# Properties

 Label 169.4.c.f Level $169$ Weight $4$ Character orbit 169.c Analytic conductor $9.971$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{2} + (3 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{3} - 5 \beta_1 q^{4} + (5 \beta_{3} + 10) q^{5} + ( - 14 \beta_{2} - 10 \beta_1) q^{6} + (7 \beta_{2} + \beta_1) q^{7} + ( - 7 \beta_{3} + 4) q^{8} + ( - 10 \beta_{2} - 15 \beta_1) q^{9}+O(q^{10})$$ q + (b3 + 2*b2 + b1 - 2) * q^2 + (3*b3 + b2 + 3*b1 - 1) * q^3 - 5*b1 * q^4 + (5*b3 + 10) * q^5 + (-14*b2 - 10*b1) * q^6 + (7*b2 + b1) * q^7 + (-7*b3 + 4) * q^8 + (-10*b2 - 15*b1) * q^9 $$q + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{2} + (3 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{3} - 5 \beta_1 q^{4} + (5 \beta_{3} + 10) q^{5} + ( - 14 \beta_{2} - 10 \beta_1) q^{6} + (7 \beta_{2} + \beta_1) q^{7} + ( - 7 \beta_{3} + 4) q^{8} + ( - 10 \beta_{2} - 15 \beta_1) q^{9} + ( - 5 \beta_{3} - 5 \beta_1) q^{10} + ( - 15 \beta_{3} - \beta_{2} - 15 \beta_1 + 1) q^{11} + ( - 20 \beta_{3} + 60) q^{12} + (10 \beta_{3} - 18) q^{14} + (10 \beta_{3} - 50 \beta_{2} + 10 \beta_1 + 50) q^{15} + ( - 15 \beta_{3} + 36 \beta_{2} - 15 \beta_1 - 36) q^{16} + ( - 43 \beta_{2} + 16 \beta_1) q^{17} + ( - 55 \beta_{3} + 80) q^{18} + ( - 73 \beta_{2} + 5 \beta_1) q^{19} + (100 \beta_{2} - 25 \beta_1) q^{20} + (25 \beta_{3} - 19) q^{21} + (62 \beta_{2} + 46 \beta_1) q^{22} + ( - 33 \beta_{3} + 89 \beta_{2} - 33 \beta_1 - 89) q^{23} + (40 \beta_{3} + 88 \beta_{2} + 40 \beta_1 - 88) q^{24} + (75 \beta_{3} + 75) q^{25} + ( - 9 \beta_{3} + 163) q^{27} + ( - 40 \beta_{3} - 20 \beta_{2} - 40 \beta_1 + 20) q^{28} + (60 \beta_{3} - 13 \beta_{2} + 60 \beta_1 + 13) q^{29} + (60 \beta_{2} + 20 \beta_1) q^{30} + (100 \beta_{3} + 120) q^{31} + (20 \beta_{2} + 65 \beta_1) q^{32} + (181 \beta_{2} + 63 \beta_1) q^{33} + (5 \beta_{3} + 22) q^{34} + (50 \beta_{2} - 30 \beta_1) q^{35} + (125 \beta_{3} + 300 \beta_{2} + 125 \beta_1 - 300) q^{36} + (44 \beta_{3} + 73 \beta_{2} + 44 \beta_1 - 73) q^{37} + ( - 58 \beta_{3} + 126) q^{38} + ( - 15 \beta_{3} - 100) q^{40} + ( - 20 \beta_{3} - 259 \beta_{2} - 20 \beta_1 + 259) q^{41} + ( - 94 \beta_{3} - 138 \beta_{2} - 94 \beta_1 + 138) q^{42} + ( - 179 \beta_{2} - 97 \beta_1) q^{43} + (80 \beta_{3} - 300) q^{44} + (200 \beta_{2} - 25 \beta_1) q^{45} + ( - 46 \beta_{2} + 10 \beta_1) q^{46} + ( - 140 \beta_{3} - 100) q^{47} + (144 \beta_{2} - 48 \beta_1) q^{48} + (15 \beta_{3} - 290 \beta_{2} + 15 \beta_1 + 290) q^{49} + ( - 150 \beta_{3} - 150 \beta_{2} - 150 \beta_1 + 150) q^{50} + ( - 65 \beta_{3} - 149) q^{51} + ( - 165 \beta_{3} + 190) q^{53} + (190 \beta_{3} + 362 \beta_{2} + 190 \beta_1 - 362) q^{54} + ( - 70 \beta_{3} + 290 \beta_{2} - 70 \beta_1 - 290) q^{55} + (56 \beta_{2} + 60 \beta_1) q^{56} + ( - 199 \beta_{3} + 13) q^{57} + ( - 214 \beta_{2} - 167 \beta_1) q^{58} + ( - 377 \beta_{2} - 55 \beta_1) q^{59} + (200 \beta_{3} + 200) q^{60} + ( - 351 \beta_{2} + 200 \beta_1) q^{61} + ( - 180 \beta_{3} - 160 \beta_{2} - 180 \beta_1 + 160) q^{62} + ( - 130 \beta_{3} - 130 \beta_{2} - 130 \beta_1 + 130) q^{63} + (95 \beta_{3} - 588) q^{64} + (370 \beta_{3} - 614) q^{66} + ( - 91 \beta_{3} + 283 \beta_{2} - 91 \beta_1 - 283) q^{67} + (135 \beta_{3} - 320 \beta_{2} + 135 \beta_1 + 320) q^{68} + (307 \beta_{2} - 135 \beta_1) q^{69} + ( - 40 \beta_{3} + 20) q^{70} + (11 \beta_{2} + 105 \beta_1) q^{71} + ( - 460 \beta_{2} - 235 \beta_1) q^{72} + (85 \beta_{3} - 250) q^{73} + ( - 322 \beta_{2} - 205 \beta_1) q^{74} + ( - 75 \beta_{3} - 825 \beta_{2} - 75 \beta_1 + 825) q^{75} + (340 \beta_{3} - 100 \beta_{2} + 340 \beta_1 + 100) q^{76} + ( - 121 \beta_{3} + 67) q^{77} + (40 \beta_{3} + 140) q^{79} + ( - 255 \beta_{3} + 660 \beta_{2} - 255 \beta_1 - 660) q^{80} + (120 \beta_{3} + \beta_{2} + 120 \beta_1 - 1) q^{81} + (598 \beta_{2} + 319 \beta_1) q^{82} + ( - 100 \beta_{3} - 180) q^{83} + (500 \beta_{2} + 220 \beta_1) q^{84} + ( - 750 \beta_{2} + 295 \beta_1) q^{85} + ( - 470 \beta_{3} + 746) q^{86} + ( - 707 \beta_{2} - 201 \beta_1) q^{87} + ( - 172 \beta_{3} - 424 \beta_{2} - 172 \beta_1 + 424) q^{88} + (125 \beta_{3} - 523 \beta_{2} + 125 \beta_1 + 523) q^{89} + (125 \beta_{3} - 300) q^{90} + ( - 280 \beta_{3} - 660) q^{92} + ( - 40 \beta_{3} - 1080 \beta_{2} - 40 \beta_1 + 1080) q^{93} + (320 \beta_{3} + 360 \beta_{2} + 320 \beta_1 - 360) q^{94} + ( - 830 \beta_{2} + 390 \beta_1) q^{95} + (320 \beta_{3} - 800) q^{96} + (27 \beta_{2} - 469 \beta_1) q^{97} + (520 \beta_{2} + 245 \beta_1) q^{98} + (390 \beta_{3} - 910) q^{99}+O(q^{100})$$ q + (b3 + 2*b2 + b1 - 2) * q^2 + (3*b3 + b2 + 3*b1 - 1) * q^3 - 5*b1 * q^4 + (5*b3 + 10) * q^5 + (-14*b2 - 10*b1) * q^6 + (7*b2 + b1) * q^7 + (-7*b3 + 4) * q^8 + (-10*b2 - 15*b1) * q^9 + (-5*b3 - 5*b1) * q^10 + (-15*b3 - b2 - 15*b1 + 1) * q^11 + (-20*b3 + 60) * q^12 + (10*b3 - 18) * q^14 + (10*b3 - 50*b2 + 10*b1 + 50) * q^15 + (-15*b3 + 36*b2 - 15*b1 - 36) * q^16 + (-43*b2 + 16*b1) * q^17 + (-55*b3 + 80) * q^18 + (-73*b2 + 5*b1) * q^19 + (100*b2 - 25*b1) * q^20 + (25*b3 - 19) * q^21 + (62*b2 + 46*b1) * q^22 + (-33*b3 + 89*b2 - 33*b1 - 89) * q^23 + (40*b3 + 88*b2 + 40*b1 - 88) * q^24 + (75*b3 + 75) * q^25 + (-9*b3 + 163) * q^27 + (-40*b3 - 20*b2 - 40*b1 + 20) * q^28 + (60*b3 - 13*b2 + 60*b1 + 13) * q^29 + (60*b2 + 20*b1) * q^30 + (100*b3 + 120) * q^31 + (20*b2 + 65*b1) * q^32 + (181*b2 + 63*b1) * q^33 + (5*b3 + 22) * q^34 + (50*b2 - 30*b1) * q^35 + (125*b3 + 300*b2 + 125*b1 - 300) * q^36 + (44*b3 + 73*b2 + 44*b1 - 73) * q^37 + (-58*b3 + 126) * q^38 + (-15*b3 - 100) * q^40 + (-20*b3 - 259*b2 - 20*b1 + 259) * q^41 + (-94*b3 - 138*b2 - 94*b1 + 138) * q^42 + (-179*b2 - 97*b1) * q^43 + (80*b3 - 300) * q^44 + (200*b2 - 25*b1) * q^45 + (-46*b2 + 10*b1) * q^46 + (-140*b3 - 100) * q^47 + (144*b2 - 48*b1) * q^48 + (15*b3 - 290*b2 + 15*b1 + 290) * q^49 + (-150*b3 - 150*b2 - 150*b1 + 150) * q^50 + (-65*b3 - 149) * q^51 + (-165*b3 + 190) * q^53 + (190*b3 + 362*b2 + 190*b1 - 362) * q^54 + (-70*b3 + 290*b2 - 70*b1 - 290) * q^55 + (56*b2 + 60*b1) * q^56 + (-199*b3 + 13) * q^57 + (-214*b2 - 167*b1) * q^58 + (-377*b2 - 55*b1) * q^59 + (200*b3 + 200) * q^60 + (-351*b2 + 200*b1) * q^61 + (-180*b3 - 160*b2 - 180*b1 + 160) * q^62 + (-130*b3 - 130*b2 - 130*b1 + 130) * q^63 + (95*b3 - 588) * q^64 + (370*b3 - 614) * q^66 + (-91*b3 + 283*b2 - 91*b1 - 283) * q^67 + (135*b3 - 320*b2 + 135*b1 + 320) * q^68 + (307*b2 - 135*b1) * q^69 + (-40*b3 + 20) * q^70 + (11*b2 + 105*b1) * q^71 + (-460*b2 - 235*b1) * q^72 + (85*b3 - 250) * q^73 + (-322*b2 - 205*b1) * q^74 + (-75*b3 - 825*b2 - 75*b1 + 825) * q^75 + (340*b3 - 100*b2 + 340*b1 + 100) * q^76 + (-121*b3 + 67) * q^77 + (40*b3 + 140) * q^79 + (-255*b3 + 660*b2 - 255*b1 - 660) * q^80 + (120*b3 + b2 + 120*b1 - 1) * q^81 + (598*b2 + 319*b1) * q^82 + (-100*b3 - 180) * q^83 + (500*b2 + 220*b1) * q^84 + (-750*b2 + 295*b1) * q^85 + (-470*b3 + 746) * q^86 + (-707*b2 - 201*b1) * q^87 + (-172*b3 - 424*b2 - 172*b1 + 424) * q^88 + (125*b3 - 523*b2 + 125*b1 + 523) * q^89 + (125*b3 - 300) * q^90 + (-280*b3 - 660) * q^92 + (-40*b3 - 1080*b2 - 40*b1 + 1080) * q^93 + (320*b3 + 360*b2 + 320*b1 - 360) * q^94 + (-830*b2 + 390*b1) * q^95 + (320*b3 - 800) * q^96 + (27*b2 - 469*b1) * q^97 + (520*b2 + 245*b1) * q^98 + (390*b3 - 910) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{2} - 5 q^{3} - 5 q^{4} + 30 q^{5} - 38 q^{6} + 15 q^{7} + 30 q^{8} - 35 q^{9}+O(q^{10})$$ 4 * q - 5 * q^2 - 5 * q^3 - 5 * q^4 + 30 * q^5 - 38 * q^6 + 15 * q^7 + 30 * q^8 - 35 * q^9 $$4 q - 5 q^{2} - 5 q^{3} - 5 q^{4} + 30 q^{5} - 38 q^{6} + 15 q^{7} + 30 q^{8} - 35 q^{9} + 5 q^{10} + 17 q^{11} + 280 q^{12} - 92 q^{14} + 