# Properties

 Label 384.4.c.d Level $384$ Weight $4$ Character orbit 384.c Analytic conductor $22.657$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 384.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.6567334422$$ 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: $$2^{30}\cdot 3^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + 1) q^{3} + \beta_1 q^{5} + \beta_{2} q^{7} + ( - \beta_{10} - \beta_{2}) q^{9}+O(q^{10})$$ q + (-b5 + 1) * q^3 + b1 * q^5 + b2 * q^7 + (-b10 - b2) * q^9 $$q + ( - \beta_{5} + 1) q^{3} + \beta_1 q^{5} + \beta_{2} q^{7} + ( - \beta_{10} - \beta_{2}) q^{9} + (\beta_{10} - \beta_{8} - \beta_1 + 3) q^{11} + (\beta_{8} - 2 \beta_{5} + \beta_{4} + \beta_1) q^{13} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 + 8) q^{15} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} - 4 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{11} - 2 \beta_{10} - \beta_{8} - \beta_{6} - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{19} + ( - 3 \beta_{9} - 2 \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots + 13) q^{21}+ \cdots + ( - 14 \beta_{11} + 10 \beta_{10} - 4 \beta_{9} - 9 \beta_{8} - 2 \beta_{7} + \cdots - 503) q^{99}+O(q^{100})$$ q + (-b5 + 1) * q^3 + b1 * q^5 + b2 * q^7 + (-b10 - b2) * q^9 + (b10 - b8 - b1 + 3) * q^11 + (b8 - 2*b5 + b4 + b1) * q^13 + (-b7 - b5 - b4 - b3 + b2 + 3*b1 + 8) * q^15 + (b11 - b10 + b9 + b7 - 4*b5 + 2*b3 + 2*b2 + b1) * q^17 + (-2*b11 - 2*b10 - b8 - b6 - b3 - 2*b2 + 3*b1 - 1) * q^19 + (-3*b9 - 2*b8 + b6 - b5 + b4 - b3 + 2*b2 - b1 + 13) * q^21 + (2*b11 + b9 - b8 - 2*b7 + b6 - b5 + 2*b4 + b1 - 9) * q^23 + (b11 + b10 + 6*b9 + b8 - b7 - b6 + 11*b5 + 2*b4 + 2*b1 - 30) * q^25 + (-2*b11 - b10 + b9 - 2*b7 - 3*b6 + 2*b5 - 2*b4 - 3*b3 - 2*b2 - 6*b1 - 23) * q^27 + (2*b11 - 2*b10 - 6*b9 + 2*b7 + 16*b5 + 6*b3 - 4*b2 - 3*b1) * q^29 + (-6*b11 + 2*b10 + b9 - b8 - 4*b7 - b6 + b5 - 4*b3 - b2 - 9*b1 - 1) * q^31 + (b11 + 4*b10 + 5*b9 - 2*b8 + b7 + 4*b6 - 4*b5 + 2*b4 - 4*b3 - b2 - b1 - 11) * q^33 + (4*b11 + b10 + 5*b9 + b8 - 4*b7 - 2*b6 - b5 + 4*b4 + 5*b1 + 35) * q^35 + (2*b11 - 2*b10 - 6*b9 + b8 - 2*b7 - 4*b6 - 22*b5 - 3*b4 + 3*b1 + 52) * q^37 + (-2*b11 - 4*b10 + 3*b9 + b8 - 3*b7 - 5*b6 + b4 + b3 - 2*b2 + 14*b1 + 49) * q^39 + (-6*b11 + 2*b10 + 11*b9 - b8 - 4*b7 - b6 - 29*b5 + 2*b3 + b1 - 1) * q^41 + (-10*b11 + 6*b10 + 4*b9 - b8 - 8*b7 - b6 - 4*b5 - 7*b3 + 2*b2 + 11*b1 - 1) * q^43 + (2*b11 - 2*b10 - 15*b9 + 3*b8 + 2*b7 + 3*b6 - 5*b5 - 4*b4 - 9*b3 + 2*b2 + 47) * q^45 + (4*b11 + 2*b10 + 6*b9 - 2*b8 - 4*b7 - 8*b6 + 6*b5 - 4*b4 + 2*b1 - 106) * q^47 + (-5*b11 - 5*b10 + 10*b9 + 7*b8 + 5*b7 - 3*b6 + 41*b5 - 6*b4 + 2*b1 - 94) * q^49 + (-3*b10 - b9 - 7*b8 - 6*b7 + 2*b6 - 5*b5 + 2*b4 + 4*b3 + 4*b2 - 5*b1 - 85) * q^51 + (-12*b11 + 4*b10 - 10*b9 - 2*b8 - 8*b7 - 2*b6 + 38*b5 + 4*b3 - b1 - 2) * q^53 + (-14*b11 - 6*b10 + 5*b9 - 5*b8 - 4*b7 - 5*b6 - 11*b5 - 16*b3 - 6*b2 + 3*b1 - 5) * q^55 + (-4*b11 - 3*b10 + 13*b9 - 5*b8 - 6*b7 - 5*b6 - b5 - 8*b4 - 24*b3 + 5*b2 - b1 - 18) * q^57 + (8*b11 - 8*b10 - 5*b9 - 8*b8 - 8*b7 - 7*b5 - 8*b4 + 222) * q^59 + (-10*b11 + 10*b10 - 18*b9 - 3*b8 + 10*b7 + 4*b6 - 38*b5 + b4 - 13*b1 + 60) * q^61 + (-8*b11 + 6*b10 - 4*b9 - 16*b8 - 6*b7 + 2*b6 - 6*b5 + 2*b4 - 18*b3 - 5*b2 - 14*b1 + 232) * q^63 + (16*b11 + 4*b10 + 11*b9 + 5*b8 + 6*b7 + 5*b6 - 39*b5 + 38*b3 - 12*b2 - 19*b1 + 5) * q^65 + (-12*b11 + 4*b10 + 5*b9 - 2*b8 - 8*b7 - 2*b6 - 7*b5 - 24*b3 + 16*b2 + 14*b1 - 2) * q^67 + (-8*b11 + 8*b10 - 5*b9 - 5*b8 - 12*b7 - 5*b6 + 11*b5 + 4*b4 - 41*b3 - 14*b2 - b1 - 1) * q^69 + (6*b11 + 2*b10 - 9*b9 - 11*b8 - 6*b7 + b6 - 27*b5 - 2*b4 - 5*b1 - 155) * q^71 + (10*b11 + 6*b10 + 14*b9 - 18*b8 - 10*b7 + 36*b5 - 2*b4 - 8*b1 - 22) * q^73 + (-2*b11 + 13*b10 + 16*b9 + 4*b8 - 8*b7 - 5*b6 - 3*b5 + 8*b4 - 25*b3 + 14*b2 + 32*b1 - 333) * q^75 + (28*b11 + 12*b10 - 14*b9 + 10*b8 + 8*b7 + 10*b6 + 34*b5 + 52*b3 + 32*b2 + 8*b1 + 10) * q^77 + (-10*b11 - 2*b10 + 7*b9 - 3*b8 - 4*b7 - 3*b6 - 17*b5 + 14*b3 + 3*b2 + 29*b1 - 3) * q^79 + (5*b11 - 5*b10 + 8*b9 + 13*b8 + 15*b7 + b6 + 33*b5 + 4*b4 - 30*b3 + 6*b2 - 22*b1 - 106) * q^81 + (4*b11 - 7*b10 + 4*b9 + 5*b8 - 4*b7 - 14*b6 + 4*b5 - 12*b4 + 9*b1 + 381) * q^83 + (16*b11 - 8*b10 - 4*b9 - 4*b8 - 16*b7 - 4*b6 - 36*b5 + 12*b1 + 60) * q^85 + (4*b11 + 18*b10 + 22*b9 + 2*b8 - 7*b7 - 4*b6 + 3*b5 + b4 + 23*b3 + 11*b2 - 17*b1 + 338) * q^87 + (-29*b11 - 