# Properties

 Label 418.2.f.e 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{3} - \zeta_{10}^{3} q^{4} + (\zeta_{10}^{2} + 1) q^{5} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{6} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{7} + \zeta_{10}^{2} q^{8} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{9} +O(q^{10})$$ q + (z^3 - z^2 + z - 1) * q^2 + (2*z^3 - z^2 + 2*z) * q^3 - z^3 * q^4 + (z^2 + 1) * q^5 + (-2*z^2 + z - 2) * q^6 + (2*z^3 + 3*z - 3) * q^7 + z^2 * q^8 + (2*z^3 - 2*z^2 + 2*z - 2) * q^9 $$q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{3} - \zeta_{10}^{3} q^{4} + (\zeta_{10}^{2} + 1) q^{5} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{6} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{7} + \zeta_{10}^{2} q^{8} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{9} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{10} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10} - 2) q^{11} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 1) q^{12} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{13} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - 3 \zeta_{10}) q^{14} + (3 \zeta_{10}^{3} + \zeta_{10} - 1) q^{15} - \zeta_{10} q^{16} + ( - 2 \zeta_{10}^{2} + 5 \zeta_{10} - 2) q^{17} - 2 \zeta_{10}^{3} q^{18} + \zeta_{10}^{2} q^{19} + ( - \zeta_{10}^{3} + 1) q^{20} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 8) q^{21} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10} + 1) q^{22} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 3) q^{23} + (\zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{24} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10}) q^{25} + (\zeta_{10} - 1) q^{26} + (2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{27} + (3 \zeta_{10}^{2} - \zeta_{10} + 3) q^{28} + (4 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{29} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} - \zeta_{10}) q^{30} + ( - \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 1) q^{31} + q^{32} + ( - 3 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 2 \zeta_{10}) q^{33} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 3) q^{34} + (5 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 5) q^{35} + 2 \zeta_{10}^{2} q^{36} + (3 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{37} - \zeta_{10} q^{38} + ( - \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{39} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{40} + ( - 5 \zeta_{10}^{3} + \zeta_{10}^{2} - 5 \zeta_{10}) q^{41} + ( - 8 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 8) q^{42} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 6) q^{43} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10} - 3) q^{44} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{45} + (3 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 3) q^{46} + ( - 4 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 4 \zeta_{10}) q^{47} + ( - \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{48} + ( - 3 \zeta_{10}^{2} - 3 \zeta_{10} - 3) q^{49} + ( - \zeta_{10}^{2} + 4 \zeta_{10} - 1) q^{50} + ( - \zeta_{10}^{3} + 8 \zeta_{10} - 8) q^{51} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{52} + ( - 7 