# Properties

 Label 124.2.j.a Level $124$ Weight $2$ Character orbit 124.j Analytic conductor $0.990$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 124.j (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.990144985064$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}^{2} - 1) q^{2} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{3} + 2 \zeta_{20} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{5} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + \zeta_{20}^{5} + 2 \zeta_{20}^{2} + 2 \zeta_{20} - 1) q^{6} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - \zeta_{20}) q^{7} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{8} + (4 \zeta_{20}^{6} + 4 \zeta_{20}^{2}) q^{9}+O(q^{10})$$ q + (z^6 - z^4 - z^3 + z^2 - 1) * q^2 + (z^7 - 2*z^5 - 2*z) * q^3 + 2*z * q^4 + (z^6 - z^4 - 1) * q^5 + (-2*z^7 + 2*z^6 + z^5 + 2*z^2 + 2*z - 1) * q^6 + (-2*z^7 + 2*z^5 - z) * q^7 + (2*z^7 - 2*z^5 - 2*z^4 + 2*z^3 - 2*z) * q^8 + (4*z^6 + 4*z^2) * q^9 $$q + (\zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}^{2} - 1) q^{2} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{3} + 2 \zeta_{20} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{5} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + \zeta_{20}^{5} + 2 \zeta_{20}^{2} + 2 \zeta_{20} - 1) q^{6} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - \zeta_{20}) q^{7} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{8} + (4 \zeta_{20}^{6} + 4 \zeta_{20}^{2}) q^{9} + ( - \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20} + 1) q^{10} + (4 \zeta_{20}^{7} + 4 \zeta_{20}^{3} - 2 \zeta_{20}) q^{11} + ( - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2} - 2) q^{12} + ( - 2 \zeta_{20}^{6} + 3 \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{13} + ( - \zeta_{20}^{7} - 2 \zeta_{20}^{6} + 3 \zeta_{20}^{5} + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{3} - 2 \zeta_{20}^{2} + \zeta_{20}) q^{14} + ( - \zeta_{20}^{7} + 2 \zeta_{20}^{5} + \zeta_{20}^{3} + 3 \zeta_{20}) q^{15} + 4 \zeta_{20}^{2} q^{16} + (2 \zeta_{20}^{6} + \zeta_{20}^{4} - 4 \zeta_{20}^{2} + 2) q^{17} + ( - 4 \zeta_{20}^{7} - 4 \zeta_{20}^{4} - 4 \zeta_{20}^{3} + 4 \zeta_{20} - 4) q^{18} - 5 \zeta_{20}^{3} q^{19} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{20} + (\zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2} + 1) q^{21} + ( - 2 \zeta_{20}^{7} - 4 \zeta_{20}^{6} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{3} - 2 \zeta_{20} + 4) q^{22} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}) q^{23} + (4 \zeta_{20}^{7} - 2 \zeta_{20}^{6} + 4 \zeta_{20}^{4} + 4 \zeta_{20}^{3} - 2 \zeta_{20} + 4) q^{24} + ( - \zeta_{20}^{6} + \zeta_{20}^{4} - 3) q^{25} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{5} + 3 \zeta_{20}^{4} + 3 \zeta_{20}^{3} - 4 \zeta_{20}^{2} - 2 \zeta_{20} + 2) q^{26} + ( - 6 \zeta_{20}^{7} - 4 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + \zeta_{20}) q^{27} + (4 \zeta_{20}^{4} - 6 \zeta_{20}^{2} + 4) q^{28} + ( - 3 \zeta_{20}^{6} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2} - 3) q^{29} + (3 \zeta_{20}^{7} - 3 \zeta_{20}^{6} - 2 \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{3} - 2 \zeta_{20}^{2} - 4 \zeta_{20} + 1) q^{30} + (6 \zeta_{20}^{7} - 2 \zeta_{20}^{5} + 3 \zeta_{20}^{3}) q^{31} + ( - 4 \zeta_{20}^{5} - 4) q^{32} + ( - 14 \zeta_{20}^{6} + 6 \zeta_{20}^{4} - 6 \zeta_{20}^{2} + 14) q^{33} + ( - 3 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 6 \zeta_{20}^{5} - 4 \zeta_{20}^{4} - 4 \zeta_{20}^{3} + \zeta_{20}^{2} + \cdots + 2) q^{34} + \cdots + (24 \zeta_{20}^{7} - 16 \zeta_{20}^{5} + 8 \zeta_{20}^{3} - 32 \zeta_{20}) q^{99} +O(q^{100})$$ q + (z^6 - z^4 - z^3 + z^2 - 1) * q^2 + (z^7 - 2*z^5 - 2*z) * q^3 + 2*z * q^4 + (z^6 - z^4 - 1) * q^5 + (-2*z^7 + 2*z^6 + z^5 + 2*z^2 + 2*z - 1) * q^6 + (-2*z^7 + 2*z^5 - z) * q^7 + (2*z^7 - 2*z^5 - 2*z^4 + 2*z^3 - 2*z) * q^8 + (4*z^6 + 4*z^2) * q^9 + (-z^6 + z^5 + z + 1) * q^10 + (4*z^7 + 4*z^3 - 2*z) * q^11 + (-2*z^6 - 2*z^4 - 2*z^2 - 2) * q^12 + (-2*z^6 + 3*z^4 - z^2 - 1) * q^13 + (-z^7 - 2*z^6 + 3*z^5 + 3*z^4 - 3*z^3 - 2*z^2 + z) * q^14 + (-z^7 + 2*z^5 + z^3 + 3*z) * q^15 + 4*z^2 * q^16 + (2*z^6 + z^4 - 4*z^2 + 2) * q^17 + (-4*z^7 - 4*z^4 - 4*z^3 + 4*z - 4) * q^18 - 5*z^3 * q^19 + (2*z^7 - 2*z^5 - 2*z) * q^20 + (z^6 - z^4 - z^2 + 1) * q^21 + (-2*z^7 - 4*z^6 - 2*z^5 + 2*z^4 - 2*z^3 - 2*z + 4) * q^22 + (-z^7 + z^5 + z^3 - z) * q^23 + (4*z^7 - 2*z^6 + 4*z^4 + 4*z^3 - 2*z + 4) * q^24 + (-z^6 + z^4 - 