# Properties

 Label 138.3.c.a Level $138$ Weight $3$ Character orbit 138.c Analytic conductor $3.760$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 138.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.76022764817$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 10 x^{14} + 8 x^{13} - 119 x^{12} + 416 x^{11} - 774 x^{10} - 1284 x^{9} + 7956 x^{8} - 11556 x^{7} - 62694 x^{6} + 303264 x^{5} - 780759 x^{4} + \cdots + 43046721$$ x^16 - 4*x^15 + 10*x^14 + 8*x^13 - 119*x^12 + 416*x^11 - 774*x^10 - 1284*x^9 + 7956*x^8 - 11556*x^7 - 62694*x^6 + 303264*x^5 - 780759*x^4 + 472392*x^3 + 5314410*x^2 - 19131876*x + 43046721 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{8}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + \beta_1 q^{3} - 2 q^{4} - \beta_{7} q^{5} + \beta_{11} q^{6} - \beta_{6} q^{7} + 2 \beta_{3} q^{8} + (\beta_{10} - \beta_{8} + \beta_{5} - 1) q^{9}+O(q^{10})$$ q - b3 * q^2 + b1 * q^3 - 2 * q^4 - b7 * q^5 + b11 * q^6 - b6 * q^7 + 2*b3 * q^8 + (b10 - b8 + b5 - 1) * q^9 $$q - \beta_{3} q^{2} + \beta_1 q^{3} - 2 q^{4} - \beta_{7} q^{5} + \beta_{11} q^{6} - \beta_{6} q^{7} + 2 \beta_{3} q^{8} + (\beta_{10} - \beta_{8} + \beta_{5} - 1) q^{9} + ( - \beta_{8} + \beta_{4} + \beta_{2} - 1) q^{10} + (\beta_{13} + \beta_{12} + \beta_{11} - \beta_{8} + \beta_{5} - \beta_{4} + \beta_{2}) q^{11} - 2 \beta_1 q^{12} + ( - \beta_{14} + \beta_{12} - \beta_{9} + \beta_{8} + 2 \beta_{5} - 1) q^{13} + ( - \beta_{15} + \beta_{12} + \beta_{9} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{14} + (\beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{10} - 2 \beta_{9} - \beta_{6} - \beta_{3} + \beta_{2} + 2) q^{15} + 4 q^{16} + (\beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - 2 \beta_{10} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{17} + ( - \beta_{14} - \beta_{12} + \beta_{9} - \beta_1 + 1) q^{18} + ( - \beta_{14} + \beta_{12} - \beta_{11} - \beta_{9} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 + 3) q^{19} + 2 \beta_{7} q^{20} + ( - 2 \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} + 2 \beta_{9} - \beta_{6} - \beta_{5} + \cdots + \beta_{2}) q^{21}+ \cdots + ( - \beta_{15} - 2 \beta_{14} - 6 \beta_{13} - 4 \beta_{12} + 4 \beta_{10} - 5 \beta_{9} + \cdots + 1) q^{99}+O(q^{100})$$ q - b3 * q^2 + b1 * q^3 - 2 * q^4 - b7 * q^5 + b11 * q^6 - b6 * q^7 + 2*b3 * q^8 + (b10 - b8 + b5 - 1) * q^9 + (-b8 + b4 + b2 - 1) * q^10 + (b13 + b12 + b11 - b8 + b5 - b4 + b2) * q^11 - 2*b1 * q^12 + (-b14 + b12 - b9 + b8 + 2*b5 - 1) * q^13 + (-b15 + b12 + b9 + b5 - b4 - b3 + b2 - 2*b1) * q^14 + (b15 - b14 + b13 + 2*b10 - 2*b9 - b6 - b3 + b2 + 2) * q^15 + 4 * q^16 + (b15 - b14 - b13 - b12 - 2*b10 - b7 - b5 + b4 - b2 + b1) * q^17 + (-b14 - b12 + b9 - b1 + 1) * q^18 + (-b14 + b12 - b11 - b9 - b6 - b5 - b3 - b1 + 3) * q^19 + 2*b7 * q^20 + (-2*b15 + b14 - b13 - b12 + b10 + 2*b9 - b6 - b5 + 4*b3 + b2) * q^21 + (b14 + b13 + 2*b11 + b9 + 2*b6 + b3 - b1 + 2) * q^22 - b10 * q^23 - 2*b11 * q^24 + (b14 + 3*b12 + b11 + b9 + b3 + b1 - 12) * q^25 + (2*b15 - 2*b14 - b13 + b11 + 2*b10 - 2*b1 + 1) * q^26 + (b14 - b13 - 3*b12 + b10 + b9 - 2*b7 + 2*b6 - b5 + 2*b4 - 4*b3 + b1 - 6) * q^27 + 2*b6 * q^28 + (-2*b15 + 2*b14 - 2*b13 - 2*b12 + b11 - 2*b10 - b8 + 2*b7 - 2*b5 + 2*b4 + b3 - 2*b2 + 3*b1) * q^29 + (b15 + 2*b14 + 2*b13 - b12 + 2*b11 - b10 - b8 + b7 + b6 + b5 - b4 - b3 + b2 - 1) * q^30 + (2*b14 + b13 + 2*b11 + 2*b9 + b8 + 2*b6 - 2*b4 + b3 - 2*b2 - b1 + 11) * q^31 - 4*b3 * q^32 + (-b15 - b14 - b12 - 3*b11 - b10 - b9 + b8 - b7 - b6 + 3*b5 - 2*b4 - 7*b3 - b2 - 2*b1 - 6) * q^33 + (2*b12 - b8 - 2*b6 - 2*b5 + b4 + b2 - 1) * q^34 + (4*b14 + 5*b13 + 2*b12 + b11 - 4*b10 - 4*b9 + 2*b7 + 2*b5 - 2*b4 - 5*b3 + 2*b2 + 1) * q^35 + (-2*b10 + 2*b8 - 2*b5 + 2) * q^36 + (b14 - 2*b13 + b12 - 3*b11 + b9 - 3*b8 - 2*b6 - b5 + b4 - b3 + b2 + 3*b1 - 12) * q^37 + (b15 + b14 + 3*b13 + b12 + 2*b10 - 2*b9 + b5 - b4 - 2*b3 + b2 + b1) * q^38 + (2*b15 - b14 - 3*b13 - 5*b12 + b11 + 3*b10 + b9 + 3*b8 + 4*b6 - b5 + 12*b3 - 4*b2 + b1 + 12) * q^39 + (2*b8 - 2*b4 - 2*b2 + 2) * q^40 + (-2*b14 - 2*b12 - 5*b11 + 2*b10 + 2*b9 + b8 - 4*b7 - 2*b5 + 2*b4 + 7*b3 - 2*b2 + 7*b1 - 4) * q^41 + (-2*b15 - 3*b10 - b9 + b7 + b6 + 2*b5 - b4 - 2*b3 + b2 - 2*b1 + 8) * q^42 + (-2*b14 - b13 + 7*b12 - b11 - 2*b9 + 3*b8 - b6 - b5 - b4 - b2 + 2*b1 + 6) * q^43 + (-2*b13 - 2*b12 - 2*b11 + 2*b8 - 2*b5 + 2*b4 - 2*b2) * q^44 + (b15 + 3*b14 + 4*b13 - 5*b12 + b11 - b10 - 3*b9 + 3*b8 - b6 - 3*b5 + 5*b3 - 5*b2 + 4*b1 + 4) * q^45 + b12 * q^46 + (-4*b14 - 4*b13 - 2*b12 - 3*b11 - 2*b10 + 4*b9 + b8 + 2*b7 - 2*b5 + 2*b4 - 2*b2 + b1 - 2) * q^47 + 4*b1 * q^48 + (-3*b14 - 4*b13 + 3*b12 - 7*b11 - 3*b9 - 3*b3 + 5*b1 + 10) * q^49 + (-2*b15 - 2*b13 + 6*b10 + 2*b9 + 11*b3 - 2*b1) * q^50 + (3*b15 - 4*b14 - 4*b13 - 2*b11 + 2*b10 - b9 - b8 + 2*b7 - 5*b6 - 2*b5 + b4 - 6*b3 + b1 - 1) * q^51 + (2*b14 - 2*b12 + 2*b9 - 2*b8 - 4*b5 + 2) * q^52 + (-5*b15 + 4*b14 - 3*b13 - b11 - 4*b10 + b9 + 3*b8 - 3*b7 + 5*b3 - 8*b1 + 2) * q^53 + (-b14 - 3*b13 - 3*b12 - 2*b11 - 4*b10 - b9 - 2*b8 + 2*b7 - 2*b6 - 2*b5 + 4*b4 + 5*b3 + 3*b1 - 14) * q^54 + (2*b14 - 2*b12 - 4*b11 + 2*b9 - 6*b8 + 4*b6 + 2*b5 - 4*b3 - 4*b1 - 10) * q^55 + (2*b15 - 2*b12 - 2*b9 - 2*b5 + 2*b4 + 2*b3 - 2*b2 + 4*b1) * q^56 + (b14 + 2*b13 + b12 + b11 - b9 + b8 + 3*b7 - 3*b5 - 3*b4 - 5*b3 - 3*b2 + 3*b1 - 6) * q^57 + (-4*b14 + b13 + 2*b12 - 3*b11 - 4*b9 + 2*b8 - 4*b6 + 4*b5 - 2*b4 - 4*b3 - 2*b2 - 6*b1 + 1) * q^58 + (4*b15 + 3*b13 - 7*b11 - 4*b9 + 8*b8 + 2*b7 + 5*b3 - 6*b1 + 1) * q^59 + (-2*b15 + 2*b14 - 2*b13 - 4*b10 + 4*b9 + 2*b6 + 2*b3 - 2*b2 - 4) * q^60 + (-4*b14 + 2*b13 + 2*b12 + 2*b11 - 4*b9 + 4*b8 - 2*b5 - 4*b1 + 8) * q^61 + (-2*b15 - 3*b13 - 2*b12 - 3*b11 + 2*b9 + 2*b8 - 4*b7 - 2*b5 + 2*b4 - 8*b3 - 2*b2 - 1) * q^62 + (-5*b15 + 4*b14 + 5*b13 + b12 - b11 - 2*b10 + b9 - 6*b8 + 4*b7 + b6 + 3*b5 + 2*b4 + 2*b3 + 7*b2 - 3*b1 + 7) * q^63 - 8 * q^64 + (9*b15 + 2*b14 + 5*b13 + 11*b11 + 4*b10 - 11*b9 - 5*b8 + 2*b7 - b3 + 4*b1 + 6) * q^65 + (2*b15 - 3*b14 + b13 + 2*b12 - 3*b10 - b8 - 3*b7 + b6 + 2*b5 + 5*b3 + 2*b2 + b1 - 11) * q^66 + (5*b14 + 3*b13 - 4*b12 + 14*b11 + 5*b9 + 7*b8 + 7*b6 - 2*b5 - b4 + 11*b3 - b2 + 5*b1 - 11) * q^67 + (-2*b15 + 2*b14 + 2*b13 + 2*b12 + 4*b10 + 2*b7 + 2*b5 - 2*b4 + 2*b2 - 2*b1) * q^68 + (-b14 - b13 + b12 - b11 + b8 + b7 - b6 + b5 - b4 - b3 - b1) * q^69 + (2*b14 + 4*b12 + 2*b9 + 4*b6 + 8*b5 - 2*b4 - 2*b2 - 14) * q^70 + (-4*b15 - 4*b14 + b13 + 6*b12 - 4*b11 - 2*b10 + 8*b9 + b8 + 4*b7 + 6*b5 - 6*b4 + 5*b3 + 6*b2 - 9*b1 - 3) * q^71 + (2*b14 + 2*b12 - 2*b9 + 2*b1 - 2) * q^72 + (2*b14 - 2*b13 - 6*b12 + b11 + 2*b9 + 3*b8 - 4*b6 - 2*b4 + 3*b3 - 2*b2 + 7*b1 + 26) * q^73 + (-4*b15 + b14 + b13 + 2*b12 + 2*b11 + 2*b10 + 3*b9 - 4*b8 + 2*b7 + 2*b5 - 2*b4 + 7*b3 + 2*b2 + 3*b1 - 2) * q^74 + (-2*b15 - b13 + 4*b12 + 3*b11 + 4*b10 + 6*b9 - 6*b8 - 6*b7 + 2*b6 + 2*b5 - 6*b4 + b3 + 4*b2 - 13*b1 + 7) * q^75 + (2*b14 - 2*b12 + 2*b11 + 2*b9 + 2*b6 + 2*b5 + 2*b3 + 2*b1 - 6) * q^76 + (-2*b15 + 2*b14 + 2*b12 + 4*b11 + 4*b10 - 2*b8 - 4*b7 + 2*b5 - 2*b4 - 26*b3 + 2*b2 - 4*b1 + 2) * q^77 + (2*b15 - b14 - 4*b13 - 7*b12 - 2*b11 - 6*b10 - b9 - 2*b8 - 4*b7 - 4*b6 - 8*b5 + 4*b4 - 9*b3 - 4*b2 + 5*b1 + 21) * q^78 + (3*b14 + 6*b13 - b12 + b11 + 3*b9 - 10*b8 + 3*b6 + b5 + 2*b4 - 5*b3 + 2*b2 - 17*b1 + 19) * q^79 - 4*b7 * q^80 + (6*b15 + 6*b13 + 9*b11 + b10 - 6*b9 - 4*b8 + b5 + 6*b4 + 3*b3 + 6*b2 - 3*b1 + 11) * q^81 + (-2*b14 - b13 - 2*b12 + 5*b11 - 2*b9 + 4*b8 - 4*b6 - 4*b5 + 4*b4 + 6*b3 + 4*b2 + 8*b1 + 11) * q^82 + (4*b15 - 6*b14 - 5*b13 - 3*b12 + 7*b11 + 2*b9 - 7*b8 - 3*b5 + 3*b4 - 10*b3 - 3*b2 + 8*b1) * q^83 + (4*b15 - 2*b14 + 2*b13 + 2*b12 - 2*b10 - 4*b9 + 2*b6 + 2*b5 - 8*b3 - 2*b2) * q^84 + (b14 - 4*b13 - b12 - b11 + b9 - 4*b8 - 4*b6 - 2*b5 + 6*b4 + 3*b3 + 6*b2 + 11*b1 - 37) * q^85 + (3*b15 + b14 + 3*b13 + b12 + 4*b11 + 14*b10 - 4*b9 - 2*b8 - 2*b7 + b5 - b4 - 4*b3 + b2 + b1 + 2) * q^86 + (-2*b15 + b14 + 2*b13 + 5*b12 + 4*b11 + 5*b10 + 11*b9 - b8 + 6*b7 + 8*b6 + 3*b5 - 11*b3 + 4*b2 - 5*b1 - 19) * q^87 + (-2*b14 - 2*b13 - 4*b11 - 2*b9 - 4*b6 - 2*b3 + 2*b1 - 4) * q^88 + (2*b15 - 7*b14 + 4*b13 + 3*b12 - 7*b11 - 2*b10 + 5*b9 + b8 + b7 + 3*b5 - 3*b4 - 19*b3 + 3*b2 + 3*b1 - 6) * q^89 + (-2*b15 - b14 - b13 + 2*b12 - 4*b11 - 5*b10 + 6*b8 - 5*b7 - 5*b6 + 6*b5 - b4 - b3 + b2 - 5*b1) * q^90 + (-3*b14 + 8*b13 - 11*b12 + 7*b11 - 3*b9 - 6*b8 - 4*b6 - b5 + 8*b4 - b3 + 8*b2 - 17*b1 - 3) * q^91 + 2*b10 * q^92 + (-2*b15 - 6*b14 - 4*b13 + 6*b12 - 4*b11 - b10 + 2*b9 - b8 - 8*b7 - 2*b6 + b5 + 2*b4 + 2*b3 + 4*b2 + 8*b1 - 14) * q^93 + (-2*b14 - b13 + 2*b12 + b11 - 2*b9 + 6*b8 - 4*b6 - 8*b5 - 2*b4 + 2*b3 - 2*b2 + 4*b1 + 7) * q^94 + (-4*b15 - 4*b14 - 11*b13 + 2*b12 + 9*b11 - 2*b10 + 8*b9 - 4*b8 - 2*b7 + 2*b5 - 2*b4 - 7*b3 + 2*b2 - 16*b1 + 5) * q^95 + 4*b11 * q^96 + (3*b14 - 2*b13 + 3*b12 + b11 + 3*b9 + b8 + 6*b6 + 3*b5 - b4 + 3*b3 - b2 + 7*b1 + 8) * q^97 + (6*b15 + 6*b13 + 8*b11 + 6*b10 - 6*b9 - 8*b8 - 11*b3 + 14*b1) * q^98 + (-b15 - 2*b14 - 6*b13 - 4*b12 + 4*b10 - 5*b9 + 11*b8 + 4*b7 - 2*b5 + 5*b4 + 4*b3 - 4*b2 - 5*b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 4 q^{3} - 32 q^{4} - 8 q^{6} - 4 q^{9}+O(q^{10})$$ 16 * q + 4 * q^3 - 32 * q^4 - 8 * q^6 - 4 * q^9 $$16 q + 4 q^{3} - 32 q^{4} - 8 q^{6} - 4 q^{9} - 8 q^{12} - 8 q^{13} + 28 q^{15} + 64 q^{16} + 16 q^{18} + 40 q^{19} + 4 q^{21} + 16 q^{22} + 16 q^{24} - 192 q^{25} - 80 q^{27} - 24 q^{30} + 136 q^{31} - 84 q^{33} - 16 q^{34} + 8 q^{36} - 136 q^{37} + 156 q^{39} + 128 q^{42} + 72 q^{43} + 4 q^{45} + 16 q^{48} + 224 q^{49} - 4 q^{51} + 16 q^{52} - 176 q^{54} - 96 q^{55} - 160 q^{57} - 56 q^{60} + 48 q^{61} + 204 q^{63} - 128 q^{64} - 144 q^{66} - 304 q^{67} - 176 q^{70} - 32 q^{72} + 408 q^{73} + 68 q^{75} - 80 q^{76} + 328 q^{78} + 312 q^{79} + 164 q^{81} + 160 q^{82} - 8 q^{84} - 464 q^{85} - 268 q^{87} - 32 q^{88} + 32 q^{90} - 72 q^{91} - 108 q^{93} - 32 q^{96} + 168 q^{97} - 60 q^{99}+O(q^{100})$$ 16 * q + 4 * q^3 - 32 * q^4 - 8 * q^6 - 4 * q^9 - 8 * q^12 - 