# Properties

 Label 91.2.h.b Level $91$ Weight $2$ Character orbit 91.h Analytic conductor $0.727$ Analytic rank $0$ Dimension $12$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,2,Mod(16,91)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("91.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.h (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.726638658394$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{11} + 7x^{10} - 2x^{9} + 33x^{8} - 11x^{7} + 55x^{6} + 17x^{5} + 47x^{4} + x^{3} + 8x^{2} + x + 1$$ x^12 - x^11 + 7*x^10 - 2*x^9 + 33*x^8 - 11*x^7 + 55*x^6 + 17*x^5 + 47*x^4 + x^3 + 8*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 7x^{10} - 2x^{9} + 33x^{8} - 11x^{7} + 55x^{6} + 17x^{5} + 47x^{4} + x^{3} + 8x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 29696 \nu^{11} - 478424 \nu^{10} + 682506 \nu^{9} - 3846008 \nu^{8} + 2684563 \nu^{7} + \cdots - 2119374 ) / 3318773$$ (-29696*v^11 - 478424*v^10 + 682506*v^9 - 3846008*v^8 + 2684563*v^7 - 16878368*v^6 + 16008568*v^5 - 31119861*v^4 + 8363982*v^3 - 14058754*v^2 + 5624108*v - 2119374) / 3318773 $$\beta_{3}$$ $$=$$ $$( - 73788 \nu^{11} - 498559 \nu^{10} + 495146 \nu^{9} - 4188508 \nu^{8} + 1631143 \nu^{7} + \cdots - 2229034 ) / 3318773$$ (-73788*v^11 - 498559*v^10 + 495146*v^9 - 4188508*v^8 + 1631143*v^7 - 18206928*v^6 + 16328192*v^5 - 34289666*v^4 + 8704710*v^3 - 14803002*v^2 + 21668998*v - 2229034) / 3318773 $$\beta_{4}$$ $$=$$ $$( - 109660 \nu^{11} + 153752 \nu^{10} - 747485 \nu^{9} + 406680 \nu^{8} - 3276280 \nu^{7} + \cdots - 6198231 ) / 3318773$$ (-109660*v^11 + 153752*v^10 - 747485*v^9 + 406680*v^8 - 3276280*v^7 + 2259680*v^6 - 4702740*v^5 - 2183844*v^4 - 1984215*v^3 - 450388*v^2 - 133032*v - 6198231) / 3318773 $$\beta_{5}$$ $$=$$ $$( 439315 \nu^{11} - 329655 \nu^{10} + 2921453 \nu^{9} - 131145 \nu^{8} + 14090715 \nu^{7} + \cdots + 572347 ) / 3318773$$ (439315*v^11 - 329655*v^10 + 2921453*v^9 - 131145*v^8 + 14090715*v^7 - 1556185*v^6 + 21902645*v^5 + 12171095*v^4 + 22831649*v^3 + 2423530*v^2 + 646135*v + 572347) / 3318773 $$\beta_{6}$$ $$=$$ $$( 566698 \nu^{11} - 1732988 \nu^{10} + 5617249 \nu^{9} - 9944902 \nu^{8} + 24340355 \nu^{7} + \cdots - 6707921 ) / 3318773$$ (566698*v^11 - 1732988*v^10 + 5617249*v^9 - 9944902*v^8 + 24340355*v^7 - 46353032*v^6 + 58565408*v^5 - 63065800*v^4 + 27901335*v^3 - 44235433*v^2 + 12588213*v - 6707921) / 3318773 $$\beta_{7}$$ $$=$$ $$( - 572347 \nu^{11} + 1011662 \nu^{10} - 4336084 \nu^{9} + 4066147 \nu^{8} - 19018596 \nu^{7} + \cdots + 3392561 ) / 3318773$$ (-572347*v^11 + 1011662*v^10 - 4336084*v^9 + 4066147*v^8 - 19018596*v^7 + 20386532*v^6 - 33035270*v^5 + 