# Properties

 Label 980.2.o.a Level $980$ Weight $2$ Character orbit 980.o Analytic conductor $7.825$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(31,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

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

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

chi := DirichletCharacter("980.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.o (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: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} - 2 \zeta_{12} q^{4} + \zeta_{12} q^{5} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} - 1) q^{6} + (2 \zeta_{12}^{3} - 2) q^{8} +O(q^{10})$$ q + (-z^3 - z^2 + z) * q^2 + (-z^3 - z) * q^3 - 2*z * q^4 + z * q^5 + (2*z^3 - z^2 - z - 1) * q^6 + (2*z^3 - 2) * q^8 $$q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} - 2 \zeta_{12} q^{4} + \zeta_{12} q^{5} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} - 1) q^{6} + (2 \zeta_{12}^{3} - 2) q^{8} + ( - \zeta_{12}^{3} + 1) q^{10} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{11} + (4 \zeta_{12}^{2} - 2) q^{12} + (3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{13} + ( - 2 \zeta_{12}^{2} + 1) q^{15} + 4 \zeta_{12}^{2} q^{16} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 3 \zeta_{12} + 4) q^{17} + (6 \zeta_{12}^{2} - 6) q^{19} - 2 \zeta_{12}^{2} q^{20} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + \zeta_{12} - 1) q^{22} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{23} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{24} + \zeta_{12}^{2} q^{25} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + 5 \zeta_{12} + 4) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 1) q^{29} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 2) q^{30} + 6 \zeta_{12}^{2} q^{31} + ( - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{32} + (2 \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{33} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + 5 \zeta_{12} - 5) q^{34} + (4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 2 \zeta_{12} + 6) q^{37} + (6 \zeta_{12}^{3} + 6) q^{38} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{39} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{40} + (4 \zeta_{12}^{2} - 2) q^{41} - 2 \zeta_{12}^{3} q^{43} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} + 4) q^{44} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{46} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{50} + (3 \zeta_{12}^{2} - 6 \zeta_{12} + 3) q^{51} + ( - 8 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 4 \zeta_{12} + 6) q^{52} - 2 \zeta_{12}^{2} q^{53} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 6) q^{54} + (\zeta_{12}^{3} - 2 \zeta_{12} - 2) q^{55} + ( - 6 \zeta_{12}^{3} + 12 \zeta_{12}) q^{57} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 5 \zeta_{12} - 