Properties

 Label 1300.2.bs.d Level $1300$ Weight $2$ Character orbit 1300.bs Analytic conductor $10.381$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.bs (of order $$12$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$10.3805522628$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + 6399 x^{4} + 666 x^{2} + 9$$ x^20 + 30*x^18 + 371*x^16 + 2460*x^14 + 9517*x^12 + 21870*x^10 + 29001*x^8 + 20400*x^6 + 6399*x^4 + 666*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 260) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{15} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{3} + 1) q^{3} + (\beta_{8} + \beta_{7} + \beta_{5} - \beta_{2}) q^{7} + (\beta_{16} - \beta_{15} + \beta_{12} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{3}) q^{9}+O(q^{10})$$ q + (-b15 + b11 + b9 - b7 - b6 + b3 + 1) * q^3 + (b8 + b7 + b5 - b2) * q^7 + (b16 - b15 + b12 - b8 - b6 + b5 + b3) * q^9 $$q + ( - \beta_{15} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{3} + 1) q^{3} + (\beta_{8} + \beta_{7} + \beta_{5} - \beta_{2}) q^{7} + (\beta_{16} - \beta_{15} + \beta_{12} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{3}) q^{9} + (\beta_{19} + \beta_{18} + 2 \beta_{14} + \beta_{12} + \beta_{9} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots + 1) q^{11}+ \cdots + ( - 2 \beta_{19} - 3 \beta_{18} - \beta_{17} + 4 \beta_{16} + 2 \beta_{15} + \beta_{14} + 2 \beta_{13} + \cdots + 2) q^{99}+O(q^{100})$$ q + (-b15 + b11 + b9 - b7 - b6 + b3 + 1) * q^3 + (b8 + b7 + b5 - b2) * q^7 + (b16 - b15 + b12 - b8 - b6 + b5 + b3) * q^9 + (b19 + b18 + 2*b14 + b12 + b9 + b5 - b4 + 2*b3 + b2 + b1 + 1) * q^11 + (b18 - b15 - b13 + b10 + 2*b9 - b7 - 2*b6 + b4 + 3*b3 + b1 + 1) * q^13 + (b18 - b14 + b13 + b12 - b11 - b10 + b7 - b6 + b5 + b4 + 2*b3 + 2*b1) * q^17 + (b18 + b17 - b12 + b10 + b8 - b6 - b4 - 2*b2 + 2) * q^19 + (b15 - b14 - b13 - b12 + b10 - b9 + b8 + b7 + b3 - b2 + b1 - 1) * q^21 + (-b18 + b15 - 2*b14 + b11 - b9 + b6 + b5 + b4 - b3 - 2) * q^23 + (b19 - b17 - b16 + 3*b15 + b13 - 2*b11 - b10 - 3*b9 + b7 + 4*b6 + 3*b5 - b4 + b2) * q^27 + (b19 - b18 - b17 - b16 + 3*b15 + b14 + 2*b13 + 2*b12 - 2*b11 - 2*b10 - 3*b9 + 2*b7 + 4*b6 + b5 - 2*b4 - 2*b3 + b2) * q^29 + (2*b17 + 2*b14 + 2*b11 + 2*b8 + b5 - b4 - b1 + 1) * q^31 + (-2*b19 + 2*b18 + b17 + 3*b16 - 3*b15 + b14 - b13 + 2*b12 + 4*b11 + 2*b10 + 3*b9 + b8 - 3*b6 + b5 + 3*b4 + 4*b3 + 3*b1) * q^33 + (-b18 + 2*b16 - b15 + b12 + 2*b11 + 3*b10 + b7 - b6 - b5 - 2*b4 - 2*b2 + 1) * q^37 + (-3*b19 - 2*b18 - b17 + b16 + 2*b15 - 5*b14 + b13 - b10 - 4*b9 + 3*b7 + 3*b6 - b5 + 4*b4 - 3*b3 - b2 + 3*b1 - 6) * q^39 + (-b19 - b18 + b17 + 2*b16 - 3*b15 + 2*b14 + b12 + 3*b11 + b10 - b6 + b5 + 3*b4 + 2*b3 - 2*b2 + b1 + 2) * q^41 + (b19 + b18 - b13 - b12 - b11 - b10 - b7 - b6 + 2*b4 + b2 - 2) * q^43 + (b18 + 2*b17 - b15 + 4*b14 - b13 + 2*b11 + b10 + 2*b9 + b8 - b7 - b6 + b5 + b3 - b1 + 2) * q^47 + (-b16 - b15 + b14 - b12 + b11 + b10 + b9 + 2*b8 + b6 + 2*b5 - b3 - 2*b2 - 2*b1 + 2) * q^49 + (-b16 - b15 - b12 + 2*b11 + 4*b10 + 3*b9 + 4*b8 - 4*b6 + 4*b5 + 3*b4 + 5*b3 - 6*b2 + 3*b1 + 6) * q^51 + (-b15 + 6*b14 - b13 + b12 - b10 + b9 - b8 - b7 + 4*b5 - b4 + b3 + 3*b2) * q^53 + (-b19 - 3*b18 + 2*b16 - b14 + b13 - b12 + b11 + b10 - 3*b9 - b8 + 2*b6 - 4*b5 - 4*b3 - 3*b2 + 3) * q^57 + (-3*b18 - 2*b17 - b16 + 2*b15 + 2*b13 - 3*b12 - 2*b11 - 3*b9 + 2*b8 + b7 + 3*b6 - 2*b5 - 4*b4 - 4*b3 - 3*b2 - 2*b1 + 1) * q^59 + (-b19 + b18 + b17 + 2*b16 - 3*b15 - 4*b14 - b13 + 3*b12 + 4*b11 + 4*b9 - 2*b7 - 3*b6 + b5 + 6*b3 + 4*b1 - 3) * q^61 + (b18 + 2*b17 - 2*b16 + b15 - 3*b14 - b12 + 2*b11 + b10 + 2*b9 + 2*b8 - 3*b6 - 3*b5 - b4 + 2*b3 - b2 + 2*b1 + 1) * q^63 + (2*b19 - 2*b17 - b16 + 3*b14 - b11 + b9 - 2*b7 + 4*b5 + 2*b3 + b2 - b1 + 3) * q^67 + (-2*b16 - b15 - b14 - b12 - b11 + 2*b9 - b7 - 2*b6 - 3*b5 - 3*b4 - b3 - b1) * q^69 + (b19 - 2*b16 + b14 - b13 + b11 + b10 - 2*b7 - 2*b6 - b5 - 2*b4 + b3 - b1 + 2) * q^71 + (b19 + 2*b18 - b16 + b14 + 3*b12 - b11 + b10 + 2*b9 - b8 + 2*b7 + b6 + b5 + 4*b4 + 3*b3 + 2*b2 + 2*b1 + 1) * q^73 + (-4*b19 - 2*b18 + b17 + 2*b16 - 3*b15 - b14 - 3*b13 + b12 + 2*b11 - 3*b10 - 2*b8 - 3*b7 - 2*b6 - 5*b5 - b4 - b3 + 3*b2 + b1 - 4) * q^77 + (3*b19 + 2*b18 + 4*b17 - 2*b16 - b15 + 5*b14 - 2*b13 - 2*b12 + 2*b11 + b10 + 4*b9 + b8 - 2*b7 - 4*b6 - 3*b4 + 2*b3 - 2*b2 - 2*b1 + 6) * q^79 + (-b19 + b17 - 4*b16 + 2*b14 - 2*b12 + 5*b11 + 2*b10 + 4*b9 + 4*b8 + b5 + 2*b4 - b3 - b2 + b1) * q^81 + (4*b19 - b16 + 3*b15 - 2*b14 - b12 - 2*b11 + b10 - b9 + b8 - b5 - 7*b4 + 2*b3 - 3*b2 - b1 + 4) * q^83 + (4*b18 + 4*b17 - 4*b16 - 3*b15 - b14 - 2*b13 - 4*b12 + 4*b11 + 2*b10 + 7*b9 + 4*b8 - 3*b7 - 7*b6 - b5 + 3*b4 + 3*b3 - 3*b2 + 1) * q^87 + (b19 - 2*b18 - 2*b17 + 2*b16 + 2*b15 - b14 + 2*b13 + b12 - 2*b11 - 2*b10 - 7*b9 - 2*b8 + b7 + 4*b6 - 4*b5 - 3*b4 - 4*b3 + b2 - b1 - 4) * q^89 + (-b19 - b18 + 3*b17 - b16 - 3*b15 + b14 - b12 + 2*b11 + b9 - b8 - 3*b7 - 3*b6 - 3*b5 - 2*b4 - b2 - 2) * q^91 + (-2*b19 - b18 + 2*b17 + 2*b16 - 5*b15 + 2*b14 - 4*b13 - 3*b12 + 4*b11 + b10 + 4*b9 - 4*b7 - 6*b6 - 4*b5 + b3 + b2 - 3*b1 + 5) * q^93 + (-b17 - b16 + 2*b15 - 5*b14 + b13 - 3*b11 - 3*b9 + 2*b7 + 2*b6 - 2*b3 - 6) * q^97 + (-2*b19 - 3*b18 - b17 + 4*b16 + 2*b15 + b14 + 2*b13 - b12 + 2*b10 - 5*b9 + b8 + 2*b7 + 3*b6 - 2*b5 - 3*b4 - 4*b3 - 3*b2 - 2*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 2 q^{3} - 6 q^{7} - 12 q^{9}+O(q^{10})$$ 20 * q + 2 * q^3 - 6 * q^7 - 12 * q^9 $$20 q + 2 q^{3} - 6 q^{7} - 12 q^{9} - 8 q^{13} + 20 q^{19} - 12 q^{21} - 6 q^{23} + 20 q^{27} + 24 q^{29} + 8 q^{31} + 10 q^{33} + 4 q^{39} + 6 q^{41} - 38 q^{43} + 14 q^{49} - 30 q^{53} + 76 q^{57} - 24 q^{59} - 32 q^{61} + 24 q^{63} - 22 q^{67} - 16 q^{69} + 44 q^{73} + 12 q^{77} + 2 q^{81} - 38 q^{87} - 30 q^{89} - 72 q^{91} + 48 q^{93} - 46 q^{97} + 20 q^{99}+O(q^{100})$$ 20 * q + 2 * q^3 - 6 * q^7 - 12 * q^9 - 8 * q^13 + 20 * q^19 - 12 * q^21 - 6 * q^23 + 20 * q^27 + 24 * q^29 + 8 * q^31 + 10 * q^33 + 4 * q^39 + 6 * q^41 - 38 * q^43 + 14 * q^49 - 30 * q^53 + 76 * q^57 - 24 * q^59 - 32 * q^61 + 24 * q^63 - 22 * q^67 - 16 * q^69 + 44 * q^73 + 12 * q^77 + 2 * q^81 - 38 * q^87 - 30 * q^89 - 72 * q^91 + 48 * q^93 - 46 * q^97 + 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + 6399 x^{4} + 666 x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( - 3047 \nu^{18} - 80534 \nu^{16} - 834504 \nu^{14} - 4296019 \nu^{12} - 11423658 \nu^{10} - 14767004 \nu^{8} - 8552045 \nu^{6} - 4645944 \nu^{4} + \cdots + 99723 ) / 1134948$$ (-3047*v^18 - 80534*v^16 - 834504*v^14 - 4296019*v^12 - 11423658*v^10 - 14767004*v^8 - 8552045*v^6 - 4645944*v^4 - 2787912*v^2 + 1134948*v + 99723) / 1134948 $$\beta_{2}$$ $$=$$ $$( 19367 \nu^{19} + 69462 \nu^{18} + 805551 \nu^{17} + 1878198 \nu^{16} + 13356712 \nu^{15} + 19986924 \nu^{14} + 115467750 \nu^{13} + 105446208 \nu^{12} + \cdots - 217278 ) / 13619376$$ (19367*v^19 + 69462*v^18 + 805551*v^17 + 1878198*v^16 + 13356712*v^15 + 19986924*v^14 + 115467750*v^13 + 105446208*v^12 + 567305219*v^11 + 278388330*v^10 + 1610670111*v^9 + 285815274*v^8 + 2558952720*v^7 - 135046284*v^6 + 2078820744*v^5 - 420626628*v^4 + 718736751*v^3 - 175581882*v^2 + 83635389*v - 217278) / 13619376 $$\beta_{3}$$ $$=$$ $$( 15705 \nu^{19} + 7449 \nu^{18} + 402397 \nu^{17} + 223669 \nu^{16} + 3930438 \nu^{15} + 2731870 \nu^{14} + 17694998 \nu^{13} + 17453738 \nu^{12} + 30393363 \nu^{11} + \cdots + 661521 ) / 4539792$$ (15705*v^19 + 7449*v^18 + 402397*v^17 + 223669*v^16 + 3930438*v^15 + 2731870*v^14 + 17694998*v^13 + 17453738*v^12 + 30393363*v^11 + 62402747*v^10 - 29388941*v^9 + 124660699*v^8 - 180768738*v^7 + 135753310*v^6 - 223758558*v^5 + 83549682*v^4 - 88480923*v^3 + 29953629*v^2 - 7543635*v + 661521) / 4539792 $$\beta_{4}$$ $$=$$ $$( - 66482 \nu^{19} - 47115 \nu^{18} - 2012742 \nu^{17} - 1207191 \nu^{16} - 25148026 \nu^{15} - 11791314 \nu^{14} - 168552744 \nu^{13} - 53084994 \nu^{12} + \cdots + 15821217 ) / 13619376$$ (-66482*v^19 - 47115*v^18 - 2012742*v^17 - 1207191*v^16 - 25148026*v^15 - 11791314*v^14 - 168552744*v^13 - 53084994*v^12 - 658485308*v^11 - 91180089*v^10 - 1522503288*v^9 + 88166823*v^8 - 2016646506*v^7 + 542306214*v^6 - 1407545070*v^5 + 671275674*v^4 - 453293982*v^3 + 265442769*v^2 - 61004484*v + 15821217) / 13619376 $$\beta_{5}$$ $$=$$ $$( - 66482 \nu^{19} + 47115 \nu^{18} - 2012742 \nu^{17} + 1207191 \nu^{16} - 25148026 \nu^{15} + 11791314 \nu^{14} - 168552744 \nu^{13} + 53084994 \nu^{12} + \cdots - 15821217 ) / 13619376$$ (-66482*v^19 + 47115*v^18 - 2012742*v^17 + 1207191*v^16 - 25148026*v^15 + 11791314*v^14 - 168552744*v^13 + 53084994*v^12 - 658485308*v^11 + 91180089*v^10 - 1522503288*v^9 - 88166823*v^8 - 2016646506*v^7 - 542306214*v^6 - 1407545070*v^5 - 671275674*v^4 - 453293982*v^3 - 265442769*v^2 - 61004484*v - 15821217) / 13619376 $$\beta_{6}$$ $$=$$ $$( 66482 \nu^{19} - 83679 \nu^{18} + 2012742 \nu^{17} - 2173599 \nu^{16} + 25148026 \nu^{15} - 21805362 \nu^{14} + 168552744 \nu^{13} - 104637222 \nu^{12} + \cdots + 17017893 ) / 13619376$$ (66482*v^19 - 83679*v^18 + 2012742*v^17 - 2173599*v^16 + 25148026*v^15 - 21805362*v^14 + 168552744*v^13 - 104637222*v^12 + 658485308*v^11 - 228263985*v^10 + 1522503288*v^9 - 89037225*v^8 + 2016646506*v^7 + 439681674*v^6 + 1407545070*v^5 + 615524346*v^4 + 453293982*v^3 + 231987825*v^2 + 47385108*v + 17017893) / 13619376 $$\beta_{7}$$ $$=$$ $$( - 165961 \nu^{19} - 181245 \nu^{18} - 5106855 \nu^{17} - 5590593 \nu^{16} - 65333654 \nu^{15} - 71485608 \nu^{14} - 453746592 \nu^{13} - 492632442 \nu^{12} + \cdots - 45388899 ) / 13619376$$ (-165961*v^19 - 181245*v^18 - 5106855*v^17 - 5590593*v^16 - 65333654*v^15 - 71485608*v^14 - 453746592*v^13 - 492632442*v^12 - 1873735009*v^11 - 1986210429*v^10 - 4736472213*v^9 - 4737866193*v^8 - 7257146256*v^7 - 6391057812*v^6 - 6384094374*v^5 - 4308088248*v^4 - 2800812015*v^3 - 1089813825*v^2 - 406515213*v - 45388899) / 13619376 $$\beta_{8}$$ $$=$$ $$( 27888 \nu^{19} + 17400 \nu^{18} + 876466 \nu^{17} + 537244 \nu^{16} + 11480801 \nu^{15} + 6830496 \nu^{14} + 81710792 \nu^{13} + 46336394 \nu^{12} + \cdots + 4472424 ) / 2269896$$ (27888*v^19 + 17400*v^18 + 876466*v^17 + 537244*v^16 + 11480801*v^15 + 6830496*v^14 + 81710792*v^13 + 46336394*v^12 + 344915287*v^11 + 181525278*v^10 + 883607113*v^9 + 415161358*v^8 + 1347639143*v^7 + 534368214*v^6 + 1146604077*v^5 + 353459730*v^4 + 472806492*v^3 + 99118266*v^2 + 69453771*v + 4472424) / 2269896 $$\beta_{9}$$ $$=$$ $$( - 84623 \nu^{19} - 64035 \nu^{18} - 2581149 \nu^{17} - 1764837 \nu^{16} - 32554318 \nu^{15} - 19533108 \nu^{14} - 220915596 \nu^{13} - 112032558 \nu^{12} + \cdots + 12376215 ) / 6809688$$ (-84623*v^19 - 64035*v^18 - 2581149*v^17 - 1764837*v^16 - 32554318*v^15 - 19533108*v^14 - 220915596*v^13 - 112032558*v^12 - 878530535*v^11 - 358798875*v^10 - 2089554123*v^9 - 646469163*v^8 - 2907434700*v^7 - 630401196*v^6 - 2213128230*v^5 - 297574542*v^4 - 806417469*v^3 - 34492167*v^2 - 114443523*v + 12376215) / 6809688 $$\beta_{10}$$ $$=$$ $$( 27888 \nu^{19} - 17400 \nu^{18} + 876466 \nu^{17} - 537244 \nu^{16} + 11480801 \nu^{15} - 6830496 \nu^{14} + 81710792 \nu^{13} - 46336394 \nu^{12} + \cdots - 4472424 ) / 2269896$$ (27888*v^19 - 17400*v^18 + 876466*v^17 - 537244*v^16 + 11480801*v^15 - 6830496*v^14 + 81710792*v^13 - 46336394*v^12 + 344915287*v^11 - 181525278*v^10 + 883607113*v^9 - 415161358*v^8 + 1347639143*v^7 - 534368214*v^6 + 1146604077*v^5 - 353459730*v^4 + 472806492*v^3 - 99118266*v^2 + 69453771*v - 4472424) / 2269896 $$\beta_{11}$$ $$=$$ $$( - 84623 \nu^{19} + 64035 \nu^{18} - 2581149 \nu^{17} + 1764837 \nu^{16} - 32554318 \nu^{15} + 19533108 \nu^{14} - 220915596 \nu^{13} + 112032558 \nu^{12} + \cdots - 12376215 ) / 6809688$$ (-84623*v^19 + 64035*v^18 - 2581149*v^17 + 1764837*v^16 - 32554318*v^15 + 19533108*v^14 - 220915596*v^13 + 112032558*v^12 - 878530535*v^11 + 358798875*v^10 - 2089554123*v^9 + 646469163*v^8 - 2907434700*v^7 + 630401196*v^6 - 2213128230*v^5 + 297574542*v^4 - 806417469*v^3 + 34492167*v^2 - 114443523*v - 12376215) / 6809688 $$\beta_{12}$$ $$=$$ $$( 165961 \nu^{19} - 320169 \nu^{18} + 5106855 \nu^{17} - 9346989 \nu^{16} + 65333654 \nu^{15} - 111459456 \nu^{14} + 453746592 \nu^{13} - 703524858 \nu^{12} + \cdots - 31334967 ) / 13619376$$ (165961*v^19 - 320169*v^18 + 5106855*v^17 - 9346989*v^16 + 65333654*v^15 - 111459456*v^14 + 453746592*v^13 - 703524858*v^12 + 1873735009*v^11 - 2542987089*v^10 + 4736472213*v^9 - 5309496741*v^8 + 7257146256*v^7 - 6120965244*v^6 + 6384094374*v^5 - 3466834992*v^4 + 2800812015*v^3 - 738650061*v^2 + 406515213*v - 31334967) / 13619376 $$\beta_{13}$$ $$=$$ $$( 282319 \nu^{19} + 174918 \nu^{18} + 8264211 \nu^{17} + 5018946 \nu^{16} + 98942810 \nu^{15} + 58624560 \nu^{14} + 628204794 \nu^{13} + 361205580 \nu^{12} + \cdots + 42156486 ) / 13619376$$ (282319*v^19 + 174918*v^18 + 8264211*v^17 + 5018946*v^16 + 98942810*v^15 + 58624560*v^14 + 628204794*v^13 + 361205580*v^12 + 2289442933*v^11 + 1270292058*v^10 + 4826080581*v^9 + 2575434930*v^8 + 5582082666*v^7 + 2899824276*v^6 + 3035691966*v^5 + 1691072568*v^4 + 470305035*v^3 + 490895010*v^2 - 20563029*v + 42156486) / 13619376 $$\beta_{14}$$ $$=$$ $$( - 353 \nu^{19} - 10585 \nu^{17} - 130732 \nu^{15} - 864458 \nu^{13} - 3326801 \nu^{11} - 7574149 \nu^{9} - 9886332 \nu^{7} - 6763380 \nu^{5} - 1984833 \nu^{3} + \cdots - 8376 ) / 16752$$ (-353*v^19 - 10585*v^17 - 130732*v^15 - 864458*v^13 - 3326801*v^11 - 7574149*v^9 - 9886332*v^7 - 6763380*v^5 - 1984833*v^3 - 145143*v - 8376) / 16752 $$\beta_{15}$$ $$=$$ $$( 397382 \nu^{19} + 50979 \nu^{18} + 11805666 \nu^{17} + 1600971 \nu^{16} + 144498028 \nu^{15} + 20826990 \nu^{14} + 949107816 \nu^{13} + 145593894 \nu^{12} + \cdots + 24710427 ) / 13619376$$ (397382*v^19 + 50979*v^18 + 11805666*v^17 + 1600971*v^16 + 144498028*v^15 + 20826990*v^14 + 949107816*v^13 + 145593894*v^12 + 3650325002*v^11 + 592149393*v^10 + 8408441922*v^9 + 1413069225*v^8 + 11346635628*v^7 + 1885023858*v^6 + 8306063952*v^5 + 1227051090*v^4 + 2736156870*v^3 + 281619531*v^2 + 216209178*v + 24710427) / 13619376 $$\beta_{16}$$ $$=$$ $$( 210017 \nu^{19} + 54111 \nu^{18} + 6333330 \nu^{17} + 1652277 \nu^{16} + 78814861 \nu^{15} + 20796687 \nu^{14} + 526330083 \nu^{13} + 139985934 \nu^{12} + \cdots + 22008474 ) / 6809688$$ (210017*v^19 + 54111*v^18 + 6333330*v^17 + 1652277*v^16 + 78814861*v^15 + 20796687*v^14 + 526330083*v^13 + 139985934*v^12 + 2050478141*v^11 + 545967756*v^10 + 4730874033*v^9 + 1247438982*v^8 + 