# Properties

 Label 2200.2.a.y Level $2200$ Weight $2$ Character orbit 2200.a Self dual yes Analytic conductor $17.567$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.54764.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 9x^{2} + 3x + 2$$ x^4 - x^3 - 9*x^2 + 3*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + \beta_{3} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + b3 * q^7 + (b3 + b2 + b1 + 1) * q^9 $$q + \beta_1 q^{3} + \beta_{3} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} + q^{11} + 4 q^{13} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{17} + (\beta_{3} - 2 \beta_{2} - 2) q^{19} + ( - \beta_{3} + 2 \beta_1 - 2) q^{21} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{23} + (2 \beta_{2} + 3 \beta_1 + 2) q^{27} + (\beta_{3} + 2 \beta_1) q^{29} + ( - \beta_{3} + \beta_{2} + \beta_1 - 4) q^{31} + \beta_1 q^{33} + ( - 2 \beta_{2} - \beta_1 + 2) q^{37} + 4 \beta_1 q^{39} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{43} + ( - 2 \beta_{2} - 2) q^{47} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 3) q^{49} + ( - \beta_{3} + 2 \beta_1 - 6) q^{51} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{53} + ( - \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{57} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{59} + ( - \beta_{3} - 2 \beta_1 + 8) q^{61} + (2 \beta_{2} - 2 \beta_1 + 10) q^{63} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{67} + ( - 2 \beta_{3} - 3 \beta_{2} + \cdots - 6) q^{69}+ \cdots + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100})$$ q + b1 * q^3 + b3 * q^7 + (b3 + b2 + b1 + 1) * q^9 + q^11 + 4 * q^13 + (-b3 + 2*b2 - 2*b1 + 2) * q^17 + (b3 - 2*b2 - 2) * q^19 + (-b3 + 2*b1 - 2) * q^21 + (b3 - 2*b2 - b1 + 2) * q^23 + (2*b2 + 3*b1 + 2) * q^27 + (b3 + 2*b1) * q^29 + (-b3 + b2 + b1 - 4) * q^31 + b1 * q^33 + (-2*b2 - b1 + 2) * q^37 + 4*b1 * q^39 + (2*b3 - 2*b2 + 2*b1 + 2) * q^41 + (-2*b3 + 2*b2 + 2) * q^43 + (-2*b2 - 2) * q^47 + (-b3 + 2*b2 - 4*b1 + 3) * q^49 + (-b3 + 2*b1 - 6) * q^51 + (-b3 - 2*b2 + 2) * q^53 + (-b3 - 2*b2 - 4*b1 - 2) * q^57 + (-2*b3 + b2 - b1 + 2) * q^59 + (-b3 - 2*b1 + 8) * q^61 + (2*b2 - 2*b1 + 10) * q^63 + (-b3 - 2*b2 + b1 + 2) * q^67 + (-2*b3 - 3*b2 - b1 - 6) * q^69 + (b3 + b2 - 3*b1) * q^71 + (2*b3 + 4) * q^73 + b3 * q^77 + (-4*b3 + 2*b2 - 2*b1 - 4) * q^79 + (2*b2 + 6*b1 + 9) * q^81 + (-2*b2 + 4*b1 + 6) * q^83 + (b3 + 2*b2 + 4*b1 + 6) * q^87 + (-b3 - b2 - b1 + 6) * q^89 + 4*b3 * q^91 + (2*b3 + 2*b2 - 3*b1 + 6) * q^93 + (-3*b3 - b1) * q^97 + (b3 + b2 + b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} + q^{7} + 7 q^{9}+O(q^{10})$$ 4 * q + q^3 + q^7 + 7 * q^9 $$4 q + q^{3} + q^{7} + 7 q^{9} + 4 q^{11} + 16 q^{13} + 7 q^{17} - 9 q^{19} - 7 q^{21} + 6 q^{23} + 13 q^{27} + 3 q^{29} - 15 q^{31} + q^{33} + 5 q^{37} + 4 q^{39} + 10 q^{41} + 8 q^{43} - 10 q^{47} + 9 q^{49} - 23 q^{51} + 5 q^{53} - 15 q^{57} + 6 q^{59} + 29 