# Properties

 Label 576.2.r.f Level $576$ Weight $2$ Character orbit 576.r Analytic conductor $4.599$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36$$ x^12 - 16*x^8 - 24*x^7 + 96*x^5 + 304*x^4 + 384*x^3 + 288*x^2 + 144*x + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - \beta_{4}) q^{3} + (\beta_{6} + 2) q^{5} + (\beta_{11} - \beta_{4} - \beta_1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{3} - 1) q^{9}+O(q^{10})$$ q + (b5 - b4) * q^3 + (b6 + 2) * q^5 + (b11 - b4 - b1) * q^7 + (b10 - b6 - b3 - 1) * q^9 $$q + (\beta_{5} - \beta_{4}) q^{3} + (\beta_{6} + 2) q^{5} + (\beta_{11} - \beta_{4} - \beta_1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{3} - 1) q^{9} + (\beta_{4} - \beta_1) q^{11} + ( - \beta_{10} + \beta_{7}) q^{13} + (\beta_{5} - 2 \beta_{4}) q^{15} + ( - \beta_{10} - \beta_{9} + \beta_{7} + \beta_{3}) q^{17} - \beta_{8} q^{19} + (\beta_{10} + 4 \beta_{6} - 2 \beta_{3} + 2) q^{21} + (\beta_{11} - \beta_{8} + \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1) q^{23} + ( - 2 \beta_{6} - 2) q^{25} + ( - \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1) q^{27} + (2 \beta_{10} + 2 \beta_{9} - \beta_{7} + \beta_{6} - \beta_{3} - 1) q^{29} + (\beta_{11} - \beta_{8} - 3 \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1) q^{31} + (\beta_{10} + 4 \beta_{6} + \beta_{3} + 5) q^{33} + (2 \beta_{11} - \beta_{5} - \beta_{4} - \beta_1) q^{35} + ( - \beta_{10} + \beta_{9} - 2 \beta_{6} + \beta_{3} - 1) q^{37} + ( - 2 \beta_{11} + \beta_{5} - 2 \beta_{4} + 3 \beta_{2} + \beta_1) q^{39} + (\beta_{10} + \beta_{9} - 3 \beta_{6}) q^{41} + ( - \beta_{11} + 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_1) q^{43} + (\beta_{10} - 2 \beta_{6} - 2 \beta_{3} - 1) q^{45} + (\beta_{11} + 2 \beta_{8} + 4 \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{47} + (\beta_{10} - 2 \beta_{9} + \beta_{7} + 5 \beta_{6} - 2 \beta_{3}) q^{49} + (\beta_{11} + 3 \beta_{2} + \beta_1) q^{51} + (\beta_{10} + \beta_{9} + \beta_{7} - 4 \beta_{6} + \beta_{3} - 2) q^{53} + (\beta_{11} + \beta_{5} + \beta_{4} - 2 \beta_1) q^{55} + ( - \beta_{7} + 1) q^{57} + (\beta_{11} - 3 \beta_{8} + \beta_{5} + 2 \beta_{4} - 3 \beta_{2}) q^{59} + ( - 2 \beta_{10} - 2 \beta_{9} + \beta_{7} + \beta_{6} + \beta_{3} - 1) q^{61} + ( - 2 \beta_{11} - 3 \beta_{8} - 3 \beta_{4} - 3 \beta_{2} + \beta_1) q^{63} + ( - 2 \beta_{10} + \beta_{9} + \beta_{7} + \beta_{3}) q^{65} + ( - \beta_{11} + 2 \beta_{8} - \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - \beta_1) q^{67} + ( - \beta_{9} - \beta_{7} - 2 \beta_{6} - 3 \beta_{3} - 7) q^{69} + (\beta_{11} - 2 \beta_{8} - 2 \beta_{5} + 4 \beta_{4} - 4 \beta_{2} + \beta_1) q^{71} + ( - \beta_{10} - \beta_{9} + 2 \beta_{7} - \beta_{3} - 1) q^{73} + 2 \beta_{4} q^{75} + (2 \beta_{10} - \beta_{9} - \beta_{7} + 6 \beta_{6} + 2 \beta_{3} + 12) q^{77} + (2 \beta_{8} - 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1) q^{79} + (2 \beta_{9} - \beta_{7} - 6 \beta_{6} + 2 \beta_{3} - 4) q^{81} + (\beta_{11} - 2 \beta_{5} + \beta_{4} + 3 \beta_{2} + \beta_1) q^{83} + ( - \beta_{10} - \beta_{9} + 2 \beta_{7} + 2 \beta_{3}) q^{85} + ( - 3 \beta_{11} + 3 \beta_{8} - 2 \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{87} + (\beta_{10} + \beta_{9} - 2 \beta_{7} + \beta_{3} - 3) q^{89} + ( - 3 \beta_{11} - 3 \beta_{5} + 3 \beta_{4}) q^{91} + ( - 2 \beta_{10} - \beta_{9} - \beta_{7} + 8 \beta_{6} + \beta_{3} + 7) q^{93} + ( - \beta_{8} + \beta_{2}) q^{95} + ( - 3 \beta_{7} - \beta_{6} - 3 \beta_{3} - 1) q^{97} + (\beta_{11} + 3 \beta_{8} - 3 \beta_{4} + \beta_1) q^{99}+O(q^{100})$$ q + (b5 - b4) * q^3 + (b6 + 2) * q^5 + (b11 - b4 - b1) * q^7 + (b10 - b6 - b3 - 1) * q^9 + (b4 - b1) * q^11 + (-b10 + b7) * q^13 + (b5 - 2*b4) * q^15 + (-b10 - b9 + b7 + b3) * q^17 - b8 * q^19 + (b10 + 4*b6 - 2*b3 + 2) * q^21 + (b11 - b8 + b5 - 2*b4 + b2 + b1) * q^23 + (-2*b6 - 2) * q^25 + (-b11 - b8 + b5 + b4 - 2*b2 + b1) * q^27 + (2*b10 + 2*b9 - b7 + b6 - b3 - 1) * q^29 + (b11 - b8 - 3*b5 + 2*b4 + b2 - b1) * q^31 + (b10 + 4*b6 + b3 + 5) * q^33 + (2*b11 - b5 - b4 - b1) * q^35 + (-b10 + b9 - 2*b6 + b3 - 1) * q^37 + (-2*b11 + b5 - 2*b4 + 3*b2 + b1) * q^39 + (b10 + b9 - 3*b6) * q^41 + (-b11 + 2*b5 - b4 - 2*b2 - b1) * q^43 + (b10 - 2*b6 - 2*b3 - 1) * q^45 + (b11 + 2*b8 + 4*b5 + b4 + b2 + b1) * q^47 + (b10 - 2*b9 + b7 + 5*b6 - 2*b3) * q^49 + (b11 + 3*b2 + b1) * q^51 + (b10 + b9 + b7 - 4*b6 + b3 - 2) * q^53 + (b11 + b5 + b4 - 2*b1) * q^55 + (-b7 + 1) * q^57 + (b11 - 3*b8 + b5 + 2*b4 - 3*b2) * q^59 + (-2*b10 - 2*b9 + b7 + b6 + b3 - 1) * q^61 + (-2*b11 - 3*b8 - 3*b4 - 3*b2 + b1) * q^63 + (-2*b10 + b9 + b7 + b3) * q^65 + (-b11 + 2*b8 - b5 - 4*b4 + 2*b2 - b1) * q^67 + (-b9 - b7 - 2*b6 - 3*b3 - 7) * q^69 + (b11 - 2*b8 - 2*b5 + 4*b4 - 4*b2 + b1) * q^71 + (-b10 - b9 + 2*b7 - b3 - 