# Properties

 Label 370.2.q.a 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{2} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12} + 1) q^{3} + (\zeta_{12}^{2} - 1) q^{4} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} + ( - \zeta_{12}^{2} - \zeta_{12}) q^{6} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{7} + q^{8} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{9} +O(q^{10})$$ q - z^2 * q^2 + (-z^3 + z + 1) * q^3 + (z^2 - 1) * q^4 + (2*z^2 + z - 2) * q^5 + (-z^2 - z) * q^6 + (-2*z^2 + 2*z + 2) * q^7 + q^8 + (z^3 - z^2 - z + 2) * q^9 $$q - \zeta_{12}^{2} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12} + 1) q^{3} + (\zeta_{12}^{2} - 1) q^{4} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} + ( - \zeta_{12}^{2} - \zeta_{12}) q^{6} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{7} + q^{8} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{9} + ( - \zeta_{12}^{3} + 2) q^{10} - 2 \zeta_{12}^{3} q^{11} + (\zeta_{12}^{3} + \zeta_{12}^{2} - 1) q^{12} + (4 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{13} + ( - 2 \zeta_{12}^{3} - 2) q^{14} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} - 1) q^{15} - \zeta_{12}^{2} q^{16} + ( - \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} + 6) q^{17} + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{18} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12} + 1) q^{19} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12}) q^{20} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{21} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{22} + 6 q^{23} + ( - \zeta_{12}^{3} + \zeta_{12} + 1) q^{24} + (4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12}) q^{25} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 1) q^{26} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{27} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{28} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{29} + ( - 3 \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{30} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 5) q^{31} + (\zeta_{12}^{2} - 1) q^{32} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2}) q^{33} + ( - 3 \zeta_{12}^{2} - \zeta_{12} - 3) q^{34} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{35} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{36} + (3 \zeta_{12}^{2} - 7) q^{37} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 3) q^{38} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{39} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{40} + ( - 2 \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{41} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{42} - 11 q^{43} + 2 \zeta_{12} q^{44} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 3) q^{45} - 6 \zeta_{12}^{2} q^{46} + (2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{47} + ( - \zeta_{12}^{2} - \zeta_{12}) q^{48} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{49} + (3 \zeta_{12}^{2} + 4 \zeta_{12} - 3) q^{50} + ( - 7 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 7) q^{51} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12}) q^{52} + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + \zeta_{12} + 7) q^{53} + (\zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12} + 3) q^{54} + ( - 2 \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{55} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{56} + (3 \zeta_{12}^{3} + 7 \zeta_{12}^{2} + 3 \zeta_{12}) q^{57} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{58} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{59} + (\zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} - 1) q^{60} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{61} + ( - 5 \zeta_{12} - 5) q^{62} + ( - 4 \zeta_{12}^{2} + 4 \zeta_{12}) q^{63} + q^{64} + ( - 5 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 3 \zeta_{12} - 4) q^{65} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{66} + (8 \zeta_{12}^{3} + \zeta_{12}^{2} - 7 \zeta_{12} + 7) q^{67} + (\zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{68} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12} + 6) q^{69} + ( - 6 \zeta_{12}^{2} + 2 \zeta_{12} + 6) q^{70} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{71} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{72} + (4 \zeta_{12}^{2} + 3) q^{74} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} - 7 \zeta_{12} - 4) q^{75} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} - 4) q^{76} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{77} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{78} + (5 \zeta_{12}^{2} + 5 \zeta_{12} - 5) q^{79} + ( - \zeta_{12}^{3} + 2) q^{80} + (10 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 5 \zeta_{12} - 2) q^{81} + (3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{82} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 7 \zeta_{12} - 12) q^{83} + (2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{84} + ( - \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 6 \zeta_{12} - 5) q^{85} + 11 \zeta_{12}^{2} q^{86} + ( - 4 \zeta_{12}^{2} - 6 \zeta_{12} - 4) q^{87} - 2 \zeta_{12}^{3} q^{88} + (\zeta_{12}^{3} + \zeta_{12}^{2} - 7 \zeta_{12} + 6) q^{89} + (\zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} + 4) q^{90} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12} - 8) q^{91} + (6 \zeta_{12}^{2} - 6) q^{92} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 5 \zeta_{12} + 10) q^{93} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 6) q^{94} + (\zeta_{12}^{3} + 5 \zeta_{12}^{2} - 5 \zeta_{12} - 12) q^{95} + (\zeta_{12}^{3} + \zeta_{12}^{2} - 1) q^{96} + (2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{97} - \zeta_{12} q^{98} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{99} +O(q^{100})$$ q - z^2 * q^2 + (-z^3 + z + 1) * q^3 + (z^2 - 1) * q^4 + (2*z^2 + z - 2) * q^5 + (-z^2 - z) * q^6 + (-2*z^2 + 2*z + 2) * q^7 + q^8 + (z^3 - z^2 - z + 2) * q^9 + (-z^3 + 2) * q^10 - 2*z^3 * q^11 + (z^3 + z^2 - 1) * q^12 + (4*z^3 - z^2 - 2*z + 1) * q^13 + (-2*z^3 - 2) * q^14 + (2*z^3 + 2*z^2 + z - 1) * q^15 - z^2 * q^16 + (-z^3 - 3*z^2 + z + 6) * q^17 + (-z^2 + z - 1) * q^18 + (4*z^3 + 3*z^2 - z + 1) * q^19 + (z^3 - 2*z^2 - z) * q^20 + (-2*z^3 - 2*z^2 + 2*z + 4) * q^21 + (2*z^3 - 2*z) * q^22 + 6 * q^23 + (-z^3 + z + 1) * q^24 + (4*z^3 - 3*z^2 - 4*z) * q^25 + (-2*z^3 + 4*z - 1) * q^26 + (2*z^3 + 3*z^2 - 3*z - 2) * q^27 + (2*z^3 + 2*z^2 - 2*z) * q^28 + (-2*z^3 - 2*z^2 - 2*z - 2) * q^29 + (-3*z^3 - z^2 + 2*z + 2) * q^30 + (-5*z^3 - 5*z^2 + 5*z + 5) * q^31 + (z^2 - 1) * q^32 + (-2*z^3 - 2*z^2) * q^33 + (-3*z^2 - z - 3) * q^34 + (2*z^3 + 6*z^2 - 2*z) * q^35 + (-z^3 + 2*z^2 - 1) * q^36 + (3*z^2 - 7) * q^37 + (-3*z^3 - 4*z^2 + 4*z + 3) * q^38 + (3*z^3 + 3*z^2 - 2*z - 1) * q^39 + (2*z^2 + z - 2) * q^40 + (-2*z^2 - 3*z - 2) * q^41 + (-2*z^2 - 2*z - 2) * q^42 - 11 * q^43 + 2*z * q^44 + (-3*z^3 + 4*z^2 + 2*z - 3) * q^45 - 6*z^2 * q^46 + (2*z^3 - 6*z^2 - 6*z + 2) * q^47 + (-z^2 - z) * q^48 + (-z^3 + z) * q^49 + (3*z^2 + 4*z - 3) * q^50 + (-7*z^3 - 4*z^2 + 4*z + 7) * q^51 + (-2*z^3 + z^2 - 2*z) * q^52 + (6*z^3 - 6*z^2 + z + 7) * q^53 + (z^3 - z^2 + 2*z + 3) * q^54 + (-2*z^2 + 4*z + 2) * q^55 + (-2*z^2 + 2*z + 2) * q^56 + (3*z^3 + 7*z^2 + 3*z) * q^57 + (4*z^3 + 4*z^2 - 2*z - 2) * q^58 + (-5*z^3 + 5*z^2 + 3*z - 2) * q^59 + (z^3 - z^2 - 3*z - 1) * q^60 + (-2*z^3 + 2*z^2 + 4*z - 4) * q^61 + (-5*z - 5) * q^62 + (-4*z^2 + 4*z) * q^63 + q^64 + (-5*z^3 + 4*z^2 - 3*z - 4) * q^65 + (2*z^3 + 2*z^2 - 2*z - 2) * q^66 + (8*z^3 + z^2 - 7*z + 7) * q^67 + (z^3 + 6*z^2 - 3) * q^68 + (-6*z^3 + 6*z + 6) * q^69 + (-6*z^2 + 2*z + 6) * q^70 + (-8*z^3 + 4*z^2 + 4*z - 4) * q^71 + (z^3 - z^2 - z + 2) * q^72 + (4*z^2 + 3) * q^74 + (4*z^3 + z^2 - 7*z - 4) * q^75 + (-z^3 + z^2 - 3*z - 4) * q^76 + (-4*z^2 - 4*z + 4) * q^77 + (-z^3 - 2*z^2 + 3*z + 3) * q^78 + (5*z^2 + 5*z - 5) * q^79 + (-z^3 + 2) * q^80 + (10*z^3 + 2*z^2 - 5*z - 2) * q^81 + (3*z^3 + 4*z^2 - 2) * q^82 + (-5*z^3 + 5*z^2 - 7*z - 12) * q^83 + (2*z^3 + 4*z^2 - 2) * q^84 + (-z^3 + 12*z^2 + 6*z - 5) * q^85 + 11*z^2 * q^86 + (-4*z^2 - 6*z - 4) * q^87 - 2*z^3 * q^88 + (z^3 + z^2 - 7*z + 6) * q^89 + (z^3 - z^2 - 3*z + 4) * q^90 + (2*z^3 + 2*z^2 + 6*z - 8) * q^91 + (6*z^2 - 6) * q^92 + (-10*z^3 - 10*z^2 + 5*z + 10) * q^93 + (4*z^3 + 4*z^2 + 2*z - 6) * q^94 + (z^3 + 5*z^2 - 5*z - 12) * q^95 + (z^3 + z^2 - 1) * q^96 + (2*z^3 - 4*z^2 + 2) * q^97 - z * q^98 + (-2*z^3 + 2*z^2 - 2*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 4 * q^3 - 2 * q^4 - 4 * q^5 - 2 * q^6 + 4 * q^7 + 4 * q^8 + 6 * q^9 $$4 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9} + 8 q^{10} - 2 q^{12} + 2 q^{13} - 8 q^{14} - 2 q^{16} + 18 q^{17} - 6 q^{18} + 10 q^{19} - 4 q^{20} + 12 q^{21} + 24 q^{23} + 4 q^{24} - 6 q^{25} - 4 q^{26} - 2 q^{27} + 4 q^{28} - 12 q^{29} + 6 q^{30} + 10 q^{31} - 2 q^{32} - 4 q^{33} - 18 q^{34} + 12 q^{35} - 22 q^{37} + 4 q^{38} + 2 q^{39} - 4 q^{40} - 12 q^{41} - 12 q^{42} - 44 q^{43} - 4 q^{45} - 12 q^{46} - 4 q^{47} - 2 q^{48} - 6 q^{50} + 20 q^{51} + 2 q^{52} + 16 q^{53} + 10 q^{54} + 4 q^{55} + 4 q^{56} + 14 q^{57} + 2 q^{59} - 6 q^{60} - 12 q^{61} - 20 q^{62} - 8 q^{63} + 4 q^{64} - 8 q^{65} - 4 q^{66} + 30 q^{67} + 24 q^{69} + 12 q^{70} - 8 q^{71} + 6 q^{72} + 20 q^{74} - 14 q^{75} - 14 q^{76} + 8 q^{77} + 8 q^{78} - 10 q^{79} + 8 q^{80} - 4 q^{81} - 38 q^{83} + 4 q^{85} + 22 q^{86} - 24 q^{87} + 26 q^{89} + 14 q^{90} - 28 