8 * q^13 + 28 * q^15 + 64 * q^16 + 16 * q^18 + 40 * q^19 + 4 * q^21 + 16 * q^22 + 16 * q^24 - 192 * q^25 - 80 * q^27 - 24 * q^30 + 136 * q^31 - 84 * q^33 - 16 * q^34 + 8 * q^36 - 136 * q^37 + 156 * q^39 + 128 * q^42 + 72 * q^43 + 4 * q^45 + 16 * q^48 + 224 * q^49 - 4 * q^51 + 16 * q^52 - 176 * q^54 - 96 * q^55 - 160 * q^57 - 56 * q^60 + 48 * q^61 + 204 * q^63 - 128 * q^64 - 144 * q^66 - 304 * q^67 - 176 * q^70 - 32 * q^72 + 408 * q^73 + 68 * q^75 - 80 * q^76 + 328 * q^78 + 312 * q^79 + 164 * q^81 + 160 * q^82 - 8 * q^84 - 464 * q^85 - 268 * q^87 - 32 * q^88 + 32 * q^90 - 72 * q^91 - 108 * q^93 - 32 * q^96 + 168 * q^97 - 60 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 10 x^{14} + 8 x^{13} - 119 x^{12} + 416 x^{11} - 774 x^{10} - 1284 x^{9} + 7956 x^{8} - 11556 x^{7} - 62694 x^{6} + 303264 x^{5} - 780759 x^{4} + \cdots + 43046721$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 217 \nu^{15} + 3982 \nu^{14} + 2141 \nu^{13} - 71198 \nu^{12} + 44174 \nu^{11} - 255260 \nu^{10} + 1371528 \nu^{9} - 1191198 \nu^{8} + 441054 \nu^{7} + \cdots + 33461651124 ) / 5739562800$$ (-217*v^15 + 3982*v^14 + 2141*v^13 - 71198*v^12 + 44174*v^11 - 255260*v^10 + 1371528*v^9 - 1191198*v^8 + 441054*v^7 + 13141494*v^6 + 55581390*v^5 + 119343132*v^4 + 286905969*v^3 - 2262049092*v^2 - 1802116431*v + 33461651124) / 5739562800 $$\beta_{3}$$ $$=$$ $$( - 1049 \nu^{15} - 2761 \nu^{14} - 563 \nu^{13} + 46859 \nu^{12} - 119312 \nu^{11} + 380240 \nu^{10} + 322866 \nu^{9} - 756006 \nu^{8} + 11963718 \nu^{7} + \cdots - 31342795857 ) / 17218688400$$ (-1049*v^15 - 2761*v^14 - 563*v^13 + 46859*v^12 - 119312*v^11 + 380240*v^10 + 322866*v^9 - 756006*v^8 + 11963718*v^7 + 6408018*v^6 - 33057720*v^5 + 188472744*v^4 - 81861597*v^3 - 1739642589*v^2 + 3525048153*v - 31342795857) / 17218688400 $$\beta_{4}$$ $$=$$ $$( - 2257 \nu^{15} - 24758 \nu^{14} + 36191 \nu^{13} - 18308 \nu^{12} + 241214 \nu^{11} - 213620 \nu^{10} - 4661172 \nu^{9} + 16826262 \nu^{8} + \cdots - 62628196086 ) / 17218688400$$ (-2257*v^15 - 24758*v^14 + 36191*v^13 - 18308*v^12 + 241214*v^11 - 213620*v^10 - 4661172*v^9 + 16826262*v^8 + 6454494*v^7 + 7348374*v^6 - 151678170*v^5 + 1126937772*v^4 + 161853309*v^3 + 12382102908*v^2 - 21789612441*v - 62628196086) / 17218688400 $$\beta_{5}$$ $$=$$ $$( - 449 \nu^{15} - 598 \nu^{14} - 6497 \nu^{13} - 17650 \nu^{12} + 70810 \nu^{11} - 149110 \nu^{10} + 90666 \nu^{9} + 2237178 \nu^{8} - 5056056 \nu^{7} + \cdots + 6782250042 ) / 3443737680$$ (-449*v^15 - 598*v^14 - 6497*v^13 - 17650*v^12 + 70810*v^11 - 149110*v^10 + 90666*v^9 + 2237178*v^8 - 5056056*v^7 + 8100432*v^6 + 64314810*v^5 - 83681910*v^4 + 205195275*v^3 + 2727119016*v^2 - 2316551319*v + 6782250042) / 3443737680 $$\beta_{6}$$ $$=$$ $$( - 329 \nu^{15} - 1708 \nu^{14} + 3703 \nu^{13} + 2120 \nu^{12} - 22220 \nu^{11} + 275480 \nu^{10} + 28926 \nu^{9} - 281292 \nu^{8} + 5112954 \nu^{7} + \cdots - 10005971148 ) / 2295825120$$ (-329*v^15 - 1708*v^14 + 3703*v^13 + 2120*v^12 - 22220*v^11 + 275480*v^10 + 28926*v^9 - 281292*v^8 + 5112954*v^7 - 18822348*v^6 - 48507660*v^5 + 95586480*v^4 - 75221865*v^3 - 581986944*v^2 + 1600168851*v - 10005971148) / 2295825120 $$\beta_{7}$$ $$=$$ $$( - 6992 \nu^{15} - 36013 \nu^{14} + 58996 \nu^{13} - 326953 \nu^{12} + 578104 \nu^{11} + 243020 \nu^{10} - 9067572 \nu^{9} + 