# Properties

 Label 36.2.h.a Level $36$ Weight $2$ Character orbit 36.h Analytic conductor $0.287$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [36,2,Mod(11,36)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

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

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

chi := DirichletCharacter("36.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 36.h (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: $$0.287461447277$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.170772624.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16$$ x^8 - 3*x^7 + 5*x^6 - 6*x^5 + 6*x^4 - 12*x^3 + 20*x^2 - 24*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} - \nu^{6} - \nu^{5} - 2\nu^{3} - 4\nu^{2} + 4\nu + 8 ) / 8$$ (v^7 - v^6 - v^5 - 2*v^3 - 4*v^2 + 4*v + 8) / 8 $$\beta_{3}$$ $$=$$ $$( -\nu^{6} + \nu^{5} - \nu^{4} + 2\nu^{3} + 4\nu - 4 ) / 4$$ (-v^6 + v^5 - v^4 + 2*v^3 + 4*v - 4) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{7} - 3\nu^{6} + 5\nu^{5} - 2\nu^{4} + 2\nu^{3} - 8\nu^{2} + 20\nu - 24 ) / 8$$ (v^7 - 3*v^6 + 5*v^5 - 2*v^4 + 2*v^3 - 8*v^2 + 20*v - 24) / 8 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} + 3\nu^{6} - 5\nu^{5} + 4\nu^{4} - 6\nu^{3} + 16\nu^{2} - 12\nu + 16 ) / 8$$ (-3*v^7 + 3*v^6 - 5*v^5 + 4*v^4 - 6*v^3 + 16*v^2 - 12*v + 16) / 8 $$\beta_{6}$$ $$=$$ $$( \nu^{7} - 3\nu^{6} + 5\nu^{5} - 6\nu^{4} + 6\nu^{3} - 12\nu^{2} + 20\nu - 24 ) / 8$$ (v^7 - 3*v^6 + 5*v^5 - 6*v^4 + 6*v^3 - 12*v^2 + 20*v - 24) / 8 $$\beta_{7}$$ $$=$$ $$( 3\nu^{7} - 5\nu^{6} + 7\nu^{5} - 6\nu^{4} + 10\nu^{3} - 24\nu^{2} + 28\nu - 32 ) / 8$$ (3*v^7 - 5*v^6 + 7*v^5 - 6*v^4 + 10*v^3 - 24*v^2 + 28*v - 32) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{5} + \beta_{3} + \beta _1 - 1$$ -b7 - b5 + b3 + b1 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1$$ b7 - b6 - b4 + b3 - b2 + b1 $$\nu^{4}$$ $$=$$ $$2\beta_{7} - 3\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 1$$ 2*b7 - 3*b6 + b5 + b4 - b2 + 1 $$\nu^{5}$$ $$=$$ $$-\beta_{7} - \beta_{5} + 2\beta_{4} - \beta_{3} - 2\beta_{2} - \beta _1 + 5$$ -b7 - b5 + 2*b4 - b3 - 2*b2 - b1 + 5 $$\nu^{6}$$ $$=$$ $$-\beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} - 3\beta_{3} - 3\beta_{2} + 5\beta_1$$ -b7 + b6 - 2*b5 - b4 - 3*b3 - 3*b2 + 5*b1 $$\nu^{7}$$ $$=$$ $$-4\beta_{7} - \beta_{6} - 7\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 6\beta _1 - 7$$ -4*b7 - b6 - 7*b5 - b4 + 2*b3 + b2 + 6*b1 - 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$$.

