# Properties

 Label 460.2.e.a Level $460$ Weight $2$ Character orbit 460.e Analytic conductor $3.673$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: 16.0.7465802011608416256.3 Defining polynomial: $$x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256$$ x^16 - x^14 + x^12 + 8*x^10 - 20*x^8 + 32*x^6 + 16*x^4 - 64*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + \beta_{8} q^{3} + (\beta_{8} - \beta_{2}) q^{4} + \beta_{3} q^{5} + (\beta_{10} - \beta_{8} + \beta_{6} + \beta_{2} + 1) q^{6} + ( - 2 \beta_{15} - \beta_{13} + 2 \beta_{12}) q^{7} + (2 \beta_{10} - \beta_{8} + \beta_{2}) q^{8} + (\beta_{14} - \beta_{10} + \beta_{8} - \beta_{6} - \beta_{4} - \beta_{2}) q^{9}+O(q^{10})$$ q - b4 * q^2 + b8 * q^3 + (b8 - b2) * q^4 + b3 * q^5 + (b10 - b8 + b6 + b2 + 1) * q^6 + (-2*b15 - b13 + 2*b12) * q^7 + (2*b10 - b8 + b2) * q^8 + (b14 - b10 + b8 - b6 - b4 - b2) * q^9 $$q - \beta_{4} q^{2} + \beta_{8} q^{3} + (\beta_{8} - \beta_{2}) q^{4} + \beta_{3} q^{5} + (\beta_{10} - \beta_{8} + \beta_{6} + \beta_{2} + 1) q^{6} + ( - 2 \beta_{15} - \beta_{13} + 2 \beta_{12}) q^{7} + (2 \beta_{10} - \beta_{8} + \beta_{2}) q^{8} + (\beta_{14} - \beta_{10} + \beta_{8} - \beta_{6} - \beta_{4} - \beta_{2}) q^{9} + \beta_{12} q^{10} + ( - \beta_{15} + \beta_{13} + \beta_{12}) q^{11} + ( - \beta_{10} - \beta_{9} - \beta_{4} - \beta_{2} - 2) q^{12} + (\beta_{6} + 2 \beta_{4} - \beta_{2}) q^{13} + (\beta_{15} + \beta_{12} + 2 \beta_{11} - \beta_{7} - \beta_{5} - 3 \beta_{3} - \beta_1) q^{14} + (\beta_{15} + \beta_{11} - \beta_{5}) q^{15} + ( - 2 \beta_{9} - \beta_{8} - \beta_{2}) q^{16} + (2 \beta_{15} - \beta_{12} + \beta_{11} + \beta_{7} + \beta_{5} + 4 \beta_1) q^{17} + (\beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{4} + \beta_{2} + 2) q^{18} + (2 \beta_{15} + \beta_{13} - 2 \beta_{12}) q^{19} + (\beta_{12} + \beta_{11} - \beta_{5} - \beta_1) q^{20} + ( - \beta_{15} - \beta_{12} + \beta_{11} + \beta_{7} + \beta_{5} + 3 \beta_{3} + \beta_1) q^{21} + ( - \beta_{15} + 2 \beta_{12} + \beta_{11} + \beta_{7} - 2 \beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{22} + ( - \beta_{15} - \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{10} + \beta_{6} - \beta_{4} - \beta_{2}) q^{23} + (2 \beta_{14} + \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{4} + \beta_{2}) q^{24} - q^{25} + (\beta_{10} - 2 \beta_{8} - \beta_{6} + 2 \beta_{2} - 3) q^{26} + ( - \beta_{14} - \beta_{10} - 2 \beta_{9} + 2 \beta_{6} - 2 \beta_{4} - \beta_{2}) q^{27} + ( - 2 \beta_{15} - \beta_{13} - \beta_{12} + 3 \beta_{7} + 3 \beta_{5}) q^{28} + ( - \beta_{6} - \beta_{4} - 3) q^{29} + ( - \beta_{15} + \beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{30} + ( - 