# Properties

 Label 184.2.a.a Level $184$ Weight $2$ Character orbit 184.a Self dual yes Analytic conductor $1.469$ Analytic rank $1$ 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] = [184,2,Mod(1,184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(184, 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("184.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$184 = 2^{3} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 184.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: $$1.46924739719$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 4 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 - 4 * q^5 + 2 * q^7 - 2 * q^9 $$q - q^{3} - 4 q^{5} + 2 q^{7} - 2 q^{9} - 4 q^{11} - 5 q^{13} + 4 q^{15} - 2 q^{17} + 6 q^{19} - 2 q^{21} + q^{23} + 11 q^{25} + 5 q^{27} + q^{29} - 9 q^{31} + 4 q^{33} - 8 q^{35} - 4 q^{37} + 5 q^{39} + 3 q^{41} + 8 q^{43} + 8 q^{45} - 5 q^{47} - 3 q^{49} + 2 q^{51} + 6 q^{53} + 16 q^{55} - 6 q^{57} - 4 q^{59} - 10 q^{61} - 4 q^{63} + 20 q^{65} - 4 q^{67} - q^{69} - 5 q^{71} - 15 q^{73} - 11 q^{75} - 8 q^{77} - 6 q^{79} + q^{81} + 6 q^{83} + 8 q^{85} - q^{87} - 8 q^{89} - 10 q^{91} + 9 q^{93} - 24 q^{95} + 10 q^{97} + 8 q^{99}+O(q^{100})$$ q - q^3 - 4 * q^5 + 2 * q^7 - 2 * q^9 - 4 * q^11 - 5 * q^13 + 4 * q^15 - 2 * q^17 + 6 * q^19 - 2 * q^21 + q^23 + 11 * q^25 + 5 * q^27 + q^29 - 9 * q^31 + 4 * q^33 - 8 * q^35 - 4 * q^37 + 5 * q^39 + 3 * q^41 + 8 * q^43 + 8 * q^45 - 5 * q^47 - 3 * q^49 + 2 * q^51 + 6 * q^53 + 16 * q^55 - 6 * q^57 - 4 * q^59 - 10 * q^61 - 4 * q^63 + 20 * q^65 - 4 * q^67 - q^69 - 5 * q^71 - 15 * q^73 - 11 * q^75 - 8 * q^77 - 6 * q^79 + q^81 + 6 * q^83 + 8 * q^85 - q^87 - 8 * q^89 - 10 * q^91 + 9 * q^93 - 24 * q^95 + 10 * q^97 + 8 * 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
0 −1.00000 0 −4.00000 0 2.00000 0 −2.00000 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
$$2$$ $$-1$$
$$23$$ $$-1$$

## 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 184.2.a.a 1
3.b odd 2 1 1656.2.a.i 1
4.b odd 2 1 368.2.a.e 1
5.b even 2 1 4600.2.a.i 1
5.c odd 4 2 4600.2.e.e 2
7.b odd 2 1 9016.2.a.k 1
8.b even 2 1 1472.2.a.l 1
8.d odd 2 1 1472.2.a.e 1
12.b even 2 1 3312.2.a.r 1
20.d odd 2 1 9200.2.a.o 1
23.b odd 2 1 4232.2.a.f 1
92.b even 2 1 8464.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.a.a 1 1.a even 1 1 trivial
368.2.a.e 1 4.b odd 2 1
1472.2.a.e 1 8.d odd 2 1
1472.2.a.l 1 8.b even 2 1
1656.2.a.i 1 3.b odd 2 1
3312.2.a.r 1 12.b even 2 1
4232.2.a.f 1 23.b odd 2 1
4600.2.a.i 1 5.b even 2 1
4600.2.e.e 2 5.c odd 4 2
8464.2.a.p 1 92.b even 2 1
9016.2.a.k 1 7.b odd 2 1
9200.2.a.o 1 20.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(184))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{5} + 4$$ T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 4$$
$7$ $$T - 2$$
$11$ $$T + 4$$
$13$ $$T + 5$$
$17$ $$T + 2$$
$19$ $$T - 6$$
$23$ $$T - 1$$
$29$ $$T - 1$$
$31$ $$T + 9$$
$37$ $$T + 4$$
$41$ $$T - 3$$
$43$ $$T - 8$$
$47$ $$T + 5$$
$53$ $$T - 6$$
$59$ $$T + 4$$
$61$ $$T + 10$$
$67$ $$T + 4$$
$71$ $$T + 5$$
$73$ $$T + 15$$
$79$ $$T + 6$$
$83$ $$T - 6$$
$89$ $$T + 8$$
$97$ $$T - 10$$