# Properties

 Label 1840.4.a.ba Level $1840$ Weight $4$ Character orbit 1840.a Self dual yes Analytic conductor $108.564$ Analytic rank $0$ Dimension $10$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$108.563514411$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 5 x^{9} - 192 x^{8} + 762 x^{7} + 12246 x^{6} - 33828 x^{5} - 298243 x^{4} + 383603 x^{3} + 2423016 x^{2} + 864576 x + 57408$$ x^10 - 5*x^9 - 192*x^8 + 762*x^7 + 12246*x^6 - 33828*x^5 - 298243*x^4 + 383603*x^3 + 2423016*x^2 + 864576*x + 57408 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + 5 q^{5} + (\beta_{6} - \beta_1 - 1) q^{7} + (\beta_{2} + \beta_1 + 13) q^{9}+O(q^{10})$$ q - b1 * q^3 + 5 * q^5 + (b6 - b1 - 1) * q^7 + (b2 + b1 + 13) * q^9 $$q - \beta_1 q^{3} + 5 q^{5} + (\beta_{6} - \beta_1 - 1) q^{7} + (\beta_{2} + \beta_1 + 13) q^{9} + (\beta_{5} - \beta_1 + 6) q^{11} + (\beta_{7} + 5) q^{13} - 5 \beta_1 q^{15} + ( - \beta_{4} - \beta_1 + 24) q^{17} + ( - \beta_{9} + \beta_{7} + 2 \beta_{6} + \beta_{5} - 4 \beta_1 - 7) q^{19} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{2} + 3 \beta_1 + 33) q^{21} + 23 q^{23} + 25 q^{25} + (\beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 18 \beta_1 - 36) q^{27} + ( - 3 \beta_{8} - \beta_{7} - \beta_{6} + 7 \beta_1 + 29) q^{29} + ( - \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 12) q^{31} + ( - \beta_{9} + 3 \beta_{8} - \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 2 \beta_1 + 51) q^{33} + (5 \beta_{6} - 5 \beta_1 - 5) q^{35} + (4 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{2} + 8 \beta_1 + 54) q^{37} + (4 \beta_{9} - 5 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} + 2 \beta_{4} + \cdots - 7) q^{39}+ \cdots + ( - 12 \beta_{9} + 14 \beta_{8} + 16 \beta_{7} + 3 \beta_{6} + 16 \beta_{5} + \cdots - 80) q^{99}+O(q^{100})$$ q - b1 * q^3 + 5 * q^5 + (b6 - b1 - 1) * q^7 + (b2 + b1 + 13) * q^9 + (b5 - b1 + 6) * q^11 + (b7 + 5) * q^13 - 5*b1 * q^15 + (-b4 - b1 + 24) * q^17 + (-b9 + b7 + 2*b6 + b5 - 4*b1 - 7) * q^19 + (-b9 + b8 + b7 - b6 + 2*b5 + b4 + b2 + 3*b1 + 33) * q^21 + 23 * q^23 + 25 * q^25 + (b9 + b8 + 2*b7 + 2*b6 - b4 - b3 - b2 - 18*b1 - 36) * q^27 + (-3*b8 - b7 - b6 + 7*b1 + 29) * q^29 + (-b9 + b8 - b6 + b5 + 2*b3 + 2*b2 - 5*b1 + 12) * q^31 + (-b9 + 3*b8 - b7 + 4*b6 + b4 - b3 - 2*b1 + 51) * q^33 + (5*b6 - 5*b1 - 5) * q^35 + (4*b9 - 2*b8 - 2*b7 - b5 + b4 + 2*b2 + 8*b1 + 54) * q^37 + (4*b9 - 5*b8 - 4*b7 + 2*b6 - 7*b5 + 2*b4 - 2*b3 + 7*b2 - 8*b1 - 7) * q^39 + (b9 - b7 + 5*b6 + 2*b5 - 3*b4 + 3*b3 - 4*b1 + 25) * q^41 + (2*b9 - b7 + 2*b6 + b5 - 3*b4 + b3 + b2 - 23*b1 - 24) * q^43 + (5*b2 + 5*b1 + 65) * q^45 + (-2*b9 - b8 + b6 - 4*b5 + 2*b4 + b3 - b2 - 21) * q^47 + (-2*b9 - b7 + b6 - b5 - b4 - b3 - 6*b2 + 2*b1 + 64) * q^49 + (-4*b9 + b8 - 4*b7 - 5*b6 - 3*b5 + 3*b4 - 2*b3 + 2*b2 - 23*b1 + 32) * q^51 + (3*b9 - b8 + b7 + 9*b6 - 3*b5 - 6*b2 - 7*b1 - 13) * q^53 + (5*b5 - 5*b1 + 30) * q^55 + (2*b8 - 4*b7 + 7*b6 - 5*b5 + 2*b4 - 3*b3 + 6*b2 - 2*b1 + 170) * q^57 + (2*b9 + b8 + 3*b7 - 2*b6 + 7*b5 + b4 + 2*b3 + 5*b2 - 13*b1 - 22) * q^59 + (-4*b9 + 3*b8 + 7*b7 - b6 + 7*b5 + 3*b4 - 10*b2 + 12*b1 + 99) * q^61 + (4*b9 + 4*b8 - b7 - 6*b6 - 4*b4 - b3 - 2*b2 - 55*b1 - 46) * q^63 + (5*b7 + 25) * q^65 + (-5*b9 + 4*b7 - 9*b6 + 3*b5 + 2*b4 - 11*b2 + 4*b1 - 31) * q^67 - 23*b1 * q^69 + (-2*b9 + 10*b8 + 3*b7 - 11*b6 + 8*b5 + b4 + 3*b3 - 6*b2 - 36*b1 - 41) * q^71 + (7*b9 - 7*b8 - 3*b7 + 4*b6 - 7*b5 - 8*b4 + 4*b3 - b2 - 27*b1 + 46) * q^73 - 25*b1 * q^75 + (-18*b9 + 8*b8 + 5*b7 + 7*b6 - 4*b5 + 5*b4 - 7*b2 - 41*b1 + 202) * q^77 + (-9*b9 - 3*b8 - 9*b7 + 27*b6 - 12*b5 - 3*b4 + 9*b1 + 99) * q^79 + (b9 - 10*b8 - 8*b7 - 7*b6 + 11*b4 - 4*b3 + 18*b2 + 64*b1 + 318) * q^81 + (3*b9 - 6*b8 + 3*b7 - 15*b6 + 6*b5 - b4 - 7*b3 - 4*b2 - 11*b1 - 65) * q^83 + (-5*b4 - 5*b1 + 120) * q^85 + (6*b9 - 2*b8 - b6 - 19*b5 - 3*b4 - 4*b3 - 17*b2 - 28*b1 - 245) * q^87 + (4*b9 + 3*b8 - b7 - 20*b6 - 6*b4 + 7*b2 - 57*b1 + 198) * q^89 + (12*b9 - 17*b8 - 16*b7 - 2*b6 - 10*b5 + b4 - 3*b3 + 4*b2 - 5*b1 - 64) * q^91 + (-6*b9 + 12*b8 - 2*b7 + 10*b6 - 2*b5 - b4 + b3 - 13*b2 - 78*b1 + 207) * q^93 + (-5*b9 + 5*b7 + 10*b6 + 5*b5 - 20*b1 - 35) * q^95 + (8*b9 - 6*b8 - 22*b6 + b5 - 6*b4 + 6*b3 + 12*b2 - 65*b1 + 286) * q^97 + (-12*b9 + 14*b8 + 16*b7 + 3*b6 + 16*b5 - 6*b4 + 9*b3 - 39*b1 - 80) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9}+O(q^{10})$$ 10 * q - 5 * q^3 + 50 * q^5 - 14 * q^7 + 139 * q^9 $$10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9} + 56 q^{11} + 49 q^{13} - 25 q^{15} + 240 q^{17} - 88 q^{19} + 346 q^{21} + 230 q^{23} + 250 q^{25} - 449 q^{27} + 319 q^{29} + 109 q^{31} + 504 q^{33} - 70 q^{35} + 580 q^{37} - 107 q^{39} + 259 q^{41} - 330 q^{43} + 695 q^{45} - 227 q^{47} + 630 q^{49} + 192 q^{51} - 186 q^{53} + 280 q^{55} + 1708 q^{57} - 262 q^{59} + 1000 q^{61} - 722 q^{63} + 245 q^{65} - 354 q^{67} - 115 q^{69} - 599 q^{71} + 355 q^{73} - 125 q^{75} + 1776 q^{77} + 1068 q^{79} + 3490 q^{81} - 754 q^{83} + 1200 q^{85} - 2675 q^{87} + 1740 q^{89} - 690 q^{91} + 1669 q^{93} - 440 q^{95} + 2592 q^{97} - 916 q^{99}+O(q^{100})$$ 10 * q - 5 * q^3 + 50 * q^5 - 14 * q^7 + 139 * q^9 + 56 * q^11 + 49 * q^13 - 25 * q^15 + 240 * q^17 - 88 * q^19 + 346 * q^21 + 230 * q^23 + 250 * q^25 - 449 * q^27 + 319 * q^29 + 109 * q^31 + 504 * q^33 - 70 * q^35 + 580 * q^37 - 107 * q^39 + 259 * q^41 - 330 * q^43 + 695 * q^45 - 227 * q^47 + 630 * q^49 + 192 * q^51 - 186 * q^53 + 280 * q^55 + 1708 * q^57 - 262 * q^59 + 1000 * q^61 - 722 * q^63 + 245 * q^65 - 354 * q^67 - 115 * q^69 - 599 * q^71 + 355 * q^73 - 125 * q^75 + 1776 * q^77 + 1068 * q^79 + 3490 * q^81 - 754 * q^83 + 1200 * q^85 - 2675 * q^87 + 1740 * q^89 - 690 * q^91 + 1669 * q^93 - 440 * q^95 + 2592 * q^97 - 916 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 5 x^{9} - 192 x^{8} + 762 x^{7} + 12246 x^{6} - 33828 x^{5} - 298243 x^{4} + 383603 x^{3} + 2423016 x^{2} + 864576 x + 57408$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 40$$ v^2 - v - 40 $$\beta_{3}$$ $$=$$ $$( 13492675 \nu^{9} - 90064369 \nu^{8} - 2424760468 \nu^{7} + 14223759494 \nu^{6} + 137945915486 \nu^{5} - 675980973244 \nu^{4} + \cdots - 13787240781840 ) / 189638109072$$ (13492675*v^9 - 90064369*v^8 - 2424760468*v^7 + 14223759494*v^6 + 137945915486*v^5 - 675980973244*v^4 - 2615834670377*v^3 + 9534676298151*v^2 + 10772744588796*v - 13787240781840) / 189638109072 $$\beta_{4}$$ $$=$$ $$( 133928635 \nu^{9} - 932238831 \nu^{8} - 23988456900 \nu^{7} + 148152752502 \nu^{6} + 1380501328554 \nu^{5} + \cdots + 46471969427424 ) / 1706742981648$$ (133928635*v^9 - 932238831*v^8 - 23988456900*v^7 + 148152752502*v^6 + 1380501328554*v^5 - 6956017097892*v^4 - 28864153120201*v^3 + 86549522559441*v^2 + 205167246593412*v + 46471969427424) / 1706742981648 $$\beta_{5}$$ $$=$$ $$( - 74568380 \nu^{9} + 381349725 \nu^{8} + 14353347492 \nu^{7} - 60353430060 \nu^{6} - 914845733946 \nu^{5} + \cdots + 7611310191408 ) / 853371490824$$ (-74568380*v^9 + 381349725*v^8 + 14353347492*v^7 - 60353430060*v^6 - 914845733946*v^5 + 2890995982752*v^4 + 22121391694268*v^3 - 40439695871631*v^2 - 176960295677076*v + 7611310191408) / 853371490824 $$\beta_{6}$$ $$=$$ $$( 73632521 \nu^{9} - 380531223 \nu^{8} - 13939762884 \nu^{7} + 57963763818 \nu^{6} + 869421171810 \nu^{5} - 2581999841436 \nu^{4} + \cdots + 30379739386224 ) / 568914327216$$ (73632521*v^9 - 380531223*v^8 - 13939762884*v^7 + 57963763818*v^6 + 869421171810*v^5 - 2581999841436*v^4 - 20419450142675*v^3 + 30440951108049*v^2 + 159297910087380*v + 30379739386224) / 568914327216 $$\beta_{7}$$ $$=$$ $$( 157136465 \nu^{9} - 873082230 \nu^{8} - 30030328044 \nu^{7} + 136872598350 \nu^{6} + 1903576710780 \nu^{5} + \cdots + 44170193069544 ) / 853371490824$$ (157136465*v^9 - 873082230*v^8 - 30030328044*v^7 + 136872598350*v^6 + 1903576710780*v^5 - 6359378019096*v^4 - 46055313237419*v^3 + 81093834307938*v^2 + 379569923446128*v + 44170193069544) / 853371490824 $$\beta_{8}$$ $$=$$ $$( - 88362863 \nu^{9} + 431383173 \nu^{8} + 17149709286 \nu^{7} - 66106410600 \nu^{6} - 1106918855226 \nu^{5} + \cdots - 61611158517060 ) / 426685745412$$ (-88362863*v^9 + 431383173*v^8 + 17149709286*v^7 - 66106410600*v^6 - 1106918855226*v^5 + 2953956686862*v^4 + 27158690324471*v^3 - 33907955846547*v^2 - 216400233118308*v - 61611158517060) / 426685745412 $$\beta_{9}$$ $$=$$ $$( - 230763412 \nu^{9} + 1153582707 \nu^{8} + 44674875612 \nu^{7} - 177340372980 \nu^{6} - 2890571942574 \nu^{5} + \cdots - 98477372031504 ) / 853371490824$$ (-230763412*v^9 + 1153582707*v^8 + 44674875612*v^7 - 177340372980*v^6 - 2890571942574*v^5 + 8036919260232*v^4 + 71994892186300*v^3 - 98660434134705*v^2 - 592582761122988*v - 98477372031504) / 853371490824
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 40$$ b2 + b1 + 40 $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{8} - 2\beta_{7} - 2\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 72\beta _1 + 36$$ -b9 - b8 - 2*b7 - 2*b6 + b4 + b3 + b2 + 72*b1 + 36 $$\nu^{4}$$ $$=$$ $$\beta_{9} - 10\beta_{8} - 8\beta_{7} - 7\beta_{6} + 11\beta_{4} - 4\beta_{3} + 99\beta_{2} + 145\beta _1 + 2829$$ b9 - 10*b8 - 8*b7 - 7*b6 + 11*b4 - 4*b3 + 99*b2 + 145*b1 + 2829 $$\nu^{5}$$ $$=$$ $$- 157 \beta_{9} - 91 \beta_{8} - 266 \beta_{7} - 245 \beta_{6} - 9 \beta_{5} + 160 \beta_{4} + 85 \beta_{3} + 190 \beta_{2} + 6003 \beta _1 + 5097$$ -157*b9 - 91*b8 - 266*b7 - 245*b6 - 9*b5 + 160*b4 + 85*b3 + 190*b2 + 6003*b1 + 5097 $$\nu^{6}$$ $$=$$ $$34 \beta_{9} - 1390 \beta_{8} - 1430 \beta_{7} - 1243 \beta_{6} - 555 \beta_{5} + 1757 \beta_{4} - 574 \beta_{3} + 9330 \beta_{2} + 18382 \beta _1 + 232185$$ 34*b9 - 1390*b8 - 1430*b7 - 1243*b6 - 555*b5 + 1757*b4 - 574*b3 + 9330*b2 + 18382*b1 + 232185 $$\nu^{7}$$ $$=$$ $$- 17788 \beta_{9} - 7819 \beta_{8} - 30512 \beta_{7} - 24080 \beta_{6} - 3465 \beta_{5} + 20305 \beta_{4} + 5755 \beta_{3} + 26830 \beta_{2} + 536847 \beta _1 + 653064$$ -17788*b9 - 7819*b8 - 30512*b7 - 24080*b6 - 3465*b5 + 20305*b4 + 5755*b3 + 26830*b2 + 536847*b1 + 653064 $$\nu^{8}$$ $$=$$ $$- 7706 \beta_{9} - 156058 \beta_{8} - 199196 \beta_{7} - 151990 \beta_{6} - 110946 \beta_{5} + 218168 \beta_{4} - 68386 \beta_{3} + 886527 \beta_{2} + 2198497 \beta _1 + 20541678$$ -7706*b9 - 156058*b8 - 199196*b7 - 151990*b6 - 110946*b5 + 218168*b4 - 68386*b3 + 886527*b2 + 2198497*b1 + 20541678 $$\nu^{9}$$ $$=$$ $$- 1822591 \beta_{9} - 746035 \beta_{8} - 3374474 \beta_{7} - 2265212 \beta_{6} - 686178 \beta_{5} + 2362249 \beta_{4} + 321355 \beta_{3} + 3408271 \beta_{2} + 50118384 \beta _1 + 79069830$$ -1822591*b9 - 746035*b8 - 3374474*b7 - 2265212*b6 - 686178*b5 + 2362249*b4 + 321355*b3 + 3408271*b2 + 50118384*b1 + 79069830

## 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
 10.3184 8.50847 6.17442 4.57895 −0.0875767 −0.292216 −2.78233 −4.67252 −7.60722 −9.13838
