# Properties

 Label 91.2.u.b Level $91$ Weight $2$ Character orbit 91.u Analytic conductor $0.727$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.u (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.726638658394$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.2346760387617129.1 Defining polynomial: $$x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64$$ x^12 - 3*x^11 + x^10 + 10*x^9 - 15*x^8 - 10*x^7 + 45*x^6 - 20*x^5 - 60*x^4 + 80*x^3 + 16*x^2 - 96*x + 64 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 72 \nu^{8} - 91 \nu^{7} - 164 \nu^{6} + 313 \nu^{5} + 42 \nu^{4} - 620 \nu^{3} + 344 \nu^{2} + 608 \nu - 800 ) / 224$$ (v^11 - 13*v^10 - 9*v^9 + 72*v^8 - 91*v^7 - 164*v^6 + 313*v^5 + 42*v^4 - 620*v^3 + 344*v^2 + 608*v - 800) / 224 $$\beta_{3}$$ $$=$$ $$( - 9 \nu^{11} + 5 \nu^{10} + 25 \nu^{9} - 32 \nu^{8} - 21 \nu^{7} + 132 \nu^{6} - 73 \nu^{5} - 154 \nu^{4} + 260 \nu^{3} + 40 \nu^{2} - 320 \nu + 256 ) / 224$$ (-9*v^11 + 5*v^10 + 25*v^9 - 32*v^8 - 21*v^7 + 132*v^6 - 73*v^5 - 154*v^4 + 260*v^3 + 40*v^2 - 320*v + 256) / 224 $$\beta_{4}$$ $$=$$ $$( - 11 \nu^{11} + 17 \nu^{10} + 29 \nu^{9} - 78 \nu^{8} + 21 \nu^{7} + 166 \nu^{6} - 167 \nu^{5} - 140 \nu^{4} + 380 \nu^{3} - 88 \nu^{2} - 304 \nu + 288 ) / 224$$ (-11*v^11 + 17*v^10 + 29*v^9 - 78*v^8 + 21*v^7 + 166*v^6 - 167*v^5 - 140*v^4 + 380*v^3 - 88*v^2 - 304*v + 288) / 224 $$\beta_{5}$$ $$=$$ $$( - 13 \nu^{11} + 29 \nu^{10} + 5 \nu^{9} - 96 \nu^{8} + 91 \nu^{7} + 200 \nu^{6} - 289 \nu^{5} - 126 \nu^{4} + 584 \nu^{3} - 160 \nu^{2} - 512 \nu + 544 ) / 224$$ (-13*v^11 + 29*v^10 + 5*v^9 - 96*v^8 + 91*v^7 + 200*v^6 - 289*v^5 - 126*v^4 + 584*v^3 - 160*v^2 - 512*v + 544) / 224 $$\beta_{6}$$ $$=$$ $$( 8 \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 51 \nu^{8} - 42 \nu^{7} - 101 \nu^{6} + 194 \nu^{5} + 7 \nu^{4} - 340 \nu^{3} + 260 \nu^{2} + 216 \nu - 464 ) / 112$$ (8*v^11 - 13*v^10 - 9*v^9 + 51*v^8 - 42*v^7 - 101*v^6 + 194*v^5 + 7*v^4 - 340*v^3 + 260*v^2 + 216*v - 464) / 112 $$\beta_{7}$$ $$=$$ $$( 13 \nu^{11} - 57 \nu^{10} - 5 \nu^{9} + 208 \nu^{8} - 231 \nu^{7} - 396 \nu^{6} + 821 \nu^{5} + 42 \nu^{4} - 1452 \nu^{3} + 720 \nu^{2} + 1184 \nu - 1664 ) / 224$$ (13*v^11 - 57*v^10 - 5*v^9 + 208*v^8 - 231*v^7 - 396*v^6 + 821*v^5 + 42*v^4 - 1452*v^3 + 720*v^2 + 1184*v - 1664) / 224 $$\beta_{8}$$ $$=$$ $$( 2 \nu^{11} - 5 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 7 \nu^{7} - 41 \nu^{6} + 45 \nu^{5} + 35 \nu^{4} - 99 \nu^{3} + 16 \nu^{2} + 96 \nu - 88 ) / 28$$ (2*v^11 - 5*v^10 - 4*v^9 + 18*v^8 - 7*v^7 - 41*v^6 + 45*v^5 + 35*v^4 - 99*v^3 + 16*v^2 + 96*v - 88) / 28 $$\beta_{9}$$ $$=$$ $$( 3 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 20 \nu^{8} - 44 \nu^{6} + 43 \nu^{5} + 56 \nu^{4} - 82 \nu^{3} + 3 \nu^{2} + 102 \nu - 48 ) / 28$$ (3*v^11 - 4*v^10 - 6*v^9 + 20*v^8 - 44*v^6 + 43*v^5 + 56*v^4 - 82*v^3 + 3*v^2 + 102*v - 48) / 28 $$\beta_{10}$$ $$=$$ $$( - 15 \nu^{11} + 20 \nu^{10} + 30 \nu^{9} - 121 \nu^{8} + 21 \nu^{7} + 269 \nu^{6} - 271 \nu^{5} - 273 \nu^{4} + 634 \nu^{3} - 64 \nu^{2} - 664 \nu + 464 ) / 112$$ (-15*v^11 + 20*v^10 + 30*v^9 - 121*v^8 + 21*v^7 + 269*v^6 - 271*v^5 - 273*v^4 + 634*v^3 - 64*v^2 - 664*v + 464) / 112 $$\beta_{11}$$ $$=$$ $$( - 17 \nu^{11} + 39 \nu^{10} + 13 \nu^{9} - 160 \nu^{8} + 133 \nu^{7} + 310 \nu^{6} - 547 \nu^{5} - 168 \nu^{4} + 1062 \nu^{3} - 500 \nu^{2} - 872 \nu + 1056 ) / 112$$ (-17*v^11 + 39*v^10 + 13*v^9 - 160*v^8 + 133*v^7 + 310*v^6 - 547*v^5 - 168*v^4 + 1062*v^3 - 500*v^2 - 872*v + 1056) / 112
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1$$ b8 - b7 + b6 + b4 + b3 + b2 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}$$ b11 + b9 + b6 - b5 + b4 + b3 + b2 $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{2} - \beta _1 - 1$$ -b11 + b10 + b9 - b7 - b6 + b2 - b1 - 1 $$\nu^{5}$$ $$=$$ $$\beta_{10} + 2\beta_{9} - 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{3} - \beta_{2} - \beta_1$$ b10 + 2*b9 - 2*b8 + 2*b7 - b6 + b5 - 2*b3 - b2 - b1 $$\nu^{6}$$ $$=$$ $$- 4 \beta_{11} + 2 \beta_{10} - 3 \beta_{8} + \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 3 \beta _1 - 6$$ -4*b11 + 2*b10 - 3*b8 + b7 - 5*b6 + 4*b5 - 7*b4 - 2*b3 - 4*b2 + 3*b1 - 6 $$\nu^{7}$$ $$=$$ $$- \beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + \beta_1$$ -b11 - b10 - b9 + 3*b8 + b7 + b6 + 6*b5 + 4*b4 - b3 - 4*b2 + b1 $$\nu^{8}$$ $$=$$ $$-4\beta_{10} - 2\beta_{9} - \beta_{8} + 2\beta_{5} - 4\beta_{4} + 8\beta_{3} - 2\beta_{2} + 3\beta _1 - 6$$ -4*b10 - 2*b9 - b8 + 2*b5 - 4*b4 + 8*b3 - 2*b2 + 3*b1 - 6 $$\nu^{9}$$ $$=$$ $$2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + 21 \beta_{4} + 6 \beta_{3} - 3 \beta _1 + 4$$ 2*b11 - 6*b10 - 2*b9 + 6*b8 - 3*b7 + 7*b6 - 4*b5 + 21*b4 + 6*b3 - 3*b1 + 4 $$\nu^{10}$$ $$=$$ $$5 \beta_{11} - 9 \beta_{10} + \beta_{9} - 16 \beta_{8} + \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} - 6 \beta _1 + 1$$ 5*b11 - 9*b10 + b9 - 16*b8 + b7 + 3*b6 - 8*b5 + 2*b4 + 2*b3 + 7*b2 - 6*b1 + 1 $$\nu^{11}$$ $$=$$ $$- 2 \beta_{11} - \beta_{10} - 19 \beta_{8} + \beta_{7} + 4 \beta_{6} - 15 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 14 \beta_{2} + 9 \beta _1 - 5$$ -2*b11 - b10 - 19*b8 + b7 + 4*b6 - 15*b5 - 5*b4 - 13*b3 - 14*b2 + 9*b1 - 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$-\beta_{4}$$ $$-1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
