# Properties

 Label 1152.2.i.l Level $1152$ Weight $2$ Character orbit 1152.i Analytic conductor $9.199$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.19876631285$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} - 216 x^{3} + 243 x^{2} - 486 x + 729$$ x^12 - 2*x^11 + 3*x^10 - 8*x^9 + 22*x^8 - 42*x^7 + 51*x^6 - 126*x^5 + 198*x^4 - 216*x^3 + 243*x^2 - 486*x + 729 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{6} - \beta_1) q^{3} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{5} + (\beta_{10} + \beta_{9} - \beta_{6} + \beta_{5} - \beta_{2} - \beta_1 - 2) q^{7} - \beta_{9} q^{9}+O(q^{10})$$ q + (-b6 - b1) * q^3 + (b6 + b5 - b4) * q^5 + (b10 + b9 - b6 + b5 - b2 - b1 - 2) * q^7 - b9 * q^9 $$q + ( - \beta_{6} - \beta_1) q^{3} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{5} + (\beta_{10} + \beta_{9} - \beta_{6} + \beta_{5} - \beta_{2} - \beta_1 - 2) q^{7} - \beta_{9} q^{9} + (\beta_{10} - \beta_{8} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{11} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{13} + (\beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{15} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_{4} + \beta_1) q^{17} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{19} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{21}+ \cdots + (2 \beta_{11} - 2 \beta_{10} - \beta_{9} + 2 \beta_{6} - 5 \beta_{5} - \beta_{3} + 2 \beta_1 + 8) q^{99}+O(q^{100})$$ q + (-b6 - b1) * q^3 + (b6 + b5 - b4) * q^5 + (b10 + b9 - b6 + b5 - b2 - b1 - 2) * q^7 - b9 * q^9 + (b10 - b8 - 2*b6 - b5 + b4 + b3 - b2 - b1) * q^11 + (b11 + b10 + b9 + b8 - b7 - b6 - 2*b5 - b4 + b3 - b1 - 1) * q^13 + (b11 + b9 - b7 - b6 - 2*b5 - b4 - b2 - b1 - 1) * q^15 + (b11 + b10 - b9 - b8 + b6 + b4 + b1) * q^17 + (-b10 + b9 + b8 + b7 + 2*b6 - b4 + b2 + 2*b1 + 2) * q^19 + (b11 + b10 - b9 - b8 - b7 - b6 - 2*b5 + b4 - b3 - b2 + b1 - 1) * q^21 + (b10 + b8 - b7 - 2*b5 + 2*b1) * q^23 + (b9 + b8 - b7 + 2*b6 + 2*b5 - b3 - b2 + b1 - 3) * q^25 + (-b10 + b8 - b7 + b6 - b5 - b4 + b3 + b2) * q^27 + (-b11 + b10 + b9 - b6 + b4 - 2*b2 - 2*b1 - 1) * q^29 + (-b10 - 2*b9 - b8 + b7 + 2*b6 - 2*b3) * q^31 + (-b11 - b10 - b9 - b8 + b6 + b5 - b4 - 2*b3 - b1 - 1) * q^33 + (-b11 - b10 + b9 + b8 + b7 - b4 + 3*b2 + b1 + 3) * q^35 + (-b8 + b7 - b3 - b2 + b1) * q^37 + (b11 + 2*b10 - 2*b7 + b6 + 3*b5 - b4 + b3 - b2 - b1 - 5) * q^39 + (b11 + b10 + b9 + b8 - b7 - 2*b4 + b3 - b1 - 1) * q^41 + (-b11 + b7 - 3*b6 + 2*b4 - 2*b2 - 2*b1 - 1) * q^43 + (b11 + b10 + b9 + b8 - b7 + b6 + 4*b5 - b4 - b3 - 2*b2 - b1 - 5) * q^45 + (b10 + b9 + b6 - b5 - 2*b4 + b2 - b1 + 2) * q^47 + (-b9 - b7 + b6 - 3*b5 - b4 - b3 + 2*b1) * q^49 + (-2*b10 - b9 - 2*b8 + 3*b5 - b3 + 2*b2 - b1 - 4) * q^51 + (-b8 - b7 - 2*b6 - b3 - b2 - 3*b1 - 4) * q^53 + (b11 + b10 - b9 + 2*b6 + b4 + b3 - 2*b2 + 2*b1 + 1) * q^55 + (2*b10 + 3*b9 + b8 + b7 - 2*b6 - 2*b5 + b3 - b2 - b1 - 3) * q^57 + (-2*b11 + b7 + 3*b6 + 2*b5 + 3*b1 + 2) * q^59 + (b11 - b10 - b9 + b6 + 2*b5 - b4 + 2*b2 + 2*b1 - 1) * q^61 + (b11 - 2*b10 - b9 + b8 - b7 + 4*b6 + 4*b5 - b4 - b3 + 4*b2 + 2*b1 + 1) * q^63 + (3*b11 + b10 + 2*b9 + b8 - b7 - 2*b6 - 2*b5 - b3 - 2*b2 + 3*b1 - 3) * q^65 + (-2*b10 - b9 - 2*b8 + 2*b7 - b6 + b5 + 2*b4 - b3 - 3*b1) * q^67 + (b11 + b10 + b9 + 2*b8 - 2*b7 + 3*b6 - b4 + 2*b3 + 3*b2 + 2*b1 + 5) * q^69 + (-b11 + b10 - b9 - b8 - 3*b6 + b4 - 3*b2 - b1 + 1) * q^71 + (b11 + b10 - b9 - b8 - 2*b7 - b6 + b4 - 2*b2 - 3*b1 + 2) * q^73 + (b11 + 2*b10 + b9 - b7 - 3*b6 + b5 + 4*b4 + b3 - 2*b2 + 2*b1 + 1) * q^75 + (-2*b11 - 2*b9 + 5*b6 + b5 - b4 - 2*b3 + 6*b1 + 2) * q^77 + (b11 + b9 + b8 + b7 - 2*b6 + 2*b5 - b4 - b3 - 3) * q^79 + (-b11 + b10 + 2*b9 + b8 + b6 + 4*b5 - b4 + 2*b3 - 2*b2 - b1) * q^81 + (-2*b11 - 3*b10 - 2*b9 + b8 + b7 + 2*b6 - 3*b5 + b4 - b3 + b2 + 6) * q^83 + (-b11 + b10 - b9 + b8 + 3*b7 + 6*b6 + b5 - b3 + b1 + 1) * q^85 + (b10 - 2*b9 - 2*b8 - b7 - 3*b6 - 4*b5 + 4*b4 - b3 - 3*b2 - b1 + 2) * q^87 + (-b8 - b7 - 2*b6 - b3 - b2 - 3*b1 - 6) * q^89 + (2*b10 - 2*b9 + b7 + b6 + 2*b4 + 2*b3 + 2*b2 + 4*b1 - 2) * q^91 + (b11 - 3*b10 - b9 + b6 - 4*b5 + b4 + b2 - 1) * q^93 + (-2*b11 - 2*b10 - 2*b9 - 2*b8 + 2*b7 + 2*b6 + 4*b5 + 2*b4 - 2*b3 + 2*b1 + 2) * q^95 + (-b11 + 3*b10 + b9 - 2*b8 - 2*b6 + 4*b5 + 2*b3 - b2 - 4*b1 - 4) * q^97 + (2*b11 - 2*b10 - b9 + 2*b6 - 5*b5 - b3 + 2*b1 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{3} + 2 q^{5} - 6 q^{7} - 2 q^{9}+O(q^{10})$$ 12 * q + 4 * q^3 + 2 * q^5 - 6 * q^7 - 2 * q^9 $$12 q + 4 q^{3} + 2 q^{5} - 6 q^{7} - 2 q^{9} + 4 q^{11} - 10 q^{13} - 4 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} - 8 q^{23} - 14 q^{25} - 14 q^{27} + 2 q^{29} - 8 q^{31} - 10 q^{33} + 8 q^{35} - 22 q^{39} - 2 q^{41} - 2 q^{43} - 10 q^{45} + 