q^{58} + (55 \beta_{5} - 55 \beta_{4} - 7 \beta_{3} - 191 \beta_1 + 7) q^{59} + ( - 67 \beta_{5} + 260 \beta_{3} - 23 \beta_{2}) q^{61} + (18 \beta_{4} + 72 \beta_{2} + 72 \beta_1 - 110) q^{62} + 64 q^{64} + ( - 55 \beta_{5} + 177 \beta_{3} + 38 \beta_{2}) q^{65} + ( - 8 \beta_{5} + 8 \beta_{4} - 128 \beta_{3} - 263 \beta_1 + 128) q^{67} + ( - 8 \beta_{5} + 8 \beta_{4} - 32 \beta_{3} + 132 \beta_1 + 32) q^{68} + (14 \beta_{5} + 42 \beta_{3} - 28 \beta_{2}) q^{70} + ( - 155 \beta_{4} + 59 \beta_{2} + 59 \beta_1 - 19) q^{71} + ( - 88 \beta_{4} + 211 \beta_{2} + 211 \beta_1 - 39) q^{73} + (54 \beta_{5} + 266 \beta_{3} + 180 \beta_{2}) q^{74} + (16 \beta_{5} - 16 \beta_{4} - 188 \beta_{3} - 116 \beta_1 + 188) q^{76} + (35 \beta_{5} - 35 \beta_{4} - 112 \beta_{3} - 42 \beta_1 + 112) q^{77} + (68 \beta_{5} + 461 \beta_{3} + 19 \beta_{2}) q^{79} + ( - 16 \beta_{4} + 32 \beta_{2} + 32 \beta_1 - 48) q^{80} + ( - 38 \beta_{4} - 24 \beta_{2} - 24 \beta_1 + 256) q^{82} + (72 \beta_{5} + 4 \beta_{3} - 229 \beta_{2}) q^{83} + (86 \beta_{5} - 86 \beta_{4} - 365 \beta_{3} + 2 \beta_1 + 365) q^{85} + ( - 10 \beta_{5} + 10 \beta_{4} - 514 \beta_{3} - 124 \beta_1 + 514) q^{86} + ( - 40 \beta_{5} + 128 \beta_{3} - 48 \beta_{2}) q^{88} + ( - 59 \beta_{4} + 10 \beta_{2} + 10 \beta_1 + 216) q^{89} + ( - 126 \beta_{2} - 126 \beta_1 - 133) q^{91} + (68 \beta_{5} - 40 \beta_{3} - 12 \beta_{2}) q^{92} + (216 \beta_{5} - 216 \beta_{4} - 54 \beta_{3} - 18 \beta_1 + 54) q^{94} + ( - 35 \beta_{5} + 35 \beta_{4} + 438 \beta_{3} + 122 \beta_1 - 438) q^{95} + ( - 103 \beta_{5} - 247 \beta_{3} + 307 \beta_{2}) q^{97} + 98 q^{98}+O(q^{100})$$ q - 2*b3 * q^2 + (4*b3 - 4) * q^4 + (-b5 + b4 - 3*b3 - 2*b1 + 3) * q^5 - 7*b3 * q^7 + 8 * q^8 + (-2*b4 + 4*b2 + 4*b1 - 6) * q^10 + (-5*b5 + 16*b3 - 6*b2) * q^11 + (-19*b3 + 18*b1 + 19) * q^13 + (14*b3 - 14) * q^14 - 16*b3 * q^16 + (-2*b4 - 33*b2 - 33*b1 - 8) * q^17 + (4*b4 + 29*b2 + 29*b1 - 47) * q^19 + (4*b5 + 12*b3 - 8*b2) * q^20 + (10*b5 - 10*b4 - 32*b3 - 12*b1 + 32) * q^22 + (-17*b5 + 17*b4 + 10*b3 - 3*b1 - 10) * q^23 + (-5*b5 + 70*b3 - b2) * q^25 + (-36*b2 - 36*b1 - 38) * q^26 + 28 * q^28 + (-35*b5 + 39*b3 + 43*b2) * q^29 + (9*b5 - 9*b4 - 55*b3 - 36*b1 + 55) * q^31 + (32*b3 - 32) * q^32 + (4*b5 + 16*b3 + 66*b2) * q^34 + (-7*b4 + 14*b2 + 14*b1 - 21) * q^35 + (-27*b4 - 90*b2 - 90*b1 - 133) * q^37 + (-8*b5 + 94*b3 - 58*b2) * q^38 + (-8*b5 + 8*b4 - 24*b3 - 16*b1 + 24) * q^40 + (-19*b5 + 19*b4 + 128*b3 + 12*b1 - 128) * q^41 + (5*b5 + 257*b3 - 62*b2) * q^43 + (20*b4 + 24*b2 + 24*b1 - 64) * q^44 + (-34*b4 + 6*b2 + 6*b1 + 20) * q^46 + (-108*b5 + 27*b3 - 9*b2) * q^47 + (49*b3 - 49) * q^49 + (10*b5 - 10*b4 - 140*b3 - 2*b1 + 140) * q^50 + (76*b3 + 72*b2) * q^52 + (-85*b4 - 145*b2 - 145*b1 - 174) * q^53 + (-26*b4 - 67*b2 - 67*b1 + 26) * q^55 - 56*b3 * q^56 + (70*b5 - 70*b4 - 78*b3 + 86*b1 + 78) * q^58 + (55*b5 - 55*b4 - 7*b3 - 191*b1 + 7) * q^59 + (-67*b5 + 260*b3 - 23*b2) * q^61 + (18*b4 + 72*b2 + 72*b1 - 110) * q^62 + 64 * q^64 + (-55*b5 + 177*b3 + 38*b2) * q^65 + (-8*b5 + 8*b4 - 128*b3 - 263*b1 + 128) * q^67 + (-8*b5 + 8*b4 - 32*b3 + 132*b1 + 32) * q^68 + (14*b5 + 42*b3 - 28*b2) * q^70 + (-155*b4 + 59*b2 + 59*b1 - 19) * q^71 + (-88*b4 + 211*b2 + 211*b1 - 39) * q^73 + (54*b5 + 266*b3 + 180*b2) * q^74 + (16*b5 - 16*b4 - 188*b3 - 116*b1 + 188) * q^76 + (35*b5 - 35*b4 - 112*b3 - 42*b1 + 112) * q^77 + (68*b5 + 461*b3 + 19*b2) * q^79 + (-16*b4 + 32*b2 + 32*b1 - 48) * q^80 + (-38*b4 - 24*b2 - 24*b1 + 256) * q^82 + (72*b5 + 4*b3 - 229*b2) * q^83 + (86*b5 - 86*b4 - 365*b3 + 2*b1 + 365) * q^85 + (-10*b5 + 10*b4 - 514*b3 - 124*b1 + 514) * q^86 + (-40*b5 + 128*b3 - 48*b2) * q^88 + (-59*b4 + 10*b2 + 10*b1 + 216) * q^89 + (-126*b2 - 126*b1 - 133) * q^91 + (68*b5 - 40*b3 - 12*b2) * q^92 + (216*b5 - 216*b4 - 54*b3 - 18*b1 + 54) * q^94 + (-35*b5 + 35*b4 + 438*b3 + 122*b1 - 438) * q^95 + (-103*b5 - 247*b3 + 307*b2) * q^97 + 98 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{2} - 12 q^{4} + 9 q^{5} - 21 q^{7} + 48 q^{8}+O(q^{10})$$ 6 * q - 6 * q^2 - 12 * q^4 + 9 * q^5 - 21 * q^7 + 48 * q^8 $$6 q - 6 q^{2} - 12 q^{4} + 9 q^{5} - 21 q^{7} + 48 q^{8} - 36 q^{10} + 48 q^{11} + 57 q^{13} - 42 q^{14} - 48 q^{16} - 48 q^{17} - 282 q^{19} + 36 q^{20} + 96 q^{22} - 30 q^{23} + 210 q^{25} - 228 q^{26} + 168 q^{28} + 117 q^{29} + 165 q^{31} - 96 q^{32} + 48 q^{34} - 126 q^{35} - 798 q^{37} + 282 q^{38} + 72 q^{40} - 384 q^{41} + 771 q^{43} - 384 q^{44} + 120 q^{46} + 81 q^{47} - 147 q^{49} + 420 q^{50} + 228 q^{52} - 1044 q^{53} + 156 q^{55} - 168 q^{56} + 234 q^{58} + 21 q^{59} + 780 q^{61} - 660 q^{62} + 384 q^{64} + 531 q^{65} + 384 q^{67} + 96 q^{68} + 126 q^{70} - 114 q^{71} - 234 q^{73} + 798 q^{74} + 564 q^{76} + 336 q^{77} + 1383 q^{79} - 288 q^{80} + 1536 q^{82} + 12 q^{83} + 1095 q^{85} + 1542 q^{86} + 384 q^{88} + 1296 q^{89} - 798 q^{91} - 120 q^{92} + 162 q^{94} - 