3 \beta_{14} - 3 \beta_{10} + \beta_{9} + 4 \beta_{8} - 3 \beta_{2}) q^{31} + (4 \beta_{14} + 2 \beta_{10} + \beta_{8} - 2 \beta_{4} + 3 \beta_{2}) q^{32} + (\beta_{15} - 2 \beta_{12} + 2 \beta_{11} + 2 \beta_{7} + 2 \beta_{5} + 3 \beta_{3} + 5 \beta_1) q^{33} + ( - 4 \beta_{15} + \beta_{13} - 2 \beta_{12} - 3 \beta_{11} - 3 \beta_{7} - 3 \beta_1) q^{34} + ( - \beta_{9} - \beta_{6} + \beta_{4}) q^{35} + ( - 2 \beta_{14} + \beta_{10} - \beta_{9} - 2 \beta_{8} + 2 \beta_{6} - \beta_{4} - \beta_{2} + \cdots + 2) q^{36}+ \cdots + (5 \beta_{15} - 2 \beta_{13} + 4 \beta_{12} + 9 \beta_{11} + 3 \beta_{7} - 9 \beta_{5}) q^{99}+O(q^{100})$$ q - b4 * q^2 + b8 * q^3 + (b8 - b2) * q^4 + b3 * q^5 + (b10 - b8 + b6 + b2 + 1) * q^6 + (-2*b15 - b13 + 2*b12) * q^7 + (2*b10 - b8 + b2) * q^8 + (b14 - b10 + b8 - b6 - b4 - b2) * q^9 + b12 * q^10 + (-b15 + b13 + b12) * q^11 + (-b10 - b9 - b4 - b2 - 2) * q^12 + (b6 + 2*b4 - b2) * q^13 + (b15 + b12 + 2*b11 - b7 - b5 - 3*b3 - b1) * q^14 + (b15 + b11 - b5) * q^15 + (-2*b9 - b8 - b2) * q^16 + (2*b15 - b12 + b11 + b7 + b5 + 4*b1) * q^17 + (b10 + b9 + 2*b8 - b4 + b2 + 2) * q^18 + (2*b15 + b13 - 2*b12) * q^19 + (b12 + b11 - b5 - b1) * q^20 + (-b15 - b12 + b11 + b7 + b5 + 3*b3 + b1) * q^21 + (-b15 + 2*b12 + b11 + b7 - 2*b5 - 3*b3 - 2*b1) * q^22 + (-b15 - b14 - 2*b13 + b12 - b10 + b6 - b4 - b2) * q^23 + (2*b14 + b10 + b9 + 2*b8 + b4 + b2) * q^24 - q^25 + (b10 - 2*b8 - b6 + 2*b2 - 3) * q^26 + (-b14 - b10 - 2*b9 + 2*b6 - 2*b4 - b2) * q^27 + (-2*b15 - b13 - b12 + 3*b7 + 3*b5) * q^28 + (-b6 - b4 - 3) * q^29 + (-b15 + b7 + b5 + b3 + b1) * q^30 + (-3*b14 - 3*b10 + b9 + 4*b8 - 3*b2) * q^31 + (4*b14 + 2*b10 + b8 - 2*b4 + 3*b2) * q^32 + (b15 - 2*b12 + 2*b11 + 2*b7 + 2*b5 + 3*b3 + 5*b1) * q^33 + (-4*b15 + b13 - 2*b12 - 3*b11 - 3*b7 - 3*b1) * q^34 + (-b9 - b6 + b4) * q^35 + (-2*b14 + b10 - b9 - 2*b8 + 2*b6 - b4 - b2 + 2) * q^36 + (2*b15 + 2*b12 + b11 + b7 + b5 + 3*b3 + b1) * q^37 + (-b15 - b12 - 2*b11 + b7 + b5 + 3*b3 + b1) * q^38 + (-2*b14 - 2*b10 - b9 + b8 - b6 + b4 - 2*b2) * q^39 + (b12 + b11 + 2*b7 + b5 + b1) * q^40 + (b14 - b10 + b8 + 2*b4 - 3*b2 - 3) * q^41 + (-b15 + b13 + b12 - 3*b11 + 3*b5 - 3*b3) * q^42 + (-b15 + b13 - 2*b12 - 3*b11 - 3*b7 + 3*b5) * q^43 + (2*b15 + b13 - 2*b12 + 3*b11 + 3*b7 + 3*b1) * q^44 + (b12 - b11 - b7 - b5 - 2*b1) * q^45 + (2*b15 - b12 + b11 + b9 + b8 - 2*b7 - b6 + b5 + b4 - 2*b2 + b1 - 3) * q^46 + (-3*b14 - 3*b10 - 6*b8 - 5*b6 + 5*b4 - 3*b2) * q^47 + (-2*b14 + b10 - b9 - 2*b8 + 2*b6 - b4 + 3*b2 + 2) * q^48 + (3*b14 - 3*b10 + 3*b8 + 3*b6 - 3*b4 + 3*b2 + 5) * q^49 + b4 * q^50 + (b15 - 4*b13 + 2*b12 + 3*b11 + 3*b7 - 3*b5) * q^51 + (-3*b10 - b9 + b8 + 3*b4 - 