# Properties

 Label 666.2.a.d Level $666$ Weight $2$ Character orbit 666.a Self dual yes Analytic conductor $5.318$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(1,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("666.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$666 = 2 \cdot 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 666.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: $$5.31803677462$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 222) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 4 q^{5} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - 4 * q^5 - q^7 + q^8 $$q + q^{2} + q^{4} - 4 q^{5} - q^{7} + q^{8} - 4 q^{10} + q^{11} - 3 q^{13} - q^{14} + q^{16} - 3 q^{17} - 5 q^{19} - 4 q^{20} + q^{22} - 5 q^{23} + 11 q^{25} - 3 q^{26} - q^{28} - 4 q^{29} - 10 q^{31} + q^{32} - 3 q^{34} + 4 q^{35} - q^{37} - 5 q^{38} - 4 q^{40} + 6 q^{41} + 4 q^{43} + q^{44} - 5 q^{46} - 2 q^{47} - 6 q^{49} + 11 q^{50} - 3 q^{52} + 11 q^{53} - 4 q^{55} - q^{56} - 4 q^{58} + 12 q^{59} + 10 q^{61} - 10 q^{62} + q^{64} + 12 q^{65} + 14 q^{67} - 3 q^{68} + 4 q^{70} - 11 q^{73} - q^{74} - 5 q^{76} - q^{77} - 10 q^{79} - 4 q^{80} + 6 q^{82} + 9 q^{83} + 12 q^{85} + 4 q^{86} + q^{88} - 11 q^{89} + 3 q^{91} - 5 q^{92} - 2 q^{94} + 20 q^{95} + 10 q^{97} - 6 q^{98}+O(q^{100})$$ q + q^2 + q^4 - 4 * q^5 - q^7 + q^8 - 4 * q^10 + q^11 - 3 * q^13 - q^14 + q^16 - 3 * q^17 - 5 * q^19 - 4 * q^20 + q^22 - 5 * q^23 + 11 * q^25 - 3 * q^26 - q^28 - 4 * q^29 - 10 * q^31 + q^32 - 3 * q^34 + 4 * q^35 - q^37 - 5 * q^38 - 4 * q^40 + 6 * q^41 + 4 * q^43 + q^44 - 5 * q^46 - 2 * q^47 - 6 * q^49 + 11 * q^50 - 3 * q^52 + 11 * q^53 - 4 * q^55 - q^56 - 4 * q^58 + 12 * q^59 + 10 * q^61 - 10 * q^62 + q^64 + 12 * q^65 + 14 * q^67 - 3 * q^68 + 4 * q^70 - 11 * q^73 - q^74 - 5 * q^76 - q^77 - 10 * q^79 - 4 * q^80 + 6 * q^82 + 9 * q^83 + 12 * q^85 + 4 * q^86 + q^88 - 11 * q^89 + 3 * q^91 - 5 * q^92 - 2 * q^94 + 20 * q^95 + 10 * q^97 - 6 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 1.00000 −4.00000 0 −1.00000 1.00000 0 −4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$37$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.a.d 1
3.b odd 2 1 222.2.a.c 1
4.b odd 2 1 5328.2.a.b 1
12.b even 2 1 1776.2.a.e 1
15.d odd 2 1 5550.2.a.z 1
24.f even 2 1 7104.2.a.p 1
24.h odd 2 1 7104.2.a.a 1
111.d odd 2 1 8214.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.a.c 1 3.b odd 2 1
666.2.a.d 1 1.a even 1 1 trivial
1776.2.a.e 1 12.b even 2 1
5328.2.a.b 1 4.b odd 2 1
5550.2.a.z 1 15.d odd 2 1
7104.2.a.a 1 24.h odd 2 1
7104.2.a.p 1 24.f even 2 1
8214.2.a.j 1 111.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(666))$$:

 $$T_{5} + 4$$ T5 + 4 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 4$$
$7$ $$T + 1$$
$11$ $$T - 1$$
$13$ $$T + 3$$
$17$ $$T + 3$$
$19$ $$T + 5$$
$23$ $$T + 5$$
$29$ $$T + 4$$
$31$ $$T + 10$$
$37$ $$T + 1$$
$41$ $$T - 6$$
$43$ $$T - 4$$
$47$ $$T + 2$$
$53$ $$T - 11$$
$59$ $$T - 12$$
$61$ $$T - 10$$
$67$ $$T - 14$$
$71$ $$T$$
$73$ $$T + 11$$
$79$ $$T + 10$$
$83$ $$T - 9$$
$89$ $$T + 11$$
$97$ $$T - 10$$
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