# Properties

 Label 440.2.a.f Level $440$ Weight $2$ Character orbit 440.a Self dual yes Analytic conductor $3.513$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(1,440)]

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, 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("440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$440 = 2^{3} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 440.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: $$3.51341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - q^{5} + ( - \beta + 2) q^{7} + (\beta + 1) q^{9}+O(q^{10})$$ q + b * q^3 - q^5 + (-b + 2) * q^7 + (b + 1) * q^9 $$q + \beta q^{3} - q^{5} + ( - \beta + 2) q^{7} + (\beta + 1) q^{9} + q^{11} + 2 \beta q^{13} - \beta q^{15} + ( - \beta + 4) q^{17} + (\beta + 4) q^{19} + (\beta - 4) q^{21} + ( - 2 \beta + 4) q^{23} + q^{25} + ( - \beta + 4) q^{27} + ( - \beta - 2) q^{29} + ( - 3 \beta + 4) q^{31} + \beta q^{33} + (\beta - 2) q^{35} + ( - 5 \beta + 2) q^{37} + (2 \beta + 8) q^{39} - 10 q^{41} + ( - 2 \beta + 2) q^{43} + ( - \beta - 1) q^{45} + (2 \beta - 4) q^{47} + ( - 3 \beta + 1) q^{49} + (3 \beta - 4) q^{51} + ( - \beta - 6) q^{53} - q^{55} + (5 \beta + 4) q^{57} + (6 \beta - 4) q^{59} + (\beta - 2) q^{61} - 2 q^{63} - 2 \beta q^{65} + (2 \beta - 8) q^{69} + ( - \beta - 4) q^{71} + (2 \beta + 8) q^{73} + \beta q^{75} + ( - \beta + 2) q^{77} + (2 \beta + 4) q^{79} - 7 q^{81} + 10 q^{83} + (\beta - 4) q^{85} + ( - 3 \beta - 4) q^{87} + ( - 5 \beta + 2) q^{89} + (2 \beta - 8) q^{91} + (\beta - 12) q^{93} + ( - \beta - 4) q^{95} + ( - 2 \beta + 14) q^{97} + (\beta + 1) q^{99} +O(q^{100})$$ q + b * q^3 - q^5 + (-b + 2) * q^7 + (b + 1) * q^9 + q^11 + 2*b * q^13 - b * q^15 + (-b + 4) * q^17 + (b + 4) * q^19 + (b - 4) * q^21 + (-2*b + 4) * q^23 + q^25 + (-b + 4) * q^27 + (-b - 2) * q^29 + (-3*b + 4) * q^31 + b * q^33 + (b - 2) * q^35 + (-5*b + 2) * q^37 + (2*b + 8) * q^39 - 10 * q^41 + (-2*b + 2) * q^43 + (-b - 1) * q^45 + (2*b - 4) * q^47 + (-3*b + 1) * q^49 + (3*b - 4) * q^51 + (-b - 6) * q^53 - q^55 + (5*b + 4) * q^57 + (6*b - 4) * q^59 + (b - 2) * q^61 - 2 * q^63 - 2*b * q^65 + (2*b - 8) * q^69 + (-b - 4) * q^71 + (2*b + 8) * q^73 + b * q^75 + (-b + 2) * q^77 + (2*b + 4) * q^79 - 7 * q^81 + 10 * q^83 + (b - 4) * q^85 + (-3*b - 4) * q^87 + (-5*b + 2) * q^89 + (2*b - 8) * q^91 + (b - 12) * q^93 + (-b - 4) * q^95 + (-2*b + 14) * q^97 + (b + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9 $$2 q + q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - q^{15} + 7 q^{17} + 9 q^{19} - 7 q^{21} + 6 q^{23} + 2 q^{25} + 7 q^{27} - 5 q^{29} + 5 q^{31} + q^{33} - 3 q^{35} - q^{37} + 18 