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$-1$$ $$-\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.774115 − 1.18353i 1.41203 − 0.0786378i −1.02187 − 0.977642i 0.335728 + 1.37379i 0.774115 + 1.18353i 1.41203 + 0.0786378i −1.02187 + 0.977642i 0.335728 − 1.37379i
−1.41203 0.0786378i 0.637910 + 1.61030i 1.98763 + 0.222077i 0.686141 0.396143i −0.774115 2.32395i −2.35143 1.35760i −2.78912 0.469882i −2.18614 + 2.05446i −1.00000 + 0.505408i
11.2 −0.774115 1.18353i −0.637910 1.61030i −0.801492 + 1.83238i 0.686141 0.396143i −1.41203 + 2.00155i 2.35143 + 1.35760i 2.78912 0.469882i −2.18614 + 2.05446i −1.00000 0.505408i
11.3 −0.335728 + 1.37379i 1.35760 1.07561i −1.77457 0.922437i −2.18614 + 1.26217i 1.02187 + 2.22616i −1.10489 0.637910i 1.86301 2.12819i 0.686141 2.92048i −1.00000 3.42703i
11.4 1.02187 0.977642i −1.35760 + 1.07561i 0.0884324 1.99804i −2.18614 + 1.26217i −0.335728 + 2.42637i 1.10489 + 0.637910i −1.86301 2.12819i 0.686141 2.92048i −1.00000 + 3.42703i
23.1 −1.41203 + 0.0786378i 0.637910 1.61030i 1.98763 0.222077i 0.686141 + 0.396143i −0.774115 + 2.32395i −2.35143 + 1.35760i −2.78912 + 0.469882i −2.18614 2.05446i −1.00000 0.505408i
23.2 −0.774115 + 1.18353i −0.637910 + 1.61030i −0.801492 1.83238i 0.686141 + 0.396143i −1.41203 2.00155i 2.35143 1.35760i 2.78912 + 0.469882i −2.18614 2.05446i −1.00000 + 0.505408i
23.3 −0.335728 1.37379i 1.35760 + 1.07561i −1.77457 + 0.922437i −2.18614 1.26217i 1.02187 2.22616i −1.10489 + 0.637910i 1.86301 + 2.12819i 0.686141 + 2.92048i −1.00000 + 3.42703i
23.4 1.02187 + 0.977642i −1.35760 1.07561i 0.0884324 + 1.99804i −2.18614 1.26217i −0.335728 2.42637i 1.10489 0.637910i −1.86301 + 2.12819i 0.686141 + 2.92048i −1.00000 3.42703i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.2.h.a 8
3.b odd 2 1 108.2.h.a 8
4.b odd 2 1 inner 36.2.h.a 8
5.b even 2 1 900.2.r.c 8
5.c odd 4 2 900.2.o.a 16
8.b even 2 1 576.2.s.f 8
8.d odd 2 1 576.2.s.f 8
9.c even 3 1 108.2.h.a 8
9.c even 3 1 324.2.b.b 8
9.d odd 6 1 inner 36.2.h.a 8
9.d odd 6 1 324.2.b.b 8
12.b even 2 1 108.2.h.a 8
20.d odd 2 1 900.2.r.c 8
20.e even 4 2 900.2.o.a 16
24.f even 2 1 1728.2.s.f 8
24.h odd 2 1 1728.2.s.f 8
36.f odd 6 1 108.2.h.a 8
36.f odd 6 1 324.2.b.b 8
36.h even 6 1 inner 36.2.h.a 8
36.h even 6 1 324.2.b.b 8
45.h odd 6 1 900.2.r.c 8
45.l even 12 2 900.2.o.a 16
72.j odd 6 1 576.2.s.f 8
72.j odd 6 1 5184.2.c.j 8
