# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,2,Mod(36,91)]

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([0, 5]))

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

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

chi := DirichletCharacter("91.36");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.q (of order $$6$$, 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 $$\Q(\zeta_{6})$$ Coefficient field: 12.0.58891012706304.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64$$ x^12 - 5*x^10 - 2*x^9 + 15*x^8 + 2*x^7 - 30*x^6 + 4*x^5 + 60*x^4 - 16*x^3 - 80*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{11} - 5\nu^{9} - 2\nu^{8} + 15\nu^{7} + 2\nu^{6} - 30\nu^{5} + 4\nu^{4} + 60\nu^{3} - 16\nu^{2} - 80\nu ) / 32$$ (v^11 - 5*v^9 - 2*v^8 + 15*v^7 + 2*v^6 - 30*v^5 + 4*v^4 + 60*v^3 - 16*v^2 - 80*v) / 32 $$\beta_{3}$$ $$=$$ $$( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} - 4 \nu^{3} + 16 \nu^{2} - 48 \nu + 96 ) / 32$$ (-3*v^11 + 4*v^10 + 7*v^9 - 6*v^8 - 13*v^7 + 30*v^6 - 6*v^5 - 28*v^4 - 4*v^3 + 16*v^2 - 48*v + 96) / 32 $$\beta_{4}$$ $$=$$ $$( 3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + 24 \nu^{2} - 64 \nu - 64 ) / 32$$ (3*v^11 + 2*v^10 - 11*v^9 - 8*v^8 + 21*v^7 + 4*v^6 - 42*v^5 + 84*v^3 + 24*v^2 - 64*v - 64) / 32 $$\beta_{5}$$ $$=$$ $$( \nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + 60 \nu^{3} + 68 \nu^{2} - 56 \nu - 64 ) / 16$$ (v^11 + 3*v^10 - 3*v^9 - 13*v^8 + 7*v^7 + 23*v^6 - 26*v^5 - 38*v^4 + 60*v^3 + 68*v^2 - 56*v - 64) / 16 $$\beta_{6}$$ $$=$$ $$( - 5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} - 20 \nu^{3} + 48 \nu^{2} - 32 \nu + 64 ) / 32$$ (-5*v^11 + 4*v^10 + 13*v^9 - 10*v^8 - 23*v^7 + 42*v^6 + 10*v^5 - 68*v^4 - 20*v^3 + 48*v^2 - 32*v + 64) / 32 $$\beta_{7}$$ $$=$$ $$( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32 ) / 32$$ (-5*v^11 + 2*v^10 + 17*v^9 - 8*v^8 - 39*v^7 + 44*v^6 + 50*v^5 - 88*v^4 - 68*v^3 + 120*v^2 + 16*v - 32) / 32 $$\beta_{8}$$ $$=$$ $$( - \nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} - 28 \nu^{3} + 52 \nu^{2} + 24 \nu - 32 ) / 8$$ (-v^11 + v^10 + 5*v^9 - 3*v^8 - 13*v^7 + 13*v^6 + 20*v^5 - 34*v^4 - 28*v^3 + 52*v^2 + 24*v - 32) / 8 $$\beta_{9}$$ $$=$$ $$( - \nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + 8 \nu^{3} + 108 \nu^{2} - 16 \nu - 80 ) / 16$$ (-v^11 + 3*v^10 + 5*v^9 - 13*v^8 - 13*v^7 + 35*v^6 + 12*v^5 - 70*v^4 + 8*v^3 + 108*v^2 - 16*v - 80) / 16 $$\beta_{10}$$ $$=$$ $$( - 3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + 96 \nu^{2} + 96 \nu - 64 ) / 16$$ (-3*v^11 + 15*v^9 - 2*v^8 - 37*v^7 + 18*v^6 + 66*v^5 - 68*v^4 - 92*v^3 + 96*v^2 + 96*v - 64) / 16 $$\beta_{11}$$ $$=$$ $$( - \nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} - 20 \nu^{3} + 192 \nu^{2} + 16 \nu - 160 ) / 16$$ (-v^11 + 4*v^10 + 9*v^9 - 18*v^8 - 27*v^7 + 50*v^6 + 34*v^5 - 116*v^4 - 20*v^3 + 192*v^2 + 16*v - 160) / 16
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1$$ -b11 + b8 + b7 - b6 + b5 + 1 $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1$$ -b11 + b10 + b9 - b6 + b4 + b3 + b2 + b1 $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{9} + \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1$$ -b11 + b9 + b7 - 2*b6 + b5 - b4 + b3 - b2 - 1 $$\nu^{5}$$ $$=$$ $$-2\beta_{11} + 4\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta _1 + 1$$ -2*b11 + 4*b9 + b8 - b7 - b6 - b5 - b1 + 1 $$\nu^{6}$$ $$=$$ $$2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta _1 - 2$$ 2*b11 - 2*b10 + b9 - b8 - b6 - 2*b5 - 3*b4 + b3 + b2 + 4*b1 - 2 $$\nu^{7}$$ $$=$$ $$- 5 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} + \beta_{2} + 1$$ -5*b11 - 3*b10 + 5*b9 + 7*b8 - b7 - 2*b6 + 3*b5 - 7*b4 - 3*b3 + b2 + 1 $$\nu^{8}$$ $$=$$ $$-3\beta_{11} + 7\beta_{8} + \beta_{7} - 5\beta_{6} - \beta_{5} + 2\beta_{4} + 4\beta_{3} + 10\beta_{2} + 8\beta _1 - 3$$ -3*b11 + 7*b8 + b7 - 5*b6 - b5 + 2*b4 + 4*b3 + 10*b2 + 8*b1 - 3 $$\nu^{9}$$ $$=$$ $$- 7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} - 17 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} + \beta _1 - 8$$ -7*b11 + 3*b10 + 3*b9 + 6*b8 - 4*b7 - 11*b6 + 12*b5 - 17*b4 + 7*b3 + 7*b2 + b1 - 8 $$\nu^{10}$$ $$=$$ $$- 17 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} - 6 \beta _1 - 11$$ -17*b11 + 4*b10 + 13*b9 + 18*b8 - 9*b7 - 8*b6 + 3*b5 + 7*b4 + 9*b3 + 3*b2 - 6*b1 - 11 $$\nu^{11}$$ $$=$$ $$18 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 21 \beta_{7} - 11 \beta_{6} - \beta_{5} - 26 \beta_{4} + 22 \beta_{3} + 14 \beta_{2} + 3 \beta _1 - 7$$ 18*b11 + 4*b10 - 6*b9 - 13*b8 - 21*b7 - 11*b6 - b5 - 26*b4 + 22*b3 + 14*b2 + 3*b1 - 7

