# Properties

 Label 190.2.i.a Level $190$ Weight $2$ Character orbit 190.i Analytic conductor $1.517$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$190 = 2 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 190.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.51715763840$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 20 x^{18} + 270 x^{16} - 1928 x^{14} + 9835 x^{12} - 29980 x^{10} + 66046 x^{8} - 89920 x^{6} + 85425 x^{4} - 34500 x^{2} + 10000$$ x^20 - 20*x^18 + 270*x^16 - 1928*x^14 + 9835*x^12 - 29980*x^10 + 66046*x^8 - 89920*x^6 + 85425*x^4 - 34500*x^2 + 10000 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{13} - \beta_{2}) q^{2} - \beta_1 q^{3} + ( - \beta_{3} + 1) q^{4} - \beta_{14} q^{5} - \beta_{5} q^{6} + ( - \beta_{13} + \beta_{9}) q^{7} - \beta_{13} q^{8} + (\beta_{17} - \beta_{16} + \beta_{5} - \beta_{3} + 1) q^{9}+O(q^{10})$$ q + (-b13 - b2) * q^2 - b1 * q^3 + (-b3 + 1) * q^4 - b14 * q^5 - b5 * q^6 + (-b13 + b9) * q^7 - b13 * q^8 + (b17 - b16 + b5 - b3 + 1) * q^9 $$q + ( - \beta_{13} - \beta_{2}) q^{2} - \beta_1 q^{3} + ( - \beta_{3} + 1) q^{4} - \beta_{14} q^{5} - \beta_{5} q^{6} + ( - \beta_{13} + \beta_{9}) q^{7} - \beta_{13} q^{8} + (\beta_{17} - \beta_{16} + \beta_{5} - \beta_{3} + 1) q^{9} + \beta_{16} q^{10} + ( - \beta_{19} - \beta_{15} + \beta_{11} + \beta_{6} - 1) q^{11} - \beta_{9} q^{12} + ( - \beta_{17} - \beta_{16} - \beta_{10} + \beta_{6}) q^{13} + ( - \beta_{4} - \beta_{3}) q^{14} + ( - \beta_{11} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} + \beta_1) q^{15} - \beta_{3} q^{16} + (\beta_{18} - \beta_{14} + \beta_{7}) q^{17} + ( - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{10} + \beta_{9} + \beta_{6}) q^{18} + ( - \beta_{17} + \beta_{16} + \beta_{15} + \beta_{14} - \beta_{10} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 2) q^{19}+ \cdots + ( - 2 \beta_{17} + 2 \beta_{16} - \beta_{10} - \beta_{6} - 7 \beta_{5} + 3 \beta_{3} - 3) q^{99}+O(q^{100})$$ q + (-b13 - b2) * q^2 - b1 * q^3 + (-b3 + 1) * q^4 - b14 * q^5 - b5 * q^6 + (-b13 + b9) * q^7 - b13 * q^8 + (b17 - b16 + b5 - b3 + 1) * q^9 + b16 * q^10 + (-b19 - b15 + b11 + b6 - 1) * q^11 - b9 * q^12 + (-b17 - b16 - b10 + b6) * q^13 + (-b4 - b3) * q^14 + (-b11 - b9 + b7 - b6 + b5 - b2 + b1) * q^15 - b3 * q^16 + (b18 - b14 + b7) * q^17 + (-b15 + b14 - b13 - b10 + b9 + b6) * q^18 + (-b17 + b16 + b15 + b14 - b10 - b6 - b5 - b4 + b3 - 2) * q^19 + (-b14 + b10) * q^20 + (b12 + b8 + 2*b4 + 4*b3) * q^21 + (-b18 + b14 + b13 - b12 + b8 - b7 + b2) * q^22 + (-b17 - b16 - b10 + b9 + b6 + 3*b2 - b1) * q^23 + b4 * q^24 + (b17 + b16 + b11 + b10 - b9 - b7 - b6 + b3 - 2*b2 + b1 - 1) * q^25 + (-b17 + b16 + b15 + b14 - b12 - b10 - b8 - b6) * q^26 + (-b19 - b17 - b16 + b15 + 4*b13 - b12 - b11 - b9 + b8 - b6) * q^27 + (b9 + b2 - b1) * q^28 + (2*b17 - 2*b16 + b10 + b6) * q^29 + (b19 + b17 - b14 + b12 + b10 + b9 + b5 + b4 + 1) * q^30 + (b19 + b17 - b16 - b14 + b12 - b11 + b10 + b8) * q^31 + b2 * q^32 + (-2*b15 + 2*b14 + b12 - b8 + b1) * q^33 + (b19 - b18 - b11 + b10 + b7) * q^34 + (b15 + b13 + b8 - b7 + b4 + b2 - b1) * q^35 + (-b12 - b8 - b4 - b3) * q^36 + (b19 + 2*b17 + 2*b16 - b14 + b13 + 2*b12 + b11 + b10 + b9 - 2*b8) * q^37 + (b15 - b14 + 2*b13 + b12 + b10 - b8 - b6 + b2 - b1) * q^38 + (-b17 + b16 + b15 + b14 - b12 - b10 - b8 - b6 - 2) * q^39 + b8 * q^40 + (b18 - b14 - b7 + b3) * q^41 + (b10 - 2*b9 - b6 - 4*b2 + 2*b1) * q^42 + (-2*b15 + 2*b14) * q^43 + (-b19 + b18 + b11 - b7 + b6 + b3 - 1) * q^44 + (-2*b17 + b16 + b15 + 4*b13 - 2*b12 - b9 - b8 - b6 - 2*b5 - 2*b4 - 3) * q^45 + (-b17 + b16 + b15 + b14 - b12 - b10 - b8 - b6 - b5 - b4 - 3) * q^46 + (b19 - b18 + 2*b17 + 2*b16 + b11 + 2*b10 - 2*b9 - b7 - b6 + 2*b2 + 2*b1) * q^47 + (-b9 + b1) * q^48 + (-b17 + b16 - b12 - b8 - 3*b5 - 3*b4 + 