# Properties

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

# Related objects

## 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{15}^{6} - \zeta_{15}) q^{2} + (2 \zeta_{15}^{7} - 2 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - 2 \zeta_{15}^{3} + 2 \zeta_{15} - 2) q^{3} + \zeta_{15}^{7} q^{4} + (\zeta_{15}^{7} - 2 \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + 2 \zeta_{15}^{2} - \zeta_{15} - 1) q^{5} + 2 \zeta_{15}^{4} q^{6} + ( - \zeta_{15}^{7} - 2 \zeta_{15}^{6} - 2 \zeta_{15}^{3} - \zeta_{15}^{2}) q^{7} + \zeta_{15}^{3} q^{8} + \zeta_{15} q^{9}+O(q^{10})$$ q + (-z^6 - z) * q^2 + (2*z^7 - 2*z^5 + 2*z^4 - 2*z^3 + 2*z - 2) * q^3 + z^7 * q^4 + (z^7 - 2*z^6 + z^4 - z^3 + 2*z^2 - z - 1) * q^5 + 2*z^4 * q^6 + (-z^7 - 2*z^6 - 2*z^3 - z^2) * q^7 + z^3 * q^8 + z * q^9 $$q + ( - \zeta_{15}^{6} - \zeta_{15}) q^{2} + (2 \zeta_{15}^{7} - 2 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - 2 \zeta_{15}^{3} + 2 \zeta_{15} - 2) q^{3} + \zeta_{15}^{7} q^{4} + (\zeta_{15}^{7} - 2 \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + 2 \zeta_{15}^{2} - \zeta_{15} - 1) q^{5} + 2 \zeta_{15}^{4} q^{6} + ( - \zeta_{15}^{7} - 2 \zeta_{15}^{6} - 2 \zeta_{15}^{3} - \zeta_{15}^{2}) q^{7} + \zeta_{15}^{3} q^{8} + \zeta_{15} q^{9} + (2 \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 2) q^{10} + (2 \zeta_{15}^{7} + 2 \zeta_{15}^{6} - \zeta_{15}^{3} + 2 \zeta_{15}^{2}) q^{11} + 2 q^{12} + (3 \zeta_{15}^{7} - 3 \zeta_{15}^{5} + 3 \zeta_{15} - 3) q^{13} + (3 \zeta_{15}^{7} - 2 \zeta_{15}^{6} - \zeta_{15}^{5} + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{3} + \zeta_{15} - 3) q^{14} + ( - 2 \zeta_{15}^{7} - 2 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - 2) q^{15} + ( - \zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1) q^{16} + (6 \zeta_{15}^{7} - 7 \zeta_{15}^{6} + 6 \zeta_{15}^{4} - 6 \zeta_{15}^{3} + 7 \zeta_{15}^{2} - \zeta_{15} - 6) q^{17} + ( - \zeta_{15}^{7} - \zeta_{15}^{2}) q^{18} + ( - 2 \zeta_{15}^{7} + 2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} + 3 \zeta_{15}^{3} + 2 \zeta_{15}) q^{19} + (\zeta_{15}^{7} - 2 \zeta_{15}^{6} + \zeta_{15}^{2} - 1) q^{20} + (4 \zeta_{15}^{7} + 2 \zeta_{15}^{5} + 4 \zeta_{15}^{4} - 4 \zeta_{15}^{3} + 4 \zeta_{15}^{2} + 4 \zeta_{15} - 4) q^{21} + ( - \zeta_{15}^{7} - \zeta_{15}^{6} + 2 \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} + 3 \zeta_{15}^{2} - 2 \zeta_{15} + 1) q^{22} + (\zeta_{15}^{5} + 1) q^{23} + ( - 2 \zeta_{15}^{6} - 2 \zeta_{15}) q^{24} + (3 \zeta_{15}^{7} + 