# Properties

 Label 1045.2.b.c Level $1045$ Weight $2$ Character orbit 1045.b Analytic conductor $8.344$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.34436701122$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 26 x^{18} + 281 x^{16} + 1640 x^{14} + 5623 x^{12} + 11551 x^{10} + 13894 x^{8} + 9095 x^{6} + 2753 x^{4} + 276 x^{2} + 4$$ x^20 + 26*x^18 + 281*x^16 + 1640*x^14 + 5623*x^12 + 11551*x^10 + 13894*x^8 + 9095*x^6 + 2753*x^4 + 276*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{11} q^{5} + (\beta_{12} + \beta_{2} - 1) q^{6} + \beta_{3} q^{7} + ( - \beta_{7} + \beta_{6}) q^{8} + (\beta_{8} - 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + b7 * q^3 + (b2 - 1) * q^4 - b11 * q^5 + (b12 + b2 - 1) * q^6 + b3 * q^7 + (-b7 + b6) * q^8 + (b8 - 1) * q^9 $$q + \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{11} q^{5} + (\beta_{12} + \beta_{2} - 1) q^{6} + \beta_{3} q^{7} + ( - \beta_{7} + \beta_{6}) q^{8} + (\beta_{8} - 1) q^{9} + ( - \beta_{17} - \beta_{14} - \beta_{7} + \beta_{3} + \beta_1) q^{10} + q^{11} + (\beta_{4} - \beta_1) q^{12} + (\beta_{19} - \beta_{7} + \beta_{3}) q^{13} + (\beta_{18} - \beta_{15} + \beta_{14} - \beta_{8} - \beta_{2} + 1) q^{14} + ( - \beta_{18} + \beta_{15} + \beta_{11} + \beta_{8} + \beta_{7} + \beta_{4}) q^{15} + (\beta_{18} + \beta_{16} - \beta_{12} + \beta_{5} - \beta_{2}) q^{16} + ( - \beta_{13} - \beta_{11} + \beta_{10} - \beta_{4} - \beta_1) q^{17} + ( - \beta_{19} - \beta_{17} - \beta_{6} + \beta_{4}) q^{18} + q^{19} + ( - \beta_{16} - \beta_{15} + \beta_{14} - \beta_{8} + \beta_{3}) q^{20} + (\beta_{18} - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{2} - 2) q^{21} + \beta_1 q^{22} + ( - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{11} + \beta_{10} - 2 \beta_{7} - \beta_{6} - \beta_{4} + \cdots + 2 \beta_1) q^{23}+ \cdots + (\beta_{8} - 1) q^{99}+O(q^{100})$$ q + b1 * q^2 + b7 * q^3 + (b2 - 1) * q^4 - b11 * q^5 + (b12 + b2 - 1) * q^6 + b3 * q^7 + (-b7 + b6) * q^8 + (b8 - 1) * q^9 + (-b17 - b14 - b7 + b3 + b1) * q^10 + q^11 + (b4 - b1) * q^12 + (b19 - b7 + b3) * q^13 + (b18 - b15 + b14 - b8 - b2 + 1) * q^14 + (-b18 + b15 + b11 + b8 + b7 + b4) * q^15 + (b18 + b16 - b12 + b5 - b2) * q^16 + (-b13 - b11 + b10 - b4 - b1) * q^17 + (-b19 - b17 - b6 + b4) * q^18 + q^19 + (-b16 - b15 + b14 - b8 + b3) * q^20 + (b18 - b15 + b14 - b12 - b2 - 2) * q^21 + b1 * q^22 + (-b15 - b14 - b13 - b11 + b10 - 2*b7 - b6 - b4 + 2*b3 + 2*b1) * q^23 + (b18 + b9 - b8 - b2 + 2) * q^24 + (-b19 - b17 + b16 + b11 - b9 + b4) * q^25 + (b18 - b16 - b15 + b14 - b12 + b9 - b8 - b5 - b2) * q^26 + (b19 - b17 - b15 - b14 - b10 - 2*b7 + b6 - b4 + b1) * q^27 + (-b19 + b15 + b14 - b10 + 3*b7 - b6 + b4 - 2*b3) * q^28 + (-b18 + b16 + b15 - b14 - b13 + b12 + b11 + 3) * q^29 + (b19 + b18 + b17 - b15 - b13 - b12 - b11 + b10 + b9 - b8 - b7 - b4 + 2*b3 - b2 + b1 - 1) * q^30 + (-2*b18 + b15 - b14 - b13 + 2*b12 + b11 + b8 - 2*b5 + 3*b2 - 4) * q^31 + (-b19 - 2*b17 + b13 + b11 - b10 - b7 + b4) * q^32 + b7 * q^33 + (b12 - b9 + b5 + 