1) * q^73 + 2*b4 * q^75 + (2*b10 - b9 - b7 + 6*b6 + 2*b3 + 12) * q^77 + (2*b8 - 2*b5 - b4 + b2 - b1) * q^79 + (2*b9 - b7 - 6*b6 + 2*b3 - 4) * q^81 + (b11 - 2*b5 + b4 + 3*b2 + b1) * q^83 + (-b10 - b9 + 2*b7 + 2*b3) * q^85 + (-3*b11 + 3*b8 - 2*b5 + b4 - 3*b2) * q^87 + (b10 + b9 - 2*b7 + b3 - 3) * q^89 + (-3*b11 - 3*b5 + 3*b4) * q^91 + (-2*b10 - b9 - b7 + 8*b6 + b3 + 7) * q^93 + (-b8 + b2) * q^95 + (-3*b7 - b6 - 3*b3 - 1) * q^97 + (b11 + 3*b8 - 3*b4 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 18 q^{5} - 4 q^{9}+O(q^{10})$$ 12 * q + 18 * q^5 - 4 * q^9 $$12 q + 18 q^{5} - 4 q^{9} + 6 q^{13} + 6 q^{21} - 12 q^{25} - 18 q^{29} + 30 q^{33} + 18 q^{41} + 6 q^{45} - 24 q^{49} + 8 q^{57} - 18 q^{61} + 6 q^{65} - 66 q^{69} + 90 q^{77} - 20 q^{81} - 48 q^{89} + 30 q^{93} - 6 q^{97}+O(q^{100})$$ 12 * q + 18 * q^5 - 4 * q^9 + 6 * q^13 + 6 * q^21 - 12 * q^25 - 18 * q^29 + 30 * q^33 + 18 * q^41 + 6 * q^45 - 24 * q^49 + 8 * q^57 - 18 * q^61 + 6 * q^65 - 66 * q^69 + 90 * q^77 - 20 * q^81 - 48 * q^89 + 30 * q^93 - 6 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36$$ :

 $$\beta_{1}$$ $$=$$ $$( - 778 \nu^{11} + 5496 \nu^{10} - 7293 \nu^{9} + 5400 \nu^{8} + 10120 \nu^{7} - 68622 \nu^{6} - 13980 \nu^{5} + 12828 \nu^{4} + 200318 \nu^{3} + 699840 \nu^{2} + \cdots + 73584 ) / 51972$$ (-778*v^11 + 5496*v^10 - 7293*v^9 + 5400*v^8 + 10120*v^7 - 68622*v^6 - 13980*v^5 + 12828*v^4 + 200318*v^3 + 699840*v^2 + 180024*v + 73584) / 51972 $$\beta_{2}$$ $$=$$ $$( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + 11380 \nu^{5} + 57708 \nu^{4} + 98118 \nu^{3} - 132080 \nu^{2} + 49284 \nu + 15984 ) / 51972$$ (962*v^11 - 2392*v^10 + 2887*v^9 - 2220*v^8 - 13990*v^7 + 14442*v^6 + 11380*v^5 + 57708*v^4 + 98118*v^3 - 132080*v^2 + 49284*v + 15984) / 51972 $$\beta_{3}$$ $$=$$ $$( 1360 \nu^{11} - 864 \nu^{10} - 400 \nu^{9} + 2902 \nu^{8} - 25539 \nu^{7} - 16320 \nu^{6} + 27692 \nu^{5} + 92050 \nu^{4} + 330320 \nu^{3} + 266532 \nu^{2} + 143634 \nu + 219540 ) / 51972$$ (1360*v^11 - 864*v^10 - 400*v^9 + 2902*v^8 - 25539*v^7 - 16320*v^6 + 27692*v^5 + 92050*v^4 + 330320*v^3 + 266532*v^2 + 143634*v + 219540) / 51972 $$\beta_{4}$$ $$=$$ $$( 5984 \nu^{11} - 7522 \nu^{10} + 8393 \nu^{9} - 6600 \nu^{8} - 90816 \nu^{7} - 28498 \nu^{6} + 55616 \nu^{5} + 473748 \nu^{4} + 1167562 \nu^{3} + 792408 \nu^{2} + \cdots + 272448 ) / 103944$$ (5984*v^11 - 7522*v^10 + 8393*v^9 - 6600*v^8 - 