1314 q^{95} - 741 q^{97} + 588 q^{98}+O(q^{100})$$ 6 * q - 6 * q^2 - 12 * q^4 + 9 * q^5 - 21 * q^7 + 48 * q^8 - 36 * q^10 + 48 * q^11 + 57 * q^13 - 42 * q^14 - 48 * q^16 - 48 * q^17 - 282 * q^19 + 36 * q^20 + 96 * q^22 - 30 * q^23 + 210 * q^25 - 228 * q^26 + 168 * q^28 + 117 * q^29 + 165 * q^31 - 96 * q^32 + 48 * q^34 - 126 * q^35 - 798 * q^37 + 282 * q^38 + 72 * q^40 - 384 * q^41 + 771 * q^43 - 384 * q^44 + 120 * q^46 + 81 * q^47 - 147 * q^49 + 420 * q^50 + 228 * q^52 - 1044 * q^53 + 156 * q^55 - 168 * q^56 + 234 * q^58 + 21 * q^59 + 780 * q^61 - 660 * q^62 + 384 * q^64 + 531 * q^65 + 384 * q^67 + 96 * q^68 + 126 * q^70 - 114 * q^71 - 234 * q^73 + 798 * q^74 + 564 * q^76 + 336 * q^77 + 1383 * q^79 - 288 * q^80 + 1536 * q^82 + 12 * q^83 + 1095 * q^85 + 1542 * q^86 + 384 * q^88 + 1296 * q^89 - 798 * q^91 - 120 * q^92 + 162 * q^94 - 1314 * q^95 - 741 * q^97 + 588 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} - \nu^{4} + 5\nu^{3} - 2\nu^{2} ) / 3$$ (v^5 - v^4 + 5*v^3 - 2*v^2) / 3 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 4\nu^{4} - 11\nu^{3} + 17\nu^{2} - 12\nu + 3 ) / 3$$ (-v^5 + 4*v^4 - 11*v^3 + 17*v^2 - 12*v + 3) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 16\nu^{3} - 19\nu^{2} + 21\nu - 6 ) / 3$$ (2*v^5 - 5*v^4 + 16*v^3 - 19*v^2 + 21*v - 6) / 3 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2\nu^{3} + 8\nu^{2} - 7\nu + 8$$ v^4 - 2*v^3 + 8*v^2 - 7*v + 8 $$\beta_{5}$$ $$=$$ $$( 7\nu^{5} - 16\nu^{4} + 62\nu^{3} - 68\nu^{2} + 99\nu - 30 ) / 3$$ (7*v^5 - 16*v^4 + 62*v^3 - 68*v^2 + 99*v - 30) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3$$ (b3 + b2 - b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{4} + \beta_{3} - 2\beta _1 - 6 ) / 3$$ (b4 + b3 - 2*b1 - 6) / 3 $$\nu^{3}$$ $$=$$ $$( \beta_{5} + \beta_{4} - 6\beta_{3} - 4\beta_{2} + \beta _1 - 6 ) / 3$$ (b5 + b4 - 6*b3 - 4*b2 + b1 - 6) / 3 $$\nu^{4}$$ $$=$$ $$( 2\beta_{5} - 3\beta_{4} - 13\beta_{3} - \beta_{2} + 11\beta _1 + 19 ) / 3$$ (2*b5 - 3*b4 - 13*b3 - b2 + 11*b1 + 19) / 3 $$\nu^{5}$$ $$=$$ $$( -3\beta_{5} - 6\beta_{4} + 19\beta_{3} + 19\beta_{2} + 11\beta _1 + 37 ) / 3$$ (-3*b5 - 6*b4 + 19*b3 + 19*b2 + 11*b1 + 37) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-1 + \beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 − 2.05195i 0.5 + 1.41036i 0.5 − 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i 0.5 + 0.224437i
