Properties

 Label 637.4.a.d Level $637$ Weight $4$ Character orbit 637.a Self dual yes Analytic conductor $37.584$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 637.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$37.5842166737$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.5364412.1 Defining polynomial: $$x^{4} - 27x^{2} - 24x + 76$$ x^4 - 27*x^2 - 24*x + 76 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + (\beta_{3} - \beta_1 + 7) q^{4} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 10) q^{5} + (2 \beta_{3} - 3 \beta_{2} + \beta_1 + 13) q^{6} + ( - \beta_{3} + 4 \beta_{2} + 5 \beta_1 - 9) q^{8} + (2 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b2 + b1 + 1) * q^3 + (b3 - b1 + 7) * q^4 + (b3 - 2*b2 + b1 + 10) * q^5 + (2*b3 - 3*b2 + b1 + 13) * q^6 + (-b3 + 4*b2 + 5*b1 - 9) * q^8 + (2*b3 + b2 + b1 + 6) * q^9 $$q + (\beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + (\beta_{3} - \beta_1 + 7) q^{4} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 10) q^{5} + (2 \beta_{3} - 3 \beta_{2} + \beta_1 + 13) q^{6} + ( - \beta_{3} + 4 \beta_{2} + 5 \beta_1 - 9) q^{8} + (2 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{9} + ( - \beta_{3} + 10 \beta_{2} + 16 \beta_1 + 8) q^{10} + (2 \beta_{3} + \beta_{2} - 5 \beta_1 - 23) q^{11} + ( - 2 \beta_{3} + 9 \beta_{2} + 17 \beta_1 + 1) q^{12} + 13 q^{13} + (2 \beta_{3} + 4 \beta_{2} + 28 \beta_1 - 4) q^{15} + (\beta_{3} - 16 \beta_{2} - 7 \beta_1 + 19) q^{16} + (2 \beta_{2} - 20 \beta_1 + 36) q^{17} + (2 \beta_{3} + 5 \beta_{2} + 18 \beta_1 + 16) q^{18} + (3 \beta_{3} - 10 \beta_{2} - 9 \beta_1 + 16) q^{19} + (18 \beta_{3} - 18 \beta_{2} - 6 \beta_1 + 132) q^{20} + ( - 4 \beta_{3} + 5 \beta_{2} - 11 \beta_1 - 39) q^{22} + ( - \beta_{3} - 7 \beta_{2} - 18 \beta_1 - 29) q^{23} + (10 \beta_{3} - 11 \beta_{2} - 19 \beta_1 + 125) q^{24} + (19 \beta_{3} - 4 \beta_{2} + 11 \beta_1 + 137) q^{25} + (13 \beta_1 - 13) q^{26} + (2 \beta_{3} - 21 \beta_{2} + 3 \beta_1 + 27) q^{27} + ( - 9 \beta_{3} - 18 \beta_{2} + 29 \beta_1 - 110) q^{29} + (32 \beta_{3} - 4 \beta_{2} + 8 \beta_1 + 404) q^{30} + (9 \beta_{3} + \beta_{2} - 30 \beta_1 + 75) q^{31} + ( - 15 \beta_{3} + 20 \beta_{2} - 15 \beta_1 - 41) q^{32} + ( - 10 \beta_{3} - 11 \beta_{2} - 11 \beta_1 - 59) q^{33} + ( - 18 \beta_{3} - 6 \beta_{2} + 36 \beta_1 - 316) q^{34} + (7 \beta_{3} - 15 \beta_{2} + 20 \beta_1 + 196) q^{36} + ( - 18 \beta_{3} + 7 \beta_{2} - 39 \beta_1 - 63) q^{37} + ( - 19 \beta_{3} + 42 \beta_{2} + 34 \beta_1 - 130) q^{38} + (13 \beta_{2} + 13 \beta_1 + 13) q^{39} + ( - 16 \beta_{3} + 46 \beta_{2} + 112 \beta_1 - 208) q^{40} + ( - 2 \beta_{3} + 77 \beta_{2} + 37 \beta_1 + 3) q^{41} + (9 \beta_{3} + 8 \beta_{2} + 21 \beta_1 + 134) q^{43} + ( - 22 \beta_{3} - 39 \beta_{2} - 23 \beta_1 + 53) q^{44} + (29 \beta_{3} + 2 \beta_{2} + 41 \beta_1 + 206) q^{45} + ( - 25 \beta_{3} + 17 \beta_{2} - 35 \beta_1 - 227) q^{46} + (7 \beta_{3} - 31 \beta_{2} + 4 \beta_1 + 207) q^{47} + ( - 14 \beta_{3} + \beta_{2} + 49 \beta_1 - 359) q^{48} + (7 \beta_{3} + 88 \beta_{2} + 251 \beta_1 + 93) q^{50} + ( - 40 \beta_{3} + 80 \beta_{2} - 8 \beta_1 - 208) q^{51} + (13 \beta_{3} - 13 \beta_1 + 91) q^{52} + (27 \beta_{3} - 54 \beta_{2} - 59 \beta_1 - 58) q^{53} + ( - 18 \beta_{3} + 71 \beta_{2} + 39 \beta_1 + 23) q^{54} + (12 \beta_{2} - 90 \beta_1 - 192) q^{55} + ( - 18 \beta_{3} + 14 \beta_{2} + 54 \beta_1 - 266) q^{57} + (11 \beta_{3} + 18 \beta_{2} - 164 \beta_1 + 480) q^{58} + ( - 48 \beta_{3} + 50 \beta_{2} + 26 \beta_1 + 194) q^{59} + ( - 12 \beta_{3} + 108 \beta_{2} + 372 \beta_1 - 132) q^{60} + (6 \beta_{3} + 13 \beta_{2} + 9 \beta_1 + 35) q^{61} + ( - 29 \beta_{3} + 33 \beta_{2} + 129 \beta_1 - 459) q^{62} + ( - 3 \beta_{3} + 8 \beta_{2} - 75 \beta_1 - 381) q^{64} + (13 \beta_{3} - 26 \beta_{2} + 13 \beta_1 + 130) q^{65} + ( - 22 \beta_{3} - 7 \beta_{2} - 119 \beta_1 - 135) q^{66} + ( - 8 \beta_{3} - 31 \beta_{2} + 11 \beta_1 - 127) q^{67} + (30 \beta_{3} - 70 \beta_{2} - 264 \beta_1 + 460) q^{68} + ( - 36 \beta_{3} - 7 \beta_{2} - 63 \beta_1 - 415) q^{69} + (18 \beta_{3} + 32 \beta_{2} + 32 \beta_1 + 368) q^{71} + ( - 11 \beta_{3} + 33 \beta_{2} + 94 \beta_1 - 16) q^{72} + ( - 75 \beta_{3} + 5 \beta_{2} + 78 \beta_1 - 27) q^{73} + ( - 32 \beta_{3} - 93 \beta_{2} - 171 \beta_1 - 555) q^{74} + (22 \beta_{3} + 107 \beta_{2} + 395 \beta_1 + 371) q^{75} + (52 \beta_{3} - 122 \beta_{2} - 172 \beta_1 + 402) q^{76} + (26 \beta_{3} - 39 \beta_{2} + 13 \beta_1 + 169) q^{78} + ( - 37 \beta_{3} + 3 \beta_{2} + 64 \beta_1 + 237) q^{79} + (14 \beta_{3} - 58 \beta_{2} - 256 \beta_1 + 656) q^{80} + ( - 48 \beta_{3} - 48 \beta_{2} + 