# Properties

 Label 475.2.j.b Level $475$ Weight $2$ Character orbit 475.j Analytic conductor $3.793$ 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] = [475,2,Mod(49,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

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

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

chi := DirichletCharacter("475.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.j (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: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.50712647503417344.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 13x^{10} + 119x^{8} - 552x^{6} + 1863x^{4} - 2450x^{2} + 2401$$ x^12 - 13*x^10 + 119*x^8 - 552*x^6 + 1863*x^4 - 2450*x^2 + 2401 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{11} q^{3} + ( - \beta_{6} - \beta_{4} + 3 \beta_{3} + \cdots + 2) q^{4}+ \cdots + (\beta_{5} + \beta_{3} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 - b11 * q^3 + (-b6 - b4 + 3*b3 - b2 + 2) * q^4 + (-2*b6 + b5 - 2*b4 + 3*b3 - 2*b2 + 1) * q^6 + (b11 + b10 + b8) * q^7 + (-b11 - b10 - b9 + 2*b8 + b1) * q^8 + (b5 + b3 + 1) * q^9 $$q + \beta_1 q^{2} - \beta_{11} q^{3} + ( - \beta_{6} - \beta_{4} + 3 \beta_{3} + \cdots + 2) q^{4}+ \cdots + (3 \beta_{6} - 4 \beta_{5} + 3 \beta_{4} + \cdots - 2) q^{99}+O(q^{100})$$ q + b1 * q^2 - b11 * q^3 + (-b6 - b4 + 3*b3 - b2 + 2) * q^4 + (-2*b6 + b5 - 2*b4 + 3*b3 - 2*b2 + 1) * q^6 + (b11 + b10 + b8) * q^7 + (-b11 - b10 - b9 + 2*b8 + b1) * q^8 + (b5 + b3 + 1) * q^9 + (b6 - b5 + b4 - 1) * q^11 + (-b11 - b10 - 2*b9 - b8 + 2*b1) * q^12 + (-5*b8 + 5*b7) * q^13 + (-3*b3 + 2*b2) * q^14 + (-2*b6 + 2*b3 - b2) * q^16 + (-2*b11 - 2*b7 - 3*b1) * q^17 + (-b11 - b10 - 5*b8) * q^18 + (-3*b6 + b5 - 2*b4 + b3 - b2 - 1) * q^19 + (-b6 - 4*b3 + b2) * q^21 + (2*b11 + 3*b7 - b1) * q^22 + (2*b10 - 3*b9 + 3*b8 - 3*b7) * q^23 + 7*b3 * q^24 + (5*b6 - 5*b5) * q^26 + (b9 - 3*b8 - b1) * q^27 + (2*b10 + b9 + 4*b8 - 4*b7) * q^28 + (b6 - b5 + b4 - b3 + b2) * q^29 + (b6 - b5) * q^31 + (2*b10 - b9 + 3*b8 - 3*b7) * q^32 + (2*b11 + 4*b7 - b1) * q^33 + (-b6 + 4*b5 - b4 - 9*b3 - b2 - 10) * q^34 + (2*b6 + b3 - 2*b2) * q^36 + (-5*b11 - 5*b10 + 2*b9 - 3*b8 - 2*b1) * q^37 + (-3*b11 - 2*b9 + 7*b8 - 8*b7) * q^38 + (5*b6 - 5*b5 + 5*b4 + 5) * q^39 + (2*b6 - b3 - b2) * q^41 + (3*b10 + 2*b9 + 8*b8 - 8*b7) * q^42 + (b11 + b7 + 3*b1) * q^43 + (3*b6 - 3*b5 + 3*b4 - 7*b3 + 3*b2 - 4) * q^44 + (-5*b6 + 5*b5 - b4 - 10) * q^46 + (b10 - 2*b9 - b8 + b7) * q^47 + (2*b10 - 3*b9 + 2*b8 - 2*b7) * q^48 + (3*b6 - 3*b5 + 2*b4 + 4) * q^49 + (4*b6 + b5 + 4*b4 - b3 + 4*b2 + 3) * q^51 + (5*b11 + 15*b7 + 5*b1) * q^52 + (-4*b10 + b9 + 2*b8 - 2*b7) * q^53 + (4*b6 - 5*b3 + b2) * q^54 + (-2*b6 + 2*b5 - b4 + 4) * q^56 + (b11 + 3*b10 - 3*b9 + 2*b8 - 7*b7 + 3*b1) * q^57 + (2*b11 + 2*b10 + 3*b8) * q^58 + (-2*b6 - b3 - b2) * q^59 + (-2*b6 - 2*b5 - 2*b4 + 5*b3 - 2*b2 + 3) * q^61 + (b11 + 5*b7 + b1) * q^62 + (3*b10 + 4*b8 - 4*b7) * q^63 + (-3*b6 + 3*b5 - 5*b4) * q^64 + (5*b6 - 6*b5 + 5*b4 - 11*b3 + 5*b2 - 6) * q^66 + (2*b10 - 2*b9 + 8*b8 - 8*b7) * q^67 + (-b11 - b10 + 7*b9 - 14*b8 - 7*b1) * q^68 + (-2*b6 + 2*b5 + 3*b4 + 2) * q^69 + (b6 - 11*b3 + 3*b2) * q^71 + (-4*b10 + 3*b9 - 6*b8 + 6*b7) * q^72 + (-2*b11 - 9*b7 - 3*b1) * q^73 + (5*b3 - 8*b2) * q^74 + (-7*b6 + 7*b5 - 2*b4 + 7*b3 - 2*b2 - 3) * q^76 + (-3*b11 - 3*b10 - b9 - 6*b8 + b1) * q^77 + (10*b11 + 15*b7 + 5*b1) * q^78 + (4*b6 + 6*b3 + 2*b2) * q^79 + (-2*b6 - 6*b3 - b2) * q^81 + (-4*b10 + 4*b9 - 13*b8 + 13*b7) * q^82 + (-3*b9 + 2*b8 + 3*b1) * q^83 + (-7*b6 + 7*b5 - 6*b4 + 5) * q^84 + (-b6 - 2*b5 - b4 + 12*b3 - b2 + 11) * q^86 + (b11 + b10 + b9 + 4*b8 - b1) * q^87 + (2*b11 + 2*b10 + 2*b9 + 3*b8 - 2*b1) * q^88 + (-4*b6 - 4*b4 - 2*b3 - 4*b2 - 6) * q^89 + (-5*b6 + 5*b5 - 5*b4 + 5*b3 - 5*b2) * q^91 + (-2*b11 - 17*b7 - 8*b1) * q^92 + (2*b11 + 3*b7 + b1) * q^93 - 7 * q^94 + (-4*b6 + 4*b5 - b4 + 4) * q^96 + (3*b11 + 4*b1) * q^97 + (5*b11 + 11*b7 + 5*b1) * q^98 + (3*b6 - 4*b5 + 3*b4 - 5*b3 + 3*b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 14 q^{4} + 12 q^{6} + 8 q^{9}+O(q^{10})$$ 12 * q + 14 * q^4 + 12 * q^6 + 8 * q^9 $$12 q + 14 q^{4} + 12 q^{6} + 8 q^{9} - 20 q^{11} + 14 q^{14} - 6 q^{16} + 24 q^{21} - 42 q^{24} - 20 q^{26} - 4 q^{29} - 4 q^{31} - 50 q^{34} - 6 q^{36} + 20 q^{39} + 4 q^{41} - 36 q^{44} - 96 q^{46} + 28 q^{49} + 12 q^{51} + 20 q^{54} + 60 q^{56} + 12 q^{59} + 18 q^{61} + 32 q^{64} - 58 q^{66} + 20 q^{69} + 58 q^{71} - 14 q^{74} - 38 q^{76} - 48 q^{79} + 42 q^{81} + 112 q^{84} + 64 q^{86} - 28 q^{89} + 20 q^{91} - 84 q^{94} + 68 q^{96} - 26 q^{99}+O(q^{100})$$ 12 * q + 14 * q^4 + 12 * q^6 + 8 * q^9 - 20 * q^11 + 14 * q^14 - 6 * q^16 + 24 * q^21 - 42 * q^24 - 20 * q^26 - 4 * q^29 - 4 * q^31 - 50 * q^34 - 6 * q^36 + 20 * q^39 + 4 * q^41 - 36 * q^44 - 96 * q^46 + 28 * q^49 + 12 * q^51 + 20 * q^54 + 60 * q^56 + 12 * q^59 + 18 * q^61 + 32 * q^64 - 58 * q^66 + 20 * q^69 + 58 * q^71 - 14 * q^74 - 38 * q^76 - 48 * q^79 + 42 * q^81 + 112 * q^84 + 64 * q^86 - 28 * q^89 + 20 * q^91 - 84 * q^94 + 68 * q^96 - 26 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 13x^{10} + 119x^{8} - 552x^{6} + 1863x^{4} - 2450x^{2} + 2401$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 736\nu^{10} - 31772\nu^{8} + 290836\nu^{6} - 1755683\nu^{4} + 4553172\nu^{2} - 5987800 ) / 1328929$$ (736*v^10 - 31772*v^8 + 290836*v^6 - 1755683*v^4 + 4553172*v^2 - 5987800) / 1328929 $$\beta_{3}$$ $$=$$ $$( 850\nu^{10} - 9867\nu^{8} + 90321\nu^{6} - 370073\nu^{4} + 1414017\nu^{2} - 1859550 ) / 1328929$$ (850*v^10 - 9867*v^8 + 90321*v^6 - 370073*v^4 + 1414017*v^2 - 1859550) / 1328929 $$\beta_{4}$$ $$=$$ $$( 169\nu^{10} - 1547\nu^{8} + 14161\nu^{6} - 24219\nu^{4} + 31850\nu^{2} + 467838 ) / 189847$$ (169*v^10 - 1547*v^8 + 14161*v^6 - 24219*v^4 + 31850*v^2 + 467838) / 189847 $$\beta_{5}$$ $$=$$ $$( -200\nu^{10} + 3917\nu^{8} - 21252\nu^{6} + 87076\nu^{4} + 37412\nu^{2} + 124852 ) / 189847$$ (-200*v^10 + 3917*v^8 - 21252*v^6 + 87076*v^4 + 37412*v^2 + 124852) / 189847 $$\beta_{6}$$ $$=$$ $$( 9\nu^{10} - 26\nu^{8} + 238\nu^{6} + 289\nu^{4} + 3726\nu^{2} - 4900 ) / 5131$$ (9*v^10 - 26*v^8 + 238*v^6 + 289*v^4 + 3726*v^2 - 4900) / 5131 $$\beta_{7}$$ $$=$$ $$( 52\nu^{11} - 476\nu^{9} + 2271\nu^{7} - 7452\nu^{5} + 9800\nu^{3} - 164812\nu ) / 189847$$ (52*v^11 - 476*v^9 + 2271*v^7 - 7452*v^5 + 9800*v^3 - 164812*v) / 189847 $$\beta_{8}$$ $$=$$ $$( -100\nu^{11} + 859\nu^{9} - 10626\nu^{7} + 43538\nu^{5} - 200461\nu^{3} + 62426\nu ) / 251419$$ (-100*v^11 + 859*v^9 - 10626*v^7 + 43538*v^5 - 200461*v^3 + 62426*v) / 251419 $$\beta_{9}$$ $$=$$ $$( -850\nu^{11} + 9867\nu^{9} - 90321\nu^{7} + 370073\nu^{5} - 1414017\nu^{3} + 1859550\nu ) / 1328929$$ (-850*v^11 + 9867*v^9 - 90321*v^7 + 370073*v^5 - 1414017*v^3 + 1859550*v) / 1328929 $$\beta_{10}$$ $$=$$ $$( 8973\nu^{11} - 159328\nu^{9} + 1458464\nu^{7} - 7813716\nu^{5} + 22832928\nu^{3} - 30027200\nu ) / 9302503$$ (8973*v^11 - 159328*v^9 + 1458464*v^7 - 7813716*v^5 + 22832928*v^3 - 30027200*v) / 9302503 $$\beta_{11}$$ $$=$$ $$( 273\nu^{11} - 2499\nu^{9} + 18703\nu^{7} - 39123\nu^{5} + 51450\nu^{3} + 328061\nu ) / 189847$$ (273*v^11 - 2499*v^9 + 18703*v^7 - 39123*v^5 + 51450*v^3 + 328061*v) / 189847
