# Properties

 Label 847.2.f.u Level $847$ Weight $2$ Character orbit 847.f Analytic conductor $6.763$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.f (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} + 7 x^{10} - 15 x^{9} + 59 x^{8} + 118 x^{7} + 266 x^{6} + 324 x^{5} + 1036 x^{4} + 376 x^{3} + 136 x^{2} + 48 x + 16$$ x^12 - x^11 + 7*x^10 - 15*x^9 + 59*x^8 + 118*x^7 + 266*x^6 + 324*x^5 + 1036*x^4 + 376*x^3 + 136*x^2 + 48*x + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{2} - \beta_{8} q^{3} + (\beta_{5} + 3 \beta_{3}) q^{4} + \beta_{9} q^{5} + (\beta_{9} + 4 \beta_{7} - \beta_{6}) q^{6} - \beta_{3} q^{7} + (2 \beta_{11} + 3 \beta_{10} + \beta_{8}) q^{8} + (\beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{9}+O(q^{10})$$ q + (-b10 - b8 - b7 - b6 + b3 + b2 + b1 + 1) * q^2 - b8 * q^3 + (b5 + 3*b3) * q^4 + b9 * q^5 + (b9 + 4*b7 - b6) * q^6 - b3 * q^7 + (2*b11 + 3*b10 + b8) * q^8 + (b11 + b10 + b9 - 2*b8 + b7 - 2*b6 - b5 - b4 - b3 + 2*b2 + 2*b1 - 1) * q^9 $$q + ( - \beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{2} - \beta_{8} q^{3} + (\beta_{5} + 3 \beta_{3}) q^{4} + \beta_{9} q^{5} + (\beta_{9} + 4 \beta_{7} - \beta_{6}) q^{6} - \beta_{3} q^{7} + (2 \beta_{11} + 3 \beta_{10} + \beta_{8}) q^{8} + (\beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{9}+ \cdots + ( - \beta_{2} - 1) q^{98}+O(q^{100})$$ q + (-b10 - b8 - b7 - b6 + b3 + b2 + b1 + 1) * q^2 - b8 * q^3 + (b5 + 3*b3) * q^4 + b9 * q^5 + (b9 + 4*b7 - b6) * q^6 - b3 * q^7 + (2*b11 + 3*b10 + b8) * q^8 + (b11 + b10 + b9 - 2*b8 + b7 - 2*b6 - b5 - b4 - b3 + 2*b2 + 2*b1 - 1) * q^9 + (2*b4 + 2) * q^10 + (b4 + 3*b2 + 2) * q^12 + (b11 - 2*b10 + b9 + b8 - 2*b7 + b6 - b5 - b4 + 2*b3 - b2 - b1 + 2) * q^13 + (-b10 - b8) * q^14 + (-b5 - 2*b3) * q^15 + (-3*b9 - 5*b7 - 2*b6) * q^16 + (-b9 + 2*b7 - b6) * q^17 + (5*b3 - 3*b1) * q^18 + (-2*b11 + 2*b8) * q^19 + (-2*b11 - 6*b10 - 2*b9 - 2*b8 - 6*b7 - 2*b6 + 2*b5 + 2*b4 + 6*b3 + 2*b2 + 2*b1 + 6) * q^20 - b2 * q^21 + (-b4 + 2) * q^23 + (-3*b11 - 8*b10 - 3*b9 - b8 - 8*b7 - b6 + 3*b5 + 3*b4 + 8*b3 + b2 + b1 + 8) * q^24 + (b11 - b10 - 2*b8) * q^25 + (-3*b5 - 4*b3 + 3*b1) * q^26 + (b9 + 6*b7 - 2*b6) * q^27 + (b9 + 3*b7) * q^28 + (-2*b5 + 2*b1) * q^29 + (-2*b11 - 4*b10 - 2*b8) * q^30 + (-4*b10 + b8 - 4*b7 + b6 + 4*b3 - b2 - b1 + 4) * q^31 + (-4*b4 - b2 - 13) * q^32 + (-3*b4 + 3*b2 - 4) * q^34 + (b11 + b9 - b5 - b4) * q^35 + (-b11 - 5*b10 + 4*b8) * q^36 + (b5 - 6*b3 - 2*b1) * q^37 + (2*b9 - 4*b7 + 2*b6) * q^38 + (-2*b9 - 6*b7 + 4*b6) * q^39 + (2*b5 + 14*b3 + 4*b1) * q^40 + (b11 + 6*b10 + b8) * q^41 + (b11 + 4*b10 + b9 - b8 + 4*b7 - b6 - b5 - b4 - 4*b3 + b2 + b1 - 4) * q^42 + (-2*b4 - 2) * q^43 + (2*b4 - 2*b2 - 2) * q^45 + (2*b11 + 2*b9 - 2*b8 - 2*b6 - 2*b5 - 2*b4 + 2*b2 + 2*b1) * q^46 + (-b11 - 2*b10 - b8) * q^47 + (5*b5 + 14*b3 + b1) * q^48 - b7 * q^49 + (7*b7 - b6) * q^50 + (2*b5 + 6*b3 - 4*b1) * q^51 + (-b11 - 2*b10 - 5*b8) * q^52 + (-2*b11 - 4*b10 - 2*b9 - 4*b7 + 2*b5 + 2*b4 + 4*b3 + 4) * q^53 + 8*b2 * q^54 + (2*b4 + b2 + 3) * q^56 + (-4*b10 + 4*b8 - 4*b7 + 4*b6 + 4*b3 - 4*b2 - 4*b1 + 4) * q^57 + (-2*b11 + 4*b10 - 2*b8) * q^58 - b1 * q^59 + (4*b9 + 12*b7 + 2*b6) * q^60 + (-b9 - 6*b7 - b6) * q^61 + (-b5 + 5*b1) * q^62 + (b11 + b10 - 2*b8) * q^63 + (3*b11 + 15*b10 + 3*b9 + 8*b8 + 15*b7 + 8*b6 - 3*b5 - 3*b4 - 15*b3 - 8*b2 - 8*b1 - 15) * q^64 + (2*b4 - 2*b2 - 8) * q^65 + (b4 - 4*b2 - 2) * q^67 + (b11 + 2*b10 + b9 + 5*b8 + 2*b7 + 5*b6 - b5 - b4 - 2*b3 - 5*b2 - 5*b1 - 2) * q^68 + (b11 + 2*b10 - 2*b8) * q^69 + (-2*b5 - 2*b3) * q^70 + (-5*b9 - 2*b7 + 4*b6) * q^71 + (-2*b9 + b7 + 3*b6) * q^72 + (-b5 - 6*b3 - b1) * q^73 + (-12*b10 - 4*b8) * q^74 + (b11 + 6*b10 + b9 - 3*b8 + 6*b7 - 3*b6 - b5 - b4 - 6*b3 + 3*b2 + 3*b1 - 6) * q^75 + (2*b4 - 2*b2 + 8) * q^76 + (-10*b2 + 6) * q^78 + (-2*b11 - 10*b10 - 2*b9 - 10*b7 + 2*b5 + 2*b4 + 10*b3 + 10) * q^79 + (4*b11 + 22*b10 + 6*b8) * q^80 + (-2*b5 + 3*b3 - 4*b1) * q^81 + (-3*b9 - 12*b7 - 5*b6) * q^82 + (2*b9 + 4*b7 + 2*b6) * q^83 + (-b5 - 2*b3 - 3*b1) * q^84 + (-2*b11 + 8*b10 + 2*b8) * q^85 + (4*b11 + 6*b10 + 4*b9 + 2*b8 + 6*b7 + 2*b6 - 4*b5 - 4*b4 - 6*b3 - 2*b2 - 2*b1 - 6) * q^86 + (-4*b2 + 4) * q^87 + (-b4 - 2*b2 - 8) * q^89 + (-2*b11 + 6*b10 - 2*b9 + 6*b7 + 2*b5 + 2*b4 - 6*b3 - 6) * q^90 + (b11 - 2*b10 + b8) * q^91 - 2*b1 * q^92 + (-b9 - 4*b7 + 6*b6) * q^93 + (3*b9 + 8*b7 + b6) * q^94 + (4*b5 - 8*b3 - 4*b1) * q^95 + (5*b11 + 12*b10 + 11*b8) * q^96 + (-3*b11 - 4*b10 - 3*b9 + 2*b8 - 4*b7 + 2*b6 + 3*b5 + 3*b4 + 4*b3 - 2*b2 - 2*b1 + 4) * q^97 + (-b2 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 2 q^{2} + q^{3} - 8 q^{4} - q^{5} + 12 q^{6} + 3 q^{7} + 6 q^{8} - 4 q^{9}+O(q^{10})$$ 12 * q + 2 * q^2 + q^3 - 8 * q^4 - q^5 + 12 * q^6 + 3 * q^7 + 6 * q^8 - 4 * q^9 $$12 q + 2 q^{2} + q^{3} - 8 q^{4} - q^{5} + 12 q^{6} + 3 q^{7} + 6 q^{8} - 4 q^{9} + 16 q^{10} + 8 q^{12} + 8 q^{13} - 2 q^{14} + 5 q^{15} - 10 q^{16} + 8 q^{17} - 18 q^{18} + 14 q^{20} + 4 q^{21} + 28 q^{23} + 20 q^{24} - 2 q^{25} + 12 q^{26} + 19 q^{27} + 8 q^{28} - 8 q^{30} + 13 q^{31} - 136 q^{32} - 48 q^{34} + q^{35} - 18 q^{36} + 17 q^{37} - 16 q^{38} - 20 q^{39} - 36 q^{40} + 16 q^{41} - 12 q^{42} - 16 q^{43} - 24 q^{45} - 4 q^{47} - 36 q^{48} - 3 q^{49} + 22 q^{50} - 20 q^{51} + 10 q^{53} - 32 q^{54} + 24 q^{56} + 16 q^{57} + 16 q^{58} - q^{59} + 30 q^{60} - 16 q^{61} + 4 q^{62} + 4 q^{63} - 34 q^{64} - 96 q^{65} - 12 q^{67} + 7 q^{69} + 4 q^{70} - 5 q^{71} + 2 q^{72} + 16 q^{73} - 32 q^{74} - 20 q^{75} + 96 q^{76} + 112 q^{78} + 28 q^{79} + 56 q^{80} - 15 q^{81} - 28 q^{82} + 8 q^{83} + 2 q^{84} + 24 q^{85} - 12 q^{86} + 64 q^{87} - 84 q^{89} - 20 q^{90} - 8 q^{91} - 2 q^{92} - 17 q^{93} + 20 q^{94} + 24 q^{95} + 20 q^{96} + 11 q^{97} - 8 q^{98}+O(q^{100})$$ 12 * q + 2 * q^2 + q^3 - 8 * q^4 - q^5 + 12 * q^6 + 3 * q^7 + 6 * q^8 - 4 * q^9 + 16 * q^10 + 8 * q^12 + 8 * q^13 - 2 * q^14 + 5 * q^15 - 10 * q^16 + 8 * q^17 - 18 * q^18 + 14 * q^20 + 4 * q^21 + 28 * q^23 + 20 * q^24 - 2 * q^25 + 12 * q^26 + 19 * q^27 + 8 * q^28 - 8 * q^30 + 13 * q^31 - 136 * q^32 - 48 * q^34 + q^35 - 18 * q^36 + 17 * q^37 - 16 * q^38 - 20 * q^39 - 36 * q^40 + 16 * q^41 - 12 * q^42 - 16 * q^43 - 24 * q^45 - 4 * q^47 - 36 * q^48 - 3 * q^49 + 22 * q^50 - 20 * q^51 + 10 * q^53 - 32 * q^54 + 24 * q^56 + 16 * q^57 + 16 * q^58 - q^59 + 30 * q^60 - 16 * q^61 + 4 * q^62 + 4 * q^63 - 34 * q^64 - 96 * q^65 - 12 * q^67 + 7 * q^69 + 4 * q^70 - 5 * q^71 + 2 * q^72 + 16 * q^73 - 32 * q^74 - 20 * q^75 + 96 * q^76 + 112 * q^78 + 28 * q^79 + 56 * q^80 - 15 * q^81 - 28 * q^82 + 8 * q^83 + 2 * q^84 + 24 * q^85 - 12 * q^86 + 64 * q^87 - 84 * q^89 - 20 * q^90 - 8 * q^91 - 2 * q^92 - 17 * q^93 + 20 * q^94 + 24 * q^95 + 20 * q^96 + 11 * q^97 - 8 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 7 x^{10} - 15 x^{9} + 59 x^{8} + 118 x^{7} + 266 x^{6} + 324 x^{5} + 1036 x^{4} + 376 x^{3} + 136 x^{2} + 48 x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 15\nu^{10} + 5251\nu^{5} + 24224 ) / 66764$$ (15*v^10 + 5251*v^5 + 24224) / 66764 $$\beta_{3}$$ $$=$$ $$( -59\nu^{11} - 