# Properties

 Label 525.3.s.g Level $525$ Weight $3$ Character orbit 525.s Analytic conductor $14.305$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 525.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3052138789$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - 2 \beta_{5} + 3 \beta_{2}) q^{4} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 1) q^{6} + (5 \beta_{3} + 3 \beta_1) q^{7} + ( - \beta_{7} + 11 \beta_{3}) q^{8} + (3 \beta_{2} - 3) q^{9}+O(q^{10})$$ q + (-b4 + b1) * q^2 + (b3 + b1) * q^3 + (-2*b5 + 3*b2) * q^4 + (b6 - 2*b5 + 2*b2 - 1) * q^6 + (5*b3 + 3*b1) * q^7 + (-b7 + 11*b3) * q^8 + (3*b2 - 3) * q^9 $$q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - 2 \beta_{5} + 3 \beta_{2}) q^{4} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 1) q^{6} + (5 \beta_{3} + 3 \beta_1) q^{7} + ( - \beta_{7} + 11 \beta_{3}) q^{8} + (3 \beta_{2} - 3) q^{9} + (2 \beta_{5} + 2 \beta_{2}) q^{11} + ( - 4 \beta_{7} + 2 \beta_{4} + 6 \beta_{3} - 3 \beta_1) q^{12} + ( - 2 \beta_{7} + 4 \beta_{4} + 5 \beta_{3} - 10 \beta_1) q^{13} + (5 \beta_{6} - 8 \beta_{5} + 8 \beta_{2} - 5) q^{14} + (4 \beta_{6} - 4 \beta_{5} + 5 \beta_{2} - 5) q^{16} + ( - 2 \beta_{7} - 2 \beta_{4} - 6 \beta_{3} - 6 \beta_1) q^{17} + ( - 3 \beta_{7} + 3 \beta_{4} + 3 \beta_{3} - 3 \beta_1) q^{18} + ( - 6 \beta_{6} - 6 \beta_{5} - 5 \beta_{2} - 5) q^{19} + (11 \beta_{2} - 13) q^{21} - 10 \beta_{3} q^{22} + 20 \beta_1 q^{23} + (2 \beta_{6} - \beta_{5} + 11 \beta_{2} - 22) q^{24} + (7 \beta_{6} + 7 \beta_{5} - 17 \beta_{2} - 17) q^{26} + (3 \beta_{3} - 6 \beta_1) q^{27} + ( - 16 \beta_{7} + 10 \beta_{4} + 24 \beta_{3} - 15 \beta_1) q^{28} + ( - 12 \beta_{6} + 2) q^{29} + ( - 16 \beta_{6} + 8 \beta_{5}) q^{31} + ( - 5 \beta_{7} + 5 \beta_{4} - 15 \beta_{3} + 15 \beta_1) q^{32} + (4 \beta_{7} - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_1) q^{33} + ( - 4 \beta_{6} + 8 \beta_{5} + 12 \beta_{2} - 6) q^{34} + (6 \beta_{6} - 9) q^{36} + (10 \beta_{4} - 25 \beta_1) q^{37} + ( - \beta_{7} - \beta_{4} + 31 \beta_{3} + 31 \beta_1) q^{38} + (6 \beta_{5} - 15 \beta_{2}) q^{39} + ( - 14 \beta_{6} + 28 \beta_{5} - 20 \beta_{2} + 10) q^{41} + ( - 11 \beta_{7} + 13 \beta_{4} + 11 \beta_{3} - 13 \beta_1) q^{42} + ( - 4 \beta_{7} + 50 \beta_{3}) q^{43} + ( - 2 \beta_{6} + 2 \beta_{5} - 18 \beta_{2} + 18) q^{44} + ( - 20 \beta_{5} + 20 \beta_{2}) q^{46} + ( - 12 \beta_{7} + 6 \beta_{4} + 48 \beta_{3} - 24 \beta_1) q^{47} + ( - 4 \beta_{7} + 8 \beta_{4} + 5 \beta_{3} - 10 \beta_1) q^{48} + (39 \beta_{2} - 55) q^{49} + (6 \beta_{6} - 6 \beta_{5} - 18 \beta_{2} + 18) q^{51} + (16 \beta_{7} + 16 \beta_{4} - 39 \beta_{3} - 39 \beta_1) q^{52} + (30 \beta_{7} - 30 \beta_{4} - 22 \beta_{3} + 22 \beta_1) q^{53} + (3 \beta_{6} + 3 \beta_{5} - 3 \beta_{2} - 3) q^{54} + (8 \beta_{6} - 3 \beta_{5} + 33 \beta_{2} - 88) q^{56} + ( - 18 \beta_{7} - 15 \beta_{3}) q^{57} + ( - 14 \beta_{4} + 74 \beta_1) q^{58} + (36 \beta_{6} - 18 \beta_{5} - 4 \beta_{2} + 8) q^{59} + ( - 6 \beta_{6} - 6 \beta_{5} - 39 \beta_{2} - 39) q^{61} + (8 \beta_{7} - 16 \beta_{4} - 48 \beta_{3} + 96 \beta_1) q^{62} + (9 \beta_{3} - 24 \beta_1) q^{63} + ( - 26 \beta_{6} + 5) q^{64} + ( - 10 \beta_{2} + 20) q^{66} + (38 \beta_{7} - 38 \beta_{4} - \beta_{3} + \beta_1) q^{67} + (12 \beta_{7} - 6 \beta_{4} + 12 \beta_{3} - 6 \beta_1) q^{68} + (40 \beta_{2} - 20) q^{69} + ( - 20 \beta_{6} - 26) q^{71} + (3 \beta_{4} - 33 \beta_1) q^{72} + (4 \beta_{7} + 4 \beta_{4} - 15 \beta_{3} - 15 \beta_1) q^{73} + (35 \beta_{5} - 85 \beta_{2}) q^{74} + (8 \beta_{6} - 16 \beta_{5} + 114 \beta_{2} - 57) q^{76} + (16 \beta_{7} - 10 \beta_{4} + 16 \beta_{3} - 10 \beta_1) q^{77} + (21 \beta_{7} - 51 \beta_{3}) q^{78} + ( - 16 \beta_{6} + 16 \beta_{5} + 19 \beta_{2} - 19) q^{79} - 9 \beta_{2} q^{81} + (48 \beta_{7} - 24 \beta_{4} - 188 \beta_{3} + 94 \beta_1) q^{82} + ( - 10 \beta_{7} + 20 \beta_{4} - 24 \beta_{3} + 48 \beta_1) q^{83} + (22 \beta_{6} + 4 \beta_{5} - 6 \beta_{2} - 33) q^{84} + (54 \beta_{6} - 54 \beta_{5} + 74 \beta_{2} - 74) q^{86} + ( - 12 \beta_{7} - 12 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{87} + (20 \beta_{7} - 20 \beta_{4} + 10 \beta_{3} - 10 \beta_1) q^{88} + (10 \beta_{6} + 10 \beta_{5} + 12 \beta_{2} + 12) q^{89} + ( - 4 \beta_{6} + 26 \beta_{5} - 65 \beta_{2} + 10) q^{91} + ( - 40 \beta_{7} + 60 \beta_{3}) q^{92} - 24 \beta_{4} q^{93} + (60 \beta_{6} - 30 \beta_{5} + 60 \beta_{2} - 120) q^{94} + (5 \beta_{6} + 5 \beta_{5} + 15 \beta_{2} + 15) q^{96} + (24 \beta_{7} - 48 \beta_{4} + 17 \beta_{3} - 34 \beta_1) q^{97} + ( - 39 \beta_{7} + 55 \beta_{4} + 39 \beta_{3} - 55 \beta_1) q^{98} + ( - 6 \beta_{6} - 6) q^{99}+O(q^{100})$$ q + (-b4 + b1) * q^2 + (b3 + b1) * q^3 + (-2*b5 + 3*b2) * q^4 + (b6 - 2*b5 + 2*b2 - 1) * q^6 + (5*b3 + 3*b1) * q^7 + (-b7 + 11*b3) * q^8 + (3*b2 - 3) * q^9 + (2*b5 + 2*b2) * q^11 + (-4*b7 + 2*b4 + 