# Properties

 Label 847.2.f.a 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: no (minimal twist has level 77) 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.

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

## 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
−0.690983 2.12663i −2.61803 1.90211i −2.42705 + 1.76336i −0.618034 + 1.90211i −2.23607 + 6.88191i −0.809017 + 0.587785i 1.80902 + 1.31433i 2.30902 + 7.10642i 4.47214
323.1 −1.80902 1.31433i −0.381966 + 1.17557i 0.927051 + 2.85317i 1.61803 1.17557i 2.23607 1.62460i 0.309017 + 0.951057i 0.690983 2.12663i 1.19098 + 0.865300i −4.47214
372.1 −0.690983 + 2.12663i −2.61803 + 1.90211i −2.42705 1.76336i −0.618034 1.90211i −2.23607 6.88191i −0.809017 0.587785i 1.80902 1.31433i 2.30902 7.10642i 4.47214
729.1 −1.80902 + 1.31433i −0.381966 1.17557i 0.927051 2.85317i 1.61803 + 1.17557i 2.23607 + 1.62460i 0.309017 0.951057i 0.690983 + 2.12663i 1.19098 0.865300i −4.47214
 $$n$$: 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.a 4
11.b odd 2 1 847.2.f.m 4
11.c even 5 1 77.2.a.d 2
11.c even 5 1 inner 847.2.f.a 4
11.c even 5 2 847.2.f.n 4
11.d odd 10 1 847.2.a.f 2
11.d odd 10 2 847.2.f.b 4
11.d odd 10 1 847.2.f.m 4
33.f even 10 1 7623.2.a.bl 2
33.h odd 10 1 693.2.a.h 2
44.h odd 10 1 1232.2.a.m 2
55.j even 10 1 1925.2.a.r 2
55.k odd 20 2 1925.2.b.h 4
77.j odd 10 1 539.2.a.f 2
77.l even 10 1 5929.2.a.m 2
77.m even 15 2 539.2.e.i 4
77.p odd 30 2 539.2.e.j 4
88.l odd 10 1 4928.2.a.bv 2
88.o even 10 1 4928.2.a.bm 2
231.u even 10 1 4851.2.a.y 2
308.t even 10 1 8624.2.a.ce 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 11.c even 5 1
539.2.a.f 2 77.j odd 10 1
539.2.e.i 4 77.m even 15 2
539.2.e.j 4 77.p odd 30 2
693.2.a.h 2 33.h odd 10 1
847.2.a.f 2 11.d odd 10 1
847.2.f.a 4 1.a even 1 1 trivial
847.2.f.a 4 11.c even 5 1 inner
847.2.f.b 4 11.d odd 10 2
847.2.f.m 4 11.b odd 2 1
847.2.f.m 4 11.d odd 10 1
847.2.f.n 4 11.c even 5 2
1232.2.a.m 2 44.h odd 10 1
1925.2.a.r 2 55.j even 10 1
1925.2.b.h 4 55.k odd 20 2
4851.2.a.y 2 231.u even 10 1
4928.2.a.bm 2 88.o even 10 1
4928.2.a.bv 2 88.l odd 10 1
5929.2.a.m 2 77.l even 10 1
7623.2.a.bl 2 33.f even 10 1
8624.2.a.ce 2 308.t even 10 1

## Hecke kernels

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

 $$T_{2}^{4} + 5T_{2}^{3} + 15T_{2}^{2} + 25T_{2} + 25$$ T2^4 + 5*T2^3 + 15*T2^2 + 25*T2 + 25 $$T_{3}^{4} + 6T_{3}^{3} + 16T_{3}^{2} + 16T_{3} + 16$$ T3^4 + 6*T3^3 + 16*T3^2 + 16*T3 + 16 $$T_{13}^{4} + 6T_{13}^{3} + 16T_{13}^{2} + 16T_{13} + 16$$ T13^4 + 6*T13^3 + 16*T13^2 + 16*T13 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25$$
$3$ $$T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16$$
$5$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$7$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16$$
$17$ $$T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16$$
$19$ $$T^{4} - 8 T^{3} + 64 T^{2} - 192 T + 256$$
$23$ $$(T^{2} + 4 T - 16)^{2}$$
$29$ $$T^{4} - 6 T^{3} + 76 T^{2} - 56 T + 16$$
$31$ $$T^{4} + 40 T^{2} - 200 T + 400$$
$37$ $$T^{4} + 6 T^{3} + 76 T^{2} + 56 T + 16$$
$41$ $$T^{4} - 4 T^{3} + 96 T^{2} + \cdots + 5776$$
$43$ $$(T - 8)^{4}$$
$47$ $$T^{4} + 10 T^{3} + 40 T^{2} + \cdots + 400$$
$53$ $$T^{4} - 6 T^{3} + 76 T^{2} - 56 T + 16$$
$59$ $$T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16$$
$61$ $$T^{4} - 10 T^{3} + 40 T^{2} + \cdots + 400$$
$67$ $$(T^{2} - 20 T + 80)^{2}$$
$71$ $$T^{4} - 16 T^{3} + 96 T^{2} + \cdots + 256$$
$73$ $$T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16$$
$79$ $$T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400$$
$83$ $$T^{4} - 28 T^{3} + 544 T^{2} + \cdots + 30976$$
$89$ $$(T - 2)^{4}$$
$97$ $$T^{4} + 34 T^{3} + 556 T^{2} + \cdots + 26896$$