# Properties

 Label 441.2.p.c Level $441$ Weight $2$ Character orbit 441.p Analytic conductor $3.521$ Analytic rank $0$ Dimension $16$ Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(80,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("441.80");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.p (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: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{48})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - x^{8} + 1$$ x^16 - x^8 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}\cdot 7^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - \beta_{3} + \beta_1) q^{2} + (2 \beta_{8} - 2 \beta_{4} - \beta_{2} + 1) q^{4} + ( - \beta_{13} + \beta_{10}) q^{5} + ( - \beta_{6} + \beta_{5} - 3 \beta_{3}) q^{8}+O(q^{10})$$ q + (b5 - b3 + b1) * q^2 + (2*b8 - 2*b4 - b2 + 1) * q^4 + (-b13 + b10) * q^5 + (-b6 + b5 - 3*b3) * q^8 $$q + (\beta_{5} - \beta_{3} + \beta_1) q^{2} + (2 \beta_{8} - 2 \beta_{4} - \beta_{2} + 1) q^{4} + ( - \beta_{13} + \beta_{10}) q^{5} + ( - \beta_{6} + \beta_{5} - 3 \beta_{3}) q^{8} + ( - \beta_{9} - \beta_{7}) q^{10} + ( - 2 \beta_{6} + 2 \beta_1) q^{11} + ( - \beta_{12} + \beta_{9}) q^{13} - 3 \beta_{2} q^{16} + (\beta_{15} + \beta_{14} + \beta_{10}) q^{17} + 2 \beta_{12} q^{19} + (2 \beta_{15} - \beta_{13}) q^{20} - 2 q^{22} + ( - 2 \beta_{3} + 2 \beta_1) q^{23} + ( - \beta_{8} + \beta_{4} + 5 \beta_{2} - 5) q^{25} + (\beta_{14} + \beta_{13} - \beta_{10}) q^{26} + ( - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3}) q^{29} + 2 \beta_{9} q^{31} + ( - \beta_{6} + 3 \beta_1) q^{32} + ( - \beta_{12} + \beta_{9}) q^{34} + ( - \beta_{4} + 4 \beta_{2}) q^{37} + ( - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{10}) q^{38} + ( - 3 \beta_{12} + \beta_{11} - \beta_{7}) q^{40} + ( - \beta_{15} + \beta_{13}) q^{41} + ( - 6 \beta_{8} - 4) q^{43} + (2 \beta_{5} + 6 \beta_{3} - 6 \beta_1) q^{44} + (2 \beta_{8} - 2 \beta_{4} - 2 \beta_{2} + 2) q^{46} + ( - 2 \beta_{13} + 2 \beta_{10}) q^{47} + (6 \beta_{6} - 6 \beta_{5} + 7 \beta_{3}) q^{50} + (\beta_{9} + 2 \beta_{7}) q^{52} + 5 \beta_{6} q^{53} + (2 \beta_{12} + 2 \beta_{11} - 2 \beta_{9}) q^{55} + ( - 4 \beta_{4} - 6 \beta_{2}) q^{58} - 2 \beta_{10} q^{59} + ( - \beta_{12} + \beta_{11} - \beta_{7}) q^{61} + ( - 2 \beta_{15} + 2 \beta_{13}) q^{62} + (2 \beta_{8} + 7) q^{64} + (\beta_{5} - 10 \beta_{3} + 10 \beta_1) q^{65} + ( - 6 \beta_{8} + 6 \beta_{4}) q^{67} + ( - \beta_{14} + 3 \beta_{13} - 