# Properties

 Label 15.3.f.a Level $15$ Weight $3$ Character orbit 15.f Analytic conductor $0.409$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 15.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.408720396540$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} + \beta_{3} q^{3} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1 - 3) q^{6} + (\beta_{2} + 2 \beta_1 + 1) q^{7} + (\beta_{3} - 3 \beta_{2} + 3) q^{8} - 3 \beta_{2} q^{9}+O(q^{10})$$ q + (-b2 + b1 - 1) * q^2 + b3 * q^3 + (-2*b3 + b2 - 2*b1) * q^4 + (b3 + 3*b2 - 2*b1 - 1) * q^5 + (-b3 + b1 - 3) * q^6 + (b2 + 2*b1 + 1) * q^7 + (b3 - 3*b2 + 3) * q^8 - 3*b2 * q^9 $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} + \beta_{3} q^{3} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1 - 3) q^{6} + (\beta_{2} + 2 \beta_1 + 1) q^{7} + (\beta_{3} - 3 \beta_{2} + 3) q^{8} - 3 \beta_{2} q^{9} + (4 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 1) q^{10} + ( - 3 \beta_{3} + 3 \beta_1 + 4) q^{11} + (6 \beta_{2} - \beta_1 + 6) q^{12} + ( - 2 \beta_{3} + 8 \beta_{2} - 8) q^{13} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{14} + ( - \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 6) q^{15} + (4 \beta_{3} - 4 \beta_1 - 5) q^{16} + ( - 10 \beta_{2} - 6 \beta_1 - 10) q^{17} + ( - 3 \beta_{3} + 3 \beta_{2} - 3) q^{18} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{19} + ( - 6 \beta_{3} + 17 \beta_{2} + 7 \beta_1 - 9) q^{20} + (\beta_{3} - \beta_1 - 6) q^{21} + (5 \beta_{2} - 2 \beta_1 + 5) q^{22} + (2 \beta_{3} - 14 \beta_{2} + 14) q^{23} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{24} + ( - 14 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 4) q^{25} + (10 \beta_{3} - 10 \beta_1 + 22) q^{26} + 3 \beta_1 q^{27} + ( - 2 \beta_{3} - 11 \beta_{2} + 11) q^{28} + (7 \beta_{3} + 18 \beta_{2} + 7 \beta_1) q^{29} + (\beta_{3} - 12 \beta_{2} + 8 \beta_1 - 6) q^{30} + ( - 6 \beta_{3} + 6 \beta_1 - 4) q^{31} + ( - 19 \beta_{2} + 7 \beta_1 - 19) q^{32} + (4 \beta_{3} + 9 \beta_{2} - 9) q^{33} + ( - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{34} + (5 \beta_{3} - 10 \beta_{2} - 5 \beta_1 - 10) q^{35} + (6 \beta_{3} - 6 \beta_1 + 3) q^{36} + (16 \beta_{2} - 18 \beta_1 + 16) q^{37} + ( - 18 \beta_{3} + 24 \beta_{2} - 24) q^{38} + ( - 8 \beta_{3} + 6 \beta_{2} - 8 \beta_1) q^{39} + (8 \beta_{3} + 9 \beta_{2} - 6 \beta_1 + 12) q^{40} + ( - 6 \beta_{3} + 6 \beta_1 - 14) q^{41} + (3 \beta_{2} - 4 \beta_1 + 3) q^{42} + (20 \beta_{3} + 2 \beta_{2} - 2) q^{43} + ( - 5 \beta_{3} - 32 \beta_{2} - 5 \beta_1) q^{44} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 9) q^{45} + ( - 