Properties

 Label 363.3.b.m Level $363$ Weight $3$ Character orbit 363.b Analytic conductor $9.891$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 363.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$9.89103359628$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 29x^{6} + 282x^{4} + 1061x^{2} + 1331$$ x^8 + 29*x^6 + 282*x^4 + 1061*x^2 + 1331 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{3} + (\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 4) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{3} + \beta_1) q^{5} + (\beta_{4} - 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{6} + (\beta_{7} + \beta_{5} - \beta_{3} + 2 \beta_{2} + 5) q^{7} + ( - 2 \beta_{7} - 3 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{8} + (3 \beta_{7} + 4 \beta_{5} + \beta_{4} + 2 \beta_{2} + 3) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b5 + b4 - b2) * q^3 + (b6 + b4 + b3 - b2 - 4) * q^4 + (-b7 + b5 - b3 + b1) * q^5 + (b4 - 2*b3 + 3*b2 - b1 + 2) * q^6 + (b7 + b5 - b3 + 2*b2 + 5) * q^7 + (-2*b7 - 3*b5 - 2*b3 - 2*b1) * q^8 + (3*b7 + 4*b5 + b4 + 2*b2 + 3) * q^9 $$q + \beta_1 q^{2} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{3} + (\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 4) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{3} + \beta_1) q^{5} + (\beta_{4} - 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{6} + (\beta_{7} + \beta_{5} - \beta_{3} + 2 \beta_{2} + 5) q^{7} + ( - 2 \beta_{7} - 3 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{8} + (3 \beta_{7} + 4 \beta_{5} + \beta_{4} + 2 \beta_{2} + 3) q^{9} + (3 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2) q^{10} + (3 \beta_{7} - \beta_{5} - 3 \beta_{4} + \beta_{3} + 7 \beta_{2} - \beta_1 + 11) q^{12} + ( - 3 \beta_{7} - 3 \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + 5) q^{13} + (3 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 3 \beta_1) q^{14} + (3 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 4) q^{15} + (\beta_{7} - 3 \beta_{6} + \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 11 \beta_{2} + 5) q^{16} + ( - 3 \beta_{7} + 5 \beta_{6} - 4 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + 5 \beta_1) q^{17} + (3 \beta_{7} + 6 \beta_{6} - 6 \beta_{5} + \beta_{4} + 10 \beta_{3} - 6 \beta_{2} - \beta_1 - 4) q^{18} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} + 5) q^{19} + (3 \beta_{7} + 4 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} + 7 \beta_{3} - 2 \beta_1) q^{20} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{3} - 9 \beta_{2} - 