# Properties

 Label 105.3.n.a Level $105$ Weight $3$ Character orbit 105.n Analytic conductor $2.861$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 105.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.86104277578$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.523596960000.16 Defining polynomial: $$x^{8} - 2x^{7} + 13x^{6} - 2x^{5} + 91x^{4} - 50x^{3} + 190x^{2} + 100x + 100$$ x^8 - 2*x^7 + 13*x^6 - 2*x^5 + 91*x^4 - 50*x^3 + 190*x^2 + 100*x + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{4} + \beta_{6} q^{5} + (\beta_{3} - 2 \beta_1) q^{6} + (\beta_{6} - 4 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 5) q^{7} + (\beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 5) q^{8} - 3 \beta_{4} q^{9}+O(q^{10})$$ q + b1 * q^2 + (b4 - 1) * q^3 + (2*b6 + b5 - b4 - b3 + b1 - 1) * q^4 + b6 * q^5 + (b3 - 2*b1) * q^6 + (b6 - 4*b4 + b3 - b2 + b1 - 5) * q^7 + (b7 + b6 - 3*b5 - b4 - b3 - 2*b2 - 5) * q^8 - 3*b4 * q^9 $$q + \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{4} + \beta_{6} q^{5} + (\beta_{3} - 2 \beta_1) q^{6} + (\beta_{6} - 4 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 5) q^{7} + (\beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 5) q^{8} - 3 \beta_{4} q^{9} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{10} + (2 \beta_{7} - \beta_{5} + 5 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 4) q^{11} + ( - 3 \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_1 + 2) q^{12} + ( - 4 \beta_{7} + 5 \beta_{6} + 5 \beta_{5} + 2 \beta_{3} - 4 \beta_1 + 2) q^{13} + (\beta_{6} - 11 \beta_{4} - 6 \beta_{3} - \beta_{2} + \beta_1 - 5) q^{14} + ( - \beta_{6} + \beta_{5}) q^{15} + ( - 2 \beta_{7} + \beta_{6} - 3 \beta_{4} - 2 \beta_{2} - 5 \beta_1) q^{16} + (2 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + \beta_{2} + 4 \beta_1 - 3) q^{17} + ( - 3 \beta_{3} + 3 \beta_1) q^{18} + ( - 4 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + \cdots + 2) q^{19}+ \cdots + (3 \beta_{7} - 6 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} + 12) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b4 - 1) * q^3 + (2*b6 + b5 - b4 - b3 + b1 - 1) * q^4 + b6 * q^5 + (b3 - 2*b1) * q^6 + (b6 - 4*b4 + b3 - b2 + b1 - 5) * q^7 + (b7 + b6 - 3*b5 - b4 - b3 - 2*b2 - 5) * q^8 - 3*b4 * q^9 + (-b5 - b4 - b3 - b2 - b1) * q^10 + (2*b7 - b5 + 5*b4 - b3 - b2 + b1 + 4) * q^11 + (-3*b6 + b4 + 2*b3 - b1 + 2) * q^12 + (-4*b7 + 5*b6 + 5*b5 + 2*b3 - 4*b1 + 2) * q^13 + (b6 - 11*b4 - 6*b3 - b2 + b1 - 5) * q^14 + (-b6 + b5) * q^15 + (-2*b7 + b6 - 3*b4 - 2*b2 - 5*b1) * q^16 + (2*b5 + 4*b4 + 4*b3 + b2 + 4*b1 - 3) * q^17 + (-3*b3 + 3*b1) * q^18 + (-4*b7 - 4*b6 + 4*b5 + b4 + 4*b3 + 4*b2 - 2*b1 + 2) * q^19 + (-b7 - b6 - b5 + 9*b4 + b3 - 2*b1 + 5) * q^20 + (b7 - 2*b6 + 3*b4 + b2 - 3*b1 + 8) * q^21 + (b7 + 3*b6 - 5*b5 - b4 + 3*b3 - 2*b2 - 5) * q^22 + (4*b7 - 2*b6 - 12*b4 + 4*b2 + 3*b1) * q^23 + (6*b5 - 2*b4 + b3 + 3*b2 + b1 + 5) * q^24 + (5*b4 + 5) * q^25 + (b7 - 14*b6 - b5 + 10*b4 - 2*b3 - b2 + b1 + 20) * q^26 + (6*b4 + 3) * q^27 + (4*b7 - 6*b6 - 14*b5 + 4*b4 - 5*b3 - 5*b2 - 3*b1 - 13) * q^28 + (-5*b7 + 9*b6 + b5 + 5*b4 + 10*b2 - 5) * q^29 + (b7 + b5 + 2*b4 + b2 + 3*b1) * q^30 + (-7*b5 + 12*b4 - b3 + 4*b2 - b1 - 8) * q^31 + (6*b7 - 10*b6 - 8*b5 + 8*b4 - b3 - 3*b2 + b1 + 5) * q^32 + (-3*b7 + 3*b5 - 4*b4 + 2*b3 + 3*b2 - b1 - 8) * q^33 + (2*b7 + 14*b6 + 14*b5 - 38*b4 - 2*b3 + 4*b1 - 20) * q^34 + (2*b7 - 5*b6 - 7*b5 + 3*b4 - b3 - 4*b2) * q^35 + (3*b6 - 3*b5 - 3*b3 - 3) * q^36 + (-4*b7 - 3*b6 - 10*b5 + 22*b4 - 4*b2 + 8*b1) * q^37 + (22*b5 - 2*b4 + 7*b3 + 8*b2 + 7*b1 + 10) * q^38 + (8*b7 - 10*b6 - 9*b5 - 2*b4 - 6*b3 - 4*b2 + 6*b1 - 6) * q^39 + (2*b7 - 5*b6 - 2*b5 + 5*b4 + 6*b3 - 2*b2 - 3*b1 + 10) * q^40 + (-3*b7 + 12*b6 + 12*b5 + 5*b4 + 3*b3 - 6*b1 + 4) * q^41 + (b7 - 2*b6 + 17*b4 + 7*b3 + b2 + 4*b1 + 15) * q^42 + (-5*b7 + 2*b6 + 8*b5 + 5*b4 + 5*b3 + 10*b2 + 49) * q^43 + (b6 + 2*b5 + b4 - 7*b1) * q^44 - 3*b5 * q^45 + (4*b7 + 22*b6 + 9*b5 - 13*b4 - 17*b3 - 2*b2 + 17*b1 - 15) * q^46 + (b7 - 5*b6 - b5 - 7*b4 + 2*b3 - b2 - b1 - 14) * q^47 + (6*b7 - 3*b6 - 3*b5 + 4*b4 - 5*b3 + 10*b1 - 1) * q^48 + (-2*b7 - 7*b6 - 14*b5 - 4*b4 - 18*b3 + 2*b2 + 9*b1 + 11) * q^49 + 5*b3 * q^50 + (-b7 - b6 - 3*b5 - 11*b4 - b2 - 12*b1) * q^51 + (12*b5 + 10*b4 + 15*b3 - 3*b2 + 15*b1 - 13) * q^52 + (-10*b7 - 14*b6 - 2*b5 - 18*b4 + 3*b3 + 5*b2 - 3*b1 - 13) * q^53 + (6*b3 - 3*b1) * q^54 + (2*b7 + 4*b6 + 4*b5 + 2*b4 + 3*b3 - 6*b1) * q^55 + (-6*b7 - 13*b6 - 21*b5 + 12*b4 + 3*b3 - 2*b2 - 14*b1 + 35) * q^56 + (4*b7 + 4*b6 - 12*b5 - 4*b4 - 6*b3 - 8*b2 - 3) * q^57 + (-4*b7 + 10*b6 + 16*b5 - 8*b4 - 4*b2 - 32*b1) * q^58 + (14*b5 - 2*b4 - 14*b3 + b2 - 14*b1 + 3) * q^59 + (2*b7 + 2*b6 - 14*b4 - 3*b3 - b2 + 3*b1 - 15) * q^60 + (-10*b7 - 4*b6 + 10*b5 - 30*b4 + 8*b3 + 10*b2 - 4*b1 - 60) * q^61 + (-7*b7 + 5*b6 + 5*b5 + 3*b4 + 20*b3 - 40*b1 + 5) * q^62 + (-3*b7 + 3*b6 + 3*b4 - 3*b3 + 6*b1 - 9) * q^63 + (6*b7 + 3*b6 - 15*b5 - 6*b4 - b3 - 12*b2 - 9) * q^64 + (-2*b7 + 2*b6 + 2*b5 + 21*b4 - 2*b2 + 14*b1) * q^65 + (12*b5 - 2*b4 - 3*b3 + 3*b2 - 3*b1 + 5) * q^66 + (2*b7 - 20*b6 - 11*b5 + 48*b4 - 5*b3 - b2 + 5*b1 + 47) * q^67 + (12*b7 + 6*b6 - 12*b5 - 22*b4 - 40*b3 - 12*b2 + 20*b1 - 44) * q^68 + (-12*b7 + 6*b6 + 6*b5 + 28*b4 + 3*b3 - 6*b1 + 20) * q^69 + (-5*b7 - 5*b6 - 7*b5 + 3*b4 + 13*b3 + 3*b2 - 7*b1) * q^70 + (5*b7 - b6 - 9*b5 - 5*b4 + 28*b3 - 10*b2 - 1) * q^71 + (-3*b7 - 3*b6 - 9*b5 + 9*b4 - 3*b2 - 3*b1) * q^72 + (-23*b5 - 38*b4 - 11*b3 - 7*b2 - 11*b1 + 31) * q^73 + (-14*b7 + 7*b5 - 47*b4 + 31*b3 + 7*b2 - 31*b1 - 40) * q^74 + (-5*b4 - 10) * q^75 + (6*b7 + 21*b6 + 21*b5 - 56*b4 - 23*b3 + 46*b1 - 31) * q^76 + (5*b7 + 13*b6 - 7*b5 - 21*b4 + 9*b3 - 11*b2 - 13*b1 + 2) * q^77 + (-b7 + 14*b6 - 12*b5 + b4 + 3*b3 + 2*b2 - 30) * q^78 + (6*b7 + 3*b6 + 12*b5 + 52*b4 + 6*b2 - 5*b1) * q^79 + (4*b4 + 9*b3 - b2 + 9*b1 - 5) * q^80 + (-9*b4 - 9) * q^81 + (9*b7 - 15*b6 - 9*b5 + 15*b4 - 10*b3 - 9*b2 + 5*b1 + 30) * q^82 + (-5*b7 + 37*b6 + 37*b5 + 13*b4 - 12*b3 + 24*b1 + 9) * q^83 + (-3*b7 + 15*b6 + 21*b5 - 17*b4 + 2*b3 + 9*b2 + 11*b1 + 8) * q^84 + (3*b7 - 3*b6 - 3*b5 - 3*b4 - 14*b3 - 6*b2 - 5) * q^85 + (3*b7 + 15*b6 + 33*b5 - 19*b4 + 3*b2 + 48*b1) * q^86 + (12*b5 - 20*b4 - 15*b2 + 5) * q^87 + (10*b7 + 10*b6 + 20*b4 + 17*b3 - 5*b2 - 17*b1 + 15) * q^88 + (11*b7 + 11*b6 - 11*b5 + 23*b4 - 20*b3 - 11*b2 + 10*b1 + 46) * q^89 + (-3*b7 - 3*b4 + 3*b3 - 6*b1) * q^90 + (14*b7 - 48*b6 - 35*b5 - 4*b4 + 15*b3 + 6*b2 + 15*b1 - 33) * q^91 + (-3*b7 + 29*b6 - 23*b5 + 3*b4 - 53*b3 + 6*b2 - 5) * q^92 + (-4*b7 + 11*b6 + 18*b5 - 32*b4 - 4*b2 + 3*b1) * q^93 + (5*b5 - b4 - b3 + 4*b2 - b1 + 5) * q^94 + (-4*b7 + 2*b6 + 3*b5 - 22*b4 - 14*b3 + 2*b2 + 14*b1 - 20) * q^95 + (-9*b7 + 15*b6 + 9*b5 - 5*b4 + 2*b3 + 9*b2 - b1 - 10) * q^96 + (10*b7 + 16*b6 + 16*b5 + 22*b4 + 14*b3 - 28*b1 + 6) * q^97 + (-16*b7 - 14*b5 + 38*b4 + 10*b3 + 9*b2 - 19*b1 - 45) * q^98 + (3*b7 - 6*b5 - 3*b4 - 3*b3 - 6*b2 + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{2} - 12 q^{3} - 6 q^{4} - 16 q^{7} - 32 q^{8} + 12 q^{9}+O(q^{10})$$ 8 * q + 2 * q^2 - 12 * q^3 - 6 * q^4 - 16 * q^7 - 32 * q^8 + 12 * q^9 $$8 q + 2 q^{2} - 12 q^{3} - 6 q^{4} - 16 q^{7} - 32 q^{8} + 12 q^{9} + 20 q^{11} + 18 q^{12} - 16 q^{14} - 2 q^{16} - 18 q^{17} - 6 q^{18} + 48 q^{21} - 16 q^{22} + 62 q^{23} + 48 q^{24} + 20 q^{25} + 120 q^{26} - 120 q^{28} - 100 q^{29} - 126 q^{31} + 36 q^{32} - 60 q^{33} - 36 q^{36} - 80 q^{37} + 114 q^{38} - 12 q^{39} + 90 q^{40} + 90 q^{42} + 352 q^{43} - 18 q^{44} - 82 q^{46} - 72 q^{47} + 38 q^{49} + 20 q^{50} + 18 q^{51} - 48 