# Properties

 Label 768.4.f.e Level $768$ Weight $4$ Character orbit 768.f Analytic conductor $45.313$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$45.3134668844$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624$$ x^12 + 39*x^10 + 549*x^8 + 3500*x^6 + 10236*x^4 + 11952*x^2 + 4624 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{26}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 384) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{3} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - \beta_{8} - \beta_{5} + \beta_{3} - \beta_1) q^{9}+O(q^{10})$$ q + b5 * q^3 + (-b2 - 1) * q^5 + (b3 - b1) * q^7 + (-b8 - b5 + b3 - b1) * q^9 $$q + \beta_{5} q^{3} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - \beta_{8} - \beta_{5} + \beta_{3} - \beta_1) q^{9} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} + 2 \beta_1) q^{11} + ( - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - 4 \beta_{5} - \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 1) q^{13} + (\beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 4 \beta_1 - 7) q^{15} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots + \beta_1) q^{17}+ \cdots + (2 \beta_{11} - 16 \beta_{10} - 12 \beta_{9} + 11 \beta_{8} + 18 \beta_{7} + \cdots + 243) q^{99}+O(q^{100})$$ q + b5 * q^3 + (-b2 - 1) * q^5 + (b3 - b1) * q^7 + (-b8 - b5 + b3 - b1) * q^9 + (b9 + b8 + b7 + b5 + 2*b1) * q^11 + (-b10 - b9 - b8 - b7 - 4*b5 - b4 + 2*b3 - 3*b1 - 1) * q^13 + (b9 + b6 + b5 + b4 - b3 - b2 + 4*b1 - 7) * q^15 + (-b11 - b10 + b9 + b8 - 2*b7 - 2*b5 - b4 + 2*b3 + b1) * q^17 + (-2*b11 + 2*b9 - b8 + 4*b7 - b6 - b2 - 2*b1 - 15) * q^19 + (-b11 + 4*b10 + b9 - 2*b8 + b6 - b5 - 2*b4 - 2*b3 + 2*b2 - 6*b1 - 11) * q^21 + (b11 + 4*b10 - 2*b9 - b7 + 2*b6 - 10*b5 + 2*b2 - 3*b1 - 11) * q^23 + (b11 - 3*b10 + b9 + b8 - 4*b7 - 2*b6 + 6*b5 + 6*b2 + 4*b1 + 27) * q^25 + (2*b11 + b10 + b9 - b7 + b6 + 3*b5 - 2*b4 - 4*b3 - 7*b2 - 7*b1 + 12) * q^27 + (-4*b10 - 2*b9 + 2*b7 + 2*b6 + 14*b5 - 3*b2 + 4*b1 - 51) * q^29 + (-b11 - 6*b10 + 2*b9 + 2*b8 - b7 - 16*b5 - 6*b4 + b3 - 6*b1 - 5) * q^31 + (-3*b11 - b10 - b9 - 4*b8 - 2*b7 + 2*b6 - 5*b5 - b4 - b3 + 10*b2 - 22*b1 - 11) * q^33 + (-4*b11 + b10 + 3*b9 + 3*b8 - 9*b7 + 6*b5 + 4*b4 - 8*b3 - 11*b1 + 5) * q^35 + (-2*b11 + 7*b10 - b9 - b8 - 7*b7 + 20*b5 + 3*b4 + 