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{6} - \beta_{4} + 5\beta_{3} - \beta_{2} + 4$$ -b6 - b4 + 5*b3 - b2 + 4 $$\nu^{3}$$ $$=$$ $$-\beta_{11} - \beta_{10} - 5\beta_{9} + 2\beta_{8} + 5\beta_1$$ -b11 - b10 - 5*b9 + 2*b8 + 5*b1 $$\nu^{4}$$ $$=$$ $$-8\beta_{6} + 28\beta_{3} - 7\beta_{2}$$ -8*b6 + 28*b3 - 7*b2 $$\nu^{5}$$ $$=$$ $$-6\beta_{10} - 29\beta_{9} + 19\beta_{8} - 19\beta_{7}$$ -6*b10 - 29*b9 + 19*b8 - 19*b7 $$\nu^{6}$$ $$=$$ $$-13\beta_{6} + 13\beta_{5} + 41\beta_{4} - 122$$ -13*b6 + 13*b5 + 41*b4 - 122 $$\nu^{7}$$ $$=$$ $$28\beta_{11} - 147\beta_{7} - 176\beta_1$$ 28*b11 - 147*b7 - 176*b1 $$\nu^{8}$$ $$=$$ $$232\beta_{6} + 119\beta_{5} + 232\beta_{4} - 964\beta_{3} + 232\beta_{2} - 732$$ 232*b6 + 119*b5 + 232*b4 - 964*b3 + 232*b2 - 732 $$\nu^{9}$$ $$=$$ $$113\beta_{11} + 113\beta_{10} + 1083\beta_{9} - 1059\beta_{8} - 1083\beta_1$$ 113*b11 + 113*b10 + 1083*b9 - 1059*b8 - 1083*b1 $$\nu^{10}$$ $$=$$ $$2255\beta_{6} - 5754\beta_{3} + 1309\beta_{2}$$ 2255*b6 - 5754*b3 + 1309*b2 $$\nu^{11}$$ $$=$$ $$363\beta_{10} + 6700\beta_{9} - 7348\beta_{8} + 7348\beta_{7}$$ 363*b10 + 6700*b9 - 7348*b8 + 7348*b7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$-1 - \beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −2.17114 − 1.25351i −1.97899 − 1.14257i −1.05818 − 0.610938i 1.05818 + 0.610938i 1.97899 + 1.14257i 2.17114 + 1.25351i −2.17114 + 1.25351i −1.97899 + 1.14257i −1.05818 + 0.610938i 1.05818 − 0.610938i 1.97899 − 1.14257i 2.17114 − 1.25351i
−2.17114 1.25351i −1.05818 0.610938i 2.14257 + 3.71104i 0 1.53163 + 2.65287i 0.221876i 5.72889i −0.753509 1.30512i 0
49.2 −1.97899 1.14257i −2.17114 1.25351i 1.61094 + 2.79023i 0 2.86445 + 4.96137i 3.50702i 2.79216i 1.64257 + 2.84502i 0
49.3 −1.05818 0.610938i 1.97899 + 1.14257i −0.253509 0.439091i 0 −1.39608 2.41808i 1.28514i 3.06327i 1.11094 + 1.92420i 0
49.4 1.05818 + 0.610938i −1.97899 1.14257i −0.253509 0.439091i 0 −1.39608 2.41808i 1.28514i 3.06327i 1.11094 + 1.92420i 0
49.5 1.97899 + 1.14257i 2.17114 + 1.25351i 1.61094 + 2.79023i 0 2.86445 + 4.96137i 3.50702i 2.79216i 1.64257 + 2.84502i 0
49.6 2.17114 + 1.25351i 1.05818 + 0.610938i 2.14257 + 3.71104i 0 1.53163 + 2.65287i 0.221876i 5.72889i −0.753509 1.30512i 0
349.1 −2.17114 + 1.25351i −1.05818 + 0.610938i 2.14257 3.71104i 0 1.53163 2.65287i 0.221876i 5.72889i −0.753509 + 1.30512i 0
349.2 −1.97899 + 1.14257i −2.17114 + 1.25351i 1.61094 2.79023i 0 2.86445 4.96137i 3.50702i 2.79216i 1.64257 2.84502i 0
349.3 −1.05818 + 0.610938i 1.97899 1.14257i −0.253509 + 0.439091i 0 −1.39608 + 2.41808i 1.28514i 3.06327i 1.11094 1.92420i 0
349.4 1.05818 0.610938i −1.97899 + 1.14257i −0.253509 + 0.439091i 0 −1.39608 + 2.41808i 1.28514i 3.06327i 1.11094 1.92420i 0