q^{52} - 76 q^{53} + 18 q^{54} + 196 q^{56} - 40 q^{58} - 54 q^{59} - 60 q^{60} - 396 q^{61} - 96 q^{63} - 4 q^{64} - 60 q^{65} + 24 q^{66} + 184 q^{67} - 312 q^{68} + 164 q^{71} - 48 q^{72} + 348 q^{73} - 140 q^{74} - 60 q^{75} + 152 q^{77} - 240 q^{78} - 206 q^{79} - 36 q^{81} + 204 q^{82} + 132 q^{84} - 60 q^{85} + 178 q^{86} + 150 q^{87} + 124 q^{88} + 282 q^{89} - 114 q^{91} - 288 q^{92} + 126 q^{93} + 30 q^{94} - 120 q^{95} - 108 q^{96} - 592 q^{98} + 120 q^{99}+O(q^{100})$$ 8 * q + 2 * q^2 - 12 * q^3 - 6 * q^4 - 16 * q^7 - 32 * q^8 + 12 * q^9 + 20 * q^11 + 18 * q^12 - 16 * q^14 - 2 * q^16 - 18 * q^17 - 6 * q^18 + 48 * q^21 - 16 * q^22 + 62 * q^23 + 48 * q^24 + 20 * q^25 + 120 * q^26 - 120 * q^28 - 100 * q^29 - 126 * q^31 + 36 * q^32 - 60 * q^33 - 36 * q^36 - 80 * q^37 + 114 * q^38 - 12 * q^39 + 90 * q^40 + 90 * q^42 + 352 * q^43 - 18 * q^44 - 82 * q^46 - 72 * q^47 + 38 * q^49 + 20 * q^50 + 18 * q^51 - 48 * q^52 - 76 * q^53 + 18 * q^54 + 196 * q^56 - 40 * q^58 - 54 * q^59 - 60 * q^60 - 396 * q^61 - 96 * q^63 - 4 * q^64 - 60 * q^65 + 24 * q^66 + 184 * q^67 - 312 * q^68 + 164 * q^71 - 48 * q^72 + 348 * q^73 - 140 * q^74 - 60 * q^75 + 152 * q^77 - 240 * q^78 - 206 * q^79 - 36 * q^81 + 204 * q^82 + 132 * q^84 - 60 * q^85 + 178 * q^86 + 150 * q^87 + 124 * q^88 + 282 * q^89 - 114 * q^91 - 288 * q^92 + 126 * q^93 + 30 * q^94 - 120 * q^95 - 108 * q^96 - 592 * q^98 + 120 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 13x^{6} - 2x^{5} + 91x^{4} - 50x^{3} + 190x^{2} + 100x + 100$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} + 20\nu^{6} - 51\nu^{5} + 304\nu^{4} - 193\nu^{3} + 1752\nu^{2} - 2510\nu + 2630 ) / 630$$ (v^7 + 20*v^6 - 51*v^5 + 304*v^4 - 193*v^3 + 1752*v^2 - 2510*v + 2630) / 630 $$\beta_{3}$$ $$=$$ $$( -87\nu^{7} + 24\nu^{6} - 841\nu^{5} - 1276\nu^{4} - 10117\nu^{3} - 4640\nu^{2} - 2900\nu - 13700 ) / 21630$$ (-87*v^7 + 24*v^6 - 841*v^5 - 1276*v^4 - 10117*v^3 - 4640*v^2 - 2900*v - 13700) / 21630 $$\beta_{4}$$ $$=$$ $$( 137\nu^{7} - 361\nu^{6} + 1805\nu^{5} - 1115\nu^{4} + 11191\nu^{3} - 16967\nu^{2} + 21390\nu - 10830 ) / 21630$$ (137*v^7 - 361*v^6 + 1805*v^5 - 1115*v^4 + 11191*v^3 - 16967*v^2 + 21390*v - 10830) / 21630 $$\beta_{5}$$ $$=$$ $$( 472\nu^{7} - 1249\nu^{6} + 6966\nu^{5} - 8699\nu^{4} + 48092\nu^{3} - 74565\nu^{2} + 78220\nu - 131200 ) / 64890$$ (472*v^7 - 1249*v^6 + 6966*v^5 - 8699*v^4 + 48092*v^3 - 74565*v^2 + 78220*v - 131200) / 64890 $$\beta_{6}$$ $$=$$ $$( 661\nu^{7} - 2047\nu^{6} + 8793\nu^{5} - 5927\nu^{4} + 