2) * q^34 + (-b18 + b17 + b15 - b13 + b11 + 2*b7 - b5 - b3 - b2 + 1) * q^35 + (-b18 - b16 - b13 - b12 + b11 + b8) * q^36 + (-b19 - b17 + b15 + b14 + 2*b13 + 2*b11 - b10 + 3*b7 - 2*b6 + 3*b4 - b3 - 2*b1) * q^37 + b1 * q^38 + (b18 + b16 - b15 + b14 - b13 - b12 + b11 - 2*b8 + b5 - b2 + 3) * q^39 + (b19 + b18 + b17 - b13 - b11 - b8 + 2*b6 - b4 - b3 - b2 - b1 + 1) * q^40 + (-b15 + b14 - b9 + 2*b5 - 1) * q^41 + (-2*b19 - b17 + b15 + b14 - b10 + 4*b7 - b6 + b4 - 4*b3 - 3*b1) * q^42 + (-2*b19 - 2*b17 + b13 + b11 - b10 - b7 - b6 + b4) * q^43 + (b2 - 1) * q^44 + (b19 - b15 - b14 - b13 - b11 + b8 - 4*b7 + b6 - 2*b4 + 2*b3 - b2 + 3*b1 - 1) * q^45 + (b18 - b16 - 2*b15 + 2*b14 - b12 - b9 - 2*b8 + b2 - 2) * q^46 + (3*b19 + b17 - 2*b15 - 2*b14 - 2*b13 - 2*b11 + b10 - 4*b7 + 2*b6 - 2*b4 + 2*b3 + b1) * q^47 + (-b19 - b6 + b4 - b3) * q^48 + (-b18 + b16 + b15 - b14 + 3*b12 - b9 + b8 - 2*b5 + 4*b2 - 3) * q^49 + (-b17 + b14 - b12 + 2*b11 - b10 - b8 + 2*b7 - b6 + b5 + 2*b4 - b3 - 2*b2 + 3) * q^50 + (-b18 + b16 + b15 - b14 - b13 - b12 + b11 + 2*b8 - b5 + b2 - 3) * q^51 + (b17 + b15 + b14 - b13 - b11 + 2*b7 + b6 - 2*b4 - 3*b3 - 2*b1) * q^52 + (2*b19 - b15 - b14 - 3*b13 - 3*b11 + b10 - 4*b7 + 2*b6 - 3*b4 - b1) * q^53 + (-2*b18 - b16 + b15 - b14 - b13 + 2*b12 + b11 + 2*b8 - b5 + 2*b2 - 1) * q^54 - b11 * q^55 + (-b18 + b15 - b14 + 2*b12 - b5 + 3*b2 - 3) * q^56 + b7 * q^57 + (2*b19 + b17 - b15 - b14 - b13 - b11 + b10 - 3*b7 - b6 - b4 + 3*b3 + 5*b1) * q^58 + (-3*b16 - b15 + b14 + b13 - b11 + b5 - b2 + 3) * q^59 + (-b19 + b18 + 2*b14 - b12 + 3*b7 - 2*b3 - b2 - 2*b1 - 2) * q^60 + (b18 + b15 - b14 - b12 + b8 + b5 - b2) * q^61 + (b19 + b17 - b15 - b14 - 2*b13 - 2*b11 + 2*b10 - 5*b7 + 2*b6 - b4 + 2*b3 - 2*b1) * q^62 + (-b17 - 2*b10 - 3*b7 + b6 - b3) * q^63 + (b16 - 2*b13 - 2*b12 + 2*b11 + 2*b5 - 3*b2 + 4) * q^64 + (-b19 - b17 - b15 + b13 - b12 + b11 - b10 + b9 - b8 + b7 - b3 - b2 + 2*b1 + 2) * q^65 + (b12 + b2 - 1) * q^66 + (2*b19 + b17 - 2*b15 - 2*b14 - 3*b13 - 3*b11 + b10 - 5*b7 - 5*b4 + 3*b3 + 2*b1) * q^67 + (b19 + b17 + b15 + b14 + b10 + b7 - b6 + b4 - b1) * q^68 + (-b16 - 2*b15 + 2*b14 + b13 - b11 - b9 - b5 + b2 - 2) * q^69 + (b19 + 2*b17 + b16 + b15 - b14 + b12 - 2*b11 + b10 + b8 + 2*b7 - 2*b4 + 2*b2 + 2*b1 - 4) * q^70 + (-b18 + 2*b16 + b15 - b14 - b13 - b12 + b11 + 2*b8 + b5 - 2) * q^71 + (-b19 + 2*b17 + b15 + b14 - b13 - b11 + b10 + 3*b7 - b3 - 3*b1) * q^72 + (b19 + 4*b17 + b15 + b14 - b13 - b11 + b10 + 2*b7 - b6 - b4 + b3 - 2*b1) * q^73 + (-2*b18 - 2*b16 - b13 - b12 + b11 + 2*b9 - b8 - 2*b5 + b2 + 3) * q^74 + (-b19 - 2*b17 - b16 - b14 - b12 + b11 - b10 + b9 + b8 - b7 + b6 - b5 + b4 - b3 - 2*b2 + b1) * q^75 + (b2 - 1) * q^76 + b3 * q^77 + (b19 + 2*b17 + b15 + b14 - b13 - b11 - b10 + 3*b7 - b4 - 4*b3 - b1) * q^78 + (b18 - b16 + b12 + b9 - 2*b8 - b5 - 2*b2 + 6) * q^79 + (-b19 + b18 + b15 + b14 + 3*b13 + b12 + b11 - b10 + 3*b7 - b6 + b5 + b4 - 2*b2 + 1) * q^80 + (2*b18 + b16 + 2*b13 - 2*b11 + 2*b9 - 2*b8 + 2*b5) * q^81 + (2*b19 + 2*b17 + b15 + b14 - b10 + b7 + 2*b4 - b3 - 4*b1) * q^82 + (-b19 - b17 - b15 - b14 - 4*b7 + b6 - b4 + b3 + 3*b1) * q^83 + (-4*b18 + 3*b15 - 3*b14 - b13 + 2*b12 + b11 - b9 + 4*b8 + 3*b2 - 1) * q^84 + (b17 - b15 - b12 - b9 + b8 + 2*b5 + b4 - b2 + b1) * q^85 + (-3*b18 - b16 - 2*b13 + 2*b11 - b9 + 5) * q^86 + (-2*b19 + b13 + b11 + b10 + 4*b7 - 2*b6 + 2*b3) * q^87 + (-b7 + b6) * q^88 + (-b18 + b16 - b15 + b14 + b12 - 2*b9 + b5 + 3*b2 + 1) * q^89 + (-b19 + b18 - b17 - b15 + b14 - b9 + b7 - 2*b6 + b4 + b1 - 4) * q^90 + (-b18 + b16 + b15 - b14 + b13 + 3*b12 - b11 - b9 + b8 + b5 + 4*b2 - 4) * q^91 + (3*b17 + b15 + b14 - b13 - b11 + 3*b7 + 2*b6 - b4 - 4*b3 - 6*b1) * q^92 + (b19 - b17 - b15 - b14 + b13 + b11 - b10 - 5*b7 + b6 + 2*b3) * q^93 + (4*b18 - b15 + b14 + b13 - 2*b12 - b11 + b9 - b8 - 4*b2 + 1) * q^94 - b11 * q^95 + (b18 + b15 - b14 - 2*b12 + 2*b9 - 2*b8 - 2*b2 + 5) * q^96 + (b17 - b13 - b11 + b10 + 2*b7 - b6 + 2*b4 - 2*b3 - 2*b1) * q^97 + (-b19 - 4*b17 - b15 - b14 + b13 + b11 - 6*b7 + 4*b4 + 2*b3 + b1) * q^98 + (b8 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10})$$ 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9 $$20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99}+O(q^{100})$$ 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9 - 6 * q^10 + 20 * q^11 + 24 * q^14 - 6 * q^15 - 4 * q^16 + 20 * q^19 - 6 * q^20 - 30 * q^21 + 38 * q^24 + 2 * q^25 + 8 * q^26 + 50 * q^29 - 20 * q^30 - 50 * q^31 + 28 * q^34 + 6 * q^35 - 12 * q^36 + 48 * q^39 + 12 * q^40 - 34 * q^41 - 12 * q^44 - 18 * q^45 - 36 * q^46 - 6 * q^49 + 26 * q^50 - 40 * q^51 - 6 * q^54 - 40 * q^56 + 30 * q^59 - 30 * q^60 - 14 * q^61 + 36 * q^64 + 30 * q^65 - 8 * q^66 - 12 * q^69 - 54 * q^70 - 40 * q^71 + 50 * q^74 - 8 * q^75 - 12 * q^76 + 106 * q^79 + 8 * q^80 - 30 * q^84 - 22 * q^85 + 56 * q^86 + 36 * q^89 - 64 * q^90 - 56 * q^91 + 28 * q^94 + 66 * q^96 - 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 26 x^{18} + 281 x^{16} + 1640 x^{14} + 5623 x^{12} + 11551 x^{10} + 13894 x^{8} + 9095 x^{6} + 2753 x^{4} + 276 x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ v^2 + 3 $$\beta_{3}$$ $$=$$ $$( 61 \nu^{19} + 1299 \nu^{17} + 10262 \nu^{15} + 33228 \nu^{13} + 5459 \nu^{11} - 240718 \nu^{9} - 592650 \nu^{7} - 522039 \nu^{5} - 131958 \nu^{3} + 1604 \nu ) / 1076$$ (61*v^19 + 1299*v^17 + 10262*v^15 + 33228*v^13 + 5459*v^11 - 240718*v^9 - 592650*v^7 - 522039*v^5 - 131958*v^3 + 1604*v) / 1076 $$\beta_{4}$$ $$=$$ $$( 57 \nu^{19} + 1205 \nu^{17} + 9104 \nu^{15} + 23376 \nu^{13} - 49969 \nu^{11} - 428112 \nu^{9} - 946122 \nu^{7} - 866947 \nu^{5} - 288912 \nu^{3} - 27024 \nu ) / 1076$$ (57*v^19 + 1205*v^17 + 9104*v^15 + 23376*v^13 - 49969*v^11 - 428112*v^9 - 946122*v^7 - 866947*v^5 - 288912*v^3 - 27024*v) / 1076 $$\beta_{5}$$ $$=$$ $$( - 127 \nu^{18} - 3119 \nu^{16} - 31252 \nu^{14} - 164582 \nu^{12} - 489083 \nu^{10} - 819526 \nu^{8} - 735502 \nu^{6} - 319949 \nu^{4} - 59110 \nu^{2} - 1602 ) / 538$$ (-127*v^18 - 3119*v^16 - 31252*v^14 - 164582*v^12 - 489083*v^10 - 819526*v^8 - 735502*v^6 - 319949*v^4 - 59110*v^2 - 1602) / 538 $$\beta_{6}$$ $$=$$ $$( 