72 \beta_1 - 455) q^{81} + (114 \beta_{3} - 239 \beta_{2} - 9 \beta_1 + 507) q^{82} + ( - 47 \beta_{3} - 48 \beta_{2} - 13 \beta_1 + 286) q^{83} + (38 \beta_{3} - 236 \beta_{2} - 284 \beta_1 - 64) q^{85} + (29 \beta_{3} + 12 \beta_{2} + 188 \beta_1 + 196) q^{86} + (58 \beta_{3} - 204 \beta_{2} - 124 \beta_1 - 100) q^{87} + ( - 30 \beta_{3} - 11 \beta_{2} + 9 \beta_1 - 151) q^{88} + ( - 33 \beta_{3} + 16 \beta_{2} - 7 \beta_1 + 720) q^{89} + (43 \beta_{3} + 110 \beta_{2} + 380 \beta_1 + 484) q^{90} + ( - 10 \beta_{3} - 95 \beta_{2} - 233 \beta_1 - 131) q^{92} + ( - 60 \beta_{3} + 137 \beta_{2} + 121 \beta_1 - 255) q^{93} + ( - 27 \beta_{3} + 121 \beta_{2} + 249 \beta_1 - 123) q^{94} + (39 \beta_{3} - 72 \beta_{2} - 225 \beta_1 + 558) q^{95} + ( - 30 \beta_{3} + 29 \beta_{2} - 291 \beta_1 - 11) q^{96} + (13 \beta_{3} + 83 \beta_{2} + 224 \beta_1 - 693) q^{97} + ( - 76 \beta_{3} - 86 \beta_{2} - 44 \beta_1 + 130) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b2 + b1 + 1) * q^3 + (b3 - b1 + 7) * q^4 + (b3 - 2*b2 + b1 + 10) * q^5 + (2*b3 - 3*b2 + b1 + 13) * q^6 + (-b3 + 4*b2 + 5*b1 - 9) * q^8 + (2*b3 + b2 + b1 + 6) * q^9 + (-b3 + 10*b2 + 16*b1 + 8) * q^10 + (2*b3 + b2 - 5*b1 - 23) * q^11 + (-2*b3 + 9*b2 + 17*b1 + 1) * q^12 + 13 * q^13 + (2*b3 + 4*b2 + 28*b1 - 4) * q^15 + (b3 - 16*b2 - 7*b1 + 19) * q^16 + (2*b2 - 20*b1 + 36) * q^17 + (2*b3 + 5*b2 + 18*b1 + 16) * q^18 + (3*b3 - 10*b2 - 9*b1 + 16) * q^19 + (18*b3 - 18*b2 - 6*b1 + 132) * q^20 + (-4*b3 + 5*b2 - 11*b1 - 39) * q^22 + (-b3 - 7*b2 - 18*b1 - 29) * q^23 + (10*b3 - 11*b2 - 19*b1 + 125) * q^24 + (19*b3 - 4*b2 + 11*b1 + 137) * q^25 + (13*b1 - 13) * q^26 + (2*b3 - 21*b2 + 3*b1 + 27) * q^27 + (-9*b3 - 18*b2 + 29*b1 - 110) * q^29 + (32*b3 - 4*b2 + 8*b1 + 404) * q^30 + (9*b3 + b2 - 30*b1 + 75) * q^31 + (-15*b3 + 20*b2 - 15*b1 - 41) * q^32 + (-10*b3 - 11*b2 - 11*b1 - 59) * q^33 + (-18*b3 - 6*b2 + 36*b1 - 316) * q^34 + (7*b3 - 15*b2 + 20*b1 + 196) * q^36 + (-18*b3 + 7*b2 - 39*b1 - 63) * q^37 + (-19*b3 + 42*b2 + 34*b1 - 130) * q^38 + (13*b2 + 13*b1 + 13) * q^39 + (-16*b3 + 46*b2 + 112*b1 - 208) * q^40 + (-2*b3 + 77*b2 + 37*b1 + 3) * q^41 + (9*b3 + 8*b2 + 21*b1 + 134) * q^43 + (-22*b3 - 39*b2 - 23*b1 + 53) * q^44 + (29*b3 + 2*b2 + 41*b1 + 206) * q^45 + (-25*b3 + 17*b2 - 35*b1 - 227) * q^46 + (7*b3 - 31*b2 + 4*b1 + 207) * q^47 + (-14*b3 + b2 + 49*b1 - 359) * q^48 + (7*b3 + 88*b2 + 251*b1 + 93) * q^50 + (-40*b3 + 80*b2 - 8*b1 - 208) * q^51 + (13*b3 - 13*b1 + 91) * q^52 + (27*b3 - 54*b2 - 59*b1 - 58) * q^53 + (-18*b3 + 71*b2 + 39*b1 + 23) * q^54 + (12*b2 - 90*b1 - 192) * q^55 + (-18*b3 + 14*b2 + 54*b1 - 266) * q^57 + (11*b3 + 18*b2 - 164*b1 + 480) * q^58 + (-48*b3 + 50*b2 + 26*b1 + 194) * q^59 + (-12*b3 + 108*b2 + 372*b1 - 132) * q^60 + (6*b3 + 13*b2 + 9*b1 + 35) * q^61 + (-29*b3 + 33*b2 + 129*b1 - 459) * q^62 + (-3*b3 + 8*b2 - 75*b1 - 381) * q^64 + (13*b3 - 26*b2 + 13*b1 + 130) * q^65 + (-22*b3 - 7*b2 - 119*b1 - 135) * q^66 + (-8*b3 - 31*b2 + 11*b1 - 127) * q^67 + (30*b3 - 70*b2 - 264*b1 + 460) * q^68 + (-36*b3 - 7*b2 - 63*b1 - 415) * q^69 + (18*b3 + 32*b2 + 32*b1 + 368) * q^71 + (-11*b3 + 33*b2 + 94*b1 - 16) * q^72 + (-75*b3 + 5*b2 + 78*b1 - 27) * q^73 + (-32*b3 - 93*b2 - 171*b1 - 555) * q^74 + (22*b3 + 107*b2 + 395*b1 + 371) * q^75 + (52*b3 - 122*b2 - 172*b1 + 402) * q^76 + (26*b3 - 39*b2 + 13*b1 + 169) * q^78 + (-37*b3 + 3*b2 + 64*b1 + 237) * q^79 + (14*b3 - 58*b2 - 256*b1 + 656) * q^80 + (-48*b3 - 48*b2 + 72*b1 - 455) * q^81 + (114*b3 - 239*b2 - 9*b1 + 507) * q^82 + (-47*b3 - 48*b2 - 13*b1 + 286) * q^83 + (38*b3 - 236*b2 - 284*b1 - 64) * q^85 + (29*b3 + 12*b2 + 188*b1 + 196) * q^86 + (58*b3 - 204*b2 - 124*b1 - 100) * q^87 + (-30*b3 - 11*b2 + 9*b1 - 151) * q^88 + (-33*b3 + 16*b2 - 7*b1 + 720) * q^89 + (43*b3 + 110*b2 + 380*b1 + 484) * q^90 + (-10*b3 - 95*b2 - 233*b1 - 131) * q^92 + (-60*b3 + 137*b2 + 121*b1 - 255) * q^93 + (-27*b3 + 121*b2 + 249*b1 - 123) * q^94 + (39*b3 - 72*b2 - 225*b1 + 558) * q^95 + (-30*b3 + 29*b2 - 291*b1 - 11) * q^96 + (13*b3 + 83*b2 + 224*b1 - 693) * q^97 + (-76*b3 - 86*b2 - 44*b1 + 130) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 5 q^{3} + 26 q^{4} + 36 q^{5} + 45 q^{6} - 30 q^{8} + 21 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 5 * q^3 + 26 * q^4 + 36 * q^5 + 45 * q^6 - 30 * q^8 + 21 * q^9 $$4 q - 4 q^{2} + 5 q^{3} + 26 q^{4} + 36 q^{5} + 45 q^{6} - 30 q^{8} + 21 q^{9} + 44 q^{10} - 95 q^{11} + 17 q^{12} + 52 q^{13} - 16 q^{15} + 58 q^{16} + 146 q^{17} + 65 q^{18} + 48 q^{19} + 474 q^{20} - 143 q^{22} - 121 q^{23} + 469 q^{24} + 506 q^{25} - 52 q^{26} + 83 q^{27} - 440 q^{29} + 1548 q^{30} + 283 q^{31} - 114 q^{32} - 227 q^{33} - 1234 q^{34} + 755 