16203\nu^{6} + 367616\nu ) / 133528$$ (-59*v^11 - 16203*v^6 + 367616*v) / 133528 $$\beta_{4}$$ $$=$$ $$( 37\nu^{10} + 10727\nu^{5} - 104932 ) / 33382$$ (37*v^10 + 10727*v^5 - 104932) / 33382 $$\beta_{5}$$ $$=$$ $$( 133\nu^{11} + 37657\nu^{6} - 577480\nu ) / 66764$$ (133*v^11 + 37657*v^6 - 577480*v) / 66764 $$\beta_{6}$$ $$=$$ $$( 59 \nu^{11} - 413 \nu^{10} + 885 \nu^{9} - 3481 \nu^{8} + 9241 \nu^{7} - 15694 \nu^{6} - 19116 \nu^{5} - 61124 \nu^{4} - 22184 \nu^{3} - 375640 \nu^{2} - 2832 \nu - 944 ) / 133528$$ (59*v^11 - 413*v^10 + 885*v^9 - 3481*v^8 + 9241*v^7 - 15694*v^6 - 19116*v^5 - 61124*v^4 - 22184*v^3 - 375640*v^2 - 2832*v - 944) / 133528 $$\beta_{7}$$ $$=$$ $$( 48 \nu^{11} - 336 \nu^{10} + 720 \nu^{9} - 2832 \nu^{8} + 7801 \nu^{7} - 12768 \nu^{6} - 15552 \nu^{5} - 49728 \nu^{4} - 18048 \nu^{3} - 259493 \nu^{2} - 2304 \nu - 768 ) / 33382$$ (48*v^11 - 336*v^10 + 720*v^9 - 2832*v^8 + 7801*v^7 - 12768*v^6 - 15552*v^5 - 49728*v^4 - 18048*v^3 - 259493*v^2 - 2304*v - 768) / 33382 $$\beta_{8}$$ $$=$$ $$( 1093 \nu^{11} - 1093 \nu^{10} + 7563 \nu^{9} - 16395 \nu^{8} + 64487 \nu^{7} + 128974 \nu^{6} + 290738 \nu^{5} + 332228 \nu^{4} + 1132348 \nu^{3} + 410968 \nu^{2} + \cdots + 52464 ) / 133528$$ (1093*v^11 - 1093*v^10 + 7563*v^9 - 16395*v^8 + 64487*v^7 + 128974*v^6 + 290738*v^5 + 332228*v^4 + 1132348*v^3 + 410968*v^2 + 148648*v + 52464) / 133528 $$\beta_{9}$$ $$=$$ $$( - 325 \nu^{11} + 2275 \nu^{10} - 4875 \nu^{9} + 19175 \nu^{8} - 53167 \nu^{7} + 86450 \nu^{6} + 105300 \nu^{5} + 336700 \nu^{4} + 122200 \nu^{3} + 1633540 \nu^{2} + \cdots + 5200 ) / 66764$$ (-325*v^11 + 2275*v^10 - 4875*v^9 + 19175*v^8 - 53167*v^7 + 86450*v^6 + 105300*v^5 + 336700*v^4 + 122200*v^3 + 1633540*v^2 + 15600*v + 5200) / 66764 $$\beta_{10}$$ $$=$$ $$( 1514 \nu^{11} - 1514 \nu^{10} + 10583 \nu^{9} - 22710 \nu^{8} + 89326 \nu^{7} + 178652 \nu^{6} + 402724 \nu^{5} + 485285 \nu^{4} + 1568504 \nu^{3} + 569264 \nu^{2} + \cdots + 72672 ) / 66764$$ (1514*v^11 - 1514*v^10 + 10583*v^9 - 22710*v^8 + 89326*v^7 + 178652*v^6 + 402724*v^5 + 485285*v^4 + 1568504*v^3 + 569264*v^2 + 205904*v + 72672) / 66764 $$\beta_{11}$$ $$=$$ $$( - 4771 \nu^{11} + 4771 \nu^{10} - 33471 \nu^{9} + 71565 \nu^{8} - 281489 \nu^{7} - 562978 \nu^{6} - 1269086 \nu^{5} - 1567258 \nu^{4} - 4942756 \nu^{3} + \cdots - 229008 ) / 66764$$ (-4771*v^11 + 4771*v^10 - 33471*v^9 + 71565*v^8 - 281489*v^7 - 562978*v^6 - 1269086*v^5 - 1567258*v^4 - 4942756*v^3 - 1793896*v^2 - 648856*v - 229008) / 66764
