# Properties

 Label 390.2.bn.b Level $390$ Weight $2$ Character orbit 390.bn Analytic conductor $3.114$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.bn (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4$$ x^16 - 4*x^15 + 12*x^14 - 48*x^13 + 67*x^12 - 24*x^11 + 118*x^10 - 176*x^9 + 351*x^8 - 180*x^7 + 358*x^6 - 336*x^5 + 390*x^4 - 344*x^3 + 164*x^2 - 40*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( 1434138866353 \nu^{15} - 74224360591111 \nu^{14} + 217326371958939 \nu^{13} - 648847305484636 \nu^{12} + \cdots + 46\!\cdots\!94 ) / 51\!\cdots\!90$$ (1434138866353*v^15 - 74224360591111*v^14 + 217326371958939*v^13 - 648847305484636*v^12 + 2680829374033629*v^11 - 1607350163227359*v^10 - 900494454550979*v^9 - 8792384561334166*v^8 + 3536477625795241*v^7 - 17495309067416083*v^6 - 6622116382602757*v^5 - 26263502656698172*v^4 - 5327429661771194*v^3 - 19428846797347660*v^2 + 4742345058394342*v + 4680253552809794) / 5173992472766390 $$\beta_{2}$$ $$=$$ $$( - 123434729989149 \nu^{15} + 492304781090243 \nu^{14} + \cdots + 195044141171618 ) / 51\!\cdots\!90$$ (-123434729989149*v^15 + 492304781090243*v^14 - 1406992399278677*v^13 + 5707540667520213*v^12 - 7621279603788347*v^11 + 281604145705947*v^10 - 12957947975492223*v^9 + 22625006932641203*v^8 - 34533205664857133*v^7 + 18681773772251579*v^6 - 26694324268699259*v^5 + 48096185658956821*v^4 - 21876042039069938*v^3 + 47788976778038450*v^2 - 814448920872776*v + 195044141171618) / 5173992472766390 $$\beta_{3}$$ $$=$$ $$( - 137809854283897 \nu^{15} + 666808683532056 \nu^{14} + \cdots + 69\!\cdots\!44 ) / 10\!\cdots\!78$$ (-137809854283897*v^15 + 666808683532056*v^14 - 2027562782554100*v^13 + 7694802116157034*v^12 - 13885376435295071*v^11 + 7288708077624688*v^10 - 15122432396762870*v^9 + 37940274269902486*v^8 - 58228159495736079*v^7 + 55051784324946342*v^6 - 44775296775045806*v^5 + 85474895099571132*v^4 - 62657114582598388*v^3 + 77038886286558388*v^2 - 37418608795158304*v + 6950822659807744) / 1034798494553278 $$\beta_{4}$$ $$=$$ $$( 3909951128714 \nu^{15} - 13126607646508 \nu^{14} + 38145193554947 \nu^{13} - 162143292202288 \nu^{12} + \cdots - 16575048540328 ) / 28585593772190$$ (3909951128714*v^15 - 13126607646508*v^14 + 38145193554947*v^13 - 162143292202288*v^12 + 154868117754112*v^11 + 18595969360728*v^10 + 464520812336313*v^9 - 395501573216208*v^8 + 1073827229563468*v^7 + 11545660383276*v^6 + 1336974192211299*v^5 - 488205468817916*v^4 + 1087029641011808*v^3 - 622728293466680*v^2 + 162118602709476*v - 16575048540328) / 28585593772190 $$\beta_{5}$$ $$=$$ $$( - 4143762135082 \nu^{15} + 12665097411614 \nu^{14} - 36598537974476 \nu^{13} + 160755388928989 \nu^{12} + \cdots + 3631882693804 ) / 28585593772190$$ (-4143762135082*v^15 + 12665097411614*v^14 - 36598537974476*v^13 + 160755388928989*v^12 - 115488770848206*v^11 - 55417826512144*v^10 - 507559901300404*v^9 + 264781323438119*v^8 - 1058958936197574*v^7 - 327950045248708*v^6 - 1495012504742632*v^5 + 55329885176253*v^4 - 1127861763864064*v^3 + 338424533456400*v^2 - 56848696686768*v + 3631882693804) / 28585593772190 $$\beta_{6}$$ $$=$$ $$( - 14\!\cdots\!27 \nu^{15} + \cdots + 35\!\cdots\!44 ) / 51\!\cdots\!90$$ (-1439070217869327*v^15 + 5626426049162414*v^14 - 16762149286150656*v^13 + 67570735976932179*v^12 - 90330349700000641*v^11 + 26412913789425376*v^10 - 167480504119187474*v^9 + 237626790891811969*v^8 - 482812364930441289*v^7 + 215899963434715562*v^6 - 494473747233645422*v^5 + 437389184714757453*v^4 - 517428552172387524*v^3 + 446449271988400340*v^2 - 192208665603330138*v + 35411484066617244) / 5173992472766390 $$\beta_{7}$$ $$=$$ $$( 6100180981 \nu^{15} - 22259671750 \nu^{14} + 65510725326 \nu^{13} - 270219726184 \nu^{12} + 314950092835 \nu^{11} - 40514369326 \nu^{10} + \cdots - 77929949288 ) / 18265555126$$ (6100180981*v^15 - 22259671750*v^14 + 65510725326*v^13 - 270219726184*v^12 + 314950092835*v^11 - 40514369326*v^10 + 709289788986*v^9 - 820031528032*v^8 + 1863520118983*v^7 - 458012963070*v^6 + 2053184268060*v^5 - 1324040796852*v^4 + 1933409293532*v^3 - 1441782411844*v^2 + 520681411390*v - 77929949288) / 18265555126 $$\beta_{8}$$ $$=$$ $$( - 6072104849199 \nu^{15} + 21780610045393 \nu^{14} - 63731636791632 \nu^{13} + 264606089742658 \nu^{12} + \cdots + 65633529574178 ) / 16530327389030$$ (-6072104849199*v^15 + 21780610045393*v^14 - 63731636791632*v^13 + 264606089742658*v^12 - 295959094070367*v^11 + 17152993349497*v^10 - 701312474278478*v^9 + 777595182487398*v^8 - 1797695843305223*v^7 + 332786417052279*v^6 - 1990877398734524*v^5 + 1207350248071826*v^4 - 1824887125776248*v^3 + 1306467148621370*v^2 - 411232893748306*v + 65633529574178) / 16530327389030 $$\beta_{9}$$ $$=$$ $$( 20\!\cdots\!87 \nu^{15} + \cdots - 28\!\cdots\!14 ) / 51\!\cdots\!90$$ (2035054902038187*v^15 - 7578221328661024*v^14 + 22354204007810251*v^13 - 91579556495364929*v^12 + 111269852929486391*v^11 - 19058949252444566*v^10 + 235208674633202009*v^9 - 292701695999447459*v^8 + 636133145212162719*v^7 - 188961643415479902*v^6 + 683022296693330837*v^5 - 490835043167996523*v^4 + 666044503208027234*v^3 - 504441981276855820*v^2 + 198734847296148758*v - 28123111426177514) / 5173992472766390 $$\beta_{10}$$ $$=$$ $$( 30320067 \nu^{15} - 111020724 \nu^{14} + 326080511 \nu^{13} - 1344589864 \nu^{12} + 1575363211 \nu^{11} - 188871116 \nu^{10} + 3515753069 \nu^{9} + \cdots - 330526984 ) / 63923990$$ (30320067*v^15 - 111020724*v^14 + 326080511*v^13 - 1344589864*v^12 + 1575363211*v^11 - 188871116*v^10 + 3515753069*v^9 - 4158557204*v^8 + 9212296429*v^7 - 2344575592*v^6 + 10021703527*v^5 - 6825888288*v^4 + 9461086554*v^3 - 7254602740*v^2 + 2513736318*v - 330526984) / 63923990 $$\beta_{11}$$ $$=$$ $$( 567889621787826 \nu^{15} + \cdots - 77\!