# Properties

 Label 144.3.m.b Level $144$ Weight $3$ Character orbit 144.m Analytic conductor $3.924$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536$$ x^16 - 6*x^14 + 10*x^12 + 88*x^10 - 752*x^8 + 1408*x^6 + 2560*x^4 - 24576*x^2 + 65536 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + ( - \beta_{2} - 1) q^{4} + \beta_{8} q^{5} + (\beta_{12} + \beta_{11}) q^{7} + ( - \beta_{13} + \beta_{7} + \beta_{4}) q^{8}+O(q^{10})$$ q - b4 * q^2 + (-b2 - 1) * q^4 + b8 * q^5 + (b12 + b11) * q^7 + (-b13 + b7 + b4) * q^8 $$q - \beta_{4} q^{2} + ( - \beta_{2} - 1) q^{4} + \beta_{8} q^{5} + (\beta_{12} + \beta_{11}) q^{7} + ( - \beta_{13} + \beta_{7} + \beta_{4}) q^{8} + ( - \beta_{15} + \beta_{11} - \beta_{6} + \beta_{3}) q^{10} + (\beta_{13} + \beta_{9} + \beta_{5} - \beta_{4} + 2 \beta_1) q^{11} + ( - \beta_{15} - \beta_{14} - \beta_{12} + \beta_{11} - \beta_{6} + \beta_{3} - \beta_{2} - 1) q^{13} + ( - \beta_{13} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_1) q^{14} + ( - \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{6} - \beta_{3} + \beta_{2} + 3) q^{16} + (2 \beta_{13} + \beta_{10} + \beta_{8} - \beta_{7} + \beta_{5} - 3 \beta_1) q^{17} + ( - \beta_{15} + 2 \beta_{14} - \beta_{12} - \beta_{11} - 2 \beta_{3} - 3 \beta_{2} - 4) q^{19} + ( - \beta_{13} - \beta_{9} - 3 \beta_{8} + \beta_{7} - 2 \beta_{5} - \beta_{4} - \beta_1) q^{20} + (2 \beta_{15} + \beta_{14} + 2 \beta_{12} - \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + 6) q^{22} + (2 \beta_{13} + \beta_{10} + 2 \beta_{8} - 2 \beta_{7} - \beta_{5} + 4 \beta_{4} + 3 \beta_1) q^{23} + (2 \beta_{15} - 2 \beta_{14} + 4 \beta_{12} + 7 \beta_{3} - 2 \beta_{2} + 2) q^{25} + ( - 2 \beta_{13} + 2 \beta_{10} - 4 \beta_{8} + 2 \beta_{7} + \beta_{4} - \beta_1) q^{26} + ( - 2 \beta_{15} - \beta_{12} - 2 \beta_{11} + 2 \beta_{6} + 9 \beta_{3} + \beta_{2} + 1) q^{28} + (2 \beta_{13} - 2 \beta_{10} + 2 \beta_{9} - \beta_{7} + 6 \beta_{4} + 6 \beta_1) q^{29} + (\beta_{15} + 2 \beta_{14} - 4 \beta_{12} + 2 \beta_{6} - 12 \beta_{3} + 3 \beta_{2}) q^{31} + ( - \beta_{13} - 2 \beta_{10} - \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + 2 \beta_{5} + \cdots - 3 \beta_1) q^{32}+ \cdots + (6 \beta_{13} + 14 \beta_{10} + 10 \beta_{9} - 6 \beta_{8} - 2 \beta_{7} + 6 \beta_{5} + \cdots + 6 \beta_1) q^{98}+O(q^{100})$$ q - b4 * q^2 + (-b2 - 1) * q^4 + b8 * q^5 + (b12 + b11) * q^7 + (-b13 + b7 + b4) * q^8 + (-b15 + b11 - b6 + b3) * q^10 + (b13 + b9 + b5 - b4 + 2*b1) * q^11 + (-b15 - b14 - b12 + b11 - b6 + b3 - b2 - 1) * q^13 + (-b13 - b10 - b9 - b8 - b7 - b5 - b1) * q^14 + (-b14 + b12 + 2*b11 + b6 - b3 + b2 + 3) * q^16 + (2*b13 + b10 + b8 - b7 + b5 - 3*b1) * q^17 + (-b15 + 2*b14 - b12 - b11 - 2*b3 - 3*b2 - 4) * q^19 + (-b13 - b9 - 3*b8 + b7 - 2*b5 - b4 - b1) * q^20 + (2*b15 + b14 + 2*b12 - b6 - 2*b3 - 2*b2 + 6) * q^22 + (2*b13 + b10 + 2*b8 - 2*b7 - b5 + 4*b4 + 3*b1) * q^23 + (2*b15 - 2*b14 + 4*b12 + 7*b3 - 2*b2 + 2) * q^25 + (-2*b13 + 2*b10 - 4*b8 + 2*b7 + b4 - b1) * q^26 + (-2*b15 - b12 - 2*b11 + 2*b6 + 9*b3 + b2 + 1) * q^28 + (2*b13 - 2*b10 + 2*b9 - b7 + 6*b4 + 6*b1) * q^29 + (b15 + 2*b14 - 4*b12 + 2*b6 - 12*b3 + 3*b2) * q^31 + (-b13 - 2*b10 - b9 - 3*b8 - 3*b7 + 2*b5 - 3*b4 - 3*b1) * q^32 + (b14 - 4*b12 - 2*b11 - 3*b6 - 14*b3 - 4) * q^34 + (b13 - b10 - b9 + 4*b8 + 4*b5 + 5*b4 - 13*b1) * q^35 + (-b15 - 3*b14 + b12 - b11 - b6 + 7*b3 + 7*b2 + 9) * q^37 + (-2*b13 - 4*b10 + 6*b7 - 2*b5 + 2*b1) * q^38 + (2*b15 - 2*b14 - 2*b12 - 2*b11 + 4*b6 + 8*b3 - 16) * q^40 + (5*b10 + 2*b9 + b8 + b7 + 3*b5 - 2*b4 + 13*b1) * q^41 + (b15 + 4*b14 + 3*b12 - b11 + 2*b6 - 2*b3 + 7*b2) * q^43 + (-2*b13 + 2*b10 + 4*b8 - 2*b7 - 6*b5 - 2*b4 + 6*b1) * q^44 + (-2*b15 + 2*b14 - 2*b11 - 4*b6 + 6*b3 + 4*b2 - 16) * q^46 + (3*b10 + 2*b9 + 2*b8 + 2*b7 - 5*b5 - 14*b4 + b1) * q^47 + (-4*b14 - 10*b12 - 2*b11 - 2*b6 + 4*b2 + 7) * q^49 + (-2*b13 + 6*b10 - 2*b9 + 6*b8 - 2*b7 + 2*b5 + 4*b4 - 5*b1) * q^50 + (4*b15 - 2*b14 + b12 + 6*b6 - 23*b3 + b2 - 11) * q^52 + (2*b13 - 8*b10 - 2*b9 - b8 - 2*b5 - 22*b4 - 8*b1) * q^53 + (6*b14 + 8*b12 - 4*b11 + 2*b6 - 4*b2 - 16) * q^55 + (3*b13 - 4*b10 - b9 + b8 + 5*b7 + 8*b5 - 7*b4 - 13*b1) * q^56 + (3*b15 + 3*b14 + 4*b12 - b11 - 2*b6 + 27*b3 + 4*b2 + 20) * q^58 + (-2*b13 - 4*b10 - 2*b9 - 4*b7 + 6*b5 - 6*b4 + 12*b1) * q^59 + (5*b15 - 3*b14 - 3*b12 - 5*b11 + b6 + 5*b3 - 11*b2 - 1) * q^61 + (3*b13 - 7*b10 + 5*b9 - 3*b8 - 5*b7 - b5 - 2*b4 + 9*b1) * q^62 + (-2*b15 + 2*b14 - 6*b11 + 4*b6 - 10*b3 - 2*b2 + 30) * q^64 + (-6*b13 + 9*b10 - b8 + b7 + b5 + 16*b4 - 27*b1) * q^65 + (4*b15 - 4*b12 + 4*b11 + 4*b6 - 20*b3 - 4*b2 - 16) * q^67 + (2*b13 + 6*b10 + 4*b9 + 2*b7 - 6*b5 + 6*b4 + 22*b1) * q^68 + (-4*b15 - b14 - 10*b12 + 2*b11 - 5*b6 - 28*b3 + 6*b2 + 42) * q^70 + (-4*b13 + 2*b10 - 8*b8 + 8*b7 - 10*b5 + 24*b4 + 22*b1) * q^71 + (-10*b15 - 2*b14 + 4*b12 - 4*b6 + 12*b3 + 2*b2 - 2) * q^73 + (8*b13 + 8*b10 - 