# Properties

 Label 1024.2.g.h Level $1024$ Weight $2$ Character orbit 1024.g Analytic conductor $8.177$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.g (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.17668116698$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8x^{14} + 32x^{12} - 64x^{10} + 127x^{8} - 576x^{6} + 2592x^{4} - 5832x^{2} + 6561$$ x^16 - 8*x^14 + 32*x^12 - 64*x^10 + 127*x^8 - 576*x^6 + 2592*x^4 - 5832*x^2 + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{13} q^{3} + ( - \beta_{8} - \beta_{5} + \beta_{3} + 1) q^{5} + ( - \beta_{13} - \beta_{9} - \beta_{4}) q^{7} + ( - \beta_{8} - \beta_{6} - 3 \beta_{5} + 2) q^{9}+O(q^{10})$$ q + b13 * q^3 + (-b8 - b5 + b3 + 1) * q^5 + (-b13 - b9 - b4) * q^7 + (-b8 - b6 - 3*b5 + 2) * q^9 $$q + \beta_{13} q^{3} + ( - \beta_{8} - \beta_{5} + \beta_{3} + 1) q^{5} + ( - \beta_{13} - \beta_{9} - \beta_{4}) q^{7} + ( - \beta_{8} - \beta_{6} - 3 \beta_{5} + 2) q^{9} + (\beta_{15} + \beta_{9} - \beta_1) q^{11} + ( - \beta_{12} - \beta_{10} + \beta_{8} - 2 \beta_{7} + 2 \beta_{5} + \beta_{3} + 1) q^{13} + ( - \beta_{14} - 2 \beta_{11} + 3 \beta_{9} - \beta_{4} + \beta_{2} - 3 \beta_1) q^{15} + ( - \beta_{12} + \beta_{8} + \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{17} + (\beta_{15} - \beta_{14} + \beta_{11} - \beta_{9} + \beta_{4} + \beta_{2} + \beta_1) q^{19} + ( - 3 \beta_{7} + 3 \beta_{5} + \beta_{3} - 1) q^{21} + (\beta_{15} + 2 \beta_{14} + 3 \beta_{11} - 2 \beta_{9} + \beta_{2} + 4 \beta_1) q^{23} + (\beta_{12} + \beta_{10} - \beta_{8} + 2 \beta_{7} + \beta_{6} + 4 \beta_{3} + 2) q^{25} + (\beta_{13} - 3 \beta_{11} + 2 \beta_{9} + \beta_{2} - \beta_1) q^{27} + (\beta_{10} + \beta_{8} + \beta_{6} + 2 \beta_{5} + \beta_{3} - 2) q^{29} + (\beta_{14} + \beta_{11} + 2 \beta_1) q^{31} + (\beta_{12} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} + 4) q^{33} + (\beta_{15} + 4 \beta_{14} - 2 \beta_{13} + 2 \beta_{11} - 6 \beta_{9} - \beta_{4} + 4 \beta_1) q^{35} + (\beta_{12} + \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 2) q^{37} + ( - \beta_{15} - 3 \beta_{14} + 2 \beta_{13} - 2 \beta_{9} + 2 \beta_{4} + \beta_{2}) q^{39} + (2 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 1) q^{41} + ( - \beta_{15} - 2 \beta_{14} - \beta_{11} + 3 \beta_{9} - 4 \beta_1) q^{43} + (3 \beta_{12} + 2 \beta_{10} - 2 \beta_{8} - 6 \beta_{7} + \beta_{5} + 7 \beta_{3} + 2) q^{45} + (\beta_{15} - \beta_{13} - 2 \beta_{11} + 2 \beta_{9} + \beta_{4} - \beta_{2} - 2 \beta_1) q^{47} + (\beta_{12} - \beta_{10} - \beta_{8} + \beta_{6} - 2 \beta_{5} + 3 \beta_{3} - 1) q^{49} + ( - 2 \beta_{15} - \beta_{14} + 2 \beta_{11} - 2 \beta_{9} - 2 \beta_{4} - 2 \beta_{2} + 3 \beta_1) q^{51} + (\beta_{10} - 3 \beta_{7} + 3 \beta_{5} + 4 \beta_{3} - 3) q^{53} + ( - 2 \beta_{15} + \beta_{13} - \beta_{11} - \beta_{4} - 2 \beta_{2}) q^{55} + ( - 3 \beta_{12} - 3 \beta_{10} + 2 \beta_{8} + 5 \beta_{7} - 2 \beta_{6} - \beta_{3} + \cdots + 5) q^{57}+ \cdots + ( - \beta_{15} + 3 \beta_{13} - \beta_{11} - \beta_{9} + \beta_{4}) q^{99}+O(q^{100})$$ q + b13 * q^3 + (-b8 - b5 + b3 + 1) * q^5 + (-b13 - b9 - b4) * q^7 + (-b8 - b6 - 3*b5 + 2) * q^9 + (b15 + b9 - b1) * q^11 + (-b12 - b10 + b8 - 2*b7 + 2*b5 + b3 + 1) * q^13 + (-b14 - 2*b11 + 3*b9 - b4 + b2 - 3*b1) * q^15 + (-b12 + b8 + b7 - 2*b5 + 2*b3) * q^17 + (b15 - b14 + b11 - b9 + b4 + b2 + b1) * q^19 + (-3*b7 + 3*b5 + b3 - 1) * q^21 + (b15 + 2*b14 + 3*b11 - 2*b9 + b2 + 4*b1) * q^23 + (b12 + b10 - b8 + 2*b7 + b6 + 4*b3 + 2) * q^25 + (b13 - 3*b11 + 2*b9 + b2 - b1) * q^27 + (b10 + b8 + b6 + 2*b5 + b3 - 2) * q^29 + (b14 + b11 + 2*b1) * q^31 + (b12 + b8 - b7 + b5 - b3 + 4) * q^33 + (b15 + 4*b14 - 2*b13 + 2*b11 - 6*b9 - b4 + 4*b1) * q^35 + (b12 + b8 - 2*b7 - b6 - b5 + b3 + 2) * q^37 + (-b15 - 3*b14 + 2*b13 - 2*b9 + 2*b4 + b2) * q^39 + (2*b8 - 3*b7 + 2*b6 - 2*b5 - 1) * q^41 + (-b15 - 2*b14 - b11 + 3*b9 - 4*b1) * q^43 + (3*b12 + 2*b10 - 2*b8 - 6*b7 + b5 + 7*b3 + 2) * q^45 + (b15 - b13 - 2*b11 + 2*b9 + b4 - b2 - 2*b1) * q^47 + (b12 - b10 - b8 + b6 - 2*b5 + 3*b3 - 1) * q^49 + (-2*b15 - b14 + 2*b11 - 2*b9 - 2*b4 - 2*b2 + 3*b1) * q^51 + (b10 - 3*b7 + 3*b5 + 4*b3 - 3) * q^53 + (-2*b15 + b13 - b11 - b4 - 2*b2) * q^55 + (-3*b12 - 3*b10 + 2*b8 + 5*b7 - 2*b6 - b3 + 5) * q^57 + (-b13 + b11 + b9 + b4 - b2 + 2*b1) * q^59 + (-2*b10 - 2*b8 - b7 - 3*b6 - 2*b5 - 4*b3 + 2) * q^61 + (-b15 - b13 - 3*b1) * q^63 + (-2*b12 - 2*b8 + 2*b7 - 5*b5 - b3 + 4) * q^65 + (-2*b15 - b14 - b13 + 2*b9 + 2*b4 - b1) * q^67 + (-2*b12 + 2*b8 - b7 + 2*b6 + b5 + 3*b3 + 1) * q^69 + (2*b15 + 3*b14 + b13 - 2*b9 + b4 - 