# Properties

 Label 39.2.k.b Level $39$ Weight $2$ Character orbit 39.k Analytic conductor $0.311$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 39.k (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.311416567883$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: 8.0.56070144.2 Defining polynomial: $$x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13$$ x^8 - 4*x^7 + 16*x^6 - 34*x^5 + 63*x^4 - 74*x^3 + 70*x^2 - 38*x + 13 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} - 15\nu^{6} + 32\nu^{5} - 172\nu^{4} + 221\nu^{3} - 426\nu^{2} + 235\nu - 159 ) / 37$$ (-v^7 - 15*v^6 + 32*v^5 - 172*v^4 + 221*v^3 - 426*v^2 + 235*v - 159) / 37 $$\beta_{3}$$ $$=$$ $$( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 390\nu^{2} + 298\nu - 70 ) / 37$$ (-3*v^7 - 8*v^6 + 22*v^5 - 146*v^4 + 256*v^3 - 390*v^2 + 298*v - 70) / 37 $$\beta_{4}$$ $$=$$ $$( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 427\nu^{2} + 335\nu - 181 ) / 37$$ (-3*v^7 - 8*v^6 + 22*v^5 - 146*v^4 + 256*v^3 - 427*v^2 + 335*v - 181) / 37 $$\beta_{5}$$ $$=$$ $$( 3\nu^{7} - 29\nu^{6} + 89\nu^{5} - 261\nu^{4} + 373\nu^{3} - 498\nu^{2} + 294\nu - 152 ) / 37$$ (3*v^7 - 29*v^6 + 89*v^5 - 261*v^4 + 373*v^3 - 498*v^2 + 294*v - 152) / 37 $$\beta_{6}$$ $$=$$ $$( -8\nu^{7} + 28\nu^{6} - 114\nu^{5} + 215\nu^{4} - 378\nu^{3} + 366\nu^{2} - 266\nu + 97 ) / 37$$ (-8*v^7 + 28*v^6 - 114*v^5 + 215*v^4 - 378*v^3 + 366*v^2 - 266*v + 97) / 37 $$\beta_{7}$$ $$=$$ $$( 17\nu^{7} - 41\nu^{6} + 159\nu^{5} - 184\nu^{4} + 276\nu^{3} - 84\nu^{2} + 38\nu + 39 ) / 37$$ (17*v^7 - 41*v^6 + 159*v^5 - 184*v^4 + 276*v^3 - 84*v^2 + 38*v + 39) / 37
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta _1 - 3$$ -b4 + b3 + b1 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{3} - 2\beta _1 - 4$$ b7 + b6 - b5 + 2*b3 - 2*b1 - 4 $$\nu^{4}$$ $$=$$ $$2\beta_{7} + 3\beta_{6} + 6\beta_{4} - 2\beta_{3} - 2\beta_{2} - 6\beta _1 + 7$$ 2*b7 + 3*b6 + 6*b4 - 2*b3 - 2*b2 - 6*b1 + 7 $$\nu^{5}$$ $$=$$ $$-4\beta_{7} - 3\beta_{6} + 7\beta_{5} + 6\beta_{4} - 12\beta_{3} - 5\beta_{2} + \beta _1 + 26$$ -4*b7 - 3*b6 + 7*b5 + 6*b4 - 12*b3 - 5*b2 + b1 + 26 $$\nu^{6}$$ $$=$$ $$-17\beta_{7} - 25\beta_{6} + 3\beta_{5} - 24\beta_{4} - 5\beta_{3} + 7\beta_{2} + 27\beta _1 - 1$$ -17*b7 - 25*b6 + 3*b5 - 24*b4 - 5*b3 + 7*b2 + 27*b1 - 1 $$\nu^{7}$$ $$=$$ $$4\beta_{7} - 16\beta_{6} - 42\beta_{5} - 54\beta_{4} + 51\beta_{3} + 42\beta_{2} + 26\beta _1 - 122$$ 4*b7 - 16*b6 - 42*b5 - 54*b4 + 51*b3 + 42*b2 + 26*b1 - 122

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$$.