2*b9 + 2*b8 - 4*b7 + 6*b5 - 7*b4 - 5*b1) * q^74 + (4*b15 - 6*b14 + 2*b12 + 8*b11 + 2*b6 + 50*b3 + 20) * q^76 + (-6*b13 - 10*b10 - 6*b9 + 2*b7 - 8*b5 + 30*b4 + 6*b1) * q^77 + (-5*b15 + 4*b14 - 8*b12 - 4*b6 + 8*b3 - 3*b2 - 8) * q^79 + (2*b13 + 4*b9 + 4*b8 - 2*b7 + 12*b5 + 18*b4 - 12*b1) * q^80 + (2*b15 + b14 + 16*b12 + 4*b11 - b6 - 44*b3 - 4*b2 - 16) * q^82 + (-3*b13 - 5*b10 + 3*b9 - 16*b8 + 8*b5 + 9*b4 - 37*b1) * q^83 + (6*b15 - 2*b14 + 10*b12 + 6*b11 - 2*b6 + 12*b3 - 2*b2 + 12) * q^85 + (8*b13 - 10*b10 - 2*b9 + 6*b8 - 8*b7 - 4*b5 - 4*b4) * q^86 + (-8*b15 + 2*b14 + 2*b12 + 4*b11 - 2*b6 + 30*b3 - 2*b2 - 62) * q^88 + (8*b10 - 8*b9 - 2*b8 - 2*b7 - 24*b4 + 16*b1) * q^89 + (-5*b15 + b12 + 5*b11 - 2*b6 - 18*b3 + 5*b2 + 16) * q^91 + (6*b13 + 4*b10 - 2*b9 + 2*b8 + 2*b7 - 8*b5 + 14*b4 + 2*b1) * q^92 + (2*b15 - 4*b12 + 6*b11 - 2*b6 + 18*b3 - 16*b2 - 76) * q^94 + (3*b10 - 6*b9 - 10*b8 - 10*b7 - 13*b5 - 38*b4 - 7*b1) * q^95 + (-4*b14 + 4*b12 + 12*b11) * q^97 + (6*b13 + 14*b10 + 10*b9 - 6*b8 - 2*b7 + 6*b5 - b4 + 6*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 12 q^{4}+O(q^{10})$$ 16 * q - 12 * q^4 $$16 q - 12 q^{4} + 16 q^{10} + 32 q^{16} - 32 q^{19} + 104 q^{22} - 24 q^{34} + 96 q^{37} - 312 q^{40} - 32 q^{43} - 224 q^{46} + 112 q^{49} - 264 q^{52} - 256 q^{55} + 312 q^{58} - 32 q^{61} + 456 q^{64} - 256 q^{67} + 744 q^{70} + 264 q^{76} - 280 q^{82} + 160 q^{85} - 912 q^{88} + 288 q^{91} - 1104 q^{94}+O(q^{100})$$ 16 * q - 12 * q^4 + 16 * q^10 + 32 * q^16 - 32 * q^19 + 104 * q^22 - 24 * q^34 + 96 * q^37 - 312 * q^40 - 32 * q^43 - 224 * q^46 + 112 * q^49 - 264 * q^52 - 256 * q^55 + 312 * q^58 - 32 * q^61 + 456 * q^64 - 256 * q^67 + 744 * q^70 + 264 * q^76 - 280 * q^82 + 160 * q^85 - 912 * q^88 + 288 * q^91 - 1104 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ v^2 - 1 $$\beta_{3}$$ $$=$$ $$( -5\nu^{14} + 14\nu^{12} + 110\nu^{10} - 472\nu^{8} + 944\nu^{6} + 5504\nu^{4} - 24064\nu^{2} - 20480 ) / 12288$$ (-5*v^14 + 14*v^12 + 110*v^10 - 472*v^8 + 944*v^6 + 5504*v^4 - 24064*v^2 - 20480) / 12288 $$\beta_{4}$$ $$=$$ $$( 5\nu^{15} - 14\nu^{13} - 110\nu^{11} + 472\nu^{9} - 944\nu^{7} - 5504\nu^{5} + 24064\nu^{3} + 20480\nu ) / 12288$$ (5*v^15 - 14*v^13 - 110*v^11 + 472*v^9 - 944*v^7 - 5504*v^5 + 24064*v^3 + 20480*v) / 12288 $$\beta_{5}$$ $$=$$ $$( -\nu^{15} + 46\nu^{13} - 122\nu^{11} - 968\nu^{9} + 4528\nu^{7} - 8960\nu^{5} - 46592\nu^{3} + 229376\nu ) / 12288$$ (-v^15 + 46*v^13 - 