12172746*v^4 - 14729214*v^3 + 22259302*v^2 - 2155246*v + 3392561) / 3318773 $$\beta_{8}$$ $$=$$ $$( - 1035034 \nu^{11} + 1869572 \nu^{10} - 7924683 \nu^{9} + 7725614 \nu^{8} - 34760912 \nu^{7} + \cdots + 6345807 ) / 3318773$$ (-1035034*v^11 + 1869572*v^10 - 7924683*v^9 + 7725614*v^8 - 34760912*v^7 + 38513384*v^6 - 61367800*v^5 + 26529336*v^4 - 27474213*v^3 + 41650219*v^2 - 4177460*v + 6345807) / 3318773 $$\beta_{9}$$ $$=$$ $$( 1166290 \nu^{11} - 1650363 \nu^{10} + 8811506 \nu^{9} - 5639321 \nu^{8} + 40119354 \nu^{7} + \cdots + 566698 ) / 3318773$$ (1166290*v^11 - 1650363*v^10 + 8811506*v^9 - 5639321*v^8 + 40119354*v^7 - 27397018*v^6 + 72699666*v^5 - 1266529*v^4 + 44802131*v^3 - 8054629*v^2 + 7274619*v + 566698) / 3318773 $$\beta_{10}$$ $$=$$ $$( - 2686072 \nu^{11} + 3882058 \nu^{10} - 19974443 \nu^{9} + 13501144 \nu^{8} - 90433689 \nu^{7} + \cdots - 1035561 ) / 3318773$$ (-2686072*v^11 + 3882058*v^10 - 19974443*v^9 + 13501144*v^8 - 90433689*v^7 + 66981583*v^6 - 158252610*v^5 + 11027874*v^4 - 96392052*v^3 + 33752077*v^2 - 15484451*v - 1035561) / 3318773 $$\beta_{11}$$ $$=$$ $$\nu^{11} - \nu^{10} + 7\nu^{9} - 2\nu^{8} + 33\nu^{7} - 11\nu^{6} + 55\nu^{5} + 17\nu^{4} + 47\nu^{3} + \nu^{2} + 8\nu + 1$$ v^11 - v^10 + 7*v^9 - 2*v^8 + 33*v^7 - 11*v^6 + 55*v^5 + 17*v^4 + 47*v^3 + v^2 + 8*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{8} + 2\beta_{7} - \beta_{4} - 2$$ -b8 + 2*b7 - b4 - 2 $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{9} + 5\beta_{5} + \beta_{3} - \beta_{2}$$ -b11 + b9 + 5*b5 + b3 - b2 $$\nu^{4}$$ $$=$$ $$5\beta_{8} - 8\beta_{7} + \beta_{6} - \beta_{2} - \beta_1$$ 5*b8 - 8*b7 + b6 - b2 - b1 $$\nu^{5}$$ $$=$$ $$5\beta_{11} - \beta_{10} - 7\beta_{9} + \beta_{8} - \beta_{7} - 24\beta_{5} + \beta_{4} - 24\beta _1 + 1$$ 5*b11 - b10 - 7*b9 + b8 - b7 - 24*b5 + b4 - 24*b1 + 1 $$\nu^{6}$$ $$=$$ $$\beta_{11} - 7\beta_{10} - 9\beta_{9} - 7\beta_{6} - 11\beta_{5} + 24\beta_{4} - \beta_{3} + 9\beta_{2} + 36$$ b11 - 7*b10 - 9*b9 - 7*b6 - 11*b5 + 24*b4 - b3 + 9*b2 + 36 $$\nu^{7}$$ $$=$$ $$-11\beta_{8} + 12\beta_{7} - 9\beta_{6} - 24\beta_{3} + 40\beta_{2} + 117\beta_1$$ -11*b8 + 12*b7 - 9*b6 - 24*b3 + 40*b2 + 117*b1 $$\nu^{8}$$ $$=$$ $$- 11 \beta_{11} + 40 \beta_{10} + 60 \beta_{9} - 117 \beta_{8} + 170 \beta_{7} + 85 \beta_{5} + \cdots - 170$$ -11*b11 + 40*b10 + 60*b9 - 117*b8 + 170*b7 + 85*b5 - 117*b4 + 85*b1 - 170 $$\nu^{9}$$ $$=$$ $$- 117 \beta_{11} + 60 \beta_{10} + 217 \beta_{9} + 60 \beta_{6} + 581 \beta_{5} - 85 \beta_{4} + \cdots - 99$$ -117*b11 + 60*b10 + 217*b9 + 60*b6 + 581*b5 - 85*b4 + 117*b3 - 217*b2 - 99 $$\nu^{10}$$ $$=$$ $$581\beta_{8} - 828\beta_{7} + 217\beta_{6} + 85\beta_{3} - 362\beta_{2} - 571\beta_1$$ 581*b8 - 828*b7 + 217*b6 + 85*b3 - 362*b2 - 571*b1 $$\nu^{11}$$ $$=$$ $$581 \beta_{11} - 362 \beta_{10} - 1160 \beta_{9} + 571 \beta_{8} - 695 \beta_{7} - 2933 \beta_{5} + \cdots + 695$$ 581*b11 - 362*b10 - 1160*b9 + 571*b8 - 695*b7 - 2933*b5 + 571*b4 - 2933*b1 + 695

