# Properties

 Label 325.2.e.e Level $325$ Weight $2$ Character orbit 325.e Analytic conductor $2.595$ 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] = [325,2,Mod(126,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("325.126");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.e (of order $$3$$, degree $$2$$, 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: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 8x^{10} + 54x^{8} + 78x^{6} + 92x^{4} + 10x^{2} + 1$$ x^12 + 8*x^10 + 54*x^8 + 78*x^6 + 92*x^4 + 10*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + ( - \beta_{11} + \beta_{6}) q^{3} + ( - \beta_{8} + \beta_{7} + \beta_{2} - 1) q^{4} + ( - \beta_{8} + 2 \beta_{7} + \beta_{2} - 2) q^{6} + ( - \beta_{5} + \beta_1) q^{7} + (\beta_{11} + \beta_{10} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - \beta_{9} + \beta_{7} - 1) q^{9}+O(q^{10})$$ q + b6 * q^2 + (-b11 + b6) * q^3 + (-b8 + b7 + b2 - 1) * q^4 + (-b8 + 2*b7 + b2 - 2) * q^6 + (-b5 + b1) * q^7 + (b11 + b10 - 2*b6 - b5 - b4 - 2*b1) * q^8 + (-b9 + b7 - 1) * q^9 $$q + \beta_{6} q^{2} + ( - \beta_{11} + \beta_{6}) q^{3} + ( - \beta_{8} + \beta_{7} + \beta_{2} - 1) q^{4} + ( - \beta_{8} + 2 \beta_{7} + \beta_{2} - 2) q^{6} + ( - \beta_{5} + \beta_1) q^{7} + (\beta_{11} + \beta_{10} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - 2 \beta_{2} - 8) q^{99}+O(q^{100})$$ q + b6 * q^2 + (-b11 + b6) * q^3 + (-b8 + b7 + b2 - 1) * q^4 + (-b8 + 2*b7 + b2 - 2) * q^6 + (-b5 + b1) * q^7 + (b11 + b10 - 2*b6 - b5 - b4 - 2*b1) * q^8 + (-b9 + b7 - 1) * q^9 + (-b9 - b3) * q^11 + (-b11 + b10 - 3*b6 + b5 - b4 - 3*b1) * q^12 + (-b11 + b10 + b5 + b1) * q^13 + (-b2 + 4) * q^14 + (b9 + b8 - 3*b7 + b3) * q^16 + (b5 + b4) * q^17 + (b10 - b6 - b4 - b1) * q^18 + (b9 + 2*b7 - 2) * q^19 + (b3 - 2*b2) * q^21 - b4 * q^22 + (b11 - b6) * q^23 + (b9 + 2*b8 - 6*b7 + b3) * q^24 + (b9 + b8 - b7 - 2*b2 + 3) * q^26 + (b11 + b6 - b5 + b1) * q^27 + (-3*b11 - b10 + 5*b6) * q^28 - 3*b7 * q^29 + (2*b2 - 2) * q^31 + (-b5 + 2*b1) * q^32 + (-3*b5 - b1) * q^33 + (-b3 - b2 - 1) * q^34 + (-b9 + 2*b8 - b7 - b3) * q^36 + (b11 + b10) * q^37 + (-b10 - 2*b6 + b4 - 2*b1) * q^38 + (-b9 + 2*b8 + 2*b7 - b3 - 2*b2 + 2) * q^39 + (-2*b8 + 3*b7) * q^41 + (-2*b11 - b10 + 6*b6) * q^42 + (3*b5 + b1) * q^43 + (-b3 + b2) * q^44 + (b8 - 2*b7 - b2 + 2) * q^46 + (2*b11 - 2*b10 - 4*b6 - 2*b5 + 2*b4 - 4*b1) * q^47 + (4*b5 + b4 + 6*b1) * q^48 + (b9 - b7 + b3) * q^49 + (-b3 + 2) * q^51 + (-2*b11 - b10 + 7*b6 + 3*b5 + 4*b1) * q^52 + (4*b11 - b10 - b6 - 4*b5 + b4 - b1) * q^53 + (-b8 + 4*b7) * q^54 + (-b9 - 4*b8 + 4*b7 + 4*b2 - 4) * q^56 + (-b11 - b6 + b5 - b1) * q^57 + 3*b1 * q^58 + (-b9 - 2*b8 + 2*b2) * q^59 + (2*b9 - b7 + 1) * q^61 + (2*b11 + 2*b10 - 8*b6) * q^62 + (4*b11 - 2*b10) * q^63 + (2*b3 + 1) * q^64 + b2 * q^66 + (-b11 - 3*b6) * q^67 + (b11 + 2*b6) * q^68 + (b9 - 4*b7 + 4) * q^69 + (-b9 + 2*b7 - 2) * q^71 + (2*b5 - b4 + 5*b1) * q^72 + (2*b11 - b10 - 5*b6 - 2*b5 + b4 - 5*b1) * q^73 + (b9 + b8 + b7 - b2 - 1) * q^74 + (b9 + b8 - 2*b7 + b3) * q^76 + (3*b11 - 2*b10 - b6 - 3*b5 + 2*b4 - b1) * q^77 + (-2*b11 - 2*b10 + 8*b6 + 2*b5 + b4 + 4*b1) * q^78 + (2*b2 + 8) * q^79 + (-2*b9 - 2*b8 + 3*b7 - 2*b3) * q^81 + (-2*b5 - 2*b4 - 9*b1) * q^82 + (-4*b11 - 2*b10 + 4*b6 + 4*b5 + 2*b4 + 4*b1) * q^83 + (-3*b9 - 3*b8 + 16*b7 + 3*b2 - 16) * q^84 - b2 * q^86 + (3*b5 + 3*b1) * q^87 + (b11 - 2*b10 - 3*b6) * q^88 + (-b9 - 4*b8 - 2*b7 - b3) * q^89 + (-b9 + 2*b8 + 2*b3 - 4) * q^91 + (b11 - b10 + 3*b6 - b5 + b4 + 3*b1) * q^92 + (-2*b11 + 2*b10 - 6*b6) * q^93 + (-2*b9 + 2*b8 - 10*b7 - 2*b3) * q^94 + (b3 - 3*b2 + 2) * q^96 + (3*b5 + 2*b4 + 5*b1) * q^97 + (b4 + b1) * q^98 + (-2*b2 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 4 q^{4} - 10 q^{6} - 6 q^{9}+O(q^{10})$$ 12 * q - 4 * q^4 - 10 * q^6 - 6 * q^9 $$12 q - 4 q^{4} - 10 q^{6} - 6 q^{9} + 44 q^{14} - 16 q^{16} - 12 q^{19} - 8 q^{21} - 32 q^{24} + 24 q^{26} - 18 q^{29} - 16 q^{31} - 16 q^{34} - 2 q^{36} + 32 q^{39} + 14 q^{41} + 4 q^{44} + 10 q^{46} - 6 q^{49} + 24 q^{51} + 22 q^{54} - 16 q^{56} + 4 q^{59} + 6 q^{61} + 12 q^{64} + 4 q^{66} + 24 q^{69} - 12 q^{71} - 8 q^{74} - 10 q^{76} + 104 q^{79} + 14 q^{81} - 90 q^{84} - 4 q^{86} - 20 q^{89} - 44 q^{91} - 56 q^{94} + 12 q^{96} - 104 q^{99}+O(q^{100})$$ 12 * q - 4 * q^4 - 10 * q^6 - 6 * q^9 + 44 * q^14 - 16 * q^16 - 12 * q^19 - 8 * q^21 - 32 * q^24 + 24 * q^26 - 18 * q^29 - 16 * q^31 - 16 * q^34 - 2 * q^36 + 32 * q^39 + 14 * q^41 + 4 * q^44 + 10 * q^46 - 6 * q^49 + 24 * q^51 + 22 * q^54 - 16 * q^56 + 4 * q^59 + 6 * q^61 + 12 * q^64 + 4 * q^66 + 24 * q^69 - 12 * q^71 - 8 * q^74 - 10 * q^76 + 104 * q^79 + 14 * q^81 - 90 * q^84 - 4 * q^86 - 20 * q^89 - 44 * q^91 - 56 * q^94 + 12 * q^96 - 104 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 8x^{10} + 54x^{8} + 78x^{6} + 92x^{4} + 10x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -16\nu^{10} - 108\nu^{8} - 729\nu^{6} - 184\nu^{4} - 20\nu^{2} + 3531 ) / 1222$$ (-16*v^10 - 108*v^8 - 729*v^6 - 184*v^4 - 20*v^2 + 3531) / 