2*b2) * q^71 + (b12 - b10 - 3*b7 - b3 + 1) * q^73 + (-3*b15 + 2*b14 + b13 + b11 + 3*b9 - b2 - 2*b1) * q^75 + (-2*b12 - b7 - 3*b5 + 3*b3 - 1) * q^77 + (-b15 - 3*b14 + b13 - 3*b11 + 6*b9 + 3*b4 - 3*b2 - 6*b1) * q^79 + (3*b12 + 2*b10 - 3*b8 + 2*b7 - 2*b6 - 5*b5 + 3*b3 + 2) * q^81 + (-b15 - 2*b14 - 2*b11 + 2*b9 - b4 - b2 - 6*b1) * q^83 + (b12 - 2*b10 - 4*b7 + b6 + 4*b5 - 4*b3 + 1) * q^85 + (-b15 + 2*b14 - 2*b13 + 5*b11 - 2*b9 + 2*b4 - b2 + 4*b1) * q^87 + (2*b12 + 2*b10 + b8 - 2*b7 - b6 - 2) * q^89 + (-3*b13 - 5*b11 + b9 - 2*b4 - 3*b2 - 4*b1) * q^91 + (-2*b7 + 2*b6 + 2*b5 - 2) * q^93 + (2*b15 + 2*b13 - b4 - b2 + 5*b1) * q^95 + (-b12 - b8 + b7 + 2*b3 + 2) * q^97 + (-b15 + 3*b13 - b11 - b9 + b4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{5} + 16 q^{9}+O(q^{10})$$ 16 * q + 8 * q^5 + 16 * q^9 $$16 q + 8 q^{5} + 16 q^{9} + 24 q^{13} - 16 q^{21} + 32 q^{25} - 24 q^{29} + 80 q^{33} + 40 q^{37} + 16 q^{41} + 24 q^{45} - 56 q^{53} + 80 q^{57} + 8 q^{61} + 32 q^{65} + 32 q^{69} + 32 q^{73} - 32 q^{77} + 48 q^{85} - 32 q^{89} - 16 q^{93} + 16 q^{97}+O(q^{100})$$ 16 * q + 8 * q^5 + 16 * q^9 + 24 * q^13 - 16 * q^21 + 32 * q^25 - 24 * q^29 + 80 * q^33 + 40 * q^37 + 16 * q^41 + 24 * q^45 - 56 * q^53 + 80 * q^57 + 8 * q^61 + 32 * q^65 + 32 * q^69 + 32 * q^73 - 32 * q^77 + 48 * q^85 - 32 * q^89 - 16 * q^93 + 16 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8x^{14} + 32x^{12} - 64x^{10} + 127x^{8} - 576x^{6} + 2592x^{4} - 5832x^{2} + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$( - 23 \nu^{15} + 1975 \nu^{13} - 8260 \nu^{11} + 13829 \nu^{9} - 4064 \nu^{7} + 152253 \nu^{5} - 708993 \nu^{3} + 1533816 \nu ) / 269001$$ (-23*v^15 + 1975*v^13 - 8260*v^11 + 13829*v^9 - 4064*v^7 + 152253*v^5 - 708993*v^3 + 1533816*v) / 269001 $$\beta_{2}$$ $$=$$ $$( 683 \nu^{15} - 13663 \nu^{13} + 44518 \nu^{11} - 82925 \nu^{9} + 22850 \nu^{7} - 946332 \nu^{5} + 3232224 \nu^{3} - 9086985 \nu ) / 1883007$$ (683*v^15 - 13663*v^13 + 44518*v^11 - 82925*v^9 + 22850*v^7 - 946332*v^5 + 3232224*v^3 - 9086985*v) / 1883007 $$\beta_{3}$$ $$=$$ $$( 928 \nu^{14} - 2582 \nu^{12} + 1895 \nu^{10} + 5156 \nu^{8} + 70408 \nu^{6} - 270972 \nu^{4} + 206145 \nu^{2} + 654642 ) / 627669$$ (928*v^14 - 2582*v^12 + 1895*v^10 + 5156*v^8 + 70408*v^6 - 270972*v^4 + 206145*v^2 + 654642) / 