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$-1$$ $$\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.5 − 1.19293i 0.5 + 2.19293i 0.5 − 1.56488i 0.5 + 0.564882i 0.5 + 1.19293i 0.5 − 2.19293i 0.5 + 1.56488i 0.5 − 0.564882i
−0.619657 + 2.31259i 1.64914 0.529480i −3.23205 1.86603i −1.69293 1.69293i 0.202571 + 4.14187i −1.36603 + 0.366025i 2.93225 2.93225i 2.43930 1.74637i 4.96410 2.86603i
2.2 0.619657 2.31259i −1.28311 + 1.16345i −3.23205 1.86603i 1.69293 + 1.69293i 1.89551 + 3.68825i −1.36603 + 0.366025i −2.93225 + 2.93225i 0.292748 2.98568i 4.96410 2.86603i
11.1 −1.45466 0.389774i 0.239203 1.71545i 0.232051 + 0.133975i 1.06488 1.06488i −1.01660 + 2.40216i 0.366025 + 1.36603i 1.84443 + 1.84443i −2.88556 0.820682i −1.96410 + 1.13397i
11.2 1.45466 + 0.389774i −1.60523 0.650571i 0.232051 + 0.133975i −1.06488 + 1.06488i −2.08148 1.57203i 0.366025 + 1.36603i −1.84443 1.84443i 2.15351 + 2.08863i −1.96410 + 1.13397i
20.1 −0.619657 2.31259i 1.64914 + 0.529480i −3.23205 + 1.86603i −1.69293 + 1.69293i 0.202571 4.14187i −1.36603 0.366025i 2.93225 + 2.93225i 2.43930 + 1.74637i 4.96410 + 2.86603i
20.2 0.619657 + 2.31259i −1.28311 1.16345i −3.23205 + 1.86603i 1.69293 1.69293i 1.89551 3.68825i −1.36603 0.366025i −2.93225 2.93225i 0.292748 + 2.98568i 4.96410 + 2.86603i
32.1 −1.45466 + 0.389774i 0.239203 + 1.71545i 0.232051 0.133975i 1.06488 + 1.06488i −1.01660 2.40216i 0.366025 1.36603i 1.84443 1.84443i −2.88556 + 0.820682i −1.96410 1.13397i
32.2 1.45466 0.389774i −1.60523 + 0.650571i 0.232051 0.133975i −1.06488 1.06488i −2.08148 + 1.57203i 0.366025 1.36603i −1.84443 + 1.84443i 2.15351 2.08863i −1.96410 1.13397i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 32.2 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
13.f odd 12 1 inner
39.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.k.b 8
3.b odd 2 1 inner 39.2.k.b 8
4.b odd 2 1 624.2.cn.c 8
5.b even 2 1 975.2.bo.d 8
5.c odd 4 1 975.2.bp.e 8
5.c odd 4 1 975.2.bp.f 8
12.b even 2 1 624.2.cn.c 8
13.b even 2 1 507.2.k.d 8
13.c even 3 1 507.2.f.f 8
13.c even 3 1 507.2.k.e 8
13.d odd 4 1 507.2.k.e 8
13.d odd 4 1 507.2.k.f 8
13.e even 6 1 507.2.f.e 8
13.e even 6 1 507.2.k.f 8
13.f odd 12 1 inner 39.2.k.b 8
13.f odd 12 1 507.2.f.e 8
13.f odd 12 1 507.2.f.f 8
13.f odd 12 1 507.2.k.d 8
15.d odd 2 1 975.2.bo.d 8
15.e even 4 1 975.2.bp.e 8
15.e even 4 1 975.2.bp.f 8
39.d odd 2 1 507.2.k.d 8
39.f even 4 1 507.2.k.e 8
39.f even 4 1 507.2.k.f 8
39.h odd 6 1 507.2.f.e 8
39.h odd 6 1 507.2.k.f 8
39.i odd 6 1 507.2.f.f 8
39.i odd 6 1 507.2.k.e 8
39.k even 12 1 inner 39.2.k.b 8
39.k even 12 1 507.2.f.e 8
39.k even 12 1 507.2.f.f 8
39.k even 12 1 507.2.k.d 8
52.l even 12 1 624.2.cn.c 8
65.o even 12 1 975.2.bp.f 8
