Properties

 Label 1755.2.i.f Level $1755$ Weight $2$ Character orbit 1755.i Analytic conductor $14.014$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1755 = 3^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1755.i (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$14.0137455547$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + 326 x^{7} + 551 x^{6} + 859 x^{5} + 1118 x^{4} + 1215 x^{3} + 1103 x^{2} + \cdots + 268$$ x^16 - 5*x^15 + 20*x^14 - 44*x^13 + 96*x^12 - 107*x^11 + 178*x^10 - 19*x^9 + 231*x^8 + 326*x^7 + 551*x^6 + 859*x^5 + 1118*x^4 + 1215*x^3 + 1103*x^2 + 770*x + 268 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: no (minimal twist has level 585) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} - \beta_1) q^{2} + ( - \beta_{11} + \beta_{2} - 1) q^{4} + (\beta_{2} - 1) q^{5} + ( - \beta_{15} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{12} + \beta_1 + 1) q^{8}+O(q^{10})$$ q + (b6 - b1) * q^2 + (-b11 + b2 - 1) * q^4 + (b2 - 1) * q^5 + (-b15 - b6 - b4 + b3 + b2 + b1) * q^7 + (b12 + b1 + 1) * q^8 $$q + (\beta_{6} - \beta_1) q^{2} + ( - \beta_{11} + \beta_{2} - 1) q^{4} + (\beta_{2} - 1) q^{5} + ( - \beta_{15} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{12} + \beta_1 + 1) q^{8} + \beta_1 q^{10} + ( - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{5}) q^{11} + ( - \beta_{2} + 1) q^{13} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{2} + 2) q^{14} + (\beta_{14} + \beta_{13} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{16} + ( - \beta_{15} - \beta_{12} + 2 \beta_{10} - \beta_{9} - \beta_{3} - \beta_1 + 1) q^{17} + ( - \beta_{15} + \beta_{14} + \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{3} - \beta_1 - 2) q^{19} + (\beta_{11} + \beta_{8} - \beta_{2}) q^{20} + (\beta_{13} + \beta_{11} - \beta_{10} + 2 \beta_{9} - 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{2}) q^{22} + ( - \beta_{13} + 2 \beta_{10} - 2 \beta_{5} - \beta_{4} + 2) q^{23} - \beta_{2} q^{25} - \beta_1 q^{26} + (\beta_{12} - 3 \beta_{10} + 2 \beta_{8} - \beta_1 - 6) q^{28} + (\beta_{14} + \beta_{13} - 2 \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{29} + ( - \beta_{13} + \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \cdots + 5) q^{31}+ \cdots + (4 \beta_{15} - \beta_{14} + 3 \beta_{12} - 3 \beta_{10} + 4 \beta_{9} - 3 \beta_{8} - \beta_{3} + \cdots + 5) q^{98}+O(q^{100})$$ q + (b6 - b1) * q^2 + (-b11 + b2 - 1) * q^4 + (b2 - 1) * q^5 + (-b15 - b6 - b4 + b3 + b2 + b1) * q^7 + (b12 + b1 + 1) * q^8 + b1 * q^10 + (-b15 - b14 - b13 + b5) * q^11 + (-b2 + 1) * q^13 + (b11 - b10 + b9 - b7 + 2*b6 + b5 - b4 - 3*b2 + 2) * q^14 + (b14 + b13 + b6 - b5 + b4 - b3 - b2 - b1) * q^16 + (-b15 - b12 + 2*b10 - b9 - b3 - b1 + 1) * q^17 + (-b15 + b14 + b12 + b10 - b9 + b8 + b3 - b1 - 2) * q^19 + (b11 + b8 - b2) * q^20 + (b13 + b11 - b10 + 2*b9 - 2*b7 + b6 + b5 - b2) * q^22 + (-b13 + 2*b10 - 2*b5 - b4 + 2) * q^23 - b2 * q^25 - b1 * q^26 + (b12 - 3*b10 + 2*b8 - b1 - 6) * q^28 + (b14 + b13 - 2*b6 - b4 + b3 + 2*b2 + 2*b1) * q^29 + (-b13 + b11 + 2*b10 + 2*b9 - 2*b6 - 2*b5 - b4 - 3*b2 + 5) * q^31 + (-b13 - 2*b11 - b9 + b7 + b6 + b4) * q^32 + (-3*b15 + b14 + b13 - b12 - b7 + 2*b6 + 4*b5 - 3*b2 - 2*b1) * q^34 + (b15 + b9 - b3 - b1 - 1) * q^35 + (-b12 - b10 + b8 - b3 + 2*b1 + 1) * q^37 + (-2*b15 + 2*b14 + 2*b13 + b11 + b8 - 2*b6 + 2*b5 + 2*b1) * q^38 + (b7 - b6 + b2 - 1) * q^40 + (-2*b13 - b11 + b10 + b7 + b6 - b5 + b4 + 2*b2 - 1) * q^41 + (b12 - b11 - b8 + b7 + 2*b5 + 3*b4 - 3*b3 - 2*b2) * q^43 + (b15 - b14 - b12 - 4*b10 + b9 + 2*b8 - 2*b3 + b1 - 2) * q^44 + (-2*b15 + 2*b14 + 3*b10 - 2*b9 + 2*b8 - b3 - 3*b1 - 4) * q^46 + (2*b15 + 2*b12 - 2*b11 - 2*b8 + 2*b7 + 2*b6 - 2*b5 + 2*b4 - 2*b3 - b2 - 2*b1) * q^47 + (-b13 - b10 + 3*b9 - 2*b6 + b5 - b4 + b2 - 2) * q^49 - b6 * q^50 + (-b11 - b8 + b2) * q^52 + (-2*b15 + b14 + 2*b12 + 2*b10 - 2*b9 - b8 + 2*b3 - 2*b1 - 2) * q^53 + (b15 + b14 - b10 + b9 - 1) * q^55 + (b15 - 2*b14 - 2*b13 + 3*b12 - b11 - b8 + 3*b7 - 6*b6 - 2*b5 - b4 + b3 + 5*b2 + 6*b1) * q^56 + (2*b11 - b10 + 3*b6 + b5 - b4 - 7*b2 + 6) * q^58 + (-b11 - b10 - 2*b9 + b7 - b6 + b5 - b4 - 3*b2 + 2) * q^59 + (b14 + b13 + 3*b12 + b11 + b8 + 3*b7 - 4*b5 - 2*b4 + 2*b3 + 2*b2) * q^61 + (2*b14 - b12 + b10 - b3 - 6*b1 + 1) * q^62 + (-b15 + b14 + 2*b12 + b10 - b9 - 4) * q^64 + b2 * q^65 + (b13 - 2*b11 - 2*b10 - b9 + 2*b7 + b6 + 2*b5 - 3*b4 - 5*b2 + 3) * q^67 + (3*b13 + b11 - 5*b10 + 5*b9 - b7 + 3*b6 + 5*b5 - 3*b4 - 5*b2) * q^68 + (b15 + b12 - b11 - b8 + b7 - 2*b6 - b5 + b4 - b3 + 3*b2 + 2*b1) * q^70 + (2*b15 + b14 - b12 - b10 + 2*b9 - 2*b8 - 2*b1 - 2) * q^71 + (-3*b14 + 2*b12 - b10 - b8 + 2*b1 + 1) * q^73 + (b15 - 2*b14 - 2*b13 + 3*b12 + 3*b7 - b6 - 4*b2 + b1) * q^74 + (2*b11 - 4*b10 + 2*b9 - b7 + 2*b6 + 4*b5 + 2*b4 - 9*b2 + 5) * q^76 + (-2*b13 - b11 - b10 + 4*b9 + 2*b7 + b5 - 3*b4 + b2 - 2) * q^77 + (-2*b15 + b12 - b11 - b8 + b7 + 6*b6 - b5 + b4 - b3 - 2*b2 - 6*b1) * q^79 + (-b14 + b10 + b3 + b1 + 2) * q^80 + (-b15 + b12 + 4*b10 - b9 + b8 + 2*b3 + b1 - 3) * q^82 + (-2*b14 - 2*b13 - b12 - b7 + b6 + b5 + 2*b4 - 2*b3 - b2 - b1) * q^83 + (-2*b10 + b9 - b7 + b6 + 2*b5 + b4 - b2 - 1) * q^85 + (3*b13 + 3*b11 - b10 + 2*b9 + 2*b6 + b5 + 4*b4 - 8*b2 + 7) * q^86 + (b15 - 3*b14 - 3*b13 + 3*b12 - 2*b11 - 2*b8 + 3*b7 - 6*b6 - 3*b5 + b4 - b3 + 6*b2 + 6*b1) * q^88 + (-3*b15 + b14 + b12 + b10 - 3*b9 + 3*b8 + 2*b3 - b1) * q^89 + (-b15 - b9 + b3 + b1 + 1) * q^91 + (-5*b15 + b14 + b13 + 2*b12 + 2*b7 - 4*b6 + 3*b5 - b4 + b3 - 2*b2 + 4*b1) * q^92 + (-2*b11 + 6*b10 - 4*b9 + 2*b7 - b6 - 6*b5 + 4*b4 + 8*b2 - 2) * q^94 + (b13 + b11 - b10 + b9 + b7 + b6 + b5 - b4 - 3*b2 + 2) * q^95 + (b15 - 3*b14 - 3*b13 + 2*b11 + 2*b8 - 3*b6 - b4 + b3 + 6*b2 + 3*b1) * q^97 + (4*b15 - b14 + 3*b12 - 3*b10 + 4*b9 - 3*b8 - b3 + 4*b1 + 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 3 q^{2} - 9 q^{4} - 8 q^{5} + 11 q^{7} + 12 q^{8}+O(q^{10})$$ 16 * q + 3 * q^2 - 9 * q^4 - 8 * q^5 + 11 * q^7 + 12 * q^8 $$16 q + 3 q^{2} - 9 q^{4} - 8 q^{5} + 11 q^{7} + 12 q^{8} - 6 q^{10} + 6 q^{11} + 8 q^{13} + 10 q^{14} - 11 q^{16} + 4 q^{17} - 20 q^{19} - 9 q^{20} - 3 q^{22} + 6 q^{23} - 8 q^{25} + 6 