# Properties

 Label 896.2.m.h Level $896$ Weight $2$ Character orbit 896.m Analytic conductor $7.155$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$896 = 2^{7} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 896.m (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.15459602111$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.20138089353117696.1 Defining polynomial: $$x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64$$ x^12 - 3*x^10 - 2*x^9 + 2*x^8 + 4*x^7 + 2*x^6 + 8*x^5 + 8*x^4 - 16*x^3 - 48*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{3} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{5} - \beta_{3} q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q - b6 * q^3 + (-b5 - b3 - b2 - 1) * q^5 - b3 * q^7 + (-b4 - 2*b3 - b2 + b1) * q^9 $$q - \beta_{6} q^{3} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{5} - \beta_{3} q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{9} - \beta_{8} q^{11} + (\beta_{11} + \beta_{9} - \beta_{6} + \beta_1) q^{13} + (\beta_{11} + \beta_{6} + \beta_{5} + 3) q^{15} + (\beta_{10} - 1) q^{17} + (\beta_{9} + \beta_{6} + \beta_{4}) q^{19} + \beta_{5} q^{21} + ( - \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{3} - \beta_1) q^{23} + ( - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{3} - \beta_1) q^{25} + ( - \beta_{10} + \beta_{8} - \beta_{7} + 2 \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{27} + ( - \beta_{11} - \beta_1) q^{29} + ( - \beta_{6} - \beta_{5}) q^{31} + (\beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - 1) q^{33} + ( - \beta_{6} + \beta_{4} + \beta_{3} - 1) q^{35} + (\beta_{3} - 2 \beta_{2} + 1) q^{37} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_1) q^{39} + (\beta_{6} - \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_{2}) q^{41} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + 2 \beta_{3} - \beta_1 + 2) q^{43} + ( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - 3 \beta_{6} + 5 \beta_{3} - \beta_1 - 5) q^{45} + ( - 2 \beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - 2) q^{47} - q^{49} + (\beta_{11} + \beta_{3} + \beta_1 - 1) q^{51} + (\beta_{10} - 2 \beta_{8} + \beta_{7} + 2 \beta_{2}) q^{53} + (\beta_{9} - \beta_{8} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{55} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} - \beta_1) q^{57} + (\beta_{10} - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{59} + (\beta_{10} - \beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + 2) q^{61} + ( - \beta_{11} + \beta_{4} - \beta_{2} - 2) q^{63} + (\beta_{11} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{65} + ( - \beta_{11} - \beta_{9} + 