# Properties

 Label 1050.3.q.b Level $1050$ Weight $3$ Character orbit 1050.q Analytic conductor $28.610$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: 16.0.22986704741655040229376.1 Defining polynomial: $$x^{16} - 31x^{12} + 880x^{8} - 2511x^{4} + 6561$$ x^16 - 31*x^12 + 880*x^8 - 2511*x^4 + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{11} q^{2} + ( - \beta_{9} - \beta_1) q^{3} + (2 \beta_{3} + 2) q^{4} + ( - 2 \beta_{12} - \beta_{6}) q^{6} + (\beta_{15} - 2 \beta_{8} - 2 \beta_{7}) q^{7} + 2 \beta_{7} q^{8} + 3 \beta_{3} q^{9}+O(q^{10})$$ q - b11 * q^2 + (-b9 - b1) * q^3 + (2*b3 + 2) * q^4 + (-2*b12 - b6) * q^6 + (b15 - 2*b8 - 2*b7) * q^7 + 2*b7 * q^8 + 3*b3 * q^9 $$q - \beta_{11} q^{2} + ( - \beta_{9} - \beta_1) q^{3} + (2 \beta_{3} + 2) q^{4} + ( - 2 \beta_{12} - \beta_{6}) q^{6} + (\beta_{15} - 2 \beta_{8} - 2 \beta_{7}) q^{7} + 2 \beta_{7} q^{8} + 3 \beta_{3} q^{9} + ( - \beta_{14} - \beta_{12} + \beta_{10} - \beta_{3} - 1) q^{11} + ( - 4 \beta_{9} + 2 \beta_1) q^{12} + (6 \beta_{11} + \beta_{9} + 2 \beta_{8} + 4 \beta_{7} - 2 \beta_{2} - 2 \beta_1) q^{13} + (\beta_{13} + 3 \beta_{5} - 2 \beta_{3} + 1) q^{14} + 4 \beta_{3} q^{16} + ( - 3 \beta_{15} - 2 \beta_{11} + 2 \beta_{9} - \beta_{7} - 5 \beta_{4} - 5 \beta_{2} + 2 \beta_1) q^{17} + (3 \beta_{11} + 3 \beta_{7}) q^{18} + ( - \beta_{14} - 7 \beta_{12} + 8 \beta_{6} + 2 \beta_{5} - 6 \beta_{3} - 12) q^{19} + ( - \beta_{14} + \beta_{12} + 4 \beta_{10} + \beta_{6}) q^{21} + ( - 3 \beta_{9} - \beta_{7} - 2 \beta_{4} - \beta_{2}) q^{22} + ( - 2 \beta_{15} - 5 \beta_{11} + 2 \beta_{8} - \beta_{4} - 2 \beta_{2} + 15 \beta_1) q^{23} + ( - 2 \beta_{12} - 4 \beta_{6}) q^{24} + ( - 4 \beta_{14} + \beta_{12} + 3 \beta_{6} - 2 \beta_{5} - 6 \beta_{3} - 12) q^{26} + ( - 3 \beta_{9} + 6 \beta_1) q^{27} + (6 \beta_{15} + 2 \beta_{8} + 2 \beta_{7}) q^{28} + ( - 8 \beta_{14} - \beta_{13} + 4 \beta_{12} - 4 \beta_{10} - 7 \beta_{6} + \beta_{5} + 3) q^{29} + ( - 2 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} - 2 \beta_{10} - 8 \beta_{6} - 8 \beta_{3} + \cdots + 8) q^{31}+ \cdots + (6 \beta_{14} - 3 \beta_{12} + 3 \beta_{10} - 9 \beta_{6} + 3) q^{99}+O(q^{100})$$ q - b11 * q^2 + (-b9 - b1) * q^3 + (2*b3 + 2) * q^4 + (-2*b12 - b6) * q^6 + (b15 - 2*b8 - 2*b7) * q^7 + 2*b7 * q^8 + 3*b3 * q^9 + (-b14 - b12 + b10 - b3 - 1) * q^11 + (-4*b9 + 2*b1) * q^12 + (6*b11 + b9 + 2*b8 + 4*b7 - 2*b2 - 2*b1) * q^13 + (b13 + 3*b5 - 2*b3 + 1) * q^14 + 4*b3 * q^16 + (-3*b15 - 2*b11 + 2*b9 - b7 - 5*b4 - 5*b2 + 2*b1) * q^17 + (3*b11 + 3*b7) * q^18 + (-b14 - 7*b12 + 8*b6 + 2*b5 - 6*b3 - 12) * q^19 + (-b14 + b12 + 