# Properties

 Label 592.2.g.b Level $592$ Weight $2$ Character orbit 592.g Analytic conductor $4.727$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,2,Mod(369,592)]

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, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("592.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$592 = 2^{4} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 592.g (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$4.72714379966$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 37) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/592\mathbb{Z}\right)^\times$$.

 $$n$$ $$113$$ $$149$$ $$223$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
369.1
 − 1.00000i 1.00000i
0 1.00000 0 2.00000i 0 −3.00000 0 −2.00000 0
369.2 0 1.00000 0 2.00000i 0 −3.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.g.b 2
3.b odd 2 1 5328.2.h.c 2
4.b odd 2 1 37.2.b.a 2
8.b even 2 1 2368.2.g.b 2
8.d odd 2 1 2368.2.g.f 2
12.b even 2 1 333.2.c.a 2
20.d odd 2 1 925.2.c.b 2
20.e even 4 1 925.2.d.a 2
20.e even 4 1 925.2.d.d 2
37.b even 2 1 inner 592.2.g.b 2
111.d odd 2 1 5328.2.h.c 2
148.b odd 2 1 37.2.b.a 2
148.g even 4 1 1369.2.a.a 1
148.g even 4 1 1369.2.a.f 1
296.e even 2 1 2368.2.g.b 2
296.h odd 2 1 2368.2.g.f 2
444.g even 2 1 333.2.c.a 2
740.g odd 2 1 925.2.c.b 2
740.m even 4 1 925.2.d.a 2
740.m even 4 1 925.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.b.a 2 4.b odd 2 1
37.2.b.a 2 148.b odd 2 1
333.2.c.a 2 12.b even 2 1
333.2.c.a 2 444.g even 2 1
592.2.g.b 2 1.a even 1 1 trivial
592.2.g.b 2 37.b even 2 1 inner
925.2.c.b 2 20.d odd 2 1
925.2.c.b 2 740.g odd 2 1
925.2.d.a 2 20.e even 4 1
925.2.d.a 2 740.m even 4 1
925.2.d.d 2 20.e even 4 1
925.2.d.d 2 740.m even 4 1
1369.2.a.a 1 148.g even 4 1
1369.2.a.f 1 148.g even 4 1
2368.2.g.b 2 8.b even 2 1
2368.2.g.b 2 296.e even 2 1
2368.2.g.f 2 8.d odd 2 1
2368.2.g.f 2 296.h odd 2 1
5328.2.h.c 2 3.b odd 2 1
5328.2.h.c 2 111.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(592, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$(T + 3)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2} + 36$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 2T + 37$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$(T + 3)^{2}$$
$53$ $$(T - 9)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$T^{2}$$
$67$ $$(T - 12)^{2}$$
$71$ $$(T - 3)^{2}$$
$73$ $$(T - 9)^{2}$$
$79$ $$T^{2} + 36$$
$83$ $$(T + 9)^{2}$$
$89$ $$T^{2} + 196$$
$97$ $$T^{2} + 144$$
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