# Properties

 Label 184.2.a.e Level $184$ Weight $2$ Character orbit 184.a Self dual yes Analytic conductor $1.469$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 2 q^{5} + (\beta + 1) q^{9} +O(q^{10})$$ q - b * q^3 + 2 * q^5 + (b + 1) * q^9 $$q - \beta q^{3} + 2 q^{5} + (\beta + 1) q^{9} + 2 \beta q^{11} + (\beta + 2) q^{13} - 2 \beta q^{15} + ( - 2 \beta + 2) q^{17} + 2 \beta q^{19} - q^{23} - q^{25} + (\beta - 4) q^{27} + ( - \beta + 2) q^{29} + ( - \beta - 4) q^{31} + ( - 2 \beta - 8) q^{33} + ( - 4 \beta + 2) q^{37} + ( - 3 \beta - 4) q^{39} + (5 \beta - 2) q^{41} - 8 q^{43} + (2 \beta + 2) q^{45} + (3 \beta + 4) q^{47} - 7 q^{49} + 8 q^{51} + 2 q^{53} + 4 \beta q^{55} + ( - 2 \beta - 8) q^{57} + ( - 4 \beta + 4) q^{59} + (4 \beta + 2) q^{61} + (2 \beta + 4) q^{65} - 2 \beta q^{67} + \beta q^{69} + ( - \beta + 12) q^{71} + (3 \beta - 10) q^{73} + \beta q^{75} - 2 \beta q^{79} - 7 q^{81} + ( - 4 \beta + 8) q^{83} + ( - 4 \beta + 4) q^{85} + ( - \beta + 4) q^{87} + ( - 6 \beta + 2) q^{89} + (5 \beta + 4) q^{93} + 4 \beta q^{95} + ( - 6 \beta + 2) q^{97} + (4 \beta + 8) q^{99} +O(q^{100})$$ q - b * q^3 + 2 * q^5 + (b + 1) * q^9 + 2*b * q^11 + (b + 2) * q^13 - 2*b * q^15 + (-2*b + 2) * q^17 + 2*b * q^19 - q^23 - q^25 + (b - 4) * q^27 + (-b + 2) * q^29 + (-b - 4) * q^31 + (-2*b - 8) * q^33 + (-4*b + 2) * q^37 + (-3*b - 4) * q^39 + (5*b - 2) * q^41 - 8 * q^43 + (2*b + 2) * q^45 + (3*b + 4) * q^47 - 7 * q^49 + 8 * q^51 + 2 * q^53 + 4*b * q^55 + (-2*b - 8) * q^57 + (-4*b + 4) * q^59 + (4*b + 2) * q^61 + (2*b + 4) * q^65 - 2*b * q^67 + b * q^69 + (-b + 12) * q^71 + (3*b - 10) * q^73 + b * q^75 - 2*b * q^79 - 7 * q^81 + (-4*b + 8) * q^83 + (-4*b + 4) * q^85 + (-b + 4) * q^87 + (-6*b + 2) * q^89 + (5*b + 4) * q^93 + 4*b * q^95 + (-6*b + 2) * q^97 + (4*b + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 4 q^{5} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 + 4 * q^5 + 3 * q^9 $$2 q - q^{3} + 4 q^{5} + 3 q^{9} + 2 q^{11} + 5 q^{13} - 2 q^{15} + 2 q^{17} + 2 q^{19} - 2 q^{23} - 2 q^{25} - 7 q^{27} + 3 q^{29} - 9 q^{31} - 18 q^{33} - 11 q^{39} + q^{41} - 16 q^{43} + 6 q^{45} + 11 q^{47} - 14 q^{49} + 16 q^{51} + 4 q^{53} + 4 q^{55} - 18 q^{57} + 4 q^{59} + 8 q^{61} + 10 q^{65} - 2 q^{67} + q^{69} + 23 q^{71} - 17 q^{73} + q^{75} - 2 q^{79} - 14 q^{81} + 12 q^{83} + 4 q^{85} + 7 q^{87} - 2 q^{89} + 13 q^{93} + 4 q^{95} - 2 q^{97} + 20 q^{99}+O(q^{100})$$ 2 * q - q^3 + 4 * q^5 + 3 * q^9 + 2 * q^11 + 5 * q^13 - 2 * q^15 + 2 * q^17 + 2 * q^19 - 2 * q^23 - 2 * q^25 - 7 * q^27 + 3 * q^29 - 9 * q^31 - 18 * q^33 - 11 * q^39 + q^41 - 16 * q^43 + 6 * q^45 + 11 * q^47 - 14 * q^49 + 16 * q^51 + 4 * q^53 + 4 * q^55 - 18 * q^57 + 4 * q^59 + 8 * q^61 + 10 * q^65 - 2 * q^67 + q^69 + 23 * q^71 - 17 * q^73 + q^75 - 2 * q^79 - 14 * q^81 + 12 * q^83 + 4 * q^85 + 7 * q^87 - 2 * q^89 + 13 * q^93 + 4 * q^95 - 2 * q^97 + 20 * 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
 2.56155 −1.56155
0 −2.56155 0 2.00000 0 0 0 3.56155 0
1.2 0 1.56155 0 2.00000 0 0 0 −0.561553 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.e 2
3.b odd 2 1 1656.2.a.j 2
4.b odd 2 1 368.2.a.i 2
5.b even 2 1 4600.2.a.s 2
5.c odd 4 2 4600.2.e.o 4
7.b odd 2 1 9016.2.a.w 2
8.b even 2 1 1472.2.a.u 2
8.d odd 2 1 1472.2.a.p 2
12.b even 2 1 3312.2.a.t 2
20.d odd 2 1 9200.2.a.br 2
23.b odd 2 1 4232.2.a.o 2
92.b even 2 1 8464.2.a.bd 2

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

 $$T_{3}^{2} + T_{3} - 4$$ T3^2 + T3 - 4 $$T_{5} - 2$$ T5 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 4$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T - 16$$
$13$ $$T^{2} - 5T + 2$$
$17$ $$T^{2} - 2T - 16$$
$19$ $$T^{2} - 2T - 16$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} - 3T - 2$$
$31$ $$T^{2} + 9T + 16$$
$37$ $$T^{2} - 68$$
$41$ $$T^{2} - T - 106$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} - 11T - 8$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} - 4T - 64$$
$61$ $$T^{2} - 8T - 52$$
$67$ $$T^{2} + 2T - 16$$
$71$ $$T^{2} - 23T + 128$$
$73$ $$T^{2} + 17T + 34$$
$79$ $$T^{2} + 2T - 16$$
$83$ $$T^{2} - 12T - 32$$
$89$ $$T^{2} + 2T - 152$$
$97$ $$T^{2} + 2T - 152$$