16 \beta_{3} + 16 \beta_1 - 34) q^{46} + (32 \beta_{2} + 10 \beta_1 + 32) q^{47} + ( - 5 \beta_{3} - 12 \beta_{2} + 12) q^{48} + (4 \beta_{3} - 35 \beta_{2} + 4 \beta_1) q^{49} + (19 \beta_{3} - 13 \beta_{2} - 8 \beta_1 + 41) q^{50} + ( - 10 \beta_{3} + 10 \beta_1 + 18) q^{51} + ( - 20 \beta_{2} + 34 \beta_1 - 20) q^{52} + (12 \beta_{3} - 14 \beta_{2} + 14) q^{53} + ( - 3 \beta_{3} + 9 \beta_{2} - 3 \beta_1) q^{54} + (16 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 31) q^{55} + ( - 5 \beta_{3} + 5 \beta_1) q^{56} + ( - 18 \beta_{2} + 6 \beta_1 - 18) q^{57} + (4 \beta_{3} + 3 \beta_{2} - 3) q^{58} + ( - 31 \beta_{3} + 36 \beta_{2} - 31 \beta_1) q^{59} + ( - 9 \beta_{3} + 18 \beta_{2} - 17 \beta_1 - 21) q^{60} + (18 \beta_{3} - 18 \beta_1 + 50) q^{61} + (22 \beta_{2} - 16 \beta_1 + 22) q^{62} + ( - 6 \beta_{3} - 3 \beta_{2} + 3) q^{63} + ( - 10 \beta_{3} + 79 \beta_{2} - 10 \beta_1) q^{64} + ( - 22 \beta_{3} - 26 \beta_{2} + 14 \beta_1 - 28) q^{65} + (5 \beta_{3} - 5 \beta_1 + 6) q^{66} + ( - 50 \beta_{2} + 4 \beta_1 - 50) q^{67} + (34 \beta_{3} + 26 \beta_{2} - 26) q^{68} + (14 \beta_{3} - 6 \beta_{2} + 14 \beta_1) q^{69} + ( - 10 \beta_{3} + 5 \beta_{2} - 15) q^{70} - 68 q^{71} + ( - 9 \beta_{2} + 3 \beta_1 - 9) q^{72} + ( - 48 \beta_{3} - 19 \beta_{2} + 19) q^{73} + (34 \beta_{3} - 86 \beta_{2} + 34 \beta_1) q^{74} + (4 \beta_{3} + 42 \beta_{2} - 3 \beta_1 + 6) q^{75} + (18 \beta_{3} - 18 \beta_1 + 78) q^{76} + (22 \beta_{2} + 14 \beta_1 + 22) q^{77} + (22 \beta_{3} - 30 \beta_{2} + 30) q^{78} + ( - 10 \beta_{3} - 10 \beta_1) q^{79} + ( - 21 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 41) q^{80} - 9 q^{81} + (32 \beta_{2} - 26 \beta_1 + 32) q^{82} + ( - 14 \beta_{3} + 4 \beta_{2} - 4) q^{83} + (11 \beta_{3} + 6 \beta_{2} + 11 \beta_1) q^{84} + ( - 8 \beta_{3} + 16 \beta_{2} + 36 \beta_1 + 58) q^{85} + ( - 18 \beta_{3} + 18 \beta_1 - 56) q^{86} + ( - 21 \beta_{2} - 18 \beta_1 - 21) q^{87} + ( - 14 \beta_{3} - 3 \beta_{2} + 3) q^{88} + (36 \beta_{3} - 6 \beta_{2} + 36 \beta_1) q^{89} + ( - 6 \beta_{3} - 3 \beta_{2} + 12 \beta_1 - 24) q^{90} + (14 \beta_{3} - 14 \beta_1 - 4) q^{91} + (26 \beta_{2} - 58 \beta_1 + 26) q^{92} + ( - 4 \beta_{3} + 18 \beta_{2} - 18) q^{93} + (22 \beta_{3} - 34 \beta_{2} + 22 \beta_1) q^{94} + (24 \beta_{3} - 48 \beta_{2} - 18 \beta_1 + 36) q^{95} + ( - 19 \beta_{3} + 19 \beta_1 - 21) q^{96} + ( - 5 \beta_{2} - 16 \beta_1 - 5) q^{97} + ( - 43 \beta_{3} + 47 \beta_{2} - 47) q^{98} + ( - 9 \beta_{3} - 12 \beta_{2} - 9 \beta_1) q^{99}+O(q^{100})$$ q + (-b2 + b1 - 1) * q^2 + b3 * q^3 + (-2*b3 + b2 - 2*b1) * q^4 + (b3 + 3*b2 - 2*b1 - 