# Properties

 Label 440.2.b.a Level $440$ Weight $2$ Character orbit 440.b Analytic conductor $3.513$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(89,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("440.89");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$440 = 2^{3} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 440.b (of order $$2$$, degree $$1$$, 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: $$3.51341768894$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + (\beta - 1) q^{5} + \beta q^{7} - q^{9}+O(q^{10})$$ q + b * q^3 + (b - 1) * q^5 + b * q^7 - q^9 $$q + \beta q^{3} + (\beta - 1) q^{5} + \beta q^{7} - q^{9} - q^{11} + ( - \beta - 4) q^{15} - 2 \beta q^{17} - 4 q^{19} - 4 q^{21} + 3 \beta q^{23} + ( - 2 \beta - 3) q^{25} + 2 \beta q^{27} - 2 q^{29} + 8 q^{31} - \beta q^{33} + ( - \beta - 4) q^{35} - 2 \beta q^{37} - 6 q^{41} + 3 \beta q^{43} + ( - \beta + 1) q^{45} - \beta q^{47} + 3 q^{49} + 8 q^{51} + 6 \beta q^{53} + ( - \beta + 1) q^{55} - 4 \beta q^{57} - 4 q^{59} + 14 q^{61} - \beta q^{63} + 5 \beta q^{67} - 12 q^{69} + 8 q^{71} - 2 \beta q^{73} + ( - 3 \beta + 8) q^{75} - \beta q^{77} + 8 q^{79} - 11 q^{81} - \beta q^{83} + (2 \beta + 8) q^{85} - 2 \beta q^{87} + 10 q^{89} + 8 \beta q^{93} + ( - 4 \beta + 4) q^{95} - 4 \beta q^{97} + q^{99} +O(q^{100})$$ q + b * q^3 + (b - 1) * q^5 + b * q^7 - q^9 - q^11 + (-b - 4) * q^15 - 2*b * q^17 - 4 * q^19 - 4 * q^21 + 3*b * q^23 + (-2*b - 3) * q^25 + 2*b * q^27 - 2 * q^29 + 8 * q^31 - b * q^33 + (-b - 4) * q^35 - 2*b * q^37 - 6 * q^41 + 3*b * q^43 + (-b + 1) * q^45 - b * q^47 + 3 * q^49 + 8 * q^51 + 6*b * q^53 + (-b + 1) * q^55 - 4*b * q^57 - 4 * q^59 + 14 * q^61 - b * q^63 + 5*b * q^67 - 12 * q^69 + 8 * q^71 - 2*b * q^73 + (-3*b + 8) * q^75 - b * q^77 + 8 * q^79 - 11 * q^81 - b * q^83 + (2*b + 8) * q^85 - 2*b * q^87 + 10 * q^89 + 8*b * q^93 + (-4*b + 4) * q^95 - 4*b * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^9 $$2 q - 2 q^{5} - 2 q^{9} - 2 q^{11} - 8 q^{15} - 8 q^{19} - 8 q^{21} - 6 q^{25} - 4 q^{29} + 16 q^{31} - 8 q^{35} - 12 q^{41} + 2 q^{45} + 6 q^{49} + 16 q^{51} + 2 q^{55} - 8 q^{59} + 28 q^{61} - 24 q^{69} + 16 q^{71} + 16 q^{75} + 16 q^{79} - 22 q^{81} + 16 q^{85} + 20 q^{89} + 8 q^{95} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^9 - 2 * q^11 - 8 * q^15 - 8 * q^19 - 8 * q^21 - 6 * q^25 - 4 * q^29 + 16 * q^31 - 8 * q^35 - 12 * q^41 + 2 * q^45 + 6 * q^49 + 16 * q^51 + 2 * q^55 - 8 * q^59 + 28 * q^61 - 24 * q^69 + 16 * q^71 + 16 * q^75 + 16 * q^79 - 22 * q^81 + 16 * q^85 + 20 * q^89 + 8 * q^95 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$177$$ $$221$$ $$321$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 − 1.00000i 1.00000i
0 2.00000i 0 −1.00000 2.00000i 0 2.00000i 0 −1.00000 0
89.2 0 2.00000i 0 −1.00000 + 2.00000i 0 2.00000i 0 −1.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.b.a 2
3.b odd 2 1 3960.2.d.c 2
4.b odd 2 1 880.2.b.b 2
5.b even 2 1 inner 440.2.b.a 2
5.c odd 4 1 2200.2.a.c 1
5.c odd 4 1 2200.2.a.i 1
15.d odd 2 1 3960.2.d.c 2
20.d odd 2 1 880.2.b.b 2
20.e even 4 1 4400.2.a.h 1
20.e even 4 1 4400.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.a 2 1.a even 1 1 trivial
440.2.b.a 2 5.b even 2 1 inner
880.2.b.b 2 4.b odd 2 1
880.2.b.b 2 20.d odd 2 1
2200.2.a.c 1 5.c odd 4 1
2200.2.a.i 1 5.c odd 4 1
3960.2.d.c 2 3.b odd 2 1
3960.2.d.c 2 15.d odd 2 1
4400.2.a.h 1 20.e even 4 1
4400.2.a.x 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(440, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7}^{2} + 4$$ T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 16$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 144$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T - 14)^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 4$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 64$$