# Properties

 Label 1045.4.a.c Level $1045$ Weight $4$ Character orbit 1045.a Self dual yes Analytic conductor $61.657$ Analytic rank $1$ Dimension $20$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1045.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$61.6569959560$$ Analytic rank: $$1$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + \cdots + 150528$$ x^20 - x^19 - 105*x^18 + 103*x^17 + 4500*x^16 - 4345*x^15 - 101844*x^14 + 95592*x^13 + 1317797*x^12 - 1160501*x^11 - 9914845*x^10 + 7570653*x^9 + 42786958*x^8 - 23777633*x^7 - 102801526*x^6 + 28436356*x^5 + 122325928*x^4 + 411232*x^3 - 47350496*x^2 - 4782848*x + 150528 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 3) q^{4} - 5 q^{5} + (\beta_{6} + \beta_1 - 3) q^{6} + (\beta_{7} + 2) q^{7} + ( - \beta_{3} - 3 \beta_1 + 1) q^{8} + ( - \beta_{10} - \beta_{5} + 7) q^{9}+O(q^{10})$$ q - b1 * q^2 + b5 * q^3 + (b2 + 3) * q^4 - 5 * q^5 + (b6 + b1 - 3) * q^6 + (b7 + 2) * q^7 + (-b3 - 3*b1 + 1) * q^8 + (-b10 - b5 + 7) * q^9 $$q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 3) q^{4} - 5 q^{5} + (\beta_{6} + \beta_1 - 3) q^{6} + (\beta_{7} + 2) q^{7} + ( - \beta_{3} - 3 \beta_1 + 1) q^{8} + ( - \beta_{10} - \beta_{5} + 7) q^{9} + 5 \beta_1 q^{10} - 11 q^{11} + (\beta_{17} + \beta_{10} - \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{12} + (\beta_{18} - \beta_{5} - \beta_1 + 3) q^{13} + ( - \beta_{12} - \beta_{7} - 2 \beta_{2} - 2 \beta_1 - 5) q^{14} - 5 \beta_{5} q^{15} + ( - \beta_{18} - \beta_{17} + \beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{6} + \beta_{3} + 3 \beta_{2} + \cdots + 15) q^{16}+ \cdots + (11 \beta_{10} + 11 \beta_{5} - 77) q^{99}+O(q^{100})$$ q - b1 * q^2 + b5 * q^3 + (b2 + 3) * q^4 - 5 * q^5 + (b6 + b1 - 3) * q^6 + (b7 + 2) * q^7 + (-b3 - 3*b1 + 1) * q^8 + (-b10 - b5 + 7) * q^9 + 5*b1 * q^10 - 11 * q^11 + (b17 + b10 - b7 - b6 + 3*b5 + b3 - b2 + 3*b1 - 2) * q^12 + (b18 - b5 - b1 + 3) * q^13 + (-b12 - b7 - 2*b2 - 2*b1 - 5) * q^14 - 5*b5 * q^15 + (-b18 - b17 + b12 + b10 + b9 - 2*b6 + b3 + 3*b2 + 15) * q^16 + (-b11 - b9 - b7 - 2*b5 - 2*b2 + 4*b1 - 9) * q^17 + (-b18 - b17 + b13 + b11 - 3*b6 - 5*b5 + b3 - b2 - 9*b1) * q^18 + 19 * q^19 + (-5*b2 - 15) * q^20 + (-b17 - b13 + b12 + b11 + b10 - b7 - b6 + 2*b5 - b4 + b3 - 2*b2 - 2*b1 + 5) * q^21 + 11*b1 * q^22 + (-b18 - b17 - b14 + b13 - 2*b7 - b6 - 2*b5 + b4 + 2*b3 - b2 + 2*b1 - 9) * q^23 + (2*b18 + b17 - b15 + b14 - 3*b13 - b11 - 2*b10 - 2*b9 - 2*b8 + 6*b6 - 2*b5 - 3*b3 - 7*b2 + 7*b1 - 25) * q^24 + 25 * q^25 + (-b19 - 2*b18 - b17 + b15 + b12 + 3*b10 + 2*b8 - 3*b6 - 3*b5 + b4 + 2*b3 - 7*b1 + 7) * q^26 + (-b18 + b14 - b13 + b10 + b9 + b8 - b7 + 3*b5 - 5*b2 + 11*b1 - 12) * q^27 + (2*b19 - 2*b17 + 2*b15 - b14 + b12 + 3*b10 + b9 + b8 - 2*b6 - 3*b5 + b4 + 5*b3 + 3*b2 + 14*b1 + 7) * q^28 + (-3*b18 + b16 + b14 + 2*b13 + 2*b12 - b11 + 4*b10 + b9 + 2*b8 - 2*b7 - 6*b6 + 2*b5 + 2*b3 + 2*b2 + 7*b1 - 12) * q^29 + (-5*b6 - 5*b1 + 15) * q^30 + (b19 + 3*b18 + b17 - 2*b16 - b14 - b13 - 3*b12 + b11 - 2*b8 + 2*b6 + b4 - 2*b3 - 8*b2 - 8*b1 - 41) * q^31 + (-3*b19 + 3*b18 + 3*b17 - 3*b15 + b14 - 2*b13 - b12 - 6*b10 - 2*b9 - 2*b8 - b7 + 5*b6 - 12*b5 - 2*b4 - 6*b3 - 7*b2 - 6*b1 - 20) * q^32 - 11*b5 * q^33 + (b19 + b18 - b16 + b15 - b14 + b13 + b10 + b9 + 2*b7 - 4*b6 - 4*b5 - 2*b4 + 3*b3 - 4*b2 + 24*b1 - 32) * q^34 + (-5*b7 - 10) * q^35 + (b19 + 3*b18 + 2*b16 - b15 - b14 + 5*b13 - 2*b12 - b11 - 6*b10 - b9 - 4*b6 - 22*b5 - 2*b3 + 4*b2 + 3*b1 + 33) * q^36 + (-b18 + 2*b16 - b15 + b14 - b13 - 2*b12 + 2*b11 - b10 + b9 - 4*b8 + 3*b7 - 4*b6 - 10*b5 - 3*b4 - b3 - b2 + 15*b1 + 28) * q^37 - 19*b1 * q^38 + (-3*b19 + 5*b17 - b16 - b15 - 2*b14 + 3*b13 - 3*b12 + b11 + b10 - b7 - 4*b6 - 4*b5 - 3*b2 + 17*b1 - 41) * q^39 + (5*b3 + 15*b1 - 5) * q^40 + (2*b19 - 3*b18 + b16 + 2*b15 + 2*b14 + 4*b13 + b12 - 2*b11 - b9 + b8 - 5*b7 - 6*b6 - 4*b5 + 4*b4 + 4*b3 + 2*b2 + 8*b1 + 30) * q^41 + (b19 + 2*b18 + 2*b17 + b15 - b14 + 3*b13 - 3*b11 - 6*b10 + 2*b8 - 4*b7 + 3*b6 - 5*b5 - b3 - 2*b2 + 12*b1 + 8) * q^42 + (-2*b19 - b18 - b16 + b15 - b13 + 2*b12 + 2*b11 + 2*b10 + 5*b9 + b8 + 2*b6 + b5 - b4 - 2*b3 - 2*b2 + 5*b1 - 47) * q^43 + (-11*b2 - 33) * q^44 + (5*b10 + 5*b5 - 35) * q^45 + (-b19 + 4*b18 + 3*b17 - 2*b15 - b14 + 4*b13 - 3*b12 + 2*b11 - 11*b10 - b9 - b8 + b6 - 14*b5 - b4 - 4*b3 - 10*b2 + 23*b1 - 21) * q^46 + (2*b19 + b18 + b17 - 3*b16 + b15 + b14 - 2*b13 - 4*b12 - b11 - 3*b10 - 2*b9 - b8 - 4*b7 + 7*b6 - 10*b5 + 2*b4 - 2*b3 - b2 + 2*b1 - 38) * q^47 + (-b19 - 6*b18 - 5*b17 - 3*b16 + 2*b15 + b14 - b13 + 4*b12 + 9*b10 + 4*b9 + 4*b8 + 2*b7 - b6 + 18*b5 + b4 + 6*b3 + 13*b2 + 39*b1 - 21) * q^48 + (-3*b19 + b18 + 4*b17 + b16 - b15 + b14 - 5*b13 + 2*b12 + 3*b11 + 