6*b3 - 15*b1 + 9) * q^37 + (b11 + 8*b10 + b9 + 4*b8 - b7 + b6 + 5*b5 - 5*b4 - 2*b3 - 13*b2 + 4*b1 + 56) * q^39 + (4*b11 - 6*b10 + 2*b9 + 2*b8 + 14*b7 - 16*b5 - b4 - 19*b1 - 10) * q^41 + (2*b11 + 2*b10 + 6*b9 - 5*b8 - 12*b7 - b6 - 2*b5 + 15*b2 + 39) * q^43 + (-5*b11 - 9*b10 + 2*b9 - 3*b8 + b7 - b6 - 3*b5 + 5*b4 - 8*b3 + 4*b2 + 15*b1 + 58) * q^45 + (4*b11 - 10*b10 - 6*b9 + 2*b8 - 6*b7 + 4*b6 + 32*b5 + 8*b2 + 14*b1 + 102) * q^47 + (-7*b11 - 15*b10 + b9 + 5*b8 + 20*b7 - 6*b6 + 34*b5 - 14*b2 + 8*b1 - 85) * q^49 + (-8*b11 + 3*b10 + 5*b9 - b8 - 3*b7 + 2*b6 - 2*b5 - 10*b4 - 4*b3 + 10*b2 + 41*b1 + 33) * q^51 + (-4*b11 + 12*b10 - 4*b9 + 4*b8 + 16*b7 - 40*b5 - b2 - 16*b1 + 71) * q^53 + (-3*b11 + 14*b10 + 6*b9 + 6*b8 - 3*b7 + 48*b5 - 2*b4 - 6*b3 - 41*b1 + 17) * q^55 + (10*b11 - 8*b10 - 4*b9 - 3*b8 - 6*b7 + 8*b6 - 3*b5 - b4 - 5*b3 - 8*b2 + 98*b1 + 7) * q^57 + (8*b11 + 3*b10 + 24*b7 + 9*b5 + 16*b4 - 16*b3 + 117*b1 - 5) * q^59 + (-10*b11 - 15*b10 + 9*b9 + 9*b8 - 21*b7 - 36*b5 - 3*b4 + 2*b3 + 7*b1 - 5) * q^61 + (2*b11 - 22*b10 - 8*b9 + 6*b8 - 4*b7 - 2*b6 - 6*b5 - 2*b4 + 5*b3 - 34*b2 + 61*b1 - 242) * q^63 + (-6*b11 - 16*b10 - 4*b9 - 4*b8 - 22*b7 - 52*b5 + 9*b4 - 12*b3 + 139*b1 - 10) * q^65 + (-4*b11 + 5*b10 - 4*b9 - 4*b8 + 16*b7 + 8*b6 - 3*b5 - 8*b2 - 9*b1 - 199) * q^67 + (-9*b11 + 7*b10 - 4*b9 + 9*b8 + 3*b7 - 7*b6 - 5*b5 + 5*b4 - 8*b3 - 5*b2 + 7*b1 - 271) * q^69 + (b11 + 14*b10 + 4*b9 - 2*b8 - 7*b7 - 2*b6 - 42*b5 + 26*b2 - 13*b1 - 165) * q^71 + (12*b11 - 26*b10 - 8*b9 + 6*b8 - 28*b7 + 2*b6 + 74*b5 - 38*b2 + 38*b1 + 22) * q^73 + (2*b11 - 4*b10 + 5*b9 + 3*b7 - 7*b6 - 8*b5 - 16*b4 + 16*b3 + 25*b2 - 172*b1 + 211) * q^75 + (-20*b11 - 20*b10 + 12*b9 + 4*b8 + 48*b7 - 16*b6 + 40*b5 + 14*b2 - 410) * q^77 + (3*b11 - 10*b10 - 2*b9 - 2*b8 + 7*b7 - 32*b5 + 14*b4 - 3*b3 - 54*b1 - 13) * q^79 + (-3*b11 - 25*b10 + b9 + 5*b8 + 8*b7 + 4*b6 + 14*b5 + 10*b4 + 6*b3 - 28*b2 - 134*b1 - 105) * q^81 + (-4*b11 - 2*b10 - 5*b9 - 5*b8 - 17*b7 - 11*b5 + 12*b4 + 24*b3 - 208*b1 + 2) * q^83 + (-16*b11 + 8*b10 + 8*b9 + 8*b8 - 40*b7 + 32*b5 - 8*b4 - 4*b1 + 24) * q^85 + (-6*b11 + 10*b10 - 5*b9 - 18*b8 + b6 - 43*b5 + 13*b4 + 11*b3 - 13*b2 + 84*b1 + 345) * q^87 + (5*b11 - 29*b10 - 7*b9 - 7*b8 + 8*b7 - 94*b5 + 8*b4 - 2*b3 - 164*b1 - 34) * q^89 + (14*b11 - 9*b10 - 22*b9 + 17*b8 - 20*b7 + 5*b6 + 15*b5 - 75*b2 + 23*b1 - 298) * q^91 + (-10*b11 - 7*b10 - 7*b9 + 17*b8 - 5*b7 + 8*b6 + 4*b5 - 19*b4 + 2*b3 - 17*b2 - 5*b1 + 338) * q^93 + (-15*b11 + 4*b10 + 22*b9 - 16*b8 + 23*b7 - 6*b6 - 2*b5 + 58*b2 - 19*b1 + 461) * q^95 + (5*b11 - 25*b10 - 7*b9 - 5*b8 - 8*b7 + 12*b6 + 92*b5 + 28*b2 + 30*b1 - 6) * q^97 + (2*b11 - 16*b10 - 12*b9 + 11*b8 + 18*b7 - 15*b6 + 8*b5 - 6*b4 + 4*b3 - 39*b2 + 260*b1 + 243) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{3} - 12 q^{5}+O(q^{10})$$ 12 * q - 2 * q^3 - 12 * q^5 $$12 q - 2 q^{3} - 12 q^{5} - 84 q^{15} - 180 q^{19} - 156 q^{21} - 120 q^{23} + 300 q^{25} + 130 q^{27} - 588 q^{29} - 116 q^{33} + 620 q^{39} + 372 q^{43} + 740 q^{45} + 1248 q^{47} - 948 q^{49} + 360 q^{51} + 948 q^{53} + 172 q^{57} - 2744 q^{63} - 2292 q^{67} - 3280 q^{69} - 2040 q^{71} + 216 q^{73} + 2522 q^{75} - 4824 q^{77} - 1076 q^{81} + 4156 q^{87} - 3480 q^{91} + 4180 q^{93} + 5448 q^{95} - 48 q^{97} + 3048 q^{99}+O(q^{100})$$ 12 * q - 2 * q^3 - 12 * q^5 - 84 * q^15 - 180 * q^19 - 156 * q^21 - 120 * q^23 + 300 * q^25 + 130 * q^27 - 588 * q^29 - 116 * q^33 + 620 * q^39 + 372 * q^43 + 740 * q^45 + 1248 * q^47 - 948 * q^49 + 360 * q^51 + 948 * q^53 + 172 * q^57 - 2744 * q^63 - 2292 * q^67 - 3280 * q^69 - 2040 * q^71 + 216 * q^73 + 2522 * q^75 - 4824 * q^77 - 1076 * q^81 + 4156 * q^87 - 3480 * q^91 + 4180 * q^93 + 5448 * q^95 - 48 * q^97 + 3048 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{11} - 22\nu^{9} + 12\nu^{7} + 2399\nu^{5} + 11762\nu^{3} + 9468\nu ) / 612$$ (-v^11 - 22*v^9 + 12*v^7 + 2399*v^5 + 11762*v^3 + 9468*v) / 612 $$\beta_{2}$$ $$=$$ $$( 2\nu^{10} + 61\nu^{8} + 539\nu^{6} + 1091\nu^{4} - 2504\nu^{2} - 3250 ) / 54$$ (2*v^10 + 61*v^8 + 539*v^6 + 1091*v^4 - 2504*v^2 - 3250) / 54 $$\beta_{3}$$ $$=$$ $$( -25\nu^{11} - 533\nu^{9} + 521\nu^{7} + 56558\nu^{5} + 242404\nu^{3} + 79280\nu ) / 3672$$ (-25*v^11 - 533*v^9 + 521*v^7 + 56558*v^5 + 242404*v^3 + 79280*v) / 3672 $$\beta_{4}$$ $$=$$ $$( 20\nu^{11} + 559\nu^{9} + 3755\nu^{7} - 5803\nu^{5} - 90026\nu^{3} - 121156\nu ) / 1836$$ (20*v^11 + 559*v^9 + 3755*v^7 - 5803*v^5 - 90026*v^3 - 121156*v) / 1836 $$\beta_{5}$$ $$=$$ $$( - 115 \nu^{11} - 68 \nu^{10} - 4145 \nu^{9} - 2176 \nu^{8} - 51439 \nu^{7} - 21488 \nu^{6} - 268540 \nu^{5} - 68612 \nu^{4} - 568472 \nu^{3} - 34000 \nu^{2} - 