12700002 \nu^{8} + \cdots - 137983872681 ) / 34437376800$$ (-6992*v^15 - 36013*v^14 + 58996*v^13 - 326953*v^12 + 578104*v^11 + 243020*v^10 - 9067572*v^9 + 12700002*v^8 + 95667444*v^7 - 230079906*v^6 + 173132640*v^5 + 1510916652*v^4 - 8620602876*v^3 + 25348968063*v^2 - 31427294976*v - 137983872681) / 34437376800 $$\beta_{8}$$ $$=$$ $$( 667 \nu^{15} - 3082 \nu^{14} + 874 \nu^{13} - 4717 \nu^{12} + 31606 \nu^{11} - 37600 \nu^{10} + 135612 \nu^{9} - 21642 \nu^{8} + 787716 \nu^{7} + 11190096 \nu^{6} + \cdots + 1095299901 ) / 2869781400$$ (667*v^15 - 3082*v^14 + 874*v^13 - 4717*v^12 + 31606*v^11 - 37600*v^10 + 135612*v^9 - 21642*v^8 + 787716*v^7 + 11190096*v^6 - 15734250*v^5 + 68257728*v^4 - 87910839*v^3 + 399643632*v^2 - 581396454*v + 1095299901) / 2869781400 $$\beta_{9}$$ $$=$$ $$( - 4039 \nu^{15} - 3041 \nu^{14} + 49277 \nu^{13} - 178841 \nu^{12} + 468248 \nu^{11} - 1872680 \nu^{10} - 1437354 \nu^{9} + 15826074 \nu^{8} + \cdots + 72064993923 ) / 11479125600$$ (-4039*v^15 - 3041*v^14 + 49277*v^13 - 178841*v^12 + 468248*v^11 - 1872680*v^10 - 1437354*v^9 + 15826074*v^8 - 24568722*v^7 - 57363822*v^6 + 490471200*v^5 - 480603456*v^4 - 146093787*v^3 + 12571473051*v^2 - 49895401167*v + 72064993923) / 11479125600 $$\beta_{10}$$ $$=$$ $$( 6247 \nu^{15} - 15502 \nu^{14} + 37729 \nu^{13} + 59948 \nu^{12} - 164414 \nu^{11} + 519950 \nu^{10} + 360342 \nu^{9} - 11315742 \nu^{8} + \cdots - 10120762404 ) / 17218688400$$ (6247*v^15 - 15502*v^14 + 37729*v^13 + 59948*v^12 - 164414*v^11 + 519950*v^10 + 360342*v^9 - 11315742*v^8 + 30006576*v^7 + 26638416*v^6 - 415979550*v^5 + 827955918*v^4 - 1553441409*v^3 + 5980955112*v^2 + 8094377871*v - 10120762404) / 17218688400 $$\beta_{11}$$ $$=$$ $$( 773 \nu^{15} - 1103 \nu^{14} - 6139 \nu^{13} + 27127 \nu^{12} - 90736 \nu^{11} + 54340 \nu^{10} + 233658 \nu^{9} - 2256618 \nu^{8} + 634914 \nu^{7} + \cdots - 5017334481 ) / 1913187600$$ (773*v^15 - 1103*v^14 - 6139*v^13 + 27127*v^12 - 90736*v^11 + 54340*v^10 + 233658*v^9 - 2256618*v^8 + 634914*v^7 + 10980414*v^6 - 56288520*v^5 + 100097532*v^4 + 138233709*v^3 - 1011096027*v^2 + 5712459309*v - 5017334481) / 1913187600 $$\beta_{12}$$ $$=$$ $$( - 559 \nu^{15} + 2326 \nu^{14} - 199 \nu^{13} - 6164 \nu^{12} + 75908 \nu^{11} - 242444 \nu^{10} + 218034 \nu^{9} + 1818708 \nu^{8} - 5902866 \nu^{7} + \cdots + 9795520512 ) / 1147912560$$ (-559*v^15 + 2326*v^14 - 199*v^13 - 6164*v^12 + 75908*v^11 - 242444*v^10 + 218034*v^9 + 1818708*v^8 - 5902866*v^7 - 2124252*v^6 + 27924588*v^5 - 155361564*v^4 + 155580993*v^3 + 125065782*v^2 - 4627256787*v + 9795520512) / 1147912560 $$\beta_{13}$$ $$=$$ $$( 13753 \nu^{15} - 64453 \nu^{14} + 112681 \nu^{13} + 104957 \nu^{12} - 1214876 \nu^{11} + 4647440 \nu^{10} - 7222662 \nu^{9} - 14753058 \nu^{8} + \cdots - 214176568851 ) / 17218688400$$ (13753*v^15 - 64453*v^14 + 112681*v^13 + 104957*v^12 - 1214876*v^11 + 4647440*v^10 - 7222662*v^9 - 14753058*v^8 + 102614814*v^7 - 51256206*v^6 - 804558420*v^5 + 3873270312*v^4 - 9041523831*v^3 + 5760052803*v^2 + 57432297429*v - 214176568851) / 17218688400 $$\beta_{14}$$ $$=$$ $$( - 3461 \nu^{15} + 18971 \nu^{14} - 24797 \nu^{13} - 41569 \nu^{12} + 437512 \nu^{11} - 1480000 \nu^{10} + 1322274 \nu^{9} + 10243146 \nu^{8} + \cdots + 61746535467 ) / 3826375200$$ (-3461*v^15 + 18971*v^14 - 24797*v^13 - 41569*v^12 + 437512*v^11 - 1480000*v^10 + 1322274*v^9 + 10243146*v^8 - 28339758*v^7 + 3611682*v^6 + 339059520*v^5 - 1125850104*v^4 + 2185208847*v^3 + 564764319*v^2 - 24577197633*v + 61746535467) / 3826375200 $$\beta_{15}$$ $$=$$ $$( - 49627 \nu^{15} + 159457 \nu^{14} + 8801 \nu^{13} - 1090223 \nu^{12} + 5845844 \nu^{11} - 15027140 \nu^{10} + 5408478 \nu^{9} + 112033842 \nu^{8} + \cdots + 673542477549 ) / 34437376800$$ (-49627*v^15 + 159457*v^14 + 8801*v^13 - 1090223*v^12 + 5845844*v^11 - 15027140*v^10 + 5408478*v^9 + 112033842*v^8 - 318871206*v^7 - 398863386*v^6 + 4508840700*v^5 - 8900282268*v^4 + 11421710289*v^3 + 40699936593*v^2 - 354190014711*v + 673542477549) / 34437376800
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{10} - \beta_{8} + \beta_{5} - 1$$ b10 - b8 + b5 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{14} - \beta_{13} - 3 \beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + \beta _1 - 6$$ b14 - b13 - 3*b12 + b10 + b9 - 2*b7 + 2*b6 - b5 + 2*b4 - 4*b3 + b1 - 6 $$\nu^{4}$$ $$=$$ $$6 \beta_{15} + 6 \beta_{13} + 9 \beta_{11} + \beta_{10} - 6 \beta_{9} - 4 \beta_{8} + \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} - 3 \beta _1 + 11$$ 6*b15 + 6*b13 + 9*b11 + b10 - 6*b9 - 4*b8 + b5 + 6*b4 + 3*b3 + 6*b2 - 3*b1 + 11 $$\nu^{5}$$ $$=$$ $$12 \beta_{15} + 4 \beta_{14} + 11 \beta_{13} - 18 \beta_{12} + 6 \beta_{11} - 8 \beta_{10} + 4 \beta_{9} + 15 \beta_{8} - 8 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - 10 \beta_{4} + 41 \beta_{3} - 6 \beta_{2} + 16 \beta _1 + 3$$ 12*b15 + 4*b14 + 11*b13 - 18*b12 + 6*b11 - 8*b10 + 4*b9 + 15*b8 - 8*b7 - 4*b6 - 4*b5 - 10*b4 + 41*b3 - 6*b2 + 16*b1 + 3 $$\nu^{6}$$ $$=$$ $$- 18 \beta_{15} - 30 \beta_{14} - 36 \beta_{13} + 6 \beta_{12} - 90 \beta_{11} + 4 \beta_{10} - 24 \beta_{9} + 8 \beta_{8} - 24 \beta_{7} - 96 \beta_{6} + 10 \beta_{5} + 30 \beta_{4} + 156 \beta_{3} + 18 \beta_{2} - 6 \beta _1 + 5$$ -18*b15 - 30*b14 - 36*b13 + 6*b12 - 90*b11 + 4*b10 - 24*b9 + 8*b8 - 24*b7 - 96*b6 + 10*b5 + 30*b4 + 156*b3 + 18*b2 - 6*b1 + 5 $$\nu^{7}$$ $$=$$ $$- 84 \beta_{15} + 184 \beta_{14} + 2 \beta_{13} - 114 \beta_{12} - 60 \beta_{11} + 160 \beta_{10} + 40 \beta_{9} + 18 \beta_{8} + 136 \beta_{7} + 20 \beta_{6} - 112 \beta_{5} - 10 \beta_{4} + 320 \beta_{3} + 42 \beta_{2} + 103 \beta _1 + 504$$ -84*b15 + 184*b14 + 2*b13 - 114*b12 - 60*b11 + 160*b10 + 40*b9 + 18*b8 + 136*b7 + 20*b6 - 112*b5 - 10*b4 + 320*b3 + 42*b2 + 103*b1 + 504 $$\nu^{8}$$ $$=$$ $$- 330 \beta_{15} + 552 \beta_{14} + 624 \beta_{13} + 408 \beta_{12} - 18 \beta_{11} - 281 \beta_{10} + 66 \beta_{9} - 187 \beta_{8} - 48 \beta_{7} + 240 \beta_{6} + 325 \beta_{5} + 90 \beta_{4} + 174 \beta_{3} + 282 \beta_{2} + \cdots + 485$$ -330*b15 + 552*b14 + 624*b13 + 408*b12 - 18*b11 - 281*b10 + 66*b9 - 187*b8 - 48*b7 + 240*b6 + 325*b5 + 90*b4 + 174*b3 + 282*b2 + 324*b1 + 485 $$\nu^{9}$$ $$=$$ $$373 \beta_{14} + 1001 \beta_{13} + 1311 \beta_{12} + 918 \beta_{11} + 1255 \beta_{10} + 373 \beta_{9} + 138 \beta_{8} - 350 \beta_{7} + 1466 \beta_{6} + 2201 \beta_{5} - 1540 \beta_{4} + 4658 \beta_{3} + 1134 \beta_{2} + \cdots - 7086$$ 373*b14 + 1001*b13 + 1311*b12 + 918*b11 + 1255*b10 + 373*b9 + 138*b8 - 350*b7 + 1466*b6 + 2201*b5 - 1540*b4 + 4658*b3 + 1134*b2 + 253*b1 - 7086 $$\nu^{10}$$ $$=$$ $$996 \beta_{15} - 1512 \beta_{14} + 