90816*v^7 - 28498*v^6 + 55616*v^5 + 473748*v^4 + 1167562*v^3 + 792408*v^2 + 746328*v + 272448) / 103944 $$\beta_{5}$$ $$=$$ $$( - 6823 \nu^{11} - 186 \nu^{10} + 2391 \nu^{9} - 2376 \nu^{8} + 113464 \nu^{7} + 159834 \nu^{6} - 23754 \nu^{5} - 683700 \nu^{4} - 2089522 \nu^{3} - 2446620 \nu^{2} + \cdots - 548280 ) / 103944$$ (-6823*v^11 - 186*v^10 + 2391*v^9 - 2376*v^8 + 113464*v^7 + 159834*v^6 - 23754*v^5 - 683700*v^4 - 2089522*v^3 - 2446620*v^2 - 1454952*v - 548280) / 103944 $$\beta_{6}$$ $$=$$ $$( 118 \nu^{11} - 90 \nu^{10} + 53 \nu^{9} - 32 \nu^{8} - 1866 \nu^{7} - 1416 \nu^{6} + 1364 \nu^{5} + 10408 \nu^{4} + 27758 \nu^{3} + 22776 \nu^{2} + 12468 \nu + 3468 ) / 1704$$ (118*v^11 - 90*v^10 + 53*v^9 - 32*v^8 - 1866*v^7 - 1416*v^6 + 1364*v^5 + 10408*v^4 + 27758*v^3 + 22776*v^2 + 12468*v + 3468) / 1704 $$\beta_{7}$$ $$=$$ $$( 4689 \nu^{11} - 3876 \nu^{10} + 3854 \nu^{9} - 5534 \nu^{8} - 68264 \nu^{7} - 56268 \nu^{6} + 37850 \nu^{5} + 438544 \nu^{4} + 1102444 \nu^{3} + 911136 \nu^{2} + \cdots + 48876 ) / 51972$$ (4689*v^11 - 3876*v^10 + 3854*v^9 - 5534*v^8 - 68264*v^7 - 56268*v^6 + 37850*v^5 + 438544*v^4 + 1102444*v^3 + 911136*v^2 + 502080*v + 48876) / 51972 $$\beta_{8}$$ $$=$$ $$( - 1688 \nu^{11} + 1054 \nu^{10} - 840 \nu^{9} + 684 \nu^{8} + 26378 \nu^{7} + 24587 \nu^{6} - 12648 \nu^{5} - 151968 \nu^{4} - 420176 \nu^{3} - 400856 \nu^{2} + \cdots - 105120 ) / 12993$$ (-1688*v^11 + 1054*v^10 - 840*v^9 + 684*v^8 + 26378*v^7 + 24587*v^6 - 12648*v^5 - 151968*v^4 - 420176*v^3 - 400856*v^2 - 282372*v - 105120) / 12993 $$\beta_{9}$$ $$=$$ $$( - 4502 \nu^{11} + 3246 \nu^{10} - 927 \nu^{9} - 1080 \nu^{8} + 73378 \nu^{7} + 54024 \nu^{6} - 56276 \nu^{5} - 403744 \nu^{4} - 1035370 \nu^{3} - 847128 \nu^{2} + \cdots - 228796 ) / 34648$$ (-4502*v^11 + 3246*v^10 - 927*v^9 - 1080*v^8 + 73378*v^7 + 54024*v^6 - 56276*v^5 - 403744*v^4 - 1035370*v^3 - 847128*v^2 - 462516*v - 228796) / 34648 $$\beta_{10}$$ $$=$$ $$( 25054 \nu^{11} - 18618 \nu^{10} + 8389 \nu^{9} + 272 \nu^{8} - 404246 \nu^{7} - 300648 \nu^{6} + 310036 \nu^{5} + 2217896 \nu^{4} + 5869134 \nu^{3} + 4806792 \nu^{2} + \cdots + 1030836 ) / 103944$$ (25054*v^11 - 18618*v^10 + 8389*v^9 + 272*v^8 - 404246*v^7 - 300648*v^6 + 310036*v^5 + 2217896*v^4 + 5869134*v^3 + 4806792*v^2 + 2626812*v + 1030836) / 103944 $$\beta_{11}$$ $$=$$ $$( - 13366 \nu^{11} + 8843 \nu^{10} - 7504 \nu^{9} + 5514 \nu^{8} + 211707 \nu^{7} + 177716 \nu^{6} - 88792 \nu^{5} - 1193088 \nu^{4} - 3315152 \nu^{3} - 3104220 \nu^{2} + \cdots - 830016 ) / 51972$$ (-13366*v^11 + 8843*v^10 - 7504*v^9 + 5514*v^8 + 211707*v^7 + 177716*v^6 - 88792*v^5 - 1193088*v^4 - 3315152*v^3 - 3104220*v^2 - 2229918*v - 830016) / 51972
