# Properties

 Label 539.2.a.l Level $539$ Weight $2$ Character orbit 539.a Self dual yes Analytic conductor $4.304$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 539.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.30393666895$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968$$ x^10 - 26*x^8 + 245*x^6 - 1038*x^4 + 1884*x^2 - 968 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + \beta_1 q^{3} + (\beta_{9} + 2) q^{4} + \beta_{7} q^{5} + (\beta_{4} + \beta_{3}) q^{6} + ( - \beta_{9} + \beta_{8} + 2 \beta_{5} - \beta_{2} - 1) q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q + b5 * q^2 + b1 * q^3 + (b9 + 2) * q^4 + b7 * q^5 + (b4 + b3) * q^6 + (-b9 + b8 + 2*b5 - b2 - 1) * q^8 + (b2 + 2) * q^9 $$q + \beta_{5} q^{2} + \beta_1 q^{3} + (\beta_{9} + 2) q^{4} + \beta_{7} q^{5} + (\beta_{4} + \beta_{3}) q^{6} + ( - \beta_{9} + \beta_{8} + 2 \beta_{5} - \beta_{2} - 1) q^{8} + (\beta_{2} + 2) q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{10} + q^{11} + (\beta_{7} - \beta_{6} - 2 \beta_{3} + \beta_1) q^{12} + ( - \beta_{7} - \beta_{6} - \beta_{4}) q^{13} + (\beta_{9} - 2 \beta_{8} + 1) q^{15} + (2 \beta_{9} - \beta_{8} - \beta_{5} - \beta_{2} + 5) q^{16} + (\beta_{7} + \beta_{6} - \beta_{4}) q^{17} + (\beta_{5} + 2 \beta_{2}) q^{18} + (2 \beta_{6} - 2 \beta_{3}) q^{19} + ( - 2 \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_1) q^{20} + \beta_{5} q^{22} + ( - \beta_{9} + \beta_{8} + \beta_{5}) q^{23} + ( - 2 \beta_{7} + 4 \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_1) q^{24} + ( - 2 \beta_{9} - 2 \beta_{5} - \beta_{2} + 2) q^{25} + ( - \beta_{7} - 3 \beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{26} + ( - \beta_{7} - 4 \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_1) q^{27} + ( - \beta_{9} + \beta_{8} - \beta_{5} + \beta_{2} + 1) q^{29} + ( - 3 \beta_{9} + \beta_{8} + \beta_{5} - \beta_{2} - 1) q^{30} + ( - 2 \beta_{3} + \beta_1) q^{31} + ( - 2 \beta_{9} + 5 \beta_{5} - 2 \beta_{2} - 4) q^{32} + \beta_1 q^{33} + ( - \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{34} + (\beta_{9} - 2 \beta_{5} + 2 \beta_{2}) q^{36} + (\beta_{8} - \beta_{5} + \beta_{2} + 4) q^{37} + (2 \beta_{7} + 6 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{38} + ( - \beta_{9} + \beta_{8} - 3 \beta_{5} - \beta_{2} - 1) q^{39} + ( - \beta_{7} - \beta_{6} + 2 \beta_{4} + 3 \beta_{3} - 4 \beta_1) q^{40} + (\beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_1) q^{41} + (\beta_{9} + \beta_{8} + \beta_{5} + \beta_{2} - 1) q^{43} + (\beta_{9} + 2) q^{44} + (2 \beta_{7} + \beta_{6}) q^{45} + (3 \beta_{9} - \beta_{8} - \beta_{5} + \beta_{2} + 5) q^{46} + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{47} + (5 \beta_{7} + 3 \beta_{6} - 2 \beta_{4} - 6 \beta_{3} + \beta_1) q^{48} + ( - 2 \beta_{8} - \beta_{5} - 6) q^{50} + ( - \beta_{9} - 3 \beta_{8} - 3 \beta_{5} - 3 \beta_{2} + 1) q^{51} + ( - \beta_{7} - 7 \beta_{6} - 3 \beta_{4} + 2 \beta_{3}) q^{52} + ( - 2 \beta_{8} - 2 \beta_{2} + 2) q^{53} + ( - 2 \beta_{7} - 8 \beta_{6} + 2 \beta_{3} + 4 \beta_1) q^{54} + \beta_{7} q^{55} + (2 \beta_{8} - 2 \beta_{5} - 2 \beta_{2}) q^{57} + (\beta_{9} - \beta_{8} - \beta_{5} + 3 \beta_{2} - 3) q^{58} + (2 \beta_{7} + 3 \beta_{6} + \beta_{4} - 2 \beta_1) q^{59} + (3 \beta_{9} + \beta_{8} - 5 \beta_{5} + \beta_{2} + 5) q^{60} + (\beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_1) q^{61} + (4 \beta_{6} + 3 \beta_{4} + \beta_{3} - 2 \beta_1) q^{62} + (3 \beta_{9} - 4 \beta_{5} + 12) q^{64} + (3 \beta_{9} - \beta_{8} + \beta_{5} + \beta_{2} - 3) q^{65} + (\beta_{4} + \beta_{3}) q^{66} + ( - \beta_{9} + \beta_{8} - 3 \beta_{5}) q^{67} + ( - \beta_{7} - 7 \beta_{6} - 3 \beta_{4} - 2 \beta_{3}) q^{68} + ( - 3 \beta_{7} + \beta_{4} + 2 \beta_{3} + \beta_1) q^{69} + ( - \beta_{9} + \beta_{8} - 3 \beta_{5} + 4) q^{71} + ( - 3 \beta_{9} + \beta_{8} - 2 \beta_{5} - \beta_{2} - 9) q^{72} + ( - 3 \beta_{7} - 5 \beta_{6} - \beta_{4}) q^{73} + (4 \beta_{5} + 2 \beta_{2} - 4) q^{74} + ( - \beta_{7} + 6 \beta_{6} - 3 \beta_{4} + 2 \beta_1) q^{75} + (6 \beta_{7} + 10 \beta_{6} + 2 \beta_{4} - 6 \beta_{3} + 2 \beta_1) q^{76} + ( - \beta_{9} - \beta_{8} - \beta_{5} - \beta_{2} - 11) q^{78} + (\beta_{9} + \beta_{8} + \beta_{5} - \beta_{2} + 1) q^{79} + (2 \beta_{7} - 2 \beta_{6} - 3 \beta_{4} - 4 \beta_{3} + 5 \beta_1) q^{80} + (2 \beta_{9} + \beta_{8} + 5 \beta_{5} + 4 \beta_{2} - 2) q^{81} + ( - \beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{82} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4}) q^{83} + ( - \beta_{9} - 3 \beta_{8} - \beta_{5} - \beta_{2} + 9) q^{85} + (\beta_{9} + \beta_{8} + \beta_{5} + \beta_{2} + 3) q^{86} + ( - 4 \beta_{7} - 4 \beta_{6} + 2 \beta_{3} + 4 \beta_1) q^{87} + ( - \beta_{9} + \beta_{8} + 2 \beta_{5} - \beta_{2} - 