\cdots\!96 ) / 10\!\cdots\!78$$ (567889621787826*v^15 - 2163110819836755*v^14 + 6376823282252541*v^13 - 25955396623934578*v^12 + 32832120300673190*v^11 - 6275666808832427*v^10 + 64640506440221435*v^9 - 87228624024570764*v^8 + 179418986889806458*v^7 - 65452477393295353*v^6 + 182753898461327581*v^5 - 154497881200594756*v^4 + 181717723715357082*v^3 - 157676074754561598*v^2 + 53424782974620022*v - 7740432548067096) / 1034798494553278 $$\beta_{12}$$ $$=$$ $$( 52\!\cdots\!58 \nu^{15} + \cdots - 60\!\cdots\!06 ) / 51\!\cdots\!90$$ (5281395788857158*v^15 - 19095159515314451*v^14 + 56052725193796619*v^13 - 232054361068567251*v^12 + 264944456866733364*v^11 - 25993531049108449*v^10 + 615506034577358971*v^9 - 694365286936506611*v^8 + 1588081348228614396*v^7 - 344839767257990343*v^6 + 1765864153109925593*v^5 - 1105265950762762657*v^4 + 1638034897211160076*v^3 - 1197019598040248580*v^2 + 413425841726845612*v - 60388611162910706) / 5173992472766390 $$\beta_{13}$$ $$=$$ $$( - 19482487322 \nu^{15} + 71829768307 \nu^{14} - 211530176114 \nu^{13} + 869648666130 \nu^{12} - 1035106924390 \nu^{11} + \cdots + 258618081490 ) / 18265555126$$ (-19482487322*v^15 + 71829768307*v^14 - 211530176114*v^13 + 869648666130*v^12 - 1035106924390*v^11 + 152629602893*v^10 - 2258419134670*v^9 + 2719627979686*v^8 - 6018321521990*v^7 + 1643327598977*v^6 - 6516717498206*v^5 + 4492931472132*v^4 - 6274129258728*v^3 + 4768566345236*v^2 - 1753345508964*v + 258618081490) / 18265555126 $$\beta_{14}$$ $$=$$ $$( - 82631746 \nu^{15} + 300206917 \nu^{14} - 880560228 \nu^{13} + 3640243297 \nu^{12} - 4191737118 \nu^{11} + 407798693 \nu^{10} + \cdots + 791533522 ) / 63923990$$ (-82631746*v^15 + 300206917*v^14 - 880560228*v^13 + 3640243297*v^12 - 4191737118*v^11 + 407798693*v^10 - 9561674912*v^9 + 11027434227*v^8 - 24845185642*v^7 + 5661417851*v^6 - 27237589476*v^5 + 17742563129*v^4 - 25400492652*v^3 + 18964234070*v^2 - 6297003604*v + 791533522) / 63923990 $$\beta_{15}$$ $$=$$ $$( 69\!\cdots\!34 \nu^{15} + \cdots - 83\!\cdots\!68 ) / 51\!\cdots\!90$$ (6956104161848134*v^15 - 25667361154231553*v^14 + 75476948193017452*v^13 - 310349129733580443*v^12 + 369420728986424472*v^11 - 50736196725114107*v^10 + 803087087863156658*v^9 - 975207215422026373*v^8 + 2135058242437940538*v^7 - 584968875881439509*v^6 + 2300137181821316574*v^5 - 1620912261036340171*v^4 + 2200212896950517088*v^3 - 1699651856766889730*v^2 + 607998971285547366*v - 83437215315298168) / 5173992472766390