7*b10 + 12*b9 - 9*b8 - 11*b7 - 9*b6 - 25*b5 + 32*b3 + 2*b2 + 18*b1 - 9) * q^89 + (-6*b11 - 22*b10 - 7*b9 - 7*b8 + 8*b7 - 7*b6 + 13*b5 + 39*b3 - 10*b2 - 51*b1 - 7) * q^91 + (6*b11 - 6*b10 - 10*b9 + 5*b8 + 26*b7 - 4*b6 + 18*b5 + b4 - 36*b3 - 16*b2 - 14*b1 - 20) * q^93 + (6*b11 - 16*b10 + 3*b9 + 21*b8 - 6*b7 - 5*b6 - 19*b5 + 6*b4 + 27*b1 - 451) * q^95 + (-15*b11 + 5*b10 - 12*b9 + b8 + 15*b7 + 21*b6 - 3*b5 + 12*b4 - 14*b1 - 7) * q^97 + (-14*b11 + 10*b10 - 4*b9 - 9*b8 - 2*b7 + 3*b6 - 4*b5 - 2*b4 + 27*b3 - 30*b2 + 9*b1 - 503) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 10 q^{3}+O(q^{10})$$ 12 * q + 10 * q^3 $$12 q + 10 q^{3} + 36 q^{11} + 84 q^{15} + 136 q^{21} - 120 q^{23} - 300 q^{25} - 266 q^{27} - 116 q^{33} + 432 q^{35} + 528 q^{37} + 620 q^{39} + 440 q^{45} - 1248 q^{47} - 948 q^{49} - 1072 q^{51} - 172 q^{57} + 2508 q^{59} + 624 q^{61} + 2744 q^{63} - 24 q^{69} - 2040 q^{71} - 216 q^{73} - 3894 q^{75} - 1076 q^{81} + 4572 q^{83} + 480 q^{85} + 4156 q^{87} - 112 q^{93} - 5448 q^{95} - 48 q^{97} - 6044 q^{99}+O(q^{100})$$ 12 * q + 10 * q^3 + 36 * q^11 + 84 * q^15 + 136 * q^21 - 120 * q^23 - 300 * q^25 - 266 * q^27 - 116 * q^33 + 432 * q^35 + 528 * q^37 + 620 * q^39 + 440 * q^45 - 1248 * q^47 - 948 * q^49 - 1072 * q^51 - 172 * q^57 + 2508 * q^59 + 624 * q^61 + 2744 * q^63 - 24 * q^69 - 2040 * q^71 - 216 * q^73 - 3894 * q^75 - 1076 * q^81 + 4572 * q^83 + 480 * q^85 + 4156 * q^87 - 112 * q^93 - 5448 * q^95 - 48 * q^97 - 6044 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( -\nu^{11} - 29\nu^{9} - 223\nu^{7} - 82\nu^{5} + 3220\nu^{3} + 5456\nu ) / 216$$ (-v^11 - 29*v^9 - 223*v^7 - 82*v^5 + 3220*v^3 + 5456*v) / 216 $$\beta_{2}$$ $$=$$ $$( -19\nu^{11} - 401\nu^{9} + 449\nu^{7} + 42164\nu^{5} + 171832\nu^{3} + 22472\nu ) / 3672$$ (-19*v^11 - 401*v^9 + 449*v^7 + 42164*v^5 + 171832*v^3 + 22472*v) / 3672 $$\beta_{3}$$ $$=$$ $$( -\nu^{11} - 22\nu^{9} + 12\nu^{7} + 2399\nu^{5} + 11762\nu^{3} + 9468\nu ) / 153$$ (-v^11 - 22*v^9 + 12*v^7 + 2399*v^5 + 11762*v^3 + 9468*v) / 153 $$\beta_{4}$$ $$=$$ $$( - 6 \nu^{11} - 85 \nu^{10} - 285 \nu^{9} - 3026 \nu^{8} - 4977 \nu^{7} - 36958 \nu^{6} - 38697 \nu^{5} - 189499 \nu^{4} - 