3 \beta_1 - 6) q^{21} + (5 \beta_{7} - 6 \beta_{6} + 10 \beta_{5} + 6 \beta_{4} - \beta_{3} + 2 \beta_1) q^{23} + (6 \beta_{6} - 8 \beta_{5} - \beta_{4} + 10 \beta_{3} - 7 \beta_{2} + 8 \beta_1 - 1) q^{24} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 15 \beta_{2} + 13) q^{25} + ( - 9 \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 7 \beta_{3} + 11 \beta_1) q^{26} + (3 \beta_{7} + 4 \beta_{5} + \beta_{4} - 7 \beta_{2} - 9 \beta_1 - 6) q^{27} + ( - 5 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} + 10 \beta_{3} + \cdots - 30) q^{28}+ \cdots + (23 \beta_{7} + 8 \beta_{6} - \beta_{5} - 8 \beta_{4} + 31 \beta_{3} - 10 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b5 + b4 - b2) * q^3 + (b6 + b4 + b3 - b2 - 4) * q^4 + (-b7 + b5 - b3 + b1) * q^5 + (b4 - 2*b3 + 3*b2 - b1 + 2) * q^6 + (b7 + b5 - b3 + 2*b2 + 5) * q^7 + (-2*b7 - 3*b5 - 2*b3 - 2*b1) * q^8 + (3*b7 + 4*b5 + b4 + 2*b2 + 3) * q^9 + (3*b7 + b6 + 3*b5 + b4 - 2*b3 + 3*b2 + 2) * q^10 + (3*b7 - b5 - 3*b4 + b3 + 7*b2 - b1 + 11) * q^12 + (-3*b7 - 3*b5 + 3*b3 + 2*b2 + 5) * q^13 + (3*b7 + 2*b6 - 3*b5 - 2*b4 + 5*b3 + 3*b1) * q^14 + (3*b6 + b5 - b4 + b3 + 5*b2 + 2*b1 - 4) * q^15 + (b7 - 3*b6 + b5 - 3*b4 - 4*b3 + 11*b2 + 5) * q^16 + (-3*b7 + 5*b6 - 4*b5 - 5*b4 + 2*b3 + 5*b1) * q^17 + (3*b7 + 6*b6 - 6*b5 + b4 + 10*b3 - 6*b2 - b1 - 4) * q^18 + (b7 - 2*b6 + b5 - 2*b4 - 3*b3 - 9*b2 + 5) * q^19 + (3*b7 + 4*b6 - 6*b5 - 4*b4 + 7*b3 - 2*b1) * q^20 + (3*b6 + 3*b4 + 3*b3 - 9*b2 - 3*b1 - 6) * q^21 + (5*b7 - 6*b6 + 10*b5 + 6*b4 - b3 + 2*b1) * q^23 + (6*b6 - 8*b5 - b4 + 10*b3 - 7*b2 + 8*b1 - 1) * q^24 + (b7 + b6 + b5 + b4 + 15*b2 + 13) * q^25 + (-9*b7 + 2*b6 + b5 - 2*b4 - 7*b3 + 11*b1) * q^26 + (3*b7 + 4*b5 + b4 - 7*b2 - 9*b1 - 6) * q^27 + (-5*b7 + 5*b6 - 5*b5 + 5*b4 + 10*b3 - 23*b2 - 30) * q^28 + (2*b6 - 2*b5 - 2*b4 + 2*b3 + 2*b1) * q^29 + (3*b7 + 9*b6 - 8*b5 - 3*b4 + 8*b3 - 13*b2 - 8*b1 - 5) * q^30 + (5*b7 + 5*b6 + 5*b5 + 5*b4 - 15*b2 - 6) * q^31 + (b7 + 8*b6 - 12*b5 - 8*b4 + 9*b3 + b1) * q^32 + (2*b7 + 3*b6 + 2*b5 + 3*b4 + b3 - 26*b2 - 21) * q^34 + (-3*b7 + 4*b6 - 3*b5 - 4*b4 + b3 - 3*b1) * q^35 + (-9*b7 - 3*b6 - 4*b5 + 6*b4 + b3 - 23*b2 - 10*b1 - 16) * q^36 + (-9*b7 - 9*b5 + 9*b3 + 16*b2 + 21) * q^37 + (7*b7 - 11*b6 + 16*b5 + 11*b4 - 4*b3 + 7*b1) * q^38 + (-9*b6 + 12*b5 + 11*b4 - b3 + 15*b2 + b1 + 10) * q^39 + (3*b6 + 3*b4 + 3*b3 - 27*b2 - 7) * q^40 + (19*b7 - 13*b6 + 14*b5 + 13*b4 + 6*b3 - 10*b1) * q^41 + (-6*b7 - 9*b6 - 3*b5 + 3*b4 - 15*b3 + 3*b2 - 12*b1 + 24) * q^42 + (-7*b7 - 5*b6 - 7*b5 - 5*b4 + 2*b3 - 6*b2 - 15) * q^43 + (-3*b7 + 6*b6 - 