# Properties

 Label 4400.2.a.by Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(1,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, 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("4400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.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: $$35.1341768894$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x + 6$$ x^3 - x^2 - 7*x + 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2200) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q - b1 * q^3 + (-b2 - 1) * q^7 + (b2 + 2) * q^9 $$q - \beta_1 q^{3} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} + 2) q^{9} - q^{11} + q^{13} + (\beta_1 - 2) q^{17} + (2 \beta_1 - 3) q^{19} + (\beta_{2} + 3 \beta_1 - 1) q^{21} + ( - \beta_1 - 3) q^{23} + ( - \beta_{2} - \beta_1 + 1) q^{27} + ( - \beta_{2} - 2 \beta_1 + 4) q^{29} + (2 \beta_{2} + 1) q^{31} + \beta_1 q^{33} + ( - \beta_{2} + \beta_1 - 3) q^{37} - \beta_1 q^{39} + ( - \beta_{2} + \beta_1 + 1) q^{41} + (\beta_{2} - 3 \beta_1 - 2) q^{43} + (\beta_{2} + 3 \beta_1 - 1) q^{47} + (\beta_1 + 3) q^{49} + ( - \beta_{2} + 2 \beta_1 - 5) q^{51} + (3 \beta_{2} + 2 \beta_1 + 1) q^{53} + ( - 2 \beta_{2} + 3 \beta_1 - 10) q^{57} + (\beta_{2} + 3 \beta_1 - 1) q^{59} + (\beta_{2} + 2 \beta_1 - 1) q^{61} + ( - \beta_{2} - \beta_1 - 11) q^{63} + (2 \beta_{2} + 2 \beta_1 + 4) q^{67} + (\beta_{2} + 3 \beta_1 + 5) q^{69} + ( - \beta_{2} + \beta_1 - 6) q^{71} + ( - 3 \beta_{2} - 7) q^{73} + (\beta_{2} + 1) q^{77} + (2 \beta_{2} + \beta_1 - 4) q^{79} + ( - \beta_{2} + \beta_1 - 2) q^{81} + (\beta_{2} + 4 \beta_1 - 8) q^{83} + (3 \beta_{2} - 2 \beta_1 + 9) q^{87} + ( - \beta_{2} + 10) q^{89} + ( - \beta_{2} - 1) q^{91} + ( - 2 \beta_{2} - 5 \beta_1 + 2) q^{93} + (2 \beta_{2} - \beta_1 - 3) q^{97} + ( - \beta_{2} - 2) q^{99}+O(q^{100})$$ q - b1 * q^3 + (-b2 - 1) * q^7 + (b2 + 2) * q^9 - q^11 + q^13 + (b1 - 2) * q^17 + (2*b1 - 3) * q^19 + (b2 + 3*b1 - 1) * q^21 + (-b1 - 3) * q^23 + (-b2 - b1 + 1) * q^27 + (-b2 - 2*b1 + 4) * q^29 + (2*b2 + 1) * q^31 + b1 * q^33 + (-b2 + b1 - 3) * q^37 - b1 * q^39 + (-b2 + b1 + 1) * q^41 + (b2 - 3*b1 - 2) * q^43 + (b2 + 3*b1 - 1) * q^47 + (b1 + 3) * q^49 + (-b2 + 2*b1 - 5) * q^51 + (3*b2 + 2*b1 + 1) * q^53 + (-2*b2 + 3*b1 - 10) * q^57 + (b2 + 3*b1 - 1) * q^59 + (b2 + 2*b1 - 1) * q^61 + (-b2 - b1 - 11) * q^63 + (2*b2 + 2*b1 + 4) * q^67 + (b2 + 3*b1 + 5) * q^69 + (-b2 + b1 - 6) * q^71 + (-3*b2 - 7) * q^73 + (b2 + 1) * q^77 + (2*b2 + b1 - 4) * q^79 + (-b2 + b1 - 2) * q^81 + (b2 + 4*b1 - 8) * q^83 + (3*b2 - 2*b1 + 9) * q^87 + (-b2 + 10) * q^89 + (-b2 - 1) * q^91 + (-2*b2 - 5*b1 + 2) * q^93 + (2*b2 - b1 - 3) * q^97 + (-b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{7} + 6 q^{9}+O(q^{10})$$ 3 * q - q^3 - 3 * q^7 + 6 * q^9 $$3 q - q^{3} - 3 q^{7} + 6 q^{9} - 3 q^{11} + 3 q^{13} - 5 q^{17} - 7 q^{19} - 10 q^{23} + 2 q^{27} + 10 q^{29} + 3 q^{31} + q^{33} - 8 q^{37} - q^{39} + 4 q^{41} - 9 q^{43} + 10 q^{49} - 13 q^{51} + 5 q^{53} - 27 q^{57} - q^{61} - 34 q^{63} + 14 q^{67} + 18 q^{69} - 17 q^{71} - 21 q^{73} + 3 q^{77} - 11 q^{79} - 5 q^{81} - 20 q^{83} + 25 q^{87} + 30 q^{89} - 3 q^{91} + q^{93} - 10 q^{97} - 6 q^{99}+O(q^{100})$$ 3 * q - q^3 - 3 * q^7 + 6 * q^9 - 3 * q^11 + 3 * q^13 - 5 * q^17 - 7 * q^19 - 10 * q^23 + 2 * q^27 + 10 * q^29 + 3 * q^31 + q^33 - 8 * q^37 - q^39 + 4 * q^41 - 9 * q^43 + 10 * q^49 - 13 * q^51 + 5 * q^53 - 27 * q^57 - q^61 - 34 * q^63 + 14 * q^67 + 18 * q^69 - 17 * q^71 - 21 * q^73 + 3 * q^77 - 11 * q^79 - 5 * q^81 - 20 * q^83 + 25 * q^87 + 30 * q^89 - 3 * q^91 + q^93 - 10 * q^97 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7x + 6$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ b2 + 5

