# Properties

 Label 375.2.b.c Level $375$ Weight $2$ Character orbit 375.b Analytic conductor $2.994$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(124,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("375.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 375.b (of order $$2$$, degree $$1$$, 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: $$2.99439007580$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1632160000.5 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 12x^{6} + 46x^{4} + 65x^{2} + 25$$ x^8 + 12*x^6 + 46*x^4 + 65*x^2 + 25 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{7} + \beta_{4}) q^{2} - \beta_{4} q^{3} + (\beta_{6} + \beta_{5} + \beta_{2} - 1) q^{4} + ( - \beta_{2} + 1) q^{6} + 2 \beta_1 q^{7} + ( - \beta_{7} - 2 \beta_{4} + \beta_1) q^{8} - q^{9}+O(q^{10})$$ q + (b7 + b4) * q^2 - b4 * q^3 + (b6 + b5 + b2 - 1) * q^4 + (-b2 + 1) * q^6 + 2*b1 * q^7 + (-b7 - 2*b4 + b1) * q^8 - q^9 $$q + (\beta_{7} + \beta_{4}) q^{2} - \beta_{4} q^{3} + (\beta_{6} + \beta_{5} + \beta_{2} - 1) q^{4} + ( - \beta_{2} + 1) q^{6} + 2 \beta_1 q^{7} + ( - \beta_{7} - 2 \beta_{4} + \beta_1) q^{8} - q^{9} + (2 \beta_{6} + 2 \beta_{2} + 2) q^{11} + (\beta_{7} + \beta_{4} + \beta_{3} - \beta_1) q^{12} + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{13} + (4 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 2) q^{14} + (3 \beta_{6} - \beta_{2} + 3) q^{16} + ( - \beta_{4} + 3 \beta_{3}) q^{17} + ( - \beta_{7} - \beta_{4}) q^{18} + ( - \beta_{6} - 2 \beta_{2} - 2) q^{19} + 2 \beta_{5} q^{21} + ( - 2 \beta_{4} - 4 \beta_{3}) q^{22} + (2 \beta_{7} - 3 \beta_{4} + \beta_{3} - 2 \beta_1) q^{23} + (\beta_{5} + \beta_{2} - 2) q^{24} + (6 \beta_{6} - 4 \beta_{5} + 2 \beta_{2}) q^{26} + \beta_{4} q^{27} + (8 \beta_{4} - 6 \beta_{3} - 4 \beta_1) q^{28} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 2) q^{29} + ( - 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + 1) q^{31} + ( - 2 \beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{32} + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3}) q^{33} + (3 \beta_{6} - 3 \beta_{5} + 2 \beta_{2} + 1) q^{34} + ( - \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{36} + ( - 2 \beta_{7} - 6 \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{37} + ( - \beta_{7} + 2 \beta_{4} + 3 \beta_{3} - \beta_1) q^{38} + ( - 2 \beta_{6} + 2 \beta_{5} + 2) q^{39} + ( - 2 \beta_{6} - 2 \beta_{5} + 4) q^{41} + ( - 2 \beta_{7} - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_1) q^{42} + (2 \beta_{7} + 6 \beta_{4} - 4 \beta_{3} + 2 \beta_1) q^{43} + (4 \beta_{5} - 2 \beta_{2} + 6) q^{44} + ( - \beta_{6} + 3 \beta_{5} - 3) q^{46} + ( - 2 \beta_{7} - \beta_{3}) q^{47} + ( - \beta_{7} - 3 \beta_{4} + 3 \beta_{3}) q^{48} + ( - 4 \beta_{6} + 4 \beta_{5} - 5) q^{49} + ( - 3 \beta_{6} - 1) q^{51} + ( - 2 \beta_{7} + 4 \beta_{4} - 12 \beta_{3} - 4 \beta_1) q^{52} + (2 \beta_{4} + \beta_{3} + 4 \beta_1) q^{53} + (\beta_{2} - 1) q^{54} + ( - 6 \beta_{6} + 6 \beta_{5} + 2 \beta_{2} - 8) q^{56} + ( - 2 \beta_{7} + 2 \beta_{4} - \beta_{3}) q^{57} + ( - 2 \beta_{7} - 4 \beta_{4} - 8 \beta_{3} - 2 \beta_1) q^{58} + ( - 6 \beta_{6} + 2 \beta_{2} - 4) q^{59} + ( - \beta_{6} - 4 \beta_{5}) q^{61} + (2 \beta_{7} + 3 \beta_{4} + 9 \beta_{3} + 3 \beta_1) q^{62} - 2 \beta_1 q^{63} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{2} + 7) q^{64} + (4 \beta_{6} - 2) q^{66} + (4 \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{67} + (\beta_{7} - 2 \beta_{4} - 5 \beta_{3} - 4 \beta_1) q^{68} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{2} - 3) q^{69} + ( - 6 \beta_{6} - 2 \beta_{5} - 4 \beta_{2} - 6) q^{71} + (\beta_{7} + 2 \beta_{4} - \beta_1) q^{72} - 2 \beta_{7} q^{73} + ( - 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 8) q^{74} + ( - 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{2} - 5) q^{76} + (4 \beta_{4} - 8 \beta_{3} + 4 \beta_1) q^{77} + (2 \beta_{7} + 6 \beta_{3} + 4 \beta_1) q^{78} + ( - \beta_{6} - 4 \beta_{5} + 2 \beta_{2} + 2) q^{79} + q^{81} + (8 \beta_{7} + 6 \beta_{4} - 2 \beta_{3}) q^{82} + (2 \beta_{7} - 8 \beta_{4} + 5 \beta_{3}) q^{83} + (6 \beta_{6} - 4 \beta_{5} + 8) q^{84} + (2 \beta_{6} + 4 \beta_{5} - 8) q^{86} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{87} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{88} + (2 \beta_{6} - 4 \beta_{5} - 8) q^{89} + ( - 4 \beta_{6} + 4 \beta_{2} - 12) q^{91} + ( - \beta_{7} - 12 \beta_{4} + 9 \beta_{3}) q^{92} + ( - 2 \beta_{7} - \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{93} + ( - 3 \beta_{6} - \beta_{5} - \beta_{2} + 4) q^{94} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{2} + 1) q^{96} + (2 \beta_{7} - 6 \beta_{4} + 8 \beta_{3} + 4 \beta_1) q^{97} + ( - 5 \beta_{7} - 9 \beta_{4} + 12 \beta_{3} + 8 \beta_1) q^{98} + ( - 2 \beta_{6} - 2 \beta_{2} - 2) q^{99}+O(q^{100})$$ q + (b7 + b4) * q^2 - b4 * q^3 + (b6 + b5 + b2 - 1) * q^4 + (-b2 + 1) * q^6 + 2*b1 * q^7 + (-b7 - 2*b4 + b1) * q^8 - q^9 + (2*b6 + 2*b2 + 2) * q^11 + (b7 + b4 + b3 - b1) * q^12 + (2*b4 + 2*b3 + 2*b1) * q^13 + (4*b6 - 2*b5 - 2*b2 + 2) * q^14 + (3*b6 - b2 + 3) * q^16 + (-b4 + 3*b3) * q^17 + (-b7 - b4) * q^18 + (-b6 - 2*b2 - 2) * q^19 + 2*b5 * q^21 + (-2*b4 - 4*b3) * q^22 + (2*b7 - 3*b4 + b3 - 2*b1) * q^23 + (b5 + b2 - 2) * q^24 + (6*b6 - 4*b5 + 2*b2) * q^26 + b4 * q^27 + (8*b4 - 6*b3 - 4*b1) * q^28 + (2*b6 - 2*b5 + 2*b2 - 2) * q^29 + (-3*b6 + 2*b5 - 2*b2 + 1) * q^31 + (-2*b7 + b4 - 2*b3 - 2*b1) * q^32 + (2*b7 - 2*b4 + 2*b3) * q^33 + (3*b6 - 3*b5 + 2*b2 + 1) * q^34 + (-b6 - b5 - b2 + 1) * q^36 + (-2*b7 - 6*b4 + 2*b3 - 2*b1) * q^37 + (-b7 + 2*b4 + 3*b3 - b1) * q^38 + (-2*b6 + 2*b5 + 2) * q^39 + (-2*b6 - 2*b5 + 4) * q^41 + (-2*b7 - 2*b4 + 4*b3 + 2*b1) * q^42 + (2*b7 + 6*b4 - 4*b3 + 2*b1) * q^43 + (4*b5 - 2*b2 + 6) * q^44 + (-b6 + 3*b5 - 3) * q^46 + (-2*b7 - b3) * q^47 + (-b7 - 3*b4 + 3*b3) * q^48 + (-4*b6 + 4*b5 - 5) * q^49 + (-3*b6 - 1) * q^51 + (-2*b7 + 4*b4 - 12*b3 - 4*b1) * q^52 + (2*b4 + b3 + 4*b1) * q^53 + (b2 - 1) * q^54 + (-6*b6 + 6*b5 + 2*b2 - 8) * q^56 + (-2*b7 + 2*b4 - b3) * q^57 + (-2*b7 - 4*b4 - 8*b3 - 2*b1) * q^58 + (-6*b6 + 2*b2 - 4) * q^59 + (-b6 - 4*b5) * q^61 + (2*b7 + 3*b4 + 9*b3 + 3*b1) * q^62 - 2*b1 * q^63 + (-2*b6 + 2*b5 - b2 + 7) * q^64 + (4*b6 - 2) * q^66 + (4*b7 - 2*b4 - 2*b1) * q^67 + (b7 - 2*b4 - 5*b3 - 4*b1) * q^68 + (-b6 - 2*b5 - 2*b2 - 3) * q^69 + (-6*b6 - 2*b5 - 4*b2 - 6) * q^71 + (b7 + 2*b4 - b1) * q^72 - 2*b7 * q^73 + (-4*b6 - 2*b5 - 2*b2 + 8) * q^74 + (-2*b6 - 3*b5 + 2*b2 - 5) * q^76 + (4*b4 - 8*b3 + 4*b1) * q^77 + (2*b7 + 6*b3 + 4*b1) * q^78 + (-b6 - 4*b5 + 2*b2 + 2) * q^79 + q^81 + (8*b7 + 6*b4 - 2*b3) * q^82 + (2*b7 - 8*b4 + 5*b3) * q^83 + (6*b6 - 4*b5 + 8) * q^84 + (2*b6 + 4*b5 - 8) * q^86 + (2*b7 + 2*b4 + 2*b3 + 2*b1) * q^87 + (2*b7 + 2*b4 + 2*b3 + 2*b1) * q^88 + (2*b6 - 4*b5 - 8) * q^89 + (-4*b6 + 4*b2 - 12) * q^91 + (-b7 - 12*b4 + 9*b3) * q^92 + (-2*b7 - b4 - 3*b3 - 2*b1) * q^93 + (-3*b6 - b5 - b2 + 4) * q^94 + (2*b6 - 2*b5 + 2*b2 + 1) * q^96 + (2*b7 - 6*b4 + 8*b3 + 4*b1) * q^97 + (-5*b7 - 9*b4 + 12*b3 + 8*b1) * q^98 + (-2*b6 - 2*b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 14 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10})$$ 8 * q - 14 * q^4 + 6 * q^6 - 8 * q^9 $$8 q - 14 q^{4} + 6 q^{6} - 8 q^{9} + 12 q^{11} + 4 q^{14} + 10 q^{16} - 16 q^{19} - 8 q^{21} - 18 q^{24} - 4 q^{26} - 12 q^{29} + 8 q^{31} + 12 q^{34} + 14 q^{36} + 16 q^{39} + 48 q^{41} + 28 q^{44} - 32 q^{46} - 40 q^{49} + 4 