# Properties

 Label 418.2.n.a Level $418$ Weight $2$ Character orbit 418.n Analytic conductor $3.338$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.n (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{15})$$ Defining polynomial: $$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$$ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{15}^{6} + \zeta_{15}) q^{2} + (2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} + 2 \zeta_{15}) q^{3} + \zeta_{15}^{7} q^{4} + ( - \zeta_{15}^{7} - 2 \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} + 2 \zeta_{15}^{2} - 3 \zeta_{15} + 1) q^{5} + (2 \zeta_{15}^{7} + 2 \zeta_{15}) q^{6} + ( - \zeta_{15}^{7} - \zeta_{15}^{2}) q^{7} - \zeta_{15}^{3} q^{8} + (4 \zeta_{15}^{7} - 4 \zeta_{15}^{5} + 5 \zeta_{15} - 4) q^{9} +O(q^{10})$$ q + (z^6 + z) * q^2 + (2*z^6 - 2*z^5 + 2*z) * q^3 + z^7 * q^4 + (-z^7 - 2*z^6 - z^4 + z^3 + 2*z^2 - 3*z + 1) * q^5 + (2*z^7 + 2*z) * q^6 + (-z^7 - z^2) * q^7 - z^3 * q^8 + (4*z^7 - 4*z^5 + 5*z - 4) * q^9 $$q + (\zeta_{15}^{6} + \zeta_{15}) q^{2} + (2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} + 2 \zeta_{15}) q^{3} + \zeta_{15}^{7} q^{4} + ( - \zeta_{15}^{7} - 2 \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} + 2 \zeta_{15}^{2} - 3 \zeta_{15} + 1) q^{5} + (2 \zeta_{15}^{7} + 2 \zeta_{15}) q^{6} + ( - \zeta_{15}^{7} - \zeta_{15}^{2}) q^{7} - \zeta_{15}^{3} q^{8} + (4 \zeta_{15}^{7} - 4 \zeta_{15}^{5} + 5 \zeta_{15} - 4) q^{9} + ( - 2 \zeta_{15}^{5} + 3 \zeta_{15}^{4} + 3 \zeta_{15} - 2) q^{10} + ( - 2 \zeta_{15}^{7} + 4 \zeta_{15}^{6} + 3 \zeta_{15}^{3} - 2 \zeta_{15}^{2} + 2) q^{11} + (2 \zeta_{15}^{7} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{2}) q^{12} + ( - 3 \zeta_{15}^{7} + 3 \zeta_{15}^{5} - \zeta_{15} + 3) q^{13} + ( - \zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15} + 1) q^{14} + ( - 6 \zeta_{15}^{7} - 4 \zeta_{15}^{5} + 4 \zeta_{15}^{4} - 4) q^{15} + ( - \zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1) q^{16} + ( - 2 \zeta_{15}^{7} + 3 \zeta_{15}^{6} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{3} - 3 \zeta_{15}^{2} + \zeta_{15} + 2) q^{17} + (5 \zeta_{15}^{7} - 4 \zeta_{15}^{6} - 4 \zeta_{15}^{3} + 5 \zeta_{15}^{2}) q^{18} + ( - 2 \zeta_{15}^{7} + 