2) * q^49 + (-b19 + b17 - b16 - b15 + b13 + b12 + b8 + b6 + b5 + b4 + 2) * q^50 + (b19 - b18 - b17 + b16 - b11 + 2*b10 + b7 + b6 + 2*b5 - 2*b3 + 2) * q^51 + (b15 - b14 + b12 - b8) * q^52 + (-b19 + b18 - 2*b17 - 2*b16 - b11 - 2*b10 - b9 + b7 + b6 + b2 + b1) * q^53 + (b18 + b15 + b12 + b8 - b7 + b4 + 4*b3) * q^54 + (-b18 + b15 + 2*b14 - b13 - 3*b12 + 2*b4 + b3 - b2 + 3*b1) * q^55 + (-b5 - b4 - 1) * q^56 + (b19 - b18 + b17 + b16 - 2*b13 + b12 + b11 + b10 + 3*b9 - b8 - b7 - 2*b2) * q^57 + (-b17 - b16 - 2*b15 + 2*b14 - b12 - 2*b10 + b8 + 2*b6) * q^58 + (-b15 - b14 + b12 + b8 - b4 - 4*b3) * q^59 + (-b15 - b13 + b7 - b4 - b2 + b1) * q^60 + (b10 + b6 + 4*b5) * q^61 + (b18 - b15 + b7) * q^62 + (b19 - b18 + b17 + b16 + b11 - b10 + 5*b9 - b7 + 2*b6 + 5*b2 - 5*b1) * q^63 - q^64 + (b19 + 2*b16 + 2*b15 - b14 - 4*b13 + b11 + b10 - b9 - 2*b8 - 2*b6 - b5 - b4 + 2) * q^65 + (2*b17 - 2*b16 - b10 - b6 + b5) * q^66 + (-b19 + b18 + b17 + b16 - b11 - 2*b10 - b9 + b7 + b6 - 2*b2 + b1) * q^67 + (b19 - b14 + b11 + b10) * q^68 + (b15 + b14 - b10 - b6 + 4*b5 + 4*b4 + 2) * q^69 + (-b19 + b18 - b17 - b9 - b5 + b3 + b1 - 1) * q^70 + (-b15 - b14 + b12 + b8 + 2*b4 - 4*b3) * q^71 + (-b10 + b9 + b6 + b2 - b1) * q^72 + (-b18 + b14 - 4*b13 + b12 - b8 - b7 - 4*b2 - b1) * q^73 + (-b18 - 2*b15 - b14 + b7 - b4 + b3) * q^74 + (-b19 + b17 - b15 + b14 + b13 + b12 - b10 + b6 - 4*b5 - 4*b4 - 3) * q^75 + (b12 - b10 + b8 - b6 - b5 + 2*b3 - 1) * q^76 + (-b19 - 2*b17 - 2*b16 + 2*b15 - b14 + b13 - 2*b12 - b11 + b10 - b9 + 2*b8 - 2*b6) * q^77 + (b15 - b14 + 2*b13 + b12 - b8 + 2*b2) * q^78 + (-b15 - b14 - b12 - b8 + 4*b4) * q^79 + b10 * q^80 + (-b18 - 2*b15 - b14 + b7 - 4*b4 - b3) * q^81 + (-b19 + b18 - b11 - b10 + b7 - b2) * q^82 + (-2*b17 - 2*b16 - 2*b13 - 2*b12 - b9 + 2*b8) * q^83 + (b17 - b16 + b12 + b8 + 2*b5 + 2*b4 + 4) * q^84 + (-b19 + b18 + 2*b17 - b16 - b11 + 2*b9 + b7 - 2*b6 - 2*b5 - b3 - 5*b2 - 2*b1 + 1) * q^85 + (2*b17 - 2*b16) * q^86 + (-b19 - 2*b17 - 2*b16 - b15 + 2*b14 - 2*b13 - 2*b12 - b11 - 2*b10 - 2*b9 + 2*b8 + b6) * q^87 + (-b19 - b17 - b16 + b14 + b13 - b12 - b11 - b10 + b8) * q^88 + (-b17 + b16 + 2*b10 + 2*b6 + b5 - b3 + 1) * q^89 + (2*b15 - b14 + 3*b13 + b12 + b4 + 4*b3 + 3*b2 - 2*b1) * q^90 + (-2*b3 + 2) * q^91 + (b15 - b14 + 3*b13 + b12 - b8 + 3*b2 - b1) * q^92 + (b15 - b14 - 2*b13 - 2*b12 + 2*b8 - 2*b2) * q^93 + (-b19 + b17 - b16 - 2*b15 - b14 + b12 + b11 + b10 + b8 + 2*b6 + 2*b5 + 2*b4 - 2) * q^94 + (-b19 + b18 + b17 - b16 - 2*b13 + 3*b12 - b10 + b7 + b6 + b5 + 2*b4 - 5*b3 + 2*b2 + 2) * q^95 + (b5 + b4) * q^96 + (2*b13 + b12 - b8 + 2*b2 - 5*b1) * q^97 + (b15 - b14 - 2*b13 - 2*b2 - 3*b1) * q^98 + (-2*b17 + 2*b16 - b10 - b6 - 7*b5 + 3*b3 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 10 q^{4} - 2 q^{5} + 10 q^{9}+O(q^{10})$$ 20 * q + 10 * q^4 - 2 * q^5 + 10 * q^9 $$20 q + 10 q^{4} - 2 q^{5} + 10 q^{9} - 12 q^{11} - 10 q^{14} - 2 q^{15} - 10 q^{16} - 22 q^{19} - 4 q^{20} + 40 q^{21} - 6 q^{25} + 8 q^{26} - 4 q^{29} + 12 q^{30} - 16 q^{31} - 8 q^{34} - 2 q^{35} - 10 q^{36} - 32 q^{39} + 2 q^{41} - 6 q^{44} - 56 q^{45} - 52 q^{46} + 40 q^{49} + 40 q^{50} + 8 q^{51} + 36 q^{54} + 18 q^{55} - 20 q^{56} - 44 q^{59} + 2 q^{60} - 4 q^{61} - 20 q^{64} + 48 q^{65} + 4 q^{66} + 48 q^{69} - 8 q^{70} - 44 q^{71} + 10 q^{74} - 56 q^{75} + 4 q^{76} - 4 q^{79} - 2 q^{80} - 10 q^{81} + 80 q^{84} + 12 q^{85} + 2 q^{89} + 42 q^{90} + 20 q^{91} - 40 q^{94} - 4 q^{95} - 26 q^{99}+O(q^{100})$$ 20 * q + 10 * q^4 - 2 * q^5 + 10 * q^9 - 12 * q^11 - 10 * q^14 - 2 * q^15 - 10 * q^16 - 22 * q^19 - 4 * q^20 + 40 * q^21 - 6 * q^25 + 8 * q^26 - 4 * q^29 + 12 * q^30 - 16 * q^31 - 8 * q^34 - 2 * q^35 - 10 * q^36 - 32 * q^39 + 2 * q^41 - 6 * q^44 - 56 * q^45 - 52 * q^46 + 40 * q^49 + 40 * q^50 + 8 * q^51 + 36 * q^54 + 18 * q^55 - 20 * q^56 - 44 * q^59 + 