2*b2 + 2) * q^52 + (-2*b15 - 2*b12 - 4*b11 - 4*b7 - 4*b5 - 4*b1) * q^53 + (4*b14 + 2*b10 + b9 + 2*b8 + b6 - b4 + b2 - 3) * q^54 + (b9 - 2*b6 + 2*b4) * q^55 + (4*b15 + 3*b13 - 5*b12 - 4*b11 - b7 + 5*b5 + 2*b1) * q^56 + (b15 + b12 - b11 - b7 - b5 - 3*b3 - b1) * q^57 + (b8 + 3*b4 - b2 + 2) * q^58 + (-4*b14 - 4*b10 - 2*b6 + 2*b4 - 4*b2) * q^59 + (b13 - b11 - b7 - 2*b3 - b1) * q^60 + (2*b15 - b12 - 2*b11 - 2*b7 - 2*b5 - 3*b3 + b1) * q^61 + (-2*b14 + 3*b10 + 3*b9 - 4*b8 + 4*b4 - b2) * q^62 + (b15 - 4*b13 + 5*b12 + 6*b11 - 6*b5) * q^63 + (-2*b9 + 3*b8 + 4*b6 - 4*b4 + 3*b2 + 4) * q^64 + (-2*b15 - b12 - b1) * q^65 + (-5*b15 + 2*b13 - b12 - 6*b11 - 3*b7 + 3*b5 - 3*b3 - 3*b1) * q^66 + (-3*b15 + 3*b13 - 3*b11 + 3*b7 + 3*b5) * q^67 + (2*b15 - 3*b13 + 2*b12 + b11 + b7 - 2*b5 - 6*b3 + b1) * q^68 + (-2*b15 - 2*b14 + b12 - b11 + 2*b10 - 2*b8 - b7 + b6 - b5 + b4 + 2*b2 - 4*b1 + 3) * q^69 + (2*b14 + b10 - b8 + b6 - b4 + 3*b2 + 3) * q^70 + (3*b14 + 3*b10 + 3*b9 - b6 + b4 + 3*b2) * q^71 + (2*b14 + b10 - b9 - 2*b6 - b4 - 3*b2 - 6) * q^72 + (-3*b14 + 3*b10 - 3*b8 - b6 + 4*b4 - 2*b2) * q^73 + (-b15 + b13 + 4*b12 + 3*b3 - 3*b1) * q^74 - b8 * q^75 + (2*b15 + b13 + b12 - 3*b7 - 3*b5) * q^76 + (3*b6 - 3*b4 + 6*b2 + 6) * q^77 + (2*b14 + 2*b10 + 2*b9 - 2*b8 + b4 + 2*b2 + 2) * q^78 + (3*b15 + 3*b13 + 3*b11 - 3*b7 - 3*b5) * q^79 + (2*b13 - b12 - b11 + b5 - b1) * q^80 + (-b6 - 3*b4 + 2*b2 + 3) * q^81 + (3*b10 + b9 - b8 - 2*b6 + 2*b4 + 4*b2 - 2) * q^82 + (6*b15 - 6*b12 - 6*b7) * q^83 + (2*b15 - b12 + b11 - 2*b7 - 5*b5 - 5*b1) * q^84 + (b14 - b10 + b8 - b6 + b4 - 3*b2) * q^85 + (-b15 - 3*b13 + 2*b12 - 2*b11 - 2*b7 - 2*b5 - 3*b3 - 5*b1) * q^86 + (b14 + b10 - 4*b8 + b6 - b4 + b2) * q^87 + (-4*b15 + 3*b13 - 4*b12 - 5*b11 + b7 + 4*b5 + b1) * q^88 + (-2*b15 - 2*b12 - b11 - b7 - b5 + 9*b3 - b1) * q^89 + (2*b15 - b13 + 2*b12 + 3*b11 + b7 - 2*b5 + 2*b3 + b1) * q^90 + (-5*b15 - b13 - b12 - 6*b11 + 6*b7 + 6*b5) * q^91 + (-4*b15 - 2*b14 - 2*b13 + b12 - 3*b11 + 2*b10 - 2*b8 - 2*b6 + 3*b5 + 4*b4 - 3*b1) * q^92 + (b14 - b10 + b8 - 4*b6 - 3*b4 - 2*b2 - 8) * q^93 + (-6*b10 + 3*b9 + b8 - 9*b6 + 3*b4 - 4*b2 + 1) * q^94 + (b9 + b6 - b4) * q^95 + (2*b14 - 3*b10 - b9 + 2*b6 - b4 - 3*b2 - 2) * q^96 + (2*b15 + 5*b12 + 4*b11 + 4*b7 + 4*b5 - 3*b3 + b1) * q^97 + (-3*b10 + 3*b9 + 6*b8 + 6*b6 - 8*b4 + 3*b2) * q^98 + (5*b15 - 2*b13 + 4*b12 + 9*b11 + 3*b7 - 9*b5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 4 q^{4} + 14 q^{6} + 4 q^{9}+O(q^{10})$$ 16 * q + 4 * q^4 + 14 * q^6 + 4 * q^9 $$16 q + 4 q^{4} + 14 q^{6} + 4 q^{9} - 30 q^{12} + 4 q^{13} + 4 q^{16} + 30 q^{18} + 2 q^{24} - 16 q^{25} - 54 q^{26} - 48 q^{29} + 34 q^{36} - 36 q^{41} - 40 q^{46} + 18 q^{48} + 68 q^{49} + 34 