 $$\nu$$ $$=$$ $$\beta_{6} - \beta_{5} - \beta_{3}$$ b6 - b5 - b3 $$\nu^{2}$$ $$=$$ $$\beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{2}$$ b14 + b12 + b11 + b10 - b9 + b6 + b5 + b4 + 3*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{15} + 4\beta_{14} + 2\beta_{13} + 3\beta_{12} + 2\beta_{11} - 2\beta_{8} + \beta_{6} - 5\beta _1 + 10$$ 2*b15 + 4*b14 + 2*b13 + 3*b12 + 2*b11 - 2*b8 + b6 - 5*b1 + 10 $$\nu^{4}$$ $$=$$ $$- 2 \beta_{15} - 6 \beta_{14} - 11 \beta_{12} - 4 \beta_{11} - 4 \beta_{10} + 9 \beta_{9} - 10 \beta_{8} - 9 \beta_{7} - 3 \beta_{6} - 15 \beta_{5} + 9 \beta_{4} - 10 \beta_{3} - 20 \beta_{2} - 9 \beta _1 + 17$$ -2*b15 - 6*b14 - 11*b12 - 4*b11 - 4*b10 + 9*b9 - 10*b8 - 9*b7 - 3*b6 - 15*b5 + 9*b4 - 10*b3 - 20*b2 - 9*b1 + 17 $$\nu^{5}$$ $$=$$ $$- 23 \beta_{15} - 16 \beta_{14} - 22 \beta_{13} - 21 \beta_{12} - 2 \beta_{11} + 21 \beta_{10} - 2 \beta_{9} + 5 \beta_{8} + 22 \beta_{6} - 2 \beta_{5} + 39 \beta_{4} - 21 \beta_{3} + 14 \beta_{2} + 7 \beta _1 - 37$$ -23*b15 - 16*b14 - 22*b13 - 21*b12 - 2*b11 + 21*b10 - 2*b9 + 5*b8 + 22*b6 - 2*b5 + 39*b4 - 21*b3 + 14*b2 + 7*b1 - 37 $$\nu^{6}$$ $$=$$ $$28 \beta_{15} + 154 \beta_{14} + 23 \beta_{13} + 140 \beta_{12} + 89 \beta_{11} + 95 \beta_{10} - 56 \beta_{9} + 12 \beta_{8} + 89 \beta_{7} + 84 \beta_{6} + 121 \beta_{5} - 12 \beta_{4} + 56 \beta_{3} + 173 \beta_{2} - 39 \beta _1 + 93$$ 28*b15 + 154*b14 + 23*b13 + 140*b12 + 89*b11 + 95*b10 - 56*b9 + 12*b8 + 89*b7 + 84*b6 + 121*b5 - 12*b4 + 56*b3 + 173*b2 - 39*b1 + 93 $$\nu^{7}$$ $$=$$ $$163 \beta_{15} + 184 \beta_{14} + 240 \beta_{13} + 50 \beta_{12} - 94 \beta_{10} + 200 \beta_{9} - 351 \beta_{8} - 35 \beta_{7} - 219 \beta_{6} - 128 \beta_{5} - 94 \beta_{4} + 35 \beta_{3} - 440 \beta_{2} - 400 \beta _1 + 985$$ 163*b15 + 184*b14 + 240*b13 + 50*b12 - 94*b10 + 200*b9 - 351*b8 - 35*b7 - 219*b6 - 128*b5 - 94*b4 + 35*b3 - 440*b2 - 400*b1 + 985 $$\nu^{8}$$ $$=$$ $$- 645 \beta_{15} - 1488 \beta_{14} - 329 \beta_{13} - 1794 \beta_{12} - 914 \beta_{11} - 242 \beta_{10} + 914 \beta_{9} - 672 \beta_{8} - 663 \beta_{7} - 645 \beta_{6} - 1243 \beta_{5} + 914 \beta_{4} - 914 \beta_{3} + \cdots + 9$$ -645*b15 - 1488*b14 - 329*b13 - 1794*b12 - 914*b11 - 242*b10 + 914*b9 - 672*b8 - 663*b7 - 645*b6 - 1243*b5 + 914*b4 - 914*b3 - 2063*b2 - 251*b1 + 9 $$\nu^{9}$$ $$=$$ $$- 1828 \beta_{15} - 186 \beta_{14} - 1819 \beta_{13} - 195 \beta_{12} + 477 \beta_{11} + 3325 \beta_{10} - 1299 \beta_{9} + 1776 \beta_{8} + 1915 \beta_{7} + 2296 \beta_{6} + 