126186 \nu^{3} - 387056 \nu^{2} - 118836 \nu - 193936 ) / 918$$ (-6*v^11 - 85*v^10 - 285*v^9 - 3026*v^8 - 4977*v^7 - 36958*v^6 - 38697*v^5 - 189499*v^4 - 126186*v^3 - 387056*v^2 - 118836*v - 193936) / 918 $$\beta_{5}$$ $$=$$ $$( 121 \nu^{11} + 136 \nu^{10} + 4277 \nu^{9} + 4556 \nu^{8} + 51367 \nu^{7} + 49300 \nu^{6} + 254146 \nu^{5} + 195364 \nu^{4} + 497900 \nu^{3} + 223040 \nu^{2} + 281488 \nu + 48688 ) / 7344$$ (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) / 7344 $$\beta_{6}$$ $$=$$ $$( 2 \nu^{11} + 85 \nu^{10} + 95 \nu^{9} + 2839 \nu^{8} + 1659 \nu^{7} + 30753 \nu^{6} + 12899 \nu^{5} + 124372 \nu^{4} + 42062 \nu^{3} + 158848 \nu^{2} + 39612 \nu + 55590 ) / 306$$ (2*v^11 + 85*v^10 + 95*v^9 + 2839*v^8 + 1659*v^7 + 30753*v^6 + 12899*v^5 + 124372*v^4 + 42062*v^3 + 158848*v^2 + 39612*v + 55590) / 306 $$\beta_{7}$$ $$=$$ $$( 13 \nu^{11} - 544 \nu^{10} + 337 \nu^{9} - 18428 \nu^{8} + 1323 \nu^{7} - 205156 \nu^{6} - 19406 \nu^{5} - 883660 \nu^{4} - 134036 \nu^{3} - 1365440 \nu^{2} - 141648 \nu - 706384 ) / 2448$$ (13*v^11 - 544*v^10 + 337*v^9 - 18428*v^8 + 1323*v^7 - 205156*v^6 - 19406*v^5 - 883660*v^4 - 134036*v^3 - 1365440*v^2 - 141648*v - 706384) / 2448 $$\beta_{8}$$ $$=$$ $$( 27 \nu^{11} + 34 \nu^{10} + 985 \nu^{9} + 1394 \nu^{8} + 12511 \nu^{7} + 20638 \nu^{6} + 68388 \nu^{5} + 134164 \nu^{4} + 157984 \nu^{3} + 356864 \nu^{2} + 110680 \nu + 242896 ) / 612$$ (27*v^11 + 34*v^10 + 985*v^9 + 1394*v^8 + 12511*v^7 + 20638*v^6 + 68388*v^5 + 134164*v^4 + 157984*v^3 + 356864*v^2 + 110680*v + 242896) / 612 $$\beta_{9}$$ $$=$$ $$( - 121 \nu^{11} + 136 \nu^{10} - 4277 \nu^{9} + 4556 \nu^{8} - 51367 \nu^{7} + 49300 \nu^{6} - 254146 \nu^{5} + 195364 \nu^{4} - 497900 \nu^{3} + 223040 \nu^{2} + \cdots + 48688 ) / 2448$$ (-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) / 2448 $$\beta_{10}$$ $$=$$ $$( 145 \nu^{11} - 748 \nu^{10} + 5417 \nu^{9} - 25568 \nu^{8} + 71275 \nu^{7} - 289408 \nu^{6} + 408934 \nu^{5} - 1285948 \nu^{4} + 1002644 \nu^{3} - 2091680 \nu^{2} + \cdots - 1073992 ) / 3672$$ (145*v^11 - 748*v^10 + 5417*v^9 - 25568*v^8 + 71275*v^7 - 289408*v^6 + 408934*v^5 - 1285948*v^4 + 1002644*v^3 - 2091680*v^2 + 756832*v - 1073992) / 3672 $$\beta_{11}$$ $$=$$ $$( 121 \nu^{11} + 68 \nu^{10} + 4277 \nu^{9} + 2176 \nu^{8} + 51367 \nu^{7} + 21488 \nu^{6} + 254146 \nu^{5} + 68612 \nu^{4} + 497900 \nu^{3} + 34000 \nu^{2} + 296176 \nu + 4964 ) / 1836$$ (121*v^11 + 68*v^10 + 4277*v^9 + 2176*v^8 + 51367*v^7 + 21488*v^6 + 254146*v^5 + 68612*v^4 + 497900*v^3 + 34000*v^2 + 296176*v + 4964) / 1836
 $$\nu$$ $$=$$ $$( 4\beta_{11} + 3\beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{6} - 11\beta_{5} - \beta _1 + 1 ) / 48$$ (4*b11 + 3*b9 + b8 + 2*b7 + b6 - 11*b5 - b1 + 1) / 48 $$\nu^{2}$$ $$=$$ $$( -\beta_{9} + \beta_{8} + \beta_{6} - 5\beta_{5} + 2\beta_{4} + \beta _1 - 103 ) / 16$$ (-b9 + b8 + b6 - 5*b5 + 2*b4 + b1 - 103) / 16 $$\nu^{3}$$ $$=$$ $$( - 40 \beta_{11} - 12 \beta_{10} - 39 \beta_{9} - 13 \beta_{8} - 14 \beta_{7} - 13 \beta_{6} + 131 \beta_{5} + 30 \beta_{3} - 24 \beta_{2} + 37 \beta _1 - 13 ) / 48$$ (-40*b11 - 12*b10 - 39*b9 - 13*b8 - 14*b7 - 13*b6 + 131*b5 + 30*b3 - 24*b2 + 37*b1 - 13) / 48 $$\nu^{4}$$ $$=$$ $$( 4\beta_{11} + 10\beta_{9} - 9\beta_{8} - 4\beta_{7} - 12\beta_{6} + 36\beta_{5} - 17\beta_{4} - 5\beta _1 + 558 ) / 8$$ (4*b11 + 10*b9 - 9*b8 - 4*b7 - 12*b6 + 36*b5 - 17*b4 - 5*b1 + 558) / 8 $$\nu^{5}$$ $$=$$ $$( 502 \beta_{11} + 294 \beta_{10} + 591 \beta_{9} + 199 \beta_{8} + 104 \beta_{7} + 199 \beta_{6} - 1877 \beta_{5} - 753 \beta_{3} + 516 \beta_{2} - 553 \beta _1 + 199 ) / 48$$ (502*b11 + 294*b10 + 591*b9 + 199*b8 + 104*b7 + 199*b6 - 1877*b5 - 753*b3 + 516*b2 - 553*b1 + 199) / 48 $$\nu^{6}$$ $$=$$ $$( - 192 \beta_{11} - 12 \beta_{10} - 399 \beta_{9} + 285 \beta_{8} + 192 \beta_{7} + 447 \beta_{6} - 1167 \beta_{5} + 528 \beta_{4} + 93 \beta _1 - 14689 ) / 16$$ (-192*b11 - 12*b10 - 399*b9 + 285*b8 + 192*b7 + 447*b6 - 1167*b5 + 528*b4 + 93*b1 - 14689) / 16 $$\nu^{7}$$ $$=$$ $$( - 7186 \beta_{11} - 5454 \beta_{10} - 9318 \beta_{9} - 3160 \beta_{8} - 866 \beta_{7} - 3160 \beta_{6} + 28820 \beta_{5} + 14571 \beta_{3} - 9468 \beta_{2} + 7678 \beta _1 - 3160 ) / 48$$ (-7186*b11 - 5454*b10 - 9318*b9 - 3160*b8 - 866*b7 - 3160*b6 + 28820*b5 + 14571*b3 - 9468*b2 + 7678*b1 - 3160) / 48 $$\nu^{8}$$ $$=$$ $$( 3576 \beta_{11} + 372 \beta_{10} + 7477 \beta_{9} - 4417 \beta_{8} - 3576 \beta_{7} - 7729 \beta_{6} + 19793 \beta_{5} - 8198 \beta_{4} - 841 \beta _1 + 212587 ) / 16$$ (3576*b11 + 372*b10 + 7477*b9 - 4417*b8 - 3576*b7 - 7729*b6 + 19793*b5 - 8198*b4 - 841*b1 + 212587) / 16 $$\nu^{9}$$ $$=$$ $$( 109840 \beta_{11} + 93540 \beta_{10} + 149223 \beta_{9} + 50845 \beta_{8} + 8150 \beta_{7} + 50845 \beta_{6} - 455819 \beta_{5} - 257844 \beta_{3} + 164256 \beta_{2} - 107821 \beta _1 + 50845 ) / 48$$ (109840*b11 + 93540*b10 + 149223*b9 + 50845*b8 + 8150*b7 + 50845*b6 - 455819*b5 - 257844*b3 + 164256*b2 - 107821*b1 + 50845) / 48 $$\nu^{10}$$ $$=$$ $$( - 30844 \beta_{11} - 4056 \beta_{10} - 66394 \beta_{9} + 34437 \beta_{8} + 30844 \beta_{7} + 64860 \beta_{6} - 167412 \beta_{5} + 64397 \beta_{4} + 3593 \beta _1 - 1618422 ) / 8$$ (-30844*b11 - 4056*b10 - 66394*b9 + 34437*b8 + 30844*b7 + 64860*b6 - 167412*b5 + 64397*b4 + 3593*b1 - 1618422) / 8 $$\nu^{11}$$ $$=$$ $$( - 1731022 \beta_{11} - 1559166 \beta_{10} - 2407227 \beta_{9} - 822547 \beta_{8} - 85928 \beta_{7} - 822547 \beta_{6} + 7307609 \beta_{5} + 4386489 \beta_{3} - 2771652 \beta_{2} + \cdots - 822547 ) / 48$$ (-1731022*b11 - 1559166*b10 - 2407227*b9 - 822547*b8 - 85928*b7 - 822547*b6 + 7307609*b5 + 4386489*b3 - 2771652*b2 + 1563277*b1 - 822547) / 48

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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
 2.29679i − 2.29679i − 4.03251i 4.03251i − 2.36157i 2.36157i − 1.08600i 1.08600i − 3.14286i 3.14286i 0.910871i − 0.910871i
0 −5.00172 1.40813i 0 5.86626i 0 5.92149i 0 23.0343 + 14.0861i 0
383.2 0 −5.00172 + 1.40813i 0 5.86626i 0 5.92149i 0 23.0343 14.0861i 0
383.3 0 −2.52186 4.54315i 0 8.01133i 0 12.6015i 0 −14.2805 + 22.9144i 0
383.4 0 −2.52186 + 4.54315i 0 8.01133i 0 12.6015i 0 −14.2805 22.9144i 0
383.5 0 0.556921 5.16622i 0 10.6077i 0 7.90379i 0 −26.3797 5.75435i 0
383.6 0 0.556921 + 5.16622i 0 10.6077i 0 7.90379i 0 −26.3797 + 5.75435i 0
383.7 0 3.04120 4.21320i 0 9.33303i 0 36.3792i 0 −8.50216 25.6264i 0
383.8 0 3.04120 + 4.21320i 0 9.33303i 0 36.3792i 0 −8.50216 + 25.6264i 0
383.9 0 4.12202 3.16369i 0 21.4043i 0 20.9034i 0 6.98212 26.0816i 0
