Properties

 Label 222.2.a.e Level $222$ Weight $2$ Character orbit 222.a Self dual yes Analytic conductor $1.773$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [222,2,Mod(1,222)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(222, 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("222.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$222 = 2 \cdot 3 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 222.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: $$1.77267892487$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 - q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{9} + 3 q^{11} + q^{12} - q^{13} - q^{14} + q^{16} - 3 q^{17} + q^{18} - 7 q^{19} - q^{21} + 3 q^{22} + 3 q^{23} + q^{24} - 5 q^{25} - q^{26} + q^{27} - q^{28} + 2 q^{31} + q^{32} + 3 q^{33} - 3 q^{34} + q^{36} + q^{37} - 7 q^{38} - q^{39} - 6 q^{41} - q^{42} - 4 q^{43} + 3 q^{44} + 3 q^{46} + 6 q^{47} + q^{48} - 6 q^{49} - 5 q^{50} - 3 q^{51} - q^{52} + 9 q^{53} + q^{54} - q^{56} - 7 q^{57} - 10 q^{61} + 2 q^{62} - q^{63} + q^{64} + 3 q^{66} + 2 q^{67} - 3 q^{68} + 3 q^{69} + 12 q^{71} + q^{72} + 5 q^{73} + q^{74} - 5 q^{75} - 7 q^{76} - 3 q^{77} - q^{78} + 2 q^{79} + q^{81} - 6 q^{82} + 3 q^{83} - q^{84} - 4 q^{86} + 3 q^{88} - 3 q^{89} + q^{91} + 3 q^{92} + 2 q^{93} + 6 q^{94} + q^{96} + 2 q^{97} - 6 q^{98} + 3 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 - q^7 + q^8 + q^9 + 3 * q^11 + q^12 - q^13 - q^14 + q^16 - 3 * q^17 + q^18 - 7 * q^19 - q^21 + 3 * q^22 + 3 * q^23 + q^24 - 5 * q^25 - q^26 + q^27 - q^28 + 2 * q^31 + q^32 + 3 * q^33 - 3 * q^34 + q^36 + q^37 - 7 * q^38 - q^39 - 6 * q^41 - q^42 - 4 * q^43 + 3 * q^44 + 3 * q^46 + 6 * q^47 + q^48 - 6 * q^49 - 5 * q^50 - 3 * q^51 - q^52 + 9 * q^53 + q^54 - q^56 - 7 * q^57 - 10 * q^61 + 2 * q^62 - q^63 + q^64 + 3 * q^66 + 2 * q^67 - 3 * q^68 + 3 * q^69 + 12 * q^71 + q^72 + 5 * q^73 + q^74 - 5 * q^75 - 7 * q^76 - 3 * q^77 - q^78 + 2 * q^79 + q^81 - 6 * q^82 + 3 * q^83 - q^84 - 4 * q^86 + 3 * q^88 - 3 * q^89 + q^91 + 3 * q^92 + 2 * q^93 + 6 * q^94 + q^96 + 2 * q^97 - 6 * q^98 + 3 * q^99

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 1.00000 1.00000 0 1.00000 −1.00000 1.00000 1.00000 0
 $$n$$: 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 222.2.a.e 1
3.b odd 2 1 666.2.a.a 1
4.b odd 2 1 1776.2.a.c 1
5.b even 2 1 5550.2.a.h 1
8.b even 2 1 7104.2.a.g 1
8.d odd 2 1 7104.2.a.u 1
12.b even 2 1 5328.2.a.l 1
37.b even 2 1 8214.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.a.e 1 1.a even 1 1 trivial
666.2.a.a 1 3.b odd 2 1
1776.2.a.c 1 4.b odd 2 1
5328.2.a.l 1 12.b even 2 1
5550.2.a.h 1 5.b even 2 1
7104.2.a.g 1 8.b even 2 1
7104.2.a.u 1 8.d odd 2 1
8214.2.a.d 1 37.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(222))$$:

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

Hecke characteristic polynomials

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