627669 $$\beta_{4}$$ $$=$$ $$( - 998 \nu^{15} + 1630 \nu^{13} + 12497 \nu^{11} - 29215 \nu^{9} - 36395 \nu^{7} + 339264 \nu^{5} + 748116 \nu^{3} - 2899962 \nu ) / 1883007$$ (-998*v^15 + 1630*v^13 + 12497*v^11 - 29215*v^9 - 36395*v^7 + 339264*v^5 + 748116*v^3 - 2899962*v) / 1883007 $$\beta_{5}$$ $$=$$ $$( 152\nu^{14} - 606\nu^{12} + 1352\nu^{10} - 828\nu^{8} + 10513\nu^{6} - 43364\nu^{4} + 119160\nu^{2} - 72738 ) / 69741$$ (152*v^14 - 606*v^12 + 1352*v^10 - 828*v^8 + 10513*v^6 - 43364*v^4 + 119160*v^2 - 72738) / 69741 $$\beta_{6}$$ $$=$$ $$( 1529 \nu^{14} - 19279 \nu^{12} + 69988 \nu^{10} - 104255 \nu^{8} + 123065 \nu^{6} - 1456047 \nu^{4} + 6035391 \nu^{2} - 10136016 ) / 627669$$ (1529*v^14 - 19279*v^12 + 69988*v^10 - 104255*v^8 + 123065*v^6 - 1456047*v^4 + 6035391*v^2 - 10136016) / 627669 $$\beta_{7}$$ $$=$$ $$( - 536 \nu^{14} + 4297 \nu^{12} - 11716 \nu^{10} + 13856 \nu^{8} - 29444 \nu^{6} + 308016 \nu^{4} - 998244 \nu^{2} + 1458000 ) / 209223$$ (-536*v^14 + 4297*v^12 - 11716*v^10 + 13856*v^8 - 29444*v^6 + 308016*v^4 - 998244*v^2 + 1458000) / 209223 $$\beta_{8}$$ $$=$$ $$( 541 \nu^{14} + 445 \nu^{12} - 5671 \nu^{10} + 23963 \nu^{8} + 27322 \nu^{6} - 19293 \nu^{4} - 331344 \nu^{2} + 2030022 ) / 209223$$ (541*v^14 + 445*v^12 - 5671*v^10 + 23963*v^8 + 27322*v^6 - 19293*v^4 - 331344*v^2 + 2030022) / 209223 $$\beta_{9}$$ $$=$$ $$( - 545 \nu^{15} + 5749 \nu^{13} - 15376 \nu^{11} + 17417 \nu^{9} - 35612 \nu^{7} + 412023 \nu^{5} - 1523259 \nu^{3} + 2404728 \nu ) / 627669$$ (-545*v^15 + 5749*v^13 - 15376*v^11 + 17417*v^9 - 35612*v^7 + 412023*v^5 - 1523259*v^3 + 2404728*v) / 627669 $$\beta_{10}$$ $$=$$ $$( - 269 \nu^{14} + 1360 \nu^{12} - 2110 \nu^{10} + 53 \nu^{8} - 22760 \nu^{6} + 121995 \nu^{4} - 283905 \nu^{2} + 39366 ) / 89667$$ (-269*v^14 + 1360*v^12 - 2110*v^10 + 53*v^8 - 22760*v^6 + 121995*v^4 - 283905*v^2 + 39366) / 89667 $$\beta_{11}$$ $$=$$ $$( 2011 \nu^{15} - 12281 \nu^{13} + 39080 \nu^{11} - 14089 \nu^{9} + 159412 \nu^{7} - 1021203 \nu^{5} + 3232629 \nu^{3} - 2566080 \nu ) / 1883007$$ (2011*v^15 - 12281*v^13 + 39080*v^11 - 14089*v^9 + 159412*v^7 - 1021203*v^5 + 3232629*v^3 - 2566080*v) / 1883007 $$\beta_{12}$$ $$=$$ $$( 2069 \nu^{14} - 8245 \nu^{12} + 15790 \nu^{10} - 13121 \nu^{8} + 116027 \nu^{6} - 763695 \nu^{4} + 2014875 \nu^{2} - 1452897 ) / 627669$$ (2069*v^14 - 8245*v^12 + 15790*v^10 - 13121*v^8 + 116027*v^6 - 763695*v^4 + 2014875*v^2 - 1452897) / 627669 $$\beta_{13}$$ $$=$$ $$( 2318 \nu^{15} - 30910 \nu^{13} + 90646 \nu^{11} - 135176 \nu^{9} + 129686 \nu^{7} - 2182401 \nu^{5} + 7802001 \nu^{3} - 12535155 \nu ) / 1883007$$ (2318*v^15 - 30910*v^13 + 90646*v^11 - 135176*v^9 + 129686*v^7 - 2182401*v^5 + 7802001*v^3 - 12535155*v) / 1883007 $$\beta_{14}$$ $$=$$ $$( 3323 \nu^{15} - 30895 \nu^{13} + 106156 \nu^{11} - 158573 \nu^{9} + 168104 \nu^{7} - 2289825 \nu^{5} + 9057663 \nu^{3} - 16315020 \nu ) / 1883007$$ (3323*v^15 - 30895*v^13 + 106156*v^11 - 158573*v^9 + 168104*v^7 - 2289825*v^5 + 9057663*v^3 - 16315020*v) / 1883007 $$\beta_{15}$$ $$=$$ $$( - 9367 \nu^{15} + 48179 \nu^{13} - 118331 \nu^{11} + 112300 \nu^{9} - 632545 \nu^{7} + 3746691 \nu^{5} - 10649475 \nu^{3} + 11764602 \nu ) / 1883007$$ (-9367*v^15 + 48179*v^13 - 118331*v^11 + 112300*v^9 - 632545*v^7 + 3746691*v^5 - 10649475*v^3 + 11764602*v) / 1883007
 $$\nu$$ $$=$$ $$( \beta_{13} + \beta_{9} - \beta_{2} ) / 2$$ (b13 + b9 - b2) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{12} - \beta_{10} + \beta_{7} + \beta_{5} - 4\beta_{3} + 1 ) / 2$$ (b12 - b10 + b7 + b5 - 4*b3 + 1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{14} - 2\beta_{11} - 2\beta_{9} - 4\beta_{2} - \beta_1 ) / 2$$ (b14 - 2*b11 - 2*b9 - 4*b2 - b1) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{12} - 2\beta_{10} - \beta_{8} - 2\beta_{6} + 11\beta_{5} - 13\beta_{3} + 1 ) / 2$$ (-b12 - 2*b10 - b8 - 2*b6 + 11*b5 - 13*b3 + 1) / 2 $$\nu^{5}$$ $$=$$ $$( \beta_{14} + \beta_{13} - 2\beta_{11} - 7\beta_{9} + 12\beta_{4} + \beta_{2} + 11\beta_1 ) / 2$$ (b14 + b13 - 2*b11 - 7*b9 + 12*b4 + b2 + 11*b1) / 2 $$\nu^{6}$$ $$=$$ $$-6\beta_{12} - 6\beta_{10} - 2\beta_{8} + 8\beta_{7} + 2\beta_{6} + 9\beta_{5} + 2\beta_{3} - 8$$ -6*b12 - 6*b10 - 2*b8 + 8*b7 + 2*b6 + 9*b5 + 2*b3 - 8 $$\nu^{7}$$ $$=$$ $$( -21\beta_{15} - 46\beta_{14} + 4\beta_{13} - 10\beta_{11} + 12\beta_{9} + 21\beta_{4} - 47\beta_1 ) / 2$$ (-21*b15 - 46*b14 + 4*b13 - 10*b11 + 12*b9 + 21*b4 - 47*b1) / 2 $$\nu^{8}$$ $$=$$ $$( -34\beta_{12} - 11\beta_{10} + 34\beta_{8} - 11\beta_{7} + 11\beta_{6} - 13\beta_{5} - 24\beta_{3} - 138 ) / 2$$ (-34*b12 - 11*b10 + 34*b8 - 