# Properties

 Label 2200.2.a.n 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 $$q - \beta q^{3} + ( - \beta + 1) q^{7} + (\beta - 2) q^{9} - q^{11} + (2 \beta - 1) q^{13} + 3 \beta q^{17} + ( - 4 \beta + 1) q^{19} + q^{21} + (7 \beta - 5) q^{23} + (4 \beta - 1) q^{27} + (\beta - 6) q^{29} + ( - 2 \beta - 1) q^{31} + \beta q^{33} + ( - 2 \beta + 5) q^{37} + ( - \beta - 2) q^{39} + (8 \beta - 5) q^{41} + ( - 4 \beta - 2) q^{43} + ( - 6 \beta + 5) q^{47} + ( - \beta - 5) q^{49} + ( - 3 \beta - 3) q^{51} + ( - 3 \beta + 1) q^{53} + (3 \beta + 4) q^{57} + (2 \beta - 7) q^{59} + (5 \beta - 1) q^{61} + (2 \beta - 3) q^{63} + ( - 2 \beta - 7) q^{69} + (6 \beta - 10) q^{71} + ( - 5 \beta + 1) q^{73} + (\beta - 1) q^{77} + (3 \beta - 2) q^{79} + ( - 6 \beta + 2) q^{81} + (3 \beta - 6) q^{83} + (5 \beta - 1) q^{87} + ( - 9 \beta + 2) q^{89} + (\beta - 3) q^{91} + (3 \beta + 2) q^{93} + ( - 5 \beta + 13) q^{97} + ( - \beta + 2) q^{99} +O(q^{100})$$ q - b * q^3 + (-b + 1) * q^7 + (b - 2) * q^9 - q^11 + (2*b - 1) * q^13 + 3*b * q^17 + (-4*b + 1) * q^19 + q^21 + (7*b - 5) * q^23 + (4*b - 1) * q^27 + (b - 6) * q^29 + (-2*b - 1) * q^31 + b * q^33 + (-2*b + 5) * q^37 + (-b - 2) * q^39 + (8*b - 5) * q^41 + (-4*b - 2) * q^43 + (-6*b + 5) * q^47 + (-b - 5) * q^49 + (-3*b - 3) * q^51 + (-3*b + 1) * q^53 + (3*b + 4) * q^57 + (2*b - 7) * q^59 + (5*b - 1) * q^61 + (2*b - 3) * q^63 + (-2*b - 7) * q^69 + (6*b - 10) * q^71 + (-5*b + 1) * q^73 + (b - 1) * q^77 + (3*b - 2) * q^79 + (-6*b + 2) * q^81 + (3*b - 6) * q^83 + (5*b - 1) * q^87 + (-9*b + 2) * q^89 + (b - 3) * q^91 + (3*b + 2) * q^93 + (-5*b + 13) * q^97 + (-b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q - q^3 + q^7 - 3 * q^9 $$2 q - q^{3} + q^{7} - 3 q^{9} - 2 q^{11} + 3 q^{17} - 2 q^{19} + 2 q^{21} - 3 q^{23} + 2 q^{27} - 11 q^{29} - 4 q^{31} + q^{33} + 8 q^{37} - 5 q^{39} - 2 q^{41} - 8 q^{43} + 4 q^{47} - 11 q^{49} - 9 q^{51} - q^{53} + 11 q^{57} - 12 q^{59} + 3 q^{61} - 4 q^{63} - 16 q^{69} - 14 q^{71} - 3 q^{73} - q^{77} - q^{79} - 2 q^{81} - 9 q^{83} + 3 q^{87} - 5 q^{89} - 5 q^{91} + 7 q^{93} + 21 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q - q^3 + q^7 - 3 * q^9 - 2 * q^11 + 3 * q^17 - 2 * q^19 + 2 * q^21 - 3 * q^23 + 2 * q^27 - 11 * q^29 - 4 * q^31 + q^33 + 8 * q^37 - 5 * q^39 - 2 * q^41 - 8 * q^43 + 4 * q^47 - 11 * q^49 - 9 * q^51 - q^53 + 11 * q^57 - 12 * q^59 + 3 * q^61 - 4 * q^63 - 16 * q^69 - 14 * q^71 - 3 * q^73 - q^77 - q^79 - 2 * q^81 - 9 * q^83 + 3 * q^87 - 5 * q^89 - 5 * q^91 + 7 * q^93 + 21 * 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 −0.618034 0 −0.381966 0
1.2 0 0.618034 0 0 0 1.61803 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.n 2
4.b odd 2 1 4400.2.a.bq 2
5.b even 2 1 2200.2.a.r yes 2
5.c odd 4 2 2200.2.b.j 4
20.d odd 2 1 4400.2.a.bk 2
20.e even 4 2 4400.2.b.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.n 2 1.a even 1 1 trivial
2200.2.a.r yes 2 5.b even 2 1
2200.2.b.j 4 5.c odd 4 2
4400.2.a.bk 2 20.d odd 2 1
4400.2.a.bq 2 4.b odd 2 1
4400.2.b.ba 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 $$T_{7}^{2} - T_{7} - 1$$ T7^2 - T7 - 1 $$T_{13}^{2} - 5$$ T13^2 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T - 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 5$$
$17$ $$T^{2} - 3T - 9$$
$19$ $$T^{2} + 2T - 19$$
$23$ $$T^{2} + 3T - 59$$
$29$ $$T^{2} + 11T + 29$$
$31$ $$T^{2} + 4T - 1$$
$37$ $$T^{2} - 8T + 11$$
$41$ $$T^{2} + 2T - 79$$
$43$ $$T^{2} + 8T - 4$$
$47$ $$T^{2} - 4T - 41$$
$53$ $$T^{2} + T - 11$$
$59$ $$T^{2} + 12T + 31$$
$61$ $$T^{2} - 3T - 29$$
$67$ $$T^{2}$$
$71$ $$T^{2} + 14T + 4$$
$73$ $$T^{2} + 3T - 29$$
$79$ $$T^{2} + T - 11$$
$83$ $$T^{2} + 9T + 9$$
$89$ $$T^{2} + 5T - 95$$
$97$ $$T^{2} - 21T + 79$$