3 \beta_{10}) q^{68} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3}) q^{71} + ( - \beta_{9} + \beta_{7}) q^{73} + (3 \beta_{6} + 2 \beta_1) q^{74} + (2 \beta_{12} - 4 \beta_{11} - 2 \beta_{9}) q^{76} + (4 \beta_{4} + 4 \beta_{2}) q^{79} - 3 \beta_{10} q^{80} + (3 \beta_{12} - 2 \beta_{11} + 2 \beta_{7}) q^{82} - 4 \beta_{13} q^{83} + (9 \beta_{8} - 8) q^{85} + ( - 10 \beta_{5} + 16 \beta_{3} - 16 \beta_1) q^{86} + ( - 4 \beta_{8} + 4 \beta_{4} + \cdots + 2) q^{88}+ \cdots + (\beta_{12} + \beta_{11} - \beta_{9}) q^{97}+O(q^{100})$$ q + (b5 - b3 + b1) * q^2 + (2*b8 - 2*b4 - b2 + 1) * q^4 + (-b13 + b10) * q^5 + (-b6 + b5 - 3*b3) * q^8 + (-b9 - b7) * q^10 + (-2*b6 + 2*b1) * q^11 + (-b12 + b9) * q^13 - 3*b2 * q^16 + (b15 + b14 + b10) * q^17 + 2*b12 * q^19 + (2*b15 - b13) * q^20 - 2 * q^22 + (-2*b3 + 2*b1) * q^23 + (-b8 + b4 + 5*b2 - 5) * q^25 + (b14 + b13 - b10) * q^26 + (-2*b6 + 2*b5 - 2*b3) * q^29 + 2*b9 * q^31 + (-b6 + 3*b1) * q^32 + (-b12 + b9) * q^34 + (-b4 + 4*b2) * q^37 + (-2*b15 - 2*b14 + 2*b10) * q^38 + (-3*b12 + b11 - b7) * q^40 + (-b15 + b13) * q^41 + (-6*b8 - 4) * q^43 + (2*b5 + 6*b3 - 6*b1) * q^44 + (2*b8 - 2*b4 - 2*b2 + 2) * q^46 + (-2*b13 + 2*b10) * q^47 + (6*b6 - 6*b5 + 7*b3) * q^50 + (b9 + 2*b7) * q^52 + 5*b6 * q^53 + (2*b12 + 2*b11 - 2*b9) * q^55 + (-4*b4 - 6*b2) * q^58 - 2*b10 * q^59 + (-b12 + b11 - b7) * q^61 + (-2*b15 + 2*b13) * q^62 + (2*b8 + 7) * q^64 + (b5 - 10*b3 + 10*b1) * q^65 + (-6*b8 + 6*b4) * q^67 + (-b14 + 3*b13 - 3*b10) * q^68 + (2*b6 - 2*b5 + 2*b3) * q^71 + (-b9 + b7) * q^73 + (3*b6 + 2*b1) * q^74 + (2*b12 - 4*b11 - 2*b9) * q^76 + (4*b4 + 4*b2) * q^79 - 3*b10 * q^80 + (3*b12 - 2*b11 + 2*b7) * q^82 - 4*b13 * q^83 + (9*b8 - 8) * q^85 + (-10*b5 + 16*b3 - 16*b1) * q^86 + (-4*b8 + 4*b4 - 2*b2 + 2) * q^88 + (-2*b14 + b13 - b10) * q^89 + (-4*b6 + 4*b5 - 2*b3) * q^92 + (-2*b9 - 2*b7) * q^94 + (-2*b6 - 20*b1) * q^95 + (b12 + b11 - b9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4}+O(q^{10})$$ 16 * q + 8 * q^4 $$16 q + 8 q^{4} - 24 q^{16} - 32 q^{22} - 40 q^{25} + 32 q^{37} - 64 q^{43} + 16 q^{46} - 48 q^{58} + 112 q^{64} + 32 q^{79} - 128 q^{85} + 16 q^{88}+O(q^{100})$$ 16 * q + 8 * q^4 - 24 * q^16 - 32 * q^22 - 40 * q^25 + 32 * q^37 - 64 * q^43 + 16 * q^46 - 48 * q^58 + 112 * q^64 + 32 * q^79 - 128 * q^85 + 16 * q^88