2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} - \zeta_{15}^{3} + 2 \zeta_{15}) q^{19} + (3 \zeta_{15}^{7} - 2 \zeta_{15}^{6} + 3 \zeta_{15}^{2} - 3) q^{20} + ( - 2 \zeta_{15}^{7} + 2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{2} - 2 \zeta_{15} + 2) q^{21} + (\zeta_{15}^{7} - \zeta_{15}^{6} + 2 \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} - \zeta_{15}^{2} - 1) q^{22} + ( - \zeta_{15}^{5} - 1) q^{23} + (2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{2} + 2 \zeta_{15}) q^{24} + ( - 5 \zeta_{15}^{7} + 8 \zeta_{15}^{5} - 5 \zeta_{15}^{4} - 8 \zeta_{15} + 8) q^{25} + ( - \zeta_{15}^{7} + 3 \zeta_{15}^{6} + 3 \zeta_{15}^{3} - \zeta_{15}^{2}) q^{26} + (12 \zeta_{15}^{7} - 12 \zeta_{15}^{6} - 8 \zeta_{15}^{3} + 12 \zeta_{15}^{2} - 8) q^{27} + \zeta_{15}^{4} q^{28} - \zeta_{15}^{7} q^{29} + ( - 4 \zeta_{15}^{6} + 6 \zeta_{15}^{3} - 4) q^{30} + (5 \zeta_{15}^{7} + 3 \zeta_{15}^{6} + 5 \zeta_{15}^{2} - 5) q^{31} + (\zeta_{15}^{5} + 1) q^{32} + ( - 4 \zeta_{15}^{7} + 6 \zeta_{15}^{6} + 6 \zeta_{15}^{5} - 4 \zeta_{15}^{4} + 4 \zeta_{15}^{3} - 6 \zeta_{15}^{2} + \cdots + 4) q^{33} + \cdots + ( - 2 \zeta_{15}^{7} + 18 \zeta_{15}^{5} - 15 \zeta_{15}^{4} - 12 \zeta_{15} + 18) q^{99} +O(q^{100})$$ q + (z^6 + z) * q^2 + (2*z^6 - 2*z^5 + 2*z) * q^3 + z^7 * q^4 + (-z^7 - 2*z^6 - z^4 + z^3 + 2*z^2 - 3*z + 1) * q^5 + (2*z^7 + 2*z) * q^6 + (-z^7 - z^2) * q^7 - z^3 * q^8 + (4*z^7 - 4*z^5 + 5*z - 4) * q^9 + (-2*z^5 + 3*z^4 + 3*z - 2) * q^10 + (-2*z^7 + 4*z^6 + 3*z^3 - 2*z^2 + 2) * q^11 + (2*z^7 - 2*z^3 + 2*z^2) * q^12 + (-3*z^7 + 3*z^5 - z + 3) * q^13 + (-z^7 + z^5 - z^4 + z^3 - z + 1) * q^14 + (-6*z^7 - 4*z^5 + 4*z^4 - 4) * q^15 + (-z^7 + z^6 - z^4 + z^3 - z^2 + 1) * q^16 + (-2*z^7 + 3*z^6 - 2*z^4 + 2*z^3 - 3*z^2 + z + 2) * q^17 + (5*z^7 - 4*z^6 - 4*z^3 + 5*z^2) * q^18 + (-2*z^7 + 2*z^6 - 2*z^5 - 2*z^4 - z^3 + 2*z) * q^19 + (3*z^7 - 2*z^6 + 3*z^2 - 3) * q^20 + (-2*z^7 + 2*z^5 - 2*z^4 + 2*z^3 - 2*z^2 - 2*z + 2) * q^21 + (z^7 - z^6 + 2*z^5 + z^4 - z^3 - z^2 - 1) * q^22 + (-z^5 - 1) * q^23 + (2*z^6 - 2*z^5 - 2*z^2 + 2*z) * q^24 + (-5*z^7 + 8*z^5 - 5*z^4 - 8*z + 8) * q^25 + (-z^7 + 3*z^6 + 3*z^3 - z^2) * q^26 + (12*z^7 - 12*z^6 - 8*z^3 + 12*z^2 - 8) * q^27 + z^4 * q^28 - z^7 * q^29 + (-4*z^6 + 6*z^3 - 4) * q^30 + (5*z^7 + 3*z^6 + 