# Properties

 Label 539.4.a.g Level $539$ Weight $4$ Character orbit 539.a Self dual yes Analytic conductor $31.802$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 539.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.8020294931$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.522072.1 Defining polynomial: $$x^{4} - x^{3} - 12x^{2} + 5x + 1$$ x^4 - x^3 - 12*x^2 + 5*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( - \beta_1 - 4) q^{3} + ( - 2 \beta_{3} - \beta_{2} + 6) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{5} + ( - 3 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 5) q^{6} + (5 \beta_{2} - 4 \beta_1 - 4) q^{8} + (6 \beta_{3} + 4 \beta_1 + 21) q^{9}+O(q^{10})$$ q + b2 * q^2 + (-b1 - 4) * q^3 + (-2*b3 - b2 + 6) * q^4 + (b3 + b2 - 2*b1 - 3) * q^5 + (-3*b3 - 5*b2 + 2*b1 - 5) * q^6 + (5*b2 - 4*b1 - 4) * q^8 + (6*b3 + 4*b1 + 21) * q^9 $$q + \beta_{2} q^{2} + ( - \beta_1 - 4) q^{3} + ( - 2 \beta_{3} - \beta_{2} + 6) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{5} + ( - 3 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 5) q^{6} + (5 \beta_{2} - 4 \beta_1 - 4) q^{8} + (6 \beta_{3} + 4 \beta_1 + 21) q^{9} + ( - 7 \beta_{3} - 9 \beta_{2} + 6 \beta_1 - 1) q^{10} - 11 q^{11} + (13 \beta_{3} + 11 \beta_{2} - 2 \beta_1 - 13) q^{12} + (10 \beta_{3} + 10 \beta_{2} + 3 \beta_1 - 8) q^{13} + (4 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 68) q^{15} + ( - 6 \beta_{3} - 5 \beta_{2} + 8 \beta_1 + 2) q^{16} + ( - 9 \beta_{3} - 13 \beta_{2} + 5 \beta_1 - 5) q^{17} + (18 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 10) q^{18} + (7 \beta_{3} + 15 \beta_{2} + 4 \beta_1 - 55) q^{19} + (21 \beta_{3} + 27 \beta_{2} - 10 \beta_1 - 37) q^{20} - 11 \beta_{2} q^{22} + ( - 6 \beta_{3} - 2 \beta_{2} + 18 \beta_1 + 10) q^{23} + (9 \beta_{3} - 25 \beta_{2} + 14 \beta_1 + 119) q^{24} + (2 \beta_{3} - 26 \beta_{2} - 4 \beta_1 + 5) q^{25} + ( - \beta_{3} - 45 \beta_{2} + 14 \beta_1 + 105) q^{26} + ( - 54 \beta_{3} - 18 \beta_{2} - 12 \beta_1 - 122) q^{27} + ( - 8 \beta_{3} + 4 \beta_{2} + 14 \beta_1 - 90) q^{29} + (26 \beta_{3} + 66 \beta_{2} + 4 \beta_1 - 122) q^{30} + (25 \beta_{3} - 7 \beta_{2} + 9 \beta_1 + 15) q^{31} + (28 \beta_{3} - 7 \beta_{2} + 4 \beta_1 + 32) q^{32} + (11 \beta_1 + 44) q^{33} + (32 \beta_{3} + 40 \beta_{2} - 28 \beta_1 - 