44711\nu^{3} - 64485\nu^{2} + 84520\nu + 126050 ) / 64890$$ (661*v^7 - 2047*v^6 + 8793*v^5 - 5927*v^4 + 44711*v^3 - 64485*v^2 + 84520*v + 126050) / 64890 $$\beta_{7}$$ $$=$$ $$( -977\nu^{7} + 1985\nu^{6} - 15693\nu^{5} + 4657\nu^{4} - 114889\nu^{3} + 25521\nu^{2} - 381050\nu - 55810 ) / 64890$$ (-977*v^7 + 1985*v^6 - 15693*v^5 + 4657*v^4 - 114889*v^3 + 25521*v^2 - 381050*v - 55810) / 64890
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{6} + \beta_{5} - 5\beta_{4} - \beta_{3} + \beta _1 - 5$$ 2*b6 + b5 - 5*b4 - b3 + b1 - 5 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} - 9\beta_{3} - 2\beta_{2} - 5$$ b7 + b6 - 3*b5 - b4 - 9*b3 - 2*b2 - 5 $$\nu^{4}$$ $$=$$ $$-2\beta_{7} - 11\beta_{6} - 24\beta_{5} + 41\beta_{4} - 2\beta_{2} - 17\beta_1$$ -2*b7 - 11*b6 - 24*b5 + 41*b4 - 2*b2 - 17*b1 $$\nu^{5}$$ $$=$$ $$-26\beta_{7} - 42\beta_{6} - 8\beta_{5} + 72\beta_{4} + 95\beta_{3} + 13\beta_{2} - 95\beta _1 + 85$$ -26*b7 - 42*b6 - 8*b5 + 72*b4 + 95*b3 + 13*b2 - 95*b1 + 85 $$\nu^{6}$$ $$=$$ $$-34\beta_{7} - 121\beta_{6} + 189\beta_{5} + 34\beta_{4} + 243\beta_{3} + 68\beta_{2} + 475$$ -34*b7 - 121*b6 + 189*b5 + 34*b4 + 243*b3 + 68*b2 + 475 $$\nu^{7}$$ $$=$$ $$155\beta_{7} + 311\beta_{6} + 777\beta_{5} - 905\beta_{4} + 155\beta_{2} + 1081\beta_1$$ 155*b7 + 311*b6 + 777*b5 - 905*b4 + 155*b2 + 1081*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/105\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$71$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −1.26021 − 2.18275i −0.336732 − 0.583237i 0.836732 + 1.44926i 1.76021 + 3.04878i −1.26021 + 2.18275i −0.336732 + 0.583237i 0.836732 − 1.44926i 1.76021 − 3.04878i
−1.26021 2.18275i −1.50000 0.866025i −1.17628 + 2.03737i −1.93649 + 1.11803i 4.36551i −6.18050 + 3.28656i −4.15226 1.50000 + 2.59808i 4.88079 + 2.81792i
31.2 −0.336732 0.583237i −1.50000 0.866025i 1.77322 3.07131i 1.93649 1.11803i 1.16647i −6.82455 1.55742i −5.08226 1.50000 + 2.59808i −1.30416 0.752955i
31.3 0.836732 + 1.44926i −1.50000 0.866025i 0.599760 1.03881i 1.93649 1.11803i 2.89852i 4.76104 + 5.13152i 8.70121 1.50000 + 2.59808i 3.24065 + 1.87099i
31.4 1.76021 + 3.04878i −1.50000 0.866025i −4.19671 + 7.26891i −1.93649 + 1.11803i 6.09756i 0.244004 + 6.99575i −15.4667 1.50000 + 2.59808i −6.81728 3.93596i
61.1 −1.26021 + 2.18275i −1.50000 + 0.866025i −1.17628 2.03737i −1.93649 1.11803i 4.36551i −6.18050 3.28656i −4.15226 1.50000 2.59808i 4.88079 2.81792i