q^{36} - 209 q^{37} - 440 q^{38} + 65 q^{39} - 754 q^{40} + 93 q^{41} + 526 q^{43} + 217 q^{44} + 768 q^{45} - 841 q^{46} + 783 q^{47} - 1407 q^{48} + 446 q^{50} - 672 q^{51} + 338 q^{52} - 340 q^{53} + 199 q^{54} - 756 q^{55} - 1014 q^{57} + 1916 q^{58} + 922 q^{59} - 396 q^{60} + 141 q^{61} - 1745 q^{62} - 1510 q^{64} + 468 q^{65} - 503 q^{66} - 523 q^{67} + 1710 q^{68} - 1595 q^{69} + 1468 q^{71} - 9 q^{72} + 47 q^{73} - 2249 q^{74} + 1547 q^{75} + 1382 q^{76} + 585 q^{78} + 1025 q^{79} + 2538 q^{80} - 1772 q^{81} + 1561 q^{82} + 1190 q^{83} - 568 q^{85} + 738 q^{86} - 720 q^{87} - 555 q^{88} + 2962 q^{89} + 1960 q^{90} - 599 q^{92} - 763 q^{93} - 317 q^{94} + 2082 q^{95} + 45 q^{96} - 2715 q^{97} + 586 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 5 * q^3 + 26 * q^4 + 36 * q^5 + 45 * q^6 - 30 * q^8 + 21 * q^9 + 44 * q^10 - 95 * q^11 + 17 * q^12 + 52 * q^13 - 16 * q^15 + 58 * q^16 + 146 * q^17 + 65 * q^18 + 48 * q^19 + 474 * q^20 - 143 * q^22 - 121 * q^23 + 469 * q^24 + 506 * q^25 - 52 * q^26 + 83 * q^27 - 440 * q^29 + 1548 * q^30 + 283 * q^31 - 114 * q^32 - 227 * q^33 - 1234 * q^34 + 755 * q^36 - 209 * q^37 - 440 * q^38 + 65 * q^39 - 754 * q^40 + 93 * q^41 + 526 * q^43 + 217 * q^44 + 768 * q^45 - 841 * q^46 + 783 * q^47 - 1407 * q^48 + 446 * q^50 - 672 * q^51 + 338 * q^52 - 340 * q^53 + 199 * q^54 - 756 * q^55 - 1014 * q^57 + 1916 * q^58 + 922 * q^59 - 396 * q^60 + 141 * q^61 - 1745 * q^62 - 1510 * q^64 + 468 * q^65 - 503 * q^66 - 523 * q^67 + 1710 * q^68 - 1595 * q^69 + 1468 * q^71 - 9 * q^72 + 47 * q^73 - 2249 * q^74 + 1547 * q^75 + 1382 * q^76 + 585 * q^78 + 1025 * q^79 + 2538 * q^80 - 1772 * q^81 + 1561 * q^82 + 1190 * q^83 - 568 * q^85 + 738 * q^86 - 720 * q^87 - 555 * q^88 + 2962 * q^89 + 1960 * q^90 - 599 * q^92 - 763 * q^93 - 317 * q^94 + 2082 * q^95 + 45 * q^96 - 2715 * q^97 + 586 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 27x^{2} - 24x + 76$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2\nu^{2} - 19\nu + 10 ) / 4$$ (v^3 - 2*v^2 - 19*v + 10) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 14$$ v^2 - v - 14
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 + 14$$ b3 + b1 + 14 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + 4\beta_{2} + 21\beta _1 + 18$$ 2*b3 + 4*b2 + 21*b1 + 18

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