 $$\nu$$ $$=$$ $$( -2\beta_{11} + \beta_{10} + \beta_{9} - 2\beta_{7} + 4\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_1 ) / 6$$ (-2*b11 + b10 + b9 - 2*b7 + 4*b5 - 2*b4 - 2*b3 + b1) / 6 $$\nu^{2}$$ $$=$$ $$( -2\beta_{11} + 4\beta_{5} - 2\beta_{4} - 9\beta_{2} - 2\beta_1 ) / 6$$ (-2*b11 + 4*b5 - 2*b4 - 9*b2 - 2*b1) / 6 $$\nu^{3}$$ $$=$$ $$( - 4 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} - 3 \beta_{8} + 4 \beta_{7} - 12 \beta_{6} + 8 \beta_{5} - 16 \beta_{4} + 4 \beta_{3} - 6 \beta_{2} - 4 \beta _1 - 6 ) / 6$$ (-4*b11 + 4*b10 + 4*b9 - 3*b8 + 4*b7 - 12*b6 + 8*b5 - 16*b4 + 4*b3 - 6*b2 - 4*b1 - 6) / 6 $$\nu^{4}$$ $$=$$ $$( 11\beta_{10} + 5\beta_{9} + 2\beta_{7} - 30\beta_{6} - 4\beta_{3} ) / 3$$ (11*b10 + 5*b9 + 2*b7 - 30*b6 - 4*b3) / 3 $$\nu^{5}$$ $$=$$ $$( - 22 \beta_{11} + 38 \beta_{10} + 35 \beta_{9} + 42 \beta_{8} - 16 \beta_{7} - 39 \beta_{6} + 32 \beta_{5} + 38 \beta_{4} - 22 \beta_{3} + 21 \beta_{2} + 38 \beta _1 + 39 ) / 6$$ (-22*b11 + 38*b10 + 35*b9 + 42*b8 - 16*b7 - 39*b6 + 32*b5 + 38*b4 - 22*b3 + 21*b2 + 38*b1 + 39) / 6 $$\nu^{6}$$ $$=$$ $$( -56\beta_{11} + 81\beta_{8} + 64\beta_{5} + 16\beta_{4} + 40\beta_1 ) / 3$$ (-56*b11 + 81*b8 + 64*b5 + 16*b4 + 40*b1) / 3 $$\nu^{7}$$ $$=$$ $$( - 94 \beta_{11} - 35 \beta_{10} - 47 \beta_{9} + 60 \beta_{8} + 82 \beta_{7} - 108 \beta_{6} + 164 \beta_{5} - 70 \beta_{4} + 94 \beta_{3} - 60 \beta_{2} + 35 \beta _1 - 216 ) / 3$$ (-94*b11 - 35*b10 - 47*b9 + 60*b8 + 82*b7 - 108*b6 + 164*b5 - 70*b4 + 94*b3 - 60*b2 + 35*b1 - 216) / 3 $$\nu^{8}$$ $$=$$ $$( 70\beta_{10} - 35\beta_{9} + 214\beta_{7} - 669\beta_{6} + 214\beta_{3} - 669 ) / 3$$ (70*b10 - 35*b9 + 214*b7 - 669*b6 + 214*b3 - 669) / 3 $$\nu^{9}$$ $$=$$ $$( 164 \beta_{11} + 308 \beta_{10} + 164 \beta_{9} + 321 \beta_{8} + 236 \beta_{7} - 1140 \beta_{6} - 472 \beta_{5} + 800 \beta_{4} + 164 \beta_{3} + 642 \beta_{2} + 308 \beta _1 - 570 ) / 3$$ (164*b11 + 308*b10 + 164*b9 + 321*b8 + 236*b7 - 1140*b6 - 472*b5 + 800*b4 + 164*b3 + 642*b2 + 308*b1 - 570) / 3 $$\nu^{10}$$ $$=$$ $$( -328\beta_{11} + 1908\beta_{8} - 328\beta_{5} + 2228\beta_{4} + 1908\beta_{2} + 1442\beta_1 ) / 3$$ (-328*b11 + 1908*b8 - 328*b5 + 2228*b4 + 1908*b2 + 1442*b1) / 3 $$\nu^{11}$$ $$=$$ $$( - 1586 \beta_{11} - 2386 \beta_{10} - 1993 \beta_{9} + 3342 \beta_{8} + 800 \beta_{7} + 2949 \beta_{6} + 1600 \beta_{5} + 2386 \beta_{4} + 1586 \beta_{3} + 1671 \beta_{2} + 2386 \beta _1 - 2949 ) / 3$$ (-1586*b11 - 2386*b10 - 1993*b9 + 3342*b8 + 800*b7 + 2949*b6 + 1600*b5 + 2386*b4 + 1586*b3 + 1671*b2 + 2386*b1 - 2949) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$\beta_{6}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.583700 − 2.17840i −1.50511 − 0.403293i −0.673288 − 0.180407i −0.180407 + 0.673288i −0.403293 + 1.50511i 2.17840 + 0.583700i 0.583700 + 2.17840i −1.50511 + 0.403293i −0.673288 + 0.180407i −0.180407 − 0.673288i −0.403293 − 1.50511i 2.17840 − 0.583700i
0 −1.59470 0.675970i 0 1.50000 0.866025i 0 −1.80664 + 3.12920i 0 2.08613 + 2.15594i 0
97.2 0 −1.10182 1.33641i 0 1.50000 0.866025i 0 −0.495361 + 0.857990i 0 −0.571993 + 2.94497i 0
97.3 0 −0.492881 + 1.66044i 0 1.50000 0.866025i 0 −2.17731 + 3.77121i 0 −2.51414 1.63680i 0
97.4 0 0.492881 1.66044i 0 1.50000 0.866025i 0 2.17731 3.77121i 0 −2.51414 1.63680i 0
97.5 0 1.10182 + 1.33641i 0 1.50000 0.866025i 0 0.495361 0.857990i 0 −0.571993 + 2.94497i 0
97.6 0 1.59470 + 0.675970i 0 1.50000 0.866025i 0 1.80664 3.12920i 0 2.08613 + 2.15594i 0
481.1 0 −1.59470 + 0.675970i 0 1.50000 + 0.866025i 0 −1.80664 3.12920i 0 2.08613 2.15594i 0
481.2 0 −1.10182 + 1.33641i 0 1.50000 + 0.866025i 0 −0.495361 0.857990i 0 −0.571993 2.94497i 0
481.3 0 −0.492881 1.66044i 0 1.50000 + 0.866025i 0 −2.17731 3.77121i 0 −2.51414 + 1.63680i 0
481.4 0 0.492881 + 1.66044i 0 1.50000 + 0.866025i 0 2.17731 + 3.77121i 0 −2.51414 + 1.63680i 0
481.5 0 1.10182 1.33641i 0 1.50000 + 0.866025i 0 0.495361 + 0.857990i 0 −0.571993 2.94497i 0
481.6 0 1.59470 0.675970i 0 1.50000 + 0.866025i 0 1.80664 + 3.12920i 0 2.08613 2.15594i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 481.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.r.f yes 12
3.b odd 2 1 1728.2.r.e 12
4.b odd 2 1 inner 576.2.r.f yes 12
8.b even 2 1 576.2.r.e 12
8.d odd 2 1 576.2.r.e 12
9.c even 3 1 576.2.r.e 12
9.c even 3 1 5184.2.d.q 12
9.d odd 6 1 1728.2.r.f 12
9.d odd 6 1 5184.2.d.r 12
12.b even 2 1 1728.2.r.e 12
24.f even 2 1 1728.2.r.f 12
24.h odd 2 1 1728.2.r.f 12
36.f odd 6 1 576.2.r.e 12
36.f odd 6 1 5184.2.d.q 12
36.h even 6 1 1728.2.r.f 12