8) q^{58} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{59} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{60} + (2 \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{61} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{62} - 8 \zeta_{12}^{3} q^{64} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{65} + ( - 5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 5) q^{66} + (2 \zeta_{12}^{2} - 4) q^{67} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 6) q^{68} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{69} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{71} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 6 \zeta_{12} - 8) q^{73} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{74} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{75} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{76} + ( - 6 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{78} + ( - 5 \zeta_{12}^{2} + 6 \zeta_{12} - 5) q^{79} + 4 \zeta_{12}^{3} q^{80} + 9 \zeta_{12}^{2} q^{81} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{82} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 12) q^{83} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 3) q^{85} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{86} + ( - \zeta_{12}^{3} + 12 \zeta_{12}^{2} - \zeta_{12}) q^{87} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{88} + ( - 2 \zeta_{12}^{2} - 6 \zeta_{12} - 2) q^{89} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12}) q^{92} + ( - 12 \zeta_{12}^{3} + 6 \zeta_{12}) q^{93} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{94} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{95} + (4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 4) q^{96} + ( - 3 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{97} +O(q^{100})$$ q + (-z^3 - z^2 + z) * q^2 + (-z^3 - z) * q^3 - 2*z * q^4 + z * q^5 + (2*z^3 - z^2 - z - 1) * q^6 + (2*z^3 - 2) * q^8 + (-z^3 + 1) * q^10 + (2*z^3 + z^2 - 2*z - 2) * q^11 + (4*z^2 - 2) * q^12 + (3*z^3 + 4*z^2 - 2) * q^13 + (-2*z^2 + 1) * q^15 + 4*z^2 * q^16 + (3*z^3 - 2*z^2 - 3*z + 4) * q^17 + (6*z^2 - 6) * q^19 - 2*z^2 * q^20 + (2*z^3 + 3*z^2 + z - 1) * q^22 + (-2*z^2 - 2*z - 2) * q^23 + (2*z^3 - 2*z^2 + 2*z + 4) * q^24 + z^2 * q^25 + (-z^3 + z^2 + 5*z + 4) * q^26 + (3*z^3 - 6*z) * q^27 + (4*z^3 - 8*z + 1) * q^29 + (-z^3 + z^2 - z - 2) * q^30 + 6*z^2 * q^31 + (-4*z^2 + 4*z + 4) * q^32 + (2*z^2 + 3*z + 2) * q^33 + (-4*z^3 + z^2 + 5*z - 5) * q^34 + (4*z^3 - 6*z^2 - 2*z + 6) * q^37 + (6*z^3 + 6) * q^38 + (-6*z^3 - 3*z^2 + 6*z + 6) * q^39 + (2*z^2 - 2*z - 2) * q^40 + (4*z^2 - 2) * q^41 - 2*z^3 * q^43 + (-2*z^3 + 4*z + 4) * q^44 + (4*z^3 + 4*z^2 - 4*z - 4) * q^46 + (-2*z^3 + z) * q^47 + (-8*z^3 + 4*z) * q^48 + (-z^2 + z + 1) * q^50 + (3*z^2 - 6*z + 3) * q^51 + (-8*z^3 - 6*z^2 + 4*z + 6) * q^52 - 2*z^2 * q^53 + (3*z^3 + 3*z^2 + 3*z - 6) * q^54 + (z^3 - 2*z - 2) * q^55 + (-6*z^3 + 12*z) * q^57 + (3*z^3 + 3*z^2 + 5*z - 8) * q^58 + (-2*z^3 - 2*z) * q^59 + (4*z^3 - 2*z) * q^60 + (2*z^2 - 6*z + 2) * q^61 + (-6*z^2 + 6*z + 6) * q^62 - 8*z^3 * q^64 + (4*z^3 + 3*z^2 - 2*z - 3) * q^65 + (-5*z^3 - 4*z^2 + 4*z + 5) * q^66 + (2*z^2 - 4) * q^67 + (4*z^3 - 8*z + 6) * q^68 + (6*z^3 + 4*z^2 - 2) * q^69 + (-4*z^3 + 4*z^2 - 2) * q^71 + (-6*z^3 + 4*z^2 + 6*z - 8) * q^73 + (-8*z^3 + 4*z^2 + 4*z - 8) * q^74 + (-2*z^3 + z) * q^75 + (-12*z^3 + 12*z) * q^76 + (-6*z^3 - 9*z^2 - 3*z + 3) * q^78 + (-5*z^2 + 6*z - 5) * q^79 + 4*z^3 * q^80 + 9*z^2 * q^81 + (2*z^3 - 2*z^2 + 2*z + 4) * q^82 + (2*z^3 - 4*z + 12) * q^83 + (-2*z^3 + 4*z - 3) * q^85 + (2*z^3 - 2*z^2 - 2*z) * q^86 + (-z^3 + 12*z^2 - z) * q^87 + (-6*z^3 - 6*z^2 + 2*z + 4) * q^88 + (-2*z^2 - 6*z - 2) * q^89 + (4*z^3 + 4*z^2 + 4*z) * q^92 + (-12*z^3 + 6*z) * q^93 + (z^3 - 2*z^2 - 2*z + 1) * q^94 + (6*z^3 - 6*z) * q^95 + (4*z^3 - 8*z^2 - 8*z + 4) * q^96 + (-3*z^3 + 12*z^2 - 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 6 q^{6} - 8 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 - 6 * q^6 - 8 * q^8 $$4 q - 2 q^{2} - 6 q^{6} - 8 q^{8} + 4 q^{10} - 6 q^{11} + 8 q^{16} + 12 q^{17} - 12 q^{19} - 4 q^{20} + 2 q^{22} - 12 q^{23} + 12 q^{24} + 2 q^{25} + 18 q^{26} + 4 q^{29} - 6 q^{30} + 12 q^{31} + 8 q^{32} + 12 q^{33} - 18 q^{34} + 12 q^{37} + 24 q^{38} + 18 q^{39} - 4 q^{40} + 16 q^{44} - 8 q^{46} + 2 q^{50} + 18 q^{51} + 12 q^{52} - 4 q^{53} - 18 q^{54} - 8 q^{55} - 26 q^{58} + 12 q^{61} + 12 q^{62} - 6 q^{65} + 12 q^{66} - 12 q^{67} + 24 q^{68} - 24 q^{73} - 24 q^{74} - 6 q^{78} - 30 q^{79} + 18 q^{81} + 12 q^{82} + 48 q^{83} - 12 q^{85} - 4 q^{86} + 24 q^{87} + 4 q^{88} - 12 q^{89} + 8 q^{92}+O(q^{100})$$ 4 * q - 2 * q^2 - 6 * q^6 - 8 * q^8 + 4 * q^10 - 6 * q^11 + 8 * q^16 + 12 * q^17 - 12 * q^19 - 4 * q^20 + 2 * q^22 - 12 * q^23 + 12 * q^24 + 2 * q^25 + 18 * q^26 + 4 * q^29 - 6 * q^30 + 12 * q^31 + 8 * q^32 + 12 * q^33 - 18 * q^34 + 12 * q^37 + 24 * q^38 + 18 * q^39 - 4 * q^40 + 16 * q^44 - 8 * q^46 + 2 * q^50 + 18 * q^51 + 12 * q^52 - 4 * q^53 - 18 * q^54 - 8 * q^55 - 26 * q^58 + 12 * q^61 + 12 * q^62 - 6 * q^65 + 12 * q^66 - 12 * q^67 + 24 * q^68 - 24 * q^73 - 24 * q^74 - 6 * q^78 - 30 * q^79 + 18 * q^81 + 12 * q^82 + 48 * q^83 - 12 * q^85 - 4 * q^86 + 24 * q^87 + 4 * q^88 - 12 * q^89 + 8 * q^92

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$1 - \zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.36603 + 0.366025i 0.866025 1.50000i 1.73205 1.00000i −0.866025 + 0.500000i −0.633975 + 2.36603i 0 −2.00000 + 2.00000i 0 1.00000 1.00000i
31.2 0.366025 + 1.36603i −0.866025 + 1.50000i −1.73205 + 1.00000i 0.866025 0.500000i −2.36603 0.633975i 0 −2.00000 2.00000i 0 1.00000 + 1.00000i
411.1 −1.36603 0.366025i 0.866025 + 1.50000i 1.73205 + 1.00000i −0.866025 0.500000i −0.633975 2.36603i 0 −2.00000 2.00000i 0 1.00000 + 1.00000i
411.2 0.366025 1.36603i −0.866025 1.50000i −1.73205 1.00000i 0.866025 + 0.500000i −2.36603 + 0.633975i 0 −2.00000 + 2.00000i 0 1.00000 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.o.a 4