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{67} + ( - \beta_{11} + \beta_{8} - 2 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{69} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_1) q^{71} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{73} + ( - 2 \beta_{11} + \beta_{8} - 3 \beta_{5} - 6 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 6) q^{75} - \beta_{9} q^{77} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - 1) q^{79} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - 3) q^{81} + ( - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{7} + \beta_{6} + 2 \beta_{4} + 3 \beta_{3} + \cdots - 3) q^{83}+ \cdots + (2 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_1 + 3) q^{99}+O(q^{100})$$ q - b6 * q^3 + (-b5 - b3 - b2 - 1) * q^5 - b3 * q^7 + (-b4 - 2*b3 - b2 + b1) * q^9 - b8 * q^11 + (b11 + b9 - b6 + b1) * q^13 + (b11 + b6 + b5 + 3) * q^15 + (b10 - 1) * q^17 + (b9 + b6 + b4) * q^19 + b5 * q^21 + (-b9 + b8 - b7 + 2*b3 - b1) * q^23 + (-b9 + b8 - b7 - b6 + b5 + 3*b3 - b1) * q^25 + (-b10 + b8 - b7 + 2*b5 + b3 - b2 + 1) * q^27 + (-b11 - b1) * q^29 + (-b6 - b5) * q^31 + (b10 - b9 - b8 - b6 - b5 - 1) * q^33 + (-b6 + b4 + b3 - 1) * q^35 + (b3 - 2*b2 + 1) * q^37 + (-b7 - 2*b6 + 2*b5 + b1) * q^39 + (b6 - b5 - b4 - 4*b3 - b2) * q^41 + (b11 - b10 + b8 - b7 + 2*b3 - b1 + 2) * q^43 + (-b11 + b10 - b9 - b7 - 3*b6 + 5*b3 - b1 - 5) * q^45 + (-2*b11 - b9 - b8 + b6 + b5 + b4 - b2 - 2) * q^47 - q^49 + (b11 + b3 + b1 - 1) * q^51 + (b10 - 2*b8 + b7 + 2*b2) * q^53 + (b9 - b8 + b4 + 2*b3 + b2 + 2*b1) * q^55 + (b9 - b8 + b7 + b6 - b5 + 2*b4 + 6*b3 + 2*b2 - b1) * q^57 + (b10 - b8 + b7 + b5 - b3 + b2 - 1) * q^59 + (b10 - b7 + b6 - b4 - 2*b3 + 2) * q^61 + (-b11 + b4 - b2 - 2) * q^63 + (b11 + 2*b9 + 2*b8 - 2*b6 - 2*b5 + b4 - b2 + 1) * q^65 + (-b11 - b9 + 2*b4 - b3 - b1 + 1) * q^67 + (-b11 + b8 - 2*b5 - b3 - b2 + b1 - 1) * q^69 + (-b7 - b6 + b5 + b3 - 2*b1) * q^71 + (-b7 - b6 + b5 - b4 + b3 - b2 - 2*b1) * q^73 + (-2*b11 + b8 - 3*b5 - 6*b3 - 3*b2 + 2*b1 - 6) * q^75 - b9 * q^77 + (-b10 + b9 + b8 + b6 + b5 - 1) * q^79 + (-b11 - b10 + b9 + b8 + b6 + b5 - 3) * q^81 + (-2*b11 + b10 - 2*b9 - b7 + b6 + 2*b4 + 3*b3 - 2*b1 - 3) * q^83 + (b11 - b10 + 2*b8 - b7 + 2*b3 + 2*b2 - b1 + 2) * q^85 + (b9 - b8 + 2*b7 + 3*b6 - 3*b5 - b4 - 6*b3 - b2) * q^87 + (b9 - b8 + b7 - 3*b6 + 3*b5 + 3*b3 + 2*b1) * q^89 + (-b11 - b8 + b5 + b1) * q^91 + (b11 - 2*b4 - 5*b3 + b1 + 5) * q^93 + (b11 - b10 + 2*b9 + 2*b8 - 2*b6 - 2*b5 + 2*b4 - 2*b2 - 2) * q^95 + (-b10 - b9 - b8 + 2*b6 + 2*b5 + b4 - b2 + 5) * q^97 + (2*b11 + b10 + b9 - b7 - 2*b6 - 2*b4 - 3*b3 + 2*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{3} - 4 q^{5}+O(q^{10})$$ 12 * q + 4 * q^3 - 4 * q^5 $$12 q + 4 q^{3} - 4 q^{5} + 24 q^{15} - 8 