3*b10 - 2*b9 - b8 + 7*b7 - 3*b6 - 15*b5 - 3*b4 - 3*b3 - 6*b2 + 34*b1 - 4) * q^49 - 25*b1 * q^50 + (2*b19 + b18 - b17 + b16 + 4*b15 - 4*b14 - 2*b13 - 3*b12 - 2*b11 + 6*b10 + 2*b9 - 2*b8 + 4*b7 - b6 - 14*b5 + b4 - 6*b3 - 2*b2 + 7*b1 - 49) * q^51 + (4*b19 + b18 - 2*b17 - 2*b13 - 2*b12 - 2*b11 - 6*b10 + b9 + b8 + 8*b7 + 2*b6 - 16*b5 + 2*b4 - 3*b3 + 2*b2 + 16*b1 + 37) * q^52 + (-b19 + 4*b16 - 4*b15 + 2*b14 - 4*b13 + 5*b12 + b10 - 3*b9 - 2*b7 - b6 - 6*b5 - 2*b4 - 4*b3 + 3*b2 + 26*b1 - 23) * q^53 + (2*b19 + b18 - 2*b17 + b15 + 2*b14 - 7*b13 + 4*b12 - b11 + 8*b10 + 10*b7 + 5*b6 + 10*b5 + 3*b4 + 5*b3 - 7*b2 + 54*b1 - 123) * q^54 + 55 * q^55 + (-b19 + 6*b18 + 5*b17 - b16 - 2*b14 - b13 - b12 - 6*b10 - 5*b9 + 5*b8 + 4*b7 + 6*b6 - 15*b5 - 7*b3 - 26*b2 - 12*b1 - 139) * q^56 + 19*b5 * q^57 + (-b19 + 8*b18 + 2*b17 + 3*b16 - 5*b15 + 4*b14 - 7*b13 + 2*b12 - 2*b11 - 12*b10 - 5*b9 - 8*b8 + 5*b7 + 17*b6 - 15*b5 - 3*b4 - 17*b3 - 11*b2 + 19*b1 - 84) * q^58 + (b19 + 3*b17 + 3*b16 - 3*b15 - b14 + 2*b13 + 3*b12 - 2*b11 + b10 + 4*b8 - 2*b7 + 2*b6 + b5 - b3 - 13*b2 + b1 - 144) * q^59 + (-5*b17 - 5*b10 + 5*b7 + 5*b6 - 15*b5 - 5*b3 + 5*b2 - 15*b1 + 10) * q^60 + (3*b19 + b18 - b17 + b16 - 5*b15 + 4*b14 + b13 - 3*b12 + 3*b11 - 6*b10 - 3*b8 + 3*b7 + 2*b5 - 2*b4 - 5*b3 - 18*b2 + 10*b1 - 57) * q^61 + (-2*b19 - 5*b18 - b17 - 3*b16 + 2*b15 - 2*b14 - 4*b13 - b11 + 8*b10 + b9 + 5*b8 - b7 + 9*b6 - 7*b5 + 4*b4 + 12*b3 + 24*b2 + 94*b1 + 72) * q^62 + (-6*b18 - 5*b17 - 4*b16 + b13 - 3*b12 + 2*b11 - b10 + 5*b9 + 2*b7 + 13*b6 - 4*b5 + b4 - 3*b3 + 8*b2 + 10*b1 - 26) * q^63 + (-6*b18 - b16 + 4*b15 + 2*b14 + 4*b13 + 6*b12 + b11 + 11*b10 - b8 + 2*b7 - 14*b6 + 23*b5 + 2*b4 + 10*b3 + 9*b2 + 43*b1 + 14) * q^64 + (-5*b18 + 5*b5 + 5*b1 - 15) * q^65 + (-11*b6 - 11*b1 + 33) * q^66 + (-2*b18 + 4*b15 - b14 + 2*b13 + 3*b12 - b11 - 3*b10 + b8 - 3*b7 + 16*b6 - 8*b5 + 6*b4 + 4*b3 + 2*b2 + 45*b1 - 58) * q^67 + (-2*b19 + 2*b18 + b17 - b16 - 3*b14 - b13 - 7*b12 + 6*b11 - 8*b10 + 2*b9 - 3*b8 + 2*b7 + 7*b6 - 19*b5 - 7*b4 - 6*b3 - 36*b2 + 33*b1 - 238) * q^68 + (-2*b18 - 3*b17 + 5*b15 + b14 + 2*b13 + 5*b12 - 4*b11 + 8*b10 - 2*b9 + 4*b8 - b7 + 13*b6 + 3*b5 + 8*b4 + 8*b3 + 9*b2 + 24*b1 - 29) * q^69 + (5*b12 + 5*b7 + 10*b2 + 10*b1 + 25) * q^70 + (-6*b19 - 2*b17 - 5*b16 + 2*b15 + b14 + 4*b11 - 9*b10 - 2*b9 - 5*b8 - 2*b7 + 16*b6 - 20*b5 - 2*b4 - 15*b3 + 7*b2 + 24*b1 - 1) * q^71 + (-2*b19 - 9*b18 - 6*b17 + 5*b16 + 5*b15 - 3*b14 + 10*b13 + 2*b12 + b11 - b10 + 2*b9 + 3*b8 + 6*b7 - 30*b6 - 20*b5 + 9*b3 + 10*b2 - 44*b1 + 12) * q^72 + (-5*b18 - 3*b16 + 5*b15 - 4*b14 + 3*b12 + 11*b10 + b9 + 9*b8 - 5*b7 + 6*b6 - 10*b5 - 3*b4 + 10*b3 + 27*b2 + 70*b1 + 31) * q^73 + (2*b19 + b17 + 4*b16 - 2*b14 + 6*b13 - 3*b12 - 2*b11 - 9*b10 - 3*b8 + 12*b7 - 9*b6 - 26*b5 - 4*b4 - 11*b3 - 32*b2 - 77*b1 - 187) * q^74 + 25*b5 * q^75 + (19*b2 + 57) * q^76 + (-11*b7 - 22) * q^77 + (2*b19 + 4*b18 - 7*b17 + b16 - 3*b15 + b14 - 4*b13 + b12 - b11 - 12*b10 - b9 - 13*b8 + 8*b7 + 26*b6 - 32*b5 - 6*b3 - 10*b2 + 79*b1 - 210) * q^78 + (4*b19 + b18 - b17 - 9*b16 + 3*b15 - 12*b14 + 5*b13 - 6*b12 - 4*b11 + 8*b10 + 8*b9 + 12*b8 - 6*b7 + 7*b6 + 3*b5 + 12*b3 + 24*b2 + 48*b1 + 40) * q^79 + (5*b18 + 5*b17 - 5*b12 - 5*b10 - 5*b9 + 10*b6 - 5*b3 - 15*b2 - 75) * q^80 + (b19 - 3*b18 - 9*b17 + b16 - 5*b15 + 3*b14 + 6*b13 + b12 + b11 + b10 + 5*b9 + 3*b7 - 18*b6 + 13*b5 - 6*b4 - 11*b3 + 11*b2 - 46*b1 - 87) * q^81 + (-2*b19 + 9*b18 + 3*b17 + 5*b16 - 9*b15 + 4*b14 - 3*b13 - 2*b12 + 4*b11 - 20*b10 - 10*b9 - 2*b8 + 7*b7 + 10*b6 - 57*b5 - b4 - 6*b3 - 21*b2 - 40*b1 - 100) * q^82 + (3*b19 - 4*b18 + 5*b17 + 6*b16 - 6*b15 + 5*b14 + 13*b13 - 3*b12 + 2*b11 - 2*b10 + b9 - 2*b8 - 4*b7 - 8*b6 - 14*b5 - b4 - 16*b2 - 5*b1 - 217) * q^83 + (-3*b19 - 13*b18 - b17 - b16 + b15 + 9*b13 - 3*b12 + 2*b11 + 9*b10 - 2*b9 - 4*b8 + 7*b7 - 17*b6 - 5*b5 + 3*b4 + 12*b3 + 15*b2 + 29*b1 - 108) * q^84 + (5*b11 + 5*b9 + 5*b7 + 10*b5 + 10*b2 - 20*b1 + 45) * q^85 + (-7*b19 + b17 - 7*b15 + 5*b14 - 15*b13 + 8*b12 - b11 + 9*b10 - 2*b9 - 6*b8 - b7 + 6*b6 + 42*b5 - 13*b3 + 15*b2 + 95*b1 - 43) * q^86 + (5*b19 + 6*b18 - 4*b17 - 5*b15 + b14 - 6*b13 - 4*b12 - 6*b10 + 2*b9 - 7*b8 + 2*b7 + 5*b6 - 67*b5 - b4 + 28*b2 + 37*b1 - 47) * q^87 + (11*b3 + 33*b1 - 11) * q^88 + (3*b19 + 10*b18 + 5*b17 - 2*b16 - 7*b15 + 2*b14 - 3*b13 - 8*b12 - 4*b11 - 14*b10 - 10*b9 - 12*b8 + 6*b7 + 22*b6 - 58*b5 - 4*b4 - b3 - 26*b2 + 32*b1 - 250) * q^89 + (5*b18 + 5*b17 - 5*b13 - 5*b11 + 15*b6 + 25*b5 - 5*b3 + 5*b2 + 45*b1) * q^90 + (-5*b19 + 5*b18 - 4*b17 + 8*b16 - 5*b15 + 5*b14 - 11*b13 + 3*b12 - b11 + 5*b10 - 9*b9 - 5*b8 + 6*b7 + 11*b6 - 2*b5 - b4 - 10*b3 + 21*b2 - 19*b1 - 5) * q^91 + (-b19 - 14*b18 - 8*b17 + 5*b16 + 3*b15 + 7*b14 + 11*b13 + 7*b12 + 8*b11 - 3*b10 + 2*b9 - b8 + 9*b7 - 32*b6 - 18*b5 - 4*b4 + 20*b3 + 4*b2 + 44*b1 - 156) * q^92 + (-6*b19 + 4*b18 + 3*b17 + 7*b16 + 2*b15 - 3*b14 + 11*b13 + 7*b12 - 5*b11 - 6*b10 - 8*b9 + 6*b8 - 6*b7 + b6 - 63*b5 - b4 - 11*b3 + 34*b2 + 34*b1 + 43) * q^93 + (6*b19 - 2*b18 - 2*b17 - 5*b16 + 5*b15 - 3*b14 - 7*b13 + 3*b12 + 20*b10 + 2*b9 + 7*b8 + 12*b7 - 12*b6 + 18*b5 + 4*b4 + 21*b3 + 20*b2 + 20*b1 + 65) * q^94 - 95 * q^95 + (-3*b19 + 9*b18 + 7*b17 - 3*b16 - 5*b14 + 3*b13 - 5*b12 + 5*b10 + 9*b9 + 12*b8 + b7 - 9*b6 + 41*b5 - 3*b4 - 21*b3 - 12*b2 - 53*b1 - 277) * q^96 + (6*b19 + 7*b18 + b17 - 7*b16 + 3*b15 - 9*b14 + b13 + b12 - b11 + 5*b10 + 6*b9 - 20*b7 + 3*b6 - 10*b5 - 4*b4 + 9*b3 + 45*b2 - 11*b1 + 101) * q^97 + (8*b19 + 2*b18 - 5*b17 + 2*b15 + 3*b14 - b13 - 11*b11 - 3*b10 + 6*b9 - b8 - 15*b7 + 8*b6 - 7*b5 + 5*b4 + b3 - 23*b2 + 30*b1 - 396) * q^98 + (11*b10 + 11*b5 - 77) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10})$$ 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 $$20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100})$$ 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + \cdots + 150528$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 11$$ v^2 - 11 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 19\nu + 1$$ v^3 - 19*v + 1 $$\beta_{4}$$ $$=$$ $$( 17\!\cdots\!19 \nu^{19} + \cdots - 27\!\cdots\!72 ) / 54\!\cdots\!76$$ (1792075879159754310418474819*v^19 - 27420983725696714946766147752*v^18 - 169092526432807655970082784819*v^17 + 2823416111498113291504179810834*v^16 + 6062448295157676956103617219910*v^15 - 117560943520380650694457129847189*v^14 - 96147049284979095898286939882697*v^13 + 2541869673398520220180787244457015*v^12 + 426940389563288603568803602199638*v^11 - 30464718601689822625420585322636713*v^10 + 5800433014004594534132149968517088*v^9 + 200124916623940026841788658288353935*v^8 - 72890875254381189456812911755311727*v^7 - 672509861488318377750729361599361242*v^6 + 248526942059849040104304125651466348*v^5 + 1028773038660102731707656514910839368*v^4 - 291551522423973428517611638792957888*v^3 - 481278885936654605567744173286774624*v^2 + 164481970480440836310269763355569408*v - 271872674486041609972254817820672) / 549488430961654131324559417765376 $$\beta_{5}$$ $$=$$ $$( - 33\!\cdots\!79 \nu^{19} + \cdots - 18\!\cdots\!84 ) / 37\!\cdots\!56$$ (-33161798078530033948132879*v^19 + 67150080257411060458080215*v^18 + 3384583925496875436561193576*v^17 - 6891590308358822970856076506*v^16 - 139213682137955127172831878092*v^15 + 287548439005208674696048645157*v^14 + 2959214882282807359801407261512*v^13 - 6236884526103760675991260567067*v^12 - 34616427907867739159642956461387*v^11 + 74749759359042956588254217260760*v^10 + 219326636183814541855510208459520*v^9 - 486712848272390724253252217830115*v^8 - 696480414057931788329299582665890*v^7 + 1594483459654753297184317812496724*v^6 + 977921122753378741469169149740504*v^5 - 2392539448265013436856652066188432*v^4 - 325380559375304446715237583099520*v^3 + 1259719997993320554070653527853696*v^2 - 238220892728105220172503260211328*v - 18812214678091796807951403920384) / 3763619390148315967976434368256 $$\beta_{6}$$ $$=$$ $$( - 48\!\cdots\!48 \nu^{19} + \cdots + 89\!\cdots\!08 ) / 53\!\cdots\!08$$ (-4855468882697289501421048*v^19 + 13914981821254018284679817*v^18 + 496560729467175639171198567*v^17 - 1430629887918575084823725344*v^16 - 20494346621999382455915897986*v^15 + 59730754461000773944605381052*v^14 + 438125989168702524403048913957*v^13 - 1297727159232129141014958155168*v^12 - 5180779932444396237344580435483*v^11 + 15638207383729796083566475175605*v^10 + 33665197451967583879102599004304*v^9 - 103201721076087639421148012789456*v^8 - 115139199331908710069724740057331*v^7 + 347308903517625226461527951747550*v^6 + 207076964643402490687922854805644*v^5 - 533023880675642690265945815412456*v^4 - 181908170077250088141601729707232*v^3 + 258349782856907039787633336395616*v^2 + 24808062129177813806962059087360*v + 899868432754282707683536727808) / 537659912878330852568062052608 $$\beta_{7}$$ $$=$$ $$( 18\!\cdots\!01 \nu^{19} + \cdots - 22\!\cdots\!32 ) / 10\!\cdots\!52$$ (18637278135300873993523712401*v^19 - 41498658043315185550615372856*v^18 - 1933162516573638653311334012505*v^17 + 4270794713626625547788730424206*v^16 + 81410319439558952448550062458114*v^15 - 178796740378284707931959581077703*v^14 - 1793792915852595131017156592253843*v^13 + 3899851443011183969023975859891957*v^12 + 22246535304188234755797155105816130*v^11 - 47212924468062073290394668342730075*v^10 - 156347999192322115680114048766243192*v^9 + 312877516227577172259713582069915653*v^8 + 607510476087914039320329596392954747*v^7 - 1052037803173919571490606919494175142*v^6 - 1281576981451753668513341842047398764*v^5 + 1577304101918122783658006476808598168*v^4 + 1322390767222538086827039412990851968*v^3 - 688898463693991367317619931726004512*v^2 - 384887057011932990442000487545817344*v - 22620524747788430661655486267026432) / 1098976861923308262649118835530752 $$\beta_{8}$$ $$=$$ $$( - 19\!