 −4.05873 −2.63459 1.32361 5.36970
−5.05873 −6.23157 17.5908 18.8190 31.5238 0 −48.5170 11.8325 −95.2001
1.2 −3.63459 5.33748 5.21021 −11.0031 −19.3995 0 10.1397 1.48874 39.9917
1.3 0.323612 −1.75980 −7.89528 5.91876 −0.569491 0 −5.14391 −23.9031 1.91538
1.4 4.36970 7.65388 11.0943 22.2654 33.4452 0 13.5212 31.5819 97.2930
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$13$$ $$-1$$

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 637.4.a.d 4
7.b odd 2 1 91.4.a.b 4
21.c even 2 1 819.4.a.h 4
28.d even 2 1 1456.4.a.s 4
35.c odd 2 1 2275.4.a.h 4
91.b odd 2 1 1183.4.a.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.4.a.b 4 7.b odd 2 1
637.4.a.d 4 1.a even 1 1 trivial
819.4.a.h 4 21.c even 2 1
1183.4.a.e 4 91.b odd 2 1
1456.4.a.s 4 28.d even 2 1
2275.4.a.h 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2}^{4} + 4T_{2}^{3} - 21T_{2}^{2} - 74T_{2} + 26$$ T2^4 + 4*T2^3 - 21*T2^2 - 74*T2 + 26 $$T_{3}^{4} - 5T_{3}^{3} - 52T_{3}^{2} + 184T_{3} + 448$$ T3^4 - 5*T3^3 - 52*T3^2 + 184*T3 + 448

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4 T^{3} - 21 T^{2} - 74 T + 26$$
$3$ $$T^{4} - 5 T^{3} - 52 T^{2} + 184 T + 448$$
$5$ $$T^{4} - 36 T^{3} + 145 T^{2} + \cdots - 27288$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 95 T^{3} + 2128 T^{2} + \cdots - 151632$$
$13$ $$(T - 13)^{4}$$
$17$ $$T^{4} - 146 T^{3} - 3120 T^{2} + \cdots - 1065472$$
$19$ $$T^{4} - 48 T^{3} - 5327 T^{2} + \cdots + 317232$$
$23$ $$T^{4} + 121 T^{3} - 5241 T^{2} + \cdots - 2384104$$
$29$ $$T^{4} + 440 T^{3} + \cdots - 484339768$$
$31$ $$T^{4} - 283 T^{3} - 3281 T^{2} + \cdots - 1026856$$
$37$ $$T^{4} + 209 T^{3} + \cdots + 328158128$$
$41$ $$T^{4} - 93 T^{3} + \cdots + 12096773224$$
$43$ $$T^{4} - 526 T^{3} + \cdots + 18583856$$
$47$ $$T^{4} - 783 T^{3} + \cdots - 1054241384$$
$53$ $$T^{4} + 340 T^{3} + \cdots - 11218230832$$
$59$ $$T^{4} - 922 T^{3} + \cdots + 10047112192$$
$61$ $$T^{4} - 141 T^{3} - 9038 T^{2} + \cdots + 3710376$$
$67$ $$T^{4} + 523 T^{3} + \cdots - 951710544$$
$71$ $$T^{4} - 1468 T^{3} + \cdots + 2887158784$$
$73$ $$T^{4} - 47 T^{3} + \cdots + 38124898514$$
$79$ $$T^{4} - 1025 T^{3} + \cdots - 13183278632$$
$83$ $$T^{4} - 1190 T^{3} + \cdots - 11400717312$$
$89$ $$T^{4} - 2962 T^{3} + \cdots + 205066944356$$
$97$ $$T^{4} + 2715 T^{3} + \cdots - 914822530202$$