# Properties

 Label 1840.4.a.bb 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: $$x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + 1222472 x^{2} + 627136 x - 43712$$ x^10 - x^9 - 204*x^8 + 42*x^7 + 12958*x^6 + 5872*x^5 - 259871*x^4 - 149461*x^3 + 1222472*x^2 + 627136*x - 43712 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{7}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - 5 q^{5} + ( - \beta_{6} - \beta_1 - 3) q^{7} + (\beta_{2} + \beta_1 + 14) q^{9}+O(q^{10})$$ q + b1 * q^3 - 5 * q^5 + (-b6 - b1 - 3) * q^7 + (b2 + b1 + 14) * q^9 $$q + \beta_1 q^{3} - 5 q^{5} + ( - \beta_{6} - \beta_1 - 3) q^{7} + (\beta_{2} + \beta_1 + 14) q^{9} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{11} + (\beta_{6} - \beta_{5} + \beta_{4} + 2) q^{13} - 5 \beta_1 q^{15} + ( - \beta_{9} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 8) q^{17} + ( - \beta_{9} - 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 11) q^{19} + ( - \beta_{6} - 3 \beta_{4} + 2 \beta_{2} - 10 \beta_1 - 23) q^{21} - 23 q^{23} + 25 q^{25} + (\beta_{9} + \beta_{8} + 3 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + \beta_{4} + 23 \beta_1 + 40) q^{27} + (2 \beta_{9} + 2 \beta_{8} - \beta_{7} - 4 \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 6 \beta_1 - 29) q^{29} + ( - \beta_{9} - \beta_{7} - 4 \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 12) q^{31}+ \cdots + (5 \beta_{9} - 7 \beta_{8} - 7 \beta_{7} + 72 \beta_{6} + \beta_{5} + 37 \beta_{4} + \cdots - 419) q^{99}+O(q^{100})$$ q + b1 * q^3 - 5 * q^5 + (-b6 - b1 - 3) * q^7 + (b2 + b1 + 14) * q^9 + (-b5 + b4 + b3 - b2 + b1 + 2) * q^11 + (b6 - b5 + b4 + 2) * q^13 - 5*b1 * q^15 + (-b9 + b7 - b5 + b4 + b3 + 8) * q^17 + (-b9 - 2*b6 - b5 - b4 + b3 + b2 - b1 - 11) * q^19 + (-b6 - 3*b4 + 2*b2 - 10*b1 - 23) * q^21 - 23 * q^23 + 25 * q^25 + (b9 + b8 + 3*b7 - 4*b6 - 2*b5 + b4 + 23*b1 + 40) * q^27 + (2*b9 + 2*b8 - b7 - 4*b6 - b4 - b3 + b2 - 6*b1 - 29) * q^29 + (-b9 - b7 - 4*b6 - b5 + 2*b4 - 2*b3 + 2*b2 + 9*b1 + 12) * q^31 + (-b9 - b8 - 3*b7 + 7*b6 + b5 + 6*b4 + 4*b3 - 3*b2 - 3*b1 + 15) * q^33 + (5*b6 + 5*b1 + 15) * q^35 + (b9 - 2*b8 + b7 - 2*b6 - 2*b5 - 3*b2 - 9*b1 + 6) * q^37 + (-2*b8 - b7 + 8*b6 + 2*b5 + 7*b4 + 2*b3 - 5*b2 + 23*b1 - 27) * q^39 + (-b9 - b8 + 2*b7 - 4*b6 + 2*b5 + 2*b4 - b3 + 14*b1 + 74) * q^41 + (2*b9 - 3*b8 - 6*b7 - 2*b6 + 6*b5 - 4*b4 + 6*b2 + 12*b1 - 37) * q^43 + (-5*b2 - 5*b1 - 70) * q^45 + (b9 - b8 - 4*b7 - 4*b6 + 4*b5 - 3*b4 - 3*b3 + 6*b2 + 13*b1 + 28) * q^47 + (-5*b8 + 3*b6 + 8*b4 - 5*b2 + 5*b1 + 118) * q^49 + (3*b9 + 2*b8 + 4*b7 + b6 - 3*b5 + 4*b4 + 3*b3 - 4*b2 + 43*b1 - 21) * q^51 + (b9 - b7 - 7*b6 - 2*b5 - b4 + 4*b2 + 5*b1 + 54) * q^53 + (5*b5 - 5*b4 - 5*b3 + 5*b2 - 5*b1 - 10) * q^55 + (b9 + 9*b8 + b7 - 22*b6 - 2*b5 - 10*b4 - 3*b3 + 7*b2 - 14*b1 - 3) * q^57 + (-3*b9 + 2*b8 + 2*b7 + b6 - 6*b5 - 5*b4 - 3*b3 - b2 + 33*b1 - 85) * q^59 + (3*b9 + 6*b8 + 6*b5 - b4 - 3*b3 + 26*b1 + 25) * q^61 + (-b9 + 11*b8 + 3*b7 - 12*b6 + 2*b5 - 16*b4 - 12*b3 + 6*b2 - 11*b1 - 280) * q^63 + (-5*b6 + 5*b5 - 5*b4 - 10) * q^65 + (-3*b9 + 4*b8 + 6*b7 - 4*b6 - 2*b5 + 2*b4 + 9*b3 - 4*b2 + 63*b1 - 119) * q^67 - 23*b1 * q^69 + (6*b9 - 7*b8 + 2*b7 + b6 + b5 + 7*b4 + 9*b3 - 6*b2 + 13*b1 - 125) * q^71 + (-b9 + 6*b8 + b7 + 10*b6 - 4*b5 - 7*b4 + 5*b2 + 15*b1 + 196) * q^73 + 25*b1 * q^75 + (-7*b9 - 8*b8 - 5*b7 - 8*b6 + 4*b5 - 10*b4 - 13*b3 + b2 - 62*b1 + 52) * q^77 + (6*b9 + 2*b8 + 8*b7 + 3*b6 - 3*b4 + 4*b3 - 13*b2 + 40*b1 - 322) * q^79 + (-2*b9 - 8*b8 + 7*b7 - 6*b6 + 10*b5 - 12*b4 - 3*b3 + 20*b2 + 68*b1 + 609) * q^81 + (-3*b9 - 13*b8 - 18*b7 + 6*b6 + 6*b5 + 8*b4 - 3*b3 + 2*b2 + 27*b1 - 46) * q^83 + (5*b9 - 5*b7 + 5*b5 - 5*b4 - 5*b3 - 40) * q^85 + (-9*b9 - 12*b8 - 7*b7 - 5*b6 + 4*b5 - 18*b4 + 2*b3 + 14*b2 - 61*b1 - 174) * q^87 + (-4*b9 - 10*b8 - b7 - 15*b6 + 4*b5 - 7*b4 - 5*b3 + 10*b2 - 12*b1 + 469) * q^89 + (-18*b9 + 7*b8 + 7*b7 - 15*b6 - 13*b5 - 10*b4 - 6*b3 + 5*b2 - 71*b1 - 490) * q^91 + (4*b9 - 5*b8 + 17*b6 - 2*b5 - 8*b4 + 5*b3 + 13*b2 + 55*b1 + 465) * q^93 + (5*b9 + 10*b6 + 5*b5 + 5*b4 - 5*b3 - 5*b2 + 5*b1 + 55) * q^95 + (4*b9 + 6*b8 + 6*b7 + 22*b6 + b5 + 7*b4 + 21*b3 - 17*b2 - 3*b1 + 242) * q^97 + (5*b9 - 7*b8 - 7*b7 + 72*b6 + b5 + 37*b4 + 10*b3 - 50*b2 + 75*b1 - 419) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9}+O(q^{10})$$ 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9 $$10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35} + 62 q^{37} - 283 q^{39} + 767 q^{41} - 332 q^{43} - 695 q^{45} + 323 q^{47} + 1162 q^{49} - 176 q^{51} + 558 q^{53} - 70 q^{55} + 46 q^{57} - 822 q^{59} + 318 q^{61} - 2698 q^{63} - 55 q^{65} - 1152 q^{67} - 23 q^{69} - 1247 q^{71} + 1941 q^{73} + 25 q^{75} + 528 q^{77} - 3134 q^{79} + 6210 q^{81} - 482 q^{83} - 340 q^{85} - 1797 q^{87} + 4734 q^{89} - 4992 q^{91} + 4647 q^{93} + 570 q^{95} + 2326 q^{97} - 4356 q^{99}+O(q^{100})$$ 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9 + 14 * q^11 + 11 * q^13 - 5 * q^15 + 68 * q^17 - 114 * q^19 - 232 * q^21 - 230 * q^23 + 250 * q^25 + 433 * q^27 - 273 * q^29 + 129 * q^31 + 98 * q^33 + 140 * q^35 + 62 * q^37 - 283 * q^39 + 767 * q^41 - 332 * q^43 - 695 * q^45 + 323 * q^47 + 1162 * q^49 - 176 * q^51 + 558 * q^53 - 70 * q^55 + 46 * q^57 - 822 * q^59 + 318 * q^61 - 2698 * q^63 - 55 * q^65 - 1152 * q^67 - 23 * q^69 - 1247 * q^71 + 1941 * q^73 + 25 * q^75 + 528 * q^77 - 3134 * q^79 + 6210 * q^81 - 482 * q^83 - 340 * q^85 - 1797 * q^87 + 4734 * q^89 - 4992 * q^91 + 4647 * q^93 + 570 * q^95 + 2326 * q^97 - 4356 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + 1222472 x^{2} + 627136 x - 43712$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 41$$ v^2 - v - 41 $$\beta_{3}$$ $$=$$ $$( - 3854981 \nu^{9} + 89269783 \nu^{8} + 723349168 \nu^{7} - 16966937858 \nu^{6} - 52533030922 \nu^{5} + 969699259072 \nu^{4} + \cdots + 15927316040736 ) / 649381829040$$ (-3854981*v^9 + 89269783*v^8 + 723349168*v^7 - 16966937858*v^6 - 52533030922*v^5 + 969699259072*v^4 + 1863152202667*v^3 - 15186810570573*v^2 - 16921771204196*v + 15927316040736) / 649381829040 $$\beta_{4}$$ $$=$$ $$( - 4287527 \nu^{9} - 18632021 \nu^{8} + 958382440 \nu^{7} + 3713138890 \nu^{6} - 67076321674 \nu^{5} - 215575562744 \nu^{4} + \cdots - 14421069607200 ) / 389629097424$$ (-4287527*v^9 - 18632021*v^8 + 958382440*v^7 + 3713138890*v^6 - 67076321674*v^5 - 215575562744*v^4 + 1551591559273*v^3 + 3320453096103*v^2 - 11104850228084*v - 14421069607200) / 389629097424 $$\beta_{5}$$ $$=$$ $$( - 72608749 \nu^{9} + 196190847 \nu^{8} + 15013259412 \nu^{7} - 32852266542 \nu^{6} - 974051270338 \nu^{5} + \cdots + 115237989620584 ) / 2922218230680$$ (-72608749*v^9 + 196190847*v^8 + 15013259412*v^7 - 32852266542*v^6 - 974051270338*v^5 + 1813891775908*v^4 + 20114961714183*v^3 - 41141880130997*v^2 - 79734314857444*v + 115237989620584) / 2922218230680 $$\beta_{6}$$ $$=$$ $$( 41435197 \nu^{9} - 38587743 \nu^{8} - 8659056120 \nu^{7} + 1706664114 \nu^{6} + 572042196850 \nu^{5} + 211662778568 \nu^{4} + \cdots + 8867676107504 ) / 1168887292272$$ (41435197*v^9 - 38587743*v^8 - 8659056120*v^7 + 1706664114*v^6 + 572042196850*v^5 + 211662778568*v^4 - 12496323365715*v^3 - 4724945311075*v^2 + 72305838824020*v + 8867676107504) / 1168887292272 $$\beta_{7}$$ $$=$$ $$( 456055897 \nu^{9} - 514489611 \nu^{8} - 93405512736 \nu^{7} + 29305755546 \nu^{6} + 5982905567554 \nu^{5} + \cdots + 273658774052288 ) / 5844436461360$$ (456055897*v^9 - 514489611*v^8 - 93405512736*v^7 + 29305755546*v^6 + 5982905567554*v^5 + 2380312790336*v^4 - 121548442958439*v^3 - 79286193459079*v^2 + 514271154017812*v + 273658774052288) / 5844436461360 $$\beta_{8}$$ $$=$$ $$( 48402857 \nu^{9} - 20806896 \nu^{8} - 9854440296 \nu^{7} - 3859706262 \nu^{6} + 625434057344 \nu^{5} + 636739850680 \nu^{4} + \cdots + 17109836030200 ) / 584443646136$$ (48402857*v^9 - 20806896*v^8 - 9854440296*v^7 - 3859706262*v^6 + 625434057344*v^5 + 636739850680*v^4 - 12645671642919*v^3 - 11523548352956*v^2 + 66174933803024*v + 17109836030200) / 584443646136 $$\beta_{9}$$ $$=$$ $$( - 312403603 \nu^{9} + 511006659 \nu^{8} + 62814279954 \nu^{7} - 50573267814 \nu^{6} - 3913068398836 \nu^{5} + \cdots - 92807654043212 ) / 1461109115340$$ (-312403603*v^9 + 511006659*v^8 + 62814279954*v^7 - 50573267814*v^6 - 3913068398836*v^5 + 303529809586*v^4 + 76051496979801*v^3 + 11055210179941*v^2 - 340208034284128*v - 92807654043212) / 1461109115340
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 41$$ b2 + b1 + 41 $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{8} + 3\beta_{7} - 4\beta_{6} - 2\beta_{5} + \beta_{4} + 77\beta _1 + 40$$ b9 + b8 + 3*b7 - 4*b6 - 2*b5 + b4 + 77*b1 + 40 $$\nu^{4}$$ $$=$$ $$- 2 \beta_{9} - 8 \beta_{8} + 7 \beta_{7} - 6 \beta_{6} + 10 \beta_{5} - 12 \beta_{4} - 3 \beta_{3} + 101 \beta_{2} + 149 \beta _1 + 3201$$ -2*b9 - 8*b8 + 7*b7 - 6*b6 + 10*b5 - 12*b4 - 3*b3 + 101*b2 + 149*b1 + 3201 $$\nu^{5}$$ $$=$$ $$126 \beta_{9} + 165 \beta_{8} + 352 \beta_{7} - 539 \beta_{6} - 192 \beta_{5} + 29 \beta_{4} - 78 \beta_{3} + 122 \beta_{2} + 6701 \beta _1 + 6340$$ 126*b9 + 165*b8 + 352*b7 - 539*b6 - 192*b5 + 29*b4 - 78*b3 + 122*b2 + 6701*b1 + 6340 $$\nu^{6}$$ $$=$$ $$- 271 \beta_{9} - 1006 \beta_{8} + 1011 \beta_{7} - 1523 \beta_{6} + 1100 \beta_{5} - 2206 \beta_{4} - 594 \beta_{3} + 10169 \beta_{2} + 17887 \beta _1 + 282923$$ -271*b9 - 1006*b8 + 1011*b7 - 1523*b6 + 1100*b5 - 2206*b4 - 594*b3 + 10169*b2 + 17887*b1 + 282923 $$\nu^{7}$$ $$=$$ $$12628 \beta_{9} + 20071 \beta_{8} + 35963 \beta_{7} - 65369 \beta_{6} - 16334 \beta_{5} - 5653 \beta_{4} - 13587 \beta_{3} + 24290 \beta_{2} + 610325 \beta _1 + 790558$$ 12628*b9 + 20071*b8 + 35963*b7 - 65369*b6 - 16334*b5 - 5653*b4 - 13587*b3 + 24290*b2 + 610325*b1 + 790558 $$\nu^{8}$$ $$=$$ $$- 27132 \beta_{9} - 95202 \beta_{8} + 124672 \beta_{7} - 253946 \beta_{6} + 102528 \beta_{5} - 308026 \beta_{4} - 86724 \beta_{3} + 1033315 \beta_{2} + 1992345 \beta _1 + 26201647$$ -27132*b9 - 95202*b8 + 124672*b7 - 253946*b6 + 102528*b5 - 308026*b4 - 86724*b3 + 1033315*b2 + 1992345*b1 + 26201647 $$\nu^{9}$$ $$=$$ $$1197157 \beta_{9} + 2211691 \beta_{8} + 3599343 \beta_{7} - 7540678 \beta_{6} - 1366838 \beta_{5} - 1414829 \beta_{4} - 1803516 \beta_{3} + 3533338 \beta_{2} + 56981385 \beta _1 + 90602246$$ 1197157*b9 + 2211691*b8 + 3599343*b7 - 7540678*b6 - 1366838*b5 - 1414829*b4 - 1803516*b3 + 3533338*b2 + 56981385*b1 + 90602246

## 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
 −9.58471 −7.77468 −5.32629 −2.66698 −0.575045 0.0622186 2.52979 4.78870 9.27510 10.2719