14 q^{47} - 18 q^{49} - 38 q^{51} - 24 q^{53} + 16 q^{55} - 38 q^{57} + 6 q^{59} - 14 q^{61} + 16 q^{63} - 8 q^{65} + 4 q^{67} + 50 q^{69} + 28 q^{71} + 60 q^{73} + 50 q^{75} - 2 q^{77} - 16 q^{79} + 22 q^{81} + 24 q^{83} - 16 q^{85} + 36 q^{87} - 48 q^{89} - 52 q^{91} - 42 q^{93} + 20 q^{95} - 14 q^{97} + 68 q^{99}+O(q^{100})$$ 12 * q + 4 * q^3 + 2 * q^5 - 6 * q^7 - 2 * q^9 + 4 * q^11 - 10 * q^13 - 4 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 - 8 * q^23 - 14 * q^25 - 14 * q^27 + 2 * q^29 - 8 * q^31 - 10 * q^33 + 8 * q^35 - 22 * q^39 - 2 * q^41 - 2 * q^43 - 10 * q^45 + 14 * q^47 - 18 * q^49 - 38 * q^51 - 24 * q^53 + 16 * q^55 - 38 * q^57 + 6 * q^59 - 14 * q^61 + 16 * q^63 - 8 * q^65 + 4 * q^67 + 50 * q^69 + 28 * q^71 + 60 * q^73 + 50 * q^75 - 2 * q^77 - 16 * q^79 + 22 * q^81 + 24 * q^83 - 16 * q^85 + 36 * q^87 - 48 * q^89 - 52 * q^91 - 42 * q^93 + 20 * q^95 - 14 * q^97 + 68 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} - 216 x^{3} + 243 x^{2} - 486 x + 729$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{11} - 2 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} + 22 \nu^{7} - 42 \nu^{6} + 51 \nu^{5} - 126 \nu^{4} + 198 \nu^{3} - 216 \nu^{2} + 243 \nu - 486 ) / 243$$ (v^11 - 2*v^10 + 3*v^9 - 8*v^8 + 22*v^7 - 42*v^6 + 51*v^5 - 126*v^4 + 198*v^3 - 216*v^2 + 243*v - 486) / 243 $$\beta_{2}$$ $$=$$ $$( - \nu^{11} + 2 \nu^{10} - 3 \nu^{9} + 8 \nu^{8} - 13 \nu^{7} + 24 \nu^{6} - 51 \nu^{5} + 108 \nu^{4} - 81 \nu^{3} + 54 \nu^{2} - 135 \nu + 162 ) / 162$$ (-v^11 + 2*v^10 - 3*v^9 + 8*v^8 - 13*v^7 + 24*v^6 - 51*v^5 + 108*v^4 - 81*v^3 + 54*v^2 - 135*v + 162) / 162 $$\beta_{3}$$ $$=$$ $$( 4 \nu^{11} - 2 \nu^{10} + 9 \nu^{9} - 5 \nu^{8} + 40 \nu^{7} + 15 \nu^{5} - 279 \nu^{4} + 36 \nu^{3} - 324 \nu^{2} - 405 \nu - 1701 ) / 486$$ (4*v^11 - 2*v^10 + 9*v^9 - 5*v^8 + 40*v^7 + 15*v^5 - 279*v^4 + 36*v^3 - 324*v^2 - 405*v - 1701) / 486 $$\beta_{4}$$ $$=$$ $$( - 2 \nu^{11} + \nu^{10} - 3 \nu^{9} + 13 \nu^{8} - 38 \nu^{7} + 33 \nu^{6} - 69 \nu^{5} + 243 \nu^{4} - 234 \nu^{3} + 189 \nu^{2} - 297 \nu + 1215 ) / 162$$ (-2*v^11 + v^10 - 3*v^9 + 13*v^8 - 38*v^7 + 33*v^6 - 69*v^5 + 243*v^4 - 234*v^3 + 189*v^2 - 297*v + 1215) / 162 $$\beta_{5}$$ $$=$$ $$( - 11 \nu^{11} + 4 \nu^{10} - 33 \nu^{9} + 52 \nu^{8} - 179 \nu^{7} + 192 \nu^{6} - 327 \nu^{5} + 954 \nu^{4} - 693 \nu^{3} + 1404 \nu^{2} - 567 \nu + 4374 ) / 486$$ (-11*v^11 + 4*v^10 - 33*v^9 + 52*v^8 - 179*v^7 + 192*v^6 - 327*v^5 + 954*v^4 - 693*v^3 + 1404*v^2 - 567*v + 4374) / 486 $$\beta_{6}$$ $$=$$ $$( 16 \nu^{11} + \nu^{10} + 36 \nu^{9} - 29 \nu^{8} + 196 \nu^{7} - 135 \nu^{6} + 240 \nu^{5} - 1035 \nu^{4} + 306 \nu^{3} - 1377 \nu^{2} - 324 \nu - 6075 ) / 486$$ (16*v^11 + v^10 + 36*v^9 - 29*v^8 + 196*v^7 - 135*v^6 + 240*v^5 - 1035*v^4 + 306*v^3 - 1377*v^2 - 324*v - 6075) / 486 $$\beta_{7}$$ $$=$$ $$( - 2 \nu^{11} - 4 \nu^{9} + 7 \nu^{8} - 30 \nu^{7} + 26 \nu^{6} - 48 \nu^{5} + 165 \nu^{4} - 108 \nu^{3} + 234 \nu^{2} - 216 \nu + 891 ) / 54$$ (-2*v^11 - 4*v^9 + 7*v^8 - 30*v^7 + 26*v^6 - 48*v^5 + 165*v^4 - 108*v^3 + 234*v^2 - 216*v + 891) / 54 $$\beta_{8}$$ $$=$$ $$( - 17 \nu^{11} - 2 \nu^{10} - 42 \nu^{9} + 46 \nu^{8} - 275 \nu^{7} + 210 \nu^{6} - 444 \nu^{5} + 1494 \nu^{4} - 1125 \nu^{3} + 2538 \nu^{2} - 972 \nu + 9234 ) / 486$$ (-17*v^11 - 2*v^10 - 42*v^9 + 46*v^8 - 275*v^7 + 210*v^6 - 444*v^5 + 1494*v^4 - 1125*v^3 + 2538*v^2 - 972*v + 9234) / 486 $$\beta_{9}$$ $$=$$ $$( 25 \nu^{11} - 2 \nu^{10} + 78 \nu^{9} - 92 \nu^{8} + 463 \nu^{7} - 462 \nu^{6} + 870 \nu^{5} - 2430 \nu^{4} + 1845 \nu^{3} - 4482 \nu^{2} + 1944 \nu - 13122 ) / 486$$ (25*v^11 - 2*v^10 + 78*v^9 - 92*v^8 + 463*v^7 - 462*v^6 + 870*v^5 - 2430*v^4 + 1845*v^3 - 4482*v^2 + 1944*v - 13122) / 486 $$\beta_{10}$$ $$=$$ $$( 11 \nu^{11} - 4 \nu^{10} + 24 \nu^{9} - 34 \nu^{8} + 161 \nu^{7} - 138 \nu^{6} + 210 \nu^{5} - 756 \nu^{4} + 513 \nu^{3} - 918 \nu^{2} - 54 \nu - 4050 ) / 162$$ (11*v^11 - 4*v^10 + 24*v^9 - 34*v^8 + 161*v^7 - 138*v^6 + 210*v^5 - 756*v^4 + 513*v^3 - 918*v^2 - 54*v - 4050) / 162 $$\beta_{11}$$ $$=$$ $$( - 2 \nu^{11} - 4 \nu^{9} + 7 \nu^{8} - 30 \nu^{7} + 26 \nu^{6} - 48 \nu^{5} + 165 \nu^{4} - 108 \nu^{3} + 234 \nu^{2} - 135 \nu + 918 ) / 27$$ (-2*v^11 - 4*v^9 + 7*v^8 - 30*v^7 + 26*v^6 - 48*v^5 + 165*v^4 - 108*v^3 + 234*v^2 - 135*v + 918) / 27
 $$\nu$$ $$=$$ $$( \beta_{11} - 2\beta_{7} - 1 ) / 3$$ (b11 - 2*b7 - 1) / 3 $$\nu^{2}$$ $$=$$ $$( 2\beta_{11} - \beta_{7} + 3\beta_{6} - 3\beta_{4} - 3\beta_{3} - 2 ) / 3$$ (2*b11 - b7 + 3*b6 - 3*b4 - 3*b3 - 2) / 3 $$\nu^{3}$$ $$=$$ $$( \beta_{11} - 3\beta_{9} - 6\beta_{8} + \beta_{7} - 3\beta_{5} - 3\beta_{3} + 3\beta _1 + 5 ) / 3$$ (b11 - 3*b9 - 6*b8 + b7 - 3*b5 - 3*b3 + 3*b1 + 5) / 3 $$\nu^{4}$$ $$=$$ $$( - \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - 6 \beta_{8} - \beta_{7} - 9 \beta_{6} - 3 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 6 \beta _1 - 14 ) / 3$$ (-b11 + 3*b10 - 3*b9 - 6*b8 - b7 - 9*b6 - 3*b5 + 9*b4 - 3*b3 - 3*b2 - 6*b1 - 14) / 3 $$\nu^{5}$$ $$=$$ $$( - 5 \beta_{11} + 3 \beta_{10} - 9 \beta_{8} + 7 \beta_{7} - 9 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} - 9 \beta_{3} - 21 \beta_{2} - 18 \beta _1 - 7 ) / 3$$ (-5*b11 + 3*b10 - 9*b8 + 7*b7 - 9*b6 + 9*b5 + 9*b4 - 9*b3 - 21*b2 - 18*b1 - 7) / 3 $$\nu^{6}$$ $$=$$ $$( 2 \beta_{11} + 6 \beta_{10} - 6 \beta_{9} - 21 \beta_{8} - 7 \beta_{7} - 30 \beta_{6} - 6 \beta_{5} + 12 \beta_{4} + 15 \beta_{3} - 15 \beta_{2} - 39 \beta _1 + 13 ) / 3$$ (2*b11 + 6*b10 - 6*b9 - 21*b8 - 7*b7 - 30*b6 - 6*b5 + 12*b4 + 15*b3 - 15*b2 - 39*b1 + 13) / 3 $$\nu^{7}$$ $$=$$ $$( 13 \beta_{11} + 18 \beta_{10} + 21 \beta_{9} + 24 \beta_{8} - 23 \beta_{7} - 24 \beta_{6} + 21 \beta_{5} - 12 \beta_{4} + 9 \beta_{3} + 18 \beta_{2} - 48 \beta _1 + 17 ) / 3$$ (13*b11 + 18*b10 + 21*b9 + 24*b8 - 23*b7 - 24*b6 + 21*b5 - 12*b4 + 9*b3 + 18*b2 - 48*b1 + 17) / 3 $$\nu^{8}$$ $$=$$ $$( 41 \beta_{11} + 3 \beta_{10} + 27 \beta_{9} - 25 \beta_{7} + 69 \beta_{6} + 99 \beta_{5} - 33 \beta_{4} - 33 \beta_{3} - 21 \beta_{2} + 54 \beta _1 + 55 ) / 3$$ (41*b11 + 3*b10 + 27*b9 - 25*b7 + 69*b6 + 99*b5 - 33*b4 - 33*b3 - 21*b2 + 54*b1 + 55) / 3 $$\nu^{9}$$ $$=$$ $$( 130 \beta_{11} - 21 \beta_{10} - 30 \beta_{9} - 69 \beta_{8} - 95 \beta_{7} + 87 \beta_{6} - 165 \beta_{5} - 51 \beta_{4} - 45 \beta_{3} + 39 \beta_{2} + 12 \beta _1 - 94 ) / 3$$ (130*b11 - 21*b10 - 30*b9 - 69*b8 - 95*b7 + 87*b6 - 165*b5 - 51*b4 - 45*b3 + 39*b2 + 12*b1 - 94) / 3 $$\nu^{10}$$ $$=$$ $$( - 16 \beta_{11} - 66 \beta_{10} - 66 \beta_{9} - 69 \beta_{8} + 92 \beta_{7} + 165 \beta_{6} - 174 \beta_{5} - 93 \beta_{4} - 258 \beta_{3} + 129 \beta_{2} + 21 \beta _1 + 241 ) / 3$$ (-16*b11 - 66*b10 - 66*b9 - 69*b8 + 92*b7 + 165*b6 - 174*b5 - 93*b4 - 258*b3 + 129*b2 + 21*b1 + 241) / 3 $$\nu^{11}$$ $$=$$ $$( - 176 \beta_{11} + 36 \beta_{10} - 324 \beta_{9} - 450 \beta_{8} + 70 \beta_{7} - 138 \beta_{6} - 18 \beta_{5} + 498 \beta_{4} - 186 \beta_{3} - 360 \beta_{2} + 153 \beta _1 + 248 ) / 3$$ (-176*b11 + 36*b10 - 324*b9 - 450*b8 + 70*b7 - 138*b6 - 18*b5 + 498*b4 - 186*b3 - 360*b2 + 153*b1 + 248) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$-\beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