349.5 1.97899 1.14257i 2.17114 1.25351i 1.61094 2.79023i 0 2.86445 4.96137i 3.50702i 2.79216i 1.64257 2.84502i 0
349.6 2.17114 1.25351i 1.05818 0.610938i 2.14257 3.71104i 0 1.53163 2.65287i 0.221876i 5.72889i −0.753509 + 1.30512i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.j.b 12
5.b even 2 1 inner 475.2.j.b 12
5.c odd 4 1 95.2.e.b 6
5.c odd 4 1 475.2.e.d 6
15.e even 4 1 855.2.k.g 6
19.c even 3 1 inner 475.2.j.b 12
20.e even 4 1 1520.2.q.j 6
95.i even 6 1 inner 475.2.j.b 12
95.l even 12 1 1805.2.a.g 3
95.l even 12 1 9025.2.a.ba 3
95.m odd 12 1 95.2.e.b 6
95.m odd 12 1 475.2.e.d 6
95.m odd 12 1 1805.2.a.h 3
95.m odd 12 1 9025.2.a.z 3
285.v even 12 1 855.2.k.g 6
380.v even 12 1 1520.2.q.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.b 6 5.c odd 4 1
95.2.e.b 6 95.m odd 12 1
475.2.e.d 6 5.c odd 4 1
475.2.e.d 6 95.m odd 12 1
475.2.j.b 12 1.a even 1 1 trivial
475.2.j.b 12 5.b even 2 1 inner
475.2.j.b 12 19.c even 3 1 inner
475.2.j.b 12 95.i even 6 1 inner
855.2.k.g 6 15.e even 4 1
855.2.k.g 6 285.v even 12 1
1520.2.q.j 6 20.e even 4 1
1520.2.q.j 6 380.v even 12 1
1805.2.a.g 3 95.l even 12 1
1805.2.a.h 3 95.m odd 12 1
9025.2.a.z 3 95.m odd 12 1
9025.2.a.ba 3 95.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 13T_{2}^{10} + 119T_{2}^{8} - 552T_{2}^{6} + 1863T_{2}^{4} - 2450T_{2}^{2} + 2401$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 13 T^{10} + \cdots + 2401$$
$3$ $$T^{12} - 13 T^{10} + \cdots + 2401$$
$5$ $$T^{12}$$
$7$ $$(T^{6} + 14 T^{4} + 21 T^{2} + 1)^{2}$$
$11$ $$(T^{3} + 5 T^{2} + 2 T - 1)^{4}$$
$13$ $$(T^{4} - 25 T^{2} + 625)^{3}$$
$17$ $$T^{12} - 89 T^{10} + \cdots + 2401$$
$19$ $$(T^{6} - 133 T^{3} + 6859)^{2}$$
$23$ $$T^{12} - 94 T^{10} + \cdots + 5764801$$
$29$ $$(T^{6} + 2 T^{5} + 9 T^{4} + \cdots + 1)^{2}$$
$31$ $$(T^{3} + T^{2} - 6 T - 7)^{4}$$
$37$ $$(T^{6} + 242 T^{4} + \cdots + 51529)^{2}$$
$41$ $$(T^{6} - 2 T^{5} + \cdots + 1369)^{2}$$
$43$ $$T^{12} + \cdots + 214358881$$
$47$ $$T^{12} - 50 T^{10} + \cdots + 5764801$$
$53$ $$T^{12} + \cdots + 9354951841$$
$59$ $$(T^{6} - 6 T^{5} + \cdots + 2401)^{2}$$
$61$ $$(T^{6} - 9 T^{5} + \cdots + 2401)^{2}$$
$67$ $$T^{12} - 184 T^{10} + \cdots + 59969536$$
$71$ $$(T^{6} - 29 T^{5} + \cdots + 218089)^{2}$$
$73$ $$T^{12} - 250 T^{10} + \cdots + 35153041$$
$79$ $$(T^{6} + 24 T^{5} + \cdots + 61504)^{2}$$
$83$ $$(T^{6} + 117 T^{4} + \cdots + 5929)^{2}$$
$89$ $$(T^{6} + 14 T^{5} + \cdots + 3136)^{2}$$
$97$ $$T^{12} + \cdots + 214358881$$