−1.00000 + 1.73205i 0 −2.00000 3.46410i −3.21780 5.57339i 0 −3.50000 + 6.06218i 8.00000 0 12.8712
127.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 3.11273 + 5.39140i 0 −3.50000 + 6.06218i 8.00000 0 −12.4509
127.3 −1.00000 + 1.73205i 0 −2.00000 3.46410i 4.60507 + 7.97622i 0 −3.50000 + 6.06218i 8.00000 0 −18.4203
253.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −3.21780 + 5.57339i 0 −3.50000 6.06218i 8.00000 0 12.8712
253.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 3.11273 5.39140i 0 −3.50000 6.06218i 8.00000 0 −12.4509
253.3 −1.00000 1.73205i 0 −2.00000 + 3.46410i 4.60507 7.97622i 0 −3.50000 6.06218i 8.00000 0 −18.4203
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.f.a 6
3.b odd 2 1 126.4.f.a 6
9.c even 3 1 inner 378.4.f.a 6
9.c even 3 1 1134.4.a.h 3
9.d odd 6 1 126.4.f.a 6
9.d odd 6 1 1134.4.a.g 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.f.a 6 3.b odd 2 1
126.4.f.a 6 9.d odd 6 1
378.4.f.a 6 1.a even 1 1 trivial
378.4.f.a 6 9.c even 3 1 inner
1134.4.a.g 3 9.d odd 6 1
1134.4.a.h 3 9.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 9T_{5}^{5} + 123T_{5}^{4} - 360T_{5}^{3} + 5085T_{5}^{2} - 15498T_{5} + 136161$$ acting on $$S_{4}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 4)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 9 T^{5} + 123 T^{4} + \cdots + 136161$$
$7$ $$(T^{2} + 7 T + 49)^{3}$$
$11$ $$T^{6} - 48 T^{5} + 2445 T^{4} + \cdots + 79548561$$
$13$ $$T^{6} - 57 T^{5} + \cdots + 1824400369$$
$17$ $$(T^{3} + 24 T^{2} - 9519 T - 157239)^{2}$$
$19$ $$(T^{3} + 141 T^{2} - 1026 T - 106169)^{2}$$
$23$ $$T^{6} + 30 T^{5} + \cdots + 104760326889$$
$29$ $$T^{6} - 117 T^{5} + \cdots + 78328896129$$
$31$ $$T^{6} - 165 T^{5} + \cdots + 559369801$$
$37$ $$(T^{3} + 399 T^{2} - 32226 T - 18438443)^{2}$$
$41$ $$T^{6} + 384 T^{5} + \cdots + 264917119401$$
$43$ $$T^{6} - 771 T^{5} + \cdots + 40798917084409$$
$47$ $$T^{6} - 81 T^{5} + \cdots + 57\!\cdots\!61$$
$53$ $$(T^{3} + 522 T^{2} - 256497 T - 106645401)^{2}$$
$59$ $$T^{6} - 21 T^{5} + \cdots + 71\!\cdots\!61$$
$61$ $$T^{6} + \cdots + 837556418281489$$
$67$ $$T^{6} - 384 T^{5} + \cdots + 16\!\cdots\!69$$
$71$ $$(T^{3} + 57 T^{2} - 706356 T + 141832269)^{2}$$
$73$ $$(T^{3} + 117 T^{2} - 660918 T - 225440003)^{2}$$
$79$ $$T^{6} - 1383 T^{5} + \cdots + 32\!\cdots\!89$$
$83$ $$T^{6} - 12 T^{5} + \cdots + 43\!\cdots\!89$$
$89$ $$(T^{3} - 648 T^{2} + 43311 T + 21052413)^{2}$$
$97$ $$T^{6} + 741 T^{5} + \cdots + 66\!\cdots\!09$$