61.2 −0.336732 + 0.583237i −1.50000 + 0.866025i 1.77322 + 3.07131i 1.93649 + 1.11803i 1.16647i −6.82455 + 1.55742i −5.08226 1.50000 2.59808i −1.30416 + 0.752955i
61.3 0.836732 1.44926i −1.50000 + 0.866025i 0.599760 + 1.03881i 1.93649 + 1.11803i 2.89852i 4.76104 5.13152i 8.70121 1.50000 2.59808i 3.24065 1.87099i
61.4 1.76021 3.04878i −1.50000 + 0.866025i −4.19671 7.26891i −1.93649 1.11803i 6.09756i 0.244004 6.99575i −15.4667 1.50000 2.59808i −6.81728 + 3.93596i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.n.a 8
3.b odd 2 1 315.3.w.a 8
5.b even 2 1 525.3.o.l 8
5.c odd 4 2 525.3.s.h 16
7.c even 3 1 735.3.h.a 8
7.d odd 6 1 inner 105.3.n.a 8
7.d odd 6 1 735.3.h.a 8
21.g even 6 1 315.3.w.a 8
35.i odd 6 1 525.3.o.l 8
35.k even 12 2 525.3.s.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.n.a 8 1.a even 1 1 trivial
105.3.n.a 8 7.d odd 6 1 inner
315.3.w.a 8 3.b odd 2 1
315.3.w.a 8 21.g even 6 1
525.3.o.l 8 5.b even 2 1
525.3.o.l 8 35.i odd 6 1
525.3.s.h 16 5.c odd 4 2
525.3.s.h 16 35.k even 12 2
735.3.h.a 8 7.c even 3 1
735.3.h.a 8 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 2T_{2}^{7} + 13T_{2}^{6} - 2T_{2}^{5} + 91T_{2}^{4} - 50T_{2}^{3} + 190T_{2}^{2} + 100T_{2} + 100$$ acting on $$S_{3}^{\mathrm{new}}(105, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 2 T^{7} + 13 T^{6} - 2 T^{5} + \cdots + 100$$
$3$ $$(T^{2} + 3 T + 3)^{4}$$
$5$ $$(T^{4} - 5 T^{2} + 25)^{2}$$
$7$ $$T^{8} + 16 T^{7} + 109 T^{6} + \cdots + 5764801$$
$11$ $$T^{8} - 20 T^{7} + 337 T^{6} + \cdots + 196$$
$13$ $$T^{8} + 1164 T^{6} + \cdots + 1230957225$$
$17$ $$T^{8} + 18 T^{7} + \cdots + 138297600$$
$19$ $$T^{8} - 846 T^{6} + 712707 T^{4} + \cdots + 9054081$$
$23$ $$T^{8} - 62 T^{7} + \cdots + 7138560100$$
$29$ $$(T^{4} + 50 T^{3} - 2130 T^{2} + \cdots - 1825400)^{2}$$
$31$ $$T^{8} + 126 T^{7} + \cdots + 385089749136$$
$37$ $$T^{8} + 80 T^{7} + \cdots + 5596891350625$$
$41$ $$T^{8} + 3342 T^{6} + \cdots + 13887679716$$
$43$ $$(T^{4} - 176 T^{3} + 9621 T^{2} + \cdots + 762376)^{2}$$
$47$ $$T^{8} + 72 T^{7} + 2115 T^{6} + \cdots + 26010000$$
$53$ $$T^{8} + 76 T^{7} + \cdots + 98219560000$$
$59$ $$T^{8} + 54 T^{7} + \cdots + 2582886122496$$
$61$ $$T^{8} + 396 T^{7} + \cdots + 84471609600$$
$67$ $$T^{8} - 184 T^{7} + \cdots + 273278017600$$
$71$ $$(T^{4} - 82 T^{3} - 7998 T^{2} + \cdots + 22760224)^{2}$$
$73$ $$T^{8} - 348 T^{7} + \cdots + 16\!\cdots\!36$$
$79$ $$T^{8} + 206 T^{7} + \cdots + 4446784387600$$
$83$ $$T^{8} + \cdots + 111959592561216$$
$89$ $$T^{8} + \cdots + 580473805464576$$
$97$ $$T^{8} + 30696 T^{6} + \cdots + 2211287961600$$