# Properties

 Label 440.2.b.c 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: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + (i + 2) q^{5} + i q^{7} + 2 q^{9}+O(q^{10})$$ q + i * q^3 + (i + 2) * q^5 + i * q^7 + 2 * q^9 $$q + i q^{3} + (i + 2) q^{5} + i q^{7} + 2 q^{9} - q^{11} + (2 i - 1) q^{15} + i q^{17} - q^{19} - q^{21} + (4 i + 3) q^{25} + 5 i q^{27} + q^{29} - q^{31} - i q^{33} + (2 i - 1) q^{35} + i q^{37} - 6 i q^{43} + (2 i + 4) q^{45} + 8 i q^{47} + 6 q^{49} - q^{51} - 9 i q^{53} + ( - i - 2) q^{55} - i q^{57} - 4 q^{59} - 7 q^{61} + 2 i q^{63} - 4 i q^{67} + 5 q^{71} - 14 i q^{73} + (3 i - 4) q^{75} - i q^{77} - 4 q^{79} + q^{81} - 16 i q^{83} + (2 i - 1) q^{85} + i q^{87} + 7 q^{89} - i q^{93} + ( - i - 2) q^{95} - 16 i q^{97} - 2 q^{99} +O(q^{100})$$ q + i * q^3 + (i + 2) * q^5 + i * q^7 + 2 * q^9 - q^11 + (2*i - 1) * q^15 + i * q^17 - q^19 - q^21 + (4*i + 3) * q^25 + 5*i * q^27 + q^29 - q^31 - i * q^33 + (2*i - 1) * q^35 + i * q^37 - 6*i * q^43 + (2*i + 4) * q^45 + 8*i * q^47 + 6 * q^49 - q^51 - 9*i * q^53 + (-i - 2) * q^55 - i * q^57 - 4 * q^59 - 7 * q^61 + 2*i * q^63 - 4*i * q^67 + 5 * q^71 - 14*i * q^73 + (3*i - 4) * q^75 - i * q^77 - 4 * q^79 + q^81 - 16*i * q^83 + (2*i - 1) * q^85 + i * q^87 + 7 * q^89 - i * q^93 + (-i - 2) * q^95 - 16*i * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^5 + 4 * q^9 $$2 q + 4 q^{5} + 4 q^{9} - 2 q^{11} - 2 q^{15} - 2 q^{19} - 2 q^{21} + 6 q^{25} + 2 q^{29} - 2 q^{31} - 2 q^{35} + 8 q^{45} + 12 q^{49} - 2 q^{51} - 4 q^{55} - 8 q^{59} - 14 q^{61} + 10 q^{71} - 8 q^{75} - 8 q^{79} + 2 q^{81} - 2 q^{85} + 14 q^{89} - 4 q^{95} - 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^5 + 4 * q^9 - 2 * q^11 - 2 * q^15 - 2 * q^19 - 2 * q^21 + 6 * q^25 + 2 * q^29 - 2 * q^31 - 2 * q^35 + 8 * q^45 + 12 * q^49 - 2 * q^51 - 4 * q^55 - 8 * q^59 - 14 * q^61 + 10 * q^71 - 8 * q^75 - 8 * q^79 + 2 * q^81 - 2 * q^85 + 14 * q^89 - 4 * q^95 - 4 * 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 1.00000i 0 2.00000 1.00000i 0 1.00000i 0 2.00000 0
89.2 0 1.00000i 0 2.00000 + 1.00000i 0 1.00000i 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
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.c 2
3.b odd 2 1 3960.2.d.a 2
4.b odd 2 1 880.2.b.g 2
5.b even 2 1 inner 440.2.b.c 2
5.c odd 4 1 2200.2.a.d 1
5.c odd 4 1 2200.2.a.h 1
15.d odd 2 1 3960.2.d.a 2
20.d odd 2 1 880.2.b.g 2
20.e even 4 1 4400.2.a.j 1
20.e even 4 1 4400.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.c 2 1.a even 1 1 trivial
440.2.b.c 2 5.b even 2 1 inner
880.2.b.g 2 4.b odd 2 1
880.2.b.g 2 20.d odd 2 1
2200.2.a.d 1 5.c odd 4 1
2200.2.a.h 1 5.c odd 4 1
3960.2.d.a 2 3.b odd 2 1
3960.2.d.a 2 15.d odd 2 1
4400.2.a.j 1 20.e even 4 1
4400.2.a.u 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 $$S_{2}^{\mathrm{new}}(440, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 1$$ T7^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 1$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 81$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T + 7)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T - 5)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 256$$
$89$ $$(T - 7)^{2}$$
$97$ $$T^{2} + 256$$