4.b odd 2 1 980.2.o.d 4
7.b odd 2 1 980.2.o.b 4
7.c even 3 1 140.2.g.a 4
7.c even 3 1 980.2.o.c 4
7.d odd 6 1 140.2.g.b yes 4
7.d odd 6 1 980.2.o.d 4
21.g even 6 1 1260.2.c.b 4
21.h odd 6 1 1260.2.c.a 4
28.d even 2 1 980.2.o.c 4
28.f even 6 1 140.2.g.a 4
28.f even 6 1 inner 980.2.o.a 4
28.g odd 6 1 140.2.g.b yes 4
28.g odd 6 1 980.2.o.b 4
35.i odd 6 1 700.2.g.g 4
35.j even 6 1 700.2.g.f 4
35.k even 12 1 700.2.c.c 4
35.k even 12 1 700.2.c.f 4
35.l odd 12 1 700.2.c.b 4
35.l odd 12 1 700.2.c.e 4
56.j odd 6 1 2240.2.k.b 4
56.k odd 6 1 2240.2.k.b 4
56.m even 6 1 2240.2.k.a 4
56.p even 6 1 2240.2.k.a 4
84.j odd 6 1 1260.2.c.a 4
84.n even 6 1 1260.2.c.b 4
140.p odd 6 1 700.2.g.g 4
140.s even 6 1 700.2.g.f 4
140.w even 12 1 700.2.c.c 4
140.w even 12 1 700.2.c.f 4
140.x odd 12 1 700.2.c.b 4
140.x odd 12 1 700.2.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 7.c even 3 1
140.2.g.a 4 28.f even 6 1
140.2.g.b yes 4 7.d odd 6 1
140.2.g.b yes 4 28.g odd 6 1
700.2.c.b 4 35.l odd 12 1
700.2.c.b 4 140.x odd 12 1
700.2.c.c 4 35.k even 12 1
700.2.c.c 4 140.w even 12 1
700.2.c.e 4 35.l odd 12 1
700.2.c.e 4 140.x odd 12 1
700.2.c.f 4 35.k even 12 1
700.2.c.f 4 140.w even 12 1
700.2.g.f 4 35.j even 6 1
700.2.g.f 4 140.s even 6 1
700.2.g.g 4 35.i odd 6 1
700.2.g.g 4 140.p odd 6 1
980.2.o.a 4 1.a even 1 1 trivial
980.2.o.a 4 28.f even 6 1 inner
980.2.o.b 4 7.b odd 2 1
980.2.o.b 4 28.g odd 6 1
980.2.o.c 4 7.c even 3 1
980.2.o.c 4 28.d even 2 1
980.2.o.d 4 4.b odd 2 1
980.2.o.d 4 7.d odd 6 1
1260.2.c.a 4 21.h odd 6 1
1260.2.c.a 4 84.j odd 6 1
1260.2.c.b 4 21.g even 6 1
1260.2.c.b 4 84.n even 6 1
2240.2.k.a 4 56.m even 6 1
2240.2.k.a 4 56.p even 6 1
2240.2.k.b 4 56.j odd 6 1
2240.2.k.b 4 56.k odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}^{4} + 3T_{3}^{2} + 9$$ T3^4 + 3*T3^2 + 9 $$T_{11}^{4} + 6T_{11}^{3} + 11T_{11}^{2} - 6T_{11} + 1$$ T11^4 + 6*T11^3 + 11*T11^2 - 6*T11 + 1 $$T_{17}^{4} - 12T_{17}^{3} + 51T_{17}^{2} - 36T_{17} + 9$$ T17^4 - 12*T17^3 + 51*T17^2 - 36*T17 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4} - T^{2} + 1$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 6 T^{3} + 11 T^{2} - 6 T + 1$$
$13$ $$T^{4} + 42T^{2} + 9$$
$17$ $$T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9$$
$19$ $$(T^{2} + 6 T + 36)^{2}$$
$23$ $$T^{4} + 12 T^{3} + 56 T^{2} + 96 T + 64$$
$29$ $$(T^{2} - 2 T - 47)^{2}$$
$31$ $$(T^{2} - 6 T + 36)^{2}$$
$37$ $$T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$T^{4} + 3T^{2} + 9$$
$53$ $$(T^{2} + 2 T + 4)^{2}$$
$59$ $$T^{4} + 12T^{2} + 144$$
$61$ $$T^{4} - 12 T^{3} + 24 T^{2} + \cdots + 576$$
$67$ $$(T^{2} + 6 T + 12)^{2}$$
$71$ $$T^{4} + 56T^{2} + 16$$
$73$ $$T^{4} + 24 T^{3} + 204 T^{2} + \cdots + 144$$
$79$ $$T^{4} + 30 T^{3} + 339 T^{2} + \cdots + 1521$$
$83$ $$(T^{2} - 24 T + 132)^{2}$$
$89$ $$T^{4} + 12 T^{3} + 24 T^{2} + \cdots + 576$$
$97$ $$T^{4} + 234T^{2} + 9801$$