385.1
 −1.15879 − 1.28733i 1.19051 − 1.25805i −1.28252 + 1.16410i 1.73202 − 0.0102491i −0.433633 + 1.67689i 0.952418 + 1.44669i −1.15879 + 1.28733i 1.19051 + 1.25805i −1.28252 − 1.16410i 1.73202 + 0.0102491i −0.433633 − 1.67689i 0.952418 − 1.44669i
0 −1.69425 0.359877i 0 1.74260 + 3.01828i 0 −1.34988 + 2.33807i 0 2.74098 + 1.21944i 0
385.2 0 −0.494250 + 1.66004i 0 −0.268104 0.464369i 0 −2.35014 + 4.07056i 0 −2.51143 1.64095i 0
385.3 0 0.366879 1.69275i 0 1.05471 + 1.82681i 0 1.43914 2.49267i 0 −2.73080 1.24207i 0
385.4 0 0.857134 + 1.50510i 0 −0.551563 0.955334i 0 1.62490 2.81442i 0 −1.53064 + 2.58014i 0
385.5 0 1.23541 1.21398i 0 −2.22043 3.84590i 0 −1.45488 + 2.51992i 0 0.0524919 2.99954i 0
385.6 0 1.72908 + 0.101475i 0 1.24278 + 2.15256i 0 −0.909142 + 1.57468i 0 2.97941 + 0.350917i 0
769.1 0 −1.69425 + 0.359877i 0 1.74260 3.01828i 0 −1.34988 2.33807i 0 2.74098 1.21944i 0
769.2 0 −0.494250 1.66004i 0 −0.268104 + 0.464369i 0 −2.35014 4.07056i 0 −2.51143 + 1.64095i 0
769.3 0 0.366879 + 1.69275i 0 1.05471 1.82681i 0 1.43914 + 2.49267i 0 −2.73080 + 1.24207i 0
769.4 0 0.857134 1.50510i 0 −0.551563 + 0.955334i 0 1.62490 + 2.81442i 0 −1.53064 2.58014i 0
769.5 0 1.23541 + 1.21398i 0 −2.22043 + 3.84590i 0 −1.45488 2.51992i 0 0.0524919 + 2.99954i 0
769.6 0 1.72908 0.101475i 0 1.24278 2.15256i 0 −0.909142 1.57468i 0 2.97941 0.350917i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.i.l yes 12
3.b odd 2 1 3456.2.i.i 12
4.b odd 2 1 1152.2.i.j yes 12
8.b even 2 1 1152.2.i.i 12
8.d odd 2 1 1152.2.i.k yes 12
9.c even 3 1 inner 1152.2.i.l yes 12
9.d odd 6 1 3456.2.i.i 12
12.b even 2 1 3456.2.i.j 12
24.f even 2 1 3456.2.i.l 12
24.h odd 2 1 3456.2.i.k 12
36.f odd 6 1 1152.2.i.j yes 12
36.h even 6 1 3456.2.i.j 12
72.j odd 6 1 3456.2.i.k 12
72.l even 6 1 3456.2.i.l 12
72.n even 6 1 1152.2.i.i 12
72.p odd 6 1 1152.2.i.k yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.i.i 12 8.b even 2 1
1152.2.i.i 12 72.n even 6 1
1152.2.i.j yes 12 4.b odd 2 1
1152.2.i.j yes 12 36.f odd 6 1
1152.2.i.k yes 12 8.d odd 2 1
1152.2.i.k yes 12 72.p odd 6 1
1152.2.i.l yes 12 1.a even 1 1 trivial
1152.2.i.l yes 12 9.c even 3 1 inner
3456.2.i.i 12 3.b odd 2 1
3456.2.i.i 12 9.d odd 6 1
3456.2.i.j 12 12.b even 2 1