q^{52} - 40 q^{54} + 36 q^{58} + 6 q^{62} + 52 q^{64} + 40 q^{69} + 42 q^{70} - 78 q^{72} + 8 q^{73} + 72 q^{77} + 32 q^{78} + 40 q^{81} - 42 q^{82} + 12 q^{85} - 120 q^{93} + 20 q^{94} - 22 q^{96} - 18 q^{98}+O(q^{100})$$ 16 * q + 4 * q^4 + 14 * q^6 + 4 * q^9 - 30 * q^12 + 4 * q^13 + 4 * q^16 + 30 * q^18 + 2 * q^24 - 16 * q^25 - 54 * q^26 - 48 * q^29 + 34 * q^36 - 36 * q^41 - 40 * q^46 + 18 * q^48 + 68 * q^49 + 34 * q^52 - 40 * q^54 + 36 * q^58 + 6 * q^62 + 52 * q^64 + 40 * q^69 + 42 * q^70 - 78 * q^72 + 8 * q^73 + 72 * q^77 + 32 * q^78 + 40 * q^81 - 42 * q^82 + 12 * q^85 - 120 * q^93 + 20 * q^94 - 22 * q^96 - 18 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{15} + 5\nu^{13} + 3\nu^{11} + 4\nu^{9} + 172\nu^{7} - 96\nu^{5} - 112\nu^{3} - 256\nu ) / 1152$$ (-v^15 + 5*v^13 + 3*v^11 + 4*v^9 + 172*v^7 - 96*v^5 - 112*v^3 - 256*v) / 1152 $$\beta_{2}$$ $$=$$ $$( -\nu^{14} - \nu^{12} - 15\nu^{10} - 2\nu^{8} + 28\nu^{6} - 96\nu^{4} + 80\nu^{2} - 160 ) / 288$$ (-v^14 - v^12 - 15*v^10 - 2*v^8 + 28*v^6 - 96*v^4 + 80*v^2 - 160) / 288 $$\beta_{3}$$ $$=$$ $$( \nu^{15} + 7\nu^{13} + 9\nu^{11} - 16\nu^{9} + 44\nu^{7} + 144\nu^{5} - 80\nu^{3} + 640\nu ) / 1152$$ (v^15 + 7*v^13 + 9*v^11 - 16*v^9 + 44*v^7 + 144*v^5 - 80*v^3 + 640*v) / 1152 $$\beta_{4}$$ $$=$$ $$( - 2 \nu^{14} - 3 \nu^{13} + 4 \nu^{12} + 15 \nu^{11} - 12 \nu^{10} - 27 \nu^{9} - 22 \nu^{8} + 24 \nu^{7} + 32 \nu^{6} + 24 \nu^{5} - 120 \nu^{4} - 216 \nu^{3} - 128 \nu^{2} + 576 \nu + 64 ) / 576$$ (-2*v^14 - 3*v^13 + 4*v^12 + 15*v^11 - 12*v^10 - 27*v^9 - 22*v^8 + 24*v^7 + 32*v^6 + 24*v^5 - 120*v^4 - 216*v^3 - 128*v^2 + 576*v + 64) / 576 $$\beta_{5}$$ $$=$$ $$( - \nu^{15} + 6 \nu^{14} + 11 \nu^{13} - 6 \nu^{12} - 15 \nu^{11} + 30 \nu^{10} + 22 \nu^{9} + 24 \nu^{8} - 32 \nu^{7} + 96 \nu^{6} - 168 \nu^{5} + 192 \nu^{4} + 176 \nu^{3} - 576 \nu^{2} + 128 \nu + 768 ) / 1152$$ (-v^15 + 6*v^14 + 11*v^13 - 6*v^12 - 15*v^11 + 30*v^10 + 22*v^9 + 24*v^8 - 32*v^7 + 96*v^6 - 168*v^5 + 192*v^4 + 176*v^3 - 576*v^2 + 128*v + 768) / 1152 $$\beta_{6}$$ $$=$$ $$( 2 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} + 15 \nu^{11} + 12 \nu^{10} - 27 \nu^{9} + 22 \nu^{8} + 24 \nu^{7} - 32 \nu^{6} + 24 \nu^{5} + 120 \nu^{4} - 216 \nu^{3} + 128 \nu^{2} + 576 \nu - 64 ) / 576$$ (2*v^14 - 3*v^13 - 4*v^12 + 15*v^11 + 12*v^10 - 27*v^9 + 22*v^8 + 24*v^7 - 32*v^6 + 24*v^5 + 120*v^4 - 216*v^3 + 128*v^2 + 576*v - 64) / 576 $$\beta_{7}$$ $$=$$ $$( \nu^{15} - 9\nu^{13} + 9\nu^{11} - 16\nu^{9} - 36\nu^{7} + 144\nu^{5} - 80\nu^{3} ) / 384$$ (v^15 - 9*v^13 + 9*v^11 - 16*v^9 - 36*v^7 + 144*v^5 - 80*v^3) / 384 $$\beta_{8}$$ $$=$$ $$( - 3 \nu^{15} - 4 \nu^{14} - 3 \nu^{13} + 8 \nu^{12} + 3 \nu^{11} - 24 \nu^{10} - 54 \nu^{9} - 44 \nu^{8} - 12 \nu^{7} + 64 \nu^{6} + 48 \nu^{5} - 240 \nu^{4} - 672 \nu^{3} - 256 \nu^{2} + 128 ) / 1152$$ (-3*v^15 - 4*v^14 - 3*v^13 + 8*v^12 + 3*v^11 - 24*v^10 - 54*v^9 - 44*v^8 - 12*v^7 + 64*v^6 + 48*v^5 - 240*v^4 - 672*v^3 - 256*v^2 + 128) / 1152 $$\beta_{9}$$ $$=$$ $$( 2 \nu^{14} + 9 \nu^{13} - 4 \nu^{12} - 9 \nu^{11} + 12 \nu^{10} - 27 \nu^{9} + 22 \nu^{8} + 36 \nu^{7} - 32 \nu^{6} - 144 \nu^{5} + 120 \nu^{4} + 216 \nu^{3} + 128 \nu^{2} - 64 ) / 576$$ (2*v^14 + 9*v^13 - 4*v^12 - 9*v^11 + 12*v^10 - 27*v^9 + 22*v^8 + 36*v^7 - 32*v^6 - 144*v^5 + 120*v^4 + 216*v^3 + 128*v^2 - 64) / 576 $$\beta_{10}$$ $$=$$ $$( - 3 \nu^{15} + 8 \nu^{14} + 3 \nu^{13} - 4 \nu^{12} - 39 \nu^{11} + 36 \nu^{10} - 12 \nu^{9} - 44 \nu^{8} + 48 \nu^{7} - 80 \nu^{6} - 120 \nu^{5} + 288 \nu^{4} - 448 \nu^{2} + 896 ) / 1152$$ (-3*v^15 + 8*v^14 + 3*v^13 - 4*v^12 - 39*v^11 + 36*v^10 - 12*v^9 - 44*v^8 + 48*v^7 - 80*v^6 - 120*v^5 + 288*v^4 - 448*v^2 + 896) / 1152 $$\beta_{11}$$ $$=$$ $$( - \nu^{15} - 6 \nu^{14} + 11 \nu^{13} + 30 \nu^{12} - 15 \nu^{11} - 6 \nu^{10} + 22 \nu^{9} - 48 \nu^{8} - 32 \nu^{7} + 336 \nu^{6} - 168 \nu^{5} - 96 \nu^{4} + 176 \nu^{3} + 192 \nu^{2} + 128 \nu + 1152 ) / 1152$$ (-v^15 - 6*v^14 + 11*v^13 + 30*v^12 - 15*v^11 - 6*v^10 + 22*v^9 - 48*v^8 - 32*v^7 + 336*v^6 - 168*v^5 - 96*v^4 + 176*v^3 + 192*v^2 + 128*v + 1152) / 1152 $$\beta_{12}$$ $$=$$ $$( \nu^{15} - 4\nu^{12} + 8\nu^{10} + 5\nu^{9} - 8\nu^{8} - 12\nu^{6} + 32\nu^{4} - 8\nu^{3} - 128\nu^{2} + 64 ) / 192$$ (v^15 - 4*v^12 + 8*v^10 + 5*v^9 - 8*v^8 - 12*v^6 + 32*v^4 - 8*v^3 - 128*v^2 + 64) / 192 $$\beta_{13}$$ $$=$$ $$( \nu^{15} + 4\nu^{12} + 5\nu^{9} + 20\nu^{6} - 96\nu^{4} - 8\nu^{3} + 96\nu^{2} + 64 ) / 192$$ (v^15 + 4*v^12 + 5*v^9 + 20*v^6 - 96*v^4 - 8*v^3 + 96*v^2 + 64) / 192 $$\beta_{14}$$ $$=$$ $$( 3 \nu^{15} + 8 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} + 39 \nu^{11} + 36 \nu^{10} + 12 \nu^{9} - 44 \nu^{8} - 48 \nu^{7} - 80 \nu^{6} + 120 \nu^{5} + 288 \nu^{4} - 448 \nu^{2} + 896 ) / 1152$$ (3*v^15 + 8*v^14 - 3*v^13 - 4*v^12 + 39*v^11 + 36*v^10 + 12*v^9 - 44*v^8 - 48*v^7 - 80*v^6 + 120*v^5 + 288*v^4 - 448*v^2 + 896) / 1152 $$\beta_{15}$$ $$=$$ $$( -\nu^{15} - 4\nu^{12} + 8\nu^{10} - 5\nu^{9} - 8\nu^{8} - 12\nu^{6} + 32\nu^{4} + 8\nu^{3} - 128\nu^{2} + 64 ) / 192$$ (-v^15 - 4*v^12 + 8*v^10 - 5*v^9 - 8*v^8 - 12*v^6 + 32*v^4 + 8*v^3 - 128*v^2 + 64) / 192