11*b5 - 10*b4 + 7*b3 + 14*b2 - 7*b1 + 2) * q^45 + (11*b6 + 11*b4 + 11*b3 + 11*b2 - 33) * q^46 + (-14*b7 + 18*b6 - 15*b5 - 18*b4 + 4*b3 + 23*b1) * q^47 + (-9*b7 + 3*b6 - 15*b5 + 6*b3 - 12*b2 - 6*b1 - 45) * q^48 + (5*b7 - 4*b6 + 5*b5 - 4*b4 - 9*b3 + 12*b2 - 8) * q^49 + (b7 + 16*b6 - 20*b5 - 16*b4 + 17*b3 + 9*b1) * q^50 + (-15*b7 + 9*b6 - 11*b5 - 9*b4 - b3 + 14*b2 + 4*b1 + 13) * q^51 + (7*b7 + 5*b6 + 7*b5 + 5*b4 - 2*b3 + 25*b2 + 2) * q^52 + (6*b6 - 15*b5 - 6*b4 + 6*b3 + 7*b1) * q^53 + (3*b7 - 12*b6 + 3*b5 + b4 - 8*b3 + 3*b2 - 10*b1 + 68) * q^54 + (-13*b7 - 10*b6 - 4*b5 + 10*b4 - 23*b3 - 18*b1) * q^56 + (-6*b7 + 3*b6 + 7*b5 + b4 - 12*b3 - 28*b2 + 12*b1 - 15) * q^57 + (-2*b7 + 2*b6 - 2*b5 + 2*b4 + 4*b3 - 16*b2 - 18) * q^58 + (18*b7 - 11*b6 + 12*b5 + 11*b4 + 7*b3 + 11*b1) * q^59 + (-12*b7 - 9*b6 - 9*b5 - 5*b4 - 5*b3 - 15*b2 - 7*b1 + 23) * q^60 + (2*b7 - 11*b6 + 2*b5 - 11*b4 - 13*b3 + 17*b2 + 15) * q^61 + (5*b7 - 10*b6 - 10*b5 + 10*b4 - 5*b3 - 26*b1) * q^62 + (9*b7 + 9*b5 - 6*b4 + 3*b3 + 6*b1 + 33) * q^63 + (-10*b7 - 14*b6 - 10*b5 - 14*b4 - 4*b3 - 14*b2 - 15) * q^64 + (-3*b7 - 12*b6 + 13*b5 + 12*b4 - 15*b3 + 13*b1) * q^65 + (3*b7 - 12*b6 + 3*b5 - 12*b4 - 15*b3 - 21*b2 + 17) * q^67 + (-12*b7 - 3*b6 - 4*b5 + 3*b4 - 15*b3 - 11*b1) * q^68 + (18*b7 - 15*b6 + 25*b5 + 15*b4 - 19*b3 + 11*b2 - 17*b1 - 11) * q^69 + (3*b7 - 5*b6 + 3*b5 - 5*b4 - 8*b3 - 11*b2 + 44) * q^70 + (-5*b7 + 2*b6 - 11*b5 - 2*b4 - 3*b3 + 3*b1) * q^71 + (-3*b7 - 12*b6 + 7*b5 + 11*b4 - 26*b3 + 32*b2 - 10*b1 + 80) * q^72 + (2*b7 + 3*b6 + 2*b5 + 3*b4 + b3 - 18*b2 - 56) * q^73 + (-27*b7 + 16*b6 - 7*b5 - 16*b4 - 11*b3 + 39*b1) * q^74 + (3*b7 + 3*b6 - 3*b5 + 12*b4 + 18*b3 - 18*b1 - 9) * q^75 + (6*b7 + 11*b6 + 6*b5 + 11*b4 + 5*b3 - 6*b2 - 59) * q^76 + (-6*b7 + 27*b6 - 7*b5 - 5*b4 + 5*b3 + 43*b2 + 16*b1 - 8) * q^78 + (9*b7 - 8*b6 + 9*b5 - 8*b4 - 17*b3 + 48*b2 - 1) * q^79 + (6*b7 - 8*b6 - 9*b5 + 8*b4 - 2*b3 - 21*b1) * q^80 + (3*b7 + 4*b5 - 17*b4 + 9*b3 - 34*b2 + 9*b1 - 15) * q^81 + (-24*b7 + 10*b6 - 24*b5 + 10*b4 + 34*b3 + 5*b2 - 37) * q^82 + (13*b7 - 7*b6 + 10*b5 + 7*b4 + 6*b3 - 12*b1) * q^83 + (15*b7 - 15*b6 + 18*b5 - 15*b4 - 18*b3 + 57*b2 + 18*b1 + 93) * q^84 + (-7*b7 - 8*b6 - 7*b5 - 8*b4 - b3 + 28*b2 - 7) * q^85 + (-11*b7 - 11*b6 + 33*b5 + 11*b4 - 22*b3 + 9*b1) * q^86 + (-6*b7 - 4*b5 - 2*b4 - 4*b3 + 4*b2 - 2*b1 + 10) * q^87 + (b7 + 5*b6 - 8*b5 - 5*b4 + 6*b3 + 16*b1) * q^89 + (-9*b7 + 3*b6 - 11*b5 - 21*b4 + 8*b3 - 43*b2 + 10*b1 + 52) * q^90 + (-3*b7 + 4*b6 - 3*b5 + 4*b4 + 7*b3 + 28*b2 - 7) * q^91 + (-2*b7 - 2*b6 - 15*b5 + 2*b4 - 4*b3 - 47*b1) * q^92 + (15*b7 + 15*b6 + 4*b5 - 11*b4 - 19*b2 + 45) * q^93 + (13*b7 + 12*b6 + 13*b5 + 12*b4 - b3 - 82*b2 - 95) * q^94 + (-14*b7 - 4*b6 + 27*b5 + 4*b4 - 18*b3 + 19*b1) * q^95 + (-24*b7 - 3*b6 - 23*b5 - 10*b4 - 5*b3 - 10*b2 - 4*b1 + 41) * q^96 + (-2*b7 + 8*b6 - 2*b5 + 8*b4 + 10*b3 + 5*b2 - 41) * q^97 + (23*b7 + 8*b6 - b5 - 8*b4 + 31*b3 - 10*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 5 q^{3} - 26 q^{4} + q^{6} + 28 q^{7} + 11 q^{9}+O(q^{10})$$ 8 * q + 5 * q^3 - 26 * q^4 + q^6 + 28 * q^7 + 11 * q^9 $$8 q + 5 q^{3} - 26 q^{4} + q^{6} + 28 q^{7} + 11 q^{9} - 6 q^{10} + 53 q^{12} + 44 q^{13} - 54 q^{15} - 14 q^{16} + q^{18} + 68 q^{19} - 6 q^{21} + 33 q^{24} + 42 q^{25} - 25 q^{27} - 118 q^{28} + 10 q^{30} + 2 q^{31} - 66 q^{34} - 7 q^{36} + 140 q^{37} + 38 q^{39} + 58 q^{40} + 174 q^{42} - 78 q^{43} - 36 q^{45} - 286 q^{46} - 285 q^{48} - 140 q^{49} + 58 q^{51} - 102 q^{52} + 523 q^{54} - 22 q^{57} - 68 q^{58} + 262 q^{60} + 22 q^{61} + 246 q^{63} - 52 q^{64} + 184 q^{67} - 176 q^{69} + 374 q^{70} + 489 q^{72} - 378 q^{73} - 33 q^{75} - 450 q^{76} - 246 q^{78} - 252 q^{79} + 11 q^{81} - 200 q^{82} + 450 q^{84} - 156 q^{85} + 66 q^{87} + 598 q^{90} - 148 q^{91} + 380 q^{93} - 460 q^{94} + 399 q^{96} - 324 q^{97}+O(q^{100})$$ 8 * q + 5 * q^3 - 26 * q^4 + q^6 + 28 * q^7 + 11 * q^9 - 6 * q^10 + 53 * q^12 + 44 * q^13 - 54 * q^15 - 14 * q^16 + q^18 + 68 * q^19 - 6 * q^21 + 33 * q^24 + 42 * q^25 - 25 * q^27 - 118 * q^28 + 10 * q^30 + 2 * q^31 - 66 * q^34 - 7 * q^36 + 140 * q^37 + 38 * q^39 + 58 * q^40 + 174 * q^42 - 78 * q^43 - 36 * q^45 - 286 * q^46 - 285 * q^48 - 140 * q^49 + 58 * q^51 - 102 * q^52 + 523 * q^54 - 22 * q^57 - 68 * q^58 + 262 * q^60 + 22 * q^61 + 246 * q^63 - 52 * q^64 + 184 * q^67 - 176 * q^69 + 374 * q^70 + 489 * q^72 - 378 * q^73 - 33 * q^75 - 450 * q^76 - 246 * q^78 - 252 * q^79 + 11 * q^81 - 200 * q^82 + 450 * q^84 - 156 * q^85 + 66 * q^87 + 598 * q^90 - 148 * q^91 + 380 * q^93 - 460 * q^94 + 399 * q^96 - 324 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 29x^{6} + 282x^{4} + 1061x^{2} + 1331$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{6} + 30\nu^{4} + 260\nu^{2} + 541 ) / 52$$ (v^6 + 30*v^4 + 260*v^2 + 541) / 52 $$\beta_{3}$$ $$=$$ $$( -2\nu^{7} + 22\nu^{6} - 47\nu^{5} + 517\nu^{4} - 377\nu^{3} + 3575\nu^{2} - 1121\nu + 6897 ) / 286$$ (-2*v^7 + 22*v^6 - 47*v^5 + 517*v^4 - 377*v^3 + 3575*v^2 - 1121*v + 6897) / 286 $$\beta_{4}$$ $$=$$ $$( -27\nu^{7} + 11\nu^{6} - 706\nu^{5} + 330\nu^{4} - 5304\nu^{3} + 3432\nu^{2} - 9199\nu + 10527 ) / 1144$$ (-27*v^7 + 11*v^6 - 706*v^5 + 330*v^4 - 