6*b3 - 3*b1) * q^12 + (-2*b7 + 4*b4 + 5*b3 - 10*b1) * q^13 + (5*b6 - 8*b5 + 8*b2 - 5) * q^14 + (4*b6 - 4*b5 + 5*b2 - 5) * q^16 + (-2*b7 - 2*b4 - 6*b3 - 6*b1) * q^17 + (-3*b7 + 3*b4 + 3*b3 - 3*b1) * q^18 + (-6*b6 - 6*b5 - 5*b2 - 5) * q^19 + (11*b2 - 13) * q^21 - 10*b3 * q^22 + 20*b1 * q^23 + (2*b6 - b5 + 11*b2 - 22) * q^24 + (7*b6 + 7*b5 - 17*b2 - 17) * q^26 + (3*b3 - 6*b1) * q^27 + (-16*b7 + 10*b4 + 24*b3 - 15*b1) * q^28 + (-12*b6 + 2) * q^29 + (-16*b6 + 8*b5) * q^31 + (-5*b7 + 5*b4 - 15*b3 + 15*b1) * q^32 + (4*b7 - 2*b4 + 4*b3 - 2*b1) * q^33 + (-4*b6 + 8*b5 + 12*b2 - 6) * q^34 + (6*b6 - 9) * q^36 + (10*b4 - 25*b1) * q^37 + (-b7 - b4 + 31*b3 + 31*b1) * q^38 + (6*b5 - 15*b2) * q^39 + (-14*b6 + 28*b5 - 20*b2 + 10) * q^41 + (-11*b7 + 13*b4 + 11*b3 - 13*b1) * q^42 + (-4*b7 + 50*b3) * q^43 + (-2*b6 + 2*b5 - 18*b2 + 18) * q^44 + (-20*b5 + 20*b2) * q^46 + (-12*b7 + 6*b4 + 48*b3 - 24*b1) * q^47 + (-4*b7 + 8*b4 + 5*b3 - 10*b1) * q^48 + (39*b2 - 55) * q^49 + (6*b6 - 6*b5 - 18*b2 + 18) * q^51 + (16*b7 + 16*b4 - 39*b3 - 39*b1) * q^52 + (30*b7 - 30*b4 - 22*b3 + 22*b1) * q^53 + (3*b6 + 3*b5 - 3*b2 - 3) * q^54 + (8*b6 - 3*b5 + 33*b2 - 88) * q^56 + (-18*b7 - 15*b3) * q^57 + (-14*b4 + 74*b1) * q^58 + (36*b6 - 18*b5 - 4*b2 + 8) * q^59 + (-6*b6 - 6*b5 - 39*b2 - 39) * q^61 + (8*b7 - 16*b4 - 48*b3 + 96*b1) * q^62 + (9*b3 - 24*b1) * q^63 + (-26*b6 + 5) * q^64 + (-10*b2 + 20) * q^66 + (38*b7 - 38*b4 - b3 + b1) * q^67 + (12*b7 - 6*b4 + 12*b3 - 6*b1) * q^68 + (40*b2 - 20) * q^69 + (-20*b6 - 26) * q^71 + (3*b4 - 33*b1) * q^72 + (4*b7 + 4*b4 - 15*b3 - 15*b1) * q^73 + (35*b5 - 85*b2) * q^74 + (8*b6 - 16*b5 + 114*b2 - 57) * q^76 + (16*b7 - 10*b4 + 16*b3 - 10*b1) * q^77 + (21*b7 - 51*b3) * q^78 + (-16*b6 + 16*b5 + 19*b2 - 19) * q^79 - 9*b2 * q^81 + (48*b7 - 24*b4 - 188*b3 + 94*b1) * q^82 + (-10*b7 + 20*b4 - 24*b3 + 48*b1) * q^83 + (22*b6 + 4*b5 - 6*b2 - 33) * q^84 + (54*b6 - 54*b5 + 74*b2 - 74) * q^86 + (-12*b7 - 12*b4 + 2*b3 + 2*b1) * q^87 + (20*b7 - 20*b4 + 10*b3 - 10*b1) * q^88 + (10*b6 + 10*b5 + 12*b2 + 12) * q^89 + (-4*b6 + 26*b5 - 65*b2 + 10) * q^91 + (-40*b7 + 60*b3) * q^92 - 24*b4 * q^93 + (60*b6 - 30*b5 + 60*b2 - 120) * q^94 + (5*b6 + 5*b5 + 15*b2 + 15) * q^96 + (24*b7 - 48*b4 + 17*b3 - 34*b1) * q^97 + (-39*b7 + 55*b4 + 39*b3 - 55*b1) * q^98 + (-6*b6 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 12 q^{4} - 12 q^{9}+O(q^{10})$$ 8 * q + 12 * q^4 - 12 * q^9 $$8 q + 12 q^{4} - 12 q^{9} + 8 q^{11} - 8 