Properties

 Label 378.4.g.d Level $378$ Weight $4$ Character orbit 378.g Analytic conductor $22.303$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 378.g (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$22.3027219822$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 66x^{6} + 59x^{5} + 3770x^{4} + 721x^{3} + 29779x^{2} + 1374x + 209764$$ x^8 - x^7 + 66*x^6 + 59*x^5 + 3770*x^4 + 721*x^3 + 29779*x^2 + 1374*x + 209764 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_1 q^{2} + ( - 4 \beta_1 - 4) q^{4} + ( - \beta_{6} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{4} - \beta_{3} + 3) q^{7} + 8 q^{8}+O(q^{10})$$ q + 2*b1 * q^2 + (-4*b1 - 4) * q^4 + (-b6 - b1) * q^5 + (-b7 - b4 - b3 + 3) * q^7 + 8 * q^8 $$q + 2 \beta_1 q^{2} + ( - 4 \beta_1 - 4) q^{4} + ( - \beta_{6} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{4} - \beta_{3} + 3) q^{7} + 8 q^{8} + (2 \beta_{2} + 2 \beta_1 + 2) q^{10} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 14 \beta_1 + 14) q^{11} + (\beta_{7} - 3 \beta_{6} + \beta_{5} + 3 \beta_{2} - 2) q^{13} + (2 \beta_{4} + 6 \beta_1) q^{14} + 16 \beta_1 q^{16} + ( - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 30 \beta_1 - 30) q^{17} + (3 \beta_{7} + 2 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 10 \beta_1) q^{19} + (4 \beta_{6} - 4 \beta_{2} - 4) q^{20} + (4 \beta_{6} - 4 \beta_{5} + 2 \beta_{3} - 4 \beta_{2} - 28) q^{22} + ( - 8 \beta_{7} + 2 \beta_{6} - 8 \beta_{4} - 48 \beta_1) q^{23} + (3 \beta_{5} - 8 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 24 \beta_1 - 24) q^{25} + ( - 2 \beta_{7} + 6 \beta_{6} - 2 \beta_{4} - 4 \beta_1) q^{26} + (4 \beta_{7} + 4 \beta_{3} - 12 \beta_1 - 12) q^{28} + ( - 6 \beta_{7} - 11 \beta_{6} + 2 \beta_{5} - 4 \beta_{3} + 11 \beta_{2} + \cdots - 67) q^{29}+ \cdots + ( - 2 \beta_{7} + 28 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 42 \beta_{2} + \cdots + 258) q^{98}+O(q^{100})$$ q + 2*b1 * q^2 + (-4*b1 - 4) * q^4 + (-b6 - b1) * q^5 + (-b7 - b4 - b3 + 3) * q^7 + 8 * q^8 + (2*b2 + 2*b1 + 2) * q^10 + (b5 + b3 + 2*b2 + 14*b1 + 14) * q^11 + (b7 - 3*b6 + b5 + 3*b2 - 2) * q^13 + (2*b4 + 6*b1) * q^14 + 16*b1 * q^16 + (-3*b5 + 2*b4 - b3 - 2*b2 - 30*b1 - 30) * q^17 + (3*b7 + 2*b5 + 3*b4 - 4*b3 - 10*b1) * q^19 + (4*b6 - 4*b2 - 4) * q^20 + (4*b6 - 4*b5 + 2*b3 - 4*b2 - 28) * q^22 + (-8*b7 + 2*b6 - 8*b4 - 48*b1) * q^23 + (3*b5 - 8*b4 - 5*b3 - 3*b2 - 24*b1 - 24) * q^25 + (-2*b7 + 6*b6 - 2*b4 - 4*b1) * q^26 + (4*b7 + 4*b3 - 12*b1 - 12) * q^28 + (-6*b7 - 11*b6 + 2*b5 - 4*b3 + 11*b2 - 67) * q^29 + (7*b5 - 8*b4 - b3 - 12*b2 + 71*b1 + 71) * q^31 + (-32*b1 - 32) * q^32 + (4*b7 - 4*b6 + 8*b5 - 2*b3 + 4*b2 + 60) * q^34 + (-4*b7 + 7*b5 - 6*b4 - 4*b3 - 14*b2 - 20*b1 + 26) * q^35 + (-3*b7 + 24*b6 - 3*b5 - 3*b4 + 6*b3 + 47*b1) * q^37 + (10*b5 - 6*b4 + 4*b3 + 20*b1 + 20) * q^38 + (-8*b6 - 8*b1) * q^40 + (-22*b7 + 2*b6 + 8*b5 - 15*b3 - 2*b2 + 2) * q^41 + (6*b7 + 14*b5 - 4*b3 - 193) * q^43 + (-8*b6 + 4*b5 - 8*b3 - 56*b1) * q^44 + (-16*b5 + 16*b4 - 4*b2 + 96*b1 + 96) * q^46 + (-18*b7 + 3*b6 + 10*b5 - 18*b4 - 20*b3 - 3*b1) * q^47 + (2*b7 - 21*b6 + b4 - 12*b3 - 129*b1 - 209) * q^49 + (-16*b7 - 6*b6 + 4*b5 - 10*b3 + 6*b2 + 48) * q^50 + (-4*b5 + 4*b4 - 12*b2 + 8*b1 + 8) * q^52 + (16*b5 - 8*b4 + 8*b3 + 28*b2 + 122*b1 + 122) * q^53 + (-4*b7 - 27*b6 - 2*b5 - b3 + 27*b2 + 181) * q^55 + (-8*b7 - 8*b4 - 8*b3 + 24) * q^56 + (12*b7 + 22*b6 - 8*b5 + 12*b4 + 16*b3 - 134*b1) * q^58 + (9*b5 + 8*b4 + 17*b3 - 12*b2 - 76*b1 - 76) * q^59 + (-34*b7 + 27*b6 + 6*b5 - 34*b4 - 12*b3 - 110*b1) * q^61 + (-16*b7 - 24*b6 - 12*b5 - 2*b3 + 24*b2 - 142) * q^62 + 64 * q^64 + (-14*b7 + 4*b6 + 11*b5 - 14*b4 - 22*b3 - 382*b1) * q^65 + (10*b5 - b4 + 9*b3 + 45*b2 + 330*b1 + 330) * q^67 + (-8*b7 + 8*b6 - 4*b5 - 8*b4 + 8*b3 + 120*b1) * q^68 + (-4*b7 - 28*b6 - 14*b5 + 8*b4 + 10*b3 + 28*b2 + 92*b1 + 40) * q^70 + (-14*b7 + 39*b6 - 2*b5 - 6*b3 - 39*b2 - 235) * q^71 + (-15*b5 + 17*b4 + 2*b3 - 36*b2 - 360*b1 - 360) * q^73 + (-12*b5 + 6*b4 - 6*b3 - 48*b2 - 94*b1 - 94) * q^74 + (-12*b7 - 28*b5 + 8*b3 - 40) * q^76 + (-28*b7 + 14*b6 - 28*b5 + 8*b4 + 14*b3 + 7*b2 - 193*b1 + 49) * q^77 + (-25*b7 - 12*b6 + 27*b5 - 25*b4 - 54*b3 - 185*b1) * q^79 + (16*b2 + 16*b1 + 16) * q^80 + (44*b7 - 4*b6 - 30*b5 + 44*b4 + 60*b3 + 4*b1) * q^82 + (14*b7 + 22*b6 + 6*b5 + 4*b3 - 22*b2 - 204) * q^83 + (-16*b7 + 27*b6 - 2*b5 - 7*b3 - 27*b2 - 77) * q^85 + (-12*b7 - 8*b5 - 12*b4 + 16*b3 - 386*b1) * q^86 + (8*b5 + 8*b3 + 16*b2 + 112*b1 + 112) * q^88 + (-8*b7 - 32*b6 + 3*b5 - 8*b4 - 6*b3 + 218*b1) * q^89 + (-21*b7 - 21*b6 - 8*b4 + 14*b3 + 207*b1 + 7) * q^91 + (32*b7 - 8*b6 + 32*b5 + 8*b2 - 192) * q^92 + (-16*b5 + 36*b4 + 20*b3 - 6*b2 + 6*b1 + 6) * q^94 + (-24*b5 + 54*b4 + 30*b3 - 4*b2 + 428*b1 + 428) * q^95 + (-16*b7 - 9*b6 + 42*b5 - 29*b3 + 9*b2 + 132) * q^97 + (-2*b7 + 28*b5 - 4*b4 - 2*b3 + 42*b2 - 160*b1 + 258) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{2} - 16 q^{4} + 4 q^{5} + 25 q^{7} + 64 q^{8}+O(q^{10})$$ 8 * q - 8 * q^2 - 16 * q^4 + 4 * q^5 + 25 * q^7 + 64 * q^8 $$8 q - 8 q^{2} - 16 q^{4} + 4 q^{5} + 25 q^{7} + 64 q^{8} + 8 q^{10} + 56 q^{11} - 18 q^{13} - 22 q^{14} - 64 q^{16} - 118 q^{17} + 37 q^{19} - 32 q^{20} - 224 q^{22} + 200 q^{23} - 104 q^{25} + 18 q^{26} - 56 q^{28} - 524 q^{29} + 276 q^{31} - 128 q^{32} + 472 q^{34} + 290 q^{35} - 185 q^{37} + 74 q^{38} + 32 q^{40} + 60 q^{41} - 1556 q^{43} + 224 q^{44} + 400 q^{46} + 30 q^{47} - 1159 q^{49} + 416 q^{50} + 36 q^{52} + 480 q^{53} + 1456 q^{55} + 