30.1
 1.32725 − 0.488273i −1.38488 − 0.286553i 0.655911 + 1.25291i −1.18541 + 0.771231i 0.874681 − 1.11128i 1.21245 + 0.727987i 1.32725 + 0.488273i −1.38488 + 0.286553i 0.655911 − 1.25291i −1.18541 − 0.771231i 0.874681 + 1.11128i 1.21245 − 0.727987i
−2.24179 + 1.29430i 0.518466 2.35043 4.07106i 1.39608 + 0.806027i −1.16229 + 0.671051i 2.62954 + 0.292422i 6.99143i −2.73119 −4.17296
30.2 −1.19430 + 0.689527i 2.88120 −0.0491037 + 0.0850501i 0.697972 + 0.402974i −3.44101 + 1.98667i −2.25549 1.38302i 2.89354i 5.30133 −1.11145
30.3 −0.156598 + 0.0904119i −1.82601 −0.983651 + 1.70373i 2.32670 + 1.34332i 0.285950 0.165093i −0.393717 + 2.61629i 0.717383i 0.334323 −0.485809
30.4 0.433001 0.249993i 0.849601 −0.875007 + 1.51556i 0.902810 + 0.521238i 0.367878 0.212395i 1.52469 2.16225i 1.87496i −2.27818 0.521224
30.5 1.16500 0.672613i 2.05010 −0.0951832 + 0.164862i −3.08979 1.78389i 2.38837 1.37893i −2.09638 + 1.61406i 2.94654i 1.20292 −4.79947
30.6 1.99469 1.15163i −1.47336 1.65252 2.86225i −0.733776 0.423646i −2.93889 + 1.69677i 2.09135 + 1.62057i 3.00585i −0.829208 −1.95154
88.1 −2.24179 1.29430i 0.518466 2.35043 + 4.07106i 1.39608 0.806027i −1.16229 0.671051i 2.62954 0.292422i 6.99143i −2.73119 −4.17296
88.2 −1.19430 0.689527i 2.88120 −0.0491037 0.0850501i 0.697972 0.402974i −3.44101 1.98667i −2.25549 + 1.38302i 2.89354i 5.30133 −1.11145
88.3 −0.156598 0.0904119i −1.82601 −0.983651 1.70373i 2.32670 1.34332i 0.285950 + 0.165093i −0.393717 2.61629i 0.717383i 0.334323 −0.485809
88.4 0.433001 + 0.249993i 0.849601 −0.875007 1.51556i 0.902810 0.521238i 0.367878 + 0.212395i 1.52469 + 2.16225i 1.87496i −2.27818 0.521224
88.5 1.16500 + 0.672613i 2.05010 −0.0951832 0.164862i −3.08979 + 1.78389i 2.38837 + 1.37893i −2.09638 1.61406i 2.94654i 1.20292 −4.79947
88.6 1.99469 + 1.15163i −1.47336 1.65252 + 2.86225i −0.733776 + 0.423646i −2.93889 1.69677i 2.09135 1.62057i 3.00585i −0.829208 −1.95154
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 88.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.u.b yes 12
3.b odd 2 1 819.2.do.e 12
7.b odd 2 1 637.2.u.g 12
7.c even 3 1 91.2.k.b 12
7.c even 3 1 637.2.q.g 12
7.d odd 6 1 637.2.k.i 12
7.d odd 6 1 637.2.q.i 12
13.e even 6 1 91.2.k.b 12