11*b7 + 11*b6 - 13*b5 - 24*b3 - 138) / 2 $$\nu^{9}$$ $$=$$ $$( 36\beta_{15} + 18\beta_{14} - 64\beta_{13} + 102\beta_{11} - 115\beta_{9} + 64\beta_{2} + 69\beta_1 ) / 2$$ (36*b15 + 18*b14 - 64*b13 + 102*b11 - 115*b9 + 64*b2 + 69*b1) / 2 $$\nu^{10}$$ $$=$$ $$( -115\beta_{12} + 115\beta_{10} + 84\beta_{8} - 7\beta_{7} + 84\beta_{6} - 31\beta_{5} + 238\beta_{3} - 7 ) / 2$$ (-115*b12 + 115*b10 + 84*b8 - 7*b7 + 84*b6 - 31*b5 + 238*b3 - 7) / 2 $$\nu^{11}$$ $$=$$ $$( 69\beta_{15} + 68\beta_{14} + 344\beta_{11} + 185\beta_{9} + 69\beta_{4} + 184\beta_{2} + 160\beta_1 ) / 2$$ (69*b15 + 68*b14 + 344*b11 + 185*b9 + 69*b4 + 184*b2 + 160*b1) / 2 $$\nu^{12}$$ $$=$$ $$92\beta_{12} + 172\beta_{10} + 92\beta_{8} + 219\beta_{7} + 172\beta_{6} - 196\beta_{5} + 368\beta_{3} - 92$$ 92*b12 + 172*b10 + 92*b8 + 219*b7 + 172*b6 - 196*b5 + 368*b3 - 92 $$\nu^{13}$$ $$=$$ $$( 389\beta_{14} - 415\beta_{13} + 137\beta_{11} + 643\beta_{9} - 600\beta_{4} - 415\beta_{2} - 806\beta_1 ) / 2$$ (389*b14 - 415*b13 + 137*b11 + 643*b9 - 600*b4 - 415*b2 - 806*b1) / 2 $$\nu^{14}$$ $$=$$ $$( 1332 \beta_{12} + 1332 \beta_{10} + 163 \beta_{8} - 142 \beta_{7} - 163 \beta_{6} + 669 \beta_{5} - 163 \beta_{3} + 142 ) / 2$$ (1332*b12 + 1332*b10 + 163*b8 - 142*b7 - 163*b6 + 669*b5 - 163*b3 + 142) / 2 $$\nu^{15}$$ $$=$$ $$( 576\beta_{15} + 3727\beta_{14} - 1516\beta_{13} - 269\beta_{11} - 489\beta_{9} - 576\beta_{4} + 3371\beta_1 ) / 2$$ (576*b15 + 3727*b14 - 1516*b13 - 269*b11 - 489*b9 - 576*b4 + 3371*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$1023$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 1.59056 + 0.685641i −0.639878 + 1.60952i 0.639878 − 1.60952i −1.59056 − 0.685641i −1.66798 − 0.466730i −1.50947 − 0.849413i 1.50947 + 0.849413i 1.66798 + 0.466730i −1.66798 + 0.466730i −1.50947 + 0.849413i 1.50947 − 0.849413i 1.66798 − 0.466730i 1.59056 − 0.685641i −0.639878 − 1.60952i 0.639878 + 1.60952i −1.59056 + 0.685641i
0 −0.942835 + 2.27621i 0 −2.49877 + 1.03503i 0 1.37128 1.37128i 0 −2.17085 2.17085i 0
129.2 0 −0.401639 + 0.969643i 0 3.49877 1.44924i 0 3.21904 3.21904i 0 1.34243 + 1.34243i 0
129.3 0 0.401639 0.969643i 0 3.49877 1.44924i 0 −3.21904 + 3.21904i 0 1.34243 + 1.34243i 0
129.4 0 0.942835 2.27621i 0 −2.49877 + 1.03503i 0 −1.37128 + 1.37128i 0 −2.17085 2.17085i 0