2 * q^60 - 4 * q^61 - 20 * q^64 + 48 * q^65 + 4 * q^66 + 48 * q^69 - 8 * q^70 - 44 * q^71 + 10 * q^74 - 56 * q^75 + 4 * q^76 - 4 * q^79 - 2 * q^80 - 10 * q^81 + 80 * q^84 + 12 * q^85 + 2 * q^89 + 42 * q^90 + 20 * q^91 - 40 * q^94 - 4 * q^95 - 26 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 20 x^{18} + 270 x^{16} - 1928 x^{14} + 9835 x^{12} - 29980 x^{10} + 66046 x^{8} - 89920 x^{6} + 85425 x^{4} - 34500 x^{2} + 10000$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 67521462807 \nu^{19} + 3129986432010 \nu^{17} - 52636278341640 \nu^{15} + 583092465086296 \nu^{13} + \cdots + 25\!\cdots\!00 \nu ) / 99\!\cdots\!00$$ (-67521462807*v^19 + 3129986432010*v^17 - 52636278341640*v^15 + 583092465086296*v^13 - 3709100447264955*v^11 + 16414690458001860*v^9 - 41552708308853972*v^7 + 69104511948824710*v^5 - 62032454787180475*v^3 + 25455372239291500*v) / 9957351510065000 $$\beta_{3}$$ $$=$$ $$( - 54067767641 \nu^{18} + 1052563531840 \nu^{16} - 14058915653070 \nu^{14} + 97142278802248 \nu^{12} + \cdots + 15\!\cdots\!00 ) / 995735151006500$$ (-54067767641*v^18 + 1052563531840*v^16 - 14058915653070*v^14 + 97142278802248*v^12 - 485110638968595*v^10 + 1395350556245180*v^8 - 2989442526333086*v^6 + 3686209593721840*v^4 - 3736670324512425*v^2 + 1502788932814500) / 995735151006500 $$\beta_{4}$$ $$=$$ $$( - 1689612 \nu^{18} + 26942635 \nu^{16} - 323563365 \nu^{14} + 1511616336 \nu^{12} - 4864178705 \nu^{10} - 4819884240 \nu^{8} + \cdots - 30857811000 ) / 22157483500$$ (-1689612*v^18 + 26942635*v^16 - 323563365*v^14 + 1511616336*v^12 - 4864178705*v^10 - 4819884240*v^8 + 31400754873*v^6 - 87629121165*v^4 + 72562221150*v^2 - 30857811000) / 22157483500 $$\beta_{5}$$ $$=$$ $$( - 102026243919 \nu^{18} + 2229773264110 \nu^{16} - 30750494419705 \nu^{14} + 236572159532307 \nu^{12} + \cdots + 13\!\cdots\!00 ) / 995735151006500$$ (-102026243919*v^18 + 2229773264110*v^16 - 30750494419705*v^14 + 236572159532307*v^12 - 1220448373480805*v^10 + 3925919355491645*v^8 - 7714416724559674*v^6 + 9564408458723185*v^4 - 5573461833504850*v^2 + 1319197705722000) / 995735151006500 $$\beta_{6}$$ $$=$$ $$( 244351455749 \nu^{19} - 1553004285275 \nu^{18} - 7051030576145 \nu^{17} + 30222412702750 \nu^{16} + 107681048716605 \nu^{15} + \cdots + 14\!\cdots\!00 ) / 99\!\cdots\!00$$ (244351455749*v^19 - 1553004285275*v^18 - 7051030576145*v^17 + 30222412702750*v^16 + 107681048716605*v^15 - 403507493841375*v^14 - 1020126168778872*v^13 + 2787911259052075*v^12 + 6085604333806910*v^11 - 13920603313595625*v^10 - 24769624245471020*v^9 + 40174551627595125*v^8 + 61102611369853779*v^7 - 85775748439887650*v^6 - 100088150548870545*v^5 + 105765732867681625*v^4 + 89077189670499575*v^3 - 82674556465981000*v^2 - 36463203971865500*v + 14548369045450000) / 9957351510065000 $$\beta_{7}$$ $$=$$ $$( 149747246509 \nu^{19} - 1149846030380 \nu^{18} - 2721738955500 \nu^{17} + 19810551139225 \nu^{16} + 35828758140480 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 49\!\cdots\!00$$ (149747246509*v^19 - 1149846030380*v^18 - 2721738955500*v^17 + 19810551139225*v^16 + 35828758140480*v^15 - 248406186805725*v^14 - 230227796742002*v^13 + 1400052307666640*v^12 + 1138391333481600*v^11 - 5790156943291675*v^10 - 2934356933309220*v^9 + 8518486506772400*v^8 + 6465395719130639*v^7 - 9043065764527855*v^6 - 4450950438561000*v^5 - 7249999966229525*v^4 + 1829435517540000*v^3 + 3250501619144750*v^2 + 153417896376500*v - 2039151855515000) / 4978675755032500 $$\beta_{8}$$ $$=$$ $$( - 446324990017 \nu^{19} + 1317043616260 \nu^{18} + 9062355730875 \nu^{17} - 24408237958125 \nu^{16} + \cdots - 20\!\cdots\!00 ) / 99\!