## Character values

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

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$-1 + \beta_{7}$$ $$-1 + \beta_{7}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 −0.181721 − 0.314749i 1.16700 + 2.02131i 0.756174 + 1.30973i −0.437442 − 0.757672i −1.02197 − 1.77010i 0.217953 + 0.377506i −0.181721 + 0.314749i 1.16700 − 2.02131i 0.756174 − 1.30973i −0.437442 + 0.757672i −1.02197 + 1.77010i 0.217953 − 0.377506i
−2.38804 1.37574 2.38285i 3.70272 −0.491140 + 0.850679i −3.28532 + 5.69033i −0.911766 2.48368i −4.06616 −2.28532 3.95828i 1.17286 2.03145i
16.2 −1.90556 −0.214224 + 0.371047i 1.63116 0.736565 1.27577i 0.408216 0.707051i 1.58334 + 2.11968i 0.702849 1.40822 + 2.43910i −1.40357 + 2.43105i
16.3 −0.851125 −0.330612 + 0.572636i −1.27559 −1.72074 + 2.98041i 0.281392 0.487385i −2.57273 0.617304i 2.78793 1.28139 + 2.21944i 1.46456 2.53670i
16.4 −0.268125 0.571504 0.989875i −1.92811 1.28088 2.21854i −0.153235 + 0.265410i 1.80416 1.93520i 1.05323 0.846765 + 1.46664i −0.343436 + 0.594848i
16.5 1.55469 0.244626 0.423704i 0.417051 0.595756 1.03188i 0.380316 0.658727i −2.44127 + 1.01990i −2.46099 1.38032 + 2.39078i 0.926214 1.60425i
16.6 1.85816 −1.14703 + 1.98672i 1.45276 0.0986811 0.170921i −2.13137 + 3.69165i 1.03826 2.43352i −1.01686 −1.13137 1.95960i 0.183365 0.317598i
74.1 −2.38804 1.37574 + 2.38285i 3.70272 −0.491140 0.850679i −3.28532 5.69033i −0.911766 + 2.48368i −4.06616 −2.28532 + 3.95828i 1.17286 + 2.03145i
74.2 −1.90556 −0.214224 0.371047i 1.63116 0.736565 + 1.27577i 0.408216 + 0.707051i 1.58334 2.11968i 0.702849 1.40822 2.43910i −1.40357 2.43105i
74.3 −0.851125 −0.330612 0.572636i −1.27559 −1.72074 2.98041i 0.281392 + 0.487385i −2.57273 + 0.617304i 2.78793 1.28139 2.21944i 1.46456 + 2.53670i
74.4 −0.268125 0.571504 + 0.989875i −1.92811 1.28088 + 2.21854i −0.153235 0.265410i 1.80416 + 1.93520i 1.05323 0.846765 1.46664i −0.343436 0.594848i
74.5 1.55469 0.244626 + 0.423704i 0.417051 0.595756 + 1.03188i 0.380316 + 0.658727i −2.44127 1.01990i −2.46099 1.38032 2.39078i 0.926214 + 1.60425i
74.6 1.85816 −1.14703 1.98672i 1.45276 0.0986811 + 0.170921i −2.13137 3.69165i 1.03826 + 2.43352i −1.01686 −1.13137 + 1.95960i 0.183365 + 0.317598i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.6 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.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.h.b yes 12