0 −9.58471 0 −5.00000 0 −21.8969 0 64.8667 0
1.2 0 −7.77468 0 −5.00000 0 16.1209 0 33.4457 0
1.3 0 −5.32629 0 −5.00000 0 −14.9292 0 1.36937 0
1.4 0 −2.66698 0 −5.00000 0 32.5699 0 −19.8872 0
1.5 0 −0.575045 0 −5.00000 0 24.8747 0 −26.6693 0
1.6 0 0.0622186 0 −5.00000 0 −14.4792 0 −26.9961 0
1.7 0 2.52979 0 −5.00000 0 −24.3398 0 −20.6002 0
1.8 0 4.78870 0 −5.00000 0 −0.156416 0 −4.06831 0
1.9 0 9.27510 0 −5.00000 0 −33.0838 0 59.0275 0
1.10 0 10.2719 0 −5.00000 0 7.31988 0 78.5119 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.bb 10
4.b odd 2 1 920.4.a.g 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.g 10 4.b odd 2 1
1840.4.a.bb 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{10} - T_{3}^{9} - 204 T_{3}^{8} + 42 T_{3}^{7} + 12958 T_{3}^{6} + 5872 T_{3}^{5} - 259871 T_{3}^{4} - 149461 T_{3}^{3} + 1222472 T_{3}^{2} + 627136 T_{3} - 43712$$ T3^10 - T3^9 - 204*T3^8 + 42*T3^7 + 12958*T3^6 + 5872*T3^5 - 259871*T3^4 - 149461*T3^3 + 1222472*T3^2 + 627136*T3 - 43712 $$T_{7}^{10} + 28 T_{7}^{9} - 1904 T_{7}^{8} - 56306 T_{7}^{7} + 1019676 T_{7}^{6} + 34617872 T_{7}^{5} - 127663083 T_{7}^{4} - 7079160712 T_{7}^{3} - 7124849036 T_{7}^{2} + \cdots + 56995990528$$ T7^10 + 28*T7^9 - 1904*T7^8 - 56306*T7^7 + 1019676*T7^6 + 34617872*T7^5 - 127663083*T7^4 - 7079160712*T7^3 - 7124849036*T7^2 + 363444632936*T7 + 56995990528

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10} - T^{9} - 204 T^{8} + \cdots - 43712$$
$5$ $$(T + 5)^{10}$$
$7$ $$T^{10} + 28 T^{9} + \cdots + 56995990528$$
$11$ $$T^{10} - 14 T^{9} + \cdots - 16\!\cdots\!56$$
$13$ $$T^{10} - 11 T^{9} + \cdots - 63035886761000$$
$17$ $$T^{10} - 68 T^{9} + \cdots + 11\!\cdots\!64$$
$19$ $$T^{10} + 114 T^{9} + \cdots - 10\!\cdots\!68$$
$23$ $$(T + 23)^{10}$$
$29$ $$T^{10} + 273 T^{9} + \cdots - 29\!\cdots\!32$$
$31$ $$T^{10} - 129 T^{9} + \cdots + 17\!\cdots\!12$$
$37$ $$T^{10} - 62 T^{9} + \cdots - 37\!\cdots\!76$$
$41$ $$T^{10} - 767 T^{9} + \cdots + 21\!\cdots\!22$$
$43$ $$T^{10} + 332 T^{9} + \cdots - 13\!\cdots\!48$$
$47$ $$T^{10} - 323 T^{9} + \cdots - 51\!\cdots\!00$$
$53$ $$T^{10} - 558 T^{9} + \cdots + 11\!\cdots\!00$$
$59$ $$T^{10} + 822 T^{9} + \cdots + 38\!\cdots\!36$$
$61$ $$T^{10} - 318 T^{9} + \cdots - 32\!\cdots\!00$$
$67$ $$T^{10} + 1152 T^{9} + \cdots + 37\!\cdots\!92$$
$71$ $$T^{10} + 1247 T^{9} + \cdots + 30\!\cdots\!00$$
$73$ $$T^{10} - 1941 T^{9} + \cdots + 31\!\cdots\!00$$
$79$ $$T^{10} + 3134 T^{9} + \cdots - 25\!\cdots\!64$$
$83$ $$T^{10} + 482 T^{9} + \cdots + 60\!\cdots\!04$$
$89$ $$T^{10} - 4734 T^{9} + \cdots + 13\!\cdots\!00$$
$97$ $$T^{10} - 2326 T^{9} + \cdots - 41\!\cdots\!00$$