200 q^{56} + 524 q^{58} - 296 q^{59} + 474 q^{61} - 1104 q^{62} + 512 q^{64} + 1542 q^{65} + 1319 q^{67} - 472 q^{68} - 32 q^{70} - 1852 q^{71} - 1423 q^{73} - 370 q^{74} - 296 q^{76} + 1228 q^{77} + 765 q^{79} + 64 q^{80} - 60 q^{82} - 1660 q^{83} - 584 q^{85} + 1556 q^{86} + 448 q^{88} - 864 q^{89} - 738 q^{91} - 1600 q^{92} + 60 q^{94} + 1766 q^{95} + 1088 q^{97} + 2704 q^{98}+O(q^{100})$$ 8 * q - 8 * q^2 - 16 * q^4 + 4 * q^5 + 25 * q^7 + 64 * q^8 + 8 * q^10 + 56 * q^11 - 18 * q^13 - 22 * q^14 - 64 * q^16 - 118 * q^17 + 37 * q^19 - 32 * q^20 - 224 * q^22 + 200 * q^23 - 104 * q^25 + 18 * q^26 - 56 * q^28 - 524 * q^29 + 276 * q^31 - 128 * q^32 + 472 * q^34 + 290 * q^35 - 185 * q^37 + 74 * q^38 + 32 * q^40 + 60 * q^41 - 1556 * q^43 + 224 * q^44 + 400 * q^46 + 30 * q^47 - 1159 * q^49 + 416 * q^50 + 36 * q^52 + 480 * q^53 + 1456 * q^55 + 200 * q^56 + 524 * q^58 - 296 * q^59 + 474 * q^61 - 1104 * q^62 + 512 * q^64 + 1542 * q^65 + 1319 * q^67 - 472 * q^68 - 32 * q^70 - 1852 * q^71 - 1423 * q^73 - 370 * q^74 - 296 * q^76 + 1228 * q^77 + 765 * q^79 + 64 * q^80 - 60 * q^82 - 1660 * q^83 - 584 * q^85 + 1556 * q^86 + 448 * q^88 - 864 * q^89 - 738 * q^91 - 1600 * q^92 + 60 * q^94 + 1766 * q^95 + 1088 * q^97 + 2704 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 66x^{6} + 59x^{5} + 3770x^{4} + 721x^{3} + 29779x^{2} + 1374x + 209764$$ :

 $$\beta_{1}$$ $$=$$ $$( - 1049071 \nu^{7} + 115038401 \nu^{6} - 163698896 \nu^{5} + 6652953189 \nu^{4} + 3920788192 \nu^{3} + 430332705877 \nu^{2} + \cdots + 368827486562 ) / 3033579204498$$ (-1049071*v^7 + 115038401*v^6 - 163698896*v^5 + 6652953189*v^4 + 3920788192*v^3 + 430332705877*v^2 + 23392170037*v + 368827486562) / 3033579204498 $$\beta_{2}$$ $$=$$ $$( - 1696542203 \nu^{7} + 16739134507 \nu^{6} - 23819679472 \nu^{5} + 926039756877 \nu^{4} + 570510371744 \nu^{3} + \cdots + 495081144679420 ) / 63705163294458$$ (-1696542203*v^7 + 16739134507*v^6 - 23819679472*v^5 + 926039756877*v^4 + 570510371744*v^3 + 62617325900039*v^2 + 232323910657043*v + 495081144679420) / 63705163294458 $$\beta_{3}$$ $$=$$ $$( 863252840 \nu^{7} - 9761813387 \nu^{6} + 75308363320 \nu^{5} - 543267423074 \nu^{4} + 3215243331501 \nu^{3} + \cdots - 128122572731930 ) / 10617527215743$$ (863252840*v^7 - 9761813387*v^6 + 75308363320*v^5 - 543267423074*v^4 + 3215243331501*v^3 - 28158496126236*v^2 + 49965354534078*v - 128122572731930) / 10617527215743 $$\beta_{4}$$ $$=$$ $$( - 7051508536 \nu^{7} + 31228732907 \nu^{6} - 412942482080 \nu^{5} + 868174142577 \nu^{4} - 20223346800686 \nu^{3} + \cdots - 39943270058320 ) / 63705163294458$$ (-7051508536*v^7 + 31228732907*v^6 - 412942482080*v^5 + 868174142577*v^4 - 20223346800686*v^3 + 66670171571461*v^2 + 216104308469206*v - 