# Properties

 Label 532.2.a.d Level $532$ Weight $2$ Character orbit 532.a Self dual yes Analytic conductor $4.248$ 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] = [532,2,Mod(1,532)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(532, 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("532.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$532 = 2^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 532.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: $$4.24804138753$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( - 2 \beta - 1) q^{5} - q^{7} + (\beta - 2) q^{9}+O(q^{10})$$ q + b * q^3 + (-2*b - 1) * q^5 - q^7 + (b - 2) * q^9 $$q + \beta q^{3} + ( - 2 \beta - 1) q^{5} - q^{7} + (\beta - 2) q^{9} + ( - \beta - 1) q^{11} + (4 \beta - 3) q^{13} + ( - 3 \beta - 2) q^{15} + ( - 3 \beta - 1) q^{17} + q^{19} - \beta q^{21} + ( - 2 \beta - 5) q^{23} + 8 \beta q^{25} + ( - 4 \beta + 1) q^{27} + ( - \beta - 3) q^{29} + ( - 3 \beta + 1) q^{31} + ( - 2 \beta - 1) q^{33} + (2 \beta + 1) q^{35} + ( - 2 \beta - 5) q^{37} + (\beta + 4) q^{39} + (9 \beta - 3) q^{41} + (4 \beta - 2) q^{43} + \beta q^{45} + (4 \beta + 1) q^{47} + q^{49} + ( - 4 \beta - 3) q^{51} + (5 \beta - 2) q^{53} + (5 \beta + 3) q^{55} + \beta q^{57} + 11 q^{59} + ( - 2 \beta - 1) q^{61} + ( - \beta + 2) q^{63} + ( - 6 \beta - 5) q^{65} + ( - 9 \beta + 7) q^{67} + ( - 7 \beta - 2) q^{69} + (2 \beta - 7) q^{71} + (\beta + 2) q^{73} + (8 \beta + 8) q^{75} + (\beta + 1) q^{77} + ( - 4 \beta + 2) q^{79} + ( - 6 \beta + 2) q^{81} + (9 \beta - 4) q^{83} + (11 \beta + 7) q^{85} + ( - 4 \beta - 1) q^{87} + ( - 6 \beta - 6) q^{89} + ( - 4 \beta + 3) q^{91} + ( - 2 \beta - 3) q^{93} + ( - 2 \beta - 1) q^{95} + ( - 4 \beta + 3) q^{97} + q^{99} +O(q^{100})$$ q + b * q^3 + (-2*b - 1) * q^5 - q^7 + (b - 2) * q^9 + (-b - 1) * q^11 + (4*b - 3) * q^13 + (-3*b - 2) * q^15 + (-3*b - 1) * q^17 + q^19 - b * q^21 + (-2*b - 5) * q^23 + 8*b * q^25 + (-4*b + 1) * q^27 + (-b - 3) * q^29 + (-3*b + 1) * q^31 + (-2*b - 1) * q^33 + (2*b + 1) * q^35 + (-2*b - 5) * q^37 + (b + 4) * q^39 + (9*b - 3) * q^41 + (4*b - 2) * q^43 + b * q^45 + (4*b + 1) * q^47 + q^49 + (-4*b - 3) * q^51 + (5*b - 2) * q^53 + (5*b + 3) * q^55 + b * q^57 + 11 * q^59 + (-2*b - 1) * q^61 + (-b + 2) * q^63 + (-6*b - 5) * q^65 + (-9*b + 7) * q^67 + (-7*b - 2) * q^69 + (2*b - 7) * q^71 + (b + 2) * q^73 + (8*b + 8) * q^75 + (b + 1) * q^77 + (-4*b + 2) * q^79 + (-6*b + 2) * q^81 + (9*b - 4) * q^83 + (11*b + 7) * q^85 + (-4*b - 1) * q^87 + (-6*b - 6) * q^89 + (-4*b + 3) * q^91 + (-2*b - 