385.1 0 −2.90008 + 1.20125i 0 1.21229 2.92673i 0 0.933460 + 0.933460i 0 4.84613 4.84613i 0
385.2 0 −1.59352 + 0.660056i 0 −0.212292 + 0.512517i 0 1.69883 + 1.69883i 0 −0.0177021 + 0.0177021i 0
385.3 0 1.59352 0.660056i 0 −0.212292 + 0.512517i 0 −1.69883 1.69883i 0 −0.0177021 + 0.0177021i 0
385.4 0 2.90008 1.20125i 0 1.21229 2.92673i 0 −0.933460 0.933460i 0 4.84613 4.84613i 0
641.1 0 −2.90008 1.20125i 0 1.21229 + 2.92673i 0 0.933460 0.933460i 0 4.84613 + 4.84613i 0
641.2 0 −1.59352 0.660056i 0 −0.212292 0.512517i 0 1.69883 1.69883i 0 −0.0177021 0.0177021i 0
641.3 0 1.59352 + 0.660056i 0 −0.212292 0.512517i 0 −1.69883 + 1.69883i 0 −0.0177021 0.0177021i 0
641.4 0 2.90008 + 1.20125i 0 1.21229 + 2.92673i 0 −0.933460 + 0.933460i 0 4.84613 + 4.84613i 0
897.1 0 −0.942835 2.27621i 0 −2.49877 1.03503i 0 1.37128 + 1.37128i 0 −2.17085 + 2.17085i 0
897.2 0 −0.401639 0.969643i 0 3.49877 + 1.44924i 0 3.21904 + 3.21904i 0 1.34243 1.34243i 0
897.3 0 0.401639 + 0.969643i 0 3.49877 + 1.44924i 0 −3.21904 3.21904i 0 1.34243 1.34243i 0
897.4 0 0.942835 + 2.27621i 0 −2.49877 1.03503i 0 −1.37128 1.37128i 0 −2.17085 + 2.17085i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 897.4 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
32.g even 8 1 inner
32.h odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.g.h yes 16
4.b odd 2 1 inner 1024.2.g.h yes 16
8.b even 2 1 1024.2.g.c yes 16
8.d odd 2 1 1024.2.g.c yes 16
16.e even 4 1 1024.2.g.b 16
16.e even 4 1 1024.2.g.e yes 16
16.f odd 4 1 1024.2.g.b 16
16.f odd 4 1 1024.2.g.e yes 16
32.g even 8 1 1024.2.g.b 16
32.g even 8 1 1024.2.g.c yes 16
32.g even 8 1 1024.2.g.e yes 16
32.g even 8 1 inner 1024.2.g.h yes 16
32.h odd 8 1 1024.2.g.b 16
32.h odd 8 1 1024.2.g.c yes 16
32.h odd 8 1 1024.2.g.e yes 16
32.h odd 8 1 inner 1024.2.g.h yes 16
64.i even 16 1 4096.2.a.n 8
64.i even 16 1 4096.2.a.o 8
64.j odd 16 1 4096.2.a.n 8
64.j odd 16 1 4096.2.a.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1024.2.g.b 16 16.e even 4 1
1024.2.g.b 16 16.f odd 4 1
1024.2.g.b 16 32.g even 8 1
1024.2.g.b 16 32.h odd 8 1
1024.2.g.c yes 16 8.b even 2 1
1024.2.g.c yes 16 8.d odd 2 1
1024.2.g.c yes 16 32.g even 8 1
1024.2.g.c yes 16 32.h odd 8 1
1024.2.g.e yes 16 16.e even 4 1
1024.2.g.e yes 16 16.f odd 4 1