122*v^11 - 968*v^9 + 4528*v^7 - 8960*v^5 - 46592*v^3 + 229376*v) / 12288 $$\beta_{6}$$ $$=$$ $$( \nu^{14} - 6\nu^{12} + 10\nu^{10} + 88\nu^{8} - 752\nu^{6} + 1408\nu^{4} + 2560\nu^{2} - 22528 ) / 2048$$ (v^14 - 6*v^12 + 10*v^10 + 88*v^8 - 752*v^6 + 1408*v^4 + 2560*v^2 - 22528) / 2048 $$\beta_{7}$$ $$=$$ $$( -17\nu^{15} - 34\nu^{13} + 326\nu^{11} - 424\nu^{9} - 1360\nu^{7} + 26624\nu^{5} + 9728\nu^{3} - 167936\nu ) / 24576$$ (-17*v^15 - 34*v^13 + 326*v^11 - 424*v^9 - 1360*v^7 + 26624*v^5 + 9728*v^3 - 167936*v) / 24576 $$\beta_{8}$$ $$=$$ $$( -11\nu^{15} + 14\nu^{13} + 170\nu^{11} - 784\nu^{9} + 1328\nu^{7} + 9536\nu^{5} - 30208\nu^{3} - 26624\nu ) / 12288$$ (-11*v^15 + 14*v^13 + 170*v^11 - 784*v^9 + 1328*v^7 + 9536*v^5 - 30208*v^3 - 26624*v) / 12288 $$\beta_{9}$$ $$=$$ $$( -11\nu^{15} + 14\nu^{13} + 170\nu^{11} - 784\nu^{9} + 1328\nu^{7} + 9536\nu^{5} - 17920\nu^{3} - 38912\nu ) / 12288$$ (-11*v^15 + 14*v^13 + 170*v^11 - 784*v^9 + 1328*v^7 + 9536*v^5 - 17920*v^3 - 38912*v) / 12288 $$\beta_{10}$$ $$=$$ $$( \nu^{15} - 16\nu^{13} + 38\nu^{11} + 308\nu^{9} - 1696\nu^{7} + 3296\nu^{5} + 13568\nu^{3} - 72704\nu ) / 3072$$ (v^15 - 16*v^13 + 38*v^11 + 308*v^9 - 1696*v^7 + 3296*v^5 + 13568*v^3 - 72704*v) / 3072 $$\beta_{11}$$ $$=$$ $$( -5\nu^{14} + 38\nu^{12} - 34\nu^{10} - 1000\nu^{8} + 4592\nu^{6} - 1792\nu^{4} - 45568\nu^{2} + 182272 ) / 6144$$ (-5*v^14 + 38*v^12 - 34*v^10 - 1000*v^8 + 4592*v^6 - 1792*v^4 - 45568*v^2 + 182272) / 6144 $$\beta_{12}$$ $$=$$ $$( - 11 \nu^{14} + 146 \nu^{12} - 142 \nu^{10} - 2344 \nu^{8} + 11600 \nu^{6} - 16768 \nu^{4} - 119296 \nu^{2} + 348160 ) / 12288$$ (-11*v^14 + 146*v^12 - 142*v^10 - 2344*v^8 + 11600*v^6 - 16768*v^4 - 119296*v^2 + 348160) / 12288 $$\beta_{13}$$ $$=$$ $$( -13\nu^{15} + 86\nu^{13} + 14\nu^{11} - 1704\nu^{9} + 7280\nu^{7} - 2304\nu^{5} - 76288\nu^{3} + 176128\nu ) / 8192$$ (-13*v^15 + 86*v^13 + 14*v^11 - 1704*v^9 + 7280*v^7 - 2304*v^5 - 76288*v^3 + 176128*v) / 8192 $$\beta_{14}$$ $$=$$ $$( -5\nu^{14} + 62\nu^{12} - 82\nu^{10} - 1336\nu^{8} + 6128\nu^{6} - 8320\nu^{4} - 62464\nu^{2} + 246784 ) / 3072$$ (-5*v^14 + 62*v^12 - 82*v^10 - 1336*v^8 + 6128*v^6 - 8320*v^4 - 62464*v^2 + 246784) / 3072 $$\beta_{15}$$ $$=$$ $$( 47\nu^{14} - 122\nu^{12} - 554\nu^{10} + 4072\nu^{8} - 10640\nu^{6} - 33152\nu^{4} + 162304\nu^{2} - 77824 ) / 12288$$ (47*v^14 - 122*v^12 - 554*v^10 + 4072*v^8 - 10640*v^6 - 33152*v^4 + 162304*v^2 - 77824) / 12288