13.f odd 12 2 1183.2.e.j 24
21.h odd 6 1 819.2.bm.f 12
39.h odd 6 1 819.2.bm.f 12
91.k even 6 1 637.2.q.g 12
91.l odd 6 1 637.2.q.i 12
91.p odd 6 1 637.2.u.g 12
91.t odd 6 1 637.2.k.i 12
91.u even 6 1 inner 91.2.u.b yes 12
91.w even 12 2 8281.2.a.co 12
91.x odd 12 2 1183.2.e.j 24
91.bd odd 12 2 8281.2.a.cp 12
273.x odd 6 1 819.2.do.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.b 12 7.c even 3 1
91.2.k.b 12 13.e even 6 1
91.2.u.b yes 12 1.a even 1 1 trivial
91.2.u.b yes 12 91.u even 6 1 inner
637.2.k.i 12 7.d odd 6 1
637.2.k.i 12 91.t odd 6 1
637.2.q.g 12 7.c even 3 1
637.2.q.g 12 91.k even 6 1
637.2.q.i 12 7.d odd 6 1
637.2.q.i 12 91.l odd 6 1
637.2.u.g 12 7.b odd 2 1
637.2.u.g 12 91.p odd 6 1
819.2.bm.f 12 21.h odd 6 1
819.2.bm.f 12 39.h odd 6 1
819.2.do.e 12 3.b odd 2 1
819.2.do.e 12 273.x odd 6 1
1183.2.e.j 24 13.f odd 12 2
1183.2.e.j 24 91.x odd 12 2
8281.2.a.co 12 91.w even 12 2
8281.2.a.cp 12 91.bd odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 8T_{2}^{10} + 52T_{2}^{8} - 18T_{2}^{7} - 91T_{2}^{6} + 36T_{2}^{5} + 130T_{2}^{4} - 72T_{2}^{3} + 6T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 8 T^{10} + 52 T^{8} - 18 T^{7} + \cdots + 1$$
$3$ $$(T^{6} - 3 T^{5} - 5 T^{4} + 16 T^{3} + 4 T^{2} + \cdots + 7)^{2}$$
$5$ $$T^{12} - 3 T^{11} - 8 T^{10} + 33 T^{9} + \cdots + 121$$
$7$ $$T^{12} - 3 T^{11} + 3 T^{9} + \cdots + 117649$$
$11$ $$T^{12} + 62 T^{10} + 1355 T^{8} + \cdots + 85849$$
$13$ $$T^{12} + 2 T^{11} - 18 T^{10} + \cdots + 4826809$$
$17$ $$T^{12} - 17 T^{11} + 193 T^{10} + \cdots + 361$$
$19$ $$T^{12} + 79 T^{10} + 1984 T^{8} + \cdots + 1$$
$23$ $$T^{12} - 3 T^{11} + 59 T^{10} + \cdots + 628849$$
$29$ $$T^{12} + T^{11} + 87 T^{10} + \cdots + 16072081$$
$31$ $$T^{12} + 18 T^{11} + \cdots + 241274089$$
$37$ $$T^{12} - 15 T^{11} + 39 T^{10} + \cdots + 123201$$
$41$ $$T^{12} + 6 T^{11} - 159 T^{10} + \cdots + 389707081$$
$43$ $$T^{12} - 11 T^{11} + \cdots + 418898089$$
$47$ $$T^{12} - 15 T^{11} + 92 T^{10} + \cdots + 121$$
$53$ $$T^{12} + 8 T^{11} + 102 T^{10} + \cdots + 289$$
$59$ $$T^{12} - 27 T^{11} + \cdots + 35582408689$$
$61$ $$(T^{6} + 5 T^{5} - 75 T^{4} - 354 T^{3} + \cdots + 1777)^{2}$$
$67$ $$T^{12} + 439 T^{10} + \cdots + 5708255809$$
$71$ $$T^{12} - 30 T^{11} + \cdots + 639230089$$
$73$ $$T^{12} + 42 T^{11} + \cdots + 484396081$$
$79$ $$T^{12} + 35 T^{11} + \cdots + 65086724641$$
$83$ $$T^{12} + 463 T^{10} + \cdots + 402363481$$
$89$ $$T^{12} - 48 T^{11} + \cdots + 145033849$$
$97$ $$T^{12} + 3 T^{11} - 176 T^{10} + \cdots + 1681$$