0 −10.3184 0 5.00000 0 −13.9200 0 79.4692 0
1.2 0 −8.50847 0 5.00000 0 −14.9751 0 45.3940 0
1.3 0 −6.17442 0 5.00000 0 21.9366 0 11.1235 0
1.4 0 −4.57895 0 5.00000 0 −22.7390 0 −6.03324 0
1.5 0 0.0875767 0 5.00000 0 28.3995 0 −26.9923 0
1.6 0 0.292216 0 5.00000 0 −23.7015 0 −26.9146 0
1.7 0 2.78233 0 5.00000 0 11.8146 0 −19.2586 0
1.8 0 4.67252 0 5.00000 0 −26.1910 0 −5.16760 0
1.9 0 7.60722 0 5.00000 0 20.6846 0 30.8698 0
1.10 0 9.13838 0 5.00000 0 4.69126 0 56.5099 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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 1840.4.a.ba 10
4.b odd 2 1 920.4.a.h 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.h 10 4.b odd 2 1
1840.4.a.ba 10 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{10} + 5 T_{3}^{9} - 192 T_{3}^{8} - 762 T_{3}^{7} + 12246 T_{3}^{6} + 33828 T_{3}^{5} - 298243 T_{3}^{4} - 383603 T_{3}^{3} + 2423016 T_{3}^{2} - 864576 T_{3} + 57408$$ T3^10 + 5*T3^9 - 192*T3^8 - 762*T3^7 + 12246*T3^6 + 33828*T3^5 - 298243*T3^4 - 383603*T3^3 + 2423016*T3^2 - 864576*T3 + 57408 $$T_{7}^{10} + 14 T_{7}^{9} - 1932 T_{7}^{8} - 26286 T_{7}^{7} + 1329984 T_{7}^{6} + 16807960 T_{7}^{5} - 395834563 T_{7}^{4} - 4276110998 T_{7}^{3} + 48504293896 T_{7}^{2} + \cdots - 2101566628800$$ T7^10 + 14*T7^9 - 1932*T7^8 - 26286*T7^7 + 1329984*T7^6 + 16807960*T7^5 - 395834563*T7^4 - 4276110998*T7^3 + 48504293896*T7^2 + 344614205160*T7 - 2101566628800

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10} + 5 T^{9} - 192 T^{8} + \cdots + 57408$$
$5$ $$(T - 5)^{10}$$
$7$ $$T^{10} + 14 T^{9} + \cdots - 2101566628800$$
$11$ $$T^{10} - 56 T^{9} + \cdots + 80040845736576$$
$13$ $$T^{10} + \cdots - 923318227263000$$
$17$ $$T^{10} - 240 T^{9} + \cdots - 83\!\cdots\!48$$
$19$ $$T^{10} + 88 T^{9} + \cdots + 65\!\cdots\!48$$
$23$ $$(T - 23)^{10}$$
$29$ $$T^{10} - 319 T^{9} + \cdots + 25\!\cdots\!64$$
$31$ $$T^{10} - 109 T^{9} + \cdots - 90\!\cdots\!16$$
$37$ $$T^{10} - 580 T^{9} + \cdots - 71\!\cdots\!28$$
$41$ $$T^{10} - 259 T^{9} + \cdots - 66\!\cdots\!14$$
$43$ $$T^{10} + 330 T^{9} + \cdots + 42\!\cdots\!00$$
$47$ $$T^{10} + 227 T^{9} + \cdots - 34\!\cdots\!68$$
$53$ $$T^{10} + 186 T^{9} + \cdots + 10\!\cdots\!32$$
$59$ $$T^{10} + 262 T^{9} + \cdots - 43\!\cdots\!92$$
$61$ $$T^{10} - 1000 T^{9} + \cdots - 21\!\cdots\!68$$
$67$ $$T^{10} + 354 T^{9} + \cdots + 16\!\cdots\!24$$
$71$ $$T^{10} + 599 T^{9} + \cdots + 79\!\cdots\!88$$
$73$ $$T^{10} - 355 T^{9} + \cdots + 41\!\cdots\!12$$
$79$ $$T^{10} - 1068 T^{9} + \cdots - 30\!\cdots\!68$$
$83$ $$T^{10} + 754 T^{9} + \cdots - 83\!\cdots\!12$$
$89$ $$T^{10} - 1740 T^{9} + \cdots - 56\!\cdots\!08$$
$97$ $$T^{10} - 2592 T^{9} + \cdots + 10\!\cdots\!96$$