5*z^2 - 5) * q^31 + (z^5 + 1) * q^32 + (-4*z^7 + 6*z^6 + 6*z^5 - 4*z^4 + 4*z^3 - 6*z^2 + 2*z + 4) * q^33 + (3*z^5 - z^4 - z + 3) * q^34 + (2*z^6 - 3*z^5 - 3*z^2 + 2*z) * q^35 + (z^7 + 4*z^6 - 5*z^5 + z^4 - z^3 + 5*z - 1) * q^36 + (3*z^7 - z^6 - z^3 + 3*z^2) * q^37 + (z^7 + z^6 - z^4 + 3*z^3 - z^2 + 2*z + 3) * q^38 + (-8*z^7 + 8*z^6 + 6*z^3 - 8*z^2 + 6) * q^39 + (3*z^7 - 3*z^6 - 3*z^5 + 3*z^4 - 3*z^3 + 2*z^2 - 3) * q^40 + (-z^7 - 2*z^6 + 3*z^5 - z^4 + z^3 - 3*z + 1) * q^41 + (-2*z^7 + 2*z^5 - 2*z + 2) * q^42 + (-4*z^7 + 9*z^5 - 4*z^4 + 4*z^3 - 4*z^2 - 4*z + 4) * q^43 + (-2*z^7 + z^5 - 2*z^4 - 4*z + 1) * q^44 + (-11*z^7 + 11*z^3 - 11*z^2 - 11) * q^45 - z^6 * q^46 + (z^7 - z^5 + z^4 + z - 1) * q^47 + (2*z^5 - 2*z^4 + 2) * q^48 + (-6*z^7 + 6*z^6 + 6*z^3 - 6*z^2 + 6) * q^49 + (-8*z^7 + 8*z^6 + 5*z^3 - 8*z^2 + 5) * q^50 + (2*z^7 + 6*z^5 - 6*z^4 + 6) * q^51 + (2*z^7 - 3*z^6 + z^5 + 2*z^4 - 2*z^3 - z - 2) * q^52 + (2*z^7 - 2*z^5 - 8*z - 2) * q^53 + (4*z^7 - 12*z^5 + 4*z^4 - 4*z^3 + 4*z^2 + 4*z - 4) * q^54 + (z^7 + 9*z^6 - 5*z^5 + z^4 - z^3 + 4*z^2 + 10*z - 1) * q^55 - q^56 + (4*z^7 - 2*z^6 - 6*z^5 - 2*z^2 + 10*z - 4) * q^57 + z^3 * q^58 + (8*z^7 - 8*z^6 - 8*z^5 + 8*z^4 - 8*z^3 + 9*z^2 - 8) * q^59 + (6*z^7 - 10*z^6 + 6*z^4 - 6*z^3 + 10*z^2 - 4*z - 6) * q^60 + (-3*z^7 - 9*z^4 - 3*z) * q^61 + (5*z^7 - 5*z^6 - 5*z^5 + 5*z^4 - 5*z^3 - 3*z^2 - 5) * q^62 + (-z^7 + 5*z^5 - z^4 - 5*z + 5) * q^63 + z^6 * q^64 + 11 * q^65 + (6*z^5 - 2*z^4 - 8*z + 6) * q^66 + (9*z^5 - 7*z^4 - 7*z + 9) * q^67 + (-z^7 + 3*z^6 - z^2 + 1) * q^68 + (-2*z^6 - 2) * q^69 + (-z^7 + 3*z^5 - 3*z^4 + 3) * q^70 + (14*z^7 - 15*z^6 + 14*z^4 - 14*z^3 + 15*z^2 - z - 14) * q^71 + (4*z^7 - z^4 + 4*z) * q^72 + (8*z^7 - 8*z^6 - 8*z^5 + 8*z^4 - 8*z^3 + 2*z^2 - 8) * q^73 + (2*z^7 + z^6 - 3*z^5 + 2*z^4 - 2*z^3 + 3*z - 2) * q^74 + (-16*z^7 + 22*z^6 - 16*z^2 + 16) * q^75 + (4*z^7 + z^5 + 2*z^4 - 4*z^3 + 4*z^2 + 2*z - 1) * q^76 + (-2*z^6 + 2*z^3 + 1) * q^77 + (-2*z^7 + 8*z^5 - 2*z^4 + 2*z^3 - 2*z^2 - 2*z + 2) * q^78 + (12*z^6 - 4*z^5 - 4*z^2 + 12*z) * q^79 + (-z^7 - 2*z^5 - z^4 + 2*z - 2) * q^80 + (12*z^7 - 12*z^6 - 12*z^5 + 12*z^4 - 12*z^3 + 17*z^2 - 12) * q^81 + (-2*z^7 + z^4 - 2*z) * q^82 + (5*z^7 - 5*z^6 - 5*z^3 + 5*z^2 - 5) * q^83 + (-2*z^7 + 2*z^6 + 2*z^3 - 2*z^2) * q^84 + (z^7 - 9*z^5 + z^4 + 9*z - 9) * q^85 + (-4*z^7 + 4*z^5 - 9*z + 4) * q^86 + (-2*z^7 + 2*z^3 - 2*z^2) * q^87 + (-4*z^7 + z^6 + 2*z^3 - 4*z^2 + 2) * q^88 + (-z^5 + 8*z^4 + 8*z - 1) * q^89 + (-22*z^6 + 11*z^5 + 11*z^2 - 22*z) * q^90 + (-2*z^7 - z^5 - 2*z^4 + z - 1) * q^91 + z^2 * q^92 + (10*z^7 - 4*z^6 + 10*z^4 - 10*z^3 + 4*z^2 + 6*z - 10) * q^93 + (z^7 - z^6 - z^3 + z^2 - 1) * q^94 + (-9*z^7 - 5*z^6 - 7*z^5 + 7*z^4 - 5*z^3 - 4*z^2 - 7) * q^95 + (2*z^6 + 2) * q^96 + (-10*z^6 - 2*z^5 - 2*z^2 - 10*z) * q^97 + 6*z^5 * q^98 + (-2*z^7 + 18*z^5 - 15*z^4 - 12*z + 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - q^{2} + 6 q^{3} + q^{4} + 7 q^{5} + 4 q^{6} - 2 q^{7} + 2 q^{8} - 7 q^{9}+O(q^{10})$$ 8 * q - q^2 + 6 * q^3 + q^4 + 7 * q^5 + 4 * q^6 - 2 * q^7 + 2 * q^8 - 7 * q^9 $$8 q - q^{2} + 6 q^{3} + q^{4} + 7 q^{5} + 4 q^{6} - 2 q^{7} + 2 q^{8} - 7 q^{9} - 2 q^{10} - 2 q^{11} + 8 q^{12} + 8 q^{13} - q^{14} - 18 q^{15} + q^{16} + 26 q^{18} + 4 q^{19} - 14 q^{20} - 4 q^{21} - 11 q^{22} - 4 q^{23} + 4 q^{24} + 14 q^{25} - 14 q^{26} + q^{28} - q^{29} - 36 q^{30} - 36 q^{31} + 4 q^{32} - 24 q^{33} + 10 q^{34} + 7 q^{35} + 13 q^{36} + 10 q^{37} + 17 q^{38} + 4 q^{39} + 8 q^{40} - 7 q^{41} + 4 q^{42} - 28 q^{43} - 4 q^{44} - 132 q^{45} + 2 q^{46} - q^{47} + 6 q^{48} + 12 q^{49} - 2 q^{50} + 20 q^{51} - 7 q^{52} - 14 q^{53} + 40 q^{54} + 12 q^{55} - 8 q^{56} + 8 q^{57} - 2 q^{58} + 25 q^{59} + 2 q^{60} - 15 q^{61} + 7 q^{62} + 13 q^{63} - 2 q^{64} + 88 q^{65} + 14 q^{66} + 22 q^{67} - 12 q^{69} + 8 q^{70} - 12 q^{71} + 7 q^{72} + 18 q^{73} + 5 q^{74} + 52 q^{75} + 8 q^{76} + 8 q^{77} - 28 q^{78} - 8 q^{80} + 41 q^{81} - 3 q^{82} - 10 q^{83} - 12 q^{84} - 25 q^{85} + 3 q^{86} - 8 q^{87} + 2 q^{88} + 12 q^{89} - 11 q^{90} - 7 q^{91} + q^{92} - 22 q^{93} - 2 q^{94} - 14 q^{95} + 12 q^{96} + 16 q^{97} - 24 q^{98} + 43 q^{99}+O(q^{100})$$ 8 * q - q^2 + 6 * q^3 + q^4 + 7 * q^5 + 4 * q^6 - 2 * q^7 + 2 * q^8 - 7 * q^9 - 2 * q^10 - 2 * q^11 + 8 * q^12 + 8 * q^13 - q^14 - 18 * q^15 + q^16 + 26 * q^18 + 4 * q^19 - 14 * q^20 - 4 * q^21 - 11 * q^22 - 4 * q^23 + 4 * q^24 + 14 * q^25 - 14 * q^26 + q^28 - q^29 - 36 * q^30 - 36 * q^31 + 4 * q^32 - 24 * q^33 + 10 * q^34 + 7 * q^35 + 13 * q^36 + 10 * q^37 + 17 * q^38 + 4 * q^39 + 8 * q^40 - 7 * q^41 + 4 * q^42 - 28 * q^43 - 4 * q^44 - 132 * q^45 + 2 * q^46 - q^47 + 6 * q^48 + 12 * q^49 - 2 * q^50 + 20 * q^51 - 7 * q^52 - 14 * q^53 + 40 * q^54 + 12 * q^55 - 8 * q^56 + 8 * q^57 - 2 * q^58 + 25 * q^59 + 2 * q^60 - 15 * q^61 + 7 * q^62 + 13 * q^63 - 2 * q^64 + 88 * q^65 + 14 * q^66 + 22 * q^67 - 12 * q^69 + 8 * q^70 - 12 * q^71 + 7 * q^72 + 18 * q^73 + 5 * q^74 + 52 * q^75 + 8 * q^76 + 8 * q^77 - 28 * q^78 - 8 * q^80 + 41 * q^81 - 3 * q^82 - 10 * q^83 - 12 * q^84 - 25 * q^85 + 3 * q^86 - 8 * q^87 + 2 * q^88 + 12 * q^89 - 11 * q^90 - 7 * q^91 + q^92 - 22 * q^93 - 2 * q^94 - 14 * q^95 + 12 * q^96 + 16 * q^97 - 24 * q^98 + 43 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/418\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$-1 - \zeta_{15}^{5}$$ $$-\zeta_{15}^{2} - \zeta_{15}^{7}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.669131 − 0.743145i −0.104528 + 0.994522i −0.978148 − 0.207912i −0.978148 + 0.207912i 0.913545 + 0.406737i 0.913545 − 0.406737i 0.669131 + 0.743145i −0.104528 − 0.994522i
0.978148 + 0.207912i 2.95630 1.31623i 0.913545 + 0.406737i −2.57890 2.86416i 3.16535 0.672816i −0.809017 + 0.587785i 0.809017 + 0.587785i 4.99983 5.55288i −1.92705 3.33775i
125.1 −0.913545 + 0.406737i −0.827091 0.918578i 0.669131 0.743145i −0.298335 2.83847i 1.12920 + 0.502754i 0.309017 + 0.951057i −0.309017 + 0.951057i 0.153880 1.46407i 1.42705 + 2.47172i
159.1 −0.669131 + 0.743145i −0.338261 + 3.21834i −0.104528 0.994522i 3.76988 0.801313i −2.16535 2.40487i −0.809017 + 0.587785i 0.809017 + 0.587785i −7.30885 1.55354i −1.92705 + 3.33775i
163.1 −0.669131 0.743145i −0.338261 3.21834i −0.104528 + 0.994522i 3.76988 + 0.801313i −2.16535 + 2.40487i −0.809017 0.587785i 0.809017 0.587785i −7.30885 + 1.55354i −1.92705 3.33775i
201.1 0.104528 + 0.994522i 1.20906 + 0.256993i −0.978148 + 0.207912i 2.60735 1.16087i −0.129204 + 1.22930i 0.309017 0.951057i −0.309017 0.951057i −1.34486 0.598772i 1.42705 + 2.47172i
235.1 0.104528 0.994522i 1.20906 0.256993i −0.978148 0.207912i 2.60735 + 1.16087i −0.129204 1.22930i 0.309017 + 0.951057i −0.309017 + 0.951057i −1.34486 + 0.598772i 1.42705 2.47172i
273.1 0.978148 0.207912i 2.95630 + 1.31623i 0.913545 0.406737i −2.57890 + 2.86416i 3.16535 + 0.672816i −0.809017 0.587785i 0.809017 0.587785i 4.99983 + 5.55288i −1.92705 + 3.33775i