3456.2.i.j 12 36.h even 6 1
3456.2.i.k 12 24.h odd 2 1
3456.2.i.k 12 72.j odd 6 1
3456.2.i.l 12 24.f even 2 1
3456.2.i.l 12 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{12} - 2 T_{5}^{11} + 24 T_{5}^{10} - 60 T_{5}^{9} + 465 T_{5}^{8} - 948 T_{5}^{7} + 2928 T_{5}^{6} - 1674 T_{5}^{5} + 4665 T_{5}^{4} + 1720 T_{5}^{3} + 9424 T_{5}^{2} + 4224 T_{5} + 2304$$ T5^12 - 2*T5^11 + 24*T5^10 - 60*T5^9 + 465*T5^8 - 948*T5^7 + 2928*T5^6 - 1674*T5^5 + 4665*T5^4 + 1720*T5^3 + 9424*T5^2 + 4224*T5 + 2304 $$T_{7}^{12} + 6 T_{7}^{11} + 48 T_{7}^{10} + 152 T_{7}^{9} + 861 T_{7}^{8} + 2400 T_{7}^{7} + 10164 T_{7}^{6} + 21192 T_{7}^{5} + 67353 T_{7}^{4} + 117452 T_{7}^{3} + 294516 T_{7}^{2} + 324048 T_{7} + 394384$$ T7^12 + 6*T7^11 + 48*T7^10 + 152*T7^9 + 861*T7^8 + 2400*T7^7 + 10164*T7^6 + 21192*T7^5 + 67353*T7^4 + 117452*T7^3 + 294516*T7^2 + 324048*T7 + 394384

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} - 4 T^{11} + 9 T^{10} - 10 T^{9} + \cdots + 729$$
$5$ $$T^{12} - 2 T^{11} + 24 T^{10} + \cdots + 2304$$
$7$ $$T^{12} + 6 T^{11} + 48 T^{10} + \cdots + 394384$$
$11$ $$T^{12} - 4 T^{11} + 47 T^{10} + \cdots + 229441$$
$13$ $$T^{12} + 10 T^{11} + 108 T^{10} + \cdots + 6533136$$
$17$ $$(T^{6} - 2 T^{5} - 83 T^{4} + 176 T^{3} + \cdots + 1812)^{2}$$
$19$ $$(T^{6} - 2 T^{5} - 65 T^{4} + 80 T^{3} + \cdots - 3408)^{2}$$
$23$ $$T^{12} + 8 T^{11} + 128 T^{10} + \cdots + 204304$$
$29$ $$T^{12} - 2 T^{11} + 88 T^{10} + \cdots + 229704336$$
$31$ $$T^{12} + 8 T^{11} + \cdots + 1021953024$$
$37$ $$(T^{6} - 60 T^{4} - 68 T^{3} + 876 T^{2} + \cdots - 128)^{2}$$
$41$ $$T^{12} + 2 T^{11} + 85 T^{10} + \cdots + 2259009$$
$43$ $$T^{12} + 2 T^{11} + 103 T^{10} + \cdots + 16621929$$
$47$ $$T^{12} - 14 T^{11} + 216 T^{10} + \cdots + 2178576$$
$53$ $$(T^{6} + 12 T^{5} - 24 T^{4} - 516 T^{3} + \cdots + 1728)^{2}$$
$59$ $$T^{12} - 6 T^{11} + \cdots + 4100737369$$
$61$ $$T^{12} + 14 T^{11} + 200 T^{10} + \cdots + 60715264$$
$67$ $$T^{12} - 4 T^{11} + \cdots + 3020711521$$
$71$ $$(T^{6} - 14 T^{5} - 72 T^{4} + 1608 T^{3} + \cdots + 1728)^{2}$$
$73$ $$(T^{6} - 30 T^{5} + 225 T^{4} + \cdots + 39892)^{2}$$
$79$ $$T^{12} + 16 T^{11} + \cdots + 674337024$$
$83$ $$T^{12} - 24 T^{11} + \cdots + 734843664$$
$89$ $$(T^{6} + 24 T^{5} + 156 T^{4} - 68 T^{3} + \cdots - 2864)^{2}$$
$97$ $$T^{12} + 14 T^{11} + \cdots + 78140934369$$