3 \zeta_{15}^{4}) q^{25} + ( - 3 \zeta_{15}^{7} + 3 \zeta_{15}^{6} + 3 \zeta_{15}^{3} - 3 \zeta_{15}^{2}) q^{26} + ( - 4 \zeta_{15}^{7} + 4 \zeta_{15}^{6} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{2} + 4) q^{27} + (2 \zeta_{15}^{7} + 3 \zeta_{15}^{4} + 2 \zeta_{15}) q^{28} + (\zeta_{15}^{7} - 2 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - 2) q^{29} + (2 \zeta_{15}^{6} - 2 \zeta_{15}^{3} + 2) q^{30} + (3 \zeta_{15}^{7} + 5 \zeta_{15}^{6} + 3 \zeta_{15}^{2} - 3) q^{31} + ( - \zeta_{15}^{5} - 1) q^{32} + ( - 4 \zeta_{15}^{7} + 6 \zeta_{15}^{6} - 4 \zeta_{15}^{5} - 4 \zeta_{15}^{4} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{2} + \cdots + 4) q^{33}+ \cdots + (4 \zeta_{15}^{7} - 2 \zeta_{15}^{5} + \zeta_{15}^{4} + 2 \zeta_{15} - 2) q^{99}+O(q^{100})$$ q + (-z^6 - z) * q^2 + (2*z^7 - 2*z^5 + 2*z^4 - 2*z^3 + 2*z - 2) * q^3 + z^7 * q^4 + (z^7 - 2*z^6 + z^4 - z^3 + 2*z^2 - z - 1) * q^5 + 2*z^4 * q^6 + (-z^7 - 2*z^6 - 2*z^3 - z^2) * q^7 + z^3 * q^8 + z * q^9 + (2*z^5 - z^4 - z + 2) * q^10 + (2*z^7 + 2*z^6 - z^3 + 2*z^2) * q^11 + 2 * q^12 + (3*z^7 - 3*z^5 + 3*z - 3) * q^13 + (3*z^7 - 2*z^6 - z^5 + 3*z^4 - 3*z^3 + z - 3) * q^14 + (-2*z^7 - 2*z^5 + 2*z^4 - 2) * q^15 + (-z^7 + z^6 - z^4 + z^3 - z^2 + 1) * q^16 + (6*z^7 - 7*z^6 + 6*z^4 - 6*z^3 + 7*z^2 - z - 6) * q^17 + (-z^7 - z^2) * q^18 + (-2*z^7 + 2*z^6 - 2*z^5 - 2*z^4 + 3*z^3 + 2*z) * q^19 + (z^7 - 2*z^6 + z^2 - 1) * q^20 + (4*z^7 + 2*z^5 + 4*z^4 - 4*z^3 + 4*z^2 + 4*z - 4) * q^21 + (-z^7 - z^6 + 2*z^5 - z^4 + z^3 + 3*z^2 - 2*z + 1) * q^22 + (z^5 + 1) * q^23 + (-2*z^6 - 2*z) * q^24 + (3*z^7 + 3*z^4) * q^25 + (-3*z^7 + 3*z^6 + 3*z^3 - 3*z^2) * q^26 + (-4*z^7 + 4*z^6 + 4*z^3 - 4*z^2 + 4) * q^27 + (2*z^7 + 3*z^4 + 2*z) * q^28 + (z^7 - 2*z^5 + 2*z^4 - 2) * q^29 + (2*z^6 - 2*z^3 + 2) * q^30 + (3*z^7 + 5*z^6 + 3*z^2 - 3) * q^31 + (-z^5 - 1) * q^32 + (-4*z^7 + 6*z^6 - 4*z^5 - 4*z^4 + 4*z^3 - 4*z^2 + 2*z + 4) * q^33 + (7*z^5 - z^4 - z + 7) * q^34 + (z^5 + z^2) * q^35 + (z^7 - z^5 + z^4 - z^3 + z - 1) * q^36 + (5*z^7 - z^6 - z^3 + 5*z^2) * q^37 + (-5*z^7 + 3*z^6 - 3*z^4 + z^3 - 3*z^2 - 2*z + 1) * q^38 + (6*z^7 - 6*z^6 + 6*z^2) * q^39 + (-z^7 + z^6 + z^5 - z^4 + z^3 - 2*z^2 + 1) * q^40 + (5*z^7 - 8*z^6 + 3*z^5 + 5*z^4 - 5*z^3 - 3*z - 5) * q^41 + (-4*z^7 + 4*z^5 + 2*z + 4) * q^42 + (-6*z^7 - z^5 - 6*z^4 + 6*z^3 - 6*z^2 - 6*z + 6) * q^43 + (-2*z^7 + 3*z^5 - 4*z^4 - 2*z + 3) * q^44 + (-z^7 + z^3 - z^2 - 1) * q^45 - z^6 * q^46 + (-z^7 + z^5 - z^4 - z + 1) * q^47 + 2*z^7 * q^48 + (-2*z^7 + 2*z^6 - 6*z^3 - 2*z^2 - 6) * q^49 + (3*z^3 + 3) * q^50 + (-12*z^7 - 2*z^5 + 2*z^4 - 2) * q^51 + (3*z^6 - 3*z^5 + 3*z) * q^52 + (2*z^7 - 2*z^5 + 10*z - 2) * q^53 - 4*z^5 * q^54 + (3*z^7 - 5*z^6 - 3*z^5 + 3*z^4 - 3*z^3 + 6*z^2 - 2*z - 3) * q^55 + (-2*z^7 + 2*z^3 - 2*z^2 + 3) * q^56 + (8*z^7 - 6*z^6 - 4*z^5 + 4*z^2 - 2*z - 8) * q^57 + (2*z^6 + z^3 + 2) * q^58 + (-6*z^7 + 6*z^6 + 6*z^5 - 6*z^4 + 6*z^3 - z^2 + 6) * q^59 + (2*z^7 - 4*z^6 + 2*z^4 - 2*z^3 + 4*z^2 - 2*z - 2) * q^60 + (3*z^7 - 9*z^4 + 3*z) * q^61 + (-3*z^7 + 3*z^6 + 3*z^5 - 3*z^4 + 3*z^3 + 5*z^2 + 3) * q^62 + (-3*z^7 + z^5 - 3*z^4 - z + 1) * q^63 + z^6 * q^64 + (-6*z^7 + 6*z^3 - 6*z^2 - 3) * q^65 + (-2*z^7 - 4*z^5 - 4*z - 4) * q^66 + (-z^5 - 7*z^4 - 7*z - 1) * q^67 + (z^7 - 7*z^6 + z^2 - 1) * q^68 - 2*z^3 * q^69 + (-z^7 + z^5 - z^4 + 1) * q^70 + (2*z^7 + 5*z^6 + 2*z^4 - 2*z^3 - 5*z^2 + 7*z - 2) * q^71 + z^4 * q^72 + (2*z^7 - 2*z^6 - 2*z^5 + 2*z^4 - 2*z^3 - 8*z^2 - 2) * q^73 + (-4*z^7 - z^6 + 5*z^5 - 4*z^4 + 4*z^3 - 5*z + 4) * q^74 + (-6*z^7 - 6*z^2 + 6) * q^75 + (4*z^7 - 3*z^5 + 2*z^4 - 4*z^3 + 4*z^2 + 2*z - 5) * q^76 + (6*z^6 + 10*z^3 + 7) * q^77 + (-6*z^7 + 6*z^5 - 6*z^4 + 6*z^3 - 6*z^2 - 6*z + 6) * q^78 + (6*z^6 - 6*z^5 - 6*z^2 + 6*z) * q^79 + (z^7 - 2*z^5 + z^4 + 2*z - 2) * q^80 - 11*z^2 * q^81 + (8*z^7 + 5*z^4 + 8*z) * q^82 + (5*z^7 - 5*z^6 - 9*z^3 + 5*z^2 - 9) * q^83 + (-2*z^7 - 4*z^6 - 4*z^3 - 2*z^2) * q^84 + (-7*z^7 + 15*z^5 - 7*z^4 - 15*z + 15) * q^85 + (6*z^7 - 6*z^5 - z - 6) * q^86 + (-4*z^7 + 4*z^3 - 4*z^2 + 2) * q^87 + (2*z^7 - 3*z^6 - 2*z^3 + 2*z^2 - 4) * q^88 + (-5*z^5 - 6*z^4 - 6*z - 5) * q^89 + (2*z^6 - z^5 - z^2 + 2*z) * q^90 + (6*z^7 - 3*z^5 + 6*z^4 + 3*z - 3) * q^91 - z^2 * q^92 + (-16*z^7 + 10*z^6 - 16*z^4 + 16*z^3 - 10*z^2 - 6*z + 16) * q^93 + (z^7 - z^6 - z^3 + z^2 - 1) * q^94 + (-7*z^7 + 5*z^6 - z^5 + z^4 + 5*z^3 - 8*z^2 - 1) * q^95 + 2*z^3 * q^96 + (10*z^5 + 10*z^2) * q^97 + (8*z^7 - 2*z^5 + 8*z^4 - 8*z^3 + 8*z^2 + 8*z - 8) * q^98 + (4*z^7 - 2*z^5 + z^4 + 2*z - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} + 2 q^{3} + q^{4} + q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + q^{9}+O(q^{10})$$ 8 * q + q^2 + 2 * q^3 + q^4 + q^5 + 2 * q^6 + 6 * q^7 - 2 * q^8 + q^9 $$8 q + q^{2} + 2 q^{3} + q^{4} + q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + q^{9} + 6 q^{10} + 2 q^{11} + 16 q^{12} - 6 q^{13} - 3 q^{14} - 8 q^{15} + q^{16} - 4 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{20} - 16 q^{21} - q^{22} + 4 q^{23} + 2 q^{24} + 6 q^{25} - 18 q^{26} + 8 q^{27} + 7 q^{28} - 5 