6231655746*v^7 + 1606404909*v^6 + 4213523391*v^5 + 1060506387*v^4 + 1133458272*v^3 + 298541511*v^2 + 54298512*v + 22008474) / 6809688 $$\beta_{17}$$ $$=$$ $$( 67147 \nu^{19} - 27215 \nu^{18} + 2045559 \nu^{17} - 784089 \nu^{16} + 25759345 \nu^{15} - 9210622 \nu^{14} + 174392772 \nu^{13} - 57243966 \nu^{12} + \cdots + 7414695 ) / 2269896$$ (67147*v^19 - 27215*v^18 + 2045559*v^17 - 784089*v^16 + 25759345*v^15 - 9210622*v^14 + 174392772*v^13 - 57243966*v^12 + 690311702*v^11 - 204225755*v^10 + 1624006860*v^9 - 423532263*v^8 + 2196992015*v^7 - 487184352*v^6 + 1551072951*v^5 - 263519850*v^4 + 455334531*v^3 - 30017193*v^2 + 30631614*v + 7414695) / 2269896 $$\beta_{18}$$ $$=$$ $$( 938081 \nu^{19} + 181527 \nu^{18} + 28105911 \nu^{17} + 5420187 \nu^{16} + 346829194 \nu^{15} + 66687072 \nu^{14} + 2291248896 \nu^{13} + 439458966 \nu^{12} + \cdots - 35447103 ) / 13619376$$ (938081*v^19 + 181527*v^18 + 28105911*v^17 + 5420187*v^16 + 346829194*v^15 + 66687072*v^14 + 2291248896*v^13 + 439458966*v^12 + 8805945593*v^11 + 1683203871*v^10 + 19989440181*v^9 + 3780968571*v^8 + 25876910064*v^7 + 4712105964*v^6 + 17296540278*v^5 + 2752673256*v^4 + 4803350499*v^3 + 417021795*v^2 + 364306149*v - 35447103) / 13619376 $$\beta_{19}$$ $$=$$ $$( - 1080005 \nu^{19} - 98094 \nu^{18} - 32065821 \nu^{17} - 2808162 \nu^{16} - 391463830 \nu^{15} - 32618304 \nu^{14} - 2554928178 \nu^{13} + \cdots - 15698898 ) / 13619376$$ (-1080005*v^19 - 98094*v^18 - 32065821*v^17 - 2808162*v^16 - 391463830*v^15 - 32618304*v^14 - 2554928178*v^13 - 198678888*v^12 - 9697129103*v^11 - 683329482*v^10 - 21783539727*v^9 - 1324902402*v^8 - 28119598350*v^7 - 1342717644*v^6 - 19138877934*v^5 - 555775416*v^4 - 5742061569*v^3 - 16176762*v^2 - 548621685*v - 15698898) / 13619376
 $$\nu$$ $$=$$ $$( -\beta_{6} - \beta_{5} + \beta_1 ) / 2$$ (-b6 - b5 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( - \beta_{19} - \beta_{18} + \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - 6 ) / 2$$ (-b19 - b18 + b16 - b15 - b14 - b13 - b12 + b10 + b9 - b8 - b7 - b6 - b5 - 6) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{19} + \beta_{18} + 2 \beta_{17} - \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - 3 \beta_{12} + 3 \beta_{10} + \beta_{9} + 3 \beta_{8} + \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - 5 \beta _1 + 4 ) / 2$$ (b19 + b18 + 2*b17 - b16 - b15 - b14 - b13 - 3*b12 + 3*b10 + b9 + 3*b8 + b7 + 2*b6 + 4*b5 - 2*b4 + 2*b3 - 6*b2 - 5*b1 + 4) / 2 $$\nu^{4}$$ $$=$$ $$( 9 \beta_{19} + 7 \beta_{18} - 13 \beta_{16} + 15 \beta_{15} + 9 \beta_{14} + 11 \beta_{13} + 5 \beta_{12} - 6 \beta_{11} - 9 \beta_{10} - 11 \beta_{9} + 9 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + 5 \beta_{5} - 4 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} - 6 \beta _1 + 34 ) / 2$$ (9*b19 + 7*b18 - 13*b16 + 15*b15 + 9*b14 + 11*b13 + 5*b12 - 6*b11 - 9*b10 - 11*b9 + 9*b8 + 11*b7 + 11*b6 + 5*b5 - 4*b4 - 8*b3 - 2*b2 - 6*b1 + 34) / 2 $$\nu^{5}$$ $$=$$ $$( - 13 \beta_{19} - 11 \beta_{18} - 22 \beta_{17} + 11 \beta_{16} + 9 \beta_{15} + 17 \beta_{14} + 11 \beta_{13} + 29 \beta_{12} - 4 \beta_{11} - 29 \beta_{10} - 15 \beta_{9} - 29 \beta_{8} - 7 \beta_{7} + 3 \beta_{6} - 15 \beta_{5} + 30 \beta_{4} - 24 \beta_{3} + \cdots - 40 ) / 2$$ (-13*b19 - 11*b18 - 22*b17 + 11*b16 + 9*b15 + 17*b14 + 11*b13 + 29*b12 - 4*b11 - 29*b10 - 15*b9 - 29*b8 - 7*b7 + 3*b6 - 15*b5 + 30*b4 - 24*b3 + 60*b2 + 30*b1 - 40) / 2 $$\nu^{6}$$ $$=$$ $$- 38 \beta_{19} - 29 \beta_{18} + 58 \beta_{16} - 67 \beta_{15} - 38 \beta_{14} - 47 \beta_{13} - 16 \beta_{12} + 34 \beta_{11} + 35 \beta_{10} + 42 \beta_{9} - 35 \beta_{8} - 45 \beta_{7} - 40 \beta_{6} - 13 \beta_{5} + 20 \beta_{4} + 38 \beta_{3} + \cdots - 119$$ -38*b19 - 29*b18 + 58*b16 - 67*b15 - 38*b14 - 47*b13 - 16*b12 + 34*b11 + 35*b10 + 42*b9 - 35*b8 - 45*b7 - 40*b6 - 13*b5 + 20*b4 + 38*b3 + 9*b2 + 36*b1 - 119 $$\nu^{7}$$ $$=$$ $$( 118 \beta_{19} + 106 \beta_{18} + 212 \beta_{17} - 106 \beta_{16} - 94 \beta_{15} - 166 \beta_{14} - 106 \beta_{13} - 242 \beta_{12} + 74 \beta_{11} + 246 \beta_{10} + 180 \beta_{9} + 246 \beta_{8} + 30 \beta_{7} - 119 \beta_{6} + \cdots + 352 ) / 2$$ (118*b19 + 106*b18 + 212*b17 - 106*b16 - 94*b15 - 166*b14 - 106*b13 - 242*b12 + 74*b11 + 246*b10 + 180*b9 + 246*b8 + 30*b7 - 119*b6 + 29*b5 - 316*b4 + 240*b3 - 512*b2 - 199*b1 + 352) / 2 $$\nu^{8}$$ $$=$$ $$( 637 \beta_{19} + 499 \beta_{18} - 951 \beta_{16} + 1089 \beta_{15} + 637 \beta_{14} + 775 \beta_{13} + 247 \beta_{12} - 624 \beta_{11} - 537 \beta_{10} - 603 \beta_{9} + 537 \beta_{8} + 699 \beta_{7} + 541 \beta_{6} + 151 \beta_{5} + \cdots + 1830 ) / 2$$ (637*b19 + 499*b18 - 951*b16 + 1089*b15 + 637*b14 + 775*b13 + 247*b12 - 624*b11 - 537*b10 - 603*b9 + 537*b8 + 699*b7 + 541*b6 + 151*b5 - 338*b4 - 590*b3 - 138*b2 - 686*b1 + 1830) / 2 $$\nu^{9}$$ $$=$$ $$( - 975 \beta_{19} - 971 \beta_{18} - 1942 \beta_{17} + 971 \beta_{16} + 967 \beta_{15} + 1443 \beta_{14} + 971 \beta_{13} + 1969 \beta_{12} - 912 \beta_{11} - 2041 \beta_{10} - 1883 \beta_{9} - 2041 \beta_{8} - 27 \beta_{7} + \cdots - 2992 ) / 2$$ (-975*b19 - 971*b18 - 1942*b17 + 971*b16 + 967*b15 + 1443*b14 + 971*b13 + 1969*b12 - 912*b11 - 2041*b10 - 1883*b9 - 2041*b8 - 27*b7 + 1508*b6 + 248*b5 + 2914*b4 - 2228*b3 + 4224*b2 + 1405*b1 - 2992) / 2 $$\nu^{10}$$ $$=$$ $$( - 5317 \beta_{19} - 4287 \beta_{18} + 7635 \beta_{16} - 8665 \beta_{15} - 5317 \beta_{14} - 6347 \beta_{13} - 2055 \beta_{12} + 5386 \beta_{11} + 4159 \beta_{10} + 4309 \beta_{9} - 4159 \beta_{8} - 5403 \beta_{7} + \cdots - 14638 ) / 2$$ (-5317*b19 - 4287*b18 + 7635*b16 - 8665*b15 - 5317*b14 - 6347*b13 - 2055*b12 + 5386*b11 + 4159*b10 + 4309*b9 - 4159*b8 - 5403*b7 - 3627*b6 - 981*b5 + 2782*b4 + 4398*b3 + 1050*b2 + 6068*b1 - 14638) / 2 $$\nu^{11}$$ $$=$$ $$( 7845 \beta_{19} + 8595 \beta_{18} + 17190 \beta_{17} - 8615 \beta_{16} - 9365 \beta_{15} - 12173 \beta_{14} - 8595 \beta_{13} - 15989 \beta_{12} + 9474 \beta_{11} + 16869 \beta_{10} + 18089 \beta_{9} + \cdots + 25044 ) / 2$$ (7845*b19 + 8595*b18 + 17190*b17 - 8615*b16 - 9365*b15 - 12173*b14 - 8595*b13 - 15989*b12 + 9474*b11 + 16869*b10 + 18089*b9 + 16869*b8 - 1221*b7 - 15451*b6 - 4575*b5 - 25378*b4 + 19814*b3 - 34582*b2 - 10354*b1 + 25044) / 2 $$\nu^{12}$$ $$=$$ $$22087 \beta_{19} + 18229 \beta_{18} - 30559 \beta_{16} + 34417 \beta_{15} + 22087 \beta_{14} + 25945 \beta_{13} + 8707 \beta_{12} - 22676 \beta_{11} - 16317 \beta_{10} - 15599 \beta_{9} + 16317 \beta_{8} + \cdots + 59485$$ 22087*b19 + 18229*b18 - 30559*b16 + 34417*b15 + 22087*b14 + 25945*b13 + 8707*b12 - 22676*b11 - 16317*b10 - 15599*b9 + 16317*b8 + 21037*b7 + 12295*b6 + 3455*b5 - 11436*b4 - 16418*b3 - 4088*b2 - 25980*b1 + 59485 $$\nu^{13}$$ $$=$$ $$( - 62818 \beta_{19} - 74282 \beta_{18} - 148564 \beta_{17} + 74742 \beta_{16} + 86206 \beta_{15} + 101842 \beta_{14} + 74282 \beta_{13} + 130194 \beta_{12} - 90060 \beta_{11} - 139346 \beta_{10} + \cdots - 207904 ) / 2$$ (-62818*b19 - 74282*b18 - 148564*b17 + 74742*b16 + 86206*b15 + 101842*b14 + 74282*b13 + 130194*b12 - 90060*b11 - 139346*b10 - 164802*b9 - 139346*b8 + 18830*b7 + 144617*b6 + 51109*b5 + 215096*b4 - 171504*b3 + 282868*b2 + 78689*b1 - 207904) / 2 $$\nu^{14}$$ $$=$$ $$( - 