112) q^{34} + ( - 32 \beta_{3} - 67 \beta_{2} - 4 \beta_1 - 140) q^{36} + ( - 32 \beta_{3} + 14 \beta_{2} - 16 \beta_1 + 20) q^{37} + ( - 11 \beta_{3} - 87 \beta_{2} + 6 \beta_1 + 195) q^{38} + ( - 98 \beta_{3} - 80 \beta_{2} - 2 \beta_1 - 144) q^{39} + ( - 7 \beta_{3} - 65 \beta_{2} + 14 \beta_1 + 231) q^{40} + ( - 23 \beta_{3} - 51 \beta_{2} + 23 \beta_1 - 27) q^{41} + ( - 74 \beta_{2} + 2 \beta_1 + 66) q^{43} + (22 \beta_{3} + 11 \beta_{2} - 66) q^{44} + ( - 35 \beta_{3} + \beta_{2} - 42 \beta_1 - 227) q^{45} + (52 \beta_{3} + 48 \beta_{2} - 48 \beta_1 + 92) q^{46} + ( - \beta_{3} + 15 \beta_{2} + 39 \beta_1 + 25) q^{47} + ( - 3 \beta_{3} + 43 \beta_{2} + 6 \beta_1 - 221) q^{48} + (42 \beta_{3} + 21 \beta_{2} + 12 \beta_1 - 394) q^{50} + (54 \beta_{3} + 92 \beta_{2} + 6 \beta_1 - 48) q^{51} + (51 \beta_{3} + 87 \beta_{2} - 54 \beta_1 - 491) q^{52} + (10 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 150) q^{53} + ( - 54 \beta_{3} + 46 \beta_{2} - 84 \beta_1 - 42) q^{54} + ( - 11 \beta_{3} - 11 \beta_{2} + 22 \beta_1 + 33) q^{55} + ( - 104 \beta_{3} - 96 \beta_{2} + 64 \beta_1 - 4) q^{57} + (26 \beta_{3} - 56 \beta_{2} - 44 \beta_1 + 166) q^{58} + (24 \beta_{3} - 124 \beta_{2} - 3 \beta_1 - 4) q^{59} + ( - 126 \beta_{3} - 198 \beta_{2} + 28 \beta_1 + 270) q^{60} + ( - 120 \beta_{3} - 24 \beta_{2} - 29 \beta_1 + 14) q^{61} + (66 \beta_{3} - 44 \beta_{2} + 32 \beta_1 - 178) q^{62} + (102 \beta_{3} - \beta_{2} - 16 \beta_1 - 234) q^{64} + ( - 142 \beta_{3} - 146 \beta_{2} + 54 \beta_1 - 54) q^{65} + (33 \beta_{3} + 55 \beta_{2} - 22 \beta_1 + 55) q^{66} + (124 \beta_{3} + 120 \beta_{2} - 62 \beta_1 + 364) q^{67} + ( - 60 \beta_{3} - 172 \beta_{2} + 80 \beta_1 + 300) q^{68} + ( - 72 \beta_{3} + 28 \beta_{2} + 4 \beta_1 - 588) q^{69} + (96 \beta_{3} + 92 \beta_{2} - 26 \beta_1 + 480) q^{71} + ( - 54 \beta_{3} - 37 \beta_{2} - 88 \beta_1 - 718) q^{72} + ( - 61 \beta_{3} - 217 \beta_{2} + 47 \beta_1 - 449) q^{73} + ( - 108 \beta_{3} + 86 \beta_{2} - 32 \beta_1 + 276) q^{74} + (92 \beta_{3} + 124 \beta_{2} - 63 \beta_1 + 232) q^{75} + (125 \beta_{3} + 201 \beta_{2} - 66 \beta_1 - 693) q^{76} + (56 \beta_{3} + 228 \beta_{2} - 192 \beta_1 - 640) q^{78} + ( - 114 \beta_{3} + 28 \beta_{2} - 6 \beta_1 - 320) q^{79} + ( - 3 \beta_{3} + 115 \beta_{2} + 38 \beta_1 - 509) q^{80} + (234 \beta_{3} + 