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ b2 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{8} + \beta_1$$ b9 - b8 + b1 $$\nu^{4}$$ $$=$$ $$-\beta_{14} + \beta_{12} + 2\beta_{11} + \beta_{6} - \beta_{3} + \beta_{2} + 3$$ -b14 + b12 + 2*b11 + b6 - b3 + b2 + 3 $$\nu^{5}$$ $$=$$ $$\beta_{13} + 2\beta_{10} - \beta_{9} - 3\beta_{8} + 3\beta_{7} + 2\beta_{5} - \beta_{4} + \beta_1$$ b13 + 2*b10 - b9 - 3*b8 + 3*b7 + 2*b5 - b4 + b1 $$\nu^{6}$$ $$=$$ $$2\beta_{15} - 2\beta_{14} + 6\beta_{11} - 4\beta_{6} + 10\beta_{3} + 2\beta_{2} - 30$$ 2*b15 - 2*b14 + 6*b11 - 4*b6 + 10*b3 + 2*b2 - 30 $$\nu^{7}$$ $$=$$ $$-2\beta_{13} - 12\beta_{10} - 4\beta_{9} - 12\beta_{8} + 10\beta_{7} - 8\beta_{5} - 18\beta_{4} - 32\beta_1$$ -2*b13 - 12*b10 - 4*b9 - 12*b8 + 10*b7 - 8*b5 - 18*b4 - 32*b1 $$\nu^{8}$$ $$=$$ $$4\beta_{15} - 14\beta_{14} + 14\beta_{12} + 16\beta_{11} - 14\beta_{6} + 14\beta_{3} - 38\beta_{2} + 110$$ 4*b15 - 14*b14 + 14*b12 + 16*b11 - 14*b6 + 14*b3 - 38*b2 + 110 $$\nu^{9}$$ $$=$$ $$10\beta_{13} - 20\beta_{10} - 54\beta_{9} + 14\beta_{8} + 14\beta_{7} - 44\beta_{5} - 18\beta_{4} + 118\beta_1$$ 10*b13 - 20*b10 - 54*b9 + 14*b8 + 14*b7 - 44*b5 - 18*b4 + 118*b1 $$\nu^{10}$$ $$=$$ $$36\beta_{15} + 24\beta_{14} - 36\beta_{12} - 44\beta_{11} - 68\beta_{6} + 328\beta_{3} + 64\beta_{2} + 488$$ 36*b15 + 24*b14 - 36*b12 - 44*b11 - 68*b6 + 328*b3 + 64*b2 + 488 $$\nu^{11}$$ $$=$$ $$-72\beta_{13} - 176\beta_{10} + 108\beta_{9} - 92\beta_{8} - 88\beta_{7} - 168\beta_{5} - 288\beta_{4} + 540\beta_1$$ -72*b13 - 176*b10 + 108*b9 - 92*b8 - 88*b7 - 168*b5 - 288*b4 + 540*b1 $$\nu^{12}$$ $$=$$ $$-164\beta_{14} + 396\beta_{12} + 40\beta_{11} + 196\beta_{6} - 60\beta_{3} + 444\beta_{2} + 3268$$ -164*b14 + 396*b12 + 40*b11 + 196*b6 - 60*b3 + 444*b2 + 3268 $$\nu^{13}$$ $$=$$ $$396 \beta_{13} + 680 \beta_{10} + 404 \beta_{9} - 484 \beta_{8} - 316 \beta_{7} + 104 \beta_{5} + 308 \beta_{4} + 3196 \beta_1$$ 396*b13 + 680*b10 + 404*b9 - 484*b8 - 316*b7 + 104*b5 + 308*b4 + 3196*b1 $$\nu^{14}$$ $$=$$ $$792\beta_{15} - 88\beta_{14} + 96\beta_{12} + 968\beta_{11} + 720\beta_{6} + 4056\beta_{3} + 2904\beta_{2} - 1768$$ 792*b15 - 88*b14 + 96*b12 + 968*b11 + 720*b6 + 4056*b3 + 2904*b2 - 1768 $$\nu^{15}$$ $$=$$ $$- 696 \beta_{13} - 144 \beta_{10} + 1936 \beta_{9} - 5456 \beta_{8} + 1048 \beta_{7} + 1440 \beta_{5} - 5816 \beta_{4} - 4160 \beta_1$$ -696*b13 - 144*b10 + 1936*b9 - 5456*b8 + 1048*b7 + 1440*b5 - 5816*b4 - 