q^{51} - 6 q^{54} - 60 q^{56} - 4 q^{59} + 20 q^{61} + 54 q^{64} - 32 q^{66} - 16 q^{69} - 24 q^{71} + 84 q^{74} - 16 q^{76} + 40 q^{79} + 8 q^{81} + 56 q^{84} - 88 q^{86} - 56 q^{89} - 72 q^{91} + 46 q^{94} + 12 q^{96} - 12 q^{99}+O(q^{100})$$ 8 * q - 14 * q^4 + 6 * q^6 - 8 * q^9 + 12 * q^11 + 4 * q^14 + 10 * q^16 - 16 * q^19 - 8 * q^21 - 18 * q^24 - 4 * q^26 - 12 * q^29 + 8 * q^31 + 12 * q^34 + 14 * q^36 + 16 * q^39 + 48 * q^41 + 28 * q^44 - 32 * q^46 - 40 * q^49 + 4 * q^51 - 6 * q^54 - 60 * q^56 - 4 * q^59 + 20 * q^61 + 54 * q^64 - 32 * q^66 - 16 * q^69 - 24 * q^71 + 84 * q^74 - 16 * q^76 + 40 * q^79 + 8 * q^81 + 56 * q^84 - 88 * q^86 - 56 * q^89 - 72 * q^91 + 46 * q^94 + 12 * q^96 - 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 12x^{6} + 46x^{4} + 65x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{6} + 7\nu^{4} + 6\nu^{2} - 5 ) / 5$$ (v^6 + 7*v^4 + 6*v^2 - 5) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 17\nu^{5} + 81\nu^{3} + 95\nu ) / 25$$ (v^7 + 17*v^5 + 81*v^3 + 95*v) / 25 $$\beta_{4}$$ $$=$$ $$( -3\nu^{7} - 26\nu^{5} - 43\nu^{3} + 15\nu ) / 25$$ (-3*v^7 - 26*v^5 - 43*v^3 + 15*v) / 25 $$\beta_{5}$$ $$=$$ $$( -2\nu^{6} - 19\nu^{4} - 42\nu^{2} - 15 ) / 5$$ (-2*v^6 - 19*v^4 - 42*v^2 - 15) / 5 $$\beta_{6}$$ $$=$$ $$( -2\nu^{6} - 19\nu^{4} - 47\nu^{2} - 30 ) / 5$$ (-2*v^6 - 19*v^4 - 47*v^2 - 30) / 5 $$\beta_{7}$$ $$=$$ $$( 2\nu^{7} + 19\nu^{5} + 47\nu^{3} + 30\nu ) / 5$$ (2*v^7 + 19*v^5 + 47*v^3 + 30*v) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{5} - 3$$ -b6 + b5 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3\beta_{4} - \beta_{3} - 4\beta_1$$ b7 + 3*b4 - b3 - 4*b1 $$\nu^{4}$$ $$=$$ $$6\beta_{6} - 7\beta_{5} - 2\beta_{2} + 13$$ 6*b6 - 7*b5 - 2*b2 + 13 $$\nu^{5}$$ $$=$$ $$-8\beta_{7} - 23\beta_{4} + 11\beta_{3} + 20\beta_1$$ -8*b7 - 23*b4 + 11*b3 + 20*b1 $$\nu^{6}$$ $$=$$ $$-36\beta_{6} + 43\beta_{5} + 19\beta_{2} - 68$$ -36*b6 + 43*b5 + 19*b2 - 68 $$\nu^{7}$$ $$=$$ $$55\beta_{7} + 148\beta_{4} - 81\beta_{3} - 111\beta_1$$ 55*b7 + 148*b4 - 81*b3 - 111*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 2.46673i − 0.777484i − 1.77748i − 1.46673i 1.46673i 1.77748i 0.777484i − 2.46673i
2.52452i 1.00000i −4.37322 0 2.52452 4.93346i 5.99126i −1.00000 0
124.2 2.25800i 1.00000i −3.09855 0 2.25800 1.55497i 2.48051i −1.00000 0
124.3 1.87603i 1.00000i −1.51949 0 −1.87603 3.55497i 0.901454i −1.00000 0
124.4 0.0935099i 1.00000i 1.99126 0 0.0935099 2.93346i 0.373222i −1.00000 0