1) q^{88} + \beta_{7} q^{89} + ( - \beta_{7} + 3 \beta_{6} + \beta_{3}) q^{90} + ( - 3 \beta_{9} + \beta_{8} + 7 \beta_{5} - \beta_{2} - 7) q^{92} + (2 \beta_{9} + 2 \beta_{8} - 2 \beta_{5} + \beta_{2} + 5) q^{93} + (4 \beta_{7} - 2 \beta_{6} + \beta_{4} - 3 \beta_{3} + 4 \beta_1) q^{94} + (4 \beta_{9} + 2 \beta_{2} - 6) q^{95} + (10 \beta_{6} + 3 \beta_{4} + 5 \beta_{3} - 6 \beta_1) q^{96} + ( - \beta_{7} + 2 \beta_{3} - 2 \beta_1) q^{97} + (\beta_{2} + 2) q^{99}+O(q^{100})$$ q + b5 * q^2 + b1 * q^3 + (b9 + 2) * q^4 + b7 * q^5 + (b4 + b3) * q^6 + (-b9 + b8 + 2*b5 - b2 - 1) * q^8 + (b2 + 2) * q^9 + (-b7 + b6 + b3) * q^10 + q^11 + (b7 - b6 - 2*b3 + b1) * q^12 + (-b7 - b6 - b4) * q^13 + (b9 - 2*b8 + 1) * q^15 + (2*b9 - b8 - b5 - b2 + 5) * q^16 + (b7 + b6 - b4) * q^17 + (b5 + 2*b2) * q^18 + (2*b6 - 2*b3) * q^19 + (-2*b6 - b4 - 2*b3 + b1) * q^20 + b5 * q^22 + (-b9 + b8 + b5) * q^23 + (-2*b7 + 4*b6 + b4 + b3 - 2*b1) * q^24 + (-2*b9 - 2*b5 - b2 + 2) * q^25 + (-b7 - 3*b6 - b4 + 2*b3 - 2*b1) * q^26 + (-b7 - 4*b6 + b4 + 2*b3 + b1) * q^27 + (-b9 + b8 - b5 + b2 + 1) * q^29 + (-3*b9 + b8 + b5 - b2 - 1) * q^30 + (-2*b3 + b1) * q^31 + (-2*b9 + 5*b5 - 2*b2 - 4) * q^32 + b1 * q^33 + (-b7 + b6 - b4 + 2*b3 - 2*b1) * q^34 + (b9 - 2*b5 + 2*b2) * q^36 + (b8 - b5 + b2 + 4) * q^37 + (2*b7 + 6*b6 + 2*b4 - 2*b3 - 2*b1) * q^38 + (-b9 + b8 - 3*b5 - b2 - 1) * q^39 + (-b7 - b6 + 2*b4 + 3*b3 - 4*b1) * q^40 + (b7 - b6 + b4 - 2*b1) * q^41 + (b9 + b8 + b5 + b2 - 1) * q^43 + (b9 + 2) * q^44 + (2*b7 + b6) * q^45 + (3*b9 - b8 - b5 + b2 + 5) * q^46 + (-b7 + 2*b6 + b4 + 2*b3 + 2*b1) * q^47 + (5*b7 + 3*b6 - 2*b4 - 6*b3 + b1) * q^48 + (-2*b8 - b5 - 6) * q^50 + (-b9 - 3*b8 - 3*b5 - 3*b2 + 1) * q^51 + (-b7 - 7*b6 - 3*b4 + 2*b3) * q^52 + (-2*b8 - 2*b2 + 2) * q^53 + (-2*b7 - 8*b6 + 2*b3 + 4*b1) * q^54 + b7 * q^55 + (2*b8 - 2*b5 - 2*b2) * q^57 + (b9 - b8 - b5 + 3*b2 - 3) * q^58 + (2*b7 + 3*b6 + b4 - 2*b1) * q^59 + (3*b9 + b8 - 5*b5 + b2 + 5) * q^60 + (b7 + b6 + b4 - 2*b1) * q^61 + (4*b6 + 3*b4 + b3 - 2*b1) * q^62 + (3*b9 - 4*b5 + 12) * q^64 + (3*b9 - b8 + b5 + b2 - 3) * q^65 + (b4 + b3) * q^66 + (-b9 + b8 - 3*b5) * q^67 + (-b7 - 7*b6 - 3*b4 - 2*b3) * q^68 + (-3*b7 + b4 + 2*b3 + b1) * q^69 + (-b9 + b8 - 3*b5 + 4) * q^71 + (-3*b9 + b8 - 2*b5 - b2 - 9) * q^72 + (-3*b7 - 5*b6 - b4) * q^73 + (4*b5 + 2*b2 - 4) * q^74 + (-b7 + 6*b6 - 3*b4 + 2*b1) * q^75 + (6*b7 + 10*b6 + 2*b4 - 6*b3 + 