3.b odd 2 1 819.2.s.d 12
7.b odd 2 1 637.2.h.l 12
7.c even 3 1 91.2.g.b 12
7.c even 3 1 637.2.f.k 12
7.d odd 6 1 637.2.f.j 12
7.d odd 6 1 637.2.g.l 12
13.c even 3 1 91.2.g.b 12
13.c even 3 1 1183.2.e.h 12
13.e even 6 1 1183.2.e.g 12
21.h odd 6 1 819.2.n.d 12
39.i odd 6 1 819.2.n.d 12
91.g even 3 1 637.2.f.k 12
91.g even 3 1 1183.2.e.h 12
91.h even 3 1 inner 91.2.h.b yes 12
91.h even 3 1 8281.2.a.bz 6
91.k even 6 1 8281.2.a.ce 6
91.l odd 6 1 8281.2.a.cf 6
91.m odd 6 1 637.2.f.j 12
91.n odd 6 1 637.2.g.l 12
91.u even 6 1 1183.2.e.g 12
91.v odd 6 1 637.2.h.l 12
91.v odd 6 1 8281.2.a.ca 6
273.s odd 6 1 819.2.s.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.b 12 7.c even 3 1
91.2.g.b 12 13.c even 3 1
91.2.h.b yes 12 1.a even 1 1 trivial
91.2.h.b yes 12 91.h even 3 1 inner
637.2.f.j 12 7.d odd 6 1
637.2.f.j 12 91.m odd 6 1
637.2.f.k 12 7.c even 3 1
637.2.f.k 12 91.g even 3 1
637.2.g.l 12 7.d odd 6 1
637.2.g.l 12 91.n odd 6 1
637.2.h.l 12 7.b odd 2 1
637.2.h.l 12 91.v odd 6 1
819.2.n.d 12 21.h odd 6 1
819.2.n.d 12 39.i odd 6 1
819.2.s.d 12 3.b odd 2 1
819.2.s.d 12 273.s odd 6 1
1183.2.e.g 12 13.e even 6 1
1183.2.e.g 12 91.u even 6 1
1183.2.e.h 12 13.c even 3 1
1183.2.e.h 12 91.g even 3 1
8281.2.a.bz 6 91.h even 3 1
8281.2.a.ca 6 91.v odd 6 1
8281.2.a.ce 6 91.k even 6 1
8281.2.a.cf 6 91.l odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 2T_{2}^{5} - 6T_{2}^{4} - 11T_{2}^{3} + 8T_{2}^{2} + 14T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} + 2 T^{5} - 6 T^{4} + \cdots + 3)^{2}$$
$3$ $$T^{12} - T^{11} + \cdots + 1$$
$5$ $$T^{12} - T^{11} + \cdots + 9$$
$7$ $$T^{12} + 3 T^{11} + \cdots + 117649$$
$11$ $$T^{12} - 4 T^{11} + \cdots + 6561$$
$13$ $$T^{12} + 2 T^{11} + \cdots + 4826809$$
$17$ $$(T^{6} + 5 T^{5} - 12 T^{4} + \cdots - 9)^{2}$$
$19$ $$T^{12} + T^{11} + \cdots + 762129$$
$23$ $$(T^{6} - T^{5} + \cdots - 24387)^{2}$$
$29$ $$T^{12} - 3 T^{11} + \cdots + 40401$$
$31$ $$T^{12} - 16 T^{11} + \cdots + 6135529$$
$37$ $$(T^{6} - 13 T^{5} + \cdots - 13477)^{2}$$
$41$ $$T^{12} + 8 T^{11} + \cdots + 4173849$$
$43$ $$T^{12} + 11 T^{11} + \cdots + 1369$$
$47$ $$T^{12} + \cdots + 318515409$$
$53$ $$T^{12} + 2 T^{11} + \cdots + 4761$$
$59$ $$(T^{6} + 13 T^{5} + \cdots + 9123)^{2}$$
$61$ $$T^{12} + \cdots + 1055015361$$
$67$ $$T^{12} + \cdots + 276324129$$
$71$ $$T^{12} + \cdots + 530979849$$
$73$ $$T^{12} + \cdots + 196812841$$
$79$ $$T^{12} + \cdots + 110859841$$
$83$ $$(T^{6} + 27 T^{5} + \cdots + 2673)^{2}$$
$89$ $$(T^{6} + 4 T^{5} + \cdots - 304479)^{2}$$
$97$ $$T^{12} + 35 T^{11} + \cdots + 15202201$$