252 \beta_{2} + 140 \beta_1 + 557) q^{81} + (148 \beta_{3} + 116 \beta_{2} - 92 \beta_1 - 484) q^{82} + (43 \beta_{3} - 25 \beta_{2} + 76 \beta_1 - 87) q^{83} + (82 \beta_{3} + 186 \beta_{2} - 2 \beta_1 - 342) q^{85} + (154 \beta_{3} + 142 \beta_{2} - 4 \beta_1 - 1026) q^{86} + ( - 56 \beta_{3} + 4 \beta_{2} + 122 \beta_1 - 84) q^{87} + ( - 55 \beta_{2} + 44 \beta_1 + 44) q^{88} + (104 \beta_{3} - 44 \beta_{2} + 48 \beta_1 + 770) q^{89} + ( - 163 \beta_{3} - 165 \beta_{2} + 14 \beta_1 - 21) q^{90} + ( - 140 \beta_{3} - 144 \beta_{2} + 56 \beta_1 + 92) q^{92} + ( - 158 \beta_{3} - 40 \beta_{2} - 104 \beta_1 - 388) q^{93} + (86 \beta_{3} + 52 \beta_{2} - 80 \beta_1 + 410) q^{94} + ( - 220 \beta_{3} - 216 \beta_{2} + 196 \beta_1 - 4) q^{95} + ( - 143 \beta_{3} - 49 \beta_{2} - 130 \beta_1 - 305) q^{96} + ( - 32 \beta_{3} + 176 \beta_{2} - 154 \beta_1 + 174) q^{97} + ( - 66 \beta_{3} - 44 \beta_1 - 231) q^{99}+O(q^{100})$$ q + b2 * q^2 + (-b1 - 4) * q^3 + (-2*b3 - b2 + 6) * q^4 + (b3 + b2 - 2*b1 - 3) * q^5 + (-3*b3 - 5*b2 + 2*b1 - 5) * q^6 + (5*b2 - 4*b1 - 4) * q^8 + (6*b3 + 4*b1 + 21) * q^9 + (-7*b3 - 9*b2 + 6*b1 - 1) * q^10 - 11 * q^11 + (13*b3 + 11*b2 - 2*b1 - 13) * q^12 + (10*b3 + 10*b2 + 3*b1 - 8) * q^13 + (4*b3 - 8*b2 + 2*b1 + 68) * q^15 + (-6*b3 - 5*b2 + 8*b1 + 2) * q^16 + (-9*b3 - 13*b2 + 5*b1 - 5) * q^17 + (18*b3 + 7*b2 + 4*b1 - 10) * q^18 + (7*b3 + 15*b2 + 4*b1 - 55) * q^19 + (21*b3 + 27*b2 - 10*b1 - 37) * q^20 - 11*b2 * q^22 + (-6*b3 - 2*b2 + 18*b1 + 10) * q^23 + (9*b3 - 25*b2 + 14*b1 + 119) * q^24 + (2*b3 - 26*b2 - 4*b1 + 5) * q^25 + (-b3 - 45*b2 + 14*b1 + 105) * q^26 + (-54*b3 - 18*b2 - 12*b1 - 122) * q^27 + (-8*b3 + 4*b2 + 14*b1 - 90) * q^29 + (26*b3 + 66*b2 + 4*b1 - 122) * q^30 + (25*b3 - 7*b2 + 9*b1 + 15) * q^31 + (28*b3 - 7*b2 + 4*b1 + 32) * q^32 + (11*b1 + 44) * q^33 + (32*b3 + 40*b2 - 28*b1 - 112) * q^34 + (-32*b3 - 67*b2 - 4*b1 - 140) * q^36 + (-32*b3 + 14*b2 - 16*b1 + 20) * q^37 + (-11*b3 - 87*b2 + 6*b1 + 195) * q^38 + (-98*b3 - 80*b2 - 2*b1 - 144) * q^39 + (-7*b3 - 65*b2 + 14*b1 + 231) * q^40 + (-23*b3 - 51*b2 + 23*b1 - 27) * q^41 + (-74*b2 + 2*b1 + 66) * q^43 + (22*b3 + 11*b2 - 66) * q^44 + (-35*b3 + b2 - 42*b1 - 227) * q^45 + (52*b3 + 48*b2 - 48*b1 + 92) * q^46 + (-b3 + 15*b2 + 39*b1 + 25) * q^47 + (-3*b3 + 43*b2 + 6*b1 - 221) * q^48 + (42*b3 + 21*b2 + 12*b1 - 