424 \nu^{19} + 10233 \nu^{17} + 99883 \nu^{15} + 504698 \nu^{13} + 1398132 \nu^{11} + 2051391 \nu^{9} + 1360162 \nu^{7} + 189834 \nu^{5} - 78267 \nu^{3} - 8898 \nu ) / 1076$$ (424*v^19 + 10233*v^17 + 99883*v^15 + 504698*v^13 + 1398132*v^11 + 2051391*v^9 + 1360162*v^7 + 189834*v^5 - 78267*v^3 - 8898*v) / 1076 $$\beta_{7}$$ $$=$$ $$( 424 \nu^{19} + 10233 \nu^{17} + 99883 \nu^{15} + 504698 \nu^{13} + 1398132 \nu^{11} + 2051391 \nu^{9} + 1360162 \nu^{7} + 189834 \nu^{5} - 79343 \nu^{3} - 13202 \nu ) / 1076$$ (424*v^19 + 10233*v^17 + 99883*v^15 + 504698*v^13 + 1398132*v^11 + 2051391*v^9 + 1360162*v^7 + 189834*v^5 - 79343*v^3 - 13202*v) / 1076 $$\beta_{8}$$ $$=$$ $$( - 119 \nu^{18} - 2931 \nu^{16} - 29474 \nu^{14} - 155907 \nu^{12} - 465652 \nu^{10} - 783140 \nu^{8} - 697023 \nu^{6} - 282189 \nu^{4} - 36260 \nu^{2} - 29 ) / 269$$ (-119*v^18 - 2931*v^16 - 29474*v^14 - 155907*v^12 - 465652*v^10 - 783140*v^8 - 697023*v^6 - 282189*v^4 - 36260*v^2 - 29) / 269 $$\beta_{9}$$ $$=$$ $$( - 284 \nu^{18} - 6943 \nu^{16} - 69037 \nu^{14} - 358938 \nu^{12} - 1043638 \nu^{10} - 1682291 \nu^{8} - 1400840 \nu^{6} - 516802 \nu^{4} - 68197 \nu^{2} - 1100 ) / 538$$ (-284*v^18 - 6943*v^16 - 69037*v^14 - 358938*v^12 - 1043638*v^10 - 1682291*v^8 - 1400840*v^6 - 516802*v^4 - 68197*v^2 - 1100) / 538 $$\beta_{10}$$ $$=$$ $$( 169 \nu^{19} + 4644 \nu^{17} + 53633 \nu^{15} + 338506 \nu^{13} + 1269599 \nu^{11} + 2874857 \nu^{9} + 3799744 \nu^{7} + 2655663 \nu^{5} + 772083 \nu^{3} + 44494 \nu ) / 1076$$ (169*v^19 + 4644*v^17 + 53633*v^15 + 338506*v^13 + 1269599*v^11 + 2874857*v^9 + 3799744*v^7 + 2655663*v^5 + 772083*v^3 + 44494*v) / 1076 $$\beta_{11}$$ $$=$$ $$( - 563 \nu^{19} - 1152 \nu^{18} - 13634 \nu^{17} - 28148 \nu^{16} - 133533 \nu^{15} - 279704 \nu^{14} - 676778 \nu^{13} - 1452564 \nu^{12} - 1878649 \nu^{11} - 4211192 \nu^{10} + \cdots + 2676 ) / 2152$$ (-563*v^19 - 1152*v^18 - 13634*v^17 - 28148*v^16 - 133533*v^15 - 279704*v^14 - 676778*v^13 - 1452564*v^12 - 1878649*v^11 - 4211192*v^10 - 2756833*v^9 - 6729844*v^8 - 1827220*v^7 - 5449516*v^6 - 280745*v^5 - 1815624*v^4 + 55629*v^3 - 151708*v^2 - 2730*v + 2676) / 2152 $$\beta_{12}$$ $$=$$ $$( - 791 \nu^{18} - 19261 \nu^{16} - 190662 \nu^{14} - 986020 \nu^{12} - 2846233 \nu^{10} - 4530894 \nu^{8} - 3666446 \nu^{6} - 1246615 \nu^{4} - 131302 \nu^{2} + \cdots - 3848 ) / 1076$$ (-791*v^18 - 19261*v^16 - 190662*v^14 - 986020*v^12 - 2846233*v^10 - 4530894*v^8 - 3666446*v^6 - 1246615*v^4 - 131302*v^2 - 3848) / 1076 $$\beta_{13}$$ $$=$$ $$( - 563 \nu^{19} + 1152 \nu^{18} - 13634 \nu^{17} + 28148 \nu^{16} - 133533 \nu^{15} + 279704 \nu^{14} - 676778 \nu^{13} + 1452564 \nu^{12} - 1878649 \nu^{11} + 4211192 \nu^{10} + \cdots - 2676 ) / 2152$$ (-563*v^19 + 1152*v^18 - 13634*v^17 + 28148*v^16 - 133533*v^15 + 279704*v^14 - 676778*v^13 + 1452564*v^12 - 1878649*v^11 + 4211192*v^10 - 2756833*v^9 + 6729844*v^8 - 1827220*v^7 + 5449516*v^6 - 280745*v^5 + 1815624*v^4 + 55629*v^3 + 151708*v^2 - 2730*v - 2676) / 2152 $$\beta_{14}$$ $$=$$ $$( 269 \nu^{19} + 502 \nu^{18} + 6994 \nu^{17} + 12335 \nu^{16} + 75589 \nu^{15} + 123271 \nu^{14} + 441160 \nu^{13} + 643550 \nu^{12} + 1512587 \nu^{11} + 1873190 \nu^{10} + \cdots + 1126 ) / 1076$$ (269*v^19 + 502*v^18 + 6994*v^17 + 12335*v^16 + 75589*v^15 + 123271*v^14 + 441160*v^13 + 643550*v^12 + 1512587*v^11 + 1873190*v^10 + 3107219*v^9 + 2997551*v^8 + 3737486*v^7 + 2419870*v^6 + 2446555*v^5 + 802784*v^4 + 740557*v^3 + 76329*v^2 + 74244*v + 1126) / 1076 $$\beta_{15}$$ $$=$$ $$( 269 \nu^{19} - 502 \nu^{18} + 6994 \nu^{17} - 12335 \nu^{16} + 75589 \nu^{15} - 123271 \nu^{14} + 441160 \nu^{13} - 643550 \nu^{12} + 1512587 \nu^{11} - 1873190 \nu^{10} + \cdots - 1126 ) / 1076$$ (269*v^19 - 502*v^18 + 6994*v^17 - 12335*v^16 + 75589*v^15 - 123271*v^14 + 441160*v^13 - 643550*v^12 + 1512587*v^11 - 1873190*v^10 + 3107219*v^9 - 2997551*v^8 + 3737486*v^7 - 2419870*v^6 + 2446555*v^5 - 802784*v^4 + 740557*v^3 - 76329*v^2 + 74244*v - 1126) / 1076 $$\beta_{16}$$ $$=$$ $$( 615 \nu^{18} + 15125 \nu^{16} + 151546 \nu^{14} + 795708 \nu^{12} + 2343125 \nu^{10} + 3838002 \nu^{8} + 3254612 \nu^{6} + 1214287 \nu^{4} + 153152 \nu^{2} + 3674 ) / 538$$ (615*v^18 + 15125*v^16 + 151546*v^14 + 795708*v^12 + 2343125*v^10 + 3838002*v^8 + 3254612*v^6 + 1214287*v^4 + 153152*v^2 + 3674) / 538 $$\beta_{17}$$ $$=$$ $$( - 604 \nu^{19} - 15001 \nu^{17} - 152531 \nu^{15} - 819456 \nu^{13} - 2505428 \nu^{11} - 4382125 \nu^{9} - 4207528 \nu^{7} - 2033120 \nu^{5} - 434513 \nu^{3} + \cdots - 28512 \nu ) / 538$$ (-604*v^19 - 15001*v^17 - 152531*v^15 - 819456*v^13 - 2505428*v^11 - 4382125*v^9 - 4207528*v^7 - 2033120*v^5 - 434513*v^3 - 28512*v) / 538 $$\beta_{18}$$ $$=$$ $$( - 1767 \nu^{18} - 43273 \nu^{16} - 431250 \nu^{14} - 2248272 \nu^{12} - 6554317 \nu^{10} - 10567846 \nu^{8} - 8704666 \nu^{6} - 3034215 \nu^{4} - 311854 \nu^{2} + \cdots - 460 ) / 1076$$ (-1767*v^18 - 43273*v^16 - 431250*v^14 - 2248272*v^12 - 6554317*v^10 - 10567846*v^8 - 8704666*v^6 - 3034215*v^4 - 311854*v^2 - 460) / 1076 $$\beta_{19}$$ $$=$$ $$( 1317 \nu^{19} + 32698 \nu^{17} + 332179 \nu^{15} + 1781218 \nu^{13} + 5425363 \nu^{11} + 9417307 \nu^{9} + 8896864 \nu^{7} + 4138215 \nu^{5} + 803421 \nu^{3} + \cdots + 40090 \nu ) / 1076$$ (1317*v^19 + 32698*v^17 + 332179*v^15 + 1781218*v^13 + 5425363*v^11 + 9417307*v^9 + 8896864*v^7 + 4138215*v^5 + 803421*v^3 + 40090*v) / 1076
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ b2 - 3 $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} - 4\beta_1$$ -b7 + b6 - 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{18} + \beta_{16} - \beta_{12} + \beta_{5} - 7\beta_{2} + 14$$ b18 + b16 - b12 + b5 - 7*b2 + 14 $$\nu^{5}$$ $$=$$ $$-\beta_{19} - 2\beta_{17} + \beta_{13} + \beta_{11} - \beta_{10} + 7\beta_{7} - 8\beta_{6} + \beta_{4} + 20\beta_1$$ -b19 - 2*b17 + b13 + b11 - b10 + 7*b7 - 8*b6 + b4 + 20*b1 $$\nu^{6}$$ $$=$$ $$-10\beta_{18} - 9\beta_{16} - 2\beta_{13} + 8\beta_{12} + 2\beta_{11} - 8\beta_{5} + 43\beta_{2} - 72$$ -10*b18 - 9*b16 - 2*b13 + 8*b12 + 2*b11 - 8*b5 + 43*b2 - 72 $$\nu^{7}$$ $$=$$ $$12 \beta_{19} + 24 \beta_{17} + \beta_{15} + \beta_{14} - 11 \beta_{13} - 11 \beta_{11} + 10 \beta_{10} - 41 \beta_{7} + 54 \beta_{6} - 12 \beta_{4} - \beta_{3} - 111 \beta_1$$ 12*b19 + 24*b17 + b15 + b14 - 11*b13 - 11*b11 + 10*b10 - 41*b7 + 54*b6 - 12*b4 - b3 - 111*b1 $$\nu^{8}$$ $$=$$ $$75 \beta_{18} + 66 \beta_{16} + 2 \beta_{15} - 2 \beta_{14} + 24 \beta_{13} - 51 \beta_{12} - 24 \beta_{11} + 3 \beta_{8} + 52 \beta_{5} - 257 \beta_{2} + 389$$ 75*b18 + 66*b16 + 2*b15 - 2*b14 + 24*b13 - 51*b12 - 24*b11 + 3*b8 + 52*b5 - 257*b2 + 389 $$\nu^{9}$$ $$=$$ $$- 101 \beta_{19} - 206 \beta_{17} - 15 \beta_{15} - 15 \beta_{14} + 88 \beta_{13} + 88 \beta_{11} - 75 \beta_{10} + 226 \beta_{7} - 350 \beta_{6} + 102 \beta_{4} + 20 \beta_{3} + 651 \beta_1$$ -101*b19 - 206*b17 - 15*b15 - 15*b14 + 88*b13 + 88*b11 - 75*b10 + 226*b7 - 350*b6 + 102*b4 + 20*b3 + 651*b1 $$\nu^{10}$$ $$=$$ $$- 509 \beta_{18} - 455 \beta_{16} - 33 \beta_{15} + 33 \beta_{14} - 206 \beta_{13} + 303 \beta_{12} + 206 \beta_{11} + \beta_{9} - 47 \beta_{8} - 324 \beta_{5} + 1533 \beta_{2} - 2175$$ -509*b18 - 455*b16 - 33*b15 + 33*b14 - 206*b13 + 303*b12 + 206*b11 + b9 - 47*b8 - 324*b5 + 1533*b2 - 2175 $$\nu^{11}$$ $$=$$ $$741 \beta_{19} + 1554 \beta_{17} + 151 \beta_{15} + 151 \beta_{14} - 631 \beta_{13} - 631 \beta_{11} + 510 \beta_{10} - 1211 \beta_{7} + 2241 \beta_{6} - 764 \beta_{4} - 238 \beta_{3} - 3940 \beta_1$$ 741*b19 + 1554*b17 + 151*b15 + 151*b14 - 631*b13 - 631*b11 + 510*b10 - 1211*b7 + 2241*b6 - 764*b4 - 238*b3 - 3940*b1 $$\nu^{12}$$ $$=$$ $$3303 \beta_{18} + 3054 \beta_{16} + 359 \beta_{15} - 359 \beta_{14} + 1554 \beta_{13} - 1747 \beta_{12} - 1554 \beta_{11} - 23 \beta_{9} + 492 \beta_{8} + 2010 \beta_{5} - 9190 \beta_{2} + 12486$$ 3303*b18 + 3054*b16 + 359*b15 - 359*b14 + 1554*b13 - 1747*b12 - 1554*b11 - 23*b9 + 492*b8 + 2010*b5 - 9190*b2 + 12486 $$\nu^{13}$$ $$=$$ $$- 5088 \beta_{19} - 11001 \beta_{17} - 1282 \beta_{15} - 1282 \beta_{14} + 4316 \beta_{13} + 4316 \beta_{11} - 3326 \beta_{10} + 6386 \beta_{7} - 14292 \beta_{6} + 5397 \beta_{4} + 2261 \beta_{3} + \cdots + 24301 \beta_1$$ -5088*b19 - 11001*b17 - 1282*b15 - 1282*b14 + 4316*b13 + 4316*b11 - 3326*b10 + 6386*b7 - 14292*b6 + 5397*b4 + 2261*b3 + 24301*b1 $$\nu^{14}$$ $$=$$ $$- 20961 \beta_{18} - 20205 \beta_{16} - 3251 \beta_{15} + 3251 \beta_{14} - 11001 \beta_{13} + 9919 \beta_{12} + 11001 \beta_{11} + 309 \beta_{9} - 4332 \beta_{8} - 12530 \beta_{5} + 55455 \beta_{2} + \cdots - 73183$$ -20961*b18 - 20205*b16 - 3251*b15 + 3251*b14 - 11001*b13 + 9919*b12 + 11001*b11 + 309*b9 - 4332*b8 - 12530*b5 + 55455*b2 - 73183 $$\nu^{15}$$ $$=$$ $$33724 \beta_{19} + 75175 \beta_{17} + 9936 \beta_{15} + 9936 \beta_{14} - 28837 \beta_{13} - 28837 \beta_{11} + 21270 \beta_{10} - 33281 \beta_{7} + 91034 \beta_{6} - 36953 \beta_{4} - 19008 \beta_{3} + \cdots - 151663 \beta_1$$ 33724*b19 + 75175*b17 + 9936*b15 + 9936*b14 - 28837*b13 - 28837*b11 + 21270*b10 - 33281*b7 + 91034*b6 - 36953*b4 - 19008*b3 - 151663*b1 $$\nu^{16}$$ $$=$$ $$131518 \beta_{18} + 