3) * q^25 + (-z^7 - z^6 - z^5 + 3*z^4 + 3*z^3 - 4*z^2 - 2*z + 2) * q^26 + (-6*z^7 - 4*z^5 - 2*z^3 + z) * q^27 + (4*z^4 - 6*z^2 + 4) * q^28 + (-3*z^6 - 2*z^4 - 2*z^2 - 3) * q^29 + (3*z^7 - 3*z^6 - 2*z^5 - z^4 + z^3 - 2*z^2 - 4*z + 1) * q^30 + (6*z^7 - 2*z^5 + 3*z^3) * q^31 + (-4*z^5 - 4) * q^32 + (-14*z^6 + 6*z^4 - 6*z^2 + 14) * q^33 + (-3*z^7 + 2*z^6 + 6*z^5 - 4*z^4 - 4*z^3 + z^2 + 2*z + 2) * q^34 + (-z^7 + z^5 - z) * q^35 + (8*z^7 + 8*z^3) * q^36 + (6*z^6 + 6*z^2 - 3) * q^37 + (5*z^6 + 5*z) * q^38 + (-2*z^7 + 3*z^5 - 3*z^3 + 2*z) * q^39 + (-2*z^7 + 2*z^6 + 2*z^2 + 2*z) * q^40 + (-4*z^6 + 2*z^4 - 2*z^2 + 4) * q^41 + (z^6 + 2*z^5 - 2*z^4 - 2*z^3 + 2*z^2 + z) * q^42 + (4*z^7 - 3*z^5 + 2*z^3 - z) * q^43 + (8*z^6 + 4*z^2 - 8) * q^44 + (-4*z^6 - 4*z^4 - 4*z^2) * q^45 + (-z^7 - 2*z^6 + 2*z^5 + 2*z^4 - 2*z^3 - z^2) * q^46 + (5*z^7 + z^3 - z) * q^47 + (-4*z^7 - 4*z^5 - 4*z^3 - 4*z) * q^48 + (-8*z^4 + 6*z^2 - 8) * q^49 + (-3*z^6 - z^5 + 4*z^4 + 4*z^3 - 4*z^2 - z + 3) * q^50 + 5*z * q^51 + (-4*z^7 + 6*z^5 - 2*z^3 - 2*z) * q^52 + (6*z^6 - 3*z^4 - 6) * q^53 + (z^7 + 6*z^6 + 5*z^5 - 5*z^4 + 5*z^3 + 4*z^2 + z - 10) * q^54 + (-6*z^7 - 2*z^5 - 4*z^3 + 2*z) * q^55 + (-4*z^7 + 4*z^6 + 6*z^5 - 4*z^4 - 4*z^3 + 2) * q^56 + (10*z^6 + 10*z^2 - 5) * q^57 + (5*z^7 - 3*z^6 - z^5 + 6*z^4 + 6*z^3 - z^2 - 3*z + 5) * q^58 + (-3*z^7 - 4*z^3 + 4*z) * q^59 + (2*z^6 + 4*z^4 + 4*z^2 + 2) * q^60 + (2*z^6 - 6*z^4 + 8*z^2 - 4) * q^61 + (-z^6 - 6*z^5 - 2*z^4 + 2*z^3 + 2*z^2 - 3*z + 4) * q^62 + (-4*z^7 + 8*z^5 - 4*z^3) * q^63 + 8*z^3 * q^64 + (-2*z^6 + 2*z^4 - z^2) * q^65 + (8*z^7 + 14*z^6 - 8*z^5 + 8*z^2 - 14*z - 8) * q^66 + (2*z^7 - 9*z^5 + 2*z^3) * q^67 + (4*z^7 + 2*z^5 - 8*z^3 + 4*z) * q^68 + (-z^6 - 2*z^2 + 2) * q^69 + (-z^7 - z^6 + 2*z^5 + 2*z^4 - 2*z^3 - z^2 + z) * q^70 + (-6*z^7 + 5*z^5 - 5*z^3 + 6*z) * q^71 + (-8*z^6 - 8*z^5 - 8*z + 8) * q^72 + (-z^2 - 1) * q^73 + (-6*z^7 - 3*z^6 - 3*z^4 - 3*z^3 - 3*z^2 + 6*z - 3) * q^74 + (-3*z^7 + 6*z^5 - z^3 + 5*z) * q^75 - 10*z^4 * q^76 + (4*z^6 - 4*z^4 + 2*z^2) * q^77 + (2*z^7 + z^4 - z^3 - 3*z^2 + z + 1) * q^78 + (-4*z^7 + 6*z^5 + 6*z^3 - 4*z) * q^79 + (-4*z^4 - 4) * q^80 + (11*z^6 - 7*z^4 + 7*z^2 - 11) * q^81 + (2*z^7 + 4*z^6 - 2*z^5 + 2*z^2 - 4*z - 2) * q^82 + (-2*z^7 - 3*z^5 - z^3 + 