# Properties

 Label 4400.2.a.bn 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 55) 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 2 q^{7} + 5 q^{9}+O(q^{10})$$ q + b * q^3 - 2 * q^7 + 5 * q^9 $$q + \beta q^{3} - 2 q^{7} + 5 q^{9} - q^{11} + (\beta + 4) q^{13} + (\beta - 4) q^{17} - 2 \beta q^{21} + \beta q^{23} + 2 \beta q^{27} + (2 \beta + 2) q^{29} - \beta q^{33} + ( - 2 \beta + 2) q^{37} + (4 \beta + 8) q^{39} + 6 q^{41} - 6 q^{43} - \beta q^{47} - 3 q^{49} + ( - 4 \beta + 8) q^{51} + (2 \beta - 6) q^{53} + (2 \beta + 4) q^{59} + (4 \beta + 2) q^{61} - 10 q^{63} + ( - 3 \beta + 4) q^{67} + 8 q^{69} + 4 \beta q^{71} + (\beta + 4) q^{73} + 2 q^{77} - 4 q^{79} + q^{81} - 6 q^{83} + (2 \beta + 16) q^{87} + (4 \beta - 2) q^{89} + ( - 2 \beta - 8) q^{91} + (2 \beta + 2) q^{97} - 5 q^{99} +O(q^{100})$$ q + b * q^3 - 2 * q^7 + 5 * q^9 - q^11 + (b + 4) * q^13 + (b - 4) * q^17 - 2*b * q^21 + b * q^23 + 2*b * q^27 + (2*b + 2) * q^29 - b * q^33 + (-2*b + 2) * q^37 + (4*b + 8) * q^39 + 6 * q^41 - 6 * q^43 - b * q^47 - 3 * q^49 + (-4*b + 8) * q^51 + (2*b - 6) * q^53 + (2*b + 4) * q^59 + (4*b + 2) * q^61 - 10 * q^63 + (-3*b + 4) * q^67 + 8 * q^69 + 4*b * q^71 + (b + 4) * q^73 + 2 * q^77 - 4 * q^79 + q^81 - 6 * q^83 + (2*b + 16) * q^87 + (4*b - 2) * q^89 + (-2*b - 8) * q^91 + (2*b + 2) * q^97 - 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} + 10 q^{9}+O(q^{10})$$ 2 * q - 4 * q^7 + 10 * q^9 $$2 q - 4 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} - 8 q^{17} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 12 q^{41} - 12 q^{43} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 8 q^{59} + 4 q^{61} - 20 q^{63} + 8 q^{67} + 16 q^{69} + 8 q^{73} + 4 q^{77} - 8 q^{79} + 2 q^{81} - 12 q^{83} + 32 q^{87} - 4 q^{89} - 16 q^{91} + 4 q^{97} - 10 q^{99}+O(q^{100})$$ 2 * q - 4 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^13 - 8 * q^17 + 4 * q^29 + 4 * q^37 + 16 * q^39 + 12 * q^41 - 12 * q^43 - 6 * q^49 + 16 * q^51 - 12 * q^53 + 8 * q^59 + 4 * q^61 - 20 * q^63 + 8 * q^67 + 16 * q^69 + 8 * q^73 + 4 * q^77 - 8 * q^79 + 2 * q^81 - 12 * q^83 + 32 * q^87 - 4 * q^89 - 16 * q^91 + 4 * q^97 - 10 * 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
 −1.41421 1.41421
0 −2.82843 0 0 0 −2.00000 0 5.00000 0
1.2 0 2.82843 0 0 0 −2.00000 0 5.00000 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.bn 2
4.b odd 2 1 275.2.a.c 2
5.b even 2 1 880.2.a.m 2
5.c odd 4 2 4400.2.b.q 4
12.b even 2 1 2475.2.a.x 2
15.d odd 2 1 7920.2.a.ch 2
20.d odd 2 1 55.2.a.b 2
20.e even 4 2 275.2.b.d 4
40.e odd 2 1 3520.2.a.bn 2
40.f even 2 1 3520.2.a.bo 2
44.c even 2 1 3025.2.a.o 2
55.d odd 2 1 9680.2.a.bn 2
60.h even 2 1 495.2.a.b 2
60.l odd 4 2 2475.2.c.l 4
140.c even 2 1 2695.2.a.f 2
220.g even 2 1 605.2.a.d 2
220.n odd 10 4 605.2.g.f 8
220.o even 10 4 605.2.g.l 8
260.g odd 2 1 9295.2.a.g 2
660.g odd 2 1 5445.2.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 20.d odd 2 1
275.2.a.c 2 4.b odd 2 1
275.2.b.d 4 20.e even 4 2
495.2.a.b 2 60.h even 2 1
605.2.a.d 2 220.g even 2 1
605.2.g.f 8 220.n odd 10 4
605.2.g.l 8 220.o even 10 4
880.2.a.m 2 5.b even 2 1
2475.2.a.x 2 12.b even 2 1
2475.2.c.l 4 60.l odd 4 2
2695.2.a.f 2 140.c even 2 1
3025.2.a.o 2 44.c even 2 1
3520.2.a.bn 2 40.e odd 2 1
3520.2.a.bo 2 40.f even 2 1
4400.2.a.bn 2 1.a even 1 1 trivial
4400.2.b.q 4 5.c odd 4 2
5445.2.a.y 2 660.g odd 2 1
7920.2.a.ch 2 15.d odd 2 1
9295.2.a.g 2 260.g odd 2 1
9680.2.a.bn 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 8$$
$5$ $$T^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 8T + 8$$
$17$ $$T^{2} + 8T + 8$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 8$$
$29$ $$T^{2} - 4T - 28$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 4T - 28$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} - 8$$
$53$ $$T^{2} + 12T + 4$$
$59$ $$T^{2} - 8T - 16$$
$61$ $$T^{2} - 4T - 124$$
$67$ $$T^{2} - 8T - 56$$
$71$ $$T^{2} - 128$$
$73$ $$T^{2} - 8T + 8$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 4T - 124$$
$97$ $$T^{2} - 4T - 28$$