# Properties

 Label 532.2.a.b 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{3} - q^{5} + q^{7} + (3 \beta - 1) q^{9}+O(q^{10})$$ q + (-b - 1) * q^3 - q^5 + q^7 + (3*b - 1) * q^9 $$q + ( - \beta - 1) q^{3} - q^{5} + q^{7} + (3 \beta - 1) q^{9} + (5 \beta - 2) q^{11} + ( - 2 \beta + 1) q^{13} + (\beta + 1) q^{15} + ( - \beta - 6) q^{17} - q^{19} + ( - \beta - 1) q^{21} + ( - 6 \beta + 1) q^{23} - 4 q^{25} + ( - 2 \beta + 1) q^{27} + (3 \beta - 4) q^{29} + (\beta - 4) q^{31} + ( - 8 \beta - 3) q^{33} - q^{35} + ( - 6 \beta + 3) q^{37} + (3 \beta + 1) q^{39} + (5 \beta - 8) q^{41} + (4 \beta + 2) q^{43} + ( - 3 \beta + 1) q^{45} + (6 \beta - 5) q^{47} + q^{49} + (8 \beta + 7) q^{51} + ( - 7 \beta - 3) q^{53} + ( - 5 \beta + 2) q^{55} + (\beta + 1) q^{57} + ( - 6 \beta - 1) q^{59} + (4 \beta - 1) q^{61} + (3 \beta - 1) q^{63} + (2 \beta - 1) q^{65} + ( - 5 \beta + 6) q^{67} + (11 \beta + 5) q^{69} + ( - 10 \beta + 5) q^{71} + (5 \beta - 9) q^{73} + (4 \beta + 4) q^{75} + (5 \beta - 2) q^{77} + ( - 8 \beta + 6) q^{79} + ( - 6 \beta + 4) q^{81} + (\beta + 13) q^{83} + (\beta + 6) q^{85} + ( - 2 \beta + 1) q^{87} + ( - 10 \beta + 4) q^{89} + ( - 2 \beta + 1) q^{91} + (2 \beta + 3) q^{93} + q^{95} + (6 \beta - 1) q^{97} + (4 \beta + 17) q^{99}+O(q^{100})$$ q + (-b - 1) * q^3 - q^5 + q^7 + (3*b - 1) * q^9 + (5*b - 2) * q^11 + (-2*b + 1) * q^13 + (b + 1) * q^15 + (-b - 6) * q^17 - q^19 + (-b - 1) * q^21 + (-6*b + 1) * q^23 - 4 * q^25 + (-2*b + 1) * q^27 + (3*b - 4) * q^29 + (b - 4) * q^31 + (-8*b - 3) * q^33 - q^35 + (-6*b + 3) * q^37 + (3*b + 1) * q^39 + (5*b - 8) * q^41 + (4*b + 2) * q^43 + (-3*b + 1) * q^45 + (6*b - 5) * q^47 + q^49 + (8*b + 7) * q^51 + (-7*b - 3) * q^53 + (-5*b + 2) * q^55 + (b + 1) * q^57 + (-6*b - 1) * q^59 + (4*b - 1) * q^61 + (3*b - 1) * q^63 + (2*b - 1) * q^65 + (-5*b + 6) * q^67 + (11*b + 5) * q^69 + (-10*b + 5) * q^71 + (5*b - 9) * q^73 + (4*b + 4) * q^75 + (5*b - 2) * q^77 + (-8*b + 6) * q^79 + (-6*b + 4) * q^81 + (b + 13) * q^83 + (b + 6) * q^85 + (-2*b + 1) * q^87 + (-10*b + 4) * q^89 + (-2*b + 1) * q^91 + (2*b + 3) * q^93 + q^95 + (6*b - 1) * q^97 + (4*b + 17) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 2 q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - 2 * q^5 + 2 * q^7 + q^9 $$2 q - 3 q^{3} - 2 q^{5} + 2 q^{7} + q^{9} + q^{11} + 3 q^{15} - 13 q^{17} - 2 q^{19} - 3 q^{21} - 4 q^{23} - 8 q^{25} - 5 q^{29} - 7 q^{31} - 14 q^{33} - 2 q^{35} + 5 q^{39} - 11 q^{41} + 8 q^{43} - q^{45} - 4 q^{47} + 2 q^{49} + 22 q^{51} - 13 q^{53} - q^{55} + 3 q^{57} - 8 q^{59} + 2 q^{61} + q^{63} + 7 q^{67} + 21 q^{69} - 13 q^{73} + 12 q^{75} + q^{77} + 4 q^{79} + 2 q^{81} + 27 q^{83} + 13 q^{85} - 2 q^{89} + 8 q^{93} + 2 q^{95} + 4 q^{97} + 38 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 - 2 * q^5 + 2 * q^7 + q^9 + q^11 + 3 * q^15 - 13 * q^17 - 2 * q^19 - 3 * q^21 - 4 * q^23 - 8 * q^25 - 5 * q^29 - 7 * q^31 - 14 * q^33 - 2 * q^35 + 5 * q^39 - 11 * q^41 + 8 * q^43 - q^45 - 4 * q^47 + 2 * q^49 + 22 * q^51 - 13 * q^53 - q^55 + 3 * q^57 - 8 * q^59 + 2 * q^61 + q^63 + 7 * q^67 + 21 * q^69 - 13 * q^73 + 12 * q^75 + q^77 + 4 * q^79 + 2 * q^81 + 27 * q^83 + 13 * q^85 - 2 * q^89 + 8 * q^93 + 2 * q^95 + 4 * q^97 + 38 * 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
 1.61803 −0.618034
0 −2.61803 0 −1.00000 0 1.00000 0 3.85410 0
1.2 0 −0.381966 0 −1.00000 0 1.00000 0 −2.85410 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.b 2
3.b odd 2 1 4788.2.a.l 2
4.b odd 2 1 2128.2.a.m 2
7.b odd 2 1 3724.2.a.g 2
8.b even 2 1 8512.2.a.bg 2
8.d odd 2 1 8512.2.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.a.b 2 1.a even 1 1 trivial
2128.2.a.m 2 4.b odd 2 1
3724.2.a.g 2 7.b odd 2 1
4788.2.a.l 2 3.b odd 2 1
8512.2.a.k 2 8.d odd 2 1
8512.2.a.bg 2 8.b even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - T - 31$$
$13$ $$T^{2} - 5$$
$17$ $$T^{2} + 13T + 41$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 4T - 41$$
$29$ $$T^{2} + 5T - 5$$
$31$ $$T^{2} + 7T + 11$$
$37$ $$T^{2} - 45$$
$41$ $$T^{2} + 11T - 1$$
$43$ $$T^{2} - 8T - 4$$
$47$ $$T^{2} + 4T - 41$$
$53$ $$T^{2} + 13T - 19$$
$59$ $$T^{2} + 8T - 29$$
$61$ $$T^{2} - 2T - 19$$
$67$ $$T^{2} - 7T - 19$$
$71$ $$T^{2} - 125$$
$73$ $$T^{2} + 13T + 11$$
$79$ $$T^{2} - 4T - 76$$
$83$ $$T^{2} - 27T + 181$$
$89$ $$T^{2} + 2T - 124$$
$97$ $$T^{2} - 4T - 41$$