## Character values

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

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$1 - \beta_{9}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
36.1
 −1.12906 − 0.851598i 0.759479 − 1.19298i 1.34408 + 0.439820i −1.08105 + 0.911778i 1.40744 + 0.138282i −1.30089 + 0.554694i −1.12906 + 0.851598i 0.759479 + 1.19298i 1.34408 − 0.439820i −1.08105 − 0.911778i 1.40744 − 0.138282i −1.30089 − 0.554694i
−2.34104 + 1.35160i 0.172975 + 0.299601i 2.65363 4.59623i 3.25812i −0.809880 0.467584i 0.866025 + 0.500000i 8.94020i 1.44016 2.49443i 4.40367 + 7.62739i
36.2 −1.20027 + 0.692976i 1.41289 + 2.44719i −0.0395678 + 0.0685334i 0.518957i −3.39169 1.95819i −0.866025 0.500000i 2.88158i −2.49250 + 4.31714i 0.359625 + 0.622889i
36.3 −0.104235 + 0.0601799i 0.291146 + 0.504280i −0.992757 + 1.71951i 1.68817i −0.0606950 0.0350423i 0.866025 + 0.500000i 0.479696i 1.33047 2.30444i −0.101594 0.175965i
36.4 0.713220 0.411778i −1.33015 2.30388i −0.660878 + 1.14467i 3.16209i −1.89737 1.09545i 0.866025 + 0.500000i 2.73565i −2.03858 + 3.53092i −1.30208 2.25527i
36.5 1.10554 0.638282i 0.583963 + 1.01145i −0.185192 + 0.320762i 1.81487i 1.29118 + 0.745466i −0.866025 0.500000i 3.02595i 0.817975 1.41677i −1.15840 2.00641i
36.6 1.82678 1.05469i −1.13082 1.95864i 1.22476 2.12135i 3.60178i −4.13154 2.38535i −0.866025 0.500000i 0.948212i −1.05753 + 1.83169i 3.79878 + 6.57967i
43.1 −2.34104 1.35160i 0.172975 0.299601i 2.65363 + 4.59623i 3.25812i −0.809880 + 0.467584i 0.866025 0.500000i 8.94020i 1.44016 + 2.49443i 4.40367 7.62739i
43.2 −1.20027 0.692976i 1.41289 2.44719i −0.0395678 0.0685334i 0.518957i −3.39169 + 1.95819i −0.866025 + 0.500000i 2.88158i −2.49250 4.31714i 0.359625 0.622889i
43.3 −0.104235 0.0601799i 0.291146 0.504280i −0.992757 1.71951i 1.68817i −0.0606950 + 0.0350423i 0.866025 0.500000i 0.479696i 1.33047 + 2.30444i −0.101594 + 0.175965i
43.4 0.713220 + 0.411778i −1.33015 + 2.30388i −0.660878 1.14467i 3.16209i −1.89737 + 1.09545i 0.866025 0.500000i 2.73565i −2.03858 3.53092i −1.30208 + 2.25527i
43.5 1.10554 + 0.638282i 0.583963 1.01145i −0.185192 0.320762i 1.81487i 1.29118 0.745466i −0.866025 + 0.500000i 3.02595i 0.817975 + 1.41677i −1.15840 + 2.00641i
43.6 1.82678 + 1.05469i −1.13082 + 1.95864i 1.22476 + 2.12135i 3.60178i −4.13154 + 2.38535i −0.866025 + 0.500000i 0.948212i −1.05753 1.83169i 3.79878 6.57967i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 36.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.q.a 12
3.b odd 2 1 819.2.ct.a 12
4.b odd 2 1 1456.2.cc.c 12
7.b odd 2 1 637.2.q.h 12