65.s odd 12 1 975.2.bo.d 8
65.t even 12 1 975.2.bp.e 8
156.v odd 12 1 624.2.cn.c 8
195.bc odd 12 1 975.2.bp.e 8
195.bh even 12 1 975.2.bo.d 8
195.bn odd 12 1 975.2.bp.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.b 8 1.a even 1 1 trivial
39.2.k.b 8 3.b odd 2 1 inner
39.2.k.b 8 13.f odd 12 1 inner
39.2.k.b 8 39.k even 12 1 inner
507.2.f.e 8 13.e even 6 1
507.2.f.e 8 13.f odd 12 1
507.2.f.e 8 39.h odd 6 1
507.2.f.e 8 39.k even 12 1
507.2.f.f 8 13.c even 3 1
507.2.f.f 8 13.f odd 12 1
507.2.f.f 8 39.i odd 6 1
507.2.f.f 8 39.k even 12 1
507.2.k.d 8 13.b even 2 1
507.2.k.d 8 13.f odd 12 1
507.2.k.d 8 39.d odd 2 1
507.2.k.d 8 39.k even 12 1
507.2.k.e 8 13.c even 3 1
507.2.k.e 8 13.d odd 4 1
507.2.k.e 8 39.f even 4 1
507.2.k.e 8 39.i odd 6 1
507.2.k.f 8 13.d odd 4 1
507.2.k.f 8 13.e even 6 1
507.2.k.f 8 39.f even 4 1
507.2.k.f 8 39.h odd 6 1
624.2.cn.c 8 4.b odd 2 1
624.2.cn.c 8 12.b even 2 1
624.2.cn.c 8 52.l even 12 1
624.2.cn.c 8 156.v odd 12 1
975.2.bo.d 8 5.b even 2 1
975.2.bo.d 8 15.d odd 2 1
975.2.bo.d 8 65.s odd 12 1
975.2.bo.d 8 195.bh even 12 1
975.2.bp.e 8 5.c odd 4 1
975.2.bp.e 8 15.e even 4 1
975.2.bp.e 8 65.t even 12 1
975.2.bp.e 8 195.bc odd 12 1
975.2.bp.f 8 5.c odd 4 1
975.2.bp.f 8 15.e even 4 1
975.2.bp.f 8 65.o even 12 1
975.2.bp.f 8 195.bn odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 6T_{2}^{6} - T_{2}^{4} - 78T_{2}^{2} + 169$$ acting on $$S_{2}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 6 T^{6} - T^{4} - 78 T^{2} + \cdots + 169$$
$3$ $$T^{8} + 2 T^{7} - 4 T^{5} - 5 T^{4} + \cdots + 81$$
$5$ $$T^{8} + 38T^{4} + 169$$
$7$ $$(T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4)^{2}$$
$11$ $$T^{8} + 24 T^{6} + 140 T^{4} + \cdots + 2704$$
$13$ $$(T^{4} - 4 T^{3} + 3 T^{2} - 52 T + 169)^{2}$$
$17$ $$T^{8} + 30 T^{6} + 783 T^{4} + \cdots + 13689$$
$19$ $$(T^{4} + 8 T^{3} + 20 T^{2} + 16 T + 16)^{2}$$
$23$ $$T^{8}$$
$29$ $$T^{8} - 82 T^{6} + 5151 T^{4} + \cdots + 2474329$$
$31$ $$(T^{4} - 4 T^{3} + 8 T^{2} + 88 T + 484)^{2}$$
$37$ $$(T^{4} + 14 T^{3} + 113 T^{2} + 592 T + 1369)^{2}$$
$41$ $$T^{8} - 54 T^{6} + 959 T^{4} + \cdots + 169$$
$43$ $$(T^{4} - 18 T^{3} + 126 T^{2} - 324 T + 324)^{2}$$
$47$ $$T^{8} + 9728 T^{4} + \cdots + 11075584$$
$53$ $$(T^{4} + 22 T^{2} + 13)^{2}$$
$59$ $$T^{8} + 24 T^{6} - 16 T^{4} + \cdots + 43264$$
$61$ $$(T^{2} - 7 T + 49)^{4}$$
$67$ $$(T^{4} + 20 T^{3} + 164 T^{2} + 832 T + 2704)^{2}$$
$71$ $$T^{8} - 24 T^{6} - 16 T^{4} + \cdots + 43264$$
$73$ $$(T^{4} + 14 T^{3} + 98 T^{2} + 154 T + 121)^{2}$$
$79$ $$(T - 2)^{8}$$
$83$ $$T^{8} + 296T^{4} + 2704$$
$89$ $$T^{8} - 24 T^{6} - 8596 T^{4} + \cdots + 77228944$$
$97$ $$(T^{4} - 10 T^{3} + 194 T^{2} - 572 T + 484)^{2}$$