365529 \beta_{19} - 307025 \beta_{18} + 490501 \beta_{16} - 549005 \beta_{15} - 365529 \beta_{14} - 424033 \beta_{13} - 147445 \beta_{12} + 377384 \beta_{11} + 259089 \beta_{10} + \cdots - 973110 ) / 2$$ (-365529*b19 - 307025*b18 + 490501*b16 - 549005*b15 - 365529*b14 - 424033*b13 - 147445*b12 + 377384*b11 + 259089*b10 + 230125*b9 - 259089*b8 - 330921*b7 - 169921*b6 - 51253*b5 + 188460*b4 + 248624*b3 + 65148*b2 + 437588*b1 - 973110) / 2 $$\nu^{15}$$ $$=$$ $$( 504149 \beta_{19} + 631589 \beta_{18} + 1263178 \beta_{17} - 638233 \beta_{16} - 765673 \beta_{15} - 849325 \beta_{14} - 631589 \beta_{13} - 1063299 \beta_{12} + 813496 \beta_{11} + \cdots + 1717868 ) / 2$$ (504149*b19 + 631589*b18 + 1263178*b17 - 638233*b16 - 765673*b15 - 849325*b14 - 631589*b13 - 1063299*b12 + 813496*b11 + 1150767*b10 + 1451729*b9 + 1150767*b8 - 206523*b7 - 1289888*b6 - 493334*b5 - 1797930*b4 + 1458746*b3 - 2315522*b2 - 611523*b1 + 1717868) / 2 $$\nu^{16}$$ $$=$$ $$( 3015847 \beta_{19} + 2566165 \beta_{18} - 3952079 \beta_{16} + 4401761 \beta_{15} + 3015847 \beta_{14} + 3465529 \beta_{13} + 1241503 \beta_{12} - 3121006 \beta_{11} - 2076279 \beta_{10} + \cdots + 7979906 ) / 2$$ (3015847*b19 + 2566165*b18 - 3952079*b16 + 4401761*b15 + 3015847*b14 + 3465529*b13 + 1241503*b12 - 3121006*b11 - 2076279*b10 - 1730437*b9 + 2076279*b8 + 2627417*b7 + 1200685*b6 + 392727*b5 - 1555096*b4 - 1913372*b3 - 527458*b2 - 3650758*b1 + 7979906) / 2 $$\nu^{17}$$ $$=$$ $$( - 4062775 \beta_{19} - 5311317 \beta_{18} - 10622634 \beta_{17} + 5389093 \beta_{16} + 6637635 \beta_{15} + 7067203 \beta_{14} + 5311317 \beta_{13} + 8703539 \beta_{12} + \cdots - 14156328 ) / 2$$ (-4062775*b19 - 5311317*b18 - 10622634*b17 + 5389093*b16 + 6637635*b15 + 7067203*b14 + 5311317*b13 + 8703539*b12 - 7121556*b11 - 9499063*b10 - 12510649*b9 - 9499063*b8 + 1996871*b7 + 11182311*b6 + 4457801*b5 + 14915902*b4 - 12266320*b3 + 18972988*b2 + 4829416*b1 - 14156328) / 2 $$\nu^{18}$$ $$=$$ $$- 12417331 \beta_{19} - 10666792 \beta_{18} + 15983777 \beta_{16} - 17734316 \beta_{15} - 12417331 \beta_{14} - 14167870 \beta_{13} - 5196791 \beta_{12} + 12862622 \beta_{11} + \cdots - 32752987$$ -12417331*b19 - 10666792*b18 + 15983777*b16 - 17734316*b15 - 12417331*b14 - 14167870*b13 - 5196791*b12 + 12862622*b11 + 8378044*b10 + 6622233*b9 - 8378044*b8 - 10513776*b7 - 4341311*b6 - 1536752*b5 + 6418306*b4 + 7472326*b3 + 2155341*b2 + 15143544*b1 - 32752987 $$\nu^{19}$$ $$=$$ $$( 32875304 \beta_{19} + 44334764 \beta_{18} + 88669528 \beta_{17} - 45144368 \beta_{16} - 56603828 \beta_{15} - 58682048 \beta_{14} - 44334764 \beta_{13} - 71348356 \beta_{12} + \cdots + 116477984 ) / 2$$ (32875304*b19 + 44334764*b18 + 88669528*b17 - 45144368*b16 - 56603828*b15 - 58682048*b14 - 44334764*b13 - 71348356*b12 + 61104178*b11 + 78361056*b10 + 106248546*b9 + 78361056*b8 - 18130776*b7 - 95233717*b6 - 38878721*b5 - 123222836*b4 + 102367764*b3 - 155585344*b2 - 38580179*b1 + 116477984) / 2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$651$$ $$677$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$ $$\beta_{4} + \beta_{5}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 2.44766i − 1.14923i − 2.86589i 1.70974i 0.125665i 0.402430i 0.676406i 2.27790i 1.86950i − 1.49418i − 2.44766i 1.14923i 2.86589i − 1.70974i − 0.125665i − 0.402430i − 0.676406i − 2.27790i − 1.86950i 1.49418i
0 −0.877081 + 3.27331i 0 0 0 −2.27273 1.31216i 0 −7.34722 4.24192i 0
193.2 0 −0.387600 + 1.44654i 0 0 0 −2.10545 1.21558i 0 0.655821 + 0.378639i 0
193.3 0 −0.355132 + 1.32537i 0 0 0 3.40883 + 1.96809i 0 0.967585 + 0.558635i 0