90 q^{15} - 57 q^{16} - 70 q^{17} + 430 q^{18} - 141 q^{19} + 175 q^{20} - 126 q^{21} + 170 q^{22} - 145 q^{23} - 216 q^{24} + 150 q^{25} + 670 q^{27} + 80 q^{28} - 34 q^{29} + 140 q^{30} + 280 q^{31} + 105 q^{32} + 425 q^{33} + 78 q^{34} + 70 q^{35} - 725 q^{36} - 190 q^{37} + 620 q^{38} - 370 q^{40} + 538 q^{41} + 370 q^{42} - 455 q^{43} - 1360 q^{44} + 375 q^{45} - 82 q^{46} - 120 q^{47} + 240 q^{48} + 565 q^{49} + 450 q^{50} - 466 q^{51} + 1090 q^{53} - 914 q^{54} - 510 q^{55} + 172 q^{56} + 450 q^{57} - 595 q^{58} - 809 q^{59} + 400 q^{60} - 502 q^{61} + 500 q^{62} + 390 q^{63} - 2542 q^{64} - 3196 q^{66} - 475 q^{67} + 505 q^{68} + 479 q^{69} + 160 q^{70} + 127 q^{71} - 1155 q^{72} - 1170 q^{73} - 849 q^{74} + 1725 q^{75} - 140 q^{76} + 510 q^{77} + 480 q^{79} - 1065 q^{80} - 122 q^{81} + 1515 q^{82} - 520 q^{83} + 1220 q^{84} - 1205 q^{85} + 3924 q^{86} - 1615 q^{87} + 1020 q^{88} + 921 q^{89} - 1450 q^{90} - 2080 q^{92} + 2200 q^{93} - 1040 q^{94} - 1270 q^{95} - 3840 q^{96} - 415 q^{97} + 1285 q^{98} - 4420 q^{99}+O(q^{100})$$ 4 * q - 5 * q^2 - 5 * q^3 - 5 * q^4 + 30 * q^5 - 38 * q^6 + 15 * q^7 + 30 * q^8 - 35 * q^9 + 5 * q^10 + 17 * q^11 + 280 * q^12 - 92 * q^14 + 90 * q^15 - 57 * q^16 - 70 * q^17 + 430 * q^18 - 141 * q^19 + 175 * q^20 - 126 * q^21 + 170 * q^22 - 145 * q^23 - 216 * q^24 + 150 * q^25 + 670 * q^27 + 80 * q^28 - 34 * q^29 + 140 * q^30 + 280 * q^31 + 105 * q^32 + 425 * q^33 + 78 * q^34 + 70 * q^35 - 725 * q^36 - 190 * q^37 + 620 * q^38 - 370 * q^40 + 538 * q^41 + 370 * q^42 - 455 * q^43 - 1360 * q^44 + 375 * q^45 - 82 * q^46 - 120 * q^47 + 240 * q^48 + 565 * q^49 + 450 * q^50 - 466 * q^51 + 1090 * q^53 - 914 * q^54 - 510 * q^55 + 172 * q^56 + 450 * q^57 - 595 * q^58 - 809 * q^59 + 400 * q^60 - 502 * q^61 + 500 * q^62 + 390 * q^63 - 2542 * q^64 - 3196 * q^66 - 475 * q^67 + 505 * q^68 + 479 * q^69 + 160 * q^70 + 127 * q^71 - 1155 * q^72 - 1170 * q^73 - 849 * q^74 + 1725 * q^75 - 140 * q^76 + 510 * q^77 + 480 * q^79 - 1065 * q^80 - 122 * q^81 + 1515 * q^82 - 520 * q^83 + 1220 * q^84 - 1205 * q^85 + 3924 * q^86 - 1615 * q^87 + 1020 * q^88 + 921 * q^89 - 1450 * q^90 - 2080 * q^92 + 2200 * q^93 - 1040 * q^94 - 1270 * q^95 - 3840 * q^96 - 415 * q^97 + 1285 * q^98 - 4420 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 1.28078 + 2.21837i −0.780776 − 1.35234i 1.28078 − 2.21837i −0.780776 + 1.35234i
−2.28078 + 3.95042i −4.34233 + 7.52113i −6.40388 11.0918i −2.80776 −19.8078 34.3081i 4.78078 + 8.28055i 21.9309 −24.2116 41.9358i 6.40388 11.0918i