\cdots\!00$$ (-446324990017*v^19 + 1317043616260*v^18 + 9062355730875*v^17 - 24408237958125*v^16 - 119296140068240*v^15 + 318379497312075*v^14 + 828828557891851*v^13 - 2049268929759280*v^12 - 3790410246400800*v^11 + 9667163825331875*v^10 + 9770292745110985*v^9 - 23916420959174800*v^8 - 12707879205563957*v^7 + 46855437547688585*v^6 + 14819972405224250*v^5 - 49624022863167125*v^4 - 6091324597145000*v^3 + 51637617199916750*v^2 + 27137894626863000*v - 20418794431595000) / 9957351510065000 $$\beta_{9}$$ $$=$$ $$( 54067767641 \nu^{19} - 1052563531840 \nu^{17} + 14058915653070 \nu^{15} - 97142278802248 \nu^{13} + \cdots - 507053781808000 \nu ) / 995735151006500$$ (54067767641*v^19 - 1052563531840*v^17 + 14058915653070*v^15 - 97142278802248*v^13 + 485110638968595*v^11 - 1395350556245180*v^9 + 2989442526333086*v^7 - 3686209593721840*v^5 + 3736670324512425*v^3 - 507053781808000*v) / 995735151006500 $$\beta_{10}$$ $$=$$ $$( - 244351455749 \nu^{19} - 1553004285275 \nu^{18} + 7051030576145 \nu^{17} + 30222412702750 \nu^{16} + \cdots + 14\!\cdots\!00 ) / 99\!\cdots\!00$$ (-244351455749*v^19 - 1553004285275*v^18 + 7051030576145*v^17 + 30222412702750*v^16 - 107681048716605*v^15 - 403507493841375*v^14 + 1020126168778872*v^13 + 2787911259052075*v^12 - 6085604333806910*v^11 - 13920603313595625*v^10 + 24769624245471020*v^9 + 40174551627595125*v^8 - 61102611369853779*v^7 - 85775748439887650*v^6 + 100088150548870545*v^5 + 105765732867681625*v^4 - 89077189670499575*v^3 - 82674556465981000*v^2 + 36463203971865500*v + 14548369045450000) / 9957351510065000 $$\beta_{11}$$ $$=$$ $$( 203038749068 \nu^{19} - 1607322513115 \nu^{18} - 3807391628420 \nu^{17} + 31253605576875 \nu^{16} + 50221730714535 \nu^{15} + \cdots + 60\!\cdots\!00 ) / 49\!\cdots\!00$$ (203038749068*v^19 - 1607322513115*v^18 - 3807391628420*v^17 + 31253605576875*v^16 + 50221730714535*v^15 - 411420012550800*v^14 - 333029982590554*v^13 + 2776792115716845*v^12 + 1639801849219460*v^11 - 13072096299636000*v^10 - 4533820841544040*v^9 + 33695088219276825*v^8 + 9989686502483928*v^7 - 55933105556144665*v^6 - 12287062395469070*v^5 + 51110062986716250*v^4 + 6388839899846325*v^3 - 21007325474025000*v^2 - 1688006956684000*v + 6025736267915000) / 4978675755032500 $$\beta_{12}$$ $$=$$ $$( 446324990017 \nu^{19} + 1317043616260 \nu^{18} - 9062355730875 \nu^{17} - 24408237958125 \nu^{16} + 119296140068240 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 99\!\cdots\!00$$ (446324990017*v^19 + 1317043616260*v^18 - 9062355730875*v^17 - 24408237958125*v^16 + 119296140068240*v^15 + 318379497312075*v^14 - 828828557891851*v^13 - 2049268929759280*v^12 + 3790410246400800*v^11 + 9667163825331875*v^10 - 9770292745110985*v^9 - 23916420959174800*v^8 + 12707879205563957*v^7 + 46855437547688585*v^6 - 14819972405224250*v^5 - 49624022863167125*v^4 + 6091324597145000*v^3 + 51637617199916750*v^2 - 27137894626863000*v - 20418794431595000) / 9957351510065000 $$\beta_{13}$$ $$=$$ $$( 30857811 \nu^{19} - 634052340 \nu^{17} + 8601035320 \nu^{15} - 62729493258 \nu^{13} + 318602734545 \nu^{11} - 973758960830 \nu^{9} + \cdots - 338972268000 \nu ) / 221574835000$$ (30857811*v^19 - 634052340*v^17 + 8601035320*v^15 - 62729493258*v^13 + 318602734545*v^11 - 973758960830*v^9 + 1989836142906*v^7 - 2460726816390*v^5 + 1759737293025*v^3 - 338972268000*v) / 221574835000 $$\beta_{14}$$ $$=$$ $$( 1748616823509 \nu^{19} - 824090674470 \nu^{18} - 33524641658625 \nu^{17} + 16617984808375 \nu^{16} + 441315753410480 \nu^{15} + \cdots + 29\!\cdots\!00 ) / 99\!\cdots\!00$$ (1748616823509*v^19 - 824090674470*v^18 - 33524641658625*v^17 + 16617984808375*v^16 + 441315753410480*v^15 - 224419872053775*v^14 - 2949303034590877*v^13 + 1614060863412160*v^12 + 14021977179381600*v^11 - 8230431307703625*v^10 - 36143534077323595*v^9 + 25507367166390600*v^8 + 64220585227787889*v^7 - 56492909891037245*v^6 - 54823963992404750*v^5 + 83517958451080375*v^4 + 22533817961915000*v^3 - 73530246893056000*v^2 + 15128311906211500*v + 29718570766840000) / 9957351510065000 $$\beta_{15}$$ $$=$$ $$( - 1748616823509 \nu^{19} - 824090674470 \nu^{18} + 33524641658625 \nu^{17} + 16617984808375 \nu^{16} + \cdots + 29\!