# Properties

 Label 4400.2.a.bt Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ 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] = [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: $$0$$ 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: 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + (\beta + 2) q^{7} + (\beta + 1) q^{9}+O(q^{10})$$ q + b * q^3 + (b + 2) * q^7 + (b + 1) * q^9 $$q + \beta q^{3} + (\beta + 2) q^{7} + (\beta + 1) q^{9} + q^{11} + (2 \beta - 4) q^{13} + 3 \beta q^{17} + (\beta - 4) q^{19} + (3 \beta + 4) q^{21} + (2 \beta - 4) q^{23} + ( - \beta + 4) q^{27} + (\beta + 6) q^{29} + ( - \beta + 4) q^{31} + \beta q^{33} + (\beta - 10) q^{37} + ( - 2 \beta + 8) q^{39} + ( - 4 \beta - 2) q^{41} + ( - 2 \beta + 2) q^{43} + ( - 6 \beta + 4) q^{47} + (5 \beta + 1) q^{49} + (3 \beta + 12) q^{51} + ( - 3 \beta - 2) q^{53} + ( - 3 \beta + 4) q^{57} + ( - 2 \beta + 4) q^{59} + ( - \beta - 10) q^{61} + (4 \beta + 6) q^{63} + ( - 2 \beta + 8) q^{69} + ( - 3 \beta + 4) q^{71} + ( - 2 \beta + 4) q^{73} + (\beta + 2) q^{77} + (6 \beta - 4) q^{79} - 7 q^{81} - 6 q^{83} + (7 \beta + 4) q^{87} + (3 \beta + 2) q^{89} + 2 \beta q^{91} + (3 \beta - 4) q^{93} + ( - 2 \beta + 10) q^{97} + (\beta + 1) q^{99}+O(q^{100})$$ q + b * q^3 + (b + 2) * q^7 + (b + 1) * q^9 + q^11 + (2*b - 4) * q^13 + 3*b * q^17 + (b - 4) * q^19 + (3*b + 4) * q^21 + (2*b - 4) * q^23 + (-b + 4) * q^27 + (b + 6) * q^29 + (-b + 4) * q^31 + b * q^33 + (b - 10) * q^37 + (-2*b + 8) * q^39 + (-4*b - 2) * q^41 + (-2*b + 2) * q^43 + (-6*b + 4) * q^47 + (5*b + 1) * q^49 + (3*b + 12) * q^51 + (-3*b - 2) * q^53 + (-3*b + 4) * q^57 + (-2*b + 4) * q^59 + (-b - 10) * q^61 + (4*b + 6) * q^63 + (-2*b + 8) * q^69 + (-3*b + 4) * q^71 + (-2*b + 4) * q^73 + (b + 2) * q^77 + (6*b - 4) * q^79 - 7 * q^81 - 6 * q^83 + (7*b + 4) * q^87 + (3*b + 2) * q^89 + 2*b * q^91 + (3*b - 4) * q^93 + (-2*b + 10) * q^97 + (b + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 5 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + 5 * q^7 + 3 * q^9 $$2 q + q^{3} + 5 q^{7} + 3 q^{9} + 2 q^{11} - 6 q^{13} + 3 q^{17} - 7 q^{19} + 11 q^{21} - 6 q^{23} + 7 q^{27} + 13 q^{29} + 7 q^{31} + q^{33} - 19 q^{37} + 14 q^{39} - 8 q^{41} + 2 q^{43} + 2 q^{47} + 7 q^{49} + 27 q^{51} - 7 q^{53} + 5 q^{57} + 6 q^{59} - 21 q^{61} + 16 q^{63} + 14 q^{69} + 5 q^{71} + 6 q^{73} + 5 q^{77} - 2 q^{79} - 14 q^{81} - 12 q^{83} + 15 q^{87} + 7 q^{89} + 2 q^{91} - 5 q^{93} + 18 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q + q^3 + 5 * q^7 + 3 * q^9 + 2 * q^11 - 6 * q^13 + 3 * q^17 - 7 * q^19 + 11 * q^21 - 6 * q^23 + 7 * q^27 + 13 * q^29 + 7 * q^31 + q^33 - 19 * q^37 + 14 * q^39 - 8 * q^41 + 2 * q^43 + 2 * q^47 + 7 * q^49 + 27 * q^51 - 7 * q^53 + 5 * q^57 + 6 * q^59 - 21 * q^61 + 16 * q^63 + 14 * q^69 + 5 * q^71 + 6 * q^73 + 5 * q^77 - 2 * q^79 - 14 * q^81 - 12 * q^83 + 15 * q^87 + 7 * q^89 + 2 * q^91 - 5 * q^93 + 18 * 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 0 0 0.438447 0 −0.561553 0
1.2 0 2.56155 0 0 0 4.56155 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 4400.2.a.bt 2
4.b odd 2 1 2200.2.a.l 2
5.b even 2 1 880.2.a.k 2
5.c odd 4 2 4400.2.b.w 4
15.d odd 2 1 7920.2.a.by 2
20.d odd 2 1 440.2.a.g 2
20.e even 4 2 2200.2.b.f 4
40.e odd 2 1 3520.2.a.bm 2
40.f even 2 1 3520.2.a.br 2
55.d odd 2 1 9680.2.a.bm 2
60.h even 2 1 3960.2.a.bf 2
220.g even 2 1 4840.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.g 2 20.d odd 2 1
880.2.a.k 2 5.b even 2 1
2200.2.a.l 2 4.b odd 2 1
2200.2.b.f 4 20.e even 4 2
3520.2.a.bm 2 40.e odd 2 1
3520.2.a.br 2 40.f even 2 1
3960.2.a.bf 2 60.h even 2 1
4400.2.a.bt 2 1.a even 1 1 trivial
4400.2.b.w 4 5.c odd 4 2
4840.2.a.m 2 220.g even 2 1
7920.2.a.by 2 15.d odd 2 1
9680.2.a.bm 2 55.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

 $$T_{3}^{2} - T_{3} - 4$$ T3^2 - T3 - 4 $$T_{7}^{2} - 5T_{7} + 2$$ T7^2 - 5*T7 + 2 $$T_{13}^{2} + 6T_{13} - 8$$ T13^2 + 6*T13 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 5T + 2$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 6T - 8$$
$17$ $$T^{2} - 3T - 36$$
$19$ $$T^{2} + 7T + 8$$
$23$ $$T^{2} + 6T - 8$$
$29$ $$T^{2} - 13T + 38$$
$31$ $$T^{2} - 7T + 8$$
$37$ $$T^{2} + 19T + 86$$
$41$ $$T^{2} + 8T - 52$$
$43$ $$T^{2} - 2T - 16$$
$47$ $$T^{2} - 2T - 152$$
$53$ $$T^{2} + 7T - 26$$
$59$ $$T^{2} - 6T - 8$$
$61$ $$T^{2} + 21T + 106$$
$67$ $$T^{2}$$
$71$ $$T^{2} - 5T - 32$$
$73$ $$T^{2} - 6T - 8$$
$79$ $$T^{2} + 2T - 152$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 7T - 26$$
$97$ $$T^{2} - 18T + 64$$