# Properties

 Label 370.2.q.b Level $370$ Weight $2$ Character orbit 370.q Analytic conductor $2.954$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.q (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$\zeta_{12}$$ $$\zeta_{12}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
−0.500000 + 0.866025i 0.633975 2.36603i −0.500000 0.866025i −1.23205 + 1.86603i 1.73205 + 1.73205i 0.366025 1.36603i 1.00000 −2.59808 1.50000i −1.00000 2.00000i
103.1 −0.500000 0.866025i 0.633975 + 2.36603i −0.500000 + 0.866025i −1.23205 1.86603i 1.73205 1.73205i 0.366025 + 1.36603i 1.00000 −2.59808 + 1.50000i −1.00000 + 2.00000i
267.1 −0.500000 0.866025i 2.36603 0.633975i −0.500000 + 0.866025i 2.23205 + 0.133975i −1.73205 1.73205i −1.36603 + 0.366025i 1.00000 2.59808 1.50000i −1.00000 2.00000i
273.1 −0.500000 + 0.866025i 2.36603 + 0.633975i −0.500000 0.866025i 2.23205 0.133975i −1.73205 + 1.73205i −1.36603 0.366025i 1.00000 2.59808 + 1.50000i −1.00000 + 2.00000i
 $$n$$: 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
185.p even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.q.b 4
5.c odd 4 1 370.2.r.b yes 4
37.g odd 12 1 370.2.r.b yes 4
185.p even 12 1 inner 370.2.q.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.q.b 4 1.a even 1 1 trivial
370.2.q.b 4 185.p even 12 1 inner
370.2.r.b yes 4 5.c odd 4 1
370.2.r.b yes 4 37.g odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 6T_{3}^{3} + 18T_{3}^{2} - 36T_{3} + 36$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4} - 6 T^{3} + 18 T^{2} - 36 T + 36$$
$5$ $$T^{4} - 2 T^{3} - T^{2} - 10 T + 25$$
$7$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$11$ $$T^{4} + 42T^{2} + 9$$
$13$ $$T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1$$
$17$ $$T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4$$
$19$ $$T^{4} - 4 T^{3} + 5 T^{2} - 2 T + 1$$
$23$ $$(T^{2} + 4 T + 1)^{2}$$
$29$ $$T^{4} + 8 T^{3} + 32 T^{2} - 128 T + 256$$
$31$ $$T^{4} + 4 T^{3} + 8 T^{2} - 208 T + 2704$$
$37$ $$(T^{2} - T + 37)^{2}$$
$41$ $$T^{4} + 12 T^{3} + 24 T^{2} + \cdots + 576$$
$43$ $$(T - 4)^{4}$$
$47$ $$T^{4} + 2 T^{3} + 2 T^{2} - 26 T + 169$$
$53$ $$T^{4} + 10 T^{3} + 146 T^{2} + \cdots + 14884$$
$59$ $$T^{4} + 8 T^{3} + 17 T^{2} + 22 T + 121$$
$61$ $$T^{4} - 22 T^{3} + 122 T^{2} + \cdots + 676$$
$67$ $$T^{4} + 6 T^{3} + 90 T^{2} + \cdots + 4356$$
$71$ $$(T^{2} - 6 T + 36)^{2}$$
$73$ $$T^{4} + 4 T^{3} + 8 T^{2} - 376 T + 8836$$
$79$ $$T^{4} - 24 T^{3} + 144 T^{2} + \cdots + 2304$$
$83$ $$T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 1024$$
$89$ $$T^{4} - 16 T^{3} + 185 T^{2} + \cdots + 32041$$
$97$ $$T^{4} + 96T^{2} + 576$$