# Properties

 Label 2184.2.a.n Level $2184$ Weight $2$ Character orbit 2184.a Self dual yes Analytic conductor $17.439$ Analytic rank $1$ 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] = [2184,2,Mod(1,2184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2184, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2184.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2184.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: $$17.4393278014$$ Analytic rank: $$1$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( - \beta - 1) q^{5} - q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (-b - 1) * q^5 - q^7 + q^9 $$q - q^{3} + ( - \beta - 1) q^{5} - q^{7} + q^{9} + (2 \beta - 2) q^{11} + q^{13} + (\beta + 1) q^{15} + ( - 2 \beta + 4) q^{17} + ( - \beta + 1) q^{19} + q^{21} + (3 \beta + 1) q^{23} + 3 \beta q^{25} - q^{27} + ( - \beta + 3) q^{29} + ( - \beta - 3) q^{31} + ( - 2 \beta + 2) q^{33} + (\beta + 1) q^{35} + (2 \beta - 4) q^{37} - q^{39} - 2 q^{41} + ( - \beta - 7) q^{43} + ( - \beta - 1) q^{45} + (\beta + 7) q^{47} + q^{49} + (2 \beta - 4) q^{51} + (3 \beta - 1) q^{53} + ( - 2 \beta - 6) q^{55} + (\beta - 1) q^{57} - 8 q^{59} - 2 q^{61} - q^{63} + ( - \beta - 1) q^{65} + ( - 2 \beta - 2) q^{67} + ( - 3 \beta - 1) q^{69} - 4 \beta q^{71} + (\beta - 11) q^{73} - 3 \beta q^{75} + ( - 2 \beta + 2) q^{77} + (3 \beta + 1) q^{79} + q^{81} + (\beta - 13) q^{83} + 4 q^{85} + (\beta - 3) q^{87} + (5 \beta - 3) q^{89} - q^{91} + (\beta + 3) q^{93} + (\beta + 3) q^{95} + ( - 3 \beta + 1) q^{97} + (2 \beta - 2) q^{99} +O(q^{100})$$ q - q^3 + (-b - 1) * q^5 - q^7 + q^9 + (2*b - 2) * q^11 + q^13 + (b + 1) * q^15 + (-2*b + 4) * q^17 + (-b + 1) * q^19 + q^21 + (3*b + 1) * q^23 + 3*b * q^25 - q^27 + (-b + 3) * q^29 + (-b - 3) * q^31 + (-2*b + 2) * q^33 + (b + 1) * q^35 + (2*b - 4) * q^37 - q^39 - 2 * q^41 + (-b - 7) * q^43 + (-b - 1) * q^45 + (b + 7) * q^47 + q^49 + (2*b - 4) * q^51 + (3*b - 1) * q^53 + (-2*b - 6) * q^55 + (b - 1) * q^57 - 8 * q^59 - 2 * q^61 - q^63 + (-b - 1) * q^65 + (-2*b - 2) * q^67 + (-3*b - 1) * q^69 - 4*b * q^71 + (b - 11) * q^73 - 3*b * q^75 + (-2*b + 2) * q^77 + (3*b + 1) * q^79 + q^81 + (b - 13) * q^83 + 4 * q^85 + (b - 3) * q^87 + (5*b - 3) * q^89 - q^91 + (b + 3) * q^93 + (b + 3) * q^95 + (-3*b + 1) * q^97 + (2*b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 3 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 3 * q^5 - 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} - 3 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 3 q^{15} + 6 q^{17} + q^{19} + 2 q^{21} + 5 q^{23} + 3 q^{25} - 2 q^{27} + 5 q^{29} - 7 q^{31} + 2 q^{33} + 3 q^{35} - 6 q^{37} - 2 q^{39} - 4 q^{41} - 15 q^{43} - 3 q^{45} + 15 q^{47} + 2 q^{49} - 6 q^{51} + q^{53} - 14 q^{55} - q^{57} - 16 q^{59} - 4 q^{61} - 2 q^{63} - 3 q^{65} - 6 q^{67} - 5 q^{69} - 4 q^{71} - 21 q^{73} - 3 q^{75} + 2 q^{77} + 5 q^{79} + 2 q^{81} - 25 q^{83} + 8 q^{85} - 5 q^{87} - q^{89} - 2 q^{91} + 7 q^{93} + 7 q^{95} - q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 3 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 + 3 * q^15 + 6 * q^17 + q^19 + 2 * q^21 + 5 * q^23 + 3 * q^25 - 2 * q^27 + 5 * q^29 - 7 * q^31 + 2 * q^33 + 3 * q^35 - 6 * q^37 - 2 * q^39 - 4 * q^41 - 15 * q^43 - 3 * q^45 + 15 * q^47 + 2 * q^49 - 6 * q^51 + q^53 - 14 * q^55 - q^57 - 16 * q^59 - 4 * q^61 - 2 * q^63 - 3 * q^65 - 6 * q^67 - 5 * q^69 - 4 * q^71 - 21 * q^73 - 3 * q^75 + 2 * q^77 + 5 * q^79 + 2 * q^81 - 25 * q^83 + 8 * q^85 - 5 * q^87 - q^89 - 2 * q^91 + 7 * q^93 + 7 * q^95 - q^97 - 2 * 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 −1.00000 0 −3.56155 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 0.561553 0 −1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$
$$13$$ $$-1$$

## 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 2184.2.a.n 2
3.b odd 2 1 6552.2.a.bk 2
4.b odd 2 1 4368.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2184.2.a.n 2 1.a even 1 1 trivial
4368.2.a.bf 2 4.b odd 2 1
6552.2.a.bk 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2184))$$:

 $$T_{5}^{2} + 3T_{5} - 2$$ T5^2 + 3*T5 - 2 $$T_{11}^{2} + 2T_{11} - 16$$ T11^2 + 2*T11 - 16 $$T_{17}^{2} - 6T_{17} - 8$$ T17^2 - 6*T17 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 3T - 2$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 2T - 16$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 6T - 8$$
$19$ $$T^{2} - T - 4$$
$23$ $$T^{2} - 5T - 32$$
$29$ $$T^{2} - 5T + 2$$
$31$ $$T^{2} + 7T + 8$$
$37$ $$T^{2} + 6T - 8$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 15T + 52$$
$47$ $$T^{2} - 15T + 52$$
$53$ $$T^{2} - T - 38$$
$59$ $$(T + 8)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 6T - 8$$
$71$ $$T^{2} + 4T - 64$$
$73$ $$T^{2} + 21T + 106$$
$79$ $$T^{2} - 5T - 32$$
$83$ $$T^{2} + 25T + 152$$
$89$ $$T^{2} + T - 106$$
$97$ $$T^{2} + T - 38$$