## 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
 2.75153 0.841083 −2.59261
0 −2.75153 0 0 0 −3.57093 0 4.57093 0
1.2 0 −0.841083 0 0 0 3.29258 0 −2.29258 0
1.3 0 2.59261 0 0 0 −2.72165 0 3.72165 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 4400.2.a.by 3
4.b odd 2 1 2200.2.a.v yes 3
5.b even 2 1 4400.2.a.bz 3
5.c odd 4 2 4400.2.b.bb 6
20.d odd 2 1 2200.2.a.u 3
20.e even 4 2 2200.2.b.m 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.u 3 20.d odd 2 1
2200.2.a.v yes 3 4.b odd 2 1
2200.2.b.m 6 20.e even 4 2
4400.2.a.by 3 1.a even 1 1 trivial
4400.2.a.bz 3 5.b even 2 1
4400.2.b.bb 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

 $$T_{3}^{3} + T_{3}^{2} - 7T_{3} - 6$$ T3^3 + T3^2 - 7*T3 - 6 $$T_{7}^{3} + 3T_{7}^{2} - 11T_{7} - 32$$ T7^3 + 3*T7^2 - 11*T7 - 32 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 7T - 6$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 3 T^{2} + \cdots - 32$$
$11$ $$(T + 1)^{3}$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} + 5T^{2} + T - 4$$
$19$ $$T^{3} + 7 T^{2} + \cdots - 27$$
$23$ $$T^{3} + 10 T^{2} + \cdots + 9$$
$29$ $$T^{3} - 10 T^{2} + \cdots + 201$$
$31$ $$T^{3} - 3 T^{2} + \cdots + 207$$
$37$ $$T^{3} + 8T^{2} - T - 44$$
$41$ $$T^{3} - 4 T^{2} + \cdots + 24$$
$43$ $$T^{3} + 9 T^{2} + \cdots - 508$$
$47$ $$T^{3} - 77T - 192$$
$53$ $$T^{3} - 5 T^{2} + \cdots + 142$$
$59$ $$T^{3} - 77T - 192$$
$61$ $$T^{3} + T^{2} + \cdots - 114$$
$67$ $$T^{3} - 14 T^{2} + \cdots + 96$$
$71$ $$T^{3} + 17 T^{2} + \cdots + 52$$
$73$ $$T^{3} + 21 T^{2} + \cdots - 1052$$
$79$ $$T^{3} + 11 T^{2} + \cdots - 144$$
$83$ $$T^{3} + 20 T^{2} + \cdots - 829$$
$89$ $$T^{3} - 30 T^{2} + \cdots - 879$$
$97$ $$T^{3} + 10 T^{2} + \cdots - 23$$