 $$\nu$$ $$=$$ $$( -\beta_{14} + \beta_{10} - \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} + 2\beta_{4} + \beta_{3} - 2\beta_1 ) / 4$$ (-b14 + b10 - b9 - 2*b8 - b7 + b6 + 2*b4 + b3 - 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{11} + \beta_{10} - \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + 3 \beta_{2} - \beta _1 + 1 ) / 4$$ (b15 + b14 + b13 + 2*b11 + b10 - b7 + 4*b6 - 4*b5 - 4*b4 + 3*b2 - b1 + 1) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{15} - \beta_{12} + \beta_{9} - 3\beta_{8} + \beta_{7} - 2\beta_{6} + 2\beta_{4} ) / 2$$ (b15 - b12 + b9 - 3*b8 + b7 - 2*b6 + 2*b4) / 2 $$\nu^{4}$$ $$=$$ $$( - 5 \beta_{15} + \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + \beta_{10} + \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta _1 - 1 ) / 4$$ (-5*b15 + b14 - 7*b13 + 2*b12 + 4*b11 + b10 + b7 + 2*b6 - 2*b5 - 2*b4 - b2 + b1 - 1) / 4 $$\nu^{5}$$ $$=$$ $$( 2 \beta_{15} - \beta_{14} - 2 \beta_{12} + \beta_{10} - 3 \beta_{9} + 5 \beta_{7} - 3 \beta_{6} - 6 \beta_{4} + 17 \beta_{3} + 2 \beta_1 ) / 4$$ (2*b15 - b14 - 2*b12 + b10 - 3*b9 + 5*b7 - 3*b6 - 6*b4 + 17*b3 + 2*b1) / 4 $$\nu^{6}$$ $$=$$ $$( \beta_{15} + \beta_{12} + 5\beta_{11} + 5\beta_{7} + 3\beta_{6} + 5\beta_{5} - 3\beta_{4} + 6\beta_{2} + 5\beta _1 - 5 ) / 2$$ (b15 + b12 + 5*b11 + 5*b7 + 3*b6 + 5*b5 - 3*b4 + 6*b2 + 5*b1 - 5) / 2 $$\nu^{7}$$ $$=$$ $$( -5\beta_{14} + 5\beta_{10} - \beta_{9} - 10\beta_{8} + 7\beta_{7} - 3\beta_{6} + 6\beta_{4} + 5\beta_{3} + 26\beta_1 ) / 4$$ (-5*b14 + 5*b10 - b9 - 10*b8 + 7*b7 - 3*b6 + 6*b4 + 5*b3 + 26*b1) / 4 $$\nu^{8}$$ $$=$$ $$( - 9 \beta_{15} - 25 \beta_{14} - \beta_{13} - 8 \beta_{12} - 18 \beta_{11} - 25 \beta_{10} + 9 \beta_{7} - 4 \beta_{6} + 36 \beta_{5} + 4 \beta_{4} - 27 \beta_{2} + 9 \beta _1 + 23 ) / 4$$ (-9*b15 - 25*b14 - b13 - 8*b12 - 18*b11 - 25*b10 + 9*b7 - 4*b6 + 36*b5 + 4*b4 - 27*b2 + 9*b1 + 23) / 4 $$\nu^{9}$$ $$=$$ $$( -\beta_{15} + \beta_{12} - 21\beta_{9} - 9\beta_{8} - 13\beta_{7} + 6\beta_{6} - 6\beta_{4} ) / 2$$ (-b15 + b12 - 21*b9 - 9*b8 - 13*b7 + 6*b6 - 6*b4) / 2 $$\nu^{10}$$ $$=$$ $$( 49 \beta_{15} - 13 \beta_{14} + 43 \beta_{13} + 6 \beta_{12} + 12 \beta_{11} - 13 \beta_{10} + 3 \beta_{7} + 22 \beta_{6} - 6 \beta_{5} - 22 \beta_{4} - 35 \beta_{2} + 3 \beta _1 - 35 ) / 4$$ (49*b15 - 13*b14 + 43*b13 + 6*b12 + 12*b11 - 13*b10 + 3*b7 + 22*b6 - 6*b5 - 22*b4 - 35*b2 + 3*b1 - 35) / 4 $$\nu^{11}$$ $$=$$ $$( 22 \beta_{15} + 61 \beta_{14} - 22 \beta_{12} - 61 \beta_{10} + 7 \beta_{9} - 17 \beta_{7} + 7 \beta_{6} + 14 \beta_{4} - 29 \beta_{3} + 22 \beta_1 ) / 4$$ (22*b15 + 61*b14 - 22*b12 - 61*b10 + 7*b9 - 17*b7 + 7*b6 + 14*b4 - 29*b3 + 22*b1) / 4 $$\nu^{12}$$ $$=$$ $$( - 29 \beta_{15} - 29 \beta_{12} - \beta_{11} - \beta_{7} - 39 \beta_{6} - \beta_{5} + 39 \beta_{4} - 78 \beta_{2} - \beta _1 - 31 ) / 2$$ (-29*b15 - 29*b12 - b11 - b7 - 39*b6 - b5 + 39*b4 - 78*b2 - b1 - 31) / 2 $$\nu^{13}$$ $$=$$ $$( 65 \beta_{14} - 65 \beta_{10} + 45 \beta_{9} + 130 \beta_{8} - 91 \beta_{7} - 25 \beta_{6} - 110 \beta_{4} + 223 \beta_{3} - 50 \beta_1 ) / 4$$ (65*b14 - 65*b10 + 45*b9 + 130*b8 - 91*b7 - 25*b6 - 110*b4 + 223*b3 - 50*b1) / 4 $$\nu^{14}$$ $$=$$ $$( - 43 \beta_{15} + 229 \beta_{14} + 109 \beta_{13} - 152 \beta_{12} - 86 \beta_{11} + 229 \beta_{10} + 43 \beta_{7} + 52 \beta_{6} + 172 \beta_{5} - 52 \beta_{4} + 255 \beta_{2} + 43 \beta _1 - 203 ) / 4$$ (-43*b15 + 229*b14 + 109*b13 - 152*b12 - 86*b11 + 229*b10 + 43*b7 + 52*b6 + 172*b5 - 52*b4 + 255*b2 + 43*b1 - 203) / 4 $$\nu^{15}$$ $$=$$ $$( -179\beta_{15} + 179\beta_{12} + 113\beta_{9} + 21\beta_{8} + 73\beta_{7} - 46\beta_{6} + 46\beta_{4} ) / 2$$ (-179*b15 + 179*b12 + 113*b9 + 21*b8 + 73*b7 - 46*b6 + 46*b4) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$$.

 $$n$$ $$231$$ $$277$$ $$281$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 1.18353 − 0.774115i −0.0786378 + 1.41203i −0.0786378 − 1.41203i 1.18353 + 0.774115i 1.37379 − 0.335728i 0.977642 − 1.02187i 0.977642 + 1.02187i 1.37379 + 0.335728i −0.977642 + 1.02187i −1.37379 + 0.335728i −1.37379 − 0.335728i −0.977642 − 1.02187i 0.0786378 − 1.41203i −1.18353 + 0.774115i −1.18353 − 0.774115i 0.0786378 + 1.41203i
−1.35760 0.396143i 1.47175i 1.68614 + 1.07561i 1.00000i 0.583024 1.99804i −3.53986 −1.86301 2.12819i 0.833952 −0.396143 + 1.35760i
91.2 −1.35760 0.396143i 1.47175i 1.68614 + 1.07561i 1.00000i 0.583024 1.99804i 3.53986 −1.86301 2.12819i 0.833952 0.396143 1.35760i
91.3 −1.35760 + 0.396143i 1.47175i 1.68614 1.07561i 1.00000i 0.583024 + 1.99804i 3.53986 −1.86301 + 2.12819i 0.833952 0.396143 + 1.35760i
91.4 −1.35760 + 0.396143i 1.47175i 1.68614 1.07561i 1.00000i 0.583024 + 1.99804i −3.53986 −1.86301 + 2.12819i 0.833952 −0.396143 1.35760i
91.5 −0.637910 1.26217i 2.87247i −1.18614 + 1.61030i 1.00000i 3.62554 1.83238i −2.68161 2.78912 + 0.469882i −5.25109 −1.26217 + 0.637910i
91.6 −0.637910 1.26217i 2.87247i −1.18614 + 1.61030i 1.00000i 3.62554 1.83238i 2.68161 2.78912 + 0.469882i −5.25109 1.26217 0.637910i
91.7 −0.637910 + 1.26217i 2.87247i −1.18614 1.61030i 1.00000i 3.62554 + 1.83238i 2.68161 2.78912 0.469882i −5.25109 1.26217 + 0.637910i
91.8 −0.637910 + 1.26217i 2.87247i −1.18614 1.61030i 1.00000i 3.62554 + 1.83238i −2.68161 2.78912 0.469882i −5.25109 −1.26217 0.637910i
91.9 0.637910 1.26217i 0.348132i −1.18614 1.61030i 1.00000i −0.439402 0.222077i −4.89140 −2.78912 + 0.469882i 2.87880 −1.26217 0.637910i
91.10 0.637910 1.26217i 0.348132i −1.18614 1.61030i 1.00000i −0.439402 0.222077i 4.89140 −2.78912 + 0.469882i 2.87880 1.26217 + 0.637910i
91.11 0.637910 + 1.26217i 0.348132i −1.18614 + 1.61030i 1.00000i −0.439402 + 0.222077i 4.89140 −2.78912 0.469882i 2.87880 1.26217 0.637910i