394) * q^50 + (54*b3 + 92*b2 + 6*b1 - 48) * q^51 + (51*b3 + 87*b2 - 54*b1 - 491) * q^52 + (10*b3 - 4*b2 - 8*b1 + 150) * q^53 + (-54*b3 + 46*b2 - 84*b1 - 42) * q^54 + (-11*b3 - 11*b2 + 22*b1 + 33) * q^55 + (-104*b3 - 96*b2 + 64*b1 - 4) * q^57 + (26*b3 - 56*b2 - 44*b1 + 166) * q^58 + (24*b3 - 124*b2 - 3*b1 - 4) * q^59 + (-126*b3 - 198*b2 + 28*b1 + 270) * q^60 + (-120*b3 - 24*b2 - 29*b1 + 14) * q^61 + (66*b3 - 44*b2 + 32*b1 - 178) * q^62 + (102*b3 - b2 - 16*b1 - 234) * q^64 + (-142*b3 - 146*b2 + 54*b1 - 54) * q^65 + (33*b3 + 55*b2 - 22*b1 + 55) * q^66 + (124*b3 + 120*b2 - 62*b1 + 364) * q^67 + (-60*b3 - 172*b2 + 80*b1 + 300) * q^68 + (-72*b3 + 28*b2 + 4*b1 - 588) * q^69 + (96*b3 + 92*b2 - 26*b1 + 480) * q^71 + (-54*b3 - 37*b2 - 88*b1 - 718) * q^72 + (-61*b3 - 217*b2 + 47*b1 - 449) * q^73 + (-108*b3 + 86*b2 - 32*b1 + 276) * q^74 + (92*b3 + 124*b2 - 63*b1 + 232) * q^75 + (125*b3 + 201*b2 - 66*b1 - 693) * q^76 + (56*b3 + 228*b2 - 192*b1 - 640) * q^78 + (-114*b3 + 28*b2 - 6*b1 - 320) * q^79 + (-3*b3 + 115*b2 + 38*b1 - 509) * q^80 + (234*b3 + 252*b2 + 140*b1 + 557) * q^81 + (148*b3 + 116*b2 - 92*b1 - 484) * q^82 + (43*b3 - 25*b2 + 76*b1 - 87) * q^83 + (82*b3 + 186*b2 - 2*b1 - 342) * q^85 + (154*b3 + 142*b2 - 4*b1 - 1026) * q^86 + (-56*b3 + 4*b2 + 122*b1 - 84) * q^87 + (-55*b2 + 44*b1 + 44) * q^88 + (104*b3 - 44*b2 + 48*b1 + 770) * q^89 + (-163*b3 - 165*b2 + 14*b1 - 21) * q^90 + (-140*b3 - 144*b2 + 56*b1 + 92) * q^92 + (-158*b3 - 40*b2 - 104*b1 - 388) * q^93 + (86*b3 + 52*b2 - 80*b1 + 410) * q^94 + (-220*b3 - 216*b2 + 196*b1 - 4) * q^95 + (-143*b3 - 49*b2 - 130*b1 - 305) * q^96 + (-32*b3 + 176*b2 - 154*b1 + 174) * q^97 + (-66*b3 - 44*b1 - 231) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 14 q^{3} + 26 q^{4} - 10 q^{5} - 14 q^{6} - 18 q^{8} + 76 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 14 * q^3 + 26 * q^4 - 10 * q^5 - 14 * q^6 - 18 * q^8 + 76 * q^9 $$4 q - 2 q^{2} - 14 q^{3} + 26 q^{4} - 10 q^{5} - 14 q^{6} - 18 q^{8} + 76 q^{9} + 2 q^{10} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 284 q^{15} + 2 q^{16} - 4 q^{17} - 62 q^{18} - 258 q^{19} - 182 q^{20} + 22 q^{22} + 8 q^{23} + 498 q^{24} + 80 q^{25} + 482 q^{26} - 428 q^{27} - 396 q^{29} - 628 q^{30} + 56 q^{31} + 134 q^{32} + 154 q^{33} - 472 q^{34} - 418 q^{36} + 84 q^{37} + 942 q^{38} - 412 q^{39} + 1026 q^{40} - 52 