# Properties

 Label 2200.2.a.p Level $2200$ Weight $2$ Character orbit 2200.a Self dual yes Analytic conductor $17.567$ Analytic rank $1$ 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] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2200.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: $$17.5670884447$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + (\beta + 1) q^{7} + (\beta - 2) q^{9} +O(q^{10})$$ q - b * q^3 + (b + 1) * q^7 + (b - 2) * q^9 $$q - \beta q^{3} + (\beta + 1) q^{7} + (\beta - 2) q^{9} + q^{11} + (4 \beta - 3) q^{13} + ( - 5 \beta + 2) q^{17} + ( - 2 \beta - 5) q^{19} + ( - 2 \beta - 1) q^{21} + ( - \beta - 1) q^{23} + (4 \beta - 1) q^{27} + ( - 7 \beta + 4) q^{29} + (6 \beta - 5) q^{31} - \beta q^{33} + (2 \beta + 1) q^{37} + ( - \beta - 4) q^{39} + (6 \beta - 7) q^{41} + (8 \beta - 2) q^{43} + ( - 2 \beta + 1) q^{47} + (3 \beta - 5) q^{49} + (3 \beta + 5) q^{51} + (\beta - 3) q^{53} + (7 \beta + 2) q^{57} + (6 \beta - 7) q^{59} + ( - 5 \beta - 5) q^{61} - q^{63} + 12 q^{67} + (2 \beta + 1) q^{69} + ( - 10 \beta + 2) q^{71} + (\beta - 11) q^{73} + (\beta + 1) q^{77} + ( - \beta - 8) q^{79} + ( - 6 \beta + 2) q^{81} + ( - 9 \beta + 8) q^{83} + (3 \beta + 7) q^{87} + (7 \beta - 10) q^{89} + (5 \beta + 1) q^{91} + ( - \beta - 6) q^{93} + ( - \beta - 11) q^{97} + (\beta - 2) q^{99} +O(q^{100})$$ q - b * q^3 + (b + 1) * q^7 + (b - 2) * q^9 + q^11 + (4*b - 3) * q^13 + (-5*b + 2) * q^17 + (-2*b - 5) * q^19 + (-2*b - 1) * q^21 + (-b - 1) * q^23 + (4*b - 1) * q^27 + (-7*b + 4) * q^29 + (6*b - 5) * q^31 - b * q^33 + (2*b + 1) * q^37 + (-b - 4) * q^39 + (6*b - 7) * q^41 + (8*b - 2) * q^43 + (-2*b + 1) * q^47 + (3*b - 5) * q^49 + (3*b + 5) * q^51 + (b - 3) * q^53 + (7*b + 2) * q^57 + (6*b - 7) * q^59 + (-5*b - 5) * q^61 - q^63 + 12 * q^67 + (2*b + 1) * q^69 + (-10*b + 2) * q^71 + (b - 11) * q^73 + (b + 1) * q^77 + (-b - 8) * q^79 + (-6*b + 2) * q^81 + (-9*b + 8) * q^83 + (3*b + 7) * q^87 + (7*b - 10) * q^89 + (5*b + 1) * q^91 + (-b - 6) * q^93 + (-b - 11) * q^97 + (b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 3 q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q - q^3 + 3 * q^7 - 3 * q^9 $$2 q - q^{3} + 3 q^{7} - 3 q^{9} + 2 q^{11} - 2 q^{13} - q^{17} - 12 q^{19} - 4 q^{21} - 3 q^{23} + 2 q^{27} + q^{29} - 4 q^{31} - q^{33} + 4 q^{37} - 9 q^{39} - 8 q^{41} + 4 q^{43} - 7 q^{49} + 13 q^{51} - 5 q^{53} + 11 q^{57} - 8 q^{59} - 15 q^{61} - 2 q^{63} + 24 q^{67} + 4 q^{69} - 6 q^{71} - 21 q^{73} + 3 q^{77} - 17 q^{79} - 2 q^{81} + 7 q^{83} + 17 q^{87} - 13 q^{89} + 7 q^{91} - 13 q^{93} - 23 q^{97} - 3 q^{99}+O(q^{100})$$ 2 * q - q^3 + 3 * q^7 - 3 * q^9 + 2 * q^11 - 2 * q^13 - q^17 - 12 * q^19 - 4 * q^21 - 3 * q^23 + 2 * q^27 + q^29 - 4 * q^31 - q^33 + 4 * q^37 - 9 * q^39 - 8 * q^41 + 4 * q^43 - 7 * q^49 + 13 * q^51 - 5 * q^53 + 11 * q^57 - 8 * q^59 - 15 * q^61 - 2 * q^63 + 24 * q^67 + 4 * q^69 - 6 * q^71 - 21 * q^73 + 3 * q^77 - 17 * q^79 - 2 * q^81 + 7 * q^83 + 17 * q^87 - 13 * q^89 + 7 * q^91 - 13 * q^93 - 23 * 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
 1.61803 −0.618034
0 −1.61803 0 0 0 2.61803 0 −0.381966 0
1.2 0 0.618034 0 0 0 0.381966 0 −2.61803 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 2200.2.a.p 2
4.b odd 2 1 4400.2.a.bo 2
5.b even 2 1 2200.2.a.q yes 2
5.c odd 4 2 2200.2.b.k 4
20.d odd 2 1 4400.2.a.bm 2
20.e even 4 2 4400.2.b.z 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.p 2 1.a even 1 1 trivial
2200.2.a.q yes 2 5.b even 2 1
2200.2.b.k 4 5.c odd 4 2
4400.2.a.bm 2 20.d odd 2 1
4400.2.a.bo 2 4.b odd 2 1
4400.2.b.z 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2200))$$:

 $$T_{3}^{2} + T_{3} - 1$$ T3^2 + T3 - 1 $$T_{7}^{2} - 3T_{7} + 1$$ T7^2 - 3*T7 + 1 $$T_{13}^{2} + 2T_{13} - 19$$ T13^2 + 2*T13 - 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 3T + 1$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 2T - 19$$
$17$ $$T^{2} + T - 31$$
$19$ $$T^{2} + 12T + 31$$
$23$ $$T^{2} + 3T + 1$$
$29$ $$T^{2} - T - 61$$
$31$ $$T^{2} + 4T - 41$$
$37$ $$T^{2} - 4T - 1$$
$41$ $$T^{2} + 8T - 29$$
$43$ $$T^{2} - 4T - 76$$
$47$ $$T^{2} - 5$$
$53$ $$T^{2} + 5T + 5$$
$59$ $$T^{2} + 8T - 29$$
$61$ $$T^{2} + 15T + 25$$
$67$ $$(T - 12)^{2}$$
$71$ $$T^{2} + 6T - 116$$
$73$ $$T^{2} + 21T + 109$$
$79$ $$T^{2} + 17T + 71$$
$83$ $$T^{2} - 7T - 89$$
$89$ $$T^{2} + 13T - 19$$
$97$ $$T^{2} + 23T + 131$$