311.1 −0.913545 0.406737i −0.827091 + 0.918578i 0.669131 + 0.743145i −0.298335 + 2.83847i 1.12920 0.502754i 0.309017 0.951057i −0.309017 0.951057i 0.153880 + 1.46407i 1.42705 2.47172i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 311.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
19.c even 3 1 inner
209.n even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.n.a 8
11.c even 5 1 inner 418.2.n.a 8
19.c even 3 1 inner 418.2.n.a 8
209.n even 15 1 inner 418.2.n.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.n.a 8 1.a even 1 1 trivial
418.2.n.a 8 11.c even 5 1 inner
418.2.n.a 8 19.c even 3 1 inner
418.2.n.a 8 209.n even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 6T_{3}^{7} + 20T_{3}^{6} - 64T_{3}^{5} + 144T_{3}^{4} - 64T_{3}^{3} - 256T_{3} + 256$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1$$
$3$ $$T^{8} - 6 T^{7} + 20 T^{6} - 64 T^{5} + \cdots + 256$$
$5$ $$T^{8} - 7 T^{7} + 15 T^{6} + \cdots + 14641$$
$7$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$11$ $$(T^{4} + T^{3} + 21 T^{2} + 11 T + 121)^{2}$$
$13$ $$T^{8} - 8 T^{7} + 30 T^{6} + \cdots + 14641$$
$17$ $$T^{8} - 10 T^{6} + 50 T^{5} + \cdots + 625$$
$19$ $$T^{8} - 4 T^{7} - 3 T^{6} + \cdots + 130321$$
$23$ $$(T^{2} + T + 1)^{4}$$
$29$ $$T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1$$
$31$ $$(T^{4} + 18 T^{3} + 124 T^{2} + 7 T + 1)^{2}$$
$37$ $$(T^{4} - 5 T^{3} + 10 T^{2} + 25)^{2}$$
$41$ $$T^{8} + 7 T^{7} + 30 T^{6} + 127 T^{5} + \cdots + 1$$
$43$ $$(T^{4} + 14 T^{3} + 167 T^{2} + 406 T + 841)^{2}$$
$47$ $$T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1$$
$53$ $$T^{8} + 14 T^{7} + 100 T^{6} + \cdots + 33362176$$
$59$ $$T^{8} - 25 T^{7} + 360 T^{6} + \cdots + 9150625$$
$61$ $$T^{8} + 15 T^{7} + 135 T^{6} + \cdots + 4100625$$
$67$ $$(T^{4} - 11 T^{3} + 152 T^{2} + 341 T + 961)^{2}$$
$71$ $$T^{8} + 12 T^{7} + \cdots + 1908029761$$
$73$ $$T^{8} - 18 T^{7} + 80 T^{6} + \cdots + 33362176$$
$79$ $$T^{8} - 160 T^{6} + \cdots + 40960000$$
$83$ $$(T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625)^{2}$$
$89$ $$(T^{4} - 6 T^{3} + 107 T^{2} + 426 T + 5041)^{2}$$
$97$ $$T^{8} - 16 T^{7} + \cdots + 181063936$$
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