1222 $$\beta_{3}$$ $$=$$ $$( 92\nu^{10} + 621\nu^{8} + 4039\nu^{6} + 1058\nu^{4} + 115\nu^{2} - 3959 ) / 1222$$ (92*v^10 + 621*v^8 + 4039*v^6 + 1058*v^4 + 115*v^2 - 3959) / 1222 $$\beta_{4}$$ $$=$$ $$( -92\nu^{11} - 621\nu^{9} - 4039\nu^{7} - 1058\nu^{5} - 115\nu^{3} + 3959\nu ) / 1222$$ (-92*v^11 - 621*v^9 - 4039*v^7 - 1058*v^5 - 115*v^3 + 3959*v) / 1222 $$\beta_{5}$$ $$=$$ $$( 108\nu^{11} + 729\nu^{9} + 4768\nu^{7} + 1242\nu^{5} + 135\nu^{3} - 11156\nu ) / 1222$$ (108*v^11 + 729*v^9 + 4768*v^7 + 1242*v^5 + 135*v^3 - 11156*v) / 1222 $$\beta_{6}$$ $$=$$ $$( -135\nu^{11} - 1064\nu^{9} - 7182\nu^{7} - 9801\nu^{5} - 12236\nu^{3} - 1330\nu ) / 1222$$ (-135*v^11 - 1064*v^9 - 7182*v^7 - 9801*v^5 - 12236*v^3 - 1330*v) / 1222 $$\beta_{7}$$ $$=$$ $$( -135\nu^{10} - 1064\nu^{8} - 7182\nu^{6} - 9801\nu^{4} - 12236\nu^{2} - 108 ) / 1222$$ (-135*v^10 - 1064*v^8 - 7182*v^6 - 9801*v^4 - 12236*v^2 - 108) / 1222 $$\beta_{8}$$ $$=$$ $$( -405\nu^{10} - 3192\nu^{8} - 21546\nu^{6} - 29403\nu^{4} - 35486\nu^{2} - 324 ) / 1222$$ (-405*v^10 - 3192*v^8 - 21546*v^6 - 29403*v^4 - 35486*v^2 - 324) / 1222 $$\beta_{9}$$ $$=$$ $$( 428\nu^{10} + 3500\nu^{8} + 23625\nu^{6} + 36694\nu^{4} + 40250\nu^{2} + 4375 ) / 1222$$ (428*v^10 + 3500*v^8 + 23625*v^6 + 36694*v^4 + 40250*v^2 + 4375) / 1222 $$\beta_{10}$$ $$=$$ $$( 428\nu^{11} + 3500\nu^{9} + 23625\nu^{7} + 36694\nu^{5} + 40250\nu^{3} + 4375\nu ) / 1222$$ (428*v^11 + 3500*v^9 + 23625*v^7 + 36694*v^5 + 40250*v^3 + 4375*v) / 1222 $$\beta_{11}$$ $$=$$ $$-\nu^{11} - 8\nu^{9} - 54\nu^{7} - 78\nu^{5} - 92\nu^{3} - 10\nu$$ -v^11 - 8*v^9 - 54*v^7 - 78*v^5 - 92*v^3 - 10*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} - 3\beta_{7}$$ b8 - 3*b7 $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{10} - 6\beta_{6} - \beta_{5} - \beta_{4} - 6\beta_1$$ b11 + b10 - 6*b6 - b5 - b4 - 6*b1 $$\nu^{4}$$ $$=$$ $$-\beta_{9} - 7\beta_{8} + 17\beta_{7} + 7\beta_{2} - 17$$ -b9 - 7*b8 + 17*b7 + 7*b2 - 17 $$\nu^{5}$$ $$=$$ $$-7\beta_{11} - 8\beta_{10} + 38\beta_{6}$$ -7*b11 - 8*b10 + 38*b6 $$\nu^{6}$$ $$=$$ $$-8\beta_{3} - 46\beta_{2} + 107$$ -8*b3 - 46*b2 + 107 $$\nu^{7}$$ $$=$$ $$46\beta_{5} + 54\beta_{4} + 245\beta_1$$ 46*b5 + 54*b4 + 245*b1 $$\nu^{8}$$ $$=$$ $$54\beta_{9} + 299\beta_{8} - 689\beta_{7} + 54\beta_{3}$$ 54*b9 + 299*b8 - 689*b7 + 54*b3 $$\nu^{9}$$ $$=$$ $$299\beta_{11} + 353\beta_{10} - 1586\beta_{6} - 299\beta_{5} - 353\beta_{4} - 1586\beta_1$$ 299*b11 + 353*b10 - 1586*b6 - 299*b5 - 353*b4 - 1586*b1 $$\nu^{10}$$ $$=$$ $$-353\beta_{9} - 1939\beta_{8} + 4459\beta_{7} + 1939\beta_{2} - 4459$$ -353*b9 - 1939*b8 + 4459*b7 + 1939*b2 - 4459 $$\nu^{11}$$ $$=$$ $$-1939\beta_{11} - 2292\beta_{10} + 10276\beta_{6}$$ -1939*b11 - 2292*b10 + 10276*b6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{7}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