q^{41} + 408 q^{43} - 286 q^{44} - 826 q^{45} + 368 q^{46} - 8 q^{47} - 982 q^{48} - 1642 q^{50} - 388 q^{51} - 2030 q^{52} + 624 q^{53} - 92 q^{54} + 110 q^{55} + 48 q^{57} + 864 q^{58} + 238 q^{59} + 1420 q^{60} + 162 q^{61} - 688 q^{62} - 902 q^{64} - 32 q^{65} + 154 q^{66} + 1340 q^{67} + 1384 q^{68} - 2416 q^{69} + 1788 q^{71} - 2622 q^{72} - 1456 q^{73} + 996 q^{74} + 806 q^{75} - 3042 q^{76} - 2632 q^{78} - 1324 q^{79} - 2342 q^{80} + 1444 q^{81} - 1984 q^{82} - 450 q^{83} - 1736 q^{85} - 4380 q^{86} - 588 q^{87} + 198 q^{88} + 3072 q^{89} + 218 q^{90} + 544 q^{92} - 1264 q^{93} + 1696 q^{94} + 24 q^{95} - 862 q^{96} + 652 q^{97} - 836 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 - 14 * q^3 + 26 * q^4 - 10 * q^5 - 14 * q^6 - 18 * q^8 + 76 * q^9 + 2 * q^10 - 44 * q^11 - 70 * q^12 - 58 * q^13 + 284 * q^15 + 2 * q^16 - 4 * q^17 - 62 * q^18 - 258 * q^19 - 182 * q^20 + 22 * q^22 + 8 * q^23 + 498 * q^24 + 80 * q^25 + 482 * q^26 - 428 * q^27 - 396 * q^29 - 628 * q^30 + 56 * q^31 + 134 * q^32 + 154 * q^33 - 472 * q^34 - 418 * q^36 + 84 * q^37 + 942 * q^38 - 412 * q^39 + 1026 * q^40 - 52 * q^41 + 408 * q^43 - 286 * q^44 - 826 * q^45 + 368 * q^46 - 8 * q^47 - 982 * q^48 - 1642 * q^50 - 388 * q^51 - 2030 * q^52 + 624 * q^53 - 92 * q^54 + 110 * q^55 + 48 * q^57 + 864 * q^58 + 238 * q^59 + 1420 * q^60 + 162 * q^61 - 688 * q^62 - 902 * q^64 - 32 * q^65 + 154 * q^66 + 1340 * q^67 + 1384 * q^68 - 2416 * q^69 + 1788 * q^71 - 2622 * q^72 - 1456 * q^73 + 996 * q^74 + 806 * q^75 - 3042 * q^76 - 2632 * q^78 - 1324 * q^79 - 2342 * q^80 + 1444 * q^81 - 1984 * q^82 - 450 * q^83 - 1736 * q^85 - 4380 * q^86 - 588 * q^87 + 198 * q^88 + 3072 * q^89 + 218 * q^90 + 544 * q^92 - 1264 * q^93 + 1696 * q^94 + 24 * q^95 - 862 * q^96 + 652 * q^97 - 836 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 12x^{2} + 5x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - 12\nu - 3$$ v^3 - 12*v - 3 $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 11\nu + 3$$ v^3 - v^2 - 11*v + 3 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 13\nu - 4$$ -v^3 + v^2 + 13*v - 4
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + 1 ) / 2$$ (b3 + b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - \beta_{2} + 2\beta _1 + 13 ) / 2$$ (b3 - b2 + 2*b1 + 13) / 2 $$\nu^{3}$$ $$=$$ $$6\beta_{3} + 6\beta_{2} + \beta _1 + 9$$ 6*b3 + 6*b2 + b1 + 9