4*b10 + b6) * q^21 + (-3*b9 - b7 - 2*b4 - b2) * q^22 + (-2*b15 - 5*b11 + 2*b8 - b4 - 2*b2 + 15*b1) * q^23 + (-2*b12 - 4*b6) * q^24 + (-4*b14 + b12 + 3*b6 - 2*b5 - 6*b3 - 12) * q^26 + (-3*b9 + 6*b1) * q^27 + (6*b15 + 2*b8 + 2*b7) * q^28 + (-8*b14 - b13 + 4*b12 - 4*b10 - 7*b6 + b5 + 3) * q^29 + (-2*b14 + 4*b13 - 4*b12 - 2*b10 - 8*b6 - 8*b3 + 8) * q^31 + (4*b11 + 4*b7) * q^32 + (3*b15 + 3*b11 + 2*b9 + 3*b8 + 6*b7 - b1) * q^33 + (-3*b13 + 4*b12 + 10*b10 - 3*b6 - 3*b5 + 2*b3 + 1) * q^34 - 6 * q^36 + (-2*b11 - 5*b4 - 10*b2 + 6*b1) * q^37 + (4*b15 + 8*b11 - 15*b9 - 4*b7 - b4 - b2 - 15*b1) * q^38 + (2*b14 + 2*b13 + 8*b12 - 2*b10 + 4*b5 + 3*b3 + 3) * q^39 + (8*b13 + 5*b12 - 7*b10 + 6*b6 + 8*b5 - 2*b3 - 1) * q^41 + (b9 - 5*b4 - b2 - 5*b1) * q^42 + (16*b15 + 8*b11 - 14*b9 + 8*b8 - 10*b7 - 8*b4 - 4*b2) * q^43 + (2*b14 - 4*b12 + 4*b10 - 6*b6 - 2*b3) * q^44 + (-2*b14 - 2*b13 + 16*b12 + 2*b10 - 4*b5 + 8*b3 + 8) * q^46 + (-13*b15 - 5*b11 - 14*b9 - 13*b8 - 10*b7 - 10*b4 + 7*b1) * q^47 + (-4*b9 + 8*b1) * q^48 + (-8*b13 - 3*b5 + 30*b3 + 48) * q^49 + (-3*b14 + 10*b13 - 6*b10 + 3*b6 + 5*b5 - 6*b3) * q^51 + (-4*b15 + 4*b11 - 2*b9 - 8*b7 - 4*b4 - 4*b2 - 2*b1) * q^52 + (9*b15 - 5*b11 + 11*b9 + 18*b8 + 4*b7 + 2*b4 - 2*b2 - 11*b1) * q^53 + (3*b12 - 3*b6) * q^54 + (6*b13 + 4*b5 + 2*b3 + 6) * q^56 + (2*b15 + b11 + 18*b9 + b8 + 24*b7 - 4*b4 - 2*b2) * q^57 + (2*b15 - 2*b11 - 2*b8 - 4*b4 - 8*b2 + 26*b1) * q^58 + (-21*b14 - 5*b13 + 12*b12 - 21*b10 + 24*b6 + 26*b3 - 26) * q^59 + (-10*b14 + 8*b12 + 2*b6 - 10*b5 - 17*b3 - 34) * q^61 + (-16*b11 - 10*b9 + 8*b8 - 4*b7 - 2*b2 + 20*b1) * q^62 + (6*b15 + 9*b8 + 9*b7) * q^63 - 8 * q^64 + (3*b13 + b12 + 2*b6 + 3*b3 - 3) * q^66 + (-3*b15 - 18*b11 - 30*b9 - 6*b8 - 21*b7 + 30*b1) * q^67 + (-6*b15 - 2*b11 + 8*b9 - 6*b8 - 4*b7 - 10*b4 - 4*b1) * q^68 + (3*b13 - 8*b12 - 6*b10 - b6 + 3*b5 - 30*b3 - 15) * q^69 + (8*b14 + 12*b13 - 4*b12 + 4*b10 - 4*b6 - 12*b5 - 8) * q^71 + 6*b11 * q^72 + (-10*b15 - 10*b11 + 7*b9 - 24*b4 - 24*b2 + 7*b1) * q^73 + (-10*b14 + 11*b12 + 10*b10 + 4*b3 + 4) * q^74 + (4*b13 - 30*b12 + 2*b10 - 16*b6 + 4*b5 - 24*b3 - 12) * q^76 + (-3*b15 - 23*b9 - b8 - b7 - 4*b4 - 5*b2 - 11*b1) * q^77 + (8*b15 + 4*b11 + 18*b9 + 4*b8 + 9*b7 + 4*b4 + 2*b2) * q^78 + (-b14 - 24*b13 + 6*b12 - 2*b10 + 7*b6 - 12*b5 - 36*b3) * q^79 + (-9*b3 - 9) * q^81 + (16*b15 + 7*b11 + 10*b9 + 16*b8 + 14*b7 + 7*b4 - 5*b1) * q^82 + (33*b11 - 16*b9 + 7*b8 + 20*b7 + b2 + 32*b1) * q^83 + (8*b14 - 8*b12 + 10*b10 - 8*b6) * q^84 + (8*b14 + 16*b13 - 18*b12 + 16*b10 - 26*b6 + 8*b5 - 44*b3) * q^86 + (12*b15 + 19*b11 - 3*b9 - 7*b7 - 3*b4 - 3*b2 - 3*b1) * q^87 + (-2*b11 - 6*b9 - 2*b7 - 2*b4 + 2*b2 + 6*b1) * q^88 + (-11*b14 - 10*b12 + 21*b6 + b5 + 28*b3 + 56) * q^89 + (-7*b14 + 6*b13 - 7*b12 - 7*b10 + 35*b6 - 10*b5 + 2*b3 - 36) * q^91 + (-8*b15 - 4*b11 + 30*b9 - 4*b8 + 2*b7 - 4*b4 - 2*b2) * q^92 + (2*b15 + 16*b11 - 2*b8 + 4*b4 + 8*b2 - 24*b1) * q^93 + (20*b14 - 13*b13 - 17*b12 + 20*b10 - 34*b6 + 3*b3 - 3) * q^94 + (4*b12 - 4*b6) * q^96 + (26*b11 + 24*b9 - 26*b8 + 3*b2 - 48*b1) * q^97 + (-6*b15 - 21*b11 - 16*b8 + 19*b7) * q^98 + (6*b14 - 3*b12 + 3*b10 - 9*b6 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 16 q^{4} - 24 q^{9}+O(q^{10})$$ 16 * q + 16 * q^4 - 24 * q^9 $$16 q + 16 q^{4} - 24 q^{9} - 8 q^{11} + 32 q^{14} - 32 q^{16} - 144 q^{19} - 144 q^{26} + 48 q^{29} + 192 q^{31} - 96 q^{36} + 24 q^{39} + 16 q^{44} + 64 q^{46} + 528 q^{49} + 48 q^{51} + 80 q^{56} - 624 q^{59} - 408 q^{61} - 128 q^{64} - 72 q^{66} - 128 q^{71} + 32 q^{74} + 288 q^{79} - 72 q^{81} + 352 q^{86} + 672 q^{89} - 592 q^{91} - 72 q^{94} + 48 q^{99}+O(q^{100})$$ 16 * q + 16 * q^4 - 24 * q^9 - 8 * q^11 + 32 * q^14 - 32 * q^16 - 144 * q^19 - 144 * q^26 + 48 * q^29 + 192 * q^31 - 96 * q^36 + 24 * q^39 + 16 * q^44 + 64 * q^46 + 528 * q^49 + 48 * q^51 + 80 * q^56 - 624 * q^59 - 408 * q^61 - 128 * q^64 - 72 * q^66 - 128 * q^71 + 32 * q^74 + 288 * q^79 - 72 * q^81 + 352 * q^86 + 672 * q^89 - 592 * q^91 - 72 * q^94 + 48 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 31x^{12} + 880x^{8} - 2511x^{4} + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{14} + 32689\nu^{2} ) / 55440$$ (v^14 + 32689*v^2) / 55440 $$\beta_{2}$$ $$=$$ $$( \nu^{12} + 11129 ) / 3080$$ (v^12 + 11129) / 3080 $$\beta_{3}$$ $$=$$ $$( 31\nu^{12} - 880\nu^{8} + 27280\nu^{4} - 77841 ) / 71280$$ (31*v^12 - 880*v^8 + 27280*v^4 - 77841) / 71280 $$\beta_{4}$$ $$=$$ $$( 799\nu^{12} - 27280\nu^{8} + 703120\nu^{4} - 2006289 ) / 498960$$ (799*v^12 - 27280*v^8 + 703120*v^4 - 2006289) / 498960 $$\beta_{5}$$ $$=$$ $$( \nu^{14} + 16849\nu^{2} ) / 7920$$ (v^14 + 16849*v^2) / 7920 $$\beta_{6}$$ $$=$$ $$( -9\nu^{15} + 127\nu^{13} - 4180\nu^{9} + 111760\nu^{5} - 169461\nu^{3} - 318897\nu ) / 374220$$ (-9*v^15 + 127*v^13 - 4180*v^9 + 111760*v^5 - 169461*v^3 - 318897*v) / 374220 $$\beta_{7}$$ $$=$$ $$( 9\nu^{15} + 127\nu^{13} - 4180\nu^{9} + 111760\nu^{5} + 169461\nu^{3} - 318897\nu ) / 374220$$ (9*v^15 + 127*v^13 - 4180*v^9 + 111760*v^5 + 169461*v^3 - 318897*v) / 374220 $$\beta_{8}$$ $$=$$ $$( -3\nu^{15} + 31\nu^{13} - 880\nu^{9} + 27280\nu^{5} - 74307\nu^{3} - 77841\nu ) / 71280$$ (-3*v^15 + 31*v^13 - 880*v^9 + 27280*v^5 - 74307*v^3 - 77841*v) / 71280 $$\beta_{9}$$ $$=$$ $$( 31\nu^{14} - 880\nu^{10} + 