91.12 0.637910 + 1.26217i 0.348132i −1.18614 + 1.61030i 1.00000i −0.439402 + 0.222077i −4.89140 −2.78912 0.469882i 2.87880 −1.26217 + 0.637910i
91.13 1.35760 0.396143i 0.679463i 1.68614 1.07561i 1.00000i −0.269165 0.922437i 1.16300 1.86301 2.12819i 2.53833 −0.396143 1.35760i
91.14 1.35760 0.396143i 0.679463i 1.68614 1.07561i 1.00000i −0.269165 0.922437i −1.16300 1.86301 2.12819i 2.53833 0.396143 + 1.35760i
91.15 1.35760 + 0.396143i 0.679463i 1.68614 + 1.07561i 1.00000i −0.269165 + 0.922437i −1.16300 1.86301 + 2.12819i 2.53833 0.396143 1.35760i
91.16 1.35760 + 0.396143i 0.679463i 1.68614 + 1.07561i 1.00000i −0.269165 + 0.922437i 1.16300 1.86301 + 2.12819i 2.53833 −0.396143 + 1.35760i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.e.a 16
4.b odd 2 1 inner 460.2.e.a 16
23.b odd 2 1 inner 460.2.e.a 16
92.b even 2 1 inner 460.2.e.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.e.a 16 1.a even 1 1 trivial
460.2.e.a 16 4.b odd 2 1 inner
460.2.e.a 16 23.b odd 2 1 inner
460.2.e.a 16 92.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 11T_{3}^{6} + 24T_{3}^{4} + 11T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} - T^{6} - 4 T^{2} + 16)^{2}$$
$3$ $$(T^{8} + 11 T^{6} + 24 T^{4} + 11 T^{2} + \cdots + 1)^{2}$$
$5$ $$(T^{2} + 1)^{8}$$
$7$ $$(T^{8} - 45 T^{6} + 621 T^{4} - 2916 T^{2} + \cdots + 2916)^{2}$$
$11$ $$(T^{8} - 45 T^{6} + 621 T^{4} - 2916 T^{2} + \cdots + 2916)^{2}$$
$13$ $$(T^{4} - T^{3} - 24 T^{2} - 25 T + 31)^{4}$$
$17$ $$(T^{8} + 99 T^{6} + 2889 T^{4} + \cdots + 11664)^{2}$$
$19$ $$(T^{8} - 45 T^{6} + 621 T^{4} - 2916 T^{2} + \cdots + 2916)^{2}$$
$23$ $$(T^{8} - 16 T^{6} - 66 T^{4} + \cdots + 279841)^{2}$$
$29$ $$(T^{4} + 12 T^{3} + 45 T^{2} + 54 T + 12)^{4}$$
$31$ $$(T^{8} + 207 T^{6} + 14892 T^{4} + \cdots + 4124961)^{2}$$
$37$ $$(T^{8} + 108 T^{6} + 2484 T^{4} + \cdots + 46656)^{2}$$
$41$ $$(T^{4} + 9 T^{3} - 30 T^{2} - 387 T - 681)^{4}$$
$43$ $$(T^{8} - 180 T^{6} + 7668 T^{4} + \cdots + 46656)^{2}$$
$47$ $$(T^{8} + 362 T^{6} + 41457 T^{4} + \cdots + 163216)^{2}$$
$53$ $$(T^{4} + 180 T^{2} + 6912)^{4}$$
$59$ $$(T^{4} + 76 T^{2} + 256)^{4}$$
$61$ $$(T^{8} + 243 T^{6} + 22005 T^{4} + \cdots + 13089924)^{2}$$
$67$ $$(T^{8} - 288 T^{6} + 26244 T^{4} + \cdots + 6718464)^{2}$$
$71$ $$(T^{8} + 215 T^{6} + 8940 T^{4} + \cdots + 6889)^{2}$$
$73$ $$(T^{4} - 2 T^{3} - 123 T^{2} + 718 T - 908)^{4}$$
$79$ $$(T^{8} - 324 T^{6} + 22356 T^{4} + \cdots + 3779136)^{2}$$
$83$ $$(T^{4} - 252 T^{2} + 5184)^{4}$$
$89$ $$(T^{8} + 396 T^{6} + 46548 T^{4} + \cdots + 13483584)^{2}$$
$97$ $$(T^{8} + 531 T^{6} + 88749 T^{4} + \cdots + 29746116)^{2}$$
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