\cdots\!41 \nu^{19} + \cdots + 10\!\cdots\!48 ) / 10\!\cdots\!52$$ (-19154818243334479077103808141*v^19 + 58352248074339773104434952616*v^18 + 1930345182879855243597409744437*v^17 - 6000559254739860730971769616902*v^16 - 77920815456853432795656119155770*v^15 + 250561157459321508614810531965579*v^14 + 1606939334403664227475796199065239*v^13 - 5444083799429536604425411913730769*v^12 - 17819206009099246409092441803503626*v^11 + 65608019714458432500154405648560975*v^10 + 101411615937990258593232139712573768*v^9 - 433116152824393010886597340149311761*v^8 - 246694322311664327532866610161097407*v^7 + 1460641350282716200981138619228541822*v^6 + 111054161816037080113914594514844284*v^5 - 2269374101048798302059291047612708536*v^4 + 352736008070464788061810903649882624*v^3 + 1169474695665422913563590571905103264*v^2 - 371596507550888213540486537565325568*v + 1064656275509394714885242788200448) / 1098976861923308262649118835530752 $$\beta_{9}$$ $$=$$ $$( 19\!\cdots\!57 \nu^{19} + \cdots + 80\!\cdots\!48 ) / 10\!\cdots\!52$$ (19758477371528064228704499457*v^19 - 13086147915833226818965480436*v^18 - 2028545068380558903928206909469*v^17 + 1362366675836381450076081055646*v^16 + 84121603138394081391619144280154*v^15 - 58584459640273411283984481465191*v^14 - 1810513571378641384539445094166031*v^13 + 1316264988956193932244973901802453*v^12 + 21593758611977362245179212904005078*v^11 - 16269504067829577222190015594007623*v^10 - 140857507685617878180899233759112360*v^9 + 107550118727814825864094034901472725*v^8 + 462377989284860096171609614799379327*v^7 - 347556793862218221766215115525850750*v^6 - 621712609855560436913938800078033372*v^5 + 536910050864295427334680299415104888*v^4 + 67101734219905188226625693364717568*v^3 - 441559811638169365434974755713448352*v^2 + 141041759297341830693694483020164352*v + 80305934574445165520007678561051648) / 1098976861923308262649118835530752 $$\beta_{10}$$ $$=$$ $$( - 22\!\cdots\!35 \nu^{19} + \cdots - 14\!\cdots\!28 ) / 97\!\cdots\!56$$ (-2255232375229393285453835*v^19 + 7700077704922051464715772*v^18 + 232518179348748555386995943*v^17 - 791166828677976629154119538*v^16 - 9708214497676132214301321326*v^15 + 32980723232827475053516083453*v^14 + 211096913521421422955789857317*v^13 - 715274854232661547276622843807*v^12 - 2564414374941391832954933061938*v^11 + 8608414145343594918422354561413*v^10 + 17482867210147834703272240025232*v^9 - 56819063938461296442720838027335*v^8 - 65794051682294441058898314385829*v^7 + 191695881311338815054762152744610*v^6 + 141652555220586796023302738114356*v^5 - 293944521456982425764405605625768*v^4 - 163514599357849991562329261589888*v^3 + 139084144161479572977633688625120*v^2 + 51245418826744070222772805107712*v - 1411440533044883904539854870528) / 97756347796060155012374918656 $$\beta_{11}$$ $$=$$ $$( 27\!\cdots\!89 \nu^{19} + \cdots + 62\!\cdots\!84 ) / 10\!\cdots\!52$$ (27699944044106316829727935889*v^19 - 3631568734237926329838070012*v^18 - 2816644487911163655978649627525*v^17 + 385294068310659215091418494078*v^16 + 115177377823479149689527498099658*v^15 - 17873969268576749162843968223799*v^14 - 2427528369791763940860475866128743*v^13 + 419580422578816520583955030012613*v^12 + 28008056022161672323015221826944494*v^11 - 4811603327193286242469616510580127*v^10 - 172433833238607727592113333479522760*v^9 + 20736463654310580116789251577485253*v^8 + 502287686983352882255928246684238119*v^7 + 20114761869516934643528997360754162*v^6 - 458885420884519916701799690903225660*v^5 - 134333284283188535533634176793839240*v^4 - 426017508918493869965071921491348608*v^3 - 175954501610790635017002896556175008*v^2 + 389832849138410304565230288683726080*v + 62256610406659067324182521247606784) / 1098976861923308262649118835530752 $$\beta_{12}$$ $$=$$ $$( - 51\!\cdots\!07 \nu^{19} + \cdots + 48\!\cdots\!36 ) / 13\!\cdots\!44$$ (-5187332255414398193826921607*v^19 + 8156293209533537694908895307*v^18 + 535539697783034272470890257426*v^17 - 841028360365200758761921721324*v^16 - 22403510789995170359831139144184*v^15 + 35387347367158973488903994323913*v^14 + 489008708419262244156526717038676*v^13 - 776683169211004382408798548122053*v^12 - 5978859982369438559479195210434163*v^11 + 9456331151142143952574903982122466*v^10 + 41016143724131165042266919622048874*v^9 - 62849934618637493232946917522828383*v^8 - 152049739955215635551710663202325382*v^7 + 210800181814691277330652103681837538*v^6 + 291113100930430579022545792322443272*v^5 - 316841959748796708412477646659397416*v^4 - 252369184509833187894845511001868064*v^3 + 148037227094332906647146222057228320*v^2 + 56425725089876509294815237761061120*v + 4812212399666762645774171136343936) / 137372107740413532831139854441344 $$\beta_{13}$$ $$=$$ $$( - 15\!