1) * q^5 + (-b3 + b1 - 3) * q^6 + (b2 + 2*b1 + 1) * q^7 + (b3 - 3*b2 + 3) * q^8 - 3*b2 * q^9 + (4*b3 - 8*b2 + 2*b1 + 1) * q^10 + (-3*b3 + 3*b1 + 4) * q^11 + (6*b2 - b1 + 6) * q^12 + (-2*b3 + 8*b2 - 8) * q^13 + (-b3 + 4*b2 - b1) * q^14 + (-b3 - 3*b2 - 3*b1 + 6) * q^15 + (4*b3 - 4*b1 - 5) * q^16 + (-10*b2 - 6*b1 - 10) * q^17 + (-3*b3 + 3*b2 - 3) * q^18 + (6*b3 - 6*b2 + 6*b1) * q^19 + (-6*b3 + 17*b2 + 7*b1 - 9) * q^20 + (b3 - b1 - 6) * q^21 + (5*b2 - 2*b1 + 5) * q^22 + (2*b3 - 14*b2 + 14) * q^23 + (3*b3 - 3*b2 + 3*b1) * q^24 + (-14*b3 + 3*b2 - 2*b1 + 4) * q^25 + (10*b3 - 10*b1 + 22) * q^26 + 3*b1 * q^27 + (-2*b3 - 11*b2 + 11) * q^28 + (7*b3 + 18*b2 + 7*b1) * q^29 + (b3 - 12*b2 + 8*b1 - 6) * q^30 + (-6*b3 + 6*b1 - 4) * q^31 + (-19*b2 + 7*b1 - 19) * q^32 + (4*b3 + 9*b2 - 9) * q^33 + (-4*b3 + 2*b2 - 4*b1) * q^34 + (5*b3 - 10*b2 - 5*b1 - 10) * q^35 + (6*b3 - 6*b1 + 3) * q^36 + (16*b2 - 18*b1 + 16) * q^37 + (-18*b3 + 24*b2 - 24) * q^38 + (-8*b3 + 6*b2 - 8*b1) * q^39 + (8*b3 + 9*b2 - 6*b1 + 12) * q^40 + (-6*b3 + 6*b1 - 14) * q^41 + (3*b2 - 4*b1 + 3) * q^42 + (20*b3 + 2*b2 - 2) * q^43 + (-5*b3 - 32*b2 - 5*b1) * q^44 + (6*b3 + 3*b2 + 3*b1 + 9) * q^45 + (-16*b3 + 16*b1 - 34) * q^46 + (32*b2 + 10*b1 + 32) * q^47 + (-5*b3 - 12*b2 + 12) * q^48 + (4*b3 - 35*b2 + 4*b1) * q^49 + (19*b3 - 13*b2 - 8*b1 + 41) * q^50 + (-10*b3 + 10*b1 + 18) * q^51 + (-20*b2 + 34*b1 - 20) * q^52 + (12*b3 - 14*b2 + 14) * q^53 + (-3*b3 + 9*b2 - 3*b1) * q^54 + (16*b3 + 3*b2 - 2*b1 - 31) * q^55 + (-5*b3 + 5*b1) * q^56 + (-18*b2 + 6*b1 - 18) * q^57 + (4*b3 + 3*b2 - 3) * q^58 + (-31*b3 + 36*b2 - 31*b1) * q^59 + (-9*b3 + 18*b2 - 17*b1 - 21) * q^60 + (18*b3 - 18*b1 + 50) * q^61 + (22*b2 - 16*b1 + 22) * q^62 + (-6*b3 - 3*b2 + 3) * q^63 + (-10*b3 + 79*b2 - 10*b1) * q^64 + (-22*b3 - 26*b2 + 14*b1 - 28) * q^65 + (5*b3 - 5*b1 + 6) * q^66 + (-50*b2 + 4*b1 - 50) * q^67 + (34*b3 + 26*b2 - 26) * q^68 + (14*b3 - 6*b2 + 14*b1) * q^69 + (-10*b3 + 5*b2 - 15) * q^70 - 68 * q^71 + (-9*b2 + 3*b1 - 9) * q^72 + (-48*b3 - 19*b2 + 19) * q^73 + (34*b3 - 86*b2 + 34*b1) * q^74 + (4*b3 + 42*b2 - 3*b1 + 6) * q^75 + (18*b3 - 18*b1 + 78) * q^76 + (22*b2 + 14*b1 + 22) * q^77 + (22*b3 - 30*b2 + 30) * q^78 + (-10*b3 - 10*b1) * q^79 + (-21*b3 - 3*b2 + 2*b1 + 41) * q^80 - 9 * q^81 + (32*b2 - 26*b1 + 32) * q^82 + (-14*b3 + 4*b2 - 4) * q^83 + (11*b3 + 6*b2 + 11*b1) * q^84 + (-8*b3 + 16*b2 + 36*b1 + 58) * q^85 + (-18*b3 + 18*b1 - 56) * q^86 + (-21*b2 - 18*b1 - 21) * q^87 + (-14*b3 - 3*b2 + 3) * q^88 + (36*b3 - 6*b2 + 36*b1) * q^89 + (-6*b3 - 3*b2 + 12*b1 - 24) * q^90 + (14*b3 - 14*b1 - 4) * q^91 + (26*b2 - 58*b1 + 26) * q^92 + (-4*b3 + 18*b2 - 18) * q^93 + (22*b3 - 34*b2 + 22*b1) * q^94 + (24*b3 - 48*b2 - 18*b1 + 36) * q^95 + (-19*b3 + 19*b1 - 21) * q^96 + (-5*b2 - 16*b1 - 5) * q^97 + (-43*b3 + 47*b2 - 47) * q^98 + (-9*b3 - 12*b2 - 9*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 4 q^{5} - 12 q^{6} + 4 q^{7} + 12 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 - 4 * q^5 - 12 * q^6 + 4 * q^7 + 12 * q^8 $$4 q - 4 q^{2} - 4 q^{5} - 12 q^{6} + 4 q^{7} + 12 q^{8} + 4 q^{10} + 16 q^{11} + 24 q^{12} - 32 q^{13} + 24 q^{15} - 20 q^{16} - 40 q^{17} - 12 q^{18} - 36 q^{20} - 24 q^{21} + 20 q^{22} + 56 q^{23} + 16 q^{25} + 88 q^{26} + 44 q^{28} - 24 q^{30} - 16 q^{31} - 76 q^{32} - 36 q^{33} - 40 q^{35} + 12 q^{36} + 64 q^{37} - 96 q^{38} + 48 q^{40} - 56 q^{41} + 12 q^{42} - 8 q^{43} + 36 q^{45} - 136 q^{46} + 128 q^{47} + 48 q^{48} + 164 q^{50} + 72 q^{51} - 80 q^{52} + 56 q^{53} - 124 q^{55} - 72 q^{57} - 12 q^{58} - 84 q^{60} + 200 q^{61} + 88 q^{62} + 12 q^{63} - 112 q^{65} + 24 q^{66} - 200 q^{67} - 104 q^{68} - 60 q^{70} - 272 q^{71} - 36 q^{72} + 76 q^{73} + 24 q^{75} + 312 q^{76} + 88 q^{77} + 120 q^{78} + 164 q^{80} - 36 q^{81} + 128 q^{82} - 16 q^{83} + 232 q^{85} - 224 q^{86} - 84 q^{87} + 12 q^{88} - 96 q^{90} - 16 q^{91} + 104 q^{92} - 72 q^{93} + 144 q^{95} - 84 q^{96} - 20 q^{97} - 188 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 - 4 * q^5 - 12 * q^6 + 4 * q^7 + 12 * q^8 + 4 * q^10 + 16 * q^11 + 24 * q^12 - 32 * q^13 + 24 * q^15 - 20 * q^16 - 40 * q^17 - 12 * q^18 - 36 * q^20 - 24 * q^21 + 20 * q^22 + 56 * q^23 + 16 * q^25 + 88 * q^26 + 44 * q^28 - 24 * q^30 - 16 * q^31 - 76 * q^32 - 36 * q^33 - 40 * q^35 + 12 * q^36 + 64 * q^37 - 96 * q^38 + 48 * q^40 - 56 * q^41 + 12 * q^42 - 8 * q^43 + 36 * q^45 - 136 * q^46 + 128 * q^47 + 48 * q^48 + 164 * q^50 + 72 * q^51 - 80 * q^52 + 56 * q^53 - 124 * q^55 - 72 * q^57 - 12 * q^58 - 84 * q^60 + 200 * q^61 + 88 * q^62 + 12 * q^63 - 112 * q^65 + 24 * q^66 - 200 * q^67 - 104 * q^68 - 60 * q^70 - 272 * q^71 - 36 * q^72 + 76 * q^73 + 24 * q^75 + 312 * q^76 + 88 * q^77 + 120 * q^78 + 164 * q^80 - 36 * q^81 + 128 * q^82 - 16 * q^83 + 232 * q^85 - 224 * q^86 - 84 * q^87 + 12 * q^88 - 96 * q^90 - 16 * q^91 + 104 * q^92 - 72 * q^93 + 144 * q^95 - 84 * q^96 - 20 * q^97 - 188 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−2.22474 2.22474i 1.22474 1.22474i 5.89898i 2.67423 + 4.22474i −5.44949 −1.44949 1.44949i 4.22474 4.22474i 3.00000i 3.44949 15.3485i
7.2 0.224745 + 0.224745i −1.22474 + 1.22474i 3.89898i −4.67423 + 1.77526i −0.550510 3.44949 + 3.44949i 1.77526 1.77526i 3.00000i −1.44949 0.651531i