q^{17} - 4 q^{21} + 4 q^{27} + 4 q^{29} + 8 q^{31} - 4 q^{35} + 20 q^{37} + 16 q^{43} - 40 q^{45} - 16 q^{47} - 12 q^{49} - 16 q^{51} - 4 q^{53} - 16 q^{59} + 20 q^{61} - 12 q^{63} + 32 q^{65} + 24 q^{67} + 4 q^{69} - 40 q^{75} - 24 q^{79} - 44 q^{81} - 20 q^{83} + 8 q^{85} + 48 q^{93} + 48 q^{97} + 32 q^{99}+O(q^{100})$$ 12 * q + 4 * q^3 - 4 * q^5 + 24 * q^15 - 8 * q^17 - 4 * q^21 + 4 * q^27 + 4 * q^29 + 8 * q^31 - 4 * q^35 + 20 * q^37 + 16 * q^43 - 40 * q^45 - 16 * q^47 - 12 * q^49 - 16 * q^51 - 4 * q^53 - 16 * q^59 + 20 * q^61 - 12 * q^63 + 32 * q^65 + 24 * q^67 + 4 * q^69 - 40 * q^75 - 24 * q^79 - 44 * q^81 - 20 * q^83 + 8 * q^85 + 48 * q^93 + 48 * q^97 + 32 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{11} + 14 \nu^{10} + \nu^{9} - 20 \nu^{8} - 54 \nu^{7} + 16 \nu^{6} + 34 \nu^{5} + 36 \nu^{4} + 192 \nu^{3} + 240 \nu^{2} - 144 \nu - 608 ) / 128$$ (v^11 + 14*v^10 + v^9 - 20*v^8 - 54*v^7 + 16*v^6 + 34*v^5 + 36*v^4 + 192*v^3 + 240*v^2 - 144*v - 608) / 128 $$\beta_{2}$$ $$=$$ $$( 3 \nu^{11} + 10 \nu^{10} + 3 \nu^{9} - 28 \nu^{8} - 34 \nu^{7} - 16 \nu^{6} - 26 \nu^{5} + 44 \nu^{4} + 192 \nu^{3} + 208 \nu^{2} - 176 \nu - 416 ) / 128$$ (3*v^11 + 10*v^10 + 3*v^9 - 28*v^8 - 34*v^7 - 16*v^6 - 26*v^5 + 44*v^4 + 192*v^3 + 208*v^2 - 176*v - 416) / 128 $$\beta_{3}$$ $$=$$ $$( 5 \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 4 \nu^{8} - 14 \nu^{7} - 16 \nu^{6} - 22 \nu^{5} - 12 \nu^{4} + 64 \nu^{3} + 112 \nu^{2} + 48 \nu + 32 ) / 128$$ (5*v^11 + 6*v^10 + 5*v^9 - 4*v^8 - 14*v^7 - 16*v^6 - 22*v^5 - 12*v^4 + 64*v^3 + 112*v^2 + 48*v + 32) / 128 $$\beta_{4}$$ $$=$$ $$( - 11 \nu^{11} - 10 \nu^{10} + 21 \nu^{9} + 44 \nu^{8} + 18 \nu^{7} - 16 \nu^{6} + 10 \nu^{5} - 108 \nu^{4} - 256 \nu^{3} - 80 \nu^{2} + 304 \nu + 416 ) / 128$$ (-11*v^11 - 10*v^10 + 21*v^9 + 44*v^8 + 18*v^7 - 16*v^6 + 10*v^5 - 108*v^4 - 256*v^3 - 80*v^2 + 304*v + 416) / 128 $$\beta_{5}$$ $$=$$ $$( - 5 \nu^{11} - 14 \nu^{10} + 3 \nu^{9} + 28 \nu^{8} + 38 \nu^{7} + 16 \nu^{6} + 6 \nu^{5} - 36 \nu^{4} - 176 \nu^{3} - 176 \nu^{2} + 80 \nu + 352 ) / 64$$ (-5*v^11 - 14*v^10 + 3*v^9 + 28*v^8 + 38*v^7 + 16*v^6 + 6*v^5 - 36*v^4 - 176*v^3 - 176*v^2 + 80*v + 352) / 64 $$\beta_{6}$$ $$=$$ $$( -\nu^{11} - \nu^{10} + 2\nu^{9} + 5\nu^{8} + 3\nu^{7} - 4\nu^{5} - 14\nu^{4} - 26\nu^{3} - 8\nu^{2} + 40\nu + 48 ) / 8$$ (-v^11 - v^10 + 2*v^9 + 5*v^8 + 3*v^7 - 4*v^5 - 14*v^4 - 26*v^3 - 8*v^2 + 40*v + 48) / 8 $$\beta_{7}$$ $$=$$ $$( 21 \nu^{11} + 54 \nu^{10} + 21 \nu^{9} - 52 \nu^{8} - 78 \nu^{7} - 112 \nu^{6} - 118 \nu^{5} + 84 \nu^{4} + 448 \nu^{3} + 752 \nu^{2} + 176 \nu - 480 ) / 128$$ (21*v^11 + 54*v^10 + 21*v^9 - 52*v^8 - 78*v^7 - 112*v^6 - 118*v^5 + 84*v^4 + 448*v^3 + 752*v^2 + 176*v - 480) / 128 $$\beta_{8}$$ $$=$$ $$( - 23 \nu^{11} - 18 \nu^{10} + 41 \nu^{9} + 60 \nu^{8} + 58 \nu^{7} + 48 \nu^{6} - 14 \nu^{5} - 220 \nu^{4} - 448 \nu^{3} - 144 \nu^{2} + 624 \nu + 544 ) / 128$$ (-23*v^11 - 18*v^10 + 41*v^9 + 60*v^8 + 58*v^7 + 48*v^6 - 14*v^5 - 220*v^4 - 448*v^3 - 144*v^2 + 624*v + 544) / 128 $$\beta_{9}$$ $$=$$ $$( - 23 \nu^{11} - 34 \nu^{10} + 41 \nu^{9} + 108 \nu^{8} + 90 \nu^{7} + 16 \nu^{6} - 78 \nu^{5} - 252 \nu^{4} - 576 \nu^{3} - 528 \nu^{2} + 880 \nu + 1312 ) / 128$$ (-23*v^11 - 34*v^10 + 41*v^9 + 108*v^8 + 90*v^7 + 16*v^6 - 78*v^5 - 252*v^4 - 576*v^3 - 528*v^2 + 880*v + 1312) / 128 $$\beta_{10}$$ $$=$$ $$( - 17 \nu^{11} - 22 \nu^{10} + 31 \nu^{9} + 76 \nu^{8} + 54 \nu^{7} - 34 \nu^{5} - 180 \nu^{4} - 416 \nu^{3} - 176 \nu^{2} + 720 \nu + 800 ) / 64$$ (-17*v^11 - 22*v^10 + 31*v^9 + 76*v^8 + 54*v^7 - 34*v^5 - 180*v^4 - 416*v^3 - 176*v^2 + 720*v + 800) / 64 $$\beta_{11}$$ $$=$$ $$( 19 \nu^{11} + 18 \nu^{10} - 29 \nu^{9} - 68 \nu^{8} - 66 \nu^{7} - 16 \nu^{6} + 38 \nu^{5} + 220 \nu^{4} + 352 \nu^{3} + 240 \nu^{2} - 560 \nu - 736 ) / 64$$ (19*v^11 + 18*v^10 - 29*v^9 - 68*v^8 - 66*v^7 - 16*v^6 + 38*v^5 + 220*v^4 + 352*v^3 + 240*v^2 - 560*v - 736) / 64
 $$\nu$$ $$=$$ $$( \beta_{10} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} - 1 ) / 4$$ (b10 - b6 - b5 - b4 - b2 - 1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{10} - 2\beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 3 ) / 4$$ (b10 - 2*b9 + b6 + b5 - b4 + b2 + 3) / 4 $$\nu^{3}$$ $$=$$ $$( -2\beta_{11} + \beta_{10} - 2\beta_{8} - 3\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 4$$ (-2*b11 + b10 - 2*b8 - 3*b6 + b5 - b4 + 2*b3 + b2 + 2*b1 + 1) / 4 $$\nu^{4}$$ $$=$$ $$( 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} - 8 \beta_{3} + \beta_{2} + 2 \beta _1 + 5 ) / 4$$ (2*b11 + b10 + 2*b9 + 2*b7 - b6 + 3*b5 - b4 - 8*b3 + b2 + 2*b1 + 5) / 4 $$\nu^{5}$$ $$=$$ $$( -2\beta_{11} + \beta_{10} - 2\beta_{9} - 7\beta_{6} + \beta_{5} + 5\beta_{4} + 6\beta_{3} - 5\beta_{2} + 2\beta _1 - 3 ) / 4$$ (-2*b11 + b10 - 2*b9 - 7*b6 + b5 + 5*b4 + 6*b3 - 5*b2 + 2*b1 - 3) / 4 $$\nu^{6}$$ $$=$$ $$( 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 3 \beta_{6} + 3 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 6 \beta _1 + 11 ) / 4$$ (2*b11 - b10 + 2*b9 + 4*b8 + 3*b6 + 3*b5 - 9*b4 + 2*b3 - 3*b2 + 6*b1 + 11) / 4 $$\nu^{7}$$ $$=$$ $$( - 6 \beta_{11} + \beta_{10} - 6 \beta_{9} + 4 \beta_{7} - 7 \beta_{6} + 9 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta _1 - 15 ) / 4$$ (-6*b11 + b10 - 6*b9 + 4*b7 - 7*b6 + 9*b5 - 3*b4 - 2*b3 - b2 - 2*b1 - 15) / 4 $$\nu^{8}$$ $$=$$ $$( - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - 12 \beta_{8} + 4 \beta_{7} + 7 \beta_{6} + 15 \beta_{5} - \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + 18 \beta _1 - 9 ) / 4$$ (-2*b11 - b10 + 2*b9 - 12*b8 + 4*b7 + 7*b6 + 15*b5 - b4 - 6*b3 - 3*b2 + 18*b1 - 9) / 4 $$\nu^{9}$$ $$=$$ $$( 2 \beta_{11} - 7 \beta_{10} + 14 \beta_{9} + 12 \beta_{8} + 4 \beta_{7} - 23 \beta_{6} + 17 \beta_{5} + 17 \beta_{4} + 30 \beta_{3} + 11 \beta_{2} + 6 \beta _1 - 23 ) / 4$$ (2*b11 - 7*b10 + 14*b9 + 12*b8 + 4*b7 - 23*b6 + 17*b5 + 17*b4 + 30*b3 + 11*b2 + 6*b1 - 23) / 4 $$\nu^{10}$$ $$=$$ $$( - 2 \beta_{11} - 21 \beta_{10} + 10 \beta_{9} + 4 \beta_{8} + 16 \beta_{7} + 15 \beta_{6} - \beta_{5} + 7 \beta_{4} - 50 \beta_{3} - 35 \beta_{2} + 10 \beta _1 + 19 ) / 4$$ (-2*b11 - 21*b10 + 10*b9 + 4*b8 + 16*b7 + 15*b6 - b5 + 7*b4 - 50*b3 - 35*b2 + 10*b1 + 19) / 4 $$\nu^{11}$$ $$=$$ $$( 10 \beta_{11} - 7 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} - 4 \beta_{7} - 7 \beta_{6} + 17 \beta_{5} + \beta_{4} + 110 \beta_{3} - 29 \beta_{2} - 2 \beta _1 - 111 ) / 4$$ (10*b11 - 7*b10 + 6*b9 + 12*b8 - 4*b7 - 7*b6 + 17*b5 + b4 + 110*b3 - 29*b2 - 2*b1 - 111) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$645$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
225.1
 −1.40471 + 0.163666i 1.37925 + 0.312504i −1.12465 + 0.857418i 0.402577 + 1.35570i −0.605558 − 1.27801i 1.35309 − 0.411286i −1.40471 − 0.163666i 1.37925 − 0.312504i −1.12465 − 0.857418i 0.402577 − 1.35570i −0.605558 + 1.27801i 1.35309 + 0.411286i
0 −2.05500 + 2.05500i 0 −2.72766 2.72766i 0 1.00000i 0 5.44602i 0
225.2 0 −0.599978 + 0.599978i 0 −0.974969 0.974969i 0 1.00000i 0 2.28005i 0
225.3 0 0.416854 0.416854i 0 1.13169 + 1.13169i 0 1.00000i 0 2.65247i 0
225.4 0 0.631188 0.631188i 0 2.34259 + 2.34259i 0 1.00000i 0 2.20320i 0
225.5 0 1.39123 1.39123i 0 −2.16478 2.16478i 0 1.00000i 0 0.871066i 0
225.6 0 2.21570 2.21570i 0 0.393125 + 0.393125i 0 1.00000i 0 6.81864i 0
673.1 0 −2.05500 2.05500i 0 −2.72766 + 2.72766i 0 1.00000i 0 5.44602i 0
673.2 0 −0.599978 0.599978i 0 −0.974969 + 0.974969i 0 1.00000i 0 2.28005i 0
673.3 0 0.416854 + 0.416854i 0 1.13169 1.13169i 0 1.00000i 0 2.65247i 0
673.4 0 0.631188 + 0.631188i 0 2.34259 2.34259i 0 1.00000i 0 2.20320i 0
673.5 0 1.39123 + 1.39123i 0 −2.16478 + 2.16478i 0 1.00000i 0 0.871066i 0
673.6 0 2.21570 + 2.21570i 0 0.393125 0.393125i 0 1.00000i 0 6.81864i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.m.h 12
4.b odd 2 1 896.2.m.g 12
8.b even 2 1 448.2.m.d 12
8.d odd 2 1 112.2.m.d 12
16.e even 4 1 448.2.m.d 12
16.e even 4 1 inner 896.2.m.h 12
16.f odd 4 1 112.2.m.d 12
16.f odd 4 1 896.2.m.g 12
32.g even 8 2 7168.2.a.bi 12
32.h odd 8 2 7168.2.a.bj 12
56.e even 2 1 784.2.m.h 12
56.k odd 6 2 784.2.x.l 24
56.m even 6 2 784.2.x.m 24