2*b1) * q^76 + (-b9 - b8 - b5 - b2 - 11) * q^78 + (b9 + b8 + b5 - b2 + 1) * q^79 + (2*b7 - 2*b6 - 3*b4 - 4*b3 + 5*b1) * q^80 + (2*b9 + b8 + 5*b5 + 4*b2 - 2) * q^81 + (-b7 + b6 - b4 - 2*b3 + 2*b1) * q^82 + (-2*b7 - 2*b6 - 2*b4) * q^83 + (-b9 - 3*b8 - b5 - b2 + 9) * q^85 + (b9 + b8 + b5 + b2 + 3) * q^86 + (-4*b7 - 4*b6 + 2*b3 + 4*b1) * q^87 + (-b9 + b8 + 2*b5 - b2 - 1) * q^88 + b7 * q^89 + (-b7 + 3*b6 + b3) * q^90 + (-3*b9 + b8 + 7*b5 - b2 - 7) * q^92 + (2*b9 + 2*b8 - 2*b5 + b2 + 5) * q^93 + (4*b7 - 2*b6 + b4 - 3*b3 + 4*b1) * q^94 + (4*b9 + 2*b2 - 6) * q^95 + (10*b6 + 3*b4 + 5*b3 - 6*b1) * q^96 + (-b7 + 2*b3 - 2*b1) * q^97 + (b2 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9}+O(q^{10})$$ 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9 $$10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + 10 q^{11} + 8 q^{15} + 42 q^{16} + 6 q^{18} + 2 q^{22} + 4 q^{23} + 18 q^{25} + 12 q^{29} - 4 q^{30} - 30 q^{32} - 2 q^{36} + 40 q^{37} - 16 q^{39} - 8 q^{43} + 18 q^{44} + 44 q^{46} - 62 q^{50} + 16 q^{53} - 8 q^{57} - 28 q^{58} + 36 q^{60} + 106 q^{64} - 32 q^{65} - 4 q^{67} + 36 q^{71} - 90 q^{72} - 28 q^{74} - 112 q^{78} + 8 q^{79} - 6 q^{81} + 88 q^{85} + 32 q^{86} - 6 q^{88} - 52 q^{92} + 44 q^{93} - 64 q^{95} + 22 q^{99}+O(q^{100})$$ 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9 + 10 * q^11 + 8 * q^15 + 42 * q^16 + 6 * q^18 + 2 * q^22 + 4 * q^23 + 18 * q^25 + 12 * q^29 - 4 * q^30 - 30 * q^32 - 2 * q^36 + 40 * q^37 - 16 * q^39 - 8 * q^43 + 18 * q^44 + 44 * q^46 - 62 * q^50 + 16 * q^53 - 8 * q^57 - 28 * q^58 + 36 * q^60 + 106 * q^64 - 32 * q^65 - 4 * q^67 + 36 * q^71 - 90 * q^72 - 28 * q^74 - 112 * q^78 + 8 * q^79 - 6 * q^81 + 88 * q^85 + 32 * q^86 - 6 * q^88 - 52 * q^92 + 44 * q^93 - 64 * q^95 + 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5 $$\beta_{3}$$ $$=$$ $$( 7\nu^{9} - 160\nu^{7} + 1187\nu^{5} - 3108\nu^{3} + 1572\nu ) / 88$$ (7*v^9 - 160*v^7 + 1187*v^5 - 3108*v^3 + 1572*v) / 88 $$\beta_{4}$$ $$=$$ $$( 15\nu^{9} - 324\nu^{7} + 2267\nu^{5} - 5912\nu^{3} + 3972\nu ) / 88$$ (15*v^9 - 324*v^7 + 2267*v^5 - 5912*v^3 + 3972*v) / 88 $$\beta_{5}$$ $$=$$ $$( \nu^{8} - 22\nu^{6} + 157\nu^{4} - 410\nu^{2} + 252 ) / 4$$ (v^8 - 22*v^6 + 157*v^4 - 410*v^2 + 252) / 4 $$\beta_{6}$$ $$=$$ $$( 25\nu^{9} - 540\nu^{7} + 3749\nu^{5} - 9472\nu^{3} + 5652\nu ) / 88$$ (25*v^9 - 540*v^7 + 3749*v^5 - 9472*v^3 + 5652*v) / 88 $$\beta_{7}$$ $$=$$ $$( -71\nu^{9} + 1516\nu^{7} - 10355\nu^{5} + 25672\nu^{3} - 14876\nu ) / 88$$ (-71*v^9 + 1516*v^7 - 10355*v^5 + 25672*v^3 - 14876*v) / 88 $$\beta_{8}$$ $$=$$ $$( 5\nu^{8} - 106\nu^{6} + 717\nu^{4} - 1762\nu^{2} + 1024 ) / 4$$ (5*v^8 - 106*v^6 + 717*v^4 - 1762*v^2 + 1024) / 4 $$\beta_{9}$$ $$=$$ $$( -5\nu^{8} + 108\nu^{6} - 749\nu^{4} + 1880\nu^{2} - 1080 ) / 4$$ (-5*v^8 + 108*v^6 - 749*v^4 + 1880*v^2 - 1080) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ b2 + 5 $$\nu^{3}$$ $$=$$ $$-\beta_{7} - 4\beta_{6} + \beta_{4} + 2\beta_{3} + 7\beta_1$$ -b7 - 4*b6 + b4 + 2*b3 + 7*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{9} + \beta_{8} + 5\beta_{5} + 13\beta_{2} + 34$$ 2*b9 + b8 + 5*b5 + 13*b2 + 34 $$\nu^{5}$$ $$=$$ $$-13\beta_{7} - 55\beta_{6} + 18\beta_{4} + 26\beta_{3} + 58\beta_1$$ -13*b7 - 55*b6 + 18*b4 + 26*b3 + 58*b1 $$\nu^{6}$$ $$=$$ $$34\beta_{9} + 18\beta_{8} + 80\beta_{5} + 149\beta_{2} + 277$$ 34*b9 + 18*b8 + 80*b5 + 149*b2 + 277 $$\nu^{7}$$ $$=$$ $$-151\beta_{7} - 648\beta_{6} + 229\beta_{4} + 292\beta_{3} + 541\beta_1$$ -151*b7 - 648*b6 + 229*b4 + 292*b3 + 541*b1 $$\nu^{8}$$ $$=$$ $$434\beta_{9} + 239\beta_{8} + 979\beta_{5} + 1647\beta_{2} + 2554$$ 434*b9 + 239*b8 + 979*b5 + 1647*b2 + 2554 $$\nu^{9}$$ $$=$$ $$-1691\beta_{7} - 7261\beta_{6} + 2626\beta_{4} + 3166\beta_{3} + 5414\beta_1$$ -1691*b7 - 7261*b6 + 2626*b4 + 3166*b3 + 5414*b1

## 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
 −2.10267 2.10267 −2.15293 2.15293 −2.32267 2.32267 −3.27614 3.27614 −0.903205 0.903205
−2.74816 −2.10267 5.55241 −2.44342 5.77848 0 −9.76260 1.42122 6.71492
1.2 −2.74816 2.10267 5.55241 2.44342 −5.77848 0 −9.76260 1.42122 −6.71492
1.3 −1.14898 −2.15293 −0.679834 3.87589 2.47369 0 3.07909 1.63513 −4.45334
1.4 −1.14898 2.15293 −0.679834 −3.87589 −2.47369 0 3.07909 1.63513 4.45334
1.5 0.566092 −2.32267 −1.67954 −3.58219 −1.31484 0 −2.08296 2.39479 −2.02785
1.6 0.566092 2.32267 −1.67954 3.58219 1.31484 0 −2.08296 2.39479 2.02785
1.7 1.70296 −3.27614 0.900071 0.246676 −5.57913 0 −1.87313 7.73309 0.420079
1.8 1.70296 3.27614 0.900071 −0.246676 5.57913 0 −1.87313 7.73309 −0.420079
1.9 2.62810 −0.903205 4.90690 0.337987 −2.37371 0 7.63960 −2.18422 0.888264
1.10 2.62810 0.903205 4.90690 −0.337987 2.37371 0 7.63960 −2.18422 −0.888264
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$
$$11$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.a.l 10