q^{26} - 68 q^{28} + 14 q^{29} + 31 q^{31} + q^{32} + 7 q^{34} - 22 q^{35} + 2 q^{37} + 9 q^{38} - 6 q^{40} - 12 q^{41} - 15 q^{43} - 32 q^{44} - 64 q^{46} - 18 q^{47} - 17 q^{49} + 3 q^{50} + 9 q^{52} - 4 q^{53} - 12 q^{55} + 16 q^{56} + 42 q^{58} + 24 q^{59} + 9 q^{61} + 40 q^{62} - 60 q^{64} + 8 q^{65} + 18 q^{67} - 14 q^{68} + 10 q^{70} - 20 q^{71} + 12 q^{73} - 37 q^{74} + 53 q^{76} - 34 q^{77} + 3 q^{79} + 22 q^{80} - 68 q^{82} - 10 q^{83} - 2 q^{85} + 60 q^{86} + 14 q^{88} + 26 q^{89} + 22 q^{91} + 5 q^{92} - 17 q^{94} + 10 q^{95} + 34 q^{97} + 60 q^{98}+O(q^{100})$$ 16 * q + 3 * q^2 - 9 * q^4 - 8 * q^5 + 11 * q^7 + 12 * q^8 - 6 * q^10 + 6 * q^11 + 8 * q^13 + 10 * q^14 - 11 * q^16 + 4 * q^17 - 20 * q^19 - 9 * q^20 - 3 * q^22 + 6 * q^23 - 8 * q^25 + 6 * q^26 - 68 * q^28 + 14 * q^29 + 31 * q^31 + q^32 + 7 * q^34 - 22 * q^35 + 2 * q^37 + 9 * q^38 - 6 * q^40 - 12 * q^41 - 15 * q^43 - 32 * q^44 - 64 * q^46 - 18 * q^47 - 17 * q^49 + 3 * q^50 + 9 * q^52 - 4 * q^53 - 12 * q^55 + 16 * q^56 + 42 * q^58 + 24 * q^59 + 9 * q^61 + 40 * q^62 - 60 * q^64 + 8 * q^65 + 18 * q^67 - 14 * q^68 + 10 * q^70 - 20 * q^71 + 12 * q^73 - 37 * q^74 + 53 * q^76 - 34 * q^77 + 3 * q^79 + 22 * q^80 - 68 * q^82 - 10 * q^83 - 2 * q^85 + 60 * q^86 + 14 * q^88 + 26 * q^89 + 22 * q^91 + 5 * q^92 - 17 * q^94 + 10 * q^95 + 34 * q^97 + 60 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + 326 x^{7} + 551 x^{6} + 859 x^{5} + 1118 x^{4} + 1215 x^{3} + 1103 x^{2} + \cdots + 268$$ :

 $$\beta_{1}$$ $$=$$ $$( - \nu^{15} + 4 \nu^{14} - 16 \nu^{13} + 28 \nu^{12} - 68 \nu^{11} + 39 \nu^{10} - 139 \nu^{9} - 120 \nu^{8} - 351 \nu^{7} - 677 \nu^{6} - 1228 \nu^{5} - 2087 \nu^{4} - 3205 \nu^{3} - 4420 \nu^{2} + \cdots - 6293 ) / 2187$$ (-v^15 + 4*v^14 - 16*v^13 + 28*v^12 - 68*v^11 + 39*v^10 - 139*v^9 - 120*v^8 - 351*v^7 - 677*v^6 - 1228*v^5 - 2087*v^4 - 3205*v^3 - 4420*v^2 - 3336*v - 6293) / 2187 $$\beta_{2}$$ $$=$$ $$( - 53 \nu^{15} + 1221 \nu^{14} - 4880 \nu^{13} + 14280 \nu^{12} - 18634 \nu^{11} + 24185 \nu^{10} + 23394 \nu^{9} - 18231 \nu^{8} + 152463 \nu^{7} + 42206 \nu^{6} + \cdots + 194118 ) / 68526$$ (-53*v^15 + 1221*v^14 - 4880*v^13 + 14280*v^12 - 18634*v^11 + 24185*v^10 + 23394*v^9 - 18231*v^8 + 152463*v^7 + 42206*v^6 + 312111*v^5 + 354893*v^4 + 509676*v^3 + 567013*v^2 + 436375*v + 194118) / 68526 $$\beta_{3}$$ $$=$$ $$( - \nu^{15} - 7 \nu^{14} + 50 \nu^{13} - 217 \nu^{12} + 469 \nu^{11} - 874 \nu^{10} + 757 \nu^{9} - 840 \nu^{8} - 447 \nu^{7} - 917 \nu^{6} - 1580 \nu^{5} - 2093 \nu^{4} - 2345 \nu^{3} - 607 \nu^{2} + \cdots + 2021 ) / 729$$ (-v^15 - 7*v^14 + 50*v^13 - 217*v^12 + 469*v^11 - 874*v^10 + 757*v^9 - 840*v^8 - 447*v^7 - 917*v^6 - 1580*v^5 - 2093*v^4 - 2345*v^3 - 607*v^2 + 70*v + 2021) / 729 $$\beta_{4}$$ $$=$$ $$( - 547 \nu^{15} - 2825 \nu^{14} + 11696 \nu^{13} - 42716 \nu^{12} + 9478 \nu^{11} + 20721 \nu^{10} - 417232 \nu^{9} + 363147 \nu^{8} - 1222965 \nu^{7} - 271472 \nu^{6} + \cdots - 568472 ) / 205578$$ (-547*v^15 - 2825*v^14 + 11696*v^13 - 42716*v^12 + 9478*v^11 + 20721*v^10 - 417232*v^9 + 363147*v^8 - 