\cdots\!00 ) / 99\!\cdots\!00$$ (-1748616823509*v^19 - 824090674470*v^18 + 33524641658625*v^17 + 16617984808375*v^16 - 441315753410480*v^15 - 224419872053775*v^14 + 2949303034590877*v^13 + 1614060863412160*v^12 - 14021977179381600*v^11 - 8230431307703625*v^10 + 36143534077323595*v^9 + 25507367166390600*v^8 - 64220585227787889*v^7 - 56492909891037245*v^6 + 54823963992404750*v^5 + 83517958451080375*v^4 - 22533817961915000*v^3 - 73530246893056000*v^2 - 15128311906211500*v + 29718570766840000) / 9957351510065000 $$\beta_{16}$$ $$=$$ $$( - 1964304297788 \nu^{19} + 571224133225 \nu^{18} + 37105178622365 \nu^{17} - 9902404316250 \nu^{16} + \cdots - 35\!\cdots\!00 ) / 99\!\cdots\!00$$ (-1964304297788*v^19 + 571224133225*v^18 + 37105178622365*v^17 - 9902404316250*v^16 - 487754598990885*v^15 + 127425840962875*v^14 + 3226296089478864*v^13 - 759984778383425*v^12 - 15521091246493545*v^11 + 3599970911967875*v^10 + 41068854338096240*v^9 - 8277414347445375*v^8 - 83798444377726023*v^7 + 21216766903863350*v^6 + 102158342714201665*v^5 - 25902149580820875*v^4 - 98123129209653650*v^3 + 41887421567691750*v^2 + 39129533465861000*v - 3545087107550000) / 9957351510065000 $$\beta_{17}$$ $$=$$ $$( - 1964304297788 \nu^{19} - 571224133225 \nu^{18} + 37105178622365 \nu^{17} + 9902404316250 \nu^{16} + \cdots + 35\!\cdots\!00 ) / 99\!\cdots\!00$$ (-1964304297788*v^19 - 571224133225*v^18 + 37105178622365*v^17 + 9902404316250*v^16 - 487754598990885*v^15 - 127425840962875*v^14 + 3226296089478864*v^13 + 759984778383425*v^12 - 15521091246493545*v^11 - 3599970911967875*v^10 + 41068854338096240*v^9 + 8277414347445375*v^8 - 83798444377726023*v^7 - 21216766903863350*v^6 + 102158342714201665*v^5 + 25902149580820875*v^4 - 98123129209653650*v^3 - 41887421567691750*v^2 + 39129533465861000*v + 3545087107550000) / 9957351510065000 $$\beta_{18}$$ $$=$$ $$( 2048111316527 \nu^{19} + 1475601386290 \nu^{18} - 38968119569625 \nu^{17} - 23003117470075 \nu^{16} + 512973269691440 \nu^{15} + \cdots + 33\!\cdots\!00 ) / 99\!\cdots\!00$$ (2048111316527*v^19 + 1475601386290*v^18 - 38968119569625*v^17 - 23003117470075*v^16 + 512973269691440*v^15 + 272392501557675*v^14 - 3409758628074881*v^13 - 1186043751921120*v^12 + 16298759846344800*v^11 + 3349882578879725*v^10 - 42012247943942035*v^9 + 8470394152845800*v^8 + 77151376666049167*v^7 - 38406778361981535*v^6 - 63725864869526750*v^5 + 98017958383539425*v^4 + 26192688996995000*v^3 - 80031250131345500*v^2 + 15435147698964500*v + 33796874477870000) / 9957351510065000 $$\beta_{19}$$ $$=$$ $$( 2399045777394 \nu^{19} + 3943558637035 \nu^{18} - 48190455491610 \nu^{17} - 76111639048125 \nu^{16} + 649440263556155 \nu^{15} + \cdots + 31\!\cdots\!00 ) / 99\!\cdots\!00$$ (2399045777394*v^19 + 3943558637035*v^18 - 48190455491610*v^17 - 76111639048125*v^16 + 649440263556155*v^15 + 1001927646889200*v^14 - 4635489168550857*v^13 - 6727434627073605*v^12 + 23387185211627430*v^11 + 31834364605164000*v^10 - 69980800005882695*v^9 - 82057360899758175*v^8 + 145302569602609524*v^7 + 141149049661139735*v^6 - 179486239332213435*v^5 - 124467900390033750*v^4 + 124388687432107225*v^3 + 51158960520975000*v^2 - 24710905979022000*v + 3118729185560000) / 9957351510065000