39943270058320) / 63705163294458 $$\beta_{5}$$ $$=$$ $$( - 1208164720 \nu^{7} - 1739457099 \nu^{6} - 58942130264 \nu^{5} - 384930665533 \nu^{4} - 3607234572013 \nu^{3} + \cdots - 212042096779230 ) / 10617527215743$$ (-1208164720*v^7 - 1739457099*v^6 - 58942130264*v^5 - 384930665533*v^4 - 3607234572013*v^3 - 14865162115186*v^2 + 19656181548311*v - 212042096779230) / 10617527215743 $$\beta_{6}$$ $$=$$ $$( 8617899467 \nu^{7} + 10568597603 \nu^{6} + 583780276750 \nu^{5} + 1638688334415 \nu^{4} + 33815120349688 \nu^{3} + \cdots + 74164049534822 ) / 63705163294458$$ (8617899467*v^7 + 10568597603*v^6 + 583780276750*v^5 + 1638688334415*v^4 + 33815120349688*v^3 + 67560799766893*v^2 + 265828073508127*v + 74164049534822) / 63705163294458 $$\beta_{7}$$ $$=$$ $$( 9748790105 \nu^{7} + 20249178479 \nu^{6} + 500910335965 \nu^{5} + 2482301067297 \nu^{4} + 34135187416753 \nu^{3} + \cdots + 219625364013092 ) / 63705163294458$$ (9748790105*v^7 + 20249178479*v^6 + 500910335965*v^5 + 2482301067297*v^4 + 34135187416753*v^3 + 90389069998933*v^2 - 109817040839384*v + 219625364013092) / 63705163294458
 $$\nu$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 6$$ (-b5 + 2*b4 + b3 - b2 + b1 + 1) / 6 $$\nu^{2}$$ $$=$$ $$( -2\beta_{7} - 5\beta_{6} - 3\beta_{5} - 2\beta_{4} + 6\beta_{3} + 197\beta_1 ) / 6$$ (-2*b7 - 5*b6 - 3*b5 - 2*b4 + 6*b3 + 197*b1) / 6 $$\nu^{3}$$ $$=$$ $$( 94\beta_{7} - 101\beta_{6} + 12\beta_{5} + 41\beta_{3} + 101\beta_{2} - 257 ) / 6$$ (94*b7 - 101*b6 + 12*b5 + 41*b3 + 101*b2 - 257) / 6 $$\nu^{4}$$ $$=$$ $$( -275\beta_{5} + 42\beta_{4} - 233\beta_{3} + 423\beta_{2} - 10311\beta _1 - 10311 ) / 6$$ (-275*b5 + 42*b4 - 233*b3 + 423*b2 - 10311*b1 - 10311) / 6 $$\nu^{5}$$ $$=$$ $$( -5158\beta_{7} + 6515\beta_{6} + 2431\beta_{5} - 5158\beta_{4} - 4862\beta_{3} - 25967\beta_1 ) / 6$$ (-5158*b7 + 6515*b6 + 2431*b5 - 5158*b4 - 4862*b3 - 25967*b1) / 6 $$\nu^{6}$$ $$=$$ $$( -3062\beta_{7} + 31417\beta_{6} + 29096\beta_{5} - 16079\beta_{3} - 31417\beta_{2} + 605185 ) / 6$$ (-3062*b7 + 31417*b6 + 29096*b5 - 16079*b3 - 31417*b2 + 605185) / 6 $$\nu^{7}$$ $$=$$ $$( -140789\beta_{5} + 295406\beta_{4} + 154617\beta_{3} - 407365\beta_{2} + 2144401\beta _1 + 2144401 ) / 6$$ (-140789*b5 + 295406*b4 + 154617*b3 - 407365*b2 + 2144401*b1 + 2144401) / 6

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$1$$ $$\beta_{1}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 4.05517 − 7.02376i −1.44566 + 2.50395i −3.50329 + 6.06788i 1.39378 − 2.41410i 4.05517 + 7.02376i −1.44566 − 2.50395i −3.50329 − 6.06788i 1.39378 + 2.41410i
−1.00000 1.73205i 0 −2.00000 + 3.46410i −8.00592 13.8667i 0 1.70902 + 18.4412i 8.00000 0 −16.0118 + 27.7333i
109.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i −1.18685 2.05569i 0 9.15909 16.0969i 8.00000 0 −2.37371 + 4.11138i