1222965*v^7 - 271472*v^6 - 2073673*v^5 - 2132363*v^4 - 3222844*v^3 - 3007411*v^2 - 2176341*v - 568472) / 205578 $$\beta_{5}$$ $$=$$ $$( 280 \nu^{15} - 2977 \nu^{14} + 16531 \nu^{13} - 56011 \nu^{12} + 137264 \nu^{11} - 241101 \nu^{10} + 338866 \nu^{9} - 309462 \nu^{8} + 265752 \nu^{7} + 470 \nu^{6} + \cdots + 492218 ) / 102789$$ (280*v^15 - 2977*v^14 + 16531*v^13 - 56011*v^12 + 137264*v^11 - 241101*v^10 + 338866*v^9 - 309462*v^8 + 265752*v^7 + 470*v^6 + 163915*v^5 + 364985*v^4 + 555574*v^3 + 651970*v^2 + 860673*v + 492218) / 102789 $$\beta_{6}$$ $$=$$ $$( 197 \nu^{15} - 633 \nu^{14} + 2996 \nu^{13} - 7602 \nu^{12} + 28186 \nu^{11} - 53597 \nu^{10} + 131148 \nu^{9} - 144885 \nu^{8} + 281373 \nu^{7} - 81968 \nu^{6} + 367611 \nu^{5} + \cdots + 53568 ) / 68526$$ (197*v^15 - 633*v^14 + 2996*v^13 - 7602*v^12 + 28186*v^11 - 53597*v^10 + 131148*v^9 - 144885*v^8 + 281373*v^7 - 81968*v^6 + 367611*v^5 + 253195*v^4 + 415092*v^3 + 423533*v^2 + 322685*v + 53568) / 68526 $$\beta_{7}$$ $$=$$ $$( - 263 \nu^{15} + 4147 \nu^{14} - 19076 \nu^{13} + 62410 \nu^{12} - 127504 \nu^{11} + 236395 \nu^{10} - 263068 \nu^{9} + 378201 \nu^{8} - 138783 \nu^{7} + 484382 \nu^{6} + \cdots + 299584 ) / 68526$$ (-263*v^15 + 4147*v^14 - 19076*v^13 + 62410*v^12 - 127504*v^11 + 236395*v^10 - 263068*v^9 + 378201*v^8 - 138783*v^7 + 484382*v^6 + 352535*v^5 + 875795*v^4 + 926282*v^3 + 1149565*v^2 + 797105*v + 299584) / 68526 $$\beta_{8}$$ $$=$$ $$( - 3 \nu^{15} + 13 \nu^{14} - 51 \nu^{13} + 97 \nu^{12} - 219 \nu^{11} + 170 \nu^{10} - 403 \nu^{9} - 207 \nu^{8} - 780 \nu^{7} - 1407 \nu^{6} - 2383 \nu^{5} - 3732 \nu^{4} - 4999 \nu^{3} + \cdots - 3302 ) / 729$$ (-3*v^15 + 13*v^14 - 51*v^13 + 97*v^12 - 219*v^11 + 170*v^10 - 403*v^9 - 207*v^8 - 780*v^7 - 1407*v^6 - 2383*v^5 - 3732*v^4 - 4999*v^3 - 6168*v^2 - 4328*v - 3302) / 729 $$\beta_{9}$$ $$=$$ $$( - 197 \nu^{15} + 1103 \nu^{14} - 3278 \nu^{13} + 4688 \nu^{12} - 2101 \nu^{11} - 8913 \nu^{10} + 25879 \nu^{9} - 39966 \nu^{8} + 54630 \nu^{7} - 39292 \nu^{6} + 92848 \nu^{5} + \cdots + 93260 ) / 34263$$ (-197*v^15 + 1103*v^14 - 3278*v^13 + 4688*v^12 - 2101*v^11 - 8913*v^10 + 25879*v^9 - 39966*v^8 + 54630*v^7 - 39292*v^6 + 92848*v^5 + 47699*v^4 + 179317*v^3 + 162604*v^2 + 216123*v + 93260) / 34263 $$\beta_{10}$$ $$=$$ $$( - 14 \nu^{15} + 44 \nu^{14} - 53 \nu^{13} - 331 \nu^{12} + 1253 \nu^{11} - 3357 \nu^{10} + 4663 \nu^{9} - 6243 \nu^{8} + 2448 \nu^{7} - 3007 \nu^{6} - 4508 \nu^{5} + 1643 \nu^{4} + \cdots + 3587 ) / 2187$$ (-14*v^15 + 44*v^14 - 53*v^13 - 331*v^12 + 1253*v^11 - 3357*v^10 + 4663*v^9 - 6243*v^8 + 2448*v^7 - 3007*v^6 - 4508*v^5 + 1643*v^4 - 1982*v^3 + 5971*v^2 + 8634*v + 3587) / 2187 $$\beta_{11}$$ $$=$$ $$( - 1076 \nu^{15} + 10427 \nu^{14} - 44171 \nu^{13} + 127925 \nu^{12} - 248173 \nu^{11} + 410358 \nu^{10} - 433283 \nu^{9} + 561027 \nu^{8} - 211347 \nu^{7} + \cdots + 610868 ) / 102789$$ (-1076*v^15 + 10427*v^14 - 44171*v^13 + 127925*v^12 - 248173*v^11 + 410358*v^10 - 433283*v^9 + 561027*v^8 - 211347*v^7 + 721403*v^6 + 552091*v^5 + 1392206*v^4 + 1685476*v^3 + 2034148*v^2 + 1540755*v + 610868) / 102789 $$\beta_{12}$$ $$=$$ $$( - 10 \nu^{15} + 48 \nu^{14} - 187 \nu^{13} + 390 \nu^{12} - 833 \nu^{11} + 826 \nu^{10} - 1425 \nu^{9} - 165 \nu^{8} - 1974 \nu^{7} - 3245 \nu^{6} - 5211 \nu^{5} - 7880 \nu^{4} - 9222 \nu^{3} + \cdots - 3849 ) / 729$$ (-10*v^15 + 48*v^14 - 187*v^13 + 390*v^12 - 833*v^11 + 826*v^10 - 1425*v^9 - 165*v^8 - 1974*v^7 - 3245*v^6 - 5211*v^5 - 7880*v^4 - 9222*v^3 - 10309*v^2 - 6526*v - 3849) / 729 $$\beta_{13}$$ $$=$$ $$( 1481 \nu^{15} - 5918 \nu^{14} + 18674 \nu^{13} - 21635 \nu^{12} + 30967 \nu^{11} + 47409 \nu^{10} - 22324 \nu^{9} + 252699 \nu^{8} + 105723 \nu^{7} + 543790 \nu^{6} + \cdots + 214288 ) / 102789$$ (1481*v^15 - 5918*v^14 + 18674*v^13 - 21635*v^12 + 30967*v^11 + 47409*v^10 - 22324*v^9 + 252699*v^8 + 105723*v^7 + 543790*v^6 + 658346*v^5 + 951484*v^4 + 1097747*v^3 + 949991*v^2 + 611973*v + 214288) / 102789 $$\beta_{14}$$ $$=$$ $$( - 70 \nu^{15} + 307 \nu^{14} - 1030 \nu^{13} + 1591 \nu^{12} - 2744 \nu^{11} + 426 \nu^{10} - 2350 \nu^{9} - 6591 \nu^{8} - 8262 \nu^{7} - 18500 \nu^{6} - 28558 \nu^{5} - 35156 \nu^{4} + \cdots - 2714 ) / 2187$$ (-70*v^15 + 307*v^14 - 1030*v^13 + 1591*v^12 - 2744*v^11 + 426*v^10 - 2350*v^9 - 6591*v^8 - 8262*v^7 - 18500*v^6 - 28558*v^5 - 35156*v^4 - 40489*v^3 - 31426*v^2 - 12516*v - 2714) / 2187 $$\beta_{15}$$ $$=$$ $$( - 1260 \nu^{15} + 5900 \nu^{14} - 21726 \nu^{13} + 42641 \nu^{12} - 89361 \nu^{11} + 88954 \nu^{10} - 167819 \nu^{9} + 22059 \nu^{8} - 276987 \nu^{7} - 278193 \nu^{6} + \cdots - 181150 ) / 34263$$ (-1260*v^15 + 5900*v^14 - 21726*v^13 + 42641*v^12 - 89361*v^11 + 88954*v^10 - 167819*v^9 + 22059*v^8 - 276987*v^7 - 278193*v^6 - 556832*v^5 - 722337*v^4 - 837740*v^3 - 710436*v^2 - 457891*v - 181150) / 34263
 $$\nu$$ $$=$$ $$( \beta_{14} + 2\beta_{13} + \beta_{6} - \beta_{2} + \beta _1 + 2 ) / 3$$ (b14 + 2*b13 + b6 - b2 + b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{14} + 2\beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} - 2\beta_{7} - 2\beta_{5} - 2 ) / 3$$ (b14 + 2*b13 - b12 + b11 + b10 - b8 - 2*b7 - 2*b5 - 2) / 3 $$\nu^{3}$$ $$=$$ $$( - \beta_{15} - \beta_{12} + 4 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 6 \beta _1 - 10 ) / 3$$ (-b15 - b12 + 4*b11 + b10 + b9 + 2*b8 - 5*b7 - 2*b5 + 2*b4 - b3 + b2 - 6*b1 - 10) / 3 $$\nu^{4}$$ $$=$$ $$( - 3 \beta_{14} - 3 \beta_{13} - \beta_{12} + 4 \beta_{11} + \beta_{10} + 11 \beta_{8} - 2 \beta_{7} + 5 \beta_{6} + \beta_{5} + 3 \beta_{4} - 2 \beta_{2} - 19 \beta _1 - 19 ) / 3$$ (-3*b14 - 3*b13 - b12 + 4*b11 + b10 + 11*b8 - 2*b7 + 5*b6 + b5 + 3*b4 - 2*b2 - 19*b1 - 19) / 3 $$\nu^{5}$$ $$=$$ $$( 5 \beta_{15} - 6 \beta_{14} - 9 \beta_{13} - 9 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 15 \beta_{8} + 18 \beta_{7} + 10 \beta_{6} + 12 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} - 21 \beta_{2} - 32 \beta _1 - 27 ) / 3$$ (5*b15 - 6*b14 - 9*b13 - 9*b11 - 6*b10 - 2*b9 + 15*b8 + 18*b7 + 10*b6 + 12*b5 - 7*b4 + 2*b3 - 21*b2 - 32*b1 - 27) / 3 $$\nu^{6}$$ $$=$$ $$( 4 \beta_{15} - 2 \beta_{14} - 31 \beta_{13} + 8 \beta_{12} - 47 \beta_{11} - 26 \beta_{10} - 7 \beta_{9} - 10 \beta_{8} + 64 \beta_{7} - 10 \beta_{6} + 31 \beta_{5} - 26 \beta_{4} + \beta_{3} - 24 \beta_{2} - 7 \beta _1 - 26 ) / 3$$ (4*b15 - 2*b14 - 31*b13 + 8*b12 - 47*b11 - 26*b10 - 7*b9 - 10*b8 + 64*b7 - 10*b6 + 31*b5 - 26*b4 + b3 - 24*b2 - 7*b1 - 26) / 3 $$\nu^{7}$$ $$=$$ $$( - 11 \beta_{15} + 9 \beta_{14} - 69 \beta_{13} + 27 \beta_{12} - 105 \beta_{11} - 57 \beta_{10} - 16 \beta_{9} - 87 \beta_{8} + 108 \beta_{7} - 106 \beta_{6} + 48 \beta_{5} - 32 \beta_{4} + 4 \beta_{3} + 108 \beta_{2} + 161 \beta _1 + 45 ) / 3$$ (-11*b15 + 9*b14 - 69*b13 + 27*b12 - 105*b11 - 57*b10 - 16*b9 - 87*b8 + 108*b7 - 106*b6 + 48*b5 - 32*b4 + 4*b3 + 108*b2 + 161*b1 + 45) / 3 $$\nu^{8}$$ $$=$$ $$( - 22 \beta_{15} + 10 \beta_{14} - 37 \beta_{13} + 58 \beta_{12} - 121 \beta_{11} - 31 \beta_{10} - 17 \beta_{9} - 230 \beta_{8} + 35 \beta_{7} - 309 \beta_{6} + 14 \beta_{5} + 8 \beta_{4} + 17 \beta_{3} + 484 \beta_{2} + 582 \beta _1 + 354 ) / 3$$ (-22*b15 + 10*b14 - 37*b13 + 58*b12 - 121*b11 - 31*b10 - 17*b9 - 230*b8 + 35*b7 - 309*b6 + 14*b5 + 8*b4 + 17*b3 + 484*b2 + 582*b1 + 354) / 3 $$\nu^{9}$$ $$=$$ $$( 14 \beta_{15} - 40 \beta_{14} + 277 \beta_{13} + 124 \beta_{12} + 98 \beta_{11} + 248 \beta_{10} + 58 \beta_{9} - 413 \beta_{8} - 379 \beta_{7} - 407 \beta_{6} - 211 \beta_{5} + 137 \beta_{4} + 23 \beta_{3} + 831 \beta_{2} + 1048 \beta _1 + 1046 ) / 3$$ (14*b15 - 40*b14 + 277*b13 + 124*b12 + 98*b11 + 248*b10 + 58*b9 - 413*b8 - 379*b7 - 407*b6 - 211*b5 + 137*b4 + 23*b3 + 831*b2 + 1048*b1 + 1046) / 3 $$\nu^{10}$$ $$=$$ $$( 98 \beta_{15} - 190 \beta_{14} + 1027 \beta_{13} + 243 \beta_{12} + 1002 \beta_{11} + 981 \beta_{10} + 313 \beta_{9} - 309 \beta_{8} - 1371 \beta_{7} + 483 \beta_{6} - 795 \beta_{5} + 398 \beta_{4} + 32 \beta_{3} - 200 \beta_{2} + \cdots + 1750 ) / 3$$ (98*b15 - 190*b14 + 1027*b13 + 243*b12 + 1002*b11 + 981*b10 + 313*b9 - 309*b8 - 1371*b7 + 483*b6 - 795*b5 + 398*b4 + 32*b3 - 200*b2 + 396*b1 + 1750) / 3 $$\nu^{11}$$ $$=$$ $$( 121 \beta_{15} - 338 \beta_{14} + 1772 \beta_{13} + 75 \beta_{12} + 3174 \beta_{11} + 1770 \beta_{10} + 617 \beta_{9} + 1314 \beta_{8} - 2733 \beta_{7} + 4206 \beta_{6} - 1491 \beta_{5} + 718 \beta_{4} + 73 \beta_{3} + \cdots + 800 ) / 3$$ (121*b15 - 338*b14 + 1772*b13 + 75*b12 + 3174*b11 + 1770*b10 + 617*b9 + 1314*b8 - 2733*b7 + 4206*b6 - 1491*b5 + 718*b4 + 73*b3 - 5503*b2 - 4101*b1 + 800) / 3 $$\nu^{12}$$ $$=$$ $$( - 103 \beta_{15} + 307 \beta_{14} + 749 \beta_{13} - 1995 \beta_{12} + 5949 \beta_{11} + 429 \beta_{10} + 163 \beta_{9} + 6813 \beta_{8} - 2553 \beta_{7} + 12074 \beta_{6} - 798 \beta_{5} + 566 \beta_{4} - 232 \beta_{3} + \cdots - 5494 ) / 3$$ (-103*b15 + 307*b14 + 749*b13 - 1995*b12 + 5949*b11 + 429*b10 + 163*b9 + 6813*b8 - 2553*b7 + 12074*b6 - 798*b5 + 566*b4 - 232*b3 - 17545*b2 - 15655*b1 - 5494) / 3 $$\nu^{13}$$ $$=$$ $$( - 890 \beta_{15} + 4039 \beta_{14} - 5440 \beta_{13} - 9317 \beta_{12} + 3422 \beta_{11} - 7912 \beta_{10} - 2575 \beta_{9} + 17143 \beta_{8} + 4739 \beta_{7} + 17608 \beta_{6} + 4415 \beta_{5} - 1616 \beta_{4} + \cdots - 23142 ) / 3$$ (-890*b15 + 4039*b14 - 5440*b13 - 9317*b12 + 3422*b11 - 7912*b10 - 2575*b9 + 17143*b8 + 4739*b7 + 17608*b6 + 4415*b5 - 1616*b4 - 2216*b3 - 28457*b2 - 31124*b1 - 23142) / 3 $$\nu^{14}$$ $$=$$ $$( - 2922 \beta_{15} + 13568 \beta_{14} - 20192 \beta_{13} - 22506 \beta_{12} - 21396 \beta_{11} - 28131 \beta_{10} - 8667 \beta_{9} + 22623 \beta_{8} + 28509 \beta_{7} - 9829 \beta_{6} + 17028 \beta_{5} + \cdots - 58190 ) / 3$$ (-2922*b15 + 13568*b14 - 20192*b13 - 22506*b12 - 21396*b11 - 28131*b10 - 8667*b9 + 22623*b8 + 28509*b7 - 9829*b6 + 17028*b5 - 8424*b4 - 6858*b3 + 4018*b2 - 23647*b1 - 58190) / 3 $$\nu^{15}$$ $$=$$ $$( - 5826 \beta_{15} + 22246 \beta_{14} - 40255 \beta_{13} - 21812 \beta_{12} - 95482 \beta_{11} - 51508 \beta_{10} - 15069 \beta_{9} - 15488 \beta_{8} + 73283 \beta_{7} - 138565 \beta_{6} + \cdots - 100389 ) / 3$$ (-5826*b15 + 22246*b14 - 40255*b13 - 21812*b12 - 95482*b11 - 51508*b10 - 15069*b9 - 15488*b8 + 73283*b7 - 138565*b6 + 33752*b5 - 20538*b4 - 8757*b3 + 181372*b2 + 80366*b1 - 100389) / 3

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1755\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$1081$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
586.1
 1.79305 + 1.53983i 1.48460 − 1.66288i 0.466399 + 1.64781i 0.252952 + 1.56266i 0.172467 − 1.52157i −0.317019 − 1.12493i −0.628312 + 0.590424i −0.724143 − 0.165319i 1.79305 − 1.53983i 1.48460 + 1.66288i 0.466399 − 1.64781i 0.252952 − 1.56266i 0.172467 + 1.52157i −0.317019 + 1.12493i −0.628312 − 0.590424i −0.724143 + 0.165319i
−1.29305 + 2.23963i 0 −2.34397 4.05987i −0.500000 0.866025i 0 2.13745 3.70217i 6.95128 0 2.58610
586.2 −0.984603 + 1.70538i 0 −0.938888 1.62620i −0.500000 0.866025i 0 1.51414 2.62256i −0.240686 0 1.96921
586.3 0.0336011 0.0581988i 0 0.997742 + 1.72814i −0.500000 0.866025i 0 −1.23179 + 2.13352i 0.268505 0 −0.0672022
586.4 0.247048 0.427900i 0 0.877935 + 1.52063i −0.500000 0.866025i 0 2.14269 3.71125i 1.85576 0 −0.494096
586.5 0.327533 0.567303i 0 0.785445 + 1.36043i −0.500000 0.866025i 0 0.388592 0.673061i 2.33917 0 −0.655066
586.6 0.817019 1.41512i 0 −0.335039 0.580304i −0.500000 0.866025i 0 −1.06506 + 1.84473i 2.17314 0 −1.63404
586.7 1.12831 1.95429i 0 −1.54617 2.67805i −0.500000 0.866025i 0 −0.353285 + 0.611908i −2.46502 0 −2.25662
586.8 1.22414 2.12028i 0 −1.99705 3.45900i −0.500000 0.866025i 0 1.96726 3.40740i −4.88214 0 −2.44829
1171.1 −1.29305 2.23963i 0 −2.34397 + 4.05987i −0.500000 + 0.866025i 0 2.13745 + 3.70217i 6.95128 0 2.58610
1171.2 −0.984603 1.70538i 0 −0.938888 + 1.62620i −0.500000 + 0.866025i 0 1.51414 + 2.62256i −0.240686 0 1.96921
1171.3 0.0336011 + 0.0581988i 0 0.997742 1.72814i −0.500000 + 0.866025i 0 −1.23179 2.13352i 0.268505 0 −0.0672022
1171.4 0.247048 + 0.427900i 0 0.877935 1.52063i −0.500000 + 0.866025i 0 2.14269 + 3.71125i 1.85576 0 −0.494096
1171.5 0.327533 + 0.567303i 0 0.785445 1.36043i −0.500000 + 0.866025i 0 0.388592 + 0.673061i 2.33917 0 −0.655066
1171.6 0.817019 + 1.41512i 0 −0.335039 + 0.580304i −0.500000 + 0.866025i 0 −1.06506 1.84473i 2.17314 0 −1.63404