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{17} - \beta_{16} + \beta_{5} - 4\beta_{3} + 4$$ b17 - b16 + b5 - 4*b3 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{19} + \beta_{17} + \beta_{16} - \beta_{15} - 4\beta_{13} + \beta_{12} + \beta_{11} + 7\beta_{9} - \beta_{8} + \beta_{6}$$ b19 + b17 + b16 - b15 - 4*b13 + b12 + b11 + 7*b9 - b8 + b6 $$\nu^{4}$$ $$=$$ $$-\beta_{18} - 2\beta_{15} - \beta_{14} - 9\beta_{12} - 9\beta_{8} + \beta_{7} - 13\beta_{4} - 28\beta_{3}$$ -b18 - 2*b15 - b14 - 9*b12 - 9*b8 + b7 - 13*b4 - 28*b3 $$\nu^{5}$$ $$=$$ $$10 \beta_{19} - 10 \beta_{18} + 10 \beta_{17} + 10 \beta_{16} + 10 \beta_{11} - 6 \beta_{10} + 61 \beta_{9} - 10 \beta_{7} + 16 \beta_{6} + 54 \beta_{2} - 61 \beta_1$$ 10*b19 - 10*b18 + 10*b17 + 10*b16 + 10*b11 - 6*b10 + 61*b9 - 10*b7 + 16*b6 + 54*b2 - 61*b1 $$\nu^{6}$$ $$=$$ $$- 16 \beta_{19} - 81 \beta_{17} + 81 \beta_{16} - 26 \beta_{15} - 10 \beta_{14} - 81 \beta_{12} + 16 \beta_{11} + 10 \beta_{10} - 81 \beta_{8} + 26 \beta_{6} - 147 \beta_{5} - 147 \beta_{4} - 244$$ -16*b19 - 81*b17 + 81*b16 - 26*b15 - 10*b14 - 81*b12 + 16*b11 + 10*b10 - 81*b8 + 26*b6 - 147*b5 - 147*b4 - 244 $$\nu^{7}$$ $$=$$ $$- 91 \beta_{18} + 189 \beta_{15} - 98 \beta_{14} + 608 \beta_{13} - 97 \beta_{12} + 97 \beta_{8} - 91 \beta_{7} + 608 \beta_{2} - 573 \beta_1$$ -91*b18 + 189*b15 - 98*b14 + 608*b13 - 97*b12 + 97*b8 - 91*b7 + 608*b2 - 573*b1 $$\nu^{8}$$ $$=$$ $$- 195 \beta_{19} + 195 \beta_{18} - 761 \beta_{17} + 761 \beta_{16} + 195 \beta_{11} + 85 \beta_{10} - 195 \beta_{7} + 280 \beta_{6} - 1571 \beta_{5} + 2304 \beta_{3} - 2304$$ -195*b19 + 195*b18 - 761*b17 + 761*b16 + 195*b11 + 85*b10 - 195*b7 + 280*b6 - 1571*b5 + 2304*b3 - 2304 $$\nu^{9}$$ $$=$$ $$- 846 \beta_{19} - 956 \beta_{17} - 956 \beta_{16} + 2046 \beta_{15} - 1200 \beta_{14} + 6454 \beta_{13} - 956 \beta_{12} - 846 \beta_{11} + 1200 \beta_{10} - 5567 \beta_{9} + 956 \beta_{8} - 2046 \beta_{6}$$ -846*b19 - 956*b17 - 956*b16 + 2046*b15 - 1200*b14 + 6454*b13 - 956*b12 - 846*b11 + 1200*b10 - 5567*b9 + 956*b8 - 2046*b6 $$\nu^{10}$$ $$=$$ $$2156 \beta_{18} + 2892 \beta_{15} + 736 \beta_{14} + 7369 \beta_{12} + 7369 \beta_{8} - 2156 \beta_{7} + 16333 \beta_{4} + 22488 \beta_{3}$$ 2156*b18 + 2892*b15 + 736*b14 + 7369*b12 + 7369*b8 - 2156*b7 + 16333*b4 + 22488*b3 $$\nu^{11}$$ $$=$$ $$- 8105 \beta_{19} + 8105 \beta_{18} - 9525 \beta_{17} - 9525 \beta_{16} - 8105 \beta_{11} + 13276 \beta_{10} - 55031 \beta_{9} + 8105 \beta_{7} - 21381 \beta_{6} - 66804 \beta_{2} + 55031 \beta_1$$ -8105*b19 + 8105*b18 - 9525*b17 - 9525*b16 - 8105*b11 + 13276*b10 - 55031*b9 + 8105*b7 - 21381*b6 - 66804*b2 + 55031*b1 $$\nu^{12}$$ $$=$$ $$22801 \beta_{19} + 72661 \beta_{17} - 72661 \beta_{16} + 29486 \beta_{15} + 6685 \beta_{14} + 72661 \beta_{12} - 22801 \beta_{11} - 6685 \beta_{10} + 72661 \beta_{8} - 29486 \beta_{6} + 167437 \beta_{5} + \cdots + 222964$$ 22801*b19 + 72661*b17 - 72661*b16 + 29486*b15 + 6685*b14 + 72661*b12 - 22801*b11 - 6685*b10 + 72661*b8 - 29486*b6 + 167437*b5 + 167437*b4 + 222964 $$\nu^{13}$$ $$=$$ $$79346 \beta_{18} - 219724 \beta_{15} + 140378 \beta_{14} - 683118 \beta_{13} + 95462 \beta_{12} - 95462 \beta_{8} + 79346 \beta_{7} - 683118 \beta_{2} + 549093 \beta_1$$ 79346*b18 - 219724*b15 + 140378*b14 - 683118*b13 + 95462*b12 - 95462*b8 + 79346*b7 - 683118*b2 + 549093*b1 $$\nu^{14}$$ $$=$$ $$235840 \beta_{19} - 235840 \beta_{18} + 723901 \beta_{17} - 723901 \beta_{16} - 235840 \beta_{11} - 63230 \beta_{10} + 235840 \beta_{7} - 299070 \beta_{6} + 1703891 \beta_{5} - 2228604 \beta_{3} + \cdots + 2228604$$ 235840*b19 - 235840*b18 + 723901*b17 - 723901*b16 - 235840*b11 - 63230*b10 + 235840*b7 - 299070*b6 + 1703891*b5 - 2228604*b3 + 2228604 $$\nu^{15}$$ $$=$$ $$787131 \beta_{19} + 959741 \beta_{17} + 959741 \beta_{16} - 2238801 \beta_{15} + 1451670 \beta_{14} - 6942024 \beta_{13} + 959741 \beta_{12} + 787131 \beta_{11} - 1451670 \beta_{10} + \cdots + 2238801 \beta_{6}$$ 787131*b19 + 959741*b17 + 959741*b16 - 2238801*b15 + 1451670*b14 - 6942024*b13 + 959741*b12 + 787131*b11 - 1451670*b10 + 5506757*b9 - 959741*b8 + 2238801*b6 $$\nu^{16}$$ $$=$$ $$- 2411411 \beta_{18} - 3025932 \beta_{15} - 