2491 \beta_{5} + 1915 \beta_{4} + \cdots - 5566$$ -1828*b15 - 186*b14 - 1819*b13 - 195*b12 + 477*b11 + 3325*b10 - 1299*b9 + 1776*b8 + 1915*b7 + 2296*b6 + 2491*b5 + 1915*b4 + 4267*b2 + 1776*b1 - 5566 $$\nu^{10}$$ $$=$$ $$6400 \beta_{15} + 16412 \beta_{14} + 6539 \beta_{13} + 14863 \beta_{12} + 7481 \beta_{11} + 6042 \beta_{10} - 3465 \beta_{9} - 1493 \beta_{8} + 8974 \beta_{7} + 1665 \beta_{6} + 11899 \beta_{5} - 7481 \beta_{4} + \cdots + 17408$$ 6400*b15 + 16412*b14 + 6539*b13 + 14863*b12 + 7481*b11 + 6042*b10 - 3465*b9 - 1493*b8 + 8974*b7 + 1665*b6 + 11899*b5 - 7481*b4 + 9507*b3 + 10946*b2 - 6042*b1 + 17408 $$\nu^{11}$$ $$=$$ $$8014 \beta_{15} - 12665 \beta_{14} + 18481 \beta_{13} - 24749 \beta_{12} - 18707 \beta_{11} - 21849 \beta_{10} + 32416 \beta_{9} - 37414 \beta_{8} - 15992 \beta_{7} - 38812 \beta_{6} - 32352 \beta_{5} + \cdots + 77085$$ 8014*b15 - 12665*b14 + 18481*b13 - 24749*b12 - 18707*b11 - 21849*b10 + 32416*b9 - 37414*b8 - 15992*b7 - 38812*b6 - 32352*b5 - 5857*b4 - 5857*b3 - 77618*b2 - 32416*b1 + 77085 $$\nu^{12}$$ $$=$$ $$- 91679 \beta_{15} - 170862 \beta_{14} - 70257 \beta_{13} - 181429 \beta_{12} - 89750 \beta_{11} + 7240 \beta_{10} + 55995 \beta_{9} - 7240 \beta_{8} - 43472 \beta_{7} - 31804 \beta_{6} + \cdots - 166538$$ -91679*b15 - 170862*b14 - 70257*b13 - 181429*b12 - 89750*b11 + 7240*b10 + 55995*b9 - 7240*b8 - 43472*b7 - 31804*b6 - 82510*b5 + 99467*b4 - 82510*b3 - 121181*b2 + 43472*b1 - 166538 $$\nu^{13}$$ $$=$$ $$- 55995 \beta_{15} + 241852 \beta_{14} - 92227 \beta_{13} + 278084 \beta_{12} + 185857 \beta_{11} + 372602 \beta_{10} - 253893 \beta_{9} + 253893 \beta_{8} + 322709 \beta_{7} + 281000 \beta_{6} + \cdots - 469625$$ -55995*b15 + 241852*b14 - 92227*b13 + 278084*b12 + 185857*b11 + 372602*b10 - 253893*b9 + 253893*b8 + 322709*b7 + 281000*b6 + 467745*b5 + 186301*b3 + 711477*b2 + 185857*b1 - 469625 $$\nu^{14}$$ $$=$$ $$880388 \beta_{15} + 1459906 \beta_{14} + 949204 \beta_{13} + 1273161 \beta_{12} + 510702 \beta_{11} + 13265 \beta_{10} + 13265 \beta_{9} - 533645 \beta_{8} + 533645 \beta_{7} - 400341 \beta_{6} + \cdots + 2650257$$ 880388*b15 + 1459906*b14 + 949204*b13 + 1273161*b12 + 510702*b11 + 13265*b10 + 13265*b9 - 533645*b8 + 533645*b7 - 400341*b6 + 656706*b5 - 1044347*b4 + 911043*b3 + 57243*b2 - 911043*b1 + 2650257 $$\nu^{15}$$ $$=$$ $$- 533645 \beta_{15} - 4295841 \beta_{14} + 533645 \beta_{13} - 5220149 \beta_{12} - 3264759 \beta_{11} - 2855121 \beta_{10} + 3769764 \beta_{9} - 3264759 \beta_{8} - 2855121 \beta_{7} + \cdots + 3840272$$ -533645*b15 - 4295841*b14 + 533645*b13 - 5220149*b12 - 3264759*b11 - 2855121*b10 + 3769764*b9 - 3264759*b8 - 2855121*b7 - 4161149*b6 - 4943250*b5 + 765039*b4 - 2090082*b3 - 9354046*b2 - 1884882*b1 + 3840272