\cdots\!86 \nu^{19} + \cdots - 20\!\cdots\!96 ) / 27\!\cdots\!88$$ (-15499680269117815944930532486*v^19 + 48067420355832495870830648403*v^18 + 1586243405145251378339830322411*v^17 - 4939242591498220018215647571468*v^16 - 65516619847552081868606887486178*v^15 + 206011509808856897765686437578598*v^14 + 1401286143921024046268714637336625*v^13 - 4470591590171749718058951956470726*v^12 - 16565268137642532748475980195671965*v^11 + 53817830168403487056993862222575193*v^10 + 107411511639260094582670275818123344*v^9 - 355052921771099332139121579029464030*v^8 - 365155661909307398886758438178977139*v^7 + 1196954661339078250116496120590465846*v^6 + 650683068435180317198479920189438380*v^5 - 1849771416129262788928238206039850616*v^4 - 566277732371025492412811320418824288*v^3 + 931961903485208275810634275830831264*v^2 + 73720920295144943856735433587629568*v - 20469439300481379236607276747101696) / 274744215480827065662279708882688 $$\beta_{14}$$ $$=$$ $$( 69\!\cdots\!63 \nu^{19} + \cdots - 87\!\cdots\!92 ) / 10\!\cdots\!52$$ (69715835015382015616750989463*v^19 - 32094468751495991555116376616*v^18 - 7228889302019211770841439234063*v^17 + 3357053990935446893969271493698*v^16 + 304279999664396233928143851753934*v^15 - 146439585604072225373576199310545*v^14 - 6706363006706829244484671912444293*v^13 + 3349015163827843443823401440440883*v^12 + 83294660889544335407922419064065518*v^11 - 42072061844246065685857991279248221*v^10 - 585911943589733000964332311449763304*v^9 + 280267374005828490580887778804928771*v^8 + 2250042332792836402918520715304228957*v^7 - 884626118506968940047328434143853226*v^6 - 4420324627836807101072708639827253812*v^5 + 1190972778545719975277895421874222696*v^4 + 3795978142545526658290576228254974976*v^3 - 541890962433968000374421062756310240*v^2 - 1046099396799597840778963122541869824*v - 87670004747503005404920980197076992) / 1098976861923308262649118835530752 $$\beta_{15}$$ $$=$$ $$( - 89\!\cdots\!99 \nu^{19} + \cdots - 25\!\cdots\!64 ) / 10\!\cdots\!52$$ (-89422663409658714611445742599*v^19 + 127088030406980145865731572064*v^18 + 9192272969665003297526050690919*v^17 - 13068193803270481334335458542658*v^16 - 382116217424735475088081244182654*v^15 + 547842887684811517917676645578433*v^14 + 8260984499466044431054778237674029*v^13 - 11942607488586701188640043680243523*v^12 - 99476659783940104617404862355574262*v^11 + 143458563828784934667024886248170181*v^10 + 665181777246593806094343268214374312*v^9 - 928277624124467712776273902706603443*v^8 - 2357552086926890001698936385463817973*v^7 + 2955641976654476832118982353782434138*v^6 + 4182666747740418337262880322476153908*v^5 - 4126041195533470013072359327797279720*v^4 - 3104827263325897521921854214084648192*v^3 + 1824863343331637425810000820129324256*v^2 + 411982462622920323584683944594681088*v - 2508174532479839702362946051390464) / 1098976861923308262649118835530752 $$\beta_{16}$$ $$=$$ $$( - 11\!\cdots\!93 \nu^{19} + \cdots + 10\!\cdots\!72 ) / 13\!\cdots\!44$$ (-11344345310434784725047094393*v^19 + 16854199349582415670675149488*v^18 + 1166843642253135685077583000699*v^17 - 1725943445264190627311485249168*v^16 - 48566139547281503360129564238490*v^15 + 71954633771245316705039190966419*v^14 + 1052700137937707858774398635980347*v^13 - 1557833563218973669003219413926707*v^12 - 12747996394350383982882543269028762*v^11 + 18558589519191212514964979306709389*v^10 + 86362054097314850301009447505732050*v^9 - 118873700205861801325992595981828965*v^8 - 316198959155134868597789192388044899*v^7 + 373520085683188873836433041884982464*v^6 + 608536531150901117421044118888756472*v^5 - 510866858475825971318591039677214568*v^4 - 551862455680805158349623058198585840*v^3 + 206441689219613519582183448640676608*v^2 + 151236861739304494040450909846827136*v + 10413162258406957659639975852127872) / 137372107740413532831139854441344 $$\beta_{17}$$ $$=$$ $$( 12\!\cdots\!71 \nu^{19} + \cdots + 43\!\cdots\!48 ) / 10\!