# Properties

 Label 55.2.e.b Level $55$ Weight $2$ Character orbit 55.e Analytic conductor $0.439$ Analytic rank $0$ Dimension $4$ CM discriminant -11 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,2,Mod(32,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 2]))

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

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

chi := DirichletCharacter("55.32");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 55.e (of order $$4$$, degree $$2$$, 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: $$0.439177211117$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1 - 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{2} - \beta_1) q^{5} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{9}+O(q^{10})$$ q + (-b3 + b1 - 1) * q^3 + 2*b2 * q^4 + (-b2 - b1) * q^5 + (2*b3 - 3*b2 + 1) * q^9 $$q + ( - \beta_{3} + \beta_1 - 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{2} - \beta_1) q^{5} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{9} + (\beta_{2} - 2 \beta_1) q^{11} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{12} + ( - 2 \beta_{3} + 4 \beta_{2} + \cdots - 3) q^{15}+ \cdots + (6 \beta_{3} - 11 \beta_{2} + 3) q^{99}+O(q^{100})$$ q + (-b3 + b1 - 1) * q^3 + 2*b2 * q^4 + (-b2 - b1) * q^5 + (2*b3 - 3*b2 + 1) * q^9 + (b2 - 2*b1) * q^11 + (2*b3 - 2*b2 + 2*b1) * q^12 + (-2*b3 + 4*b2 - b1 - 3) * q^15 - 4 * q^16 + (-2*b3 + 2) * q^20 + (-b3 - 5*b2 + b1 + 4) * q^23 + (3*b3 + 2) * q^25 + (-b3 + 6*b2 - b1 + 5) * q^27 + (-3*b2 + 6*b1) * q^31 + (-b3 + 5*b2 + b1 - 6) * q^33 + (2*b2 - 4*b1 + 6) * q^36 + (-3*b3 - 2*b2 - 3*b1 - 5) * q^37 + (-4*b3 - 2) * q^44 + (3*b3 - 7*b2 + 3*b1 - 3) * q^45 + (2*b3 - 7*b2 + 2*b1 - 5) * q^47 + (4*b3 - 4*b1 + 4) * q^48 + 7*b2 * q^49 + (4*b3 + 5*b2 - 4*b1 - 1) * q^53 + (3*b3 + 7) * q^55 + (2*b3 + 1) * q^59 + (-2*b3 - 6*b2 + 4*b1 - 8) * q^60 - 8*b2 * q^64 + (-3*b3 + 8*b2 - 3*b1 + 5) * q^67 + (-8*b3 - b2 - 4) * q^69 - 3 * q^71 + (-2*b3 + 9*b2 - b1 + 7) * q^75 + (4*b2 + 4*b1) * q^80 + (-3*b2 + 6*b1 - 2) * q^81 + 9*b2 * q^89 + (2*b3 + 8*b2 + 2*b1 + 10) * q^92 + (3*b3 - 15*b2 - 3*b1 + 18) * q^93 + (-3*b3 - 7*b2 - 3*b1 - 10) * q^97 + (6*b3 - 11*b2 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3}+O(q^{10})$$ 4 * q - 2 * q^3 $$4 q - 2 q^{3} - 4 q^{12} - 8 q^{15} - 16 q^{16} + 12 q^{20} + 18 q^{23} + 2 q^{25} + 22 q^{27} - 22 q^{33} + 24 q^{36} - 14 q^{37} - 18 q^{45} - 24 q^{47} + 8 q^{48} - 12 q^{53} + 22 q^{55} - 28 q^{60} + 26 q^{67} - 12 q^{71} + 32 q^{75} - 8 q^{81} + 36 q^{92} + 66 q^{93} - 34 q^{97}+O(q^{100})$$ 4 * q - 2 * q^3 - 4 * q^12 - 8 * q^15 - 16 * q^16 + 12 * q^20 + 18 * q^23 + 2 * q^25 + 22 * q^27 - 22 * q^33 + 24 * q^36 - 14 * q^37 - 18 * q^45 - 24 * q^47 + 8 * q^48 - 12 * q^53 + 22 * q^55 - 28 * q^60 + 26 * q^67 - 12 * q^71 + 32 * q^75 - 8 * q^81 + 36 * q^92 + 66 * q^93 - 34 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2\nu ) / 3$$ (v^3 - 2*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3$$ b3 + 3 $$\nu^{3}$$ $$=$$ $$3\beta_{2} + 2\beta_1$$ 3*b2 + 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$\beta_{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −1.65831 + 0.500000i 1.65831 + 0.500000i −1.65831 − 0.500000i 1.65831 − 0.500000i
0 −2.15831 + 2.15831i 2.00000i 1.65831 1.50000i 0 0 0 6.31662i 0
32.2 0 1.15831 1.15831i 2.00000i −1.65831 1.50000i 0 0 0 0.316625i 0
43.1 0 −2.15831 2.15831i 2.00000i 1.65831 + 1.50000i 0 0 0 6.31662i 0
43.2 0 1.15831 + 1.15831i 2.00000i −1.65831 + 1.50000i 0 0 0 0.316625i 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
5.c odd 4 1 inner
55.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.2.e.b 4
3.b odd 2 1 495.2.k.a 4
4.b odd 2 1 880.2.bd.d 4
5.b even 2 1 275.2.e.a 4
5.c odd 4 1 inner 55.2.e.b 4
5.c odd 4 1 275.2.e.a 4
11.b odd 2 1 CM 55.2.e.b 4
11.c even 5 4 605.2.m.a 16
11.d odd 10 4 605.2.m.a 16
15.e even 4 1 495.2.k.a 4
20.e even 4 1 880.2.bd.d 4
33.d even 2 1 495.2.k.a 4
44.c even 2 1 880.2.bd.d 4
55.d odd 2 1 275.2.e.a 4
55.e even 4 1 inner 55.2.e.b 4
55.e even 4 1 275.2.e.a 4
55.k odd 20 4 605.2.m.a 16
55.l even 20 4 605.2.m.a 16
165.l odd 4 1 495.2.k.a 4
220.i odd 4 1 880.2.bd.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.b 4 1.a even 1 1 trivial
55.2.e.b 4 5.c odd 4 1 inner
55.2.e.b 4 11.b odd 2 1 CM
55.2.e.b 4 55.e even 4 1 inner
275.2.e.a 4 5.b even 2 1
275.2.e.a 4 5.c odd 4 1
275.2.e.a 4 55.d odd 2 1
275.2.e.a 4 55.e even 4 1
495.2.k.a 4 3.b odd 2 1
495.2.k.a 4 15.e even 4 1
495.2.k.a 4 33.d even 2 1
495.2.k.a 4 165.l odd 4 1
605.2.m.a 16 11.c even 5 4
605.2.m.a 16 11.d odd 10 4
605.2.m.a 16 55.k odd 20 4
605.2.m.a 16 55.l even 20 4
880.2.bd.d 4 4.b odd 2 1
880.2.bd.d 4 20.e even 4 1
880.2.bd.d 4 44.c even 2 1
880.2.bd.d 4 220.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(55, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + \cdots + 25$$
$5$ $$T^{4} - T^{2} + 25$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 11)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} - 18 T^{3} + \cdots + 1225$$
$29$ $$T^{4}$$
$31$ $$(T^{2} - 99)^{2}$$
$37$ $$T^{4} + 14 T^{3} + \cdots + 625$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 24 T^{3} + \cdots + 2500$$
$53$ $$T^{4} + 12 T^{3} + \cdots + 4900$$
$59$ $$(T^{2} + 11)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4} - 26 T^{3} + \cdots + 1225$$
$71$ $$(T + 3)^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} + 81)^{2}$$
$97$ $$T^{4} + 34 T^{3} + \cdots + 9025$$