q^{29} + 16 q^{30} - 28 q^{31} - 4 q^{32} + 18 q^{33} + 26 q^{34} - 3 q^{35} + q^{36} + 14 q^{37} - 13 q^{38} + 24 q^{39} - 4 q^{40} - 19 q^{41} + 14 q^{42} + 16 q^{43} + 4 q^{44} - 12 q^{45} + 2 q^{46} + q^{47} + 2 q^{48} - 44 q^{49} + 18 q^{50} - 18 q^{51} + 9 q^{52} + 4 q^{53} + 16 q^{54} + 14 q^{55} + 16 q^{56} - 26 q^{57} + 10 q^{58} - 13 q^{59} + 2 q^{60} - 3 q^{61} - q^{62} - 3 q^{63} - 2 q^{64} - 48 q^{65} - 22 q^{66} - 18 q^{67} + 8 q^{68} + 4 q^{69} + 2 q^{70} - 16 q^{71} + q^{72} - 4 q^{73} - 7 q^{74} + 36 q^{75} - 8 q^{76} + 24 q^{77} - 12 q^{78} + 12 q^{79} - 4 q^{80} - 11 q^{81} + 21 q^{82} - 34 q^{83} + 12 q^{84} + 31 q^{85} - 19 q^{86} - 18 q^{88} - 32 q^{89} + q^{90} + 3 q^{91} - q^{92} + 28 q^{93} - 2 q^{94} - 38 q^{95} - 4 q^{96} - 30 q^{97} - 8 q^{98} - q^{99}+O(q^{100})$$ 8 * q + q^2 + 2 * q^3 + q^4 + q^5 + 2 * q^6 + 6 * q^7 - 2 * q^8 + q^9 + 6 * q^10 + 2 * q^11 + 16 * q^12 - 6 * q^13 - 3 * q^14 - 8 * q^15 + q^16 - 4 * q^17 - 2 * q^18 - 4 * q^19 - 2 * q^20 - 16 * q^21 - q^22 + 4 * q^23 + 2 * q^24 + 6 * q^25 - 18 * q^26 + 8 * q^27 + 7 * q^28 - 5 * q^29 + 16 * q^30 - 28 * q^31 - 4 * q^32 + 18 * q^33 + 26 * q^34 - 3 * q^35 + q^36 + 14 * q^37 - 13 * q^38 + 24 * q^39 - 4 * q^40 - 19 * q^41 + 14 * q^42 + 16 * q^43 + 4 * q^44 - 12 * q^45 + 2 * q^46 + q^47 + 2 * q^48 - 44 * q^49 + 18 * q^50 - 18 * q^51 + 9 * q^52 + 4 * q^53 + 16 * q^54 + 14 * q^55 + 16 * q^56 - 26 * q^57 + 10 * q^58 - 13 * q^59 + 2 * q^60 - 3 * q^61 - q^62 - 3 * q^63 - 2 * q^64 - 48 * q^65 - 22 * q^66 - 18 * q^67 + 8 * q^68 + 4 * q^69 + 2 * q^70 - 16 * q^71 + q^72 - 4 * q^73 - 7 * q^74 + 36 * q^75 - 8 * q^76 + 24 * q^77 - 12 * q^78 + 12 * q^79 - 4 * q^80 - 11 * q^81 + 21 * q^82 - 34 * q^83 + 12 * q^84 + 31 * q^85 - 19 * q^86 - 18 * q^88 - 32 * q^89 + q^90 + 3 * q^91 - q^92 + 28 * q^93 - 2 * q^94 - 38 * q^95 - 4 * q^96 - 30 * q^97 - 8 * q^98 - 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 1.82709 0.813473i 0.913545 + 0.406737i −1.75181 1.94558i −1.95630 + 0.415823i 0.190983 0.138757i −0.809017 0.587785i 0.669131 0.743145i 1.30902 + 2.26728i
125.1 0.913545 0.406737i 1.33826 + 1.48629i 0.669131 0.743145i 0.0399263 + 0.379874i 1.82709 + 0.813473i 1.30902 + 4.02874i 0.309017 0.951057i −0.104528 + 0.994522i 0.190983 + 0.330792i
159.1 0.669131 0.743145i −0.209057 + 1.98904i −0.104528 0.994522i 2.56082 0.544320i 1.33826 + 1.48629i 0.190983 0.138757i −0.809017 0.587785i −0.978148 0.207912i 1.30902 2.26728i