q^{14} - 20 q^{16} - 60 q^{19} - 60 q^{21} - 132 q^{24} - 204 q^{26} + 16 q^{29} - 72 q^{36} - 60 q^{39} + 72 q^{44} + 80 q^{46} - 284 q^{49} + 72 q^{51} - 36 q^{54} - 572 q^{56} + 48 q^{59} - 468 q^{61} + 40 q^{64} + 120 q^{66} - 208 q^{71} - 340 q^{74} - 76 q^{79} - 36 q^{81} - 288 q^{84} - 296 q^{86} + 144 q^{89} - 180 q^{91} - 720 q^{94} + 180 q^{96} - 48 q^{99}+O(q^{100})$$ 8 * q + 12 * q^4 - 12 * q^9 + 8 * q^11 - 8 * q^14 - 20 * q^16 - 60 * q^19 - 60 * q^21 - 132 * q^24 - 204 * q^26 + 16 * q^29 - 72 * q^36 - 60 * q^39 + 72 * q^44 + 80 * q^46 - 284 * q^49 + 72 * q^51 - 36 * q^54 - 572 * q^56 + 48 * q^59 - 468 * q^61 + 40 * q^64 + 120 * q^66 - 208 * q^71 - 340 * q^74 - 76 * q^79 - 36 * q^81 - 288 * q^84 - 296 * q^86 + 144 * q^89 - 180 * q^91 - 720 * q^94 + 180 * q^96 - 48 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}$$ v^7 - v^5 + v^3 + 2*v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24}$$ -v^7 + 2*v^5 + 2*v^3 - v $$\beta_{6}$$ $$=$$ $$-2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -2*v^7 + v^5 + v^3 + v $$\beta_{7}$$ $$=$$ $$2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ 2*v^7 + v^5 - v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6$$ (b7 + 2*b6 - b5 + b4) / 6 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6$$ (-b7 - b6 + 2*b5 + 2*b4) / 6 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6$$ (2*b7 + b6 + b5 - b4) / 6 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6$$ (b7 - 2*b6 + b5 + b4) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1 - \beta_{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 + 0.258819i
−2.98735 1.72474i −0.866025 1.50000i 3.94949 + 6.84072i 0 5.97469i −2.59808 6.50000i 13.4495i −1.50000 + 2.59808i 0
124.2 −1.25529 0.724745i 0.866025 + 1.50000i −0.949490 1.64456i 0 2.51059i 2.59808 + 6.50000i 8.55051i −1.50000 + 2.59808i 0
124.3 1.25529 + 0.724745i −0.866025 1.50000i −0.949490 1.64456i 0 2.51059i −2.59808 6.50000i 8.55051i −1.50000 + 2.59808i 0
124.4 2.98735 + 1.72474i 0.866025 + 1.50000i 3.94949 + 6.84072i 0 5.97469i 2.59808 + 6.50000i 13.4495i −1.50000 + 2.59808i 0
199.1 −2.98735 + 1.72474i −0.866025 + 1.50000i 3.94949 6.84072i 0 5.97469i −2.59808 + 6.50000i 13.4495i −1.50000 2.59808i 0
199.2 −1.25529 + 0.724745i 0.866025 1.50000i −0.949490 + 1.64456i 0 2.51059i 2.59808 6.50000i 8.55051i −1.50000 2.59808i 0
199.3 1.25529 0.724745i −0.866025 + 1.50000i −0.949490 + 1.64456i 0 2.51059i −2.59808 + 6.50000i 8.55051i −1.50000 2.59808i 0