126.1
 −1.27287 − 2.20467i −0.593667 − 1.02826i −0.165418 − 0.286513i 0.165418 + 0.286513i 0.593667 + 1.02826i 1.27287 + 2.20467i −1.27287 + 2.20467i −0.593667 + 1.02826i −0.165418 + 0.286513i 0.165418 − 0.286513i 0.593667 − 1.02826i 1.27287 − 2.20467i
−1.27287 + 2.20467i −1.07646 + 1.86449i −2.24039 3.88048i 0 −2.74039 4.74650i −1.46928 2.54486i 6.31544 −0.817544 1.41603i 0
126.2 −0.593667 + 1.02826i −0.172555 + 0.298874i 0.295120 + 0.511162i 0 −0.204880 0.354863i −1.01478 1.75765i −3.07548 1.44045 + 2.49493i 0
126.3 −0.165418 + 0.286513i 1.34590 2.33117i 0.945274 + 1.63726i 0 0.445274 + 0.771236i −1.67674 2.90420i −1.28714 −2.12291 3.67698i 0
126.4 0.165418 0.286513i −1.34590 + 2.33117i 0.945274 + 1.63726i 0 0.445274 + 0.771236i 1.67674 + 2.90420i 1.28714 −2.12291 3.67698i 0
126.5 0.593667 1.02826i 0.172555 0.298874i 0.295120 + 0.511162i 0 −0.204880 0.354863i 1.01478 + 1.75765i 3.07548 1.44045 + 2.49493i 0
126.6 1.27287 2.20467i 1.07646 1.86449i −2.24039 3.88048i 0 −2.74039 4.74650i 1.46928 + 2.54486i −6.31544 −0.817544 1.41603i 0
276.1 −1.27287 2.20467i −1.07646 1.86449i −2.24039 + 3.88048i 0 −2.74039 + 4.74650i −1.46928 + 2.54486i 6.31544 −0.817544 + 1.41603i 0
276.2 −0.593667 1.02826i −0.172555 0.298874i 0.295120 0.511162i 0 −0.204880 + 0.354863i −1.01478 + 1.75765i −3.07548 1.44045 2.49493i 0
276.3 −0.165418 0.286513i 1.34590 + 2.33117i 0.945274 1.63726i 0 0.445274 0.771236i −1.67674 + 2.90420i −1.28714 −2.12291 + 3.67698i 0
276.4 0.165418 + 0.286513i −1.34590 2.33117i 0.945274 1.63726i 0 0.445274 0.771236i 1.67674 2.90420i 1.28714 −2.12291 + 3.67698i 0
276.5 0.593667 + 1.02826i 0.172555 + 0.298874i 0.295120 0.511162i 0 −0.204880 + 0.354863i 1.01478 1.75765i 3.07548 1.44045 2.49493i 0
276.6 1.27287 + 2.20467i 1.07646 + 1.86449i −2.24039 + 3.88048i 0 −2.74039 + 4.74650i 1.46928 2.54486i −6.31544 −0.817544 + 1.41603i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 126.6 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
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.e.e 12
5.b even 2 1 inner 325.2.e.e 12
5.c odd 4 2 65.2.n.a 12
13.c even 3 1 inner 325.2.e.e 12
13.c even 3 1 4225.2.a.br 6