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{9} - 4\beta_{7} + 2\beta_{6}$$ -b9 - 4*b7 + 2*b6 $$\nu^{3}$$ $$=$$ $$- \beta_{11} - 6 \beta_{10} - \beta_{9} + 8 \beta_{8} - 6 \beta_{7} + 8 \beta_{6} + \beta_{5} + \beta_{4} + 6 \beta_{3} - 8 \beta_{2} - 8 \beta _1 + 6$$ -b11 - 6*b10 - b9 + 8*b8 - 6*b7 + 8*b6 + b5 + b4 + 6*b3 - 8*b2 - 8*b1 + 6 $$\nu^{4}$$ $$=$$ $$-7\beta_{11} - 30\beta_{10} + 22\beta_{8}$$ -7*b11 - 30*b10 + 22*b8 $$\nu^{5}$$ $$=$$ $$-15\beta_{4} + 74\beta_{2} - 74$$ -15*b4 + 74*b2 - 74 $$\nu^{6}$$ $$=$$ $$59\beta_{5} + 266\beta_{3} - 222\beta_1$$ 59*b5 + 266*b3 - 222*b1 $$\nu^{7}$$ $$=$$ $$163\beta_{9} + 770\beta_{7} - 710\beta_{6}$$ 163*b9 + 770*b7 - 710*b6 $$\nu^{8}$$ $$=$$ $$547 \beta_{11} + 2514 \beta_{10} + 547 \beta_{9} - 2190 \beta_{8} + 2514 \beta_{7} - 2190 \beta_{6} - 547 \beta_{5} - 547 \beta_{4} - 2514 \beta_{3} + 2190 \beta_{2} + 2190 \beta _1 - 2514$$ 547*b11 + 2514*b10 + 547*b9 - 2190*b8 + 2514*b7 - 2190*b6 - 547*b5 - 547*b4 - 2514*b3 + 2190*b2 + 2190*b1 - 2514 $$\nu^{9}$$ $$=$$ $$1643\beta_{11} + 7666\beta_{10} - 6894\beta_{8}$$ 1643*b11 + 7666*b10 - 6894*b8 $$\nu^{10}$$ $$=$$ $$5251\beta_{4} - 21454\beta_{2} + 24290$$ 5251*b4 - 21454*b2 + 24290 $$\nu^{11}$$ $$=$$ $$-16203\beta_{5} - 75314\beta_{3} + 67198\beta_1$$ -16203*b5 - 75314*b3 + 67198*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 −1.42513 + 1.03542i −0.293939 + 0.213559i 2.52809 − 1.83676i −0.965643 − 2.97194i 0.112275 + 0.345546i 0.544351 + 1.67534i −1.42513 − 1.03542i −0.293939 − 0.213559i 2.52809 + 1.83676i −0.965643 + 2.97194i 0.112275 − 0.345546i 0.544351 − 1.67534i
−0.853368 2.62640i −1.42513 1.03542i −4.55169 + 3.30700i −0.811540 + 2.49766i −1.50326 + 4.62655i 0.809017 0.587785i 8.10146 + 5.88605i 0.0318546 + 0.0980384i 7.25240
148.2 −0.421292 1.29660i −0.293939 0.213559i 0.114343 0.0830753i 0.970726 2.98759i −0.153067 + 0.471092i 0.809017 0.587785i −2.36180 1.71595i −0.886258 2.72762i −4.28267
148.3 0.656626 + 2.02089i 2.52809 + 1.83676i −2.03479 + 1.47836i 0.149831 0.461131i −2.05188 + 6.31504i 0.809017 0.587785i −0.885558 0.643395i 2.09047 + 6.43381i 1.03028