199.4 2.98735 1.72474i 0.866025 1.50000i 3.94949 6.84072i 0 5.97469i 2.59808 6.50000i 13.4495i −1.50000 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.4 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
7.d odd 6 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.s.g 8
5.b even 2 1 inner 525.3.s.g 8
5.c odd 4 1 525.3.o.j 4
5.c odd 4 1 525.3.o.k yes 4
7.d odd 6 1 inner 525.3.s.g 8
35.i odd 6 1 inner 525.3.s.g 8
35.k even 12 1 525.3.o.j 4
35.k even 12 1 525.3.o.k yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.3.o.j 4 5.c odd 4 1
525.3.o.j 4 35.k even 12 1
525.3.o.k yes 4 5.c odd 4 1
525.3.o.k yes 4 35.k even 12 1
525.3.s.g 8 1.a even 1 1 trivial
525.3.s.g 8 5.b even 2 1 inner
525.3.s.g 8 7.d odd 6 1 inner
525.3.s.g 8 35.i odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{8} - 14T_{2}^{6} + 171T_{2}^{4} - 350T_{2}^{2} + 625$$ T2^8 - 14*T2^6 + 171*T2^4 - 350*T2^2 + 625 $$T_{11}^{4} - 4T_{11}^{3} + 36T_{11}^{2} + 80T_{11} + 400$$ T11^4 - 4*T11^3 + 36*T11^2 + 80*T11 + 400 $$T_{13}^{4} - 294T_{13}^{2} + 9$$ T13^4 - 294*T13^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 14 T^{6} + 171 T^{4} + \cdots + 625$$
$3$ $$(T^{4} + 3 T^{2} + 9)^{2}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 71 T^{2} + 2401)^{2}$$
$11$ $$(T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400)^{2}$$
$13$ $$(T^{4} - 294 T^{2} + 9)^{2}$$
$17$ $$T^{8} + 360 T^{6} + 128304 T^{4} + \cdots + 1679616$$
$19$ $$(T^{4} + 30 T^{3} - 273 T^{2} + \cdots + 328329)^{2}$$
$23$ $$(T^{4} - 400 T^{2} + 160000)^{2}$$
$29$ $$(T^{2} - 4 T - 860)^{4}$$
$31$ $$(T^{4} - 1152 T^{2} + 1327104)^{2}$$
$37$ $$T^{8} - 2450 T^{6} + 6001875 T^{4} + \cdots + 390625$$
$41$ $$(T^{4} + 7656 T^{2} + 10419984)^{2}$$
$43$ $$(T^{4} + 5192 T^{2} + 5779216)^{2}$$
$47$ $$T^{8} + 4752 T^{6} + \cdots + 1360488960000$$
$53$ $$T^{8} + \cdots + 584046595707136$$
$59$ $$(T^{4} - 24 T^{3} - 5592 T^{2} + \cdots + 33454656)^{2}$$
$61$ $$(T^{4} + 234 T^{3} + 22167 T^{2} + \cdots + 15327225)^{2}$$
$67$ $$T^{8} - 17330 T^{6} + \cdots + 56\!\cdots\!61$$
$71$ $$(T^{2} + 52 T - 1724)^{4}$$
$73$ $$T^{8} + 1926 T^{6} + \cdots + 22430753361$$
$79$ $$(T^{4} + 38 T^{3} + 2619 T^{2} + \cdots + 1380625)^{2}$$
$83$ $$(T^{4} - 7056 T^{2} + 5184)^{2}$$
$89$ $$(T^{4} - 72 T^{3} + 360 T^{2} + \cdots + 1871424)^{2}$$
$97$ $$(T^{4} - 22470 T^{2} + 90269001)^{2}$$
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