5304*v^3 + 3432*v^2 - 9199*v + 10527) / 1144 $$\beta_{5}$$ $$=$$ $$( 4\nu^{7} + 94\nu^{5} + 611\nu^{3} + 812\nu ) / 143$$ (4*v^7 + 94*v^5 + 611*v^3 + 812*v) / 143 $$\beta_{6}$$ $$=$$ $$( 35\nu^{7} - 77\nu^{6} + 894\nu^{5} - 1738\nu^{4} + 6812\nu^{3} - 10868\nu^{2} + 13683\nu - 17061 ) / 1144$$ (35*v^7 - 77*v^6 + 894*v^5 - 1738*v^4 + 6812*v^3 - 10868*v^2 + 13683*v - 17061) / 1144 $$\beta_{7}$$ $$=$$ $$( -10\nu^{7} - 22\nu^{6} - 235\nu^{5} - 517\nu^{4} - 1599\nu^{3} - 3575\nu^{2} - 2745\nu - 6897 ) / 286$$ (-10*v^7 - 22*v^6 - 235*v^5 - 517*v^4 - 1599*v^3 - 3575*v^2 - 2745*v - 6897) / 286
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 8$$ b6 + b4 + b3 - b2 - 8 $$\nu^{3}$$ $$=$$ $$-2\beta_{7} - 3\beta_{5} - 2\beta_{3} - 10\beta_1$$ -2*b7 - 3*b5 - 2*b3 - 10*b1 $$\nu^{4}$$ $$=$$ $$\beta_{7} - 15\beta_{6} + \beta_{5} - 15\beta_{4} - 16\beta_{3} + 23\beta_{2} + 85$$ b7 - 15*b6 + b5 - 15*b4 - 16*b3 + 23*b2 + 85 $$\nu^{5}$$ $$=$$ $$33\beta_{7} + 8\beta_{6} + 36\beta_{5} - 8\beta_{4} + 41\beta_{3} + 113\beta_1$$ 33*b7 + 8*b6 + 36*b5 - 8*b4 + 41*b3 + 113*b1 $$\nu^{6}$$ $$=$$ $$-30\beta_{7} + 190\beta_{6} - 30\beta_{5} + 190\beta_{4} + 220\beta_{3} - 378\beta_{2} - 1011$$ -30*b7 + 190*b6 - 30*b5 + 190*b4 + 220*b3 - 378*b2 - 1011 $$\nu^{7}$$ $$=$$ $$-470\beta_{7} - 188\beta_{6} - 352\beta_{5} + 188\beta_{4} - 658\beta_{3} - 1331\beta_1$$ -470*b7 - 188*b6 - 352*b5 + 188*b4 - 658*b3 - 1331*b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
122.1
 − 3.58727i − 3.06025i − 2.00420i − 1.65816i 1.65816i 2.00420i 3.06025i 3.58727i
3.58727i −2.74329 + 1.21424i −8.86854 2.08639i 4.35581 + 9.84093i 8.86303 17.4648i 6.05125 6.66201i 7.48447
122.2 3.06025i 0.126437 2.99733i −5.36516 6.63932i −9.17261 0.386930i −3.06298 4.17774i −8.96803 0.757950i −20.3180
122.3 2.00420i 2.80061 + 1.07544i −0.0168066 5.48438i 2.15539 5.61298i 5.59084 7.98311i 6.68687 + 6.02376i 10.9918
122.4 1.65816i 2.31624 + 1.90658i 1.25051 0.698503i 3.16141 3.84069i 2.60911 8.70618i 1.72990 + 8.83218i −1.15823
122.5 1.65816i 2.31624 1.90658i 1.25051 0.698503i 3.16141 + 3.84069i 2.60911 8.70618i 1.72990 8.83218i −1.15823
122.6 2.00420i 2.80061 1.07544i −0.0168066 5.48438i 2.15539 + 5.61298i 5.59084 7.98311i 6.68687 6.02376i 10.9918
122.7 3.06025i 0.126437 + 2.99733i −5.36516 6.63932i −9.17261 + 0.386930i −3.06298 4.17774i −8.96803 + 0.757950i −20.3180
122.8 3.58727i −2.74329 1.21424i −8.86854 2.08639i 4.35581 9.84093i 8.86303 17.4648i 6.05125 + 6.66201i 7.48447
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 122.8 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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.m 8