323.1 −1.71907 1.24898i −0.965643 + 2.97194i 0.777220 + 2.39204i −0.392262 + 0.284995i 5.37189 3.90291i −0.309017 0.951057i 0.338253 1.04104i −5.47293 3.97631i 1.03028
323.2 1.10296 + 0.801344i 0.112275 0.345546i −0.0436753 0.134419i −2.54139 + 1.84643i 0.400735 0.291151i −0.309017 0.951057i 0.902127 2.77646i 2.32025 + 1.68576i −4.28267
323.3 2.23415 + 1.62320i 0.544351 1.67534i 1.73859 + 5.35083i 2.12464 1.54364i 3.93558 2.85936i −0.309017 0.951057i −3.09448 + 9.52384i −0.0833965 0.0605911i 7.25240
372.1 −0.853368 + 2.62640i −1.42513 + 1.03542i −4.55169 3.30700i −0.811540 2.49766i −1.50326 4.62655i 0.809017 + 0.587785i 8.10146 5.88605i 0.0318546 0.0980384i 7.25240
372.2 −0.421292 + 1.29660i −0.293939 + 0.213559i 0.114343 + 0.0830753i 0.970726 + 2.98759i −0.153067 0.471092i 0.809017 + 0.587785i −2.36180 + 1.71595i −0.886258 + 2.72762i −4.28267
372.3 0.656626 2.02089i 2.52809 1.83676i −2.03479 1.47836i 0.149831 + 0.461131i −2.05188 6.31504i 0.809017 + 0.587785i −0.885558 + 0.643395i 2.09047 6.43381i 1.03028
729.1 −1.71907 + 1.24898i −0.965643 2.97194i 0.777220 2.39204i −0.392262 0.284995i 5.37189 + 3.90291i −0.309017 + 0.951057i 0.338253 + 1.04104i −5.47293 + 3.97631i 1.03028
729.2 1.10296 0.801344i 0.112275 + 0.345546i −0.0436753 + 0.134419i −2.54139 1.84643i 0.400735 + 0.291151i −0.309017 + 0.951057i 0.902127 + 2.77646i 2.32025 1.68576i −4.28267
729.3 2.23415 1.62320i 0.544351 + 1.67534i 1.73859 5.35083i 2.12464 + 1.54364i 3.93558 + 2.85936i −0.309017 + 0.951057i −3.09448 9.52384i −0.0833965 + 0.0605911i 7.25240
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 729.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.u 12
11.b odd 2 1 847.2.f.t 12
11.c even 5 1 847.2.a.i 3
11.c even 5 3 inner 847.2.f.u 12
11.d odd 10 1 847.2.a.j yes 3
11.d odd 10 3 847.2.f.t 12
33.f even 10 1 7623.2.a.bz 3
33.h odd 10 1 7623.2.a.ce 3
77.j odd 10 1 5929.2.a.t 3
77.l even 10 1 5929.2.a.y 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.i 3 11.c even 5 1
847.2.a.j yes 3 11.d odd 10 1
847.2.f.t 12 11.b odd 2 1
847.2.f.t 12 11.d odd 10 3
847.2.f.u 12 1.a even 1 1 trivial
847.2.f.u 12 11.c even 5 3 inner
5929.2.a.t 3 77.j odd 10 1
5929.2.a.y 3 77.l even 10 1
7623.2.a.bz 3 33.f even 10 1