1024.2.g.e yes 16 32.g even 8 1
1024.2.g.e yes 16 32.h odd 8 1
1024.2.g.h yes 16 1.a even 1 1 trivial
1024.2.g.h yes 16 4.b odd 2 1 inner
1024.2.g.h yes 16 32.g even 8 1 inner
1024.2.g.h yes 16 32.h odd 8 1 inner
4096.2.a.n 8 64.i even 16 1
4096.2.a.n 8 64.j odd 16 1
4096.2.a.o 8 64.i even 16 1
4096.2.a.o 8 64.j odd 16 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1024, [\chi])$$:

 $$T_{3}^{16} - 8T_{3}^{14} + 32T_{3}^{12} + 272T_{3}^{10} + 2744T_{3}^{8} - 8288T_{3}^{6} + 15488T_{3}^{4} + 34496T_{3}^{2} + 38416$$ T3^16 - 8*T3^14 + 32*T3^12 + 272*T3^10 + 2744*T3^8 - 8288*T3^6 + 15488*T3^4 + 34496*T3^2 + 38416 $$T_{5}^{8} - 4T_{5}^{7} + 32T_{5}^{5} - 64T_{5}^{4} - 72T_{5}^{3} + 1008T_{5}^{2} + 432T_{5} + 324$$ T5^8 - 4*T5^7 + 32*T5^5 - 64*T5^4 - 72*T5^3 + 1008*T5^2 + 432*T5 + 324

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16} - 8 T^{14} + 32 T^{12} + \cdots + 38416$$
$5$ $$(T^{8} - 4 T^{7} + 32 T^{5} - 64 T^{4} + \cdots + 324)^{2}$$
$7$ $$T^{16} + 480 T^{12} + 22304 T^{8} + \cdots + 614656$$
$11$ $$T^{16} + 8 T^{14} + 32 T^{12} + \cdots + 104976$$
$13$ $$(T^{8} - 12 T^{7} + 80 T^{6} - 552 T^{5} + \cdots + 56644)^{2}$$
$17$ $$(T^{8} + 72 T^{6} + 1408 T^{4} + \cdots + 5184)^{2}$$
$19$ $$T^{16} + 24 T^{14} + \cdots + 35477982736$$
$23$ $$T^{16} + 2784 T^{12} + \cdots + 1679616$$
$29$ $$(T^{8} + 12 T^{7} + 32 T^{6} - 80 T^{5} + \cdots + 324)^{2}$$
$31$ $$(T^{4} - 16 T^{2} + 32)^{4}$$
$37$ $$(T^{8} - 20 T^{7} + 128 T^{6} + \cdots + 1110916)^{2}$$
$41$ $$(T^{8} - 8 T^{7} + 32 T^{6} + 592 T^{5} + \cdots + 104976)^{2}$$
$43$ $$T^{16} + 88 T^{14} + 3872 T^{12} + \cdots + 45212176$$
$47$ $$(T^{8} + 128 T^{6} + 3392 T^{4} + \cdots + 20736)^{2}$$
$53$ $$(T^{8} + 28 T^{7} + 304 T^{6} + \cdots + 93636)^{2}$$
$59$ $$T^{16} - 24 T^{14} + \cdots + 2981133747216$$
$61$ $$(T^{8} - 4 T^{7} - 144 T^{6} + \cdots + 3740356)^{2}$$
$67$ $$T^{16} - 216 T^{14} + \cdots + 688747536$$
$71$ $$T^{16} + 48480 T^{12} + \cdots + 49\!\cdots\!36$$
$73$ $$(T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 169744)^{2}$$
$79$ $$(T^{8} + 512 T^{6} + 83264 T^{4} + \cdots + 3268864)^{2}$$
$83$ $$T^{16} - 216 T^{14} + \cdots + 252047376$$
$89$ $$(T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 3111696)^{2}$$
$97$ $$(T^{4} - 4 T^{3} - 36 T^{2} - 16 T + 56)^{4}$$