q^{91} - 12 q^{92} + 20 q^{93} - 16 q^{94} - 38 q^{95} - 2 q^{96} + 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 4 * q^3 - 2 * q^4 - 4 * q^5 - 2 * q^6 + 4 * q^7 + 4 * q^8 + 6 * q^9 + 8 * q^10 - 2 * q^12 + 2 * q^13 - 8 * q^14 - 2 * q^16 + 18 * q^17 - 6 * q^18 + 10 * q^19 - 4 * q^20 + 12 * q^21 + 24 * q^23 + 4 * q^24 - 6 * q^25 - 4 * q^26 - 2 * q^27 + 4 * q^28 - 12 * q^29 + 6 * q^30 + 10 * q^31 - 2 * q^32 - 4 * q^33 - 18 * q^34 + 12 * q^35 - 22 * q^37 + 4 * q^38 + 2 * q^39 - 4 * q^40 - 12 * q^41 - 12 * q^42 - 44 * q^43 - 4 * q^45 - 12 * q^46 - 4 * q^47 - 2 * q^48 - 6 * q^50 + 20 * q^51 + 2 * q^52 + 16 * q^53 + 10 * q^54 + 4 * q^55 + 4 * q^56 + 14 * q^57 + 2 * q^59 - 6 * q^60 - 12 * q^61 - 20 * q^62 - 8 * q^63 + 4 * q^64 - 8 * q^65 - 4 * q^66 + 30 * q^67 + 24 * q^69 + 12 * q^70 - 8 * q^71 + 6 * q^72 + 20 * q^74 - 14 * q^75 - 14 * q^76 + 8 * q^77 + 8 * q^78 - 10 * q^79 + 8 * q^80 - 4 * q^81 - 38 * q^83 + 4 * q^85 + 22 * q^86 - 24 * q^87 + 26 * q^89 + 14 * q^90 - 28 * q^91 - 12 * q^92 + 20 * q^93 - 16 * q^94 - 38 * q^95 - 2 * q^96 + 4 * 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 0.133975 0.500000i −0.500000 0.866025i −1.86603 1.23205i 0.366025 + 0.366025i −0.732051 + 2.73205i 1.00000 2.36603 + 1.36603i 2.00000 1.00000i
103.1 −0.500000 0.866025i 0.133975 + 0.500000i −0.500000 + 0.866025i −1.86603 + 1.23205i 0.366025 0.366025i −0.732051 2.73205i 1.00000 2.36603 1.36603i 2.00000 + 1.00000i
267.1 −0.500000 0.866025i 1.86603 0.500000i −0.500000 + 0.866025i −0.133975 + 2.23205i −1.36603 1.36603i 2.73205 0.732051i 1.00000 0.633975 0.366025i 2.00000 1.00000i
273.1 −0.500000 + 0.866025i 1.86603 + 0.500000i −0.500000 0.866025i −0.133975 2.23205i −1.36603 + 1.36603i 2.73205 + 0.732051i 1.00000 0.633975 + 0.366025i 2.00000 + 1.00000i
 $$n$$: 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.a 4
5.c odd 4 1 370.2.r.a yes 4
37.g odd 12 1 370.2.r.a yes 4
185.p even 12 1 inner 370.2.q.a 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 4T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4} - 4 T^{3} + 5 T^{2} - 2 T + 1$$
$5$ $$T^{4} + 4 T^{3} + 11 T^{2} + 20 T + 25$$
$7$ $$T^{4} - 4 T^{3} + 8 T^{2} - 32 T + 64$$
$11$ $$(T^{2} + 4)^{2}$$
$13$ $$T^{4} - 2 T^{3} + 15 T^{2} + 22 T + 121$$
$17$ $$T^{4} - 18 T^{3} + 134 T^{2} + \cdots + 676$$
$19$ $$T^{4} - 10 T^{3} + 74 T^{2} + \cdots + 484$$
$23$ $$(T - 6)^{4}$$
$29$ $$T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 144$$
$31$ $$T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 625$$
$37$ $$(T^{2} + 11 T + 37)^{2}$$
$41$ $$T^{4} + 12 T^{3} + 51 T^{2} + 36 T + 9$$
$43$ $$(T + 11)^{4}$$
$47$ $$T^{4} + 4 T^{3} + 8 T^{2} - 208 T + 2704$$
$53$ $$T^{4} - 16 T^{3} + 233 T^{2} + \cdots + 3721$$
$59$ $$T^{4} - 2 T^{3} + 50 T^{2} - 364 T + 676$$
$61$ $$T^{4} + 12 T^{3} + 36 T^{2} + \cdots + 144$$
$67$ $$T^{4} - 30 T^{3} + 306 T^{2} + \cdots + 6084$$
$71$ $$T^{4} + 8 T^{3} + 96 T^{2} + \cdots + 1024$$
$73$ $$T^{4}$$
$79$ $$T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 2500$$
$83$ $$T^{4} + 38 T^{3} + 650 T^{2} + \cdots + 45796$$
$89$ $$T^{4} - 26 T^{3} + 194 T^{2} + \cdots + 484$$
$97$ $$T^{4} + 32T^{2} + 64$$