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$-\beta_{13}$$ $$\beta_{5} - \beta_{13}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.277956 + 0.213283i −1.09227 − 0.838128i 0.792206 − 1.03242i −0.709944 + 0.925217i 0.339278 + 0.0446668i 2.69978 + 0.355433i 0.117630 − 0.893490i −0.424637 + 3.22544i 0.277956 − 0.213283i −1.09227 + 0.838128i 0.792206 + 1.03242i −0.709944 − 0.925217i 0.339278 − 0.0446668i 2.69978 − 0.355433i 0.117630 + 0.893490i −0.424637 − 3.22544i
0.500000 + 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i −1.90502 1.17085i −0.258819 0.965926i 3.41596 + 1.97220i −1.00000 0.866025 + 0.500000i 0.0614757 2.23522i
67.2 0.500000 + 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i 2.06394 + 0.860320i −0.258819 0.965926i 0.450069 + 0.259847i −1.00000 0.866025 + 0.500000i 0.286912 + 2.21758i
67.3 0.500000 + 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i −0.0421887 2.23567i 0.258819 + 0.965926i 2.21194 + 1.27707i −1.00000 0.866025 + 0.500000i 1.91505 1.15437i
67.4 0.500000 + 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i 1.61532 + 1.54620i 0.258819 + 0.965926i 1.65408 + 0.954985i −1.00000 0.866025 + 0.500000i −0.531389 + 2.17201i
97.1 0.500000 0.866025i −0.258819 0.965926i −0.500000 0.866025i −1.66815 1.48905i −0.965926 0.258819i −0.568824 + 0.328411i −1.00000 −0.866025 + 0.500000i −2.12363 + 0.700141i
97.2 0.500000 0.866025i −0.258819 0.965926i −0.500000 0.866025i 1.50924 1.64991i −0.965926 0.258819i 2.70280 1.56046i −1.00000 −0.866025 + 0.500000i −0.674247 2.13199i
97.3 0.500000 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i −2.23607 + 0.00342112i 0.965926 + 0.258819i 3.25724 1.88057i −1.00000 −0.866025 + 0.500000i −1.11507 + 1.93820i
97.4 0.500000 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i 0.662933 + 2.13554i 0.965926 + 0.258819i −1.12326 + 0.648516i −1.00000 −0.866025 + 0.500000i 2.18090 + 0.493652i
163.1 0.500000 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i −1.90502 + 1.17085i −0.258819 + 0.965926i 3.41596 1.97220i −1.00000 0.866025 0.500000i 0.0614757 + 2.23522i
163.2 0.500000 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i 2.06394 0.860320i −0.258819 + 0.965926i 0.450069 0.259847i −1.00000 0.866025 0.500000i 0.286912 2.21758i
163.3 0.500000 0.866025i 0.965926 0.258819i −0.500000 0.866025i −0.0421887 + 2.23567i 0.258819 0.965926i 2.21194 1.27707i −1.00000 0.866025 0.500000i 1.91505 + 1.15437i
163.4 0.500000 0.866025i 0.965926 0.258819i −0.500000 0.866025i 1.61532 1.54620i 0.258819 0.965926i 1.65408 0.954985i −1.00000 0.866025 0.500000i −0.531389 2.17201i