7.c even 3 1 637.2.k.h 12
7.c even 3 1 637.2.u.h 12
7.d odd 6 1 637.2.k.g 12
7.d odd 6 1 637.2.u.i 12
13.c even 3 1 1183.2.c.i 12
13.e even 6 1 inner 91.2.q.a 12
13.e even 6 1 1183.2.c.i 12
13.f odd 12 1 1183.2.a.m 6
13.f odd 12 1 1183.2.a.p 6
39.h odd 6 1 819.2.ct.a 12
52.i odd 6 1 1456.2.cc.c 12
91.k even 6 1 637.2.u.h 12
91.l odd 6 1 637.2.u.i 12
91.p odd 6 1 637.2.k.g 12
91.t odd 6 1 637.2.q.h 12
91.u even 6 1 637.2.k.h 12
91.bc even 12 1 8281.2.a.by 6
91.bc even 12 1 8281.2.a.ch 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a 12 1.a even 1 1 trivial
91.2.q.a 12 13.e even 6 1 inner
637.2.k.g 12 7.d odd 6 1
637.2.k.g 12 91.p odd 6 1
637.2.k.h 12 7.c even 3 1
637.2.k.h 12 91.u even 6 1
637.2.q.h 12 7.b odd 2 1
637.2.q.h 12 91.t odd 6 1
637.2.u.h 12 7.c even 3 1
637.2.u.h 12 91.k even 6 1
637.2.u.i 12 7.d odd 6 1
637.2.u.i 12 91.l odd 6 1
819.2.ct.a 12 3.b odd 2 1
819.2.ct.a 12 39.h odd 6 1
1183.2.a.m 6 13.f odd 12 1
1183.2.a.p 6 13.f odd 12 1
1183.2.c.i 12 13.c even 3 1
1183.2.c.i 12 13.e even 6 1
1456.2.cc.c 12 4.b odd 2 1
1456.2.cc.c 12 52.i odd 6 1
8281.2.a.by 6 91.bc even 12 1
8281.2.a.ch 6 91.bc even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 8 T^{10} + 52 T^{8} - 36 T^{7} + \cdots + 1$$
$3$ $$T^{12} + 11 T^{10} - 4 T^{9} + 96 T^{8} + \cdots + 16$$
$5$ $$T^{12} + 40 T^{10} + 600 T^{8} + \cdots + 3481$$
$7$ $$(T^{4} - T^{2} + 1)^{3}$$
$11$ $$T^{12} - 6 T^{11} - 7 T^{10} + 114 T^{9} + \cdots + 256$$
$13$ $$T^{12} - 4 T^{11} + 21 T^{10} + \cdots + 4826809$$
$17$ $$T^{12} + 4 T^{11} + 37 T^{10} + \cdots + 241081$$
$19$ $$T^{12} - 29 T^{10} + 748 T^{8} + \cdots + 55696$$
$23$ $$T^{12} + 12 T^{11} + 164 T^{10} + \cdots + 38539264$$
$29$ $$T^{12} - 8 T^{11} + 108 T^{10} + \cdots + 10042561$$
$31$ $$T^{12} + 136 T^{10} + 5854 T^{8} + \cdots + 913936$$
$37$ $$T^{12} + 42 T^{11} + \cdots + 1755945216$$
$41$ $$T^{12} - 30 T^{11} + \cdots + 884705536$$
$43$ $$T^{12} - 2 T^{11} + 113 T^{10} + \cdots + 2408704$$
$47$ $$T^{12} + 272 T^{10} + 21782 T^{8} + \cdots + 9461776$$
$53$ $$(T^{6} + 22 T^{5} + 91 T^{4} - 700 T^{3} + \cdots - 2339)^{2}$$
$59$ $$T^{12} - 18 T^{11} + \cdots + 4571923456$$
$61$ $$T^{12} - 14 T^{11} + 283 T^{10} + \cdots + 5607424$$
$67$ $$T^{12} + 24 T^{11} + \cdots + 613651984$$
$71$ $$T^{12} + 24 T^{11} + 212 T^{10} + \cdots + 46895104$$
$73$ $$T^{12} + 334 T^{10} + \cdots + 1386221824$$
$79$ $$(T^{6} + 28 T^{5} + 212 T^{4} + 192 T^{3} + \cdots - 512)^{2}$$
$83$ $$T^{12} + 304 T^{10} + \cdots + 141324544$$
$89$ $$T^{12} + 12 T^{11} + \cdots + 1834580224$$
$97$ $$T^{12} - 6 T^{11} - 173 T^{10} + \cdots + 53465344$$