564 \beta_{13} - 1512 \beta_{12} - 6201 \beta_{11} - 809 \beta_{10} - 3804 \beta_{9} + 176 \beta_{8} - 4752 \beta_{7} + 1296 \beta_{6} + 4645 \beta_{5} - 300 \beta_{4} - 717 \beta_{3} + \cdots + 13763$$ 996*b15 - 1512*b14 + 564*b13 - 1512*b12 - 6201*b11 - 809*b10 - 3804*b9 + 176*b8 - 4752*b7 + 1296*b6 + 4645*b5 - 300*b4 - 717*b3 - 1164*b2 - 7761*b1 + 13763 $$\nu^{11}$$ $$=$$ $$12360 \beta_{15} - 3632 \beta_{14} - 2275 \beta_{13} - 5220 \beta_{12} - 4836 \beta_{11} + 6904 \beta_{10} - 18752 \beta_{9} + 24927 \beta_{8} + 352 \beta_{7} + 248 \beta_{6} - 10192 \beta_{5} + 5732 \beta_{4} + \cdots + 16527$$ 12360*b15 - 3632*b14 - 2275*b13 - 5220*b12 - 4836*b11 + 6904*b10 - 18752*b9 + 24927*b8 + 352*b7 + 248*b6 - 10192*b5 + 5732*b4 - 1741*b3 - 7476*b2 + 23632*b1 + 16527 $$\nu^{12}$$ $$=$$ $$28260 \beta_{15} + 6468 \beta_{14} + 21384 \beta_{13} + 348 \beta_{12} + 29556 \beta_{11} + 30712 \beta_{10} - 16008 \beta_{9} + 11384 \beta_{8} - 10032 \beta_{7} - 7296 \beta_{6} - 19700 \beta_{5} + \cdots + 259997$$ 28260*b15 + 6468*b14 + 21384*b13 + 348*b12 + 29556*b11 + 30712*b10 - 16008*b9 + 11384*b8 - 10032*b7 - 7296*b6 - 19700*b5 - 492*b4 + 146280*b3 - 39636*b2 + 31980*b1 + 259997 $$\nu^{13}$$ $$=$$ $$4488 \beta_{15} - 35240 \beta_{14} - 24460 \beta_{13} - 14388 \beta_{12} - 127104 \beta_{11} + 112984 \beta_{10} + 73480 \beta_{9} - 27756 \beta_{8} - 132704 \beta_{7} - 36088 \beta_{6} + \cdots - 404640$$ 4488*b15 - 35240*b14 - 24460*b13 - 14388*b12 - 127104*b11 + 112984*b10 + 73480*b9 - 27756*b8 - 132704*b7 - 36088*b6 - 106600*b5 + 27476*b4 + 208544*b3 - 2244*b2 + 307273*b1 - 404640 $$\nu^{14}$$ $$=$$ $$49764 \beta_{15} + 133464 \beta_{14} + 227136 \beta_{13} + 78888 \beta_{12} - 95004 \beta_{11} + 406537 \beta_{10} - 343644 \beta_{9} - 795625 \beta_{8} - 10704 \beta_{7} - 206544 \beta_{6} + \cdots - 1093681$$ 49764*b15 + 133464*b14 + 227136*b13 + 78888*b12 - 95004*b11 + 406537*b10 - 343644*b9 - 795625*b8 - 10704*b7 - 206544*b6 + 86509*b5 + 146124*b4 - 218364*b3 + 272748*b2 - 125208*b1 - 1093681 $$\nu^{15}$$ $$=$$ $$- 46080 \beta_{15} + 1249717 \beta_{14} + 1304399 \beta_{13} - 491403 \beta_{12} + 430236 \beta_{11} + 296113 \beta_{10} - 1355 \beta_{9} + 744564 \beta_{8} - 371762 \beta_{7} + \cdots - 4803294$$ -46080*b15 + 1249717*b14 + 1304399*b13 - 491403*b12 + 430236*b11 + 296113*b10 - 1355*b9 + 744564*b8 - 371762*b7 + 568538*b6 - 255217*b5 - 414586*b4 - 4009216*b3 - 56988*b2 - 686339*b1 - 4803294

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$97$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −2.99749 − 0.122585i −2.52762 − 1.61590i −0.752365 − 2.90413i −0.0515370 + 2.99956i 1.30755 + 2.70006i 1.62763 − 2.52009i 2.42141 − 1.77110i 2.97243 + 0.405752i −2.99749 + 0.122585i −2.52762 + 1.61590i −0.752365 + 2.90413i −0.0515370 − 2.99956i 1.30755 − 2.70006i 1.62763 + 2.52009i 2.42141 + 1.77110i 2.97243 − 0.405752i
1.41421i −2.99749 0.122585i −2.00000 4.36583i −0.173361 + 4.23910i 5.99969 2.82843i 8.96995 + 0.734895i 6.17421
47.2 1.41421i −2.52762 1.61590i −2.00000 7.30416i −2.28522 + 3.57460i −0.709880 2.82843i 3.77777 + 8.16875i −10.3296
47.3 1.41421i −0.752365 2.90413i −2.00000 9.12709i −4.10705 + 1.06400i −11.5474 2.82843i −7.86789 + 4.36992i 12.9076
47.4 1.41421i −0.0515370 + 2.99956i −2.00000 2.42657i 4.24201 + 0.0728843i −2.87957 2.82843i −8.99469 0.309176i 3.43169