109.3 −1.00000 1.73205i 0 −2.00000 + 3.46410i 2.21575 + 3.83779i 0 −11.5959 14.4407i 8.00000 0 4.43150 7.67558i
109.4 −1.00000 1.73205i 0 −2.00000 + 3.46410i 8.97703 + 15.5487i 0 13.2278 + 12.9624i 8.00000 0 17.9541 31.0973i
163.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −8.00592 + 13.8667i 0 1.70902 18.4412i 8.00000 0 −16.0118 27.7333i
163.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i −1.18685 + 2.05569i 0 9.15909 + 16.0969i 8.00000 0 −2.37371 4.11138i
163.3 −1.00000 + 1.73205i 0 −2.00000 3.46410i 2.21575 3.83779i 0 −11.5959 + 14.4407i 8.00000 0 4.43150 + 7.67558i
163.4 −1.00000 + 1.73205i 0 −2.00000 3.46410i 8.97703 15.5487i 0 13.2278 12.9624i 8.00000 0 17.9541 + 31.0973i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 163.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.g.d 8
3.b odd 2 1 378.4.g.e yes 8
7.c even 3 1 inner 378.4.g.d 8
21.h odd 6 1 378.4.g.e yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.4.g.d 8 1.a even 1 1 trivial
378.4.g.d 8 7.c even 3 1 inner
378.4.g.e yes 8 3.b odd 2 1
378.4.g.e yes 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 4T_{5}^{7} + 310T_{5}^{6} - 48T_{5}^{5} + 85860T_{5}^{4} - 155736T_{5}^{3} + 1263600T_{5}^{2} + 1850688T_{5} + 9144576$$ acting on $$S_{4}^{\mathrm{new}}(378, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 4)^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 4 T^{7} + 310 T^{6} + \cdots + 9144576$$
$7$ $$T^{8} - 25 T^{7} + \cdots + 13841287201$$
$11$ $$T^{8} - 56 T^{7} + \cdots + 943105130496$$
$13$ $$(T^{4} + 9 T^{3} - 2637 T^{2} + \cdots + 1313022)^{2}$$
$17$ $$T^{8} + 118 T^{7} + \cdots + 8822183086656$$
$19$ $$T^{8} - 37 T^{7} + \cdots + 96\!\cdots\!84$$
$23$ $$T^{8} - 200 T^{7} + \cdots + 10\!\cdots\!44$$
$29$ $$(T^{4} + 262 T^{3} - 27174 T^{2} + \cdots - 737431128)^{2}$$
$31$ $$T^{8} - 276 T^{7} + \cdots + 11\!\cdots\!69$$
$37$ $$T^{8} + 185 T^{7} + \cdots + 26\!\cdots\!16$$
$41$ $$(T^{4} - 30 T^{3} - 329454 T^{2} + \cdots + 23111436456)^{2}$$
$43$ $$(T^{4} + 778 T^{3} + 137328 T^{2} + \cdots - 309283361)^{2}$$
$47$ $$T^{8} - 30 T^{7} + \cdots + 47\!\cdots\!36$$
$53$ $$T^{8} - 480 T^{7} + \cdots + 21\!\cdots\!24$$
$59$ $$T^{8} + 296 T^{7} + \cdots + 17\!\cdots\!64$$
$61$ $$T^{8} - 474 T^{7} + \cdots + 37\!\cdots\!61$$
$67$ $$T^{8} - 1319 T^{7} + \cdots + 35\!\cdots\!64$$
$71$ $$(T^{4} + 926 T^{3} + \cdots - 55652552064)^{2}$$
$73$ $$T^{8} + 1423 T^{7} + \cdots + 42\!\cdots\!44$$
$79$ $$T^{8} - 765 T^{7} + \cdots + 25\!\cdots\!36$$
$83$ $$(T^{4} + 830 T^{3} - 10644 T^{2} + \cdots + 331327584)^{2}$$
$89$ $$T^{8} + 864 T^{7} + \cdots + 10\!\cdots\!04$$
$97$ $$(T^{4} - 544 T^{3} + \cdots - 91533471583)^{2}$$