13.1 −2.22474 + 2.22474i 1.22474 + 1.22474i 5.89898i 2.67423 4.22474i −5.44949 −1.44949 + 1.44949i 4.22474 + 4.22474i 3.00000i 3.44949 + 15.3485i
13.2 0.224745 0.224745i −1.22474 1.22474i 3.89898i −4.67423 1.77526i −0.550510 3.44949 3.44949i 1.77526 + 1.77526i 3.00000i −1.44949 + 0.651531i
 $$n$$: 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.3.f.a 4
3.b odd 2 1 45.3.g.b 4
4.b odd 2 1 240.3.bg.a 4
5.b even 2 1 75.3.f.c 4
5.c odd 4 1 inner 15.3.f.a 4
5.c odd 4 1 75.3.f.c 4
8.b even 2 1 960.3.bg.i 4
8.d odd 2 1 960.3.bg.h 4
9.c even 3 2 405.3.l.h 8
9.d odd 6 2 405.3.l.f 8
12.b even 2 1 720.3.bh.k 4
15.d odd 2 1 225.3.g.a 4
15.e even 4 1 45.3.g.b 4
15.e even 4 1 225.3.g.a 4
20.d odd 2 1 1200.3.bg.k 4
20.e even 4 1 240.3.bg.a 4
20.e even 4 1 1200.3.bg.k 4
40.i odd 4 1 960.3.bg.i 4
40.k even 4 1 960.3.bg.h 4
45.k odd 12 2 405.3.l.h 8
45.l even 12 2 405.3.l.f 8
60.l odd 4 1 720.3.bh.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a 4 1.a even 1 1 trivial
15.3.f.a 4 5.c odd 4 1 inner
45.3.g.b 4 3.b odd 2 1
45.3.g.b 4 15.e even 4 1
75.3.f.c 4 5.b even 2 1
75.3.f.c 4 5.c odd 4 1
225.3.g.a 4 15.d odd 2 1
225.3.g.a 4 15.e even 4 1
240.3.bg.a 4 4.b odd 2 1
240.3.bg.a 4 20.e even 4 1
405.3.l.f 8 9.d odd 6 2
405.3.l.f 8 45.l even 12 2
405.3.l.h 8 9.c even 3 2
405.3.l.h 8 45.k odd 12 2
720.3.bh.k 4 12.b even 2 1
720.3.bh.k 4 60.l odd 4 1
960.3.bg.h 4 8.d odd 2 1
960.3.bg.h 4 40.k even 4 1
960.3.bg.i 4 8.b even 2 1
960.3.bg.i 4 40.i odd 4 1
1200.3.bg.k 4 20.d odd 2 1
1200.3.bg.k 4 20.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4 T^{3} + 8 T^{2} - 4 T + 1$$
$3$ $$T^{4} + 9$$
$5$ $$T^{4} + 4 T^{3} + 100 T + 625$$
$7$ $$T^{4} - 4 T^{3} + 8 T^{2} + 40 T + 100$$
$11$ $$(T^{2} - 8 T - 38)^{2}$$
$13$ $$T^{4} + 32 T^{3} + 512 T^{2} + \cdots + 13456$$
$17$ $$T^{4} + 40 T^{3} + 800 T^{2} + \cdots + 8464$$
$19$ $$T^{4} + 504 T^{2} + 32400$$
$23$ $$T^{4} - 56 T^{3} + 1568 T^{2} + \cdots + 144400$$
$29$ $$T^{4} + 1236T^{2} + 900$$
$31$ $$(T^{2} + 8 T - 200)^{2}$$
$37$ $$T^{4} - 64 T^{3} + 2048 T^{2} + \cdots + 211600$$
$41$ $$(T^{2} + 28 T - 20)^{2}$$
$43$ $$T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 1420864$$
$47$ $$T^{4} - 128 T^{3} + 8192 T^{2} + \cdots + 3055504$$
$53$ $$T^{4} - 56 T^{3} + 1568 T^{2} + \cdots + 1600$$
$59$ $$T^{4} + 14124 T^{2} + \cdots + 19980900$$
$61$ $$(T^{2} - 100 T + 556)^{2}$$
$67$ $$T^{4} + 200 T^{3} + \cdots + 24522304$$
$71$ $$(T + 68)^{4}$$
$73$ $$T^{4} - 76 T^{3} + 2888 T^{2} + \cdots + 38316100$$
$79$ $$(T^{2} + 600)^{2}$$
$83$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 309136$$
$89$ $$T^{4} + 15624 T^{2} + \cdots + 59907600$$
$97$ $$T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 515524$$