\cdots\!52$$ (122064073083224968976112986871*v^19 - 288196165972624734605362735500*v^18 - 12501469410383785088674047773275*v^17 + 29606514493484479627570661383922*v^16 + 516846600367527093172426597959478*v^15 - 1235852714603023718491663886383905*v^14 - 11072569082274221848606414479826905*v^13 + 26832224683764917332904558447026051*v^12 + 131227121226497584981287045938632410*v^11 - 322531454561737428003037673245504257*v^10 - 852747105529050021985745910301513272*v^9 + 2114716788222236498838741249067066307*v^8 + 2875001355615260716061199396774927177*v^7 - 7021667886188735095651663262672359474*v^6 - 4784570370118512630169406567391995236*v^5 + 10634951345173642692271663961329527816*v^4 + 3300676713034585301200310805189917056*v^3 - 5351410059091882428073959554201832288*v^2 - 78605889794215852249807143130786560*v + 43027358791221476646056541338779648) / 1098976861923308262649118835530752 $$\beta_{18}$$ $$=$$ $$( - 37\!\cdots\!79 \nu^{19} + \cdots - 81\!\cdots\!76 ) / 27\!\cdots\!88$$ (-37327104831218602532821657279*v^19 + 92510047901626771291058008612*v^18 + 3835317758611079402810427893131*v^17 - 9504564301689095641445294184698*v^16 - 159327985485308994383211957553230*v^15 + 396739250061724351446248092987017*v^14 + 3439054409083960533607127184417945*v^13 - 8616359083209848911486154500487883*v^12 - 41278590128183541594939026845038282*v^11 + 103689849935077572521182812242502017*v^10 + 274734453407329984744849282788213824*v^9 - 681709356870164310873409284792056147*v^8 - 974497242028909250980593800141317849*v^7 + 2273939711741516531400393707753032562*v^6 + 1809422490508501051099393714332406380*v^5 - 3439849658794753338283928819765974184*v^4 - 1586505001184143388172659442188742432*v^3 + 1657817618956155754917598407960965408*v^2 + 281214632475051208728310311959031552*v - 8196731865205067949570366490680576) / 274744215480827065662279708882688 $$\beta_{19}$$ $$=$$ $$( 51\!\cdots\!03 \nu^{19} + \cdots + 13\!\cdots\!12 ) / 27\!\cdots\!88$$ (51119164122276371974782875403*v^19 - 97452850610481933134047676892*v^18 - 5243598618514244729974460695727*v^17 + 10014448398997898838236289416074*v^16 + 217310249030532884458911055708062*v^15 - 418708024458066975413838537871517*v^14 - 4675462535313471765450496762032645*v^13 + 9106808293465389136948531481744119*v^12 + 55852984681057220422694063801946946*v^11 - 109526489086147320552572656985592333*v^10 - 368498743132678746864115393572925496*v^9 + 716082983076467989907781435958352775*v^8 + 1278812482477732634072256609079591013*v^7 - 2353604450831243077836574991768502170*v^6 - 2224875629524536104218953415333418404*v^5 + 3505357781124740285039797062556538296*v^4 + 1649700237279243532006461352492903616*v^3 - 1735135764138789488208820745121611936*v^2 - 169586823739864306261728700133200768*v + 13597285350107932784635950653773312) / 274744215480827065662279708882688
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 11$$ b2 + 11 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 19\beta _1 - 1$$ b3 + 19*b1 - 1 $$\nu^{4}$$ $$=$$ $$-\beta_{18} - \beta_{17} + \beta_{12} + \beta_{10} + \beta_{9} - 2\beta_{6} + \beta_{3} + 27\beta_{2} + 215$$ -b18 - b17 + b12 + b10 + b9 - 2*b6 + b3 + 27*b2 + 215 $$\nu^{5}$$ $$=$$ $$3 \beta_{19} - 3 \beta_{18} - 3 \beta_{17} + 3 \beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} + 6 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \beta_{7} - 5 \beta_{6} + 12 \beta_{5} + 2 \beta_{4} + 38 \beta_{3} + 7 \beta_{2} + 422 \beta _1 - 12$$ 3*b19 - 3*b18 - 3*b17 + 3*b15 - b14 + 2*b13 + b12 + 6*b10 + 2*b9 + 2*b8 + b7 - 5*b6 + 12*b5 + 2*b4 + 38*b3 + 7*b2 + 422*b1 - 12 $$\nu^{6}$$ $$=$$ $$- 46 \beta_{18} - 40 \beta_{17} - \beta_{16} + 4 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 46 \beta_{12} + \beta_{11} + 51 \beta_{10} + 40 \beta_{9} - \beta_{8} + 2 \beta_{7} - 94 \beta_{6} + 23 \beta_{5} + 2 \beta_{4} + 50 \beta_{3} + 705 \beta_{2} + 43 \beta _1 + 4902$$ -46*b18 - 40*b17 - b16 + 4*b15 + 2*b14 + 4*b13 + 46*b12 + b11 + 51*b10 + 40*b9 - b8 + 2*b7 - 94*b6 + 23*b5 + 2*b4 + 50*b3 + 705*b2 + 43*b1 + 4902 $$\nu^{7}$$ $$=$$ $$130 \beta_{19} - 159 \beta_{18} - 149 \beta_{17} - 6 \beta_{16} + 147 \beta_{15} - 46 \beta_{14} + 93 \beta_{13} + 59 \beta_{12} + 13 \beta_{11} + 313 \beta_{10} + 104 \beta_{9} + 95 \beta_{8} + 72 \beta_{7} - 301 \beta_{6} + 562 \beta_{5} + \cdots + 579$$ 130*b19 - 159*b18 - 149*b17 - 6*b16 + 147*b15 - 46*b14 + 93*b13 + 59*b12 + 13*b11 + 313*b10 + 104*b9 + 95*b8 + 72*b7 - 301*b6 + 562*b5 + 95*b4 + 1206*b3 + 383*b2 + 10091*b1 + 579 $$\nu^{8}$$ $$=$$ $$11 \beta_{19} - 1661 \beta_{18} - 1292 \beta_{17} - 54 \beta_{16} + 236 \beta_{15} + 84 \beta_{14} + 223 \beta_{13} + 1607 \beta_{12} + 87 \beta_{11} + 1982 \beta_{10} + 1308 \beta_{9} - 22 \beta_{8} + 95 \beta_{7} - 3408 \beta_{6} + \cdots + 120340$$ 11*b19 - 1661*b18 - 1292*b17 - 54*b16 + 236*b15 + 84*b14 + 223*b13 + 1607*b12 + 87*b11 + 1982*b10 + 1308*b9 - 22*b8 + 95*b7 - 3408*b6 + 1660*b5 + 104*b4 + 2000*b3 + 18733*b2 + 2815*b1 + 120340 $$\nu^{9}$$ $$=$$ $$4271 \beta_{19} - 6297 \beta_{18} - 5567 \beta_{17} - 340 \beta_{16} + 5316 \beta_{15} - 1575 \beta_{14} + 3305 \beta_{13} + 2549 \beta_{12} + 795 \beta_{11} + 12086 \beta_{10} + 4017 \beta_{9} + 3441 \beta_{8} + \cdots + 45091$$ 4271*b19 - 6297*b18 - 5567*b17 - 340*b16 + 5316*b15 - 1575*b14 + 3305*b13 + 2549*b12 + 795*b11 + 12086*b10 + 4017*b9 + 3441*b8 + 2969*b7 - 12944*b6 + 20022*b5 + 3359*b4 + 36429*b3 + 16044*b2 + 253158*b1 + 45091 $$\nu^{10}$$ $$=$$ $$846 \beta_{19} - 55036 \beta_{18} - 39243 \beta_{17} - 2185 \beta_{16} + 9967 \beta_{15} + 2533 \beta_{14} + 8997 \beta_{13} + 50974 \beta_{12} + 4270 \beta_{11} + 69417 \beta_{10} + 40328 \beta_{9} + \cdots + 3094240$$ 846*b19 - 55036*b18 - 39243*b17 - 2185*b16 + 9967*b15 + 2533*b14 + 8997*b13 + 50974*b12 + 4270*b11 + 69417*b10 + 40328*b9 + 335*b8 + 3010*b7 - 113033*b6 + 77689*b5 + 4192*b4 + 73125*b3 + 507722*b2 + 130915*b1 + 3094240 $$\nu^{11}$$ $$=$$ $$128035 \beta_{19} - 223257 \beta_{18} - 187549 \beta_{17} - 13615 \beta_{16} + 172773 \beta_{15} - 48817 \beta_{14} + 108006 \beta_{13} + 96255 \beta_{12} + 33885 \beta_{11} + 416453 \beta_{10} + \cdots + 2118774$$ 128035*b19 - 223257*b18 - 187549*b17 - 13615*b16 + 172773*b15 - 48817*b14 + 108006*b13 + 96255*b12 + 33885*b11 + 416453*b10 + 139410*b9 + 112773*b8 + 100193*b7 - 483313*b6 + 654251*b5 + 107032*b4 + 1081059*b3 + 605069*b2 + 6576971*b1 + 2118774 $$\nu^{12}$$ $$=$$ $$44522 \beta_{19} - 1750196 \beta_{18} - 1168530 \beta_{17} - 79459 \beta_{16} + 370735 \beta_{15} + 66442 \beta_{14} + 320437 \beta_{13} + 1550768 \beta_{12} + 168694 \beta_{11} + \cdots + 82177470$$ 44522*b19 - 1750196*b18 - 1168530*b17 - 79459*b16 + 370735*b15 + 66442*b14 + 320437*b13 + 1550768*b12 + 168694*b11 + 2306043*b10 + 1217241*b9 + 51056*b8 + 82758*b7 - 3601153*b6 + 3033677*b5 + 155343*b4 + 2536300*b3 + 13995466*b2 + 5289799*b1 + 82177470 $$\nu^{13}$$ $$=$$ $$3699520 \beta_{19} - 7480110 \beta_{18} - 6013437 \beta_{17} - 476114 \beta_{16} + 5353570 \beta_{15} - 1450529 \beta_{14} + 3410026 \beta_{13} + 3383330 \beta_{12} + 1248054 \beta_{11} + \cdots + 84526989$$ 3699520*b19 - 7480110*b18 - 6013437*b17 - 476114*b16 + 5353570*b15 - 1450529*b14 + 3410026*b13 + 3383330*b12 + 1248054*b11 + 13571362*b10 + 4595583*b9 + 3518967*b8 + 3085952*b7 - 16732248*b6 + 20648724*b5 + 3254967*b4 + 31868126*b3 + 21518924*b2 + 175471987*b1 + 84526989 $$\nu^{14}$$ $$=$$ $$1979265 \beta_{19} - 54415322 \beta_{18} - 34629199 \beta_{17} - 2731437 \beta_{16} + 12932089 \beta_{15} + 1583656 \beta_{14} + 10731240 \beta_{13} + 46273786 \beta_{12} + \cdots + 2234952776$$ 1979265*b19 - 54415322*b18 - 34629199*b17 - 2731437*b16 + 12932089*b15 + 1583656*b14 + 10731240*b13 + 46273786*b12 + 5994451*b11 + 74268574*b10 + 36449766*b9 + 2873370*b8 + 2216123*b7 - 112339735*b6 + 107698235*b5 + 5509703*b4 + 84955450*b3 + 391175908*b2 + 197618710*b1 + 2234952776 $$\nu^{15}$$ $$=$$ $$105330882 \beta_{19} - 242415793 \beta_{18} - 187736926 \beta_{17} - 15577910 \beta_{16} + 162009101 \beta_{15} - 42255623 \beta_{14} + 105779221 \beta_{13} + \cdots + 3105704128$$ 105330882*b19 - 242415793*b18 - 187736926*b17 - 15577910*b16 + 162009101*b15 - 42255623*b14 + 105779221*b13 + 114007801*b12 + 42618649*b11 + 428907363*b10 + 147202031*b9 + 106865206*b8 + 90830278*b7 - 553637757*b6 + 641288574*b5 + 96745170*b4 + 937523228*b3 + 736670315*b2 + 4779653805*b1 + 3105704128 $$\nu^{16}$$ $$=$$ $$79748010 \beta_{19} - 1669451069 \beta_{18} - 1027307523 \beta_{17} - 90564337 \beta_{16} + 434268389 \beta_{15} + 34046648 \beta_{14} + 347412829 \beta_{13} + \cdots + 61887295479$$ 79748010*b19 - 1669451069*b18 - 1027307523*b17 - 90564337*b16 + 434268389*b15 + 34046648*b14 + 347412829*b13 + 1368553291*b12 + 200901098*b11 + 2345031162*b10 + 1088563318*b9 + 125942194*b8 + 61859116*b7 - 3462360151*b6 + 3614575653*b5 + 189587399*b4 + 2777539760*b3 + 11058415360*b2 + 7017447453*b1 + 61887295479 $$\nu^{17}$$ $$=$$ $$2985292839 \beta_{19} - 7692052264 \beta_{18} - 5772398627 \beta_{17} - 491796238 \beta_{16} + 4843493623 \beta_{15} - 1219174744 \beta_{14} + 3245700414 \beta_{13} + \cdots + 108549113772$$ 2985292839*b19 - 7692052264*b18 - 5772398627*b17 - 491796238*b16 + 4843493623*b15 - 1219174744*b14 + 3245700414*b13 + 3740267536*b12 + 1392392528*b11 + 13314754333*b10 + 4633178476*b9 + 3194635139*b8 + 2613283213*b7 - 17796810975*b6 + 19750725668*b5 + 2843341885*b4 + 27584139284*b3 + 24552647349*b2 + 132357450898*b1 + 108549113772 $$\nu^{18}$$ $$=$$ $$3010521868 \beta_{19} - 50805181552 \beta_{18} - 30570060429 \beta_{17} - 2927121960 \beta_{16} + 14224935444 \beta_{15} + 618279097 \beta_{14} + \cdots + 1737872315738$$ 