3.b odd 2 1 inner 363.3.b.m 8
11.b odd 2 1 363.3.b.l 8
11.c even 5 2 33.3.h.b 16
11.c even 5 2 363.3.h.o 16
11.d odd 10 2 363.3.h.j 16
11.d odd 10 2 363.3.h.n 16
33.d even 2 1 363.3.b.l 8
33.f even 10 2 363.3.h.j 16
33.f even 10 2 363.3.h.n 16
33.h odd 10 2 33.3.h.b 16
33.h odd 10 2 363.3.h.o 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.b 16 11.c even 5 2
33.3.h.b 16 33.h odd 10 2
363.3.b.l 8 11.b odd 2 1
363.3.b.l 8 33.d even 2 1
363.3.b.m 8 1.a even 1 1 trivial
363.3.b.m 8 3.b odd 2 1 inner
363.3.h.j 16 11.d odd 10 2
363.3.h.j 16 33.f even 10 2
363.3.h.n 16 11.d odd 10 2
363.3.h.n 16 33.f even 10 2
363.3.h.o 16 11.c even 5 2
363.3.h.o 16 33.h odd 10 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}^{8} + 29T_{2}^{6} + 282T_{2}^{4} + 1061T_{2}^{2} + 1331$$ T2^8 + 29*T2^6 + 282*T2^4 + 1061*T2^2 + 1331 $$T_{5}^{8} + 79T_{5}^{6} + 1687T_{5}^{4} + 6576T_{5}^{2} + 2816$$ T5^8 + 79*T5^6 + 1687*T5^4 + 6576*T5^2 + 2816 $$T_{7}^{4} - 14T_{7}^{3} + 35T_{7}^{2} + 138T_{7} - 396$$ T7^4 - 14*T7^3 + 35*T7^2 + 138*T7 - 396

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 29 T^{6} + 282 T^{4} + \cdots + 1331$$
$3$ $$T^{8} - 5 T^{7} + 7 T^{6} + 15 T^{5} + \cdots + 6561$$
$5$ $$T^{8} + 79 T^{6} + 1687 T^{4} + \cdots + 2816$$
$7$ $$(T^{4} - 14 T^{3} + 35 T^{2} + 138 T - 396)^{2}$$
$11$ $$T^{8}$$
$13$ $$(T^{4} - 22 T^{3} - 85 T^{2} + 2626 T + 4444)^{2}$$
$17$ $$T^{8} + 734 T^{6} + 172761 T^{4} + \cdots + 8306771$$
$19$ $$(T^{4} - 34 T^{3} + 7 T^{2} + 8694 T - 64999)^{2}$$
$23$ $$T^{8} + 1848 T^{6} + \cdots + 25255373616$$
$29$ $$T^{8} + 184 T^{6} + 7312 T^{4} + \cdots + 340736$$
$31$ $$(T^{4} - T^{3} - 1709 T^{2} - 30786 T - 97204)^{2}$$
$37$ $$(T^{4} - 70 T^{3} - 1111 T^{2} + \cdots - 698036)^{2}$$
$41$ $$T^{8} + 9433 T^{6} + \cdots + 3927187055891$$
$43$ $$(T^{4} + 39 T^{3} - 1528 T^{2} + \cdots - 684409)^{2}$$
$47$ $$T^{8} + 11237 T^{6} + \cdots + 14778028911536$$
$53$ $$T^{8} + 5633 T^{6} + \cdots + 1733384590256$$
$59$ $$T^{8} + \cdots + 162457590843251$$
$61$ $$(T^{4} - 11 T^{3} - 4549 T^{2} + \cdots + 1180476)^{2}$$
$67$ $$(T^{4} - 92 T^{3} - 3927 T^{2} + \cdots - 11977619)^{2}$$
$71$ $$T^{8} + 3227 T^{6} + \cdots + 1657701936$$
$73$ $$(T^{4} + 189 T^{3} + 12357 T^{2} + \cdots + 2336356)^{2}$$
$79$ $$(T^{4} + 126 T^{3} - 3903 T^{2} + \cdots - 27267284)^{2}$$
$83$ $$T^{8} + 4757 T^{6} + \cdots + 46820268891$$
$89$ $$T^{8} + 8529 T^{6} + \cdots + 3248250664131$$
$97$ $$(T^{4} + 162 T^{3} + 7293 T^{2} + \cdots - 3010169)^{2}$$