193.4 0 0.563138 2.10166i 0 0 0 −1.25530 0.724750i 0 −1.50178 0.867051i 0
193.5 0 0.690650 2.57754i 0 0 0 −1.87342 1.08162i 0 −3.56864 2.06036i 0
293.1 0 −2.49316 + 0.668040i 0 0 0 0.749297 0.432607i 0 3.17149 1.83106i 0
293.2 0 −0.834228 + 0.223531i 0 0 0 −2.07295 + 1.19682i 0 −1.95211 + 1.12705i 0
293.3 0 −0.243028 + 0.0651192i 0 0 0 4.30824 2.48736i 0 −2.54325 + 1.46835i 0
293.4 0 2.41732 0.647720i 0 0 0 −3.34088 + 1.92886i 0 2.82584 1.63150i 0
293.5 0 2.51912 0.674996i 0 0 0 1.45437 0.839682i 0 3.29226 1.90079i 0
357.1 0 −0.877081 3.27331i 0 0 0 −2.27273 + 1.31216i 0 −7.34722 + 4.24192i 0
357.2 0 −0.387600 1.44654i 0 0 0 −2.10545 + 1.21558i 0 0.655821 0.378639i 0
357.3 0 −0.355132 1.32537i 0 0 0 3.40883 1.96809i 0 0.967585 0.558635i 0
357.4 0 0.563138 + 2.10166i 0 0 0 −1.25530 + 0.724750i 0 −1.50178 + 0.867051i 0
357.5 0 0.690650 + 2.57754i 0 0 0 −1.87342 + 1.08162i 0 −3.56864 + 2.06036i 0
457.1 0 −2.49316 0.668040i 0 0 0 0.749297 + 0.432607i 0 3.17149 + 1.83106i 0
457.2 0 −0.834228 0.223531i 0 0 0 −2.07295 1.19682i 0 −1.95211 1.12705i 0
457.3 0 −0.243028 0.0651192i 0 0 0 4.30824 + 2.48736i 0 −2.54325 1.46835i 0
457.4 0 2.41732 + 0.647720i 0 0 0 −3.34088 1.92886i 0 2.82584 + 1.63150i 0
457.5 0 2.51912 + 0.674996i 0 0 0 1.45437 + 0.839682i 0 3.29226 + 1.90079i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 457.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o even 12 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.bs.d 20
5.b even 2 1 260.2.bk.c yes 20
5.c odd 4 1 260.2.bf.c 20
5.c odd 4 1 1300.2.bn.d 20
13.f odd 12 1 1300.2.bn.d 20
65.o even 12 1 inner 1300.2.bs.d 20
65.s odd 12 1 260.2.bf.c 20
65.t even 12 1 260.2.bk.c yes 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bf.c 20 5.c odd 4 1
260.2.bf.c 20 65.s odd 12 1
260.2.bk.c yes 20 5.b even 2 1
260.2.bk.c yes 20 65.t even 12 1
1300.2.bn.d 20 5.c odd 4 1
1300.2.bn.d 20 13.f odd 12 1
1300.2.bs.d 20 1.a even 1 1 trivial
1300.2.bs.d 20 65.o even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{20} - 2 T_{3}^{19} + 8 T_{3}^{18} - 28 T_{3}^{17} - 26 T_{3}^{16} + 106 T_{3}^{15} - 500 T_{3}^{14} + 1366 T_{3}^{13} + 2867 T_{3}^{12} - 5410 T_{3}^{11} + 14776 T_{3}^{10} - 43502 T_{3}^{9} - 9730 T_{3}^{8} - 76600 T_{3}^{7} + \cdots + 21904$$ acting on $$S_{2}^{\mathrm{new}}(1300, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$T^{20} - 2 T^{19} + 8 T^{18} + \cdots + 21904$$
$5$ $$T^{20}$$
$7$ $$T^{20} + 6 T^{19} - 24 T^{18} + \cdots + 27625536$$
$11$ $$T^{20} - 12 T^{18} + 128 T^{17} + \cdots + 8202496$$
$13$ $$T^{20} + 8 T^{19} + \cdots + 137858491849$$
$17$ $$T^{20} + 93 T^{18} + \cdots + 109980446689$$
$19$ $$T^{20} - 20 T^{19} + \cdots + 472279824$$
$23$ $$T^{20} + 6 T^{19} + 60 T^{18} + \cdots + 3139984$$
$29$ $$T^{20} - 24 T^{19} + \cdots + 341103961$$
$31$ $$T^{20} - 8 T^{19} + \cdots + 54838398976$$
$37$ $$T^{20} - 285 T^{18} + \cdots + 52\!\cdots\!69$$
$41$ $$T^{20} - 6 T^{19} + \cdots + 10017978284161$$
$43$ $$T^{20} + 38 T^{19} + \cdots + 319577657344$$
$47$ $$T^{20} + 224 T^{18} + \cdots + 5858983936$$
$53$ $$T^{20} + 30 T^{19} + \cdots + 90\!\cdots\!64$$
$59$ $$T^{20} + 24 T^{19} + \cdots + 26\!\cdots\!96$$
$61$ $$T^{20} + 32 T^{19} + \cdots + 43\!\cdots\!41$$
$67$ $$T^{20} + 22 T^{19} + \cdots + 11546104977936$$
$71$ $$T^{20} - 24 T^{18} + \cdots + 27\!\cdots\!76$$
$73$ $$(T^{10} - 22 T^{9} - 123 T^{8} + \cdots - 119575728)^{2}$$
$79$ $$T^{20} + \cdots + 207459544743936$$
$83$ $$T^{20} + 896 T^{18} + \cdots + 10\!\cdots\!36$$
$89$ $$T^{20} + \cdots + 237919334033956$$
$97$ $$T^{20} + 46 T^{19} + \cdots + 963004643584$$