25498\nu^{6} - 6561\nu^{2} ) / 112266$$ (31*v^14 - 880*v^10 + 25498*v^6 - 6561*v^2) / 112266 $$\beta_{10}$$ $$=$$ $$( -3\nu^{15} - 31\nu^{13} + 880\nu^{9} - 27280\nu^{5} - 74307\nu^{3} + 77841\nu ) / 71280$$ (-3*v^15 - 31*v^13 + 880*v^9 - 27280*v^5 - 74307*v^3 + 77841*v) / 71280 $$\beta_{11}$$ $$=$$ $$( 2683 \nu^{15} - 5301 \nu^{13} - 85360 \nu^{11} + 150480 \nu^{9} + 2361040 \nu^{7} - 4023360 \nu^{5} - 6737013 \nu^{3} + 1121931 \nu ) / 13471920$$ (2683*v^15 - 5301*v^13 - 85360*v^11 + 150480*v^9 + 2361040*v^7 - 4023360*v^5 - 6737013*v^3 + 1121931*v) / 13471920 $$\beta_{12}$$ $$=$$ $$( 3007\nu^{15} + 729\nu^{13} - 85360\nu^{11} + 2361040\nu^{7} - 636417\nu^{3} + 10358361\nu ) / 13471920$$ (3007*v^15 + 729*v^13 - 85360*v^11 + 2361040*v^7 - 636417*v^3 + 10358361*v) / 13471920 $$\beta_{13}$$ $$=$$ $$( 601\nu^{14} - 19360\nu^{10} + 528880\nu^{6} - 1509111\nu^{2} ) / 641520$$ (601*v^14 - 19360*v^10 + 528880*v^6 - 1509111*v^2) / 641520 $$\beta_{14}$$ $$=$$ $$( - 1159 \nu^{15} + 3720 \nu^{13} + 35200 \nu^{11} - 105600 \nu^{9} - 1019920 \nu^{7} + 3059760 \nu^{5} + 2910249 \nu^{3} - 787320 \nu ) / 4490640$$ (-1159*v^15 + 3720*v^13 + 35200*v^11 - 105600*v^9 - 1019920*v^7 + 3059760*v^5 + 2910249*v^3 - 787320*v) / 4490640 $$\beta_{15}$$ $$=$$ $$( 1240\nu^{15} + 243\nu^{13} - 35200\nu^{11} + 1019920\nu^{7} - 262440\nu^{3} + 7943427\nu ) / 4490640$$ (1240*v^15 + 243*v^13 - 35200*v^11 + 1019920*v^7 - 262440*v^3 + 7943427*v) / 4490640
 $$\nu$$ $$=$$ $$( \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} ) / 2$$ (b15 + b14 - b12 + b11 + b10 + b7 - b6) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} + 7\beta_1 ) / 2$$ (-b5 + 7*b1) / 2 $$\nu^{3}$$ $$=$$ $$( -4\beta_{10} - 4\beta_{8} - 7\beta_{7} + 7\beta_{6} ) / 2$$ (-4*b10 - 4*b8 - 7*b7 + 7*b6) / 2 $$\nu^{4}$$ $$=$$ $$( -7\beta_{4} + 31\beta_{3} - 7\beta_{2} + 31 ) / 2$$ (-7*b4 + 31*b3 - 7*b2 + 31) / 2 $$\nu^{5}$$ $$=$$ $$( 19\beta_{15} + 19\beta_{14} - 40\beta_{12} + 40\beta_{11} + 19\beta_{8} + 19\beta_{7} - 40\beta_{6} ) / 2$$ (19*b15 + 19*b14 - 40*b12 + 40*b11 + 19*b8 + 19*b7 - 40*b6) / 2 $$\nu^{6}$$ $$=$$ $$-20\beta_{13} + 77\beta_{9} - 20\beta_{5}$$ -20*b13 + 77*b9 - 20*b5 $$\nu^{7}$$ $$=$$ $$( 97\beta_{15} - 97\beta_{14} - 120\beta_{12} - 120\beta_{11} - 97\beta_{10} - 120\beta_{7} + 97\beta_{6} ) / 2$$ (97*b15 - 97*b14 - 120*b12 - 120*b11 - 97*b10 - 120*b7 + 97*b6) / 2 $$\nu^{8}$$ $$=$$ $$( -217\beta_{4} + 799\beta_{3} ) / 2$$ (-217*b4 + 799*b3) / 2 $$\nu^{9}$$ $$=$$ $$( -508\beta_{10} + 508\beta_{8} - 651\beta_{7} - 651\beta_{6} ) / 2$$ (-508*b10 + 508*b8 - 651*b7 - 651*b6) / 2 $$\nu^{10}$$ $$=$$ $$( -1159\beta_{13} + 4207\beta_{9} - 4207\beta_1 ) / 2$$ (-1159*b13 + 4207*b9 - 4207*b1) / 2 $$\nu^{11}$$ $$=$$ $$( 2683 \beta_{15} - 2683 \beta_{14} - 3477 \beta_{12} - 3477 \beta_{11} + 2683 \beta_{8} + 2683 \beta_{7} - 3477 \beta_{6} ) / 2$$ (2683*b15 - 2683*b14 - 3477*b12 - 3477*b11 + 2683*b8 + 2683*b7 - 3477*b6) / 2 $$\nu^{12}$$ $$=$$ $$3080\beta_{2} - 11129$$ 3080*b2 - 11129 $$\nu^{13}$$ $$=$$ $$( - 14209 \beta_{15} - 14209 \beta_{14} + 32689 \beta_{12} - 32689 \beta_{11} - 14209 \beta_{10} - 32689 \beta_{7} + 14209 \beta_{6} ) / 2$$ (-14209*b15 - 14209*b14 + 32689*b12 - 32689*b11 - 14209*b10 - 32689*b7 + 14209*b6) / 2 $$\nu^{14}$$ $$=$$ $$( 32689\beta_{5} - 117943\beta_1 ) / 2$$ (32689*b5 - 117943*b1) / 2 $$\nu^{15}$$ $$=$$ $$( 75316\beta_{10} + 75316\beta_{8} + 173383\beta_{7} - 173383\beta_{6} ) / 2$$ (75316*b10 + 75316*b8 + 173383*b7 - 173383*b6) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-1$$ $$-\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −1.25838 + 0.337183i 2.22431 − 0.596002i 0.596002 + 2.22431i −0.337183 − 1.25838i −2.22431 + 0.596002i 1.25838 − 0.337183i 0.337183 + 1.25838i −0.596002 − 2.22431i −1.25838 − 0.337183i 2.22431 + 0.596002i 0.596002 − 2.22431i −0.337183 + 1.25838i −2.22431 − 0.596002i 1.25838 + 0.337183i 0.337183 − 1.25838i −0.596002 + 2.22431i
−1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −6.98615 + 0.440173i 2.82843i −1.50000 2.59808i 0
199.2 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 5.76140 3.97571i 2.82843i −1.50000 2.59808i 0
199.3 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −6.98615 + 0.440173i 2.82843i −1.50000 2.59808i 0
199.4 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 5.76140 3.97571i 2.82843i −1.50000 2.59808i 0
199.5 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −5.76140 + 3.97571i 2.82843i −1.50000 2.59808i 0
199.6 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 6.98615 0.440173i 2.82843i −1.50000 2.59808i 0
199.7 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −5.76140 + 3.97571i 2.82843i −1.50000 2.59808i 0
199.8 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 6.98615 0.440173i 2.82843i −1.50000 2.59808i 0
649.1 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.98615 0.440173i 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 5.76140 + 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.3 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.98615 0.440173i 2.82843i −1.50000 + 2.59808i 0