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{48}^{4}$$ v^4 $$\beta_{2}$$ $$=$$ $$\zeta_{48}^{8}$$ v^8 $$\beta_{3}$$ $$=$$ $$\zeta_{48}^{12}$$ v^12 $$\beta_{4}$$ $$=$$ $$\zeta_{48}^{14} + \zeta_{48}^{2}$$ v^14 + v^2 $$\beta_{5}$$ $$=$$ $$-\zeta_{48}^{14} + \zeta_{48}^{2}$$ -v^14 + v^2 $$\beta_{6}$$ $$=$$ $$-\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6}$$ -v^14 + v^10 + v^6 $$\beta_{7}$$ $$=$$ $$-\zeta_{48}^{13} + \zeta_{48}^{11} + 3\zeta_{48}^{7} - \zeta_{48}^{3} + 3\zeta_{48}$$ -v^13 + v^11 + 3*v^7 - v^3 + 3*v $$\beta_{8}$$ $$=$$ $$-\zeta_{48}^{10} + \zeta_{48}^{6} + \zeta_{48}^{2}$$ -v^10 + v^6 + v^2 $$\beta_{9}$$ $$=$$ $$2\zeta_{48}^{13} - 2\zeta_{48}^{11} + \zeta_{48}^{7} + 2\zeta_{48}^{3} + \zeta_{48}$$ 2*v^13 - 2*v^11 + v^7 + 2*v^3 + v $$\beta_{10}$$ $$=$$ $$-\zeta_{48}^{13} - \zeta_{48}^{11} - 2\zeta_{48}^{7} + \zeta_{48}^{3} + 2\zeta_{48}$$ -v^13 - v^11 - 2*v^7 + v^3 + 2*v $$\beta_{11}$$ $$=$$ $$3\zeta_{48}^{15} - \zeta_{48}^{13} + 3\zeta_{48}^{9} + \zeta_{48}^{5} - \zeta_{48}^{3}$$ 3*v^15 - v^13 + 3*v^9 + v^5 - v^3 $$\beta_{12}$$ $$=$$ $$-\zeta_{48}^{15} - 2\zeta_{48}^{11} - \zeta_{48}^{9} + \zeta_{48}^{7} + 2\zeta_{48}^{5} + \zeta_{48}$$ -v^15 - 2*v^11 - v^9 + v^7 + 2*v^5 + v $$\beta_{13}$$ $$=$$ $$-2\zeta_{48}^{15} - \zeta_{48}^{13} + 2\zeta_{48}^{9} + \zeta_{48}^{5} + \zeta_{48}^{3}$$ -2*v^15 - v^13 + 2*v^9 + v^5 + v^3 $$\beta_{14}$$ $$=$$ $$\zeta_{48}^{15} + 3\zeta_{48}^{11} - \zeta_{48}^{9} - \zeta_{48}^{7} + 3\zeta_{48}^{5} + \zeta_{48}$$ v^15 + 3*v^11 - v^9 - v^7 + 3*v^5 + v $$\beta_{15}$$ $$=$$ $$-\zeta_{48}^{15} + 3\zeta_{48}^{13} + \zeta_{48}^{9} - 3\zeta_{48}^{5} - 3\zeta_{48}^{3}$$ -v^15 + 3*v^13 + v^9 - 3*v^5 - 3*v^3
 $$\zeta_{48}$$ $$=$$ $$( \beta_{15} + \beta_{14} + 3\beta_{10} + \beta_{9} + 2\beta_{7} ) / 14$$ (b15 + b14 + 3*b10 + b9 + 2*b7) / 14 $$\zeta_{48}^{2}$$ $$=$$ $$( \beta_{5} + \beta_{4} ) / 2$$ (b5 + b4) / 2 $$\zeta_{48}^{3}$$ $$=$$ $$( -2\beta_{15} + \beta_{13} - 3\beta_{12} - \beta_{11} + 3\beta_{9} ) / 14$$ (-2*b15 + b13 - 3*b12 - b11 + 3*b9) / 14 $$\zeta_{48}^{4}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{48}^{5}$$ $$=$$ $$( 2\beta_{14} + \beta_{13} + 3\beta_{12} + \beta_{11} - \beta_{10} - \beta_{7} ) / 14$$ (2*b14 + b13 + 3*b12 + b11 - b10 - b7) / 14 $$\zeta_{48}^{6}$$ $$=$$ $$( \beta_{8} + \beta_{6} - \beta_{5} ) / 2$$ (b8 + b6 - b5) / 2 $$\zeta_{48}^{7}$$ $$=$$ $$( -\beta_{15} - \beta_{14} - 3\beta_{10} + \beta_{9} + 2\beta_{7} ) / 14$$ (-b15 - b14 - 3*b10 + b9 + 2*b7) / 14 $$\zeta_{48}^{8}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{48}^{9}$$ $$=$$ $$( \beta_{15} + 3\beta_{13} - \beta_{12} + 2\beta_{11} + \beta_{9} ) / 14$$ (b15 + 3*b13 - b12 + 2*b11 + b9) / 14 $$\zeta_{48}^{10}$$ $$=$$ $$( -\beta_{8} + \beta_{6} + \beta_{4} ) / 2$$ (-b8 + b6 + b4) / 2 $$\zeta_{48}^{11}$$ $$=$$ $$( 2\beta_{14} + \beta_{13} - 3\beta_{12} - \beta_{11} - \beta_{10} + \beta_{7} ) / 14$$ (2*b14 + b13 - 3*b12 - b11 - b10 + b7) / 14 $$\zeta_{48}^{12}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{48}^{13}$$ $$=$$ $$( 2\beta_{15} + 2\beta_{14} - \beta_{10} + 3\beta_{9} - \beta_{7} ) / 14$$ (2*b15 + 2*b14 - b10 + 3*b9 - b7) / 14 $$\zeta_{48}^{14}$$ $$=$$ $$( -\beta_{5} + \beta_{4} ) / 2$$ (-b5 + b4) / 2 $$\zeta_{48}^{15}$$ $$=$$ $$( -\beta_{15} - 3\beta_{13} - \beta_{12} + 2\beta_{11} + \beta_{9} ) / 14$$ (-b15 - 3*b13 - b12 + 2*b11 + b9) / 14

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$\beta_{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 −0.608761 − 0.793353i 0.608761 + 0.793353i 0.130526 − 0.991445i −0.130526 + 0.991445i 0.793353 − 0.608761i −0.793353 + 0.608761i 0.991445 + 0.130526i −0.991445 − 0.130526i −0.608761 + 0.793353i 0.608761 − 0.793353i 0.130526 + 0.991445i −0.130526 − 0.991445i 0.793353 + 0.608761i −0.793353 − 0.608761i 0.991445 − 0.130526i −0.991445 + 0.130526i
−2.09077 + 1.20711i 0 1.91421 3.31552i −1.68925 2.92586i 0 0 4.41421i 0 7.06365 + 4.07820i
80.2 −2.09077 + 1.20711i 0 1.91421 3.31552i 1.68925 + 2.92586i 0 0 4.41421i 0 −7.06365 4.07820i
80.3 −0.358719 + 0.207107i 0 −0.914214 + 1.58346i −1.46508 2.53759i 0 0 1.58579i 0 1.05110 + 0.606854i
80.4 −0.358719 + 0.207107i 0 −0.914214 + 1.58346i 1.46508 + 2.53759i 0 0 1.58579i 0 −1.05110 0.606854i
80.5 0.358719 0.207107i 0 −0.914214 + 1.58346i −1.46508 2.53759i 0 0 1.58579i 0 −1.05110 0.606854i
80.6 0.358719 0.207107i 0 −0.914214 + 1.58346i 1.46508 + 2.53759i 0 0 1.58579i 0 1.05110 + 0.606854i
80.7 2.09077 1.20711i 0 1.91421 3.31552i −1.68925 2.92586i 0 0 4.41421i 0 −7.06365 4.07820i
80.8 2.09077 1.20711i 0 1.91421 3.31552i 1.68925 + 2.92586i 0 0 4.41421i 0 7.06365 + 4.07820i
215.1 −2.09077 1.20711i 0 1.91421 + 3.31552i −1.68925 + 2.92586i 0 0 4.41421i 0 7.06365 4.07820i
215.2 −2.09077 1.20711i 0 1.91421 + 3.31552i 1.68925 2.92586i 0 0 4.41421i 0 −7.06365 + 4.07820i
215.3 −0.358719 0.207107i 0 −0.914214 1.58346i −1.46508 + 2.53759i 0 0 1.58579i 0 1.05110 0.606854i
215.4 −0.358719 0.207107i 0 −0.914214 1.58346i 1.46508 2.53759i 0 0 1.58579i 0 −1.05110 + 0.606854i
215.5 0.358719 + 0.207107i 0 −0.914214 1.58346i −1.46508 + 2.53759i 0 0 1.58579i 0 −1.05110 + 0.606854i