5*z) * q^83 + (2*z^7 - 2*z^5 - 2*z^3 + 2*z) * q^84 + (-z^6 + 2*z^4 - 3*z^2 + 4) * q^85 + (-z^7 + z^6 - 3*z^5 - 2*z^4 + 2*z^3 + 3*z^2 - z + 1) * q^86 + (9*z^7 + 8*z^5 + 9*z^3) * q^87 + (-8*z^7 - 8*z^6 + 4*z^5 - 8*z^2 + 8*z + 4) * q^88 + (-6*z^6 + 12*z^4 - 11*z^2 + 1) * q^89 + (8*z^7 + 4*z^4 + 4*z^3 + 4*z^2 - 4*z + 4) * q^90 + (10*z^7 - 13*z^5 + 5*z^3 + 3*z) * q^91 + (4*z^4 - 4*z^2 + 2) * q^92 + (-14*z^6 + 6*z^4 - 4*z^2 + 11) * q^93 + (-z^7 - z^6 - 4*z^5 + z^4 - z^3 + 5) * q^94 + (5*z^5 + 5*z) * q^95 + (-4*z^7 + 8*z^6 + 8*z^5 + 4*z^2 + 8*z - 8) * q^96 + (z^6 - 4*z^2 + 4) * q^97 + (8*z^7 - 8*z^6 - 6*z^5 + 8*z^4 + 8*z^3 + 2) * q^98 + (24*z^7 - 16*z^5 + 8*z^3 - 32*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{2} - 4 q^{5} + 4 q^{8} + 16 q^{9}+O(q^{10})$$ 8 * q - 2 * q^2 - 4 * q^5 + 4 * q^8 + 16 * q^9 $$8 q - 2 q^{2} - 4 q^{5} + 4 q^{8} + 16 q^{9} + 6 q^{10} - 20 q^{12} - 20 q^{13} - 14 q^{14} + 8 q^{16} + 10 q^{17} - 24 q^{18} + 10 q^{21} + 20 q^{22} + 20 q^{24} - 28 q^{25} + 12 q^{28} - 30 q^{29} - 32 q^{32} + 60 q^{33} + 30 q^{34} + 10 q^{38} + 8 q^{40} + 16 q^{41} + 10 q^{42} - 40 q^{44} - 8 q^{45} - 10 q^{46} - 36 q^{49} + 2 q^{50} - 30 q^{53} - 50 q^{54} + 32 q^{56} + 20 q^{58} + 20 q^{60} + 38 q^{62} - 10 q^{65} - 20 q^{66} + 10 q^{69} - 8 q^{70} + 48 q^{72} - 10 q^{73} - 30 q^{74} + 20 q^{76} + 20 q^{77} - 24 q^{80} - 38 q^{81} - 4 q^{82} + 20 q^{85} + 20 q^{86} - 50 q^{89} + 32 q^{90} + 40 q^{93} + 36 q^{94} - 40 q^{96} + 26 q^{97} - 16 q^{98}+O(q^{100})$$ 8 * q - 2 * q^2 - 4 * q^5 + 4 * q^8 + 16 * q^9 + 6 * q^10 - 20 * q^12 - 20 * q^13 - 14 * q^14 + 8 * q^16 + 10 * q^17 - 24 * q^18 + 10 * q^21 + 20 * q^22 + 20 * q^24 - 28 * q^25 + 12 * q^28 - 30 * q^29 - 32 * q^32 + 60 * q^33 + 30 * q^34 + 10 * q^38 + 8 * q^40 + 16 * q^41 + 10 * q^42 - 40 * q^44 - 8 * q^45 - 10 * q^46 - 36 * q^49 + 2 * q^50 - 30 * q^53 - 50 * q^54 + 32 * q^56 + 20 * q^58 + 20 * q^60 + 38 * q^62 - 10 * q^65 - 20 * q^66 + 10 * q^69 - 8 * q^70 + 48 * q^72 - 10 * q^73 - 30 * q^74 + 20 * q^76 + 20 * q^77 - 24 * q^80 - 38 * q^81 - 4 * q^82 + 20 * q^85 + 20 * q^86 - 50 * q^89 + 32 * q^90 + 40 * q^93 + 36 * q^94 - 40 * q^96 + 26 * q^97 - 16 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/124\mathbb{Z}\right)^\times$$.