22.2 −0.219224 + 0.379706i 1.84233 3.19101i 3.90388 + 6.76172i 17.8078 0.807764 + 1.39909i 2.71922 + 4.70983i −6.93087 6.71165 + 11.6249i −3.90388 + 6.76172i
146.1 −2.28078 3.95042i −4.34233 7.52113i −6.40388 + 11.0918i −2.80776 −19.8078 + 34.3081i 4.78078 8.28055i 21.9309 −24.2116 + 41.9358i 6.40388 + 11.0918i
146.2 −0.219224 0.379706i 1.84233 + 3.19101i 3.90388 6.76172i 17.8078 0.807764 1.39909i 2.71922 4.70983i −6.93087 6.71165 11.6249i −3.90388 6.76172i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.c.f 4
13.b even 2 1 13.4.c.b 4
13.c even 3 1 169.4.a.j 2
13.c even 3 1 inner 169.4.c.f 4
13.d odd 4 2 169.4.e.g 8
13.e even 6 1 13.4.c.b 4
13.e even 6 1 169.4.a.f 2
13.f odd 12 2 169.4.b.e 4
13.f odd 12 2 169.4.e.g 8
39.d odd 2 1 117.4.g.d 4
39.h odd 6 1 117.4.g.d 4
39.h odd 6 1 1521.4.a.t 2
39.i odd 6 1 1521.4.a.l 2
52.b odd 2 1 208.4.i.e 4
52.i odd 6 1 208.4.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 13.b even 2 1
13.4.c.b 4 13.e even 6 1
117.4.g.d 4 39.d odd 2 1
117.4.g.d 4 39.h odd 6 1
169.4.a.f 2 13.e even 6 1
169.4.a.j 2 13.c even 3 1
169.4.b.e 4 13.f odd 12 2
169.4.c.f 4 1.a even 1 1 trivial
169.4.c.f 4 13.c even 3 1 inner
169.4.e.g 8 13.d odd 4 2
169.4.e.g 8 13.f odd 12 2
208.4.i.e 4 52.b odd 2 1
208.4.i.e 4 52.i odd 6 1
1521.4.a.l 2 39.i odd 6 1
1521.4.a.t 2 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 5T_{2}^{3} + 23T_{2}^{2} + 10T_{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4$$
$3$ $$T^{4} + 5 T^{3} + 57 T^{2} + \cdots + 1024$$
$5$ $$(T^{2} - 15 T - 50)^{2}$$
$7$ $$T^{4} - 15 T^{3} + 173 T^{2} + \cdots + 2704$$
$11$ $$T^{4} - 17 T^{3} + 1173 T^{2} + \cdots + 781456$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 70 T^{3} + 4763 T^{2} + \cdots + 18769$$
$19$ $$T^{4} + 141 T^{3} + \cdots + 23658496$$
$23$ $$T^{4} + 145 T^{3} + 20397 T^{2} + \cdots + 394384$$
$29$ $$T^{4} + 34 T^{3} + \cdots + 225330121$$
$31$ $$(T^{2} - 140 T - 37600)^{2}$$
$37$ $$T^{4} + 190 T^{3} + 35303 T^{2} + \cdots + 635209$$
$41$ $$T^{4} - 538 T^{3} + \cdots + 4992976921$$
$43$ $$T^{4} + 455 T^{3} + \cdots + 138485824$$
$47$ $$(T^{2} + 60 T - 82400)^{2}$$
$53$ $$(T^{2} - 545 T - 41450)^{2}$$
$59$ $$T^{4} + 809 T^{3} + \cdots + 22729783696$$
$61$ $$T^{4} + 502 T^{3} + \cdots + 11448786001$$
$67$ $$T^{4} + 475 T^{3} + \cdots + 449948944$$
$71$ $$T^{4} - 127 T^{3} + \cdots + 1833894976$$
$73$ $$(T^{2} + 585 T + 54850)^{2}$$
$79$ $$(T^{2} - 240 T + 7600)^{2}$$
$83$ $$(T^{2} + 260 T - 25600)^{2}$$
$89$ $$T^{4} - 921 T^{3} + \cdots + 21215087716$$
$97$ $$T^{4} + 415 T^{3} + \cdots + 795268001284$$