36.h even 6 1 5184.2.d.r 12
72.j odd 6 1 1728.2.r.e 12
72.j odd 6 1 5184.2.d.r 12
72.l even 6 1 1728.2.r.e 12
72.l even 6 1 5184.2.d.r 12
72.n even 6 1 inner 576.2.r.f yes 12
72.n even 6 1 5184.2.d.q 12
72.p odd 6 1 inner 576.2.r.f yes 12
72.p odd 6 1 5184.2.d.q 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.e 12 8.b even 2 1
576.2.r.e 12 8.d odd 2 1
576.2.r.e 12 9.c even 3 1
576.2.r.e 12 36.f odd 6 1
576.2.r.f yes 12 1.a even 1 1 trivial
576.2.r.f yes 12 4.b odd 2 1 inner
576.2.r.f yes 12 72.n even 6 1 inner
576.2.r.f yes 12 72.p odd 6 1 inner
1728.2.r.e 12 3.b odd 2 1
1728.2.r.e 12 12.b even 2 1
1728.2.r.e 12 72.j odd 6 1
1728.2.r.e 12 72.l even 6 1
1728.2.r.f 12 9.d odd 6 1
1728.2.r.f 12 24.f even 2 1
1728.2.r.f 12 24.h odd 2 1
1728.2.r.f 12 36.h even 6 1
5184.2.d.q 12 9.c even 3 1
5184.2.d.q 12 36.f odd 6 1
5184.2.d.q 12 72.n even 6 1
5184.2.d.q 12 72.p odd 6 1
5184.2.d.r 12 9.d odd 6 1
5184.2.d.r 12 36.h even 6 1
5184.2.d.r 12 72.j odd 6 1
5184.2.d.r 12 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3T_{5} + 3$$ acting on $$S_{2}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 2 T^{10} + 7 T^{8} + 12 T^{6} + \cdots + 729$$
$5$ $$(T^{2} - 3 T + 3)^{6}$$
$7$ $$T^{12} + 33 T^{10} + 810 T^{8} + \cdots + 59049$$
$11$ $$T^{12} - 39 T^{10} + 1350 T^{8} + \cdots + 6561$$
$13$ $$(T^{6} - 3 T^{5} - 24 T^{4} + 81 T^{3} + \cdots + 243)^{2}$$
$17$ $$(T^{3} - 24 T + 36)^{4}$$
$19$ $$(T^{2} + 4)^{6}$$
$23$ $$T^{12} + 117 T^{10} + \cdots + 2492305929$$
$29$ $$(T^{6} + 9 T^{5} - 36 T^{4} - 567 T^{3} + \cdots + 93987)^{2}$$
$31$ $$T^{12} + 117 T^{10} + \cdots + 95004009$$
$37$ $$(T^{6} + 60 T^{4} + 1008 T^{2} + \cdots + 3888)^{2}$$
$41$ $$(T^{6} - 9 T^{5} + 78 T^{4} - 45 T^{3} + \cdots + 81)^{2}$$
$43$ $$T^{12} - 123 T^{10} + \cdots + 47458321$$
$47$ $$T^{12} + 297 T^{10} + \cdots + 514609673769$$
$53$ $$(T^{6} + 180 T^{4} + 1728 T^{2} + \cdots + 432)^{2}$$
$59$ $$T^{12} - 147 T^{10} + 20898 T^{8} + \cdots + 531441$$
$61$ $$(T^{6} + 9 T^{5} - 36 T^{4} - 567 T^{3} + \cdots + 4563)^{2}$$
$67$ $$T^{12} - 231 T^{10} + \cdots + 352275361$$
$71$ $$(T^{6} - 288 T^{4} + 19872 T^{2} + \cdots - 124848)^{2}$$
$73$ $$(T^{3} - 84 T - 164)^{4}$$
$79$ $$T^{12} + 93 T^{10} + 7758 T^{8} + \cdots + 4782969$$
$83$ $$T^{12} - 171 T^{10} + \cdots + 43046721$$
$89$ $$(T^{3} + 12 T^{2} - 36 T - 108)^{4}$$
$97$ $$(T^{6} + 3 T^{5} + 222 T^{4} + \cdots + 1408969)^{2}$$