3) * q^93 + (-2*b - 1) * q^95 + (-4*b + 3) * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 4 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q + q^3 - 4 * q^5 - 2 * q^7 - 3 * q^9 $$2 q + q^{3} - 4 q^{5} - 2 q^{7} - 3 q^{9} - 3 q^{11} - 2 q^{13} - 7 q^{15} - 5 q^{17} + 2 q^{19} - q^{21} - 12 q^{23} + 8 q^{25} - 2 q^{27} - 7 q^{29} - q^{31} - 4 q^{33} + 4 q^{35} - 12 q^{37} + 9 q^{39} + 3 q^{41} + q^{45} + 6 q^{47} + 2 q^{49} - 10 q^{51} + q^{53} + 11 q^{55} + q^{57} + 22 q^{59} - 4 q^{61} + 3 q^{63} - 16 q^{65} + 5 q^{67} - 11 q^{69} - 12 q^{71} + 5 q^{73} + 24 q^{75} + 3 q^{77} - 2 q^{81} + q^{83} + 25 q^{85} - 6 q^{87} - 18 q^{89} + 2 q^{91} - 8 q^{93} - 4 q^{95} + 2 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + q^3 - 4 * q^5 - 2 * q^7 - 3 * q^9 - 3 * q^11 - 2 * q^13 - 7 * q^15 - 5 * q^17 + 2 * q^19 - q^21 - 12 * q^23 + 8 * q^25 - 2 * q^27 - 7 * q^29 - q^31 - 4 * q^33 + 4 * q^35 - 12 * q^37 + 9 * q^39 + 3 * q^41 + q^45 + 6 * q^47 + 2 * q^49 - 10 * q^51 + q^53 + 11 * q^55 + q^57 + 22 * q^59 - 4 * q^61 + 3 * q^63 - 16 * q^65 + 5 * q^67 - 11 * q^69 - 12 * q^71 + 5 * q^73 + 24 * q^75 + 3 * q^77 - 2 * q^81 + q^83 + 25 * q^85 - 6 * q^87 - 18 * q^89 + 2 * q^91 - 8 * q^93 - 4 * q^95 + 2 * 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
 −0.618034 1.61803
0 −0.618034 0 0.236068 0 −1.00000 0 −2.61803 0
1.2 0 1.61803 0 −4.23607 0 −1.00000 0 −0.381966 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
$$2$$ $$-1$$
$$7$$ $$+1$$
$$19$$ $$-1$$

## 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 532.2.a.d 2
3.b odd 2 1 4788.2.a.n 2
4.b odd 2 1 2128.2.a.e 2
7.b odd 2 1 3724.2.a.d 2
8.b even 2 1 8512.2.a.s 2
8.d odd 2 1 8512.2.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.a.d 2 1.a even 1 1 trivial
2128.2.a.e 2 4.b odd 2 1
3724.2.a.d 2 7.b odd 2 1
4788.2.a.n 2 3.b odd 2 1
8512.2.a.s 2 8.b even 2 1
8512.2.a.z 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(532))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 1$$
$5$ $$T^{2} + 4T - 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 3T + 1$$
$13$ $$T^{2} + 2T - 19$$
$17$ $$T^{2} + 5T - 5$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 12T + 31$$
$29$ $$T^{2} + 7T + 11$$
$31$ $$T^{2} + T - 11$$
$37$ $$T^{2} + 12T + 31$$
$41$ $$T^{2} - 3T - 99$$
$43$ $$T^{2} - 20$$
$47$ $$T^{2} - 6T - 11$$
$53$ $$T^{2} - T - 31$$
$59$ $$(T - 11)^{2}$$
$61$ $$T^{2} + 4T - 1$$
$67$ $$T^{2} - 5T - 95$$
$71$ $$T^{2} + 12T + 31$$
$73$ $$T^{2} - 5T + 5$$
$79$ $$T^{2} - 20$$
$83$ $$T^{2} - T - 101$$
$89$ $$T^{2} + 18T + 36$$
$97$ $$T^{2} - 2T - 19$$