# Properties

 Label 4400.2.a.bs Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ 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] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( - \beta + 3) q^{7} + \beta q^{9}+O(q^{10})$$ q + b * q^3 + (-b + 3) * q^7 + b * q^9 $$q + \beta q^{3} + ( - \beta + 3) q^{7} + \beta q^{9} + q^{11} - 5 q^{13} - 3 \beta q^{17} + q^{19} + (2 \beta - 3) q^{21} + ( - \beta - 5) q^{23} + ( - 2 \beta + 3) q^{27} + (3 \beta - 6) q^{29} + ( - 4 \beta - 1) q^{31} + \beta q^{33} + (2 \beta - 7) q^{37} - 5 \beta q^{39} + ( - 2 \beta - 1) q^{41} + (4 \beta - 2) q^{43} - 3 q^{47} + ( - 5 \beta + 5) q^{49} + ( - 3 \beta - 9) q^{51} + (\beta - 1) q^{53} + \beta q^{57} + (4 \beta + 5) q^{59} + ( - 3 \beta - 1) q^{61} + (2 \beta - 3) q^{63} - 4 q^{67} + ( - 6 \beta - 3) q^{69} + (2 \beta - 2) q^{71} + (3 \beta + 1) q^{73} + ( - \beta + 3) q^{77} + (3 \beta + 4) q^{79} + ( - 2 \beta - 6) q^{81} + ( - 5 \beta + 8) q^{83} + ( - 3 \beta + 9) q^{87} + ( - \beta + 4) q^{89} + (5 \beta - 15) q^{91} + ( - 5 \beta - 12) q^{93} + ( - \beta - 13) q^{97} + \beta q^{99} +O(q^{100})$$ q + b * q^3 + (-b + 3) * q^7 + b * q^9 + q^11 - 5 * q^13 - 3*b * q^17 + q^19 + (2*b - 3) * q^21 + (-b - 5) * q^23 + (-2*b + 3) * q^27 + (3*b - 6) * q^29 + (-4*b - 1) * q^31 + b * q^33 + (2*b - 7) * q^37 - 5*b * q^39 + (-2*b - 1) * q^41 + (4*b - 2) * q^43 - 3 * q^47 + (-5*b + 5) * q^49 + (-3*b - 9) * q^51 + (b - 1) * q^53 + b * q^57 + (4*b + 5) * q^59 + (-3*b - 1) * q^61 + (2*b - 3) * q^63 - 4 * q^67 + (-6*b - 3) * q^69 + (2*b - 2) * q^71 + (3*b + 1) * q^73 + (-b + 3) * q^77 + (3*b + 4) * q^79 + (-2*b - 6) * q^81 + (-5*b + 8) * q^83 + (-3*b + 9) * q^87 + (-b + 4) * q^89 + (5*b - 15) * q^91 + (-5*b - 12) * q^93 + (-b - 13) * q^97 + b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 5 q^{7} + q^{9}+O(q^{10})$$ 2 * q + q^3 + 5 * q^7 + q^9 $$2 q + q^{3} + 5 q^{7} + q^{9} + 2 q^{11} - 10 q^{13} - 3 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} + 4 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} - 12 q^{37} - 5 q^{39} - 4 q^{41} - 6 q^{47} + 5 q^{49} - 21 q^{51} - q^{53} + q^{57} + 14 q^{59} - 5 q^{61} - 4 q^{63} - 8 q^{67} - 12 q^{69} - 2 q^{71} + 5 q^{73} + 5 q^{77} + 11 q^{79} - 14 q^{81} + 11 q^{83} + 15 q^{87} + 7 q^{89} - 25 q^{91} - 29 q^{93} - 27 q^{97} + q^{99}+O(q^{100})$$ 2 * q + q^3 + 5 * q^7 + q^9 + 2 * q^11 - 10 * q^13 - 3 * q^17 + 2 * q^19 - 4 * q^21 - 11 * q^23 + 4 * q^27 - 9 * q^29 - 6 * q^31 + q^33 - 12 * q^37 - 5 * q^39 - 4 * q^41 - 6 * q^47 + 5 * q^49 - 21 * q^51 - q^53 + q^57 + 14 * q^59 - 5 * q^61 - 4 * q^63 - 8 * q^67 - 12 * q^69 - 2 * q^71 + 5 * q^73 + 5 * q^77 + 11 * q^79 - 14 * q^81 + 11 * q^83 + 15 * q^87 + 7 * q^89 - 25 * q^91 - 29 * q^93 - 27 * q^97 + 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
 −1.30278 2.30278
0 −1.30278 0 0 0 4.30278 0 −1.30278 0
1.2 0 2.30278 0 0 0 0.697224 0 2.30278 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.bs 2
4.b odd 2 1 275.2.a.e 2
5.b even 2 1 4400.2.a.bh 2
5.c odd 4 2 4400.2.b.y 4
12.b even 2 1 2475.2.a.t 2
20.d odd 2 1 275.2.a.f yes 2
20.e even 4 2 275.2.b.c 4
44.c even 2 1 3025.2.a.n 2
60.h even 2 1 2475.2.a.o 2
60.l odd 4 2 2475.2.c.k 4
220.g even 2 1 3025.2.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 4.b odd 2 1
275.2.a.f yes 2 20.d odd 2 1
275.2.b.c 4 20.e even 4 2
2475.2.a.o 2 60.h even 2 1
2475.2.a.t 2 12.b even 2 1
2475.2.c.k 4 60.l odd 4 2
3025.2.a.h 2 220.g even 2 1
3025.2.a.n 2 44.c even 2 1
4400.2.a.bh 2 5.b even 2 1
4400.2.a.bs 2 1.a even 1 1 trivial
4400.2.b.y 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 3$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 5T + 3$$
$11$ $$(T - 1)^{2}$$
$13$ $$(T + 5)^{2}$$
$17$ $$T^{2} + 3T - 27$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 11T + 27$$
$29$ $$T^{2} + 9T - 9$$
$31$ $$T^{2} + 6T - 43$$
$37$ $$T^{2} + 12T + 23$$
$41$ $$T^{2} + 4T - 9$$
$43$ $$T^{2} - 52$$
$47$ $$(T + 3)^{2}$$
$53$ $$T^{2} + T - 3$$
$59$ $$T^{2} - 14T - 3$$
$61$ $$T^{2} + 5T - 23$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} + 2T - 12$$
$73$ $$T^{2} - 5T - 23$$
$79$ $$T^{2} - 11T + 1$$
$83$ $$T^{2} - 11T - 51$$
$89$ $$T^{2} - 7T + 9$$
$97$ $$T^{2} + 27T + 179$$