649.4 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 5.76140 + 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.5 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −5.76140 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.6 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 6.98615 + 0.440173i 2.82843i −1.50000 + 2.59808i 0
649.7 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −5.76140 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.8 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 6.98615 + 0.440173i 2.82843i −1.50000 + 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.q.b 16
5.b even 2 1 inner 1050.3.q.b 16
5.c odd 4 1 1050.3.p.c 8
5.c odd 4 1 1050.3.p.d yes 8
7.d odd 6 1 inner 1050.3.q.b 16
35.i odd 6 1 inner 1050.3.q.b 16
35.k even 12 1 1050.3.p.c 8
35.k even 12 1 1050.3.p.d yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.p.c 8 5.c odd 4 1
1050.3.p.c 8 35.k even 12 1
1050.3.p.d yes 8 5.c odd 4 1
1050.3.p.d yes 8 35.k even 12 1
1050.3.q.b 16 1.a even 1 1 trivial
1050.3.q.b 16 5.b even 2 1 inner
1050.3.q.b 16 7.d odd 6 1 inner
1050.3.q.b 16 35.i odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{8} + 4T_{11}^{7} + 58T_{11}^{6} + 16T_{11}^{5} + 1954T_{11}^{4} + 2440T_{11}^{3} + 15940T_{11}^{2} - 16376T_{11} + 31684$$ acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 2 T^{2} + 4)^{4}$$
$3$ $$(T^{4} + 3 T^{2} + 9)^{4}$$
$5$ $$T^{16}$$
$7$ $$(T^{8} - 132 T^{6} + 8183 T^{4} + \cdots + 5764801)^{2}$$
$11$ $$(T^{8} + 4 T^{7} + 58 T^{6} + 16 T^{5} + \cdots + 31684)^{2}$$
$13$ $$(T^{8} - 540 T^{6} + 39206 T^{4} + \cdots + 3500641)^{2}$$
$17$ $$T^{16} + 1588 T^{14} + \cdots + 18\!\cdots\!96$$
$19$ $$(T^{8} + 72 T^{7} + 1584 T^{6} + \cdots + 648364369)^{2}$$
$23$ $$T^{16} - 1496 T^{14} + \cdots + 66\!\cdots\!56$$
$29$ $$(T^{4} - 12 T^{3} - 1324 T^{2} + \cdots - 109496)^{4}$$
$31$ $$(T^{8} - 96 T^{7} + 3456 T^{6} + \cdots + 641507584)^{2}$$
$37$ $$T^{16} - 4076 T^{14} + \cdots + 56\!\cdots\!61$$
$41$ $$(T^{8} + 4764 T^{6} + \cdots + 16384512004)^{2}$$
$43$ $$(T^{8} + 12144 T^{6} + \cdots + 41573196361984)^{2}$$
$47$ $$T^{16} + 10236 T^{14} + \cdots + 16\!\cdots\!76$$
$53$ $$T^{16} - 8148 T^{14} + \cdots + 24\!\cdots\!16$$
$59$ $$(T^{8} + 312 T^{7} + \cdots + 14372454374404)^{2}$$
$61$ $$(T^{8} + 204 T^{7} + \cdots + 16806974336161)^{2}$$
$67$ $$T^{16} - 6480 T^{14} + \cdots + 14\!\cdots\!41$$
$71$ $$(T^{4} + 32 T^{3} - 11488 T^{2} + \cdots + 27042304)^{4}$$
$73$ $$T^{16} + 33740 T^{14} + \cdots + 39\!\cdots\!21$$
$79$ $$(T^{8} - 144 T^{7} + \cdots + 334870712077729)^{2}$$
$83$ $$(T^{8} - 10932 T^{6} + \cdots + 29997547204)^{2}$$
$89$ $$(T^{8} - 336 T^{7} + \cdots + 9916238788036)^{2}$$
$97$ $$(T^{8} - 29012 T^{6} + \cdots + 2745758363089)^{2}$$