13.e even 6 1 4225.2.a.bq 6
15.e even 4 2 585.2.bs.a 12
20.e even 4 2 1040.2.dh.a 12
65.f even 4 2 845.2.l.f 24
65.h odd 4 2 845.2.n.e 12
65.k even 4 2 845.2.l.f 24
65.l even 6 1 4225.2.a.bq 6
65.n even 6 1 inner 325.2.e.e 12
65.n even 6 1 4225.2.a.br 6
65.o even 12 2 845.2.d.d 12
65.o even 12 2 845.2.l.f 24
65.q odd 12 2 65.2.n.a 12
65.q odd 12 2 845.2.b.d 6
65.r odd 12 2 845.2.b.e 6
65.r odd 12 2 845.2.n.e 12
65.t even 12 2 845.2.d.d 12
65.t even 12 2 845.2.l.f 24
195.bl even 12 2 585.2.bs.a 12
260.bj even 12 2 1040.2.dh.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 5.c odd 4 2
65.2.n.a 12 65.q odd 12 2
325.2.e.e 12 1.a even 1 1 trivial
325.2.e.e 12 5.b even 2 1 inner
325.2.e.e 12 13.c even 3 1 inner
325.2.e.e 12 65.n even 6 1 inner
585.2.bs.a 12 15.e even 4 2
585.2.bs.a 12 195.bl even 12 2
845.2.b.d 6 65.q odd 12 2
845.2.b.e 6 65.r odd 12 2
845.2.d.d 12 65.o even 12 2
845.2.d.d 12 65.t even 12 2
845.2.l.f 24 65.f even 4 2
845.2.l.f 24 65.k even 4 2
845.2.l.f 24 65.o even 12 2
845.2.l.f 24 65.t even 12 2
845.2.n.e 12 65.h odd 4 2
845.2.n.e 12 65.r odd 12 2
1040.2.dh.a 12 20.e even 4 2
1040.2.dh.a 12 260.bj even 12 2
4225.2.a.bq 6 13.e even 6 1
4225.2.a.bq 6 65.l even 6 1
4225.2.a.br 6 13.c even 3 1
4225.2.a.br 6 65.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 8T_{2}^{10} + 54T_{2}^{8} + 78T_{2}^{6} + 92T_{2}^{4} + 10T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 8 T^{10} + \cdots + 1$$
$3$ $$T^{12} + 12 T^{10} + \cdots + 16$$
$5$ $$T^{12}$$
$7$ $$T^{12} + 24 T^{10} + \cdots + 160000$$
$11$ $$(T^{6} + 13 T^{4} + \cdots + 64)^{2}$$
$13$ $$T^{12} + 15 T^{10} + \cdots + 4826809$$
$17$ $$T^{12} + 35 T^{10} + \cdots + 28561$$
$19$ $$(T^{6} + 6 T^{5} + \cdots + 100)^{2}$$
$23$ $$T^{12} + 12 T^{10} + \cdots + 16$$
$29$ $$(T^{2} + 3 T + 9)^{6}$$
$31$ $$(T^{3} + 4 T^{2} - 40 T + 40)^{4}$$
$37$ $$T^{12} + 35 T^{10} + \cdots + 28561$$
$41$ $$(T^{6} - 7 T^{5} + 78 T^{4} + \cdots + 25)^{2}$$
$43$ $$T^{12} + 80 T^{10} + \cdots + 65536$$
$47$ $$(T^{6} - 236 T^{4} + \cdots - 270400)^{2}$$
$53$ $$(T^{6} - 171 T^{4} + \cdots - 400)^{2}$$
$59$ $$(T^{6} - 2 T^{5} + \cdots + 18496)^{2}$$
$61$ $$(T^{6} - 3 T^{5} + \cdots + 13225)^{2}$$
$67$ $$T^{12} + \cdots + 406586896$$
$71$ $$(T^{6} + 6 T^{5} + \cdots + 676)^{2}$$
$73$ $$(T^{6} - 215 T^{4} + \cdots - 250000)^{2}$$
$79$ $$(T^{3} - 26 T^{2} + \cdots - 160)^{4}$$
$83$ $$(T^{6} - 276 T^{4} + \cdots - 640000)^{2}$$
$89$ $$(T^{6} + 10 T^{5} + \cdots + 2515396)^{2}$$
$97$ $$T^{12} + \cdots + 41740124416$$