352984 \nu - 6800 ) / 7344$$ (-115*v^11 - 68*v^10 - 4145*v^9 - 2176*v^8 - 51439*v^7 - 21488*v^6 - 268540*v^5 - 68612*v^4 - 568472*v^3 - 34000*v^2 - 352984*v - 6800) / 7344 $$\beta_{6}$$ $$=$$ $$( 55 \nu^{11} - 68 \nu^{10} + 1601 \nu^{9} - 4216 \nu^{8} + 11767 \nu^{7} - 89624 \nu^{6} - 12248 \nu^{5} - 791996 \nu^{4} - 299872 \nu^{3} - 2646832 \nu^{2} - 454712 \nu - 1819136 ) / 7344$$ (55*v^11 - 68*v^10 + 1601*v^9 - 4216*v^8 + 11767*v^7 - 89624*v^6 - 12248*v^5 - 791996*v^4 - 299872*v^3 - 2646832*v^2 - 454712*v - 1819136) / 7344 $$\beta_{7}$$ $$=$$ $$( 121 \nu^{11} + 136 \nu^{10} + 4277 \nu^{9} + 4556 \nu^{8} + 51367 \nu^{7} + 49300 \nu^{6} + 254146 \nu^{5} + 195364 \nu^{4} + 497900 \nu^{3} + 223040 \nu^{2} + 281488 \nu + 48688 ) / 3672$$ (121*v^11 + 136*v^10 + 4277*v^9 + 4556*v^8 + 51367*v^7 + 49300*v^6 + 254146*v^5 + 195364*v^4 + 497900*v^3 + 223040*v^2 + 281488*v + 48688) / 3672 $$\beta_{8}$$ $$=$$ $$( - 175 \nu^{11} + 1564 \nu^{10} - 6689 \nu^{9} + 53312 \nu^{8} - 91111 \nu^{7} + 600304 \nu^{6} - 549328 \nu^{5} + 2640508 \nu^{4} - 1436816 \nu^{3} + 4217360 \nu^{2} + \cdots + 2154784 ) / 7344$$ (-175*v^11 + 1564*v^10 - 6689*v^9 + 53312*v^8 - 91111*v^7 + 600304*v^6 - 549328*v^5 + 2640508*v^4 - 1436816*v^3 + 4217360*v^2 - 1160680*v + 2154784) / 7344 $$\beta_{9}$$ $$=$$ $$( 91 \nu^{11} - 884 \nu^{10} + 3005 \nu^{9} - 30124 \nu^{8} + 31531 \nu^{7} - 338708 \nu^{6} + 113752 \nu^{5} - 1481312 \nu^{4} + 63728 \nu^{3} - 2314720 \nu^{2} + \cdots - 1122680 ) / 3672$$ (91*v^11 - 884*v^10 + 3005*v^9 - 30124*v^8 + 31531*v^7 - 338708*v^6 + 113752*v^5 - 1481312*v^4 + 63728*v^3 - 2314720*v^2 - 122360*v - 1122680) / 3672 $$\beta_{10}$$ $$=$$ $$( - 7 \nu^{11} + 4 \nu^{10} - 249 \nu^{9} + 128 \nu^{8} - 3023 \nu^{7} + 1264 \nu^{6} - 15232 \nu^{5} + 4036 \nu^{4} - 30672 \nu^{3} + 2000 \nu^{2} - 18536 \nu + 256 ) / 144$$ (-7*v^11 + 4*v^10 - 249*v^9 + 128*v^8 - 3023*v^7 + 1264*v^6 - 15232*v^5 + 4036*v^4 - 30672*v^3 + 2000*v^2 - 18536*v + 256) / 144 $$\beta_{11}$$ $$=$$ $$( 123 \nu^{11} - 136 \nu^{10} + 4321 \nu^{9} - 4556 \nu^{8} + 51343 \nu^{7} - 49300 \nu^{6} + 249348 \nu^{5} - 195364 \nu^{4} + 474376 \nu^{3} - 223040 \nu^{2} + 262552 \nu - 47464 ) / 1224$$ (123*v^11 - 136*v^10 + 4321*v^9 - 4556*v^8 + 51343*v^7 - 49300*v^6 + 249348*v^5 - 195364*v^4 + 474376*v^3 - 223040*v^2 + 262552*v - 47464) / 1224