4160*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.136762 − 1.99532i 1.64663 − 1.13516i −1.66730 − 1.10459i 1.99750 − 0.0999235i −1.99750 + 0.0999235i 1.66730 + 1.10459i −1.64663 + 1.13516i −0.136762 + 1.99532i 0.136762 + 1.99532i 1.64663 + 1.13516i −1.66730 + 1.10459i 1.99750 + 0.0999235i −1.99750 − 0.0999235i 1.66730 − 1.10459i −1.64663 − 1.13516i −0.136762 − 1.99532i
−1.99532 0.136762i 0 3.96259 + 0.545766i −0.227650 + 0.227650i 0 3.90219 −7.83199 1.63091i 0 0.485368 0.423100i
19.2 −1.13516 1.64663i 0 −1.42280 + 3.73840i 2.41234 2.41234i 0 −11.8718 7.77089 1.90087i 0 −6.71063 1.23383i
19.3 −1.10459 + 1.66730i 0 −1.55976 3.68336i −4.23991 + 4.23991i 0 −0.262225 7.86415 + 1.46802i 0 −2.38583 11.7525i
19.4 −0.0999235 1.99750i 0 −3.98003 + 0.399195i −6.01265 + 6.01265i 0 8.23187 1.19509 + 7.91023i 0 12.6111 + 11.4095i
19.5 0.0999235 + 1.99750i 0 −3.98003 + 0.399195i 6.01265 6.01265i 0 8.23187 −1.19509 7.91023i 0 12.6111 + 11.4095i
19.6 1.10459 1.66730i 0 −1.55976 3.68336i 4.23991 4.23991i 0 −0.262225 −7.86415 1.46802i 0 −2.38583 11.7525i
19.7 1.13516 + 1.64663i 0 −1.42280 + 3.73840i −2.41234 + 2.41234i 0 −11.8718 −7.77089 + 1.90087i 0 −6.71063 1.23383i
19.8 1.99532 + 0.136762i 0 3.96259 + 0.545766i 0.227650 0.227650i 0 3.90219 7.83199 + 1.63091i 0 0.485368 0.423100i
91.1 −1.99532 + 0.136762i 0 3.96259 0.545766i −0.227650 0.227650i 0 3.90219 −7.83199 + 1.63091i 0 0.485368 + 0.423100i
91.2 −1.13516 + 1.64663i 0 −1.42280 3.73840i 2.41234 + 2.41234i 0 −11.8718 7.77089 + 1.90087i 0 −6.71063 + 1.23383i
91.3 −1.10459 1.66730i 0 −1.55976 + 3.68336i −4.23991 4.23991i 0 −0.262225 7.86415 1.46802i 0 −2.38583 + 11.7525i
91.4 −0.0999235 + 1.99750i 0 −3.98003 0.399195i −6.01265 6.01265i 0 8.23187 1.19509 7.91023i 0 12.6111 11.4095i
91.5 0.0999235 1.99750i 0 −3.98003 0.399195i 6.01265 + 6.01265i 0 8.23187 −1.19509 + 7.91023i 0 12.6111 11.4095i
91.6 1.10459 + 1.66730i 0 −1.55976 + 3.68336i 4.23991 + 4.23991i 0 −0.262225 −7.86415 + 1.46802i 0 −2.38583 + 11.7525i
91.7 1.13516 1.64663i 0 −1.42280 3.73840i −2.41234 2.41234i 0 −11.8718 −7.77089 1.90087i 0 −6.71063 + 1.23383i
91.8 1.99532 0.136762i 0 3.96259 0.545766i 0.227650 + 0.227650i 0 3.90219 7.83199 1.63091i 0 0.485368 + 0.423100i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.m.b 16
3.b odd 2 1 inner 144.3.m.b 16
4.b odd 2 1 576.3.m.b 16
8.b even 2 1 1152.3.m.e 16
8.d odd 2 1 1152.3.m.d 16
12.b even 2 1 576.3.m.b 16