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 7) q^{53} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 1) q^{54} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 3 \zeta_{10}) q^{55} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 2) q^{56} + (\zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{57} + (4 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 4 \zeta_{10}) q^{58} + (5 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{59} + (\zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{60} + (\zeta_{10}^{2} + 5 \zeta_{10} + 1) q^{61} + (\zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{62} + ( - 6 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 6 \zeta_{10}) q^{63} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + q^{65} + (3 \zeta_{10}^{2} + 6 \zeta_{10} - 2) q^{66} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 2) q^{67} + ( - 3 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 3) q^{68} + (4 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 4 \zeta_{10}) q^{69} + ( - 5 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{70} + (2 \zeta_{10}^{2} + 5 \zeta_{10} + 2) q^{71} - 2 \zeta_{10} q^{72} + (10 \zeta_{10}^{3} + 7 \zeta_{10} - 7) q^{73} + (3 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 3 \zeta_{10}) q^{74} + ( - \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} + 1) q^{75} + q^{76} + ( - 14 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 5 \zeta_{10} + 6) q^{77} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 2) q^{78} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{79} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{80} + 11 \zeta_{10}^{3} q^{81} + (5 \zeta_{10}^{2} - \zeta_{10} + 5) q^{82} + (5 \zeta_{10}^{2} + 2 \zeta_{10} + 5) q^{83} + (8 \zeta_{10}^{3} + \zeta_{10} - 1) q^{84} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 3 \zeta_{10}) q^{85} + ( - 6 \zeta_{10}^{3} + 6) q^{86} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 4) q^{87} + ( - 3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{88} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} + 7) q^{89} + ( - 2 \zeta_{10}^{3} + 2) q^{90} + ( - \zeta_{10}^{3} + 4 \zeta_{10}^{2} - \zeta_{10}) q^{91} + ( - 3 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{92} + (8 \zeta_{10}^{2} - 19 \zeta_{10} + 8) q^{93} + (4 \zeta_{10}^{2} - 9 \zeta_{10} + 4) q^{94} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{95} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{96} + ( - 3 \zeta_{10}^{3} - 9 \zeta_{10}^{2} + 9 \zeta_{10} + 3) q^{97} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 6) q^{98} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10} + 2) q^{99} +O(q^{100})$$ q + (z^3 - z^2 + z - 1) * q^2 + (2*z^3 - z^2 + 2*z) * q^3 - z^3 * q^4 + (z^2 + 1) * q^5 + (-2*z^2 + z - 2) * q^6 + (2*z^3 + 3*z - 3) * q^7 + z^2 * q^8 + (2*z^3 - 2*z^2 + 2*z - 2) * q^9 + (z^3 - z^2 - 1) * q^10 + (2*z^3 - 4*z^2 + z - 2) * q^11 + (-2*z^3 + 2*z^2 + 1) * q^12 + (-z^2 + z) * q^13 + (-3*z^3 + z^2 - 3*z) * q^14 + (3*z^3 + z - 1) * q^15 - z * q^16 + (-2*z^2 + 5*z - 2) * q^17 - 2*z^3 * q^18 + z^2 * q^19 + (-z^3 + 1) * q^20 + (z^3 - z^2 - 8) * q^21 + (-2*z^3 + 2*z + 1) * q^22 + (-2*z^3 + 2*z^2 + 3) * q^23 + (z^3 + z^2 - z - 1) * q^24 + (z^3 - 4*z^2 + z) * q^25 + (z - 1) * q^26 + (2*z^2 - z + 2) * q^27 + (3*z^2 - z + 3) * q^28 + (4*z^3 - 4*z + 4) * q^29 + (-z^3 - 2*z^2 - z) * q^30 + (-z^3 + 7*z^2 - 7*z + 1) * q^31 + q^32 + (-3*z^3 - 6*z^2 + 2*z) * q^33 + (-2*z^3 + 2*z^2 - 3) * q^34 + (5*z^3 - 3*z^2 + 3*z - 5) * q^35 + 2*z^2 * q^36 + (3*z^3 - 3*z + 3) * q^37 - z * q^38 + (-z^2 + 3*z - 1) * q^39 + (z^3 + z - 1) * q^40 + (-5*z^3 + z^2 - 5*z) * q^41 + (-8*z^3 + 7*z^2 - 7*z + 8) * q^42 + (6*z^3 - 6*z^2 - 6) * q^43 + (z^3 + z^2 + z - 3) * q^44 + (2*z^3 - 2*z^2 - 2) * q^45 + (3*z^3 - z^2 + z - 3) * q^46 + (-4*z^3 + 9*z^2 - 4*z) * q^47 + (-z^3 - 2*z + 2) * q^48 + (-3*z^2 - 3*z - 3) * q^49 + (-z^2 + 4*z - 1) * q^50 + (-z^3 + 8*z - 8) * q^51 + (-z^3 + z^2 - z) * q^52 + (-7*z^3 - z^2 + z + 7) * q^53 + (2*z^3 - 2*z^2 - 1) * q^54 + (-z^3 - 2*z^2 - 3*z) * q^55 + (3*z^3 - 3*z^2 - 2) * q^56 + (z^3 + z^2 - z - 1) * q^57 + (4*z^3 - 8*z^2 + 4*z) * q^58 + (5*z^3 - 5*z + 5) * q^59 + (z^2 + 2*z + 1) * q^60 + (z^2 + 5*z + 1) * q^61 + (z^3 - 6*z + 6) * q^62 + (-6*z^3 + 2*z^2 - 6*z) * q^63 + (z^3 - z^2 + z - 1) * q^64 + q^65 + (3*z^2 + 6*z - 2) * q^66 + (-5*z^3 + 5*z^2 + 2) * q^67 + (-3*z^3 + 5*z^2 - 5*z + 3) * q^68 + (4*z^3 + 3*z^2 + 4*z) * q^69 + (-5*z^3 - 2*z + 2) * q^70 + (2*z^2 + 5*z + 2) * q^71 - 2*z * q^72 + (10*z^3 + 7*z - 7) * q^73 + (3*z^3 - 6*z^2 + 3*z) * q^74 + (-z^3 - 6*z^2 + 6*z + 1) * q^75 + q^76 + (-14*z^3 + 7*z^2 - 5*z + 6) * q^77 + (-z^3 + z^2 - 2) * q^78 + (3*z^3 - 3*z^2 + 3*z - 3) * q^79 + (-z^3 - z) * q^80 + 11*z^3 * q^81 + (5*z^2 - z + 5) * q^82 + (5*z^2 + 2*z + 5) * q^83 + (8*z^3 + z - 1) * q^84 + (3*z^3 - 2*z^2 + 3*z) * q^85 + (-6*z^3 + 6) * q^86 + (12*z^3 - 12*z^2 + 4) * q^87 + (-3*z^3 + 2*z^2 - 4*z + 2) * q^88 + (-10*z^3 + 10*z^2 + 7) * q^89 + (-2*z^3 + 2) * q^90 + (-z^3 + 4*z^2 - z) * q^91 + (-3*z^3 - 2*z + 2) * q^92 + (8*z^2 - 19*z + 8) * q^93 + (4*z^2 - 9*z + 4) * q^94 + (z^3 + z - 1) * q^95 + (2*z^3 - z^2 + 2*z) * q^96 + (-3*z^3 - 9*z^2 + 9*z + 3) * q^97 + (-3*z^3 + 3*z^2 + 6) * q^98 + (-4*z^3 + 4*z + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} + 5 q^{3} - q^{4} + 3 q^{5} - 5 q^{6} - 7 q^{7} - q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - q^2 + 5 * q^3 - q^4 + 3 * q^5 - 5 * q^6 - 7 * q^7 - q^8 - 2 * q^9 $$4 q - q^{2} + 5 q^{3} - q^{4} + 3 q^{5} - 5 q^{6} - 7 q^{7} - q^{8} - 2 q^{9} - 2 q^{10} - q^{11} + 2 q^{13} - 7 q^{14} - q^{16} - q^{17} - 2 q^{18} - q^{19} + 3 q^{20} - 30 q^{21} + 4 q^{22} + 8 q^{23} - 5 q^{24} + 6 q^{25} - 3 q^{26} + 5 q^{27} + 8 q^{28} + 16 q^{29} - 11 q^{31} + 4 q^{32} + 5 q^{33} - 16 q^{34} - 9 q^{35} - 2 q^{36} + 12 q^{37} - q^{38} - 2 q^{40} - 11 q^{41} + 10 q^{42} - 12 q^{43} - 11 q^{44} - 4 q^{45} - 7 q^{46} - 17 q^{47} + 5 q^{48} - 12 q^{49} + q^{50} - 25 q^{51} - 3 q^{52} + 23 q^{53} - 2 q^{55} - 2 q^{56} - 5 q^{57} + 16 q^{58} + 20 q^{59} + 5 q^{60} + 8 q^{61} + 19 q^{62} - 14 q^{63} - q^{64} + 4 q^{65} - 5 q^{66} - 2 q^{67} - q^{68} + 5 q^{69} + q^{70} + 11 q^{71} - 2 q^{72} - 11 q^{73} + 12 q^{74} + 15 q^{75} + 4 q^{76} - 2 q^{77} - 10 q^{78} - 3 q^{79} - 2 q^{80} + 11 q^{81} + 14 q^{82} + 17 q^{83} + 5 q^{84} + 8 q^{85} + 18 q^{86} + 40 q^{87} - q^{88} + 8 q^{89} + 6 q^{90} - 6 q^{91} + 3 q^{92} + 5 q^{93} + 3 q^{94} - 2 q^{95} + 5 q^{96} + 27 q^{97} + 