q^{61} + 40 q^{63} + 6 q^{67} - 30 q^{69} - q^{71} + 18 q^{73} + q^{77} - 20 q^{79} + 44 q^{81} + 26 q^{83} + 31 q^{87} + 21 q^{89} + 4 q^{91} + 25 q^{93} - 4 q^{97} + 7 q^{99}+O(q^{100})$$ 4 * q + q^3 + q^7 + 7 * q^9 + 4 * q^11 + 16 * q^13 + 7 * q^17 - 9 * q^19 - 7 * q^21 + 6 * q^23 + 13 * q^27 + 3 * q^29 - 15 * q^31 + q^33 + 5 * q^37 + 4 * q^39 + 10 * q^41 + 8 * q^43 - 10 * q^47 + 9 * q^49 - 23 * q^51 + 5 * q^53 - 15 * q^57 + 6 * q^59 + 29 * q^61 + 40 * q^63 + 6 * q^67 - 30 * q^69 - q^71 + 18 * q^73 + q^77 - 20 * q^79 + 44 * q^81 + 26 * q^83 + 31 * q^87 + 21 * q^89 + 4 * q^91 + 25 * q^93 - 4 * q^97 + 7 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 9x^{2} + 3x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 9\nu - 2 ) / 2$$ (v^3 - 9*v - 2) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} + 7\nu - 6 ) / 2$$ (-v^3 + 2*v^2 + 7*v - 6) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 + 4$$ b3 + b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$2\beta_{2} + 9\beta _1 + 2$$ 2*b2 + 9*b1 + 2

## 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
 −2.67673 −0.339102 0.655762 3.36007
0 −2.67673 0 0 0 4.38559 0 4.16490 0
1.2 0 −0.339102 0 0 0 −4.05237 0 −2.88501 0
1.3 0 0.655762 0 0 0 −0.415806 0 −2.56998 0
1.4 0 3.36007 0 0 0 1.08258 0 8.29009 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.y 4
4.b odd 2 1 4400.2.a.cb 4
5.b even 2 1 2200.2.a.x 4
5.c odd 4 2 440.2.b.d 8
15.e even 4 2 3960.2.d.f 8
20.d odd 2 1 4400.2.a.ce 4
20.e even 4 2 880.2.b.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.d 8 5.c odd 4 2
880.2.b.j 8 20.e even 4 2
2200.2.a.x 4 5.b even 2 1
2200.2.a.y 4 1.a even 1 1 trivial
3960.2.d.f 8 15.e even 4 2
4400.2.a.cb 4 4.b odd 2 1
4400.2.a.ce 4 20.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{4} - T_{3}^{3} - 9T_{3}^{2} + 3T_{3} + 2$$ T3^4 - T3^3 - 9*T3^2 + 3*T3 + 2 $$T_{7}^{4} - T_{7}^{3} - 18T_{7}^{2} + 12T_{7} + 8$$ T7^4 - T7^3 - 18*T7^2 + 12*T7 + 8 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{3} - 9 T^{2} + \cdots + 2$$
$5$ $$T^{4}$$
$7$ $$T^{4} - T^{3} - 18 T^{2} + \cdots + 8$$
$11$ $$(T - 1)^{4}$$
$13$ $$(T - 4)^{4}$$
$17$ $$T^{4} - 7 T^{3} + \cdots + 32$$
$19$ $$T^{4} + 9 T^{3} + \cdots - 128$$
$23$ $$T^{4} - 6 T^{3} + \cdots + 856$$
$29$ $$T^{4} - 3 T^{3} + \cdots + 32$$
$31$ $$T^{4} + 15 T^{3} + \cdots + 16$$
$37$ $$T^{4} - 5 T^{3} + \cdots - 148$$
$41$ $$T^{4} - 10 T^{3} + \cdots - 1024$$
$43$ $$T^{4} - 8 T^{3} + \cdots + 1136$$
$47$ $$T^{4} + 10 T^{3} + \cdots - 640$$
$53$ $$T^{4} - 5 T^{3} + \cdots + 1280$$
$59$ $$T^{4} - 6 T^{3} + \cdots - 32$$
$61$ $$T^{4} - 29 T^{3} + \cdots + 160$$
$67$ $$T^{4} - 6 T^{3} + \cdots + 568$$
$71$ $$T^{4} + T^{3} + \cdots - 1336$$
$73$ $$T^{4} - 18 T^{3} + \cdots - 1024$$
$79$ $$T^{4} + 20 T^{3} + \cdots - 19456$$
$83$ $$T^{4} - 26 T^{3} + \cdots - 6176$$
$89$ $$T^{4} - 21 T^{3} + \cdots - 346$$
$97$ $$T^{4} + 4 T^{3} + \cdots + 512$$