# Properties

 Label 20691.2.a.e Level $20691$ Weight $2$ Character orbit 20691.a Self dual yes Analytic conductor $165.218$ Dimension $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20691,2,Mod(1,20691)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20691, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("20691.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$20691 = 3^{2} \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 20691.a (trivial)

## 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: yes Analytic conductor: $$165.218466822$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: not computed Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 5 q^{7}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 + 3 * q^5 + 5 * q^7 $$q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 5 q^{7} - 6 q^{10} - 2 q^{13} - 10 q^{14} - 4 q^{16} - q^{17} + q^{19} + 6 q^{20} + 4 q^{23} + 4 q^{25} + 4 q^{26} + 10 q^{28} - 2 q^{29} - 6 q^{31} + 8 q^{32} + 2 q^{34} + 15 q^{35} - 2 q^{38} + q^{43} - 8 q^{46} + 9 q^{47} + 18 q^{49} - 8 q^{50} - 4 q^{52} - 10 q^{53} + 4 q^{58} + 8 q^{59} + q^{61} + 12 q^{62} - 8 q^{64} - 6 q^{65} + 8 q^{67} - 2 q^{68} - 30 q^{70} + 12 q^{71} + 11 q^{73} + 2 q^{76} - 16 q^{79} - 12 q^{80} + 12 q^{83} - 3 q^{85} - 2 q^{86} + 6 q^{89} - 10 q^{91} + 8 q^{92} - 18 q^{94} + 3 q^{95} - 10 q^{97} - 36 q^{98}+O(q^{100})$$ q - 2 * q^2 + 2 * q^4 + 3 * q^5 + 5 * q^7 - 6 * q^10 - 2 * q^13 - 10 * q^14 - 4 * q^16 - q^17 + q^19 + 6 * q^20 + 4 * q^23 + 4 * q^25 + 4 * q^26 + 10 * q^28 - 2 * q^29 - 6 * q^31 + 8 * q^32 + 2 * q^34 + 15 * q^35 - 2 * q^38 + q^43 - 8 * q^46 + 9 * q^47 + 18 * q^49 - 8 * q^50 - 4 * q^52 - 10 * q^53 + 4 * q^58 + 8 * q^59 + q^61 + 12 * q^62 - 8 * q^64 - 6 * q^65 + 8 * q^67 - 2 * q^68 - 30 * q^70 + 12 * q^71 + 11 * q^73 + 2 * q^76 - 16 * q^79 - 12 * q^80 + 12 * q^83 - 3 * q^85 - 2 * q^86 + 6 * q^89 - 10 * q^91 + 8 * q^92 - 18 * q^94 + 3 * q^95 - 10 * q^97 - 36 * q^98

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$-1$$

## Inner twists

Inner twists of this newform have not been computed.

## Twists

Twists of this newform have not been computed.