7623.2.a.ce 3 33.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{12} - 2 T_{2}^{11} + 9 T_{2}^{10} - 20 T_{2}^{9} + 69 T_{2}^{8} - 44 T_{2}^{7} + 273 T_{2}^{6} - 214 T_{2}^{5} + 1441 T_{2}^{4} - 1768 T_{2}^{3} + 2624 T_{2}^{2} - 2560 T_{2} + 4096$$ T2^12 - 2*T2^11 + 9*T2^10 - 20*T2^9 + 69*T2^8 - 44*T2^7 + 273*T2^6 - 214*T2^5 + 1441*T2^4 - 1768*T2^3 + 2624*T2^2 - 2560*T2 + 4096 $$T_{3}^{12} - T_{3}^{11} + 7 T_{3}^{10} - 15 T_{3}^{9} + 59 T_{3}^{8} + 118 T_{3}^{7} + 266 T_{3}^{6} + 324 T_{3}^{5} + 1036 T_{3}^{4} + 376 T_{3}^{3} + 136 T_{3}^{2} + 48 T_{3} + 16$$ T3^12 - T3^11 + 7*T3^10 - 15*T3^9 + 59*T3^8 + 118*T3^7 + 266*T3^6 + 324*T3^5 + 1036*T3^4 + 376*T3^3 + 136*T3^2 + 48*T3 + 16 $$T_{13}^{12} - 8 T_{13}^{11} + 62 T_{13}^{10} - 416 T_{13}^{9} + 2692 T_{13}^{8} - 8768 T_{13}^{7} + 38136 T_{13}^{6} - 115264 T_{13}^{5} + 284688 T_{13}^{4} + 393728 T_{13}^{3} + 2113536 T_{13}^{2} + 524288 T_{13} + 16777216$$ T13^12 - 8*T13^11 + 62*T13^10 - 416*T13^9 + 2692*T13^8 - 8768*T13^7 + 38136*T13^6 - 115264*T13^5 + 284688*T13^4 + 393728*T13^3 + 2113536*T13^2 + 524288*T13 + 16777216

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 2 T^{11} + 9 T^{10} - 20 T^{9} + \cdots + 4096$$
$3$ $$T^{12} - T^{11} + 7 T^{10} - 15 T^{9} + \cdots + 16$$
$5$ $$T^{12} + T^{11} + 9 T^{10} + 13 T^{9} + \cdots + 256$$
$7$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{3}$$
$11$ $$T^{12}$$
$13$ $$T^{12} - 8 T^{11} + 62 T^{10} + \cdots + 16777216$$
$17$ $$T^{12} - 8 T^{11} + 62 T^{10} + \cdots + 16777216$$
$19$ $$T^{12} + 40 T^{10} - 64 T^{9} + \cdots + 16777216$$
$23$ $$(T^{3} - 7 T^{2} + 8 T + 8)^{4}$$
$29$ $$T^{12} + 40 T^{10} + 64 T^{9} + \cdots + 16777216$$
$31$ $$T^{12} - 13 T^{11} + 119 T^{10} + \cdots + 11316496$$
$37$ $$T^{12} - 17 T^{11} + 217 T^{10} + \cdots + 65536$$
$41$ $$T^{12} - 16 T^{11} + 190 T^{10} + \cdots + 40960000$$
$43$ $$(T^{3} + 4 T^{2} - 28 T - 32)^{4}$$
$47$ $$T^{12} + 4 T^{11} + 30 T^{10} + \cdots + 4096$$
$53$ $$T^{12} - 10 T^{11} + 100 T^{10} + \cdots + 16777216$$
$59$ $$T^{12} + T^{11} + 7 T^{10} + 15 T^{9} + \cdots + 16$$
$61$ $$T^{12} + 16 T^{11} + 190 T^{10} + \cdots + 40960000$$
$67$ $$(T^{3} + 3 T^{2} - 88 T - 424)^{4}$$
$71$ $$T^{12} + 5 T^{11} + \cdots + 4797852160000$$
$73$ $$T^{12} - 16 T^{11} + 190 T^{10} + \cdots + 40960000$$
$79$ $$T^{12} - 28 T^{11} + \cdots + 68719476736$$
$83$ $$T^{12} - 8 T^{11} + 120 T^{10} + \cdots + 16777216$$
$89$ $$(T^{3} + 21 T^{2} + 104 T + 100)^{4}$$
$97$ $$T^{12} - 11 T^{11} + \cdots + 41740124416$$