q^{39} - 20 q^{41} + 2 q^{43} - 3 q^{45} - 6 q^{47} - q^{49} - 5 q^{51} - 13 q^{53} - 2 q^{55} + 13 q^{57} - 2 q^{59} - 3 q^{61} - 4 q^{63} - 2 q^{65} - 14 q^{69} - 9 q^{71} + 18 q^{73} + q^{75} + 3 q^{77} + 10 q^{79} - 14 q^{81} + 20 q^{83} - 7 q^{85} - 11 q^{87} - q^{89} - 14 q^{91} - 23 q^{93} - 9 q^{95} + 26 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q + q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - q^15 + 7 * q^17 + 9 * q^19 - 7 * q^21 + 6 * q^23 + 2 * q^25 + 7 * q^27 - 5 * q^29 + 5 * q^31 + q^33 - 3 * q^35 - q^37 + 18 * q^39 - 20 * q^41 + 2 * q^43 - 3 * q^45 - 6 * q^47 - q^49 - 5 * q^51 - 13 * q^53 - 2 * q^55 + 13 * q^57 - 2 * q^59 - 3 * q^61 - 4 * q^63 - 2 * q^65 - 14 * q^69 - 9 * q^71 + 18 * q^73 + q^75 + 3 * q^77 + 10 * q^79 - 14 * q^81 + 20 * q^83 - 7 * q^85 - 11 * q^87 - q^89 - 14 * q^91 - 23 * q^93 - 9 * q^95 + 26 * q^97 + 3 * 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
 −1.56155 2.56155
0 −1.56155 0 −1.00000 0 3.56155 0 −0.561553 0
1.2 0 2.56155 0 −1.00000 0 −0.561553 0 3.56155 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
$$2$$ $$1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 440.2.a.f 2
3.b odd 2 1 3960.2.a.be 2
4.b odd 2 1 880.2.a.l 2
5.b even 2 1 2200.2.a.m 2
5.c odd 4 2 2200.2.b.h 4
8.b even 2 1 3520.2.a.bl 2
8.d odd 2 1 3520.2.a.bs 2
11.b odd 2 1 4840.2.a.n 2
12.b even 2 1 7920.2.a.ca 2
20.d odd 2 1 4400.2.a.br 2
20.e even 4 2 4400.2.b.u 4
44.c even 2 1 9680.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.f 2 1.a even 1 1 trivial
880.2.a.l 2 4.b odd 2 1
2200.2.a.m 2 5.b even 2 1
2200.2.b.h 4 5.c odd 4 2
3520.2.a.bl 2 8.b even 2 1
3520.2.a.bs 2 8.d odd 2 1
3960.2.a.be 2 3.b odd 2 1
4400.2.a.br 2 20.d odd 2 1
4400.2.b.u 4 20.e even 4 2
4840.2.a.n 2 11.b odd 2 1
7920.2.a.ca 2 12.b even 2 1
9680.2.a.bl 2 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(440))$$:

 $$T_{3}^{2} - T_{3} - 4$$ T3^2 - T3 - 4 $$T_{7}^{2} - 3T_{7} - 2$$ T7^2 - 3*T7 - 2 $$T_{13}^{2} - 2T_{13} - 16$$ T13^2 - 2*T13 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 4$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 3T - 2$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 2T - 16$$
$17$ $$T^{2} - 7T + 8$$
$19$ $$T^{2} - 9T + 16$$
$23$ $$T^{2} - 6T - 8$$
$29$ $$T^{2} + 5T + 2$$
$31$ $$T^{2} - 5T - 32$$
$37$ $$T^{2} + T - 106$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} - 2T - 16$$
$47$ $$T^{2} + 6T - 8$$
$53$ $$T^{2} + 13T + 38$$
$59$ $$T^{2} + 2T - 152$$
$61$ $$T^{2} + 3T - 2$$
$67$ $$T^{2}$$
$71$ $$T^{2} + 9T + 16$$
$73$ $$T^{2} - 18T + 64$$
$79$ $$T^{2} - 10T + 8$$
$83$ $$(T - 10)^{2}$$
$89$ $$T^{2} + T - 106$$
$97$ $$T^{2} - 26T + 152$$