215.6 0.358719 + 0.207107i 0 −0.914214 1.58346i 1.46508 2.53759i 0 0 1.58579i 0 1.05110 0.606854i
215.7 2.09077 + 1.20711i 0 1.91421 + 3.31552i −1.68925 + 2.92586i 0 0 4.41421i 0 −7.06365 + 4.07820i
215.8 2.09077 + 1.20711i 0 1.91421 + 3.31552i 1.68925 2.92586i 0 0 4.41421i 0 7.06365 4.07820i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 80.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.p.c 16
3.b odd 2 1 inner 441.2.p.c 16
7.b odd 2 1 inner 441.2.p.c 16
7.c even 3 1 441.2.c.b 8
7.c even 3 1 inner 441.2.p.c 16
7.d odd 6 1 441.2.c.b 8
7.d odd 6 1 inner 441.2.p.c 16
21.c even 2 1 inner 441.2.p.c 16
21.g even 6 1 441.2.c.b 8
21.g even 6 1 inner 441.2.p.c 16
21.h odd 6 1 441.2.c.b 8
21.h odd 6 1 inner 441.2.p.c 16
28.f even 6 1 7056.2.k.g 8
28.g odd 6 1 7056.2.k.g 8
84.j odd 6 1 7056.2.k.g 8
84.n even 6 1 7056.2.k.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.c.b 8 7.c even 3 1
441.2.c.b 8 7.d odd 6 1
441.2.c.b 8 21.g even 6 1
441.2.c.b 8 21.h odd 6 1
441.2.p.c 16 1.a even 1 1 trivial
441.2.p.c 16 3.b odd 2 1 inner
441.2.p.c 16 7.b odd 2 1 inner
441.2.p.c 16 7.c even 3 1 inner
441.2.p.c 16 7.d odd 6 1 inner
441.2.p.c 16 21.c even 2 1 inner
441.2.p.c 16 21.g even 6 1 inner
441.2.p.c 16 21.h odd 6 1 inner
7056.2.k.g 8 28.f even 6 1
7056.2.k.g 8 28.g odd 6 1
7056.2.k.g 8 84.j odd 6 1
7056.2.k.g 8 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 6T_{2}^{6} + 35T_{2}^{4} - 6T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} - 6 T^{6} + 35 T^{4} + \cdots + 1)^{2}$$
$3$ $$T^{16}$$
$5$ $$(T^{8} + 20 T^{6} + \cdots + 9604)^{2}$$
$7$ $$T^{16}$$
$11$ $$(T^{8} - 24 T^{6} + \cdots + 256)^{2}$$
$13$ $$(T^{4} + 20 T^{2} + 98)^{4}$$
$17$ $$(T^{8} + 52 T^{6} + \cdots + 9604)^{2}$$
$19$ $$(T^{8} - 80 T^{6} + \cdots + 2458624)^{2}$$
$23$ $$(T^{4} - 4 T^{2} + 16)^{4}$$
$29$ $$(T^{4} + 24 T^{2} + 16)^{4}$$
$31$ $$(T^{8} - 80 T^{6} + \cdots + 2458624)^{2}$$
$37$ $$(T^{4} - 8 T^{3} + \cdots + 196)^{4}$$
$41$ $$(T^{4} - 68 T^{2} + 98)^{4}$$
$43$ $$(T^{2} + 8 T - 56)^{8}$$
$47$ $$(T^{8} + 80 T^{6} + \cdots + 2458624)^{2}$$
$53$ $$(T^{4} - 50 T^{2} + 2500)^{4}$$
$59$ $$(T^{8} + 80 T^{6} + \cdots + 2458624)^{2}$$
$61$ $$(T^{8} - 68 T^{6} + \cdots + 9604)^{2}$$
$67$ $$(T^{4} + 72 T^{2} + 5184)^{4}$$
$71$ $$(T^{4} + 24 T^{2} + 16)^{4}$$
$73$ $$(T^{8} - 52 T^{6} + \cdots + 9604)^{2}$$
$79$ $$(T^{4} - 8 T^{3} + \cdots + 256)^{4}$$
$83$ $$(T^{4} - 320 T^{2} + 25088)^{4}$$
$89$ $$(T^{8} + 164 T^{6} + \cdots + 23059204)^{2}$$
$97$ $$(T^{4} + 52 T^{2} + 98)^{4}$$