 $$\nu$$ $$=$$ $$( -\beta_{11} - 2\beta_{10} - 3\beta_{7} - 6\beta_{5} - 5\beta _1 - 1 ) / 24$$ (-b11 - 2*b10 - 3*b7 - 6*b5 - 5*b1 - 1) / 24 $$\nu^{2}$$ $$=$$ $$( \beta_{11} - \beta_{9} - 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{2} + \beta _1 - 52 ) / 8$$ (b11 - b9 - 2*b7 + b6 + b5 - b2 + b1 - 52) / 8 $$\nu^{3}$$ $$=$$ $$( 10 \beta_{11} + 20 \beta_{10} + 6 \beta_{9} + 6 \beta_{8} + 36 \beta_{7} + 66 \beta_{5} - 9 \beta_{4} - 12 \beta_{3} + 107 \beta _1 + 10 ) / 24$$ (10*b11 + 20*b10 + 6*b9 + 6*b8 + 36*b7 + 66*b5 - 9*b4 - 12*b3 + 107*b1 + 10) / 24 $$\nu^{4}$$ $$=$$ $$( - 16 \beta_{11} + 8 \beta_{10} + 17 \beta_{9} + 31 \beta_{7} - 17 \beta_{6} - 41 \beta_{5} + 11 \beta_{2} - 24 \beta _1 + 565 ) / 8$$ (-16*b11 + 8*b10 + 17*b9 + 31*b7 - 17*b6 - 41*b5 + 11*b2 - 24*b1 + 565) / 8 $$\nu^{5}$$ $$=$$ $$( - 124 \beta_{11} - 251 \beta_{10} - 147 \beta_{9} - 147 \beta_{8} - 519 \beta_{7} - 900 \beta_{5} + 162 \beta_{4} + 258 \beta_{3} - 2081 \beta _1 - 127 ) / 24$$ (-124*b11 - 251*b10 - 147*b9 - 147*b8 - 519*b7 - 900*b5 + 162*b4 + 258*b3 - 2081*b1 - 127) / 24 $$\nu^{6}$$ $$=$$ $$( 255 \beta_{11} - 198 \beta_{10} - 270 \beta_{9} + 6 \beta_{8} - 495 \beta_{7} + 264 \beta_{6} + 852 \beta_{5} - 102 \beta_{2} + 453 \beta _1 - 7463 ) / 8$$ (255*b11 - 198*b10 - 270*b9 + 6*b8 - 495*b7 + 264*b6 + 852*b5 - 102*b2 + 453*b1 - 7463) / 8 $$\nu^{7}$$ $$=$$ $$( 1756 \beta_{11} + 3593 \beta_{10} + 2727 \beta_{9} + 2727 \beta_{8} + 7995 \beta_{7} + 13506 \beta_{5} - 2493 \beta_{4} - 4734 \beta_{3} + 37598 \beta _1 + 1837 ) / 24$$ (1756*b11 + 3593*b10 + 2727*b9 + 2727*b8 + 7995*b7 + 13506*b5 - 2493*b4 - 4734*b3 + 37598*b1 + 1837) / 24 $$\nu^{8}$$ $$=$$ $$( - 4129 \beta_{11} + 3762 \beta_{10} + 4285 \beta_{9} - 186 \beta_{8} + 8102 \beta_{7} - 4099 \beta_{6} - 15199 \beta_{5} + 787 \beta_{2} - 7891 \beta _1 + 108400 ) / 8$$ (-4129*b11 + 3762*b10 + 4285*b9 - 186*b8 + 8102*b7 - 4099*b6 - 15199*b5 + 787*b2 - 7891*b1 + 108400) / 8 $$\nu^{9}$$ $$=$$ $$( - 26632 \beta_{11} - 54920 \beta_{10} - 46770 \beta_{9} - 46770 \beta_{8} - 126666 \beta_{7} - 211530 \beta_{5} + 37629 \beta_{4} + 82128 \beta_{3} - 649889 \beta _1 - 28288 ) / 24$$ (-26632*b11 - 54920*b10 - 46770*b9 - 46770*b8 - 126666*b7 - 211530*b5 + 37629*b4 + 82128*b3 - 649889*b1 - 28288) / 24 $$\nu^{10}$$ $$=$$ $$( 67192 \beta_{11} - 65744 \beta_{10} - 68453 \beta_{9} + 4056 \beta_{8} - 133123 \beta_{7} + 64397 \beta_{6} + 257573 \beta_{5} - 3551 \beta_{2} + 132936 \beta _1 - 1655233 ) / 8$$ (67192*b11 - 65744*b10 - 68453*b9 + 4056*b8 - 133123*b7 + 64397*b6 + 257573*b5 - 3551*b2 + 132936*b1 - 1655233) / 8 $$\nu^{11}$$ $$=$$ $$( 417652 \beta_{11} + 865511 \beta_{10} + 779583 \beta_{9} + 779583 \beta_{8} + 2032539 \beta_{7} + 3376116 \beta_{5} - 574974 \beta_{4} - 1385826 \beta_{3} + 10952921 \beta _1 + 447859 ) / 24$$ (417652*b11 + 865511*b10 + 779583*b9 + 779583*b8 + 2032539*b7 + 3376116*b5 - 574974*b4 - 1385826*b3 + 10952921*b1 + 447859) / 24

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
383.1
 4.03251i − 4.03251i − 1.08600i 1.08600i 0.910871i − 0.910871i 2.29679i − 2.29679i 3.14286i − 3.14286i 2.36157i − 2.36157i
0 −4.54315 2.52186i 0 8.01133 0 12.6015i 0 14.2805 + 22.9144i 0
383.2 0 −4.54315 + 2.52186i 0 8.01133 0 12.6015i 0 14.2805 22.9144i 0
383.3 0 −4.21320 3.04120i 0 −9.33303 0 36.3792i 0 8.50216 + 25.6264i 0
383.4 0 −4.21320 + 3.04120i 0 −9.33303 0 36.3792i 0 8.50216 25.6264i 0
383.5 0 −1.98169 4.80343i 0 11.9846 0 22.6995i 0 −19.1458 + 19.0378i 0
383.6 0 −1.98169 + 4.80343i 0 11.9846 0 22.6995i 0 −19.1458 19.0378i 0
383.7 0 1.40813 5.00172i 0 −5.86626 0 5.92149i 0 −23.0343 14.0861i 0
383.8 0 1.40813 + 5.00172i 0 −5.86626 0 5.92149i 0 −23.0343 + 14.0861i 0
383.9 0 3.16369 4.12202i 0 −21.4043 0 20.9034i 0 −6.98212 26.0816i 0
383.10 0 3.16369 + 4.12202i 0 −21.4043 0 20.9034i 0 −6.98212 + 26.0816i 0
383.11 0 5.16622 0.556921i 0 10.6077 0 7.90379i 0 26.3797 5.75435i 0
383.12 0 5.16622 + 0.556921i 0 10.6077 0 7.90379i 0 26.3797 + 5.75435i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 383.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.f.e 12
3.b odd 2 1 768.4.f.f 12
4.b odd 2 1 768.4.f.g 12
8.b even 2 1 768.4.f.h 12
8.d odd 2 1 768.4.f.f 12
12.b even 2 1 768.4.f.h 12
16.e even 4 1 384.4.c.a 12
16.e even 4 1 384.4.c.c yes 12
16.f odd 4 1 384.4.c.b yes 12
16.f odd 4 1 384.4.c.d yes 12
24.f even 2 1 inner 768.4.f.e 12
24.h odd 2 1 768.4.f.g 12
48.i odd 4 1 384.4.c.b yes 12
48.i odd 4 1 384.4.c.d yes 12
48.k even 4 1 384.4.c.a 12
48.k even 4 1 384.4.c.c yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.c.a 12 16.e even 4 1