16.e even 4 1 576.3.m.b 16
16.e even 4 1 1152.3.m.d 16
16.f odd 4 1 inner 144.3.m.b 16
16.f odd 4 1 1152.3.m.e 16
24.f even 2 1 1152.3.m.d 16
24.h odd 2 1 1152.3.m.e 16
48.i odd 4 1 576.3.m.b 16
48.i odd 4 1 1152.3.m.d 16
48.k even 4 1 inner 144.3.m.b 16
48.k even 4 1 1152.3.m.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.m.b 16 1.a even 1 1 trivial
144.3.m.b 16 3.b odd 2 1 inner
144.3.m.b 16 16.f odd 4 1 inner
144.3.m.b 16 48.k even 4 1 inner
576.3.m.b 16 4.b odd 2 1
576.3.m.b 16 12.b even 2 1
576.3.m.b 16 16.e even 4 1
576.3.m.b 16 48.i odd 4 1
1152.3.m.d 16 8.d odd 2 1
1152.3.m.d 16 16.e even 4 1
1152.3.m.d 16 24.f even 2 1
1152.3.m.d 16 48.i odd 4 1
1152.3.m.e 16 8.b even 2 1
1152.3.m.e 16 16.f odd 4 1
1152.3.m.e 16 24.h odd 2 1
1152.3.m.e 16 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} + 6656T_{5}^{12} + 7641216T_{5}^{8} + 915505152T_{5}^{4} + 9834496$$ acting on $$S_{3}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 6 T^{14} + 10 T^{12} + \cdots + 65536$$
$3$ $$T^{16}$$
$5$ $$T^{16} + 6656 T^{12} + \cdots + 9834496$$
$7$ $$(T^{4} - 112 T^{2} + 352 T + 100)^{4}$$
$11$ $$T^{16} + 110592 T^{12} + \cdots + 12\!\cdots\!00$$
$13$ $$(T^{8} - 4352 T^{5} + 144456 T^{4} + \cdots + 35760400)^{2}$$
$17$ $$(T^{8} - 1104 T^{6} + 382592 T^{4} + \cdots + 472105984)^{2}$$
$19$ $$(T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 6629867776)^{2}$$
$23$ $$(T^{8} - 1632 T^{6} + 502272 T^{4} + \cdots + 467943424)^{2}$$
$29$ $$T^{16} + 3912960 T^{12} + \cdots + 28\!\cdots\!00$$
$31$ $$(T^{8} + 4032 T^{6} + \cdots + 11588953104)^{2}$$
$37$ $$(T^{8} - 48 T^{7} + \cdots + 137744899600)^{2}$$
$41$ $$(T^{8} + 7248 T^{6} + \cdots + 198844646400)^{2}$$
$43$ $$(T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 21036601600)^{2}$$
$47$ $$(T^{8} + 11360 T^{6} + \cdots + 15982596734976)^{2}$$
$53$ $$T^{16} + 58906368 T^{12} + \cdots + 18\!\cdots\!00$$
$59$ $$T^{16} + 140066816 T^{12} + \cdots + 11\!\cdots\!00$$
$61$ $$(T^{8} + 16 T^{7} + \cdots + 51506974970896)^{2}$$
$67$ $$(T^{8} + 128 T^{7} + \cdots + 7615833702400)^{2}$$
$71$ $$(T^{8} - 39552 T^{6} + \cdots + 51\!\cdots\!00)^{2}$$
$73$ $$(T^{8} + 20096 T^{6} + \cdots + 534697177190400)^{2}$$
$79$ $$(T^{8} + 14624 T^{6} + \cdots + 56939504400)^{2}$$
$83$ $$T^{16} + 1008676864 T^{12} + \cdots + 60\!\cdots\!36$$
$89$ $$(T^{8} + 45760 T^{6} + \cdots + 236165588582400)^{2}$$
$97$ $$(T^{4} - 12384 T^{2} - 77824 T + 29866240)^{4}$$