163.1 0.669131 + 0.743145i −0.209057 1.98904i −0.104528 + 0.994522i 2.56082 + 0.544320i 1.33826 1.48629i 0.190983 + 0.138757i −0.809017 + 0.587785i −0.978148 + 0.207912i 1.30902 + 2.26728i
201.1 −0.104528 0.994522i −1.95630 0.415823i −0.978148 + 0.207912i −0.348943 + 0.155360i −0.209057 + 1.98904i 1.30902 4.02874i 0.309017 + 0.951057i 0.913545 + 0.406737i 0.190983 + 0.330792i
235.1 −0.104528 + 0.994522i −1.95630 + 0.415823i −0.978148 0.207912i −0.348943 0.155360i −0.209057 1.98904i 1.30902 + 4.02874i 0.309017 0.951057i 0.913545 0.406737i 0.190983 0.330792i
273.1 −0.978148 + 0.207912i 1.82709 + 0.813473i 0.913545 0.406737i −1.75181 + 1.94558i −1.95630 0.415823i 0.190983 + 0.138757i −0.809017 + 0.587785i 0.669131 + 0.743145i 1.30902 2.26728i
311.1 0.913545 + 0.406737i 1.33826 1.48629i 0.669131 + 0.743145i 0.0399263 0.379874i 1.82709 0.813473i 1.30902 4.02874i 0.309017 + 0.951057i −0.104528 0.994522i 0.190983 0.330792i
 $$n$$: 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.c 8
11.c even 5 1 inner 418.2.n.c 8
19.c even 3 1 inner 418.2.n.c 8
209.n even 15 1 inner 418.2.n.c 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 2T_{3}^{7} + 8T_{3}^{5} - 16T_{3}^{4} + 32T_{3}^{3} - 128T_{3} + 256$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1$$
$3$ $$T^{8} - 2 T^{7} + 8 T^{5} - 16 T^{4} + \cdots + 256$$
$5$ $$T^{8} - T^{7} - 5 T^{6} - 14 T^{5} + \cdots + 1$$
$7$ $$(T^{4} - 3 T^{3} + 19 T^{2} - 7 T + 1)^{2}$$
$11$ $$(T^{4} - T^{3} - 9 T^{2} - 11 T + 121)^{2}$$
$13$ $$T^{8} + 6 T^{7} + 54 T^{5} + \cdots + 6561$$
$17$ $$T^{8} + 4 T^{7} - 30 T^{6} + \cdots + 2825761$$
$19$ $$T^{8} + 4 T^{7} - 3 T^{6} + \cdots + 130321$$
$23$ $$(T^{2} - T + 1)^{4}$$
$29$ $$T^{8} + 5 T^{7} + 10 T^{6} + 25 T^{5} + \cdots + 625$$
$31$ $$(T^{4} + 14 T^{3} + 76 T^{2} - 31 T + 961)^{2}$$
$37$ $$(T^{4} - 7 T^{3} + 24 T^{2} - 38 T + 361)^{2}$$
$41$ $$T^{8} + 19 T^{7} + 120 T^{6} + \cdots + 38950081$$
$43$ $$(T^{4} - 8 T^{3} + 93 T^{2} + 232 T + 841)^{2}$$
$47$ $$T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1$$
$53$ $$T^{8} - 4 T^{7} - 80 T^{6} + \cdots + 33362176$$
$59$ $$T^{8} + 13 T^{7} + 30 T^{6} + \cdots + 2825761$$
$61$ $$T^{8} + 3 T^{7} - 135 T^{6} + \cdots + 96059601$$
$67$ $$(T^{4} + 9 T^{3} + 122 T^{2} - 369 T + 1681)^{2}$$
$71$ $$T^{8} + 16 T^{7} + 70 T^{6} + \cdots + 12117361$$
$73$ $$T^{8} + 4 T^{7} - 80 T^{6} + \cdots + 33362176$$
$79$ $$T^{8} - 12 T^{7} - 432 T^{5} + \cdots + 1679616$$
$83$ $$(T^{4} + 17 T^{3} + 109 T^{2} - 87 T + 841)^{2}$$
$89$ $$(T^{4} + 16 T^{3} + 237 T^{2} + 304 T + 361)^{2}$$
$97$ $$T^{8} + 30 T^{7} + \cdots + 100000000$$