3010521868*b19 - 50805181552*b18 - 30570060429*b17 - 2927121960*b16 + 14224935444*b15 + 618279097*b14 + 11027591910*b13 + 40328797096*b12 + 6503247994*b11 + 73064881348*b10 + 32494166277*b9 + 4895561507*b8 + 1853595974*b7 - 105928178786*b6 + 117087940064*b5 + 6371745715*b4 + 89244718500*b3 + 315567136391*b2 + 240494948454*b1 + 1737872315738 $$\nu^{19}$$ $$=$$ $$84647898327 \beta_{19} - 240671770704 \beta_{18} - 175897812579 \beta_{17} - 15221997041 \beta_{16} + 143922218163 \beta_{15} - 35017992312 \beta_{14} + \cdots + 3668728755334$$ 84647898327*b19 - 240671770704*b18 - 175897812579*b17 - 15221997041*b16 + 143922218163*b15 - 35017992312*b14 + 98835719252*b13 + 120512667360*b12 + 44247618407*b11 + 408904689734*b10 + 144185574772*b9 + 94634149732*b8 + 74391093409*b7 - 561170924129*b6 + 605386684927*b5 + 83154238433*b4 + 812532498323*b3 + 802373727677*b2 + 3714437856979*b1 + 3668728755334

## 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
 5.49185 4.63319 4.18815 3.91893 2.81607 2.35645 2.10393 2.01989 0.783853 0.0251982 −0.130697 −1.09490 −1.13531 −1.56126 −1.78835 −3.55600 −3.59432 −4.38362 −4.96274 −5.13031
−5.49185 3.98529 22.1605 −5.00000 −21.8866 23.4740 −77.7672 −11.1175 27.4593
1.2 −4.63319 −7.91113 13.4664 −5.00000 36.6537 11.8333 −25.3269 35.5860 23.1659
1.3 −4.18815 8.40510 9.54062 −5.00000 −35.2018 −15.6625 −6.45234 43.6456 20.9408
1.4 −3.91893 −3.32066 7.35803 −5.00000 13.0134 −28.1182 2.51584 −15.9732 19.5947
1.5 −2.81607 3.33662 −0.0697562 −5.00000 −9.39614 35.3821 22.7250 −15.8670 14.0803
1.6 −2.35645 −6.45217 −2.44713 −5.00000 15.2042 27.1536 24.6182 14.6305 11.7823
1.7 −2.10393 −1.35130 −3.57349 −5.00000 2.84303 −13.9401 24.3498 −25.1740 10.5196
1.8 −2.01989 7.09682 −3.92003 −5.00000 −14.3348 −8.46588 24.0772 23.3649 10.0995
1.9 −0.783853 0.0603595 −7.38557 −5.00000 −0.0473130 −1.09978 12.0600 −26.9964 3.91926
1.10 −0.0251982 −6.38249 −7.99937 −5.00000 0.160827 −27.7864 0.403155 13.7361 0.125991
1.11 0.130697 8.99140 −7.98292 −5.00000 1.17515 14.2539 −2.08892 53.8454 −0.653486
1.12 1.09490 3.72657 −6.80118 −5.00000 4.08024 3.04995 −16.2059 −13.1127 −5.47452
1.13 1.13531 −7.07177 −6.71106 −5.00000 −8.02868 20.7485 −16.7017 23.0099 −5.67657
1.14 1.56126 −1.47971 −5.56246 −5.00000 −2.31022 22.4493 −21.1746 −24.8104 −7.80632
1.15 1.78835 −7.45188 −4.80180 −5.00000 −13.3266 −20.9424 −22.8941 28.5305 −8.94175
1.16 3.55600 6.62353 4.64511 −5.00000 23.5532 −2.03165 −11.9300 16.8711 −17.7800
1.17 3.59432 4.02265 4.91912 −5.00000 14.4587 10.5448 −11.0737 −10.8183 −17.9716
1.18 4.38362 −4.80730 11.2161 −5.00000 −21.0733 18.1824 14.0981 −3.88988 −21.9181
1.19 4.96274 −9.59476 16.6288 −5.00000 −47.6163 −3.47238 42.8227 65.0593 −24.8137
1.20 5.13031 1.57482 18.3201 −5.00000 8.07930 −16.5525 52.9453 −24.5200 −25.6516
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$11$$ $$1$$
$$19$$ $$-1$$

## 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 1045.4.a.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.c 20 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + T_{2}^{19} - 105 T_{2}^{18} - 103 T_{2}^{17} + 4500 T_{2}^{16} + 4345 T_{2}^{15} - 101844 T_{2}^{14} - 95592 T_{2}^{13} + 1317797 T_{2}^{12} + 1160501 T_{2}^{11} - 9914845 T_{2}^{10} - 7570653 T_{2}^{9} + \cdots + 150528$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1045))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} + T^{19} - 105 T^{18} + \cdots + 150528$$
$3$ $$T^{20} + 8 T^{19} + \cdots + 353918232320$$
$5$ $$(T + 5)^{20}$$
$7$ $$T^{20} - 49 T^{19} + \cdots + 40\!\cdots\!00$$
$11$ $$(T + 11)^{20}$$
$13$ $$T^{20} - 60 T^{19} + \cdots + 30\!\cdots\!92$$
$17$ $$T^{20} + 155 T^{19} + \cdots - 11\!\cdots\!56$$
$19$ $$(T - 19)^{20}$$
$23$ $$T^{20} + 154 T^{19} + \cdots - 79\!\cdots\!36$$
$29$ $$T^{20} + 305 T^{19} + \cdots - 23\!\cdots\!04$$
$31$ $$T^{20} + 759 T^{19} + \cdots - 22\!\cdots\!16$$
$37$ $$T^{20} - 698 T^{19} + \cdots + 15\!\cdots\!52$$
$41$ $$T^{20} - 547 T^{19} + \cdots + 26\!\cdots\!80$$
$43$ $$T^{20} + 925 T^{19} + \cdots + 57\!\cdots\!88$$
$47$ $$T^{20} + 681 T^{19} + \cdots - 30\!\cdots\!20$$
$53$ $$T^{20} + 419 T^{19} + \cdots - 58\!\cdots\!96$$
$59$ $$T^{20} + 2829 T^{19} + \cdots + 39\!\cdots\!56$$
$61$ $$T^{20} + 959 T^{19} + \cdots + 88\!\cdots\!88$$
$67$ $$T^{20} + 1020 T^{19} + \cdots - 57\!\cdots\!64$$
$71$ $$T^{20} - 106 T^{19} + \cdots - 52\!\cdots\!64$$
$73$ $$T^{20} - 558 T^{19} + \cdots - 20\!\cdots\!12$$
$79$ $$T^{20} - 536 T^{19} + \cdots + 37\!\cdots\!72$$
$83$ $$T^{20} + 4179 T^{19} + \cdots - 79\!\cdots\!76$$
$89$ $$T^{20} + 4120 T^{19} + \cdots - 28\!\cdots\!80$$
$97$ $$T^{20} - 1414 T^{19} + \cdots - 18\!\cdots\!40$$