124.5 0.0935099i 1.00000i 1.99126 0 0.0935099 2.93346i 0.373222i −1.00000 0
124.6 1.87603i 1.00000i −1.51949 0 −1.87603 3.55497i 0.901454i −1.00000 0
124.7 2.25800i 1.00000i −3.09855 0 2.25800 1.55497i 2.48051i −1.00000 0
124.8 2.52452i 1.00000i −4.37322 0 2.52452 4.93346i 5.99126i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.b.c 8
3.b odd 2 1 1125.2.b.g 8
4.b odd 2 1 6000.2.f.o 8
5.b even 2 1 inner 375.2.b.c 8
5.c odd 4 1 375.2.a.e 4
5.c odd 4 1 375.2.a.f yes 4
15.d odd 2 1 1125.2.b.g 8
15.e even 4 1 1125.2.a.h 4
15.e even 4 1 1125.2.a.l 4
20.d odd 2 1 6000.2.f.o 8
20.e even 4 1 6000.2.a.bg 4
20.e even 4 1 6000.2.a.bh 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
375.2.a.e 4 5.c odd 4 1
375.2.a.f yes 4 5.c odd 4 1
375.2.b.c 8 1.a even 1 1 trivial
375.2.b.c 8 5.b even 2 1 inner
1125.2.a.h 4 15.e even 4 1
1125.2.a.l 4 15.e even 4 1
1125.2.b.g 8 3.b odd 2 1
1125.2.b.g 8 15.d odd 2 1
6000.2.a.bg 4 20.e even 4 1
6000.2.a.bh 4 20.e even 4 1
6000.2.f.o 8 4.b odd 2 1
6000.2.f.o 8 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 15T_{2}^{6} + 73T_{2}^{4} + 115T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(375, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 15 T^{6} + 73 T^{4} + 115 T^{2} + \cdots + 1$$
$3$ $$(T^{2} + 1)^{4}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 48 T^{6} + 736 T^{4} + \cdots + 6400$$
$11$ $$(T^{4} - 6 T^{3} - 12 T^{2} + 88 T - 16)^{2}$$
$13$ $$T^{8} + 80 T^{6} + 1888 T^{4} + \cdots + 30976$$
$17$ $$(T^{4} + 23 T^{2} + 121)^{2}$$
$19$ $$(T^{4} + 8 T^{3} + T^{2} - 60 T + 25)^{2}$$
$23$ $$T^{8} + 102 T^{6} + 2491 T^{4} + \cdots + 3025$$
$29$ $$(T^{4} + 6 T^{3} - 36 T^{2} - 40 T + 80)^{2}$$
$31$ $$(T^{4} - 4 T^{3} - 51 T^{2} - 80 T + 25)^{2}$$
$37$ $$T^{8} + 188 T^{6} + 12816 T^{4} + \cdots + 4000000$$
$41$ $$(T^{4} - 24 T^{3} + 184 T^{2} - 440 T - 80)^{2}$$
$43$ $$T^{8} + 244 T^{6} + 19936 T^{4} + \cdots + 4946176$$
$47$ $$(T^{4} + 23 T^{2} + 101)^{2}$$
$53$ $$T^{8} + 182 T^{6} + 11491 T^{4} + \cdots + 2088025$$
$59$ $$(T^{4} + 2 T^{3} - 144 T^{2} - 320 T + 2320)^{2}$$
$61$ $$(T^{4} - 10 T^{3} - 53 T^{2} + 470 T + 781)^{2}$$
$67$ $$T^{8} + 256 T^{6} + 21856 T^{4} + \cdots + 1290496$$
$71$ $$(T^{4} + 12 T^{3} - 96 T^{2} - 392 T + 656)^{2}$$
$73$ $$T^{8} + 52 T^{6} + 736 T^{4} + \cdots + 6400$$
$79$ $$(T^{4} - 20 T^{3} + 25 T^{2} + 500 T + 725)^{2}$$
$83$ $$T^{8} + 270 T^{6} + 19963 T^{4} + \cdots + 2343961$$
$89$ $$(T^{4} + 28 T^{3} + 196 T^{2} - 160 T - 2320)^{2}$$
$97$ $$T^{8} + 556 T^{6} + \cdots + 82882816$$