18 q^{98} + 8 q^{99}+O(q^{100})$$ 4 * q - q^2 + 5 * q^3 - q^4 + 3 * q^5 - 5 * q^6 - 7 * q^7 - q^8 - 2 * q^9 - 2 * q^10 - q^11 + 2 * q^13 - 7 * q^14 - q^16 - q^17 - 2 * q^18 - q^19 + 3 * q^20 - 30 * q^21 + 4 * q^22 + 8 * q^23 - 5 * q^24 + 6 * q^25 - 3 * q^26 + 5 * q^27 + 8 * q^28 + 16 * q^29 - 11 * q^31 + 4 * q^32 + 5 * q^33 - 16 * q^34 - 9 * q^35 - 2 * q^36 + 12 * q^37 - q^38 - 2 * q^40 - 11 * q^41 + 10 * q^42 - 12 * q^43 - 11 * q^44 - 4 * q^45 - 7 * q^46 - 17 * q^47 + 5 * q^48 - 12 * q^49 + q^50 - 25 * q^51 - 3 * q^52 + 23 * q^53 - 2 * q^55 - 2 * q^56 - 5 * q^57 + 16 * q^58 + 20 * q^59 + 5 * q^60 + 8 * q^61 + 19 * q^62 - 14 * q^63 - q^64 + 4 * q^65 - 5 * q^66 - 2 * q^67 - q^68 + 5 * q^69 + q^70 + 11 * q^71 - 2 * q^72 - 11 * q^73 + 12 * q^74 + 15 * q^75 + 4 * q^76 - 2 * q^77 - 10 * q^78 - 3 * q^79 - 2 * q^80 + 11 * q^81 + 14 * q^82 + 17 * q^83 + 5 * q^84 + 8 * q^85 + 18 * q^86 + 40 * q^87 - q^88 + 8 * q^89 + 6 * q^90 - 6 * q^91 + 3 * q^92 + 5 * q^93 + 3 * q^94 - 2 * q^95 + 5 * q^96 + 27 * q^97 + 18 * q^98 + 8 * 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 1.80902 + 1.31433i −0.809017 + 0.587785i 0.190983 0.587785i −0.690983 + 2.12663i −2.30902 + 1.67760i −0.809017 0.587785i 0.618034 + 1.90211i 0.618034
191.1 −0.809017 0.587785i 0.690983 2.12663i 0.309017 + 0.951057i 1.30902 0.951057i −1.80902 + 1.31433i −1.19098 3.66547i 0.309017 0.951057i −1.61803 1.17557i −1.61803
229.1 0.309017 0.951057i 1.80902 1.31433i −0.809017 0.587785i 0.190983 + 0.587785i −0.690983 2.12663i −2.30902 1.67760i −0.809017 + 0.587785i 0.618034 1.90211i 0.618034
267.1 −0.809017 + 0.587785i 0.690983 + 2.12663i 0.309017 0.951057i 1.30902 + 0.951057i −1.80902 1.31433i −1.19098 + 3.66547i 0.309017 + 0.951057i −1.61803 + 1.17557i −1.61803
 $$n$$: 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.e 4
11.c even 5 1 inner 418.2.f.e 4
11.c even 5 1 4598.2.a.bi 2
11.d odd 10 1 4598.2.a.ba 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 5T_{3}^{3} + 15T_{3}^{2} - 25T_{3} + 25$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$3$ $$T^{4} - 5 T^{3} + 15 T^{2} - 25 T + 25$$
$5$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$7$ $$T^{4} + 7 T^{3} + 34 T^{2} + 88 T + 121$$
$11$ $$T^{4} + T^{3} - 9 T^{2} + 11 T + 121$$
$13$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$17$ $$T^{4} + T^{3} + 31 T^{2} - 99 T + 121$$
$19$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$23$ $$(T^{2} - 4 T - 1)^{2}$$
$29$ $$T^{4} - 16 T^{3} + 96 T^{2} + \cdots + 256$$
$31$ $$T^{4} + 11 T^{3} + 151 T^{2} + \cdots + 841$$
$37$ $$T^{4} - 12 T^{3} + 54 T^{2} + 27 T + 81$$
$41$ $$T^{4} + 11 T^{3} + 96 T^{2} + \cdots + 841$$
$43$ $$(T^{2} + 6 T - 36)^{2}$$
$47$ $$T^{4} + 17 T^{3} + 109 T^{2} + \cdots + 841$$
$53$ $$T^{4} - 23 T^{3} + 249 T^{2} + \cdots + 5041$$
$59$ $$T^{4} - 20 T^{3} + 150 T^{2} + \cdots + 625$$
$61$ $$T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841$$
$67$ $$(T^{2} + T - 31)^{2}$$
$71$ $$T^{4} - 11 T^{3} + 51 T^{2} + \cdots + 961$$
$73$ $$T^{4} + 11 T^{3} + 226 T^{2} + \cdots + 361$$
$79$ $$T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81$$
$83$ $$T^{4} - 17 T^{3} + 114 T^{2} + \cdots + 121$$
$89$ $$(T^{2} - 4 T - 121)^{2}$$
$97$ $$T^{4} - 27 T^{3} + 549 T^{2} + \cdots + 29241$$