# Properties

 Label 847.2.f.j Level $847$ Weight $2$ Character orbit 847.f Analytic conductor $6.763$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.f (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.76332905120$$ 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: $$\Z[a_1, a_2, a_3]$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 −0.309017 + 0.951057i 0.809017 − 0.587785i −0.309017 − 0.951057i 0.809017 + 0.587785i
0.809017 + 2.48990i −0.500000 0.363271i −3.92705 + 2.85317i 0.309017 0.951057i 0.500000 1.53884i 0.809017 0.587785i −6.04508 4.39201i −0.809017 2.48990i 2.61803
323.1 −0.309017 0.224514i −0.500000 + 1.53884i −0.572949 1.76336i −0.809017 + 0.587785i 0.500000 0.363271i −0.309017 0.951057i −0.454915 + 1.40008i 0.309017 + 0.224514i 0.381966
372.1 0.809017 2.48990i −0.500000 + 0.363271i −3.92705 2.85317i 0.309017 + 0.951057i 0.500000 + 1.53884i 0.809017 + 0.587785i −6.04508 + 4.39201i −0.809017 + 2.48990i 2.61803
729.1 −0.309017 + 0.224514i −0.500000 1.53884i −0.572949 + 1.76336i −0.809017 0.587785i 0.500000 + 0.363271i −0.309017 + 0.951057i −0.454915 1.40008i 0.309017 0.224514i 0.381966
 $$n$$: 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 847.2.f.j 4
11.b odd 2 1 847.2.f.d 4
11.c even 5 1 847.2.a.h yes 2
11.c even 5 2 847.2.f.c 4
11.c even 5 1 inner 847.2.f.j 4
11.d odd 10 1 847.2.a.d 2
11.d odd 10 1 847.2.f.d 4
11.d odd 10 2 847.2.f.l 4
33.f even 10 1 7623.2.a.bx 2
33.h odd 10 1 7623.2.a.t 2
77.j odd 10 1 5929.2.a.s 2
77.l even 10 1 5929.2.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.d 2 11.d odd 10 1
847.2.a.h yes 2 11.c even 5 1
847.2.f.c 4 11.c even 5 2
847.2.f.d 4 11.b odd 2 1
847.2.f.d 4 11.d odd 10 1
847.2.f.j 4 1.a even 1 1 trivial
847.2.f.j 4 11.c even 5 1 inner
847.2.f.l 4 11.d odd 10 2
5929.2.a.i 2 77.l even 10 1
5929.2.a.s 2 77.j odd 10 1
7623.2.a.t 2 33.h odd 10 1
7623.2.a.bx 2 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 1$$ T2^4 - T2^3 + 6*T2^2 + 4*T2 + 1 $$T_{3}^{4} + 2T_{3}^{3} + 4T_{3}^{2} + 3T_{3} + 1$$ T3^4 + 2*T3^3 + 4*T3^2 + 3*T3 + 1 $$T_{13}^{4} + 4T_{13}^{3} + 16T_{13}^{2} + 24T_{13} + 16$$ T13^4 + 4*T13^3 + 16*T13^2 + 24*T13 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + 6 T^{2} + 4 T + 1$$
$3$ $$T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1$$
$5$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$7$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16$$
$17$ $$T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 625$$
$19$ $$T^{4} + 9 T^{3} + 31 T^{2} - 11 T + 121$$
$23$ $$(T^{2} + T - 31)^{2}$$
$29$ $$T^{4} + T^{3} + 16 T^{2} + 66 T + 121$$
$31$ $$T^{4} - 7 T^{3} + 19 T^{2} - 3 T + 1$$
$37$ $$T^{4} - 8 T^{3} + 64 T^{2} - 192 T + 256$$
$41$ $$T^{4} - 25 T^{3} + 375 T^{2} + \cdots + 15625$$
$43$ $$(T^{2} - 5 T - 95)^{2}$$
$47$ $$T^{4} - 3 T^{3} + 34 T^{2} - 232 T + 841$$
$53$ $$T^{4} - T^{3} + 16 T^{2} - 66 T + 121$$
$59$ $$T^{4} - 7 T^{3} + 124 T^{2} - 18 T + 1$$
$61$ $$T^{4} + 9 T^{3} + 46 T^{2} + \cdots + 1681$$
$67$ $$(T^{2} + T - 61)^{2}$$
$71$ $$T^{4} - 23 T^{3} + 204 T^{2} + \cdots + 6241$$
$73$ $$T^{4} + 7 T^{3} + 159 T^{2} + \cdots + 19321$$
$79$ $$T^{4} - 5 T^{3} + 60 T^{2} + \cdots + 3025$$
$83$ $$T^{4} - 11 T^{3} + 151 T^{2} + \cdots + 841$$
$89$ $$(T^{2} - 7 T + 1)^{2}$$
$97$ $$T^{4} + 7 T^{3} + 49 T^{2} + \cdots + 2401$$