112.j even 4 1 784.2.m.h 12
112.u odd 12 2 784.2.x.l 24
112.v even 12 2 784.2.x.m 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.d 12 8.d odd 2 1
112.2.m.d 12 16.f odd 4 1
448.2.m.d 12 8.b even 2 1
448.2.m.d 12 16.e even 4 1
784.2.m.h 12 56.e even 2 1
784.2.m.h 12 112.j even 4 1
784.2.x.l 24 56.k odd 6 2
784.2.x.l 24 112.u odd 12 2
784.2.x.m 24 56.m even 6 2
784.2.x.m 24 112.v even 12 2
896.2.m.g 12 4.b odd 2 1
896.2.m.g 12 16.f odd 4 1
896.2.m.h 12 1.a even 1 1 trivial
896.2.m.h 12 16.e even 4 1 inner
7168.2.a.bi 12 32.g even 8 2
7168.2.a.bj 12 32.h odd 8 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(896, [\chi])$$:

 $$T_{3}^{12} - 4 T_{3}^{11} + 8 T_{3}^{10} - 4 T_{3}^{9} + 76 T_{3}^{8} - 288 T_{3}^{7} + 552 T_{3}^{6} - 376 T_{3}^{5} + 164 T_{3}^{4} - 144 T_{3}^{3} + 288 T_{3}^{2} - 192 T_{3} + 64$$ T3^12 - 4*T3^11 + 8*T3^10 - 4*T3^9 + 76*T3^8 - 288*T3^7 + 552*T3^6 - 376*T3^5 + 164*T3^4 - 144*T3^3 + 288*T3^2 - 192*T3 + 64 $$T_{11}^{12} + 32 T_{11}^{9} + 740 T_{11}^{8} + 896 T_{11}^{7} + 512 T_{11}^{6} + 896 T_{11}^{5} + 67968 T_{11}^{4} + 89088 T_{11}^{3} + 51200 T_{11}^{2} - 235520 T_{11} + 541696$$ T11^12 + 32*T11^9 + 740*T11^8 + 896*T11^7 + 512*T11^6 + 896*T11^5 + 67968*T11^4 + 89088*T11^3 + 51200*T11^2 - 235520*T11 + 541696

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} - 4 T^{11} + 8 T^{10} - 4 T^{9} + \cdots + 64$$
$5$ $$T^{12} + 4 T^{11} + 8 T^{10} - 12 T^{9} + \cdots + 2304$$
$7$ $$(T^{2} + 1)^{6}$$
$11$ $$T^{12} + 32 T^{9} + 740 T^{8} + \cdots + 541696$$
$13$ $$T^{12} - 20 T^{9} + 1724 T^{8} + \cdots + 3211264$$
$17$ $$(T^{6} + 4 T^{5} - 44 T^{4} - 200 T^{3} + \cdots - 96)^{2}$$
$19$ $$T^{12} - 76 T^{9} + 2444 T^{8} + \cdots + 2849344$$
$23$ $$T^{12} + 136 T^{10} + \cdots + 10137856$$
$29$ $$T^{12} - 4 T^{11} + 8 T^{10} + \cdots + 8620096$$
$31$ $$(T^{6} - 4 T^{5} - 16 T^{4} + 72 T^{3} + \cdots + 64)^{2}$$
$37$ $$T^{12} - 20 T^{11} + 200 T^{10} + \cdots + 5053504$$
$41$ $$T^{12} + 120 T^{10} + 4704 T^{8} + \cdots + 25600$$
$43$ $$T^{12} - 16 T^{11} + 128 T^{10} + \cdots + 23040000$$
$47$ $$(T^{6} + 8 T^{5} - 104 T^{4} - 1032 T^{3} + \cdots + 19776)^{2}$$
$53$ $$T^{12} + 4 T^{11} + 8 T^{10} + \cdots + 78400$$
$59$ $$T^{12} + 16 T^{11} + \cdots + 119596096$$
$61$ $$T^{12} - 20 T^{11} + 200 T^{10} + \cdots + 256$$
$67$ $$T^{12} - 24 T^{11} + 288 T^{10} + \cdots + 3686400$$
$71$ $$T^{12} + 432 T^{10} + 60960 T^{8} + \cdots + 6553600$$
$73$ $$T^{12} + 616 T^{10} + \cdots + 3050573824$$
$79$ $$(T^{6} + 12 T^{5} - 44 T^{4} - 576 T^{3} + \cdots - 2240)^{2}$$
$83$ $$T^{12} + 20 T^{11} + \cdots + 320093429824$$
$89$ $$T^{12} + 552 T^{10} + \cdots + 35476475904$$
$97$ $$(T^{6} - 24 T^{5} - 60 T^{4} + 2568 T^{3} + \cdots - 39008)^{2}$$