1171.7 1.12831 + 1.95429i 0 −1.54617 + 2.67805i −0.500000 + 0.866025i 0 −0.353285 0.611908i −2.46502 0 −2.25662
1171.8 1.22414 + 2.12028i 0 −1.99705 + 3.45900i −0.500000 + 0.866025i 0 1.96726 + 3.40740i −4.88214 0 −2.44829
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1171.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1755.2.i.f 16
3.b odd 2 1 585.2.i.e 16
9.c even 3 1 inner 1755.2.i.f 16
9.c even 3 1 5265.2.a.ba 8
9.d odd 6 1 585.2.i.e 16
9.d odd 6 1 5265.2.a.bf 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.e 16 3.b odd 2 1
585.2.i.e 16 9.d odd 6 1
1755.2.i.f 16 1.a even 1 1 trivial
1755.2.i.f 16 9.c even 3 1 inner
5265.2.a.ba 8 9.c even 3 1
5265.2.a.bf 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1755, [\chi])$$:

 $$T_{2}^{16} - 3 T_{2}^{15} + 17 T_{2}^{14} - 38 T_{2}^{13} + 158 T_{2}^{12} - 324 T_{2}^{11} + 875 T_{2}^{10} - 1368 T_{2}^{9} + 2812 T_{2}^{8} - 3873 T_{2}^{7} + 5539 T_{2}^{6} - 4627 T_{2}^{5} + 3027 T_{2}^{4} - 1114 T_{2}^{3} + \cdots + 1$$ T2^16 - 3*T2^15 + 17*T2^14 - 38*T2^13 + 158*T2^12 - 324*T2^11 + 875*T2^10 - 1368*T2^9 + 2812*T2^8 - 3873*T2^7 + 5539*T2^6 - 4627*T2^5 + 3027*T2^4 - 1114*T2^3 + 295*T2^2 - 19*T2 + 1 $$T_{7}^{16} - 11 T_{7}^{15} + 97 T_{7}^{14} - 476 T_{7}^{13} + 2127 T_{7}^{12} - 6158 T_{7}^{11} + 20644 T_{7}^{10} - 44604 T_{7}^{9} + 140146 T_{7}^{8} - 191361 T_{7}^{7} + 544052 T_{7}^{6} - 359094 T_{7}^{5} + \cdots + 395641$$ T7^16 - 11*T7^15 + 97*T7^14 - 476*T7^13 + 2127*T7^12 - 6158*T7^11 + 20644*T7^10 - 44604*T7^9 + 140146*T7^8 - 191361*T7^7 + 544052*T7^6 - 359094*T7^5 + 1614395*T7^4 - 251056*T7^3 + 856340*T7^2 - 18870*T7 + 395641

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 3 T^{15} + 17 T^{14} - 38 T^{13} + \cdots + 1$$
$3$ $$T^{16}$$
$5$ $$(T^{2} + T + 1)^{8}$$
$7$ $$T^{16} - 11 T^{15} + 97 T^{14} + \cdots + 395641$$
$11$ $$T^{16} - 6 T^{15} + 74 T^{14} + \cdots + 401956$$
$13$ $$(T^{2} - T + 1)^{8}$$
$17$ $$(T^{8} - 2 T^{7} - 78 T^{6} + 73 T^{5} + \cdots + 892)^{2}$$
$19$ $$(T^{8} + 10 T^{7} - 37 T^{6} - 541 T^{5} + \cdots - 1584)^{2}$$
$23$ $$T^{16} - 6 T^{15} + 115 T^{14} + \cdots + 144360225$$
$29$ $$T^{16} - 14 T^{15} + 191 T^{14} + \cdots + 1243225$$
$31$ $$T^{16} - 31 T^{15} + \cdots + 2547422784$$
$37$ $$(T^{8} - T^{7} - 120 T^{6} - 211 T^{5} + \cdots - 33660)^{2}$$
$41$ $$T^{16} + 12 T^{15} + \cdots + 371525625$$
$43$ $$T^{16} + 15 T^{15} + \cdots + 62415940941376$$
$47$ $$T^{16} + 18 T^{15} + \cdots + 424482219529$$
$53$ $$(T^{8} + 2 T^{7} - 282 T^{6} + \cdots + 111438)^{2}$$
$59$ $$T^{16} - 24 T^{15} + \cdots + 3524698346724$$
$61$ $$T^{16} + \cdots + 106354483234561$$
$67$ $$T^{16} + \cdots + 974523752673769$$
$71$ $$(T^{8} + 10 T^{7} - 211 T^{6} + \cdots - 127950)^{2}$$
$73$ $$(T^{8} - 6 T^{7} - 300 T^{6} + \cdots + 1196532)^{2}$$
$79$ $$T^{16} - 3 T^{15} + \cdots + 17\!\cdots\!04$$
$83$ $$T^{16} + 10 T^{15} + 215 T^{14} + \cdots + 469225$$
$89$ $$(T^{8} - 13 T^{7} - 228 T^{6} + \cdots + 1032219)^{2}$$
$97$ $$T^{16} + \cdots + 122483314259524$$