614521 \beta_{14} - 7253629 \beta_{12} - 7253629 \beta_{8} + 2411411 \beta_{7} - 17271603 \beta_{4} - 22372248 \beta_{3}$$ -2411411*b18 - 3025932*b15 - 614521*b14 - 7253629*b12 - 7253629*b8 + 2411411*b7 - 17271603*b4 - 22372248*b3 $$\nu^{17}$$ $$=$$ $$7868150 \beta_{19} - 7868150 \beta_{18} + 9665040 \beta_{17} + 9665040 \beta_{16} + 7868150 \beta_{11} - 14840796 \beta_{10} + 55380151 \beta_{9} - 7868150 \beta_{7} + \cdots - 55380151 \beta_1$$ 7868150*b19 - 7868150*b18 + 9665040*b17 + 9665040*b16 + 7868150*b11 - 14840796*b10 + 55380151*b9 - 7868150*b7 + 22708946*b6 + 70315454*b2 - 55380151*b1 $$\nu^{18}$$ $$=$$ $$- 24505836 \beta_{19} - 72913341 \beta_{17} + 72913341 \beta_{16} - 30577096 \beta_{15} - 6071260 \beta_{14} - 72913341 \beta_{12} + 24505836 \beta_{11} + 6071260 \beta_{10} + \cdots - 225114384$$ -24505836*b19 - 72913341*b17 + 72913341*b16 - 30577096*b15 - 6071260*b14 - 72913341*b12 + 24505836*b11 + 6071260*b10 - 72913341*b8 + 30577096*b6 - 174707277*b5 - 174707277*b4 - 225114384 $$\nu^{19}$$ $$=$$ $$- 78984601 \beta_{18} + 229790209 \beta_{15} - 150805608 \beta_{14} + 710971628 \beta_{13} - 97419177 \beta_{12} + 97419177 \beta_{8} - 78984601 \beta_{7} + \cdots - 557790863 \beta_1$$ -78984601*b18 + 229790209*b15 - 150805608*b14 + 710971628*b13 - 97419177*b12 + 97419177*b8 - 78984601*b7 + 710971628*b2 - 557790863*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/190\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-1 + \beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 2.02701 + 1.17030i 1.28416 + 0.741409i 0.590953 + 0.341187i −1.15118 − 0.664633i −2.75095 − 1.58826i 2.75095 + 1.58826i 1.15118 + 0.664633i −0.590953 − 0.341187i −1.28416 − 0.741409i −2.02701 − 1.17030i 2.02701 − 1.17030i 1.28416 − 0.741409i 0.590953 − 0.341187i −1.15118 + 0.664633i −2.75095 + 1.58826i 2.75095 − 1.58826i 1.15118 − 0.664633i −0.590953 + 0.341187i −1.28416 + 0.741409i −2.02701 + 1.17030i
−0.866025 0.500000i −2.02701 1.17030i 0.500000 + 0.866025i 1.07189 + 1.96241i 1.17030 + 2.02701i 1.34059i 1.00000i 1.23919 + 2.14634i 0.0529205 2.23544i
49.2 −0.866025 0.500000i −1.28416 0.741409i 0.500000 + 0.866025i 0.977310 2.01118i 0.741409 + 1.28416i 0.482818i 1.00000i −0.400626 0.693904i −1.85197 + 1.25308i
49.3 −0.866025 0.500000i −0.590953 0.341187i 0.500000 + 0.866025i −2.09613 + 0.778613i 0.341187 + 0.590953i 0.317626i 1.00000i −1.26718 2.19482i 2.20461 + 0.373767i
49.4 −0.866025 0.500000i 1.15118 + 0.664633i 0.500000 + 0.866025i −0.866610 2.06131i −0.664633 1.15118i 2.32927i 1.00000i −0.616527 1.06786i −0.280148 + 2.21845i
49.5 −0.866025 0.500000i 2.75095 + 1.58826i 0.500000 + 0.866025i 0.413539 + 2.19750i −1.58826 2.75095i 4.17652i 1.00000i 3.54514 + 6.14037i 0.740612 2.10986i
49.6 0.866025 + 0.500000i −2.75095 1.58826i 0.500000 + 0.866025i −2.10986 + 0.740612i −1.58826 2.75095i 4.17652i 1.00000i 3.54514 + 6.14037i −2.19750 0.413539i
49.7 0.866025 + 0.500000i −1.15118 0.664633i 0.500000 + 0.866025i 2.21845 0.280148i −0.664633 1.15118i 2.32927i 1.00000i −0.616527 1.06786i 2.06131 + 0.866610i
49.8 0.866025 + 0.500000i 0.590953 + 0.341187i 0.500000 + 0.866025i 0.373767 + 2.20461i 0.341187 + 0.590953i 0.317626i 1.00000i −1.26718 2.19482i −0.778613 + 2.09613i
49.9 0.866025 + 0.500000i 1.28416 + 0.741409i 0.500000 + 0.866025i 1.25308 1.85197i 0.741409 + 1.28416i 0.482818i 1.00000i −0.400626 0.693904i 2.01118 0.977310i
49.10 0.866025 + 0.500000i 2.02701 + 1.17030i 0.500000 + 0.866025i −2.23544 + 0.0529205i 1.17030 + 2.02701i 1.34059i 1.00000i 1.23919 + 2.14634i −1.96241 1.07189i
159.1 −0.866025 + 0.500000i −2.02701 + 1.17030i 0.500000 0.866025i 1.07189 1.96241i 1.17030 2.02701i 1.34059i 1.00000i 1.23919 2.14634i 0.0529205 + 2.23544i