132485 \beta_{16} + 26575 \beta_{15} - 26575 \beta_{14} + 75175 \beta_{13} - 55820 \beta_{12} - 75175 \beta_{11} - 3229 \beta_{9} + 34691 \beta_{8} + 78580 \beta_{5} - 336819 \beta_{2} + \cdots + 436023$$ 131518*b18 + 132485*b16 + 26575*b15 - 26575*b14 + 75175*b13 - 55820*b12 - 75175*b11 - 3229*b9 + 34691*b8 + 78580*b5 - 336819*b2 + 436023 $$\nu^{17}$$ $$=$$ $$- 219147 \beta_{19} - 502949 \beta_{17} - 72913 \beta_{15} - 72913 \beta_{14} + 190159 \beta_{13} + 190159 \beta_{11} - 134747 \beta_{10} + 171462 \beta_{7} - 579693 \beta_{6} + \cdots + 953706 \beta_1$$ -219147*b19 - 502949*b17 - 72913*b15 - 72913*b14 + 190159*b13 + 190159*b11 - 134747*b10 + 171462*b7 - 579693*b6 + 248365*b4 + 148300*b3 + 953706*b1 $$\nu^{18}$$ $$=$$ $$- 820724 \beta_{18} - 863495 \beta_{16} - 203712 \beta_{15} + 203712 \beta_{14} - 502949 \beta_{13} + 312428 \beta_{12} + 502949 \beta_{11} + 29218 \beta_{9} - 261918 \beta_{8} + \cdots - 2631488$$ -820724*b18 - 863495*b16 - 203712*b15 + 203712*b14 - 502949*b13 + 312428*b12 + 502949*b11 + 29218*b9 - 261918*b8 - 495293*b5 + 2058073*b2 - 2631488 $$\nu^{19}$$ $$=$$ $$1408073 \beta_{19} + 3320237 \beta_{17} + 516502 \beta_{15} + 516502 \beta_{14} - 1243813 \beta_{13} - 1243813 \beta_{11} + 849942 \beta_{10} - 871422 \beta_{7} + 3691782 \beta_{6} + \cdots - 6026704 \beta_1$$ 1408073*b19 + 3320237*b17 + 516502*b15 + 516502*b14 - 1243813*b13 - 1243813*b11 + 849942*b10 - 871422*b7 + 3691782*b6 - 1649374*b4 - 1101444*b3 - 6026704*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$$.

 $$n$$ $$496$$ $$761$$ $$837$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
419.1
 − 2.52711i − 2.33380i − 2.04146i − 2.00304i − 1.51095i − 1.23709i − 1.22468i − 0.712399i − 0.386431i − 0.131596i 0.131596i 0.386431i 0.712399i 1.22468i 1.23709i 1.51095i 2.00304i 2.04146i 2.33380i 2.52711i
2.52711i 0.473551i −4.38628 2.21995 0.267979i −1.19672 1.17543i 6.03039i 2.77575 −0.677212 5.61006i
419.2 2.33380i 1.97214i −3.44661 −1.75107 1.39059i −4.60257 4.79282i 3.37609i −0.889329 −3.24536 + 4.08665i
419.3 2.04146i 1.84076i −2.16755 −2.00596 0.987985i 3.75784 2.14151i 0.342047i −0.388411 −2.01693 + 4.09509i
419.4 2.00304i 2.62978i −2.01218 0.741511 + 2.10954i −5.26757 4.24477i 0.0243959i −3.91575 4.22550 1.48528i
419.5 1.51095i 0.900785i −0.282982 0.672383 2.13258i 1.36104 0.603482i 2.59434i 2.18859 −3.22223 1.01594i
419.6 1.23709i 2.07277i 0.469598 2.13362 0.669071i −2.56421 0.314194i 3.05513i −1.29637 −0.827703 2.63949i
419.7 1.22468i 1.51450i 0.500160 −1.12399 + 1.93304i 1.85478 0.966830i 3.06189i 0.706295 2.36736 + 1.37652i
419.8 0.712399i 3.32227i 1.49249 1.31011 + 1.81207i 2.36678 2.70450i 2.48804i −8.03747 1.29092 0.933320i
419.9 0.386431i 0.261531i 1.85067 0.0276061 2.23590i 0.101064 4.13791i 1.48802i 2.93160 −0.864019 0.0106678i
419.10 0.131596i 1.44045i 1.98268 −2.22416 0.230433i 0.189557 0.456875i 0.524103i 0.925103 −0.0303239 + 0.292690i
419.11 0.131596i 1.44045i 1.98268 −2.22416 + 0.230433i 0.189557 0.456875i 0.524103i 0.925103 −0.0303239 0.292690i