72.l even 6 1 576.2.s.f 8
72.l even 6 1 5184.2.c.j 8
72.n even 6 1 1728.2.s.f 8
72.n even 6 1 5184.2.c.j 8
72.p odd 6 1 1728.2.s.f 8
72.p odd 6 1 5184.2.c.j 8
180.n even 6 1 900.2.r.c 8
180.v odd 12 2 900.2.o.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.h.a 8 1.a even 1 1 trivial
36.2.h.a 8 4.b odd 2 1 inner
36.2.h.a 8 9.d odd 6 1 inner
36.2.h.a 8 36.h even 6 1 inner
108.2.h.a 8 3.b odd 2 1
108.2.h.a 8 9.c even 3 1
108.2.h.a 8 12.b even 2 1
108.2.h.a 8 36.f odd 6 1
324.2.b.b 8 9.c even 3 1
324.2.b.b 8 9.d odd 6 1
324.2.b.b 8 36.f odd 6 1
324.2.b.b 8 36.h even 6 1
576.2.s.f 8 8.b even 2 1
576.2.s.f 8 8.d odd 2 1
576.2.s.f 8 72.j odd 6 1
576.2.s.f 8 72.l even 6 1
900.2.o.a 16 5.c odd 4 2
900.2.o.a 16 20.e even 4 2
900.2.o.a 16 45.l even 12 2
900.2.o.a 16 180.v odd 12 2
900.2.r.c 8 5.b even 2 1
900.2.r.c 8 20.d odd 2 1
900.2.r.c 8 45.h odd 6 1
900.2.r.c 8 180.n even 6 1
1728.2.s.f 8 24.f even 2 1
1728.2.s.f 8 24.h odd 2 1
1728.2.s.f 8 72.n even 6 1
1728.2.s.f 8 72.p odd 6 1
5184.2.c.j 8 72.j odd 6 1
5184.2.c.j 8 72.l even 6 1
5184.2.c.j 8 72.n even 6 1
5184.2.c.j 8 72.p odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 3 T^{7} + 5 T^{6} + 6 T^{5} + \cdots + 16$$
$3$ $$T^{8} + 3 T^{6} + 12 T^{4} + 27 T^{2} + \cdots + 81$$
$5$ $$(T^{4} + 3 T^{3} + T^{2} - 6 T + 4)^{2}$$
$7$ $$T^{8} - 9 T^{6} + 69 T^{4} - 108 T^{2} + \cdots + 144$$
$11$ $$T^{8} + 12 T^{6} + 141 T^{4} + 36 T^{2} + \cdots + 9$$
$13$ $$(T^{4} + T^{3} + 9 T^{2} - 8 T + 64)^{2}$$
$17$ $$(T^{4} + 7 T^{2} + 4)^{2}$$
$19$ $$(T^{4} + 27 T^{2} + 108)^{2}$$
$23$ $$T^{8} + 15 T^{6} + 177 T^{4} + \cdots + 2304$$
$29$ $$(T^{4} - 3 T^{3} + T^{2} + 6 T + 4)^{2}$$
$31$ $$T^{8} - 69 T^{6} + 4569 T^{4} + \cdots + 36864$$
$37$ $$(T^{2} + 2 T - 32)^{4}$$
$41$ $$(T^{4} - 12 T^{3} + 49 T^{2} - 12 T + 1)^{2}$$
$43$ $$T^{8} - 108 T^{6} + 8781 T^{4} + \cdots + 8311689$$
$47$ $$T^{8} + 135 T^{6} + 18033 T^{4} + \cdots + 36864$$
$53$ $$(T^{4} + 76 T^{2} + 256)^{2}$$
$59$ $$T^{8} + 180 T^{6} + \cdots + 16867449$$
$61$ $$(T^{4} + T^{3} + 9 T^{2} - 8 T + 64)^{2}$$
$67$ $$T^{8} - 108 T^{6} + 8781 T^{4} + \cdots + 8311689$$
$71$ $$(T^{4} - 144 T^{2} + 432)^{2}$$
$73$ $$(T^{2} - T - 8)^{4}$$
$79$ $$T^{8} - 201 T^{6} + \cdots + 101848464$$
$83$ $$T^{8} + 111 T^{6} + 9249 T^{4} + \cdots + 9437184$$
$89$ $$(T^{4} + 172 T^{2} + 4096)^{2}$$
$97$ $$(T^{4} - 2 T^{3} + 135 T^{2} + 262 T + 17161)^{2}$$