# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.f (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
115.1
 −0.309017 + 0.951057i 0.809017 − 0.587785i −0.309017 − 0.951057i 0.809017 + 0.587785i
0.309017 + 0.951057i 1.61803 + 1.17557i −0.809017 + 0.587785i 0.500000 1.53884i −0.618034 + 1.90211i 3.92705 2.85317i −0.809017 0.587785i 0.309017 + 0.951057i 1.61803
191.1 −0.809017 0.587785i −0.618034 + 1.90211i 0.309017 + 0.951057i 0.500000 0.363271i 1.61803 1.17557i 0.572949 + 1.76336i 0.309017 0.951057i −0.809017 0.587785i −0.618034
229.1 0.309017 0.951057i 1.61803 1.17557i −0.809017 0.587785i 0.500000 + 1.53884i −0.618034 1.90211i 3.92705 + 2.85317i −0.809017 + 0.587785i 0.309017 0.951057i 1.61803
267.1 −0.809017 + 0.587785i −0.618034 1.90211i 0.309017 0.951057i 0.500000 + 0.363271i 1.61803 + 1.17557i 0.572949 1.76336i 0.309017 + 0.951057i −0.809017 + 0.587785i −0.618034
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.f.d 4
11.c even 5 1 inner 418.2.f.d 4
11.c even 5 1 4598.2.a.bb 2
11.d odd 10 1 4598.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.d 4 1.a even 1 1 trivial
418.2.f.d 4 11.c even 5 1 inner
4598.2.a.t 2 11.d odd 10 1
4598.2.a.bb 2 11.c even 5 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$3$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$5$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$7$ $$T^{4} - 9 T^{3} + 36 T^{2} - 54 T + 81$$
$11$ $$T^{4} + T^{3} + 21 T^{2} + 11 T + 121$$
$13$ $$T^{4} - 12 T^{3} + 64 T^{2} + \cdots + 256$$
$17$ $$T^{4} + 7 T^{3} + 24 T^{2} + 38 T + 361$$
$19$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$23$ $$(T^{2} + 9 T + 9)^{2}$$
$29$ $$T^{4} + 16 T^{3} + 96 T^{2} + \cdots + 256$$
$31$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$37$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$41$ $$T^{4} - 2 T^{3} + 64 T^{2} + \cdots + 1936$$
$43$ $$(T^{2} + 7 T - 19)^{2}$$
$47$ $$T^{4} + 10 T^{2} + 25 T + 25$$
$53$ $$T^{4} - 12 T^{3} + 64 T^{2} + \cdots + 1936$$
$59$ $$T^{4} + 2 T^{3} + 124 T^{2} + \cdots + 1936$$
$61$ $$T^{4} + 23 T^{3} + 304 T^{2} + \cdots + 10201$$
$67$ $$(T^{2} - 10 T + 20)^{2}$$
$71$ $$T^{4} - 24 T^{3} + 256 T^{2} + \cdots + 4096$$
$73$ $$T^{4} - 14 T^{3} + 196 T^{2} + \cdots + 38416$$
$79$ $$T^{4} - 18 T^{3} + 144 T^{2} + \cdots + 1296$$
$83$ $$T^{4} + 26 T^{3} + 456 T^{2} + \cdots + 22201$$
$89$ $$(T^{2} - 18 T + 76)^{2}$$
$97$ $$T^{4} - 2 T^{3} + 64 T^{2} + \cdots + 1936$$