3.b odd 2 1 4851.2.a.cg 10
4.b odd 2 1 8624.2.a.df 10
7.b odd 2 1 inner 539.2.a.l 10
7.c even 3 2 539.2.e.o 20
7.d odd 6 2 539.2.e.o 20
11.b odd 2 1 5929.2.a.bv 10
21.c even 2 1 4851.2.a.cg 10
28.d even 2 1 8624.2.a.df 10
77.b even 2 1 5929.2.a.bv 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.a.l 10 1.a even 1 1 trivial
539.2.a.l 10 7.b odd 2 1 inner
539.2.e.o 20 7.c even 3 2
539.2.e.o 20 7.d odd 6 2
4851.2.a.cg 10 3.b odd 2 1
4851.2.a.cg 10 21.c even 2 1
5929.2.a.bv 10 11.b odd 2 1
5929.2.a.bv 10 77.b even 2 1
8624.2.a.df 10 4.b odd 2 1
8624.2.a.df 10 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(539))$$:

 $$T_{2}^{5} - T_{2}^{4} - 9T_{2}^{3} + 9T_{2}^{2} + 12T_{2} - 8$$ T2^5 - T2^4 - 9*T2^3 + 9*T2^2 + 12*T2 - 8 $$T_{3}^{10} - 26T_{3}^{8} + 245T_{3}^{6} - 1038T_{3}^{4} + 1884T_{3}^{2} - 968$$ T3^10 - 26*T3^8 + 245*T3^6 - 1038*T3^4 + 1884*T3^2 - 968

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{5} - T^{4} - 9 T^{3} + 9 T^{2} + 12 T - 8)^{2}$$
$3$ $$T^{10} - 26 T^{8} + 245 T^{6} + \cdots - 968$$
$5$ $$T^{10} - 34 T^{8} + 365 T^{6} - 1214 T^{4} + \cdots - 8$$
$7$ $$T^{10}$$
$11$ $$(T - 1)^{10}$$
$13$ $$T^{10} - 72 T^{8} + 1696 T^{6} + \cdots - 100352$$
$17$ $$T^{10} - 136 T^{8} + 6192 T^{6} + \cdots - 430592$$
$19$ $$T^{10} - 144 T^{8} + 7312 T^{6} + \cdots - 6422528$$
$23$ $$(T^{5} - 2 T^{4} - 39 T^{3} + 126 T^{2} + \cdots - 232)^{2}$$
$29$ $$(T^{5} - 6 T^{4} - 56 T^{3} + 256 T^{2} + \cdots + 224)^{2}$$
$31$ $$T^{10} - 154 T^{8} + 8469 T^{6} + \cdots - 3998792$$
$37$ $$(T^{5} - 20 T^{4} + 113 T^{3} - 34 T^{2} + \cdots - 472)^{2}$$
$41$ $$T^{10} - 168 T^{8} + 5680 T^{6} + \cdots - 25088$$
$43$ $$(T^{5} + 4 T^{4} - 56 T^{3} - 48 T^{2} + \cdots + 256)^{2}$$
$47$ $$T^{10} - 370 T^{8} + \cdots - 1158537248$$
$53$ $$(T^{5} - 8 T^{4} - 128 T^{3} + 864 T^{2} + \cdots - 11264)^{2}$$
$59$ $$T^{10} - 234 T^{8} + \cdots - 108162632$$
$61$ $$T^{10} - 144 T^{8} + 6928 T^{6} + \cdots - 3527168$$
$67$ $$(T^{5} + 2 T^{4} - 103 T^{3} - 270 T^{2} + \cdots + 4648)^{2}$$
$71$ $$(T^{5} - 18 T^{4} + 25 T^{3} + 518 T^{2} + \cdots + 88)^{2}$$
$73$ $$T^{10} - 400 T^{8} + \cdots - 65987072$$
$79$ $$(T^{5} - 4 T^{4} - 116 T^{3} - 32 T^{2} + \cdots - 704)^{2}$$
$83$ $$T^{10} - 288 T^{8} + \cdots - 102760448$$
$89$ $$T^{10} - 34 T^{8} + 365 T^{6} - 1214 T^{4} + \cdots - 8$$
$97$ $$T^{10} - 226 T^{8} + 12541 T^{6} + \cdots - 19208$$