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.20317 0.555307 3.79597 −0.148103
−4.89098 −6.57251 15.9217 −15.5514 32.1460 0 −38.7449 16.1978 76.0614
1.2 −3.24550 5.49244 2.53327 16.0955 −17.8257 0 17.7423 3.16692 −52.2379
1.3 1.53253 −10.1459 −5.65135 −8.69995 −15.5490 0 −20.9211 75.9402 −13.3330
1.4 4.60395 −2.77399 13.1964 −1.84418 −12.7713 0 23.9238 −19.3050 −8.49053
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.4.a.g 4
7.b odd 2 1 77.4.a.d 4
21.c even 2 1 693.4.a.l 4
28.d even 2 1 1232.4.a.s 4
35.c odd 2 1 1925.4.a.p 4
77.b even 2 1 847.4.a.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.d 4 7.b odd 2 1
539.4.a.g 4 1.a even 1 1 trivial
693.4.a.l 4 21.c even 2 1
847.4.a.d 4 77.b even 2 1
1232.4.a.s 4 28.d even 2 1
1925.4.a.p 4 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(539))$$:

 $$T_{2}^{4} + 2T_{2}^{3} - 27T_{2}^{2} - 40T_{2} + 112$$ T2^4 + 2*T2^3 - 27*T2^2 - 40*T2 + 112 $$T_{3}^{4} + 14T_{3}^{3} + 6T_{3}^{2} - 436T_{3} - 1016$$ T3^4 + 14*T3^3 + 6*T3^2 - 436*T3 - 1016

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} - 27 T^{2} - 40 T + 112$$
$3$ $$T^{4} + 14 T^{3} + 6 T^{2} + \cdots - 1016$$
$5$ $$T^{4} + 10 T^{3} - 240 T^{2} + \cdots - 4016$$
$7$ $$T^{4}$$
$11$ $$(T + 11)^{4}$$
$13$ $$T^{4} + 58 T^{3} - 4926 T^{2} + \cdots - 4947656$$
$17$ $$T^{4} + 4 T^{3} - 6186 T^{2} + \cdots - 2705024$$
$19$ $$T^{4} + 258 T^{3} + \cdots - 14423904$$
$23$ $$T^{4} - 8 T^{3} - 22392 T^{2} + \cdots - 17449856$$
$29$ $$T^{4} + 396 T^{3} + \cdots + 22336464$$
$31$ $$T^{4} - 56 T^{3} - 31890 T^{2} + \cdots - 11250248$$
$37$ $$T^{4} - 84 T^{3} - 63516 T^{2} + \cdots + 11157312$$
$41$ $$T^{4} + 52 T^{3} + \cdots - 659233664$$
$43$ $$T^{4} - 408 T^{3} + \cdots + 1210397376$$
$47$ $$T^{4} + 8 T^{3} - 121650 T^{2} + \cdots - 318931592$$
$53$ $$T^{4} - 624 T^{3} + \cdots + 403923072$$
$59$ $$T^{4} - 238 T^{3} + \cdots + 17599820728$$
$61$ $$T^{4} - 162 T^{3} + \cdots + 6668930664$$
$67$ $$T^{4} - 1340 T^{3} + \cdots - 140865466496$$
$71$ $$T^{4} - 1788 T^{3} + \cdots - 72982082688$$
$73$ $$T^{4} + 1456 T^{3} + \cdots - 322052228384$$
$79$ $$T^{4} + 1324 T^{3} + \cdots - 59537293568$$
$83$ $$T^{4} + 450 T^{3} + \cdots + 21951092064$$
$89$ $$T^{4} - 3072 T^{3} + \cdots - 109303561968$$
$97$ $$T^{4} - 652 T^{3} + \cdots + 868634650768$$