Properties

 Label 1045.2.b.b Level $1045$ Weight $2$ Character orbit 1045.b Analytic conductor $8.344$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

Related objects

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: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 19x^{14} + 144x^{12} + 552x^{10} + 1119x^{8} + 1146x^{6} + 524x^{4} + 83x^{2} + 4$$ x^16 + 19*x^14 + 144*x^12 + 552*x^10 + 1119*x^8 + 1146*x^6 + 524*x^4 + 83*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 19x^{14} + 144x^{12} + 552x^{10} + 1119x^{8} + 1146x^{6} + 524x^{4} + 83x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{4}$$ $$=$$ $$\nu^{4} + 5\nu^{2} + 3$$ v^4 + 5*v^2 + 3 $$\beta_{5}$$ $$=$$ $$\nu^{5} + 6\nu^{3} + 7\nu$$ v^5 + 6*v^3 + 7*v $$\beta_{6}$$ $$=$$ $$\nu^{6} + 7\nu^{4} + 12\nu^{2} + 3$$ v^6 + 7*v^4 + 12*v^2 + 3 $$\beta_{7}$$ $$=$$ $$-\nu^{12} - 14\nu^{10} - 72\nu^{8} - 165\nu^{6} - 163\nu^{4} - 60\nu^{2} - 6$$ -v^12 - 14*v^10 - 72*v^8 - 165*v^6 - 163*v^4 - 60*v^2 - 6 $$\beta_{8}$$ $$=$$ $$\nu^{12} + 15\nu^{10} + 84\nu^{8} + 215\nu^{6} + 246\nu^{4} + 104\nu^{2} + 7$$ v^12 + 15*v^10 + 84*v^8 + 215*v^6 + 246*v^4 + 104*v^2 + 7 $$\beta_{9}$$ $$=$$ $$\nu^{12} + 15\nu^{10} + 85\nu^{8} + 225\nu^{6} + 277\nu^{4} + 134\nu^{2} + 13$$ v^12 + 15*v^10 + 85*v^8 + 225*v^6 + 277*v^4 + 134*v^2 + 13 $$\beta_{10}$$ $$=$$ $$( -\nu^{15} - 17\nu^{13} - 114\nu^{11} - 382\nu^{9} - 667\nu^{7} - 572\nu^{5} - 198\nu^{3} - 15\nu ) / 2$$ (-v^15 - 17*v^13 - 114*v^11 - 382*v^9 - 667*v^7 - 572*v^5 - 198*v^3 - 15*v) / 2 $$\beta_{11}$$ $$=$$ $$( -\nu^{15} - 19\nu^{13} - 144\nu^{11} - 552\nu^{9} - 1117\nu^{7} - 1128\nu^{5} - 476\nu^{3} - 47\nu ) / 2$$ (-v^15 - 19*v^13 - 144*v^11 - 552*v^9 - 1117*v^7 - 1128*v^5 - 476*v^3 - 47*v) / 2 $$\beta_{12}$$ $$=$$ $$( \nu^{15} + 2 \nu^{14} + 19 \nu^{13} + 34 \nu^{12} + 144 \nu^{11} + 228 \nu^{10} + 552 \nu^{9} + 764 \nu^{8} + 1119 \nu^{7} + 1334 \nu^{6} + 1146 \nu^{5} + 1144 \nu^{4} + 524 \nu^{3} + 396 \nu^{2} + 83 \nu + 30 ) / 4$$ (v^15 + 2*v^14 + 19*v^13 + 34*v^12 + 144*v^11 + 228*v^10 + 552*v^9 + 764*v^8 + 1119*v^7 + 1334*v^6 + 1146*v^5 + 1144*v^4 + 524*v^3 + 396*v^2 + 83*v + 30) / 4 $$\beta_{13}$$ $$=$$ $$( - \nu^{15} + 2 \nu^{14} - 19 \nu^{13} + 34 \nu^{12} - 144 \nu^{11} + 228 \nu^{10} - 552 \nu^{9} + 764 \nu^{8} - 1119 \nu^{7} + 1334 \nu^{6} - 1146 \nu^{5} + 1144 \nu^{4} - 524 \nu^{3} + 396 \nu^{2} + \cdots + 30 ) / 4$$ (-v^15 + 2*v^14 - 19*v^13 + 34*v^12 - 144*v^11 + 228*v^10 - 552*v^9 + 764*v^8 - 1119*v^7 + 1334*v^6 - 1146*v^5 + 1144*v^4 - 524*v^3 + 396*v^2 - 83*v + 30) / 4 $$\beta_{14}$$ $$=$$ $$-\nu^{15} - 19\nu^{13} - 143\nu^{11} - 538\nu^{9} - 1047\nu^{7} - 981\nu^{5} - 362\nu^{3} - 28\nu$$ -v^15 - 19*v^13 - 143*v^11 - 538*v^9 - 1047*v^7 - 981*v^5 - 362*v^3 - 28*v $$\beta_{15}$$ $$=$$ $$-\nu^{15} - 19\nu^{13} - 144\nu^{11} - 551\nu^{9} - 1107\nu^{7} - 1096\nu^{5} - 440\nu^{3} - 34\nu$$ -v^15 - 19*v^13 - 144*v^11 - 551*v^9 - 1107*v^7 - 1096*v^5 - 440*v^3 - 34*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4\beta_1$$ b3 - 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 5\beta_{2} + 7$$ b4 - 5*b2 + 7 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 6\beta_{3} + 17\beta_1$$ b5 - 6*b3 + 17*b1 $$\nu^{6}$$ $$=$$ $$\beta_{6} - 7\beta_{4} + 23\beta_{2} - 28$$ b6 - 7*b4 + 23*b2 - 28 $$\nu^{7}$$ $$=$$ $$-\beta_{13} + \beta_{12} + \beta_{11} - 9\beta_{5} + 30\beta_{3} - 75\beta_1$$ -b13 + b12 + b11 - 9*b5 + 30*b3 - 75*b1 $$\nu^{8}$$ $$=$$ $$\beta_{9} - \beta_{8} - 10\beta_{6} + 39\beta_{4} - 105\beta_{2} + 117$$ b9 - b8 - 10*b6 + 39*b4 - 105*b2 + 117 $$\nu^{9}$$ $$=$$ $$\beta_{15} + 10\beta_{13} - 10\beta_{12} - 12\beta_{11} + 58\beta_{5} - 144\beta_{3} + 337\beta_1$$ b15 + 10*b13 - 10*b12 - 12*b11 + 58*b5 - 144*b3 + 337*b1 $$\nu^{10}$$ $$=$$ $$-12\beta_{9} + 13\beta_{8} + \beta_{7} + 70\beta_{6} - 201\beta_{4} + 481\beta_{2} - 498$$ -12*b9 + 13*b8 + b7 + 70*b6 - 201*b4 + 481*b2 - 498 $$\nu^{11}$$ $$=$$ $$-14\beta_{15} + \beta_{14} - 70\beta_{13} + 70\beta_{12} + 96\beta_{11} - 329\beta_{5} + 684\beta_{3} - 1530\beta_1$$ -14*b15 + b14 - 70*b13 + 70*b12 + 96*b11 - 329*b5 + 684*b3 - 1530*b1 $$\nu^{12}$$ $$=$$ $$96\beta_{9} - 110\beta_{8} - 15\beta_{7} - 425\beta_{6} + 998\beta_{4} - 2214\beta_{2} + 2141$$ 96*b9 - 110*b8 - 15*b7 - 425*b6 + 998*b4 - 2214*b2 + 2141 $$\nu^{13}$$ $$=$$ $$125 \beta_{15} - 15 \beta_{14} + 425 \beta_{13} - 425 \beta_{12} - 646 \beta_{11} + \beta_{10} + 1752 \beta_{5} - 3241 \beta_{3} + 6994 \beta_1$$ 125*b15 - 15*b14 + 425*b13 - 425*b12 - 646*b11 + b10 + 1752*b5 - 3241*b3 + 6994*b1 $$\nu^{14}$$ $$=$$ $$\beta_{13} + \beta_{12} - 646 \beta_{9} + 770 \beta_{8} + 141 \beta_{7} + 2398 \beta_{6} - 4853 \beta_{4} + 10235 \beta_{2} - 9266$$ b13 + b12 - 646*b9 + 770*b8 + 141*b7 + 2398*b6 - 4853*b4 + 10235*b2 - 9266 $$\nu^{15}$$ $$=$$ $$- 911 \beta_{15} + 141 \beta_{14} - 2398 \beta_{13} + 2398 \beta_{12} + 3955 \beta_{11} - 19 \beta_{10} - 9003 \beta_{5} + 15353 \beta_{3} - 32134 \beta_1$$ -911*b15 + 141*b14 - 2398*b13 + 2398*b12 + 3955*b11 - 19*b10 - 9003*b5 + 15353*b3 - 32134*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
 − 2.18657i − 2.10564i − 2.05535i − 1.78099i − 1.19140i − 0.853553i − 0.387488i − 0.301154i 0.301154i 0.387488i 0.853553i 1.19140i 1.78099i 2.05535i 2.10564i 2.18657i
2.18657i 1.58567i −2.78109 −0.558280 + 2.16525i 3.46718 2.56330i 1.70792i 0.485645 4.73448 + 1.22072i
419.2 2.10564i 1.90142i −2.43371 1.75301 + 1.38815i −4.00370 0.116917i 0.913237i −0.615392 2.92295 3.69120i
419.3 2.05535i 2.80376i −2.22445 −2.02523 + 0.947852i −5.76269 1.02917i 0.461316i −4.86106 1.94816 + 4.16256i
419.4 1.78099i 0.213501i −1.17191 2.04104 0.913326i 0.380243 3.50934i 1.47481i 2.95442 −1.62662 3.63506i
419.5 1.19140i 2.62369i 0.580557 0.0644373 2.23514i 3.12588 0.593512i 3.07449i −3.88377 −2.66295 0.0767708i
419.6 0.853553i 1.84364i 1.27145 1.54048 1.62078i −1.57364 2.69678i 2.79235i −0.399003 −1.38342 1.31488i
419.7 0.387488i 0.494325i 1.84985 −1.95299 + 1.08895i −0.191545 2.37207i 1.49177i 2.75564 0.421956 + 0.756762i
