# Properties

 Label 592.2.a.e Level $592$ Weight $2$ Character orbit 592.a Self dual yes Analytic conductor $4.727$ Analytic rank $0$ 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] = [592,2,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, 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("592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$592 = 2^{4} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 592.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: $$4.72714379966$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 37) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 2 q^{5} + q^{7} + 6 q^{9}+O(q^{10})$$ q + 3 * q^3 - 2 * q^5 + q^7 + 6 * q^9 $$q + 3 q^{3} - 2 q^{5} + q^{7} + 6 q^{9} + 5 q^{11} - 2 q^{13} - 6 q^{15} + 3 q^{21} - 2 q^{23} - q^{25} + 9 q^{27} + 6 q^{29} + 4 q^{31} + 15 q^{33} - 2 q^{35} - q^{37} - 6 q^{39} - 9 q^{41} - 2 q^{43} - 12 q^{45} + 9 q^{47} - 6 q^{49} + q^{53} - 10 q^{55} - 8 q^{59} - 8 q^{61} + 6 q^{63} + 4 q^{65} - 8 q^{67} - 6 q^{69} - 9 q^{71} - q^{73} - 3 q^{75} + 5 q^{77} - 4 q^{79} + 9 q^{81} + 15 q^{83} + 18 q^{87} + 4 q^{89} - 2 q^{91} + 12 q^{93} + 4 q^{97} + 30 q^{99}+O(q^{100})$$ q + 3 * q^3 - 2 * q^5 + q^7 + 6 * q^9 + 5 * q^11 - 2 * q^13 - 6 * q^15 + 3 * q^21 - 2 * q^23 - q^25 + 9 * q^27 + 6 * q^29 + 4 * q^31 + 15 * q^33 - 2 * q^35 - q^37 - 6 * q^39 - 9 * q^41 - 2 * q^43 - 12 * q^45 + 9 * q^47 - 6 * q^49 + q^53 - 10 * q^55 - 8 * q^59 - 8 * q^61 + 6 * q^63 + 4 * q^65 - 8 * q^67 - 6 * q^69 - 9 * q^71 - q^73 - 3 * q^75 + 5 * q^77 - 4 * q^79 + 9 * q^81 + 15 * q^83 + 18 * q^87 + 4 * q^89 - 2 * q^91 + 12 * q^93 + 4 * q^97 + 30 * 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 3.00000 0 −2.00000 0 1.00000 0 6.00000 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
$$2$$ $$-1$$
$$37$$ $$+1$$

## 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 592.2.a.e 1
3.b odd 2 1 5328.2.a.r 1
4.b odd 2 1 37.2.a.a 1
8.b even 2 1 2368.2.a.b 1
8.d odd 2 1 2368.2.a.q 1
12.b even 2 1 333.2.a.d 1
20.d odd 2 1 925.2.a.e 1
20.e even 4 2 925.2.b.b 2
28.d even 2 1 1813.2.a.a 1
44.c even 2 1 4477.2.a.b 1
52.b odd 2 1 6253.2.a.c 1
60.h even 2 1 8325.2.a.e 1
148.b odd 2 1 1369.2.a.e 1
148.g even 4 2 1369.2.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.a.a 1 4.b odd 2 1
333.2.a.d 1 12.b even 2 1
592.2.a.e 1 1.a even 1 1 trivial
925.2.a.e 1 20.d odd 2 1
925.2.b.b 2 20.e even 4 2
1369.2.a.e 1 148.b odd 2 1
1369.2.b.c 2 148.g even 4 2
1813.2.a.a 1 28.d even 2 1
2368.2.a.b 1 8.b even 2 1
2368.2.a.q 1 8.d odd 2 1
4477.2.a.b 1 44.c even 2 1
5328.2.a.r 1 3.b odd 2 1
6253.2.a.c 1 52.b odd 2 1
8325.2.a.e 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(592))$$:

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

## Hecke characteristic polynomials

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