419.12 0.386431i 0.261531i 1.85067 0.0276061 + 2.23590i 0.101064 4.13791i 1.48802i 2.93160 −0.864019 + 0.0106678i
419.13 0.712399i 3.32227i 1.49249 1.31011 1.81207i 2.36678 2.70450i 2.48804i −8.03747 1.29092 + 0.933320i
419.14 1.22468i 1.51450i 0.500160 −1.12399 1.93304i 1.85478 0.966830i 3.06189i 0.706295 2.36736 1.37652i
419.15 1.23709i 2.07277i 0.469598 2.13362 + 0.669071i −2.56421 0.314194i 3.05513i −1.29637 −0.827703 + 2.63949i
419.16 1.51095i 0.900785i −0.282982 0.672383 + 2.13258i 1.36104 0.603482i 2.59434i 2.18859 −3.22223 + 1.01594i
419.17 2.00304i 2.62978i −2.01218 0.741511 2.10954i −5.26757 4.24477i 0.0243959i −3.91575 4.22550 + 1.48528i
419.18 2.04146i 1.84076i −2.16755 −2.00596 + 0.987985i 3.75784 2.14151i 0.342047i −0.388411 −2.01693 4.09509i
419.19 2.33380i 1.97214i −3.44661 −1.75107 + 1.39059i −4.60257 4.79282i 3.37609i −0.889329 −3.24536 4.08665i
419.20 2.52711i 0.473551i −4.38628 2.21995 + 0.267979i −1.19672 1.17543i 6.03039i 2.77575 −0.677212 + 5.61006i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.20 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.c 20
5.b even 2 1 inner 1045.2.b.c 20
5.c odd 4 2 5225.2.a.ba 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.c 20 1.a even 1 1 trivial
1045.2.b.c 20 5.b even 2 1 inner
5225.2.a.ba 20 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + 26 T_{2}^{18} + 281 T_{2}^{16} + 1640 T_{2}^{14} + 5623 T_{2}^{12} + 11551 T_{2}^{10} + 13894 T_{2}^{8} + 9095 T_{2}^{6} + 2753 T_{2}^{4} + 276 T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1045, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} + 26 T^{18} + 281 T^{16} + 1640 T^{14} + \cdots + 4$$
$3$ $$T^{20} + 35 T^{18} + 500 T^{16} + \cdots + 256$$
$5$ $$T^{20} - T^{18} + 6 T^{17} + \cdots + 9765625$$
$7$ $$T^{20} + 73 T^{18} + 2053 T^{16} + \cdots + 2304$$
$11$ $$(T - 1)^{20}$$
$13$ $$T^{20} + 137 T^{18} + \cdots + 33732864$$
$17$ $$T^{20} + 157 T^{18} + 9767 T^{16} + \cdots + 7054336$$
$19$ $$(T - 1)^{20}$$
$23$ $$T^{20} + 220 T^{18} + \cdots + 172764736$$
$29$ $$(T^{10} - 25 T^{9} + 153 T^{8} + \cdots + 4194192)^{2}$$
$31$ $$(T^{10} + 25 T^{9} + 124 T^{8} - 1350 T^{7} + \cdots - 7456)^{2}$$
$37$ $$T^{20} + 411 T^{18} + \cdots + 61309721664$$
$41$ $$(T^{10} + 17 T^{9} - 44 T^{8} + \cdots + 166848)^{2}$$
$43$ $$T^{20} + 379 T^{18} + \cdots + 7783474176$$
$47$ $$T^{20} + 606 T^{18} + \cdots + 47\!\cdots\!44$$
$53$ $$T^{20} + 641 T^{18} + \cdots + 58\!\cdots\!36$$
$59$ $$(T^{10} - 15 T^{9} - 296 T^{8} + \cdots + 86423136)^{2}$$
$61$ $$(T^{10} + 7 T^{9} - 203 T^{8} + \cdots - 26701648)^{2}$$
$67$ $$T^{20} + \cdots + 345472702914816$$
$71$ $$(T^{10} + 20 T^{9} - 73 T^{8} + \cdots - 400032)^{2}$$
$73$ $$T^{20} + 964 T^{18} + \cdots + 27\!\cdots\!84$$
$79$ $$(T^{10} - 53 T^{9} + 990 T^{8} + \cdots - 204893824)^{2}$$
$83$ $$T^{20} + 425 T^{18} + \cdots + 5917659525376$$
$89$ $$(T^{10} - 18 T^{9} - 234 T^{8} + \cdots + 14189256)^{2}$$
$97$ $$T^{20} + 1248 T^{18} + \cdots + 19285658703936$$
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