159.2 −0.866025 + 0.500000i −1.28416 + 0.741409i 0.500000 0.866025i 0.977310 + 2.01118i 0.741409 1.28416i 0.482818i 1.00000i −0.400626 + 0.693904i −1.85197 1.25308i
159.3 −0.866025 + 0.500000i −0.590953 + 0.341187i 0.500000 0.866025i −2.09613 0.778613i 0.341187 0.590953i 0.317626i 1.00000i −1.26718 + 2.19482i 2.20461 0.373767i
159.4 −0.866025 + 0.500000i 1.15118 0.664633i 0.500000 0.866025i −0.866610 + 2.06131i −0.664633 + 1.15118i 2.32927i 1.00000i −0.616527 + 1.06786i −0.280148 2.21845i
159.5 −0.866025 + 0.500000i 2.75095 1.58826i 0.500000 0.866025i 0.413539 2.19750i −1.58826 + 2.75095i 4.17652i 1.00000i 3.54514 6.14037i 0.740612 + 2.10986i
159.6 0.866025 0.500000i −2.75095 + 1.58826i 0.500000 0.866025i −2.10986 0.740612i −1.58826 + 2.75095i 4.17652i 1.00000i 3.54514 6.14037i −2.19750 + 0.413539i
159.7 0.866025 0.500000i −1.15118 + 0.664633i 0.500000 0.866025i 2.21845 + 0.280148i −0.664633 + 1.15118i 2.32927i 1.00000i −0.616527 + 1.06786i 2.06131 0.866610i
159.8 0.866025 0.500000i 0.590953 0.341187i 0.500000 0.866025i 0.373767 2.20461i 0.341187 0.590953i 0.317626i 1.00000i −1.26718 + 2.19482i −0.778613 2.09613i
159.9 0.866025 0.500000i 1.28416 0.741409i 0.500000 0.866025i 1.25308 + 1.85197i 0.741409 1.28416i 0.482818i 1.00000i −0.400626 + 0.693904i 2.01118 + 0.977310i
159.10 0.866025 0.500000i 2.02701 1.17030i 0.500000 0.866025i −2.23544 0.0529205i 1.17030 2.02701i 1.34059i 1.00000i 1.23919 2.14634i −1.96241 + 1.07189i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 159.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.2.i.a 20
3.b odd 2 1 1710.2.t.d 20
5.b even 2 1 inner 190.2.i.a 20
5.c odd 4 1 950.2.e.n 10
5.c odd 4 1 950.2.e.o 10
15.d odd 2 1 1710.2.t.d 20
19.c even 3 1 inner 190.2.i.a 20
57.h odd 6 1 1710.2.t.d 20
95.i even 6 1 inner 190.2.i.a 20
95.m odd 12 1 950.2.e.n 10
95.m odd 12 1 950.2.e.o 10
285.n odd 6 1 1710.2.t.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.i.a 20 1.a even 1 1 trivial
190.2.i.a 20 5.b even 2 1 inner
190.2.i.a 20 19.c even 3 1 inner
190.2.i.a 20 95.i even 6 1 inner
950.2.e.n 10 5.c odd 4 1
950.2.e.n 10 95.m odd 12 1
950.2.e.o 10 5.c odd 4 1
950.2.e.o 10 95.m odd 12 1
1710.2.t.d 20 3.b odd 2 1
1710.2.t.d 20 15.d odd 2 1
1710.2.t.d 20 57.h odd 6 1
1710.2.t.d 20 285.n odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(190, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{5}$$
$3$ $$T^{20} - 20 T^{18} + 270 T^{16} + \cdots + 10000$$
$5$ $$T^{20} + 2 T^{19} + 5 T^{18} + \cdots + 9765625$$
$7$ $$(T^{10} + 25 T^{8} + 144 T^{6} + 216 T^{4} + \cdots + 4)^{2}$$
$11$ $$(T^{5} + 3 T^{4} - 48 T^{3} - 176 T^{2} + \cdots + 1555)^{4}$$
$13$ $$T^{20} - 60 T^{18} + 2736 T^{16} + \cdots + 65536$$
$17$ $$T^{20} - 128 T^{18} + 11320 T^{16} + \cdots + 65536$$
$19$ $$(T^{10} + 11 T^{9} + 39 T^{8} + \cdots + 2476099)^{2}$$
$23$ $$T^{20} - 97 T^{18} + \cdots + 842290759696$$
$29$ $$(T^{10} + 2 T^{9} + 78 T^{8} + \cdots + 4000000)^{2}$$
$31$ $$(T^{5} + 4 T^{4} - 66 T^{3} - 232 T^{2} + \cdots + 1252)^{4}$$
$37$ $$(T^{10} + 233 T^{8} + 19224 T^{6} + \cdots + 60186564)^{2}$$
$41$ $$(T^{10} - T^{9} + 63 T^{8} + 258 T^{7} + \cdots + 6561)^{2}$$
$43$ $$T^{20} - 176 T^{18} + \cdots + 1761205026816$$
$47$ $$T^{20} - 240 T^{18} + \cdots + 16796160000$$
$53$ $$T^{20} - 225 T^{18} + \cdots + 39\!\cdots\!00$$
$59$ $$(T^{10} + 22 T^{9} + 338 T^{8} + \cdots + 15376)^{2}$$
$61$ $$(T^{10} + 2 T^{9} + 166 T^{8} + \cdots + 62853184)^{2}$$
$67$ $$T^{20} - 372 T^{18} + \cdots + 430467210000$$
$71$ $$(T^{10} + 22 T^{9} + 344 T^{8} + \cdots + 952576)^{2}$$
$73$ $$T^{20} - 392 T^{18} + \cdots + 13\!\cdots\!00$$
$79$ $$(T^{10} + 2 T^{9} + 216 T^{8} + \cdots + 19360000)^{2}$$
$83$ $$(T^{10} + 272 T^{8} + 22306 T^{6} + \cdots + 627264)^{2}$$
$89$ $$(T^{10} - T^{9} + 173 T^{8} + \cdots + 397922704)^{2}$$
$97$ $$T^{20} - 508 T^{18} + \cdots + 70\!\cdots\!00$$