384.4.c.a 12 48.k even 4 1
384.4.c.b yes 12 16.f odd 4 1
384.4.c.b yes 12 48.i odd 4 1
384.4.c.c yes 12 16.e even 4 1
384.4.c.c yes 12 48.k even 4 1
384.4.c.d yes 12 16.f odd 4 1
384.4.c.d yes 12 48.i odd 4 1
768.4.f.e 12 1.a even 1 1 trivial
768.4.f.e 12 24.f even 2 1 inner
768.4.f.f 12 3.b odd 2 1
768.4.f.f 12 8.d odd 2 1
768.4.f.g 12 4.b odd 2 1
768.4.f.g 12 24.h odd 2 1
768.4.f.h 12 8.b even 2 1
768.4.f.h 12 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{6} + 6T_{5}^{5} - 432T_{5}^{4} - 200T_{5}^{3} + 43968T_{5}^{2} - 26016T_{5} - 1193536$$ T5^6 + 6*T5^5 - 432*T5^4 - 200*T5^3 + 43968*T5^2 - 26016*T5 - 1193536 $$T_{19}^{6} + 90T_{19}^{5} - 18024T_{19}^{4} - 1956216T_{19}^{3} + 27304032T_{19}^{2} + 6866489760T_{19} + 146229576256$$ T19^6 + 90*T19^5 - 18024*T19^4 - 1956216*T19^3 + 27304032*T19^2 + 6866489760*T19 + 146229576256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 2 T^{11} + 2 T^{10} + \cdots + 387420489$$
$5$ $$(T^{6} + 6 T^{5} - 432 T^{4} + \cdots - 1193536)^{2}$$
$7$ $$T^{12} + \cdots + 103646082371584$$
$11$ $$T^{12} + 8628 T^{10} + \cdots + 13\!\cdots\!76$$
$13$ $$T^{12} + \cdots + 226854686396416$$
$17$ $$T^{12} + 27792 T^{10} + \cdots + 50\!\cdots\!44$$
$19$ $$(T^{6} + 90 T^{5} + \cdots + 146229576256)^{2}$$
$23$ $$(T^{6} + 60 T^{5} + \cdots - 261412163584)^{2}$$
$29$ $$(T^{6} + 294 T^{5} + \cdots + 4470831889088)^{2}$$
$31$ $$T^{12} + 169044 T^{10} + \cdots + 12\!\cdots\!84$$
$37$ $$T^{12} + 317976 T^{10} + \cdots + 71\!\cdots\!64$$
$41$ $$T^{12} + 354192 T^{10} + \cdots + 19\!\cdots\!96$$
$43$ $$(T^{6} - 186 T^{5} + \cdots - 58326025091776)^{2}$$
$47$ $$(T^{6} - 624 T^{5} + \cdots - 144142667350016)^{2}$$
$53$ $$(T^{6} - 474 T^{5} + \cdots + 13\!\cdots\!68)^{2}$$
$59$ $$T^{12} + 1799556 T^{10} + \cdots + 19\!\cdots\!04$$
$61$ $$T^{12} + 1802904 T^{10} + \cdots + 92\!\cdots\!16$$
$67$ $$(T^{6} + 1146 T^{5} + \cdots + 2132115742656)^{2}$$
$71$ $$(T^{6} + 1020 T^{5} + \cdots - 379600692064256)^{2}$$
$73$ $$(T^{6} - 108 T^{5} + \cdots - 10\!\cdots\!72)^{2}$$
$79$ $$T^{12} + 1129140 T^{10} + \cdots + 43\!\cdots\!76$$
$83$ $$T^{12} + 2756052 T^{10} + \cdots + 13\!\cdots\!44$$
$89$ $$T^{12} + 3691104 T^{10} + \cdots + 18\!\cdots\!36$$
$97$ $$(T^{6} + 24 T^{5} + \cdots + 63\!\cdots\!68)^{2}$$