47.5 1.41421i 1.30755 + 2.70006i −2.00000 8.14461i 3.81846 1.84916i 7.21035 2.82843i −5.58060 + 7.06094i −11.5182
47.6 1.41421i 1.62763 2.52009i −2.00000 1.81026i −3.56394 2.30181i 9.47484 2.82843i −3.70167 8.20351i −2.56010
47.7 1.41421i 2.42141 1.77110i −2.00000 4.98213i −2.50471 3.42438i −12.5950 2.82843i 2.72641 8.57710i −7.04580
47.8 1.41421i 2.97243 + 0.405752i −2.00000 6.32168i 0.573819 4.20366i 5.04690 2.82843i 8.67073 + 2.41214i 8.94020
47.9 1.41421i −2.99749 + 0.122585i −2.00000 4.36583i −0.173361 4.23910i 5.99969 2.82843i 8.96995 0.734895i 6.17421
47.10 1.41421i −2.52762 + 1.61590i −2.00000 7.30416i −2.28522 3.57460i −0.709880 2.82843i 3.77777 8.16875i −10.3296
47.11 1.41421i −0.752365 + 2.90413i −2.00000 9.12709i −4.10705 1.06400i −11.5474 2.82843i −7.86789 4.36992i 12.9076
47.12 1.41421i −0.0515370 2.99956i −2.00000 2.42657i 4.24201 0.0728843i −2.87957 2.82843i −8.99469 + 0.309176i 3.43169
47.13 1.41421i 1.30755 2.70006i −2.00000 8.14461i 3.81846 + 1.84916i 7.21035 2.82843i −5.58060 7.06094i −11.5182
47.14 1.41421i 1.62763 + 2.52009i −2.00000 1.81026i −3.56394 + 2.30181i 9.47484 2.82843i −3.70167 + 8.20351i −2.56010
47.15 1.41421i 2.42141 + 1.77110i −2.00000 4.98213i −2.50471 + 3.42438i −12.5950 2.82843i 2.72641 + 8.57710i −7.04580
47.16 1.41421i 2.97243 0.405752i −2.00000 6.32168i 0.573819 + 4.20366i 5.04690 2.82843i 8.67073 2.41214i 8.94020
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.16 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 138.3.c.a 16
3.b odd 2 1 inner 138.3.c.a 16
4.b odd 2 1 1104.3.g.c 16
12.b even 2 1 1104.3.g.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.3.c.a 16 1.a even 1 1 trivial
138.3.c.a 16 3.b odd 2 1 inner
1104.3.g.c 16 4.b odd 2 1
1104.3.g.c 16 12.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(138, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{8}$$
$3$ $$T^{16} - 4 T^{15} + 10 T^{14} + \cdots + 43046721$$
$5$ $$T^{16} + 296 T^{14} + \cdots + 107557761600$$
$7$ $$(T^{8} - 252 T^{6} + 664 T^{5} + \cdots + 615000)^{2}$$
$11$ $$T^{16} + 816 T^{14} + \cdots + 429981696$$
$13$ $$(T^{8} + 4 T^{7} - 1014 T^{6} + \cdots + 960369700)^{2}$$
$17$ $$T^{16} + 2056 T^{14} + \cdots + 16\!\cdots\!36$$
$19$ $$(T^{8} - 20 T^{7} - 708 T^{6} + \cdots - 780768)^{2}$$
$23$ $$(T^{2} + 23)^{8}$$
$29$ $$T^{16} + 9116 T^{14} + \cdots + 21\!\cdots\!00$$
$31$ $$(T^{8} - 68 T^{7} - 774 T^{6} + \cdots + 145917216)^{2}$$
$37$ $$(T^{8} + 68 T^{7} - 2008 T^{6} + \cdots + 44098389696)^{2}$$
$41$ $$T^{16} + 19196 T^{14} + \cdots + 69\!\cdots\!24$$
$43$ $$(T^{8} - 36 T^{7} + \cdots - 2868882576000)^{2}$$
$47$ $$T^{16} + 16508 T^{14} + \cdots + 37\!\cdots\!00$$
$53$ $$T^{16} + 27592 T^{14} + \cdots + 32\!\cdots\!00$$
$59$ $$T^{16} + 37288 T^{14} + \cdots + 30\!\cdots\!84$$
$61$ $$(T^{8} - 24 T^{7} + \cdots - 20431830161152)^{2}$$
$67$ $$(T^{8} + 152 T^{7} + \cdots + 3918307649688)^{2}$$
$71$ $$T^{16} + 52292 T^{14} + \cdots + 26\!\cdots\!76$$
$73$ $$(T^{8} - 204 T^{7} + \cdots + 171604877785344)^{2}$$
$79$ $$(T^{8} - 156 T^{7} + \cdots + 173075158110240)^{2}$$
$83$ $$T^{16} + 51824 T^{14} + \cdots + 11\!\cdots\!84$$
$89$ $$T^{16} + 64472 T^{14} + \cdots + 12\!\cdots\!00$$
$97$ $$(T^{8} - 84 T^{7} - 12312 T^{6} + \cdots - 44369259840)^{2}$$