 $$n$$ $$63$$ $$65$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{20}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 0.951057 + 0.309017i −0.951057 − 0.309017i −0.587785 + 0.809017i 0.587785 − 0.809017i −0.587785 − 0.809017i 0.587785 + 0.809017i 0.951057 − 0.309017i −0.951057 + 0.309017i
−1.39680 0.221232i −2.48990 1.80902i 1.90211 + 0.618034i −1.61803 3.07768 + 3.07768i 0.224514 + 0.0729490i −2.52015 1.28408i 2.00000 + 6.15537i 2.26007 + 0.357960i
15.2 −0.221232 + 1.39680i 2.48990 + 1.80902i −1.90211 0.618034i −1.61803 −3.07768 + 3.07768i −0.224514 0.0729490i 1.28408 2.52015i 2.00000 + 6.15537i 0.357960 2.26007i
23.1 −0.642040 1.26007i 0.224514 + 0.690983i −1.17557 + 1.61803i 0.618034 0.726543 0.726543i 2.48990 3.42705i 2.79360 + 0.442463i 2.00000 1.45309i −0.396802 0.778768i
23.2 1.26007 0.642040i −0.224514 0.690983i 1.17557 1.61803i 0.618034 −0.726543 0.726543i −2.48990 + 3.42705i 0.442463 2.79360i 2.00000 1.45309i 0.778768 0.396802i
27.1 −0.642040 + 1.26007i 0.224514 0.690983i −1.17557 1.61803i 0.618034 0.726543 + 0.726543i 2.48990 + 3.42705i 2.79360 0.442463i 2.00000 + 1.45309i −0.396802 + 0.778768i
27.2 1.26007 + 0.642040i −0.224514 + 0.690983i 1.17557 + 1.61803i 0.618034 −0.726543 + 0.726543i −2.48990 3.42705i 0.442463 + 2.79360i 2.00000 + 1.45309i 0.778768 + 0.396802i
91.1 −1.39680 + 0.221232i −2.48990 + 1.80902i 1.90211 0.618034i −1.61803 3.07768 3.07768i 0.224514 0.0729490i −2.52015 + 1.28408i 2.00000 6.15537i 2.26007 0.357960i
91.2 −0.221232 1.39680i 2.48990 1.80902i −1.90211 + 0.618034i −1.61803 −3.07768 3.07768i −0.224514 + 0.0729490i 1.28408 + 2.52015i 2.00000 6.15537i 0.357960 + 2.26007i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
31.f odd 10 1 inner
124.j even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.2.j.a 8
4.b odd 2 1 inner 124.2.j.a 8
31.f odd 10 1 inner 124.2.j.a 8
124.j even 10 1 inner 124.2.j.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.j.a 8 1.a even 1 1 trivial
124.2.j.a 8 4.b odd 2 1 inner
124.2.j.a 8 31.f odd 10 1 inner
124.2.j.a 8 124.j even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 5T_{3}^{6} + 85T_{3}^{4} + 75T_{3}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(124, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 16$$
$3$ $$T^{8} - 5 T^{6} + 85 T^{4} + 75 T^{2} + \cdots + 25$$
$5$ $$(T^{2} + T - 1)^{4}$$
$7$ $$T^{8} + 11 T^{6} + 321 T^{4} - 29 T^{2} + \cdots + 1$$
$11$ $$T^{8} + 60 T^{6} + 1360 T^{4} + \cdots + 6400$$
$13$ $$(T^{4} + 10 T^{3} + 30 T^{2} + 15 T + 5)^{2}$$
$17$ $$(T^{4} - 5 T^{3} + 25 T^{2} + 125 T + 125)^{2}$$
$19$ $$T^{8} - 25 T^{6} + 625 T^{4} + \cdots + 390625$$
$23$ $$T^{8} - 5 T^{6} + 85 T^{4} + 75 T^{2} + \cdots + 25$$
$29$ $$(T^{4} + 15 T^{3} + 120 T^{2} + 440 T + 605)^{2}$$
$31$ $$T^{8} + 79 T^{6} + 3381 T^{4} + \cdots + 923521$$
$37$ $$(T^{4} + 90 T^{2} + 405)^{2}$$
$41$ $$(T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16)^{2}$$
$43$ $$T^{8} + 25 T^{6} + 250 T^{4} + \cdots + 15625$$
$47$ $$T^{8} + T^{6} + 766 T^{4} + \cdots + 130321$$
$53$ $$(T^{4} + 15 T^{3} + 45 T^{2} - 135 T + 405)^{2}$$
$59$ $$T^{8} - T^{6} + 681 T^{4} + \cdots + 130321$$
$61$ $$(T^{4} + 100 T^{2} + 80)^{2}$$
$67$ $$(T^{4} + 138 T^{2} + 3481)^{2}$$
$71$ $$T^{8} - 59 T^{6} + 1446 T^{4} + \cdots + 707281$$
$73$ $$(T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5)^{2}$$
$79$ $$T^{8} - 60 T^{6} + 38560 T^{4} + \cdots + 93702400$$
$83$ $$T^{8} + 40 T^{6} + 1810 T^{4} + \cdots + 3258025$$
$89$ $$(T^{4} + 25 T^{3} + 400 T^{2} + 1520 T + 1805)^{2}$$
$97$ $$(T^{4} - 13 T^{3} + 69 T^{2} - 77 T + 121)^{2}$$