193.1 0.500000 + 0.866025i −0.258819 + 0.965926i −0.500000 + 0.866025i −1.66815 + 1.48905i −0.965926 + 0.258819i −0.568824 0.328411i −1.00000 −0.866025 0.500000i −2.12363 0.700141i
193.2 0.500000 + 0.866025i −0.258819 + 0.965926i −0.500000 + 0.866025i 1.50924 + 1.64991i −0.965926 + 0.258819i 2.70280 + 1.56046i −1.00000 −0.866025 0.500000i −0.674247 + 2.13199i
193.3 0.500000 + 0.866025i 0.258819 0.965926i −0.500000 + 0.866025i −2.23607 0.00342112i 0.965926 0.258819i 3.25724 + 1.88057i −1.00000 −0.866025 0.500000i −1.11507 1.93820i
193.4 0.500000 + 0.866025i 0.258819 0.965926i −0.500000 + 0.866025i 0.662933 2.13554i 0.965926 0.258819i −1.12326 0.648516i −1.00000 −0.866025 0.500000i 2.18090 0.493652i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bn.b yes 16
5.c odd 4 1 390.2.bd.b 16
13.f odd 12 1 390.2.bd.b 16
65.o even 12 1 inner 390.2.bn.b yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bd.b 16 5.c odd 4 1
390.2.bd.b 16 13.f odd 12 1
390.2.bn.b yes 16 1.a even 1 1 trivial
390.2.bn.b yes 16 65.o even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{16} - 24 T_{7}^{15} + 262 T_{7}^{14} - 1680 T_{7}^{13} + 6823 T_{7}^{12} - 17400 T_{7}^{11} + 24326 T_{7}^{10} - 5472 T_{7}^{9} - 33171 T_{7}^{8} + 17400 T_{7}^{7} + 87604 T_{7}^{6} - 136800 T_{7}^{5} + \cdots + 10000$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{8}$$
$3$ $$(T^{8} - T^{4} + 1)^{2}$$
$5$ $$T^{16} - 2 T^{14} + 16 T^{13} + \cdots + 390625$$
$7$ $$T^{16} - 24 T^{15} + 262 T^{14} + \cdots + 10000$$
$11$ $$T^{16} - 12 T^{15} + 78 T^{14} + \cdots + 10000$$
$13$ $$T^{16} - 8 T^{15} + 20 T^{14} + \cdots + 815730721$$
$17$ $$T^{16} + 4 T^{15} - 58 T^{14} + \cdots + 160000$$
$19$ $$T^{16} + 4 T^{15} + 26 T^{14} + 68 T^{13} + \cdots + 16$$
$23$ $$T^{16} - 4 T^{15} - 28 T^{14} + 64 T^{13} + \cdots + 256$$
$29$ $$T^{16} + 48 T^{15} + 1030 T^{14} + \cdots + 24010000$$
$31$ $$T^{16} - 16 T^{15} + 128 T^{14} + \cdots + 21827584$$
$37$ $$T^{16} - 12 T^{15} + 576 T^{13} + \cdots + 418609$$
$41$ $$T^{16} + 12 T^{15} + 90 T^{14} + \cdots + 85264$$
$43$ $$T^{16} - 20 T^{15} + \cdots + 151388416$$
$47$ $$T^{16} + \cdots + 610001280825616$$
$53$ $$T^{16} - 32 T^{15} + \cdots + 14952153841$$
$59$ $$T^{16} - 4 T^{15} + \cdots + 2922403926016$$
$61$ $$T^{16} - 4 T^{15} + \cdots + 1536975103504$$
$67$ $$T^{16} + 28 T^{15} + \cdots + 2412728463616$$
$71$ $$T^{16} + 36 T^{15} + \cdots + 43\!\cdots\!16$$
$73$ $$(T^{8} - 16 T^{7} - 54 T^{6} + 864 T^{5} + \cdots - 86528)^{2}$$
$79$ $$T^{16} + \cdots + 243571706822656$$
$83$ $$T^{16} + 232 T^{14} + \cdots + 64770304$$
$89$ $$T^{16} + 24 T^{15} + \cdots + 30073246952464$$
$97$ $$T^{16} + \cdots + 344958409179136$$