419.8 0.301154i 1.85377i 1.90931 0.637551 + 2.14325i 0.558272 1.94668i 1.17730i −0.436480 0.645450 0.192001i
419.9 0.301154i 1.85377i 1.90931 0.637551 2.14325i 0.558272 1.94668i 1.17730i −0.436480 0.645450 + 0.192001i
419.10 0.387488i 0.494325i 1.84985 −1.95299 1.08895i −0.191545 2.37207i 1.49177i 2.75564 0.421956 0.756762i
419.11 0.853553i 1.84364i 1.27145 1.54048 + 1.62078i −1.57364 2.69678i 2.79235i −0.399003 −1.38342 + 1.31488i
419.12 1.19140i 2.62369i 0.580557 0.0644373 + 2.23514i 3.12588 0.593512i 3.07449i −3.88377 −2.66295 + 0.0767708i
419.13 1.78099i 0.213501i −1.17191 2.04104 + 0.913326i 0.380243 3.50934i 1.47481i 2.95442 −1.62662 + 3.63506i
419.14 2.05535i 2.80376i −2.22445 −2.02523 0.947852i −5.76269 1.02917i 0.461316i −4.86106 1.94816 4.16256i
419.15 2.10564i 1.90142i −2.43371 1.75301 1.38815i −4.00370 0.116917i 0.913237i −0.615392 2.92295 + 3.69120i
419.16 2.18657i 1.58567i −2.78109 −0.558280 2.16525i 3.46718 2.56330i 1.70792i 0.485645 4.73448 1.22072i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.16 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.b 16
5.b even 2 1 inner 1045.2.b.b 16
5.c odd 4 2 5225.2.a.z 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.b 16 1.a even 1 1 trivial
1045.2.b.b 16 5.b even 2 1 inner
5225.2.a.z 16 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 19T_{2}^{14} + 144T_{2}^{12} + 552T_{2}^{10} + 1119T_{2}^{8} + 1146T_{2}^{6} + 524T_{2}^{4} + 83T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1045, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 19 T^{14} + 144 T^{12} + 552 T^{10} + \cdots + 4$$
$3$ $$T^{16} + 28 T^{14} + 316 T^{12} + \cdots + 64$$
$5$ $$T^{16} - 3 T^{15} + 8 T^{14} + \cdots + 390625$$
$7$ $$T^{16} + 37 T^{14} + 537 T^{12} + \cdots + 64$$
$11$ $$(T + 1)^{16}$$
$13$ $$T^{16} + 81 T^{14} + 2287 T^{12} + \cdots + 65536$$
$17$ $$T^{16} + 113 T^{14} + 4843 T^{12} + \cdots + 6390784$$
$19$ $$(T + 1)^{16}$$
$23$ $$T^{16} + 181 T^{14} + 11120 T^{12} + \cdots + 1936$$
$29$ $$(T^{8} - T^{7} - 59 T^{6} + 151 T^{5} + \cdots - 784)^{2}$$
$31$ $$(T^{8} + 16 T^{7} + 6 T^{6} - 972 T^{5} + \cdots - 3104)^{2}$$
$37$ $$T^{16} + 204 T^{14} + \cdots + 476636224$$
$41$ $$(T^{8} - 3 T^{7} - 68 T^{6} + 157 T^{5} + \cdots - 3152)^{2}$$
$43$ $$T^{16} + 315 T^{14} + \cdots + 277688896$$
$47$ $$T^{16} + 374 T^{14} + \cdots + 37937690176$$
$53$ $$T^{16} + 369 T^{14} + \cdots + 32319410176$$
$59$ $$(T^{8} - 12 T^{7} - 250 T^{6} + \cdots + 9973312)^{2}$$
$61$ $$(T^{8} + 21 T^{7} + 3 T^{6} - 3213 T^{5} + \cdots + 269488)^{2}$$
$67$ $$T^{16} + 479 T^{14} + \cdots + 3129530826304$$
$71$ $$(T^{8} + 23 T^{7} - 28 T^{6} + \cdots + 1022368)^{2}$$
$73$ $$T^{16} + 644 T^{14} + \cdots + 3803560873984$$
$79$ $$(T^{8} - 37 T^{7} + 328 T^{6} + \cdots + 143552)^{2}$$
$83$ $$T^{16} + 665 T^{14} + \cdots + 28036855560256$$
$89$ $$(T^{8} - 7 T^{7} - 163 T^{6} + 619 T^{5} + \cdots - 10084)^{2}$$
$97$ $$T^{16} + \cdots + 574639907835904$$