383.10 0 4.12202 + 3.16369i 0 21.4043i 0 20.9034i 0 6.98212 + 26.0816i 0
383.11 0 4.80343 1.98169i 0 11.9846i 0 22.6995i 0 19.1458 19.0378i 0
383.12 0 4.80343 + 1.98169i 0 11.9846i 0 22.6995i 0 19.1458 + 19.0378i 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
12.b even 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{11}^{6} - 18T_{11}^{5} - 4152T_{11}^{4} + 106488T_{11}^{3} + 2349600T_{11}^{2} - 67009824T_{11} + 370690624$$ T11^6 - 18*T11^5 - 4152*T11^4 + 106488*T11^3 + 2349600*T11^2 - 67009824*T11 + 370690624 $$T_{13}^{6} - 7452T_{13}^{4} + 39936T_{13}^{3} + 12415920T_{13}^{2} - 231124992T_{13} + 15061696$$ T13^6 - 7452*T13^4 + 39936*T13^3 + 12415920*T13^2 - 231124992*T13 + 15061696

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} - 10 T^{11} + \cdots + 387420489$$
$5$ $$T^{12} + 900 T^{10} + \cdots + 1424528183296$$
$7$ $$T^{12} + \cdots + 103646082371584$$
$11$ $$(T^{6} - 18 T^{5} - 4152 T^{4} + \cdots + 370690624)^{2}$$
$13$ $$(T^{6} - 7452 T^{4} + 39936 T^{3} + \cdots + 15061696)^{2}$$
$17$ $$T^{12} + 27792 T^{10} + \cdots + 50\!\cdots\!44$$
$19$ $$T^{12} + 44148 T^{10} + \cdots + 21\!\cdots\!36$$
$23$ $$(T^{6} + 60 T^{5} + \cdots - 261412163584)^{2}$$
$29$ $$T^{12} + 118884 T^{10} + \cdots + 19\!\cdots\!44$$
$31$ $$T^{12} + 169044 T^{10} + \cdots + 12\!\cdots\!84$$
$37$ $$(T^{6} - 264 T^{5} + \cdots + 847281526208)^{2}$$
$41$ $$T^{12} + 354192 T^{10} + \cdots + 19\!\cdots\!96$$
$43$ $$T^{12} + 567828 T^{10} + \cdots + 34\!\cdots\!76$$
$47$ $$(T^{6} + 624 T^{5} + \cdots - 144142667350016)^{2}$$
$53$ $$T^{12} + 1010436 T^{10} + \cdots + 17\!\cdots\!24$$
$59$ $$(T^{6} - 1254 T^{5} + \cdots - 43\!\cdots\!48)^{2}$$
$61$ $$(T^{6} - 312 T^{5} + \cdots - 96\!\cdots\!96)^{2}$$
$67$ $$T^{12} + 1322340 T^{10} + \cdots + 45\!\cdots\!36$$
$71$ $$(T^{6} + 1020 T^{5} + \cdots - 379600692064256)^{2}$$
$73$ $$(T^{6} + 108 T^{5} + \cdots - 10\!\cdots\!72)^{2}$$
$79$ $$T^{12} + 1129140 T^{10} + \cdots + 43\!\cdots\!76$$
$83$ $$(T^{6} - 2286 T^{5} + \cdots + 11\!\cdots\!12)^{2}$$
$89$ $$T^{12} + 3691104 T^{10} + \cdots + 18\!\cdots\!36$$
$97$ $$(T^{6} + 24 T^{5} + \cdots + 63\!\cdots\!68)^{2}$$