# Properties

 Label 2800.2.a.bo Level $2800$ Weight $2$ Character orbit 2800.a Self dual yes Analytic conductor $22.358$ 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] = [2800,2,Mod(1,2800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, 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("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.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: $$22.3581125660$$ 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 1400) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{7} + (\beta + 1) q^{9}+O(q^{10})$$ q + b * q^3 + q^7 + (b + 1) * q^9 $$q + \beta q^{3} + q^{7} + (\beta + 1) q^{9} + ( - 2 \beta + 3) q^{11} - 2 q^{13} + \beta q^{17} + (\beta - 2) q^{19} + \beta q^{21} + (\beta + 3) q^{23} + ( - \beta + 4) q^{27} + (\beta + 5) q^{29} + ( - 2 \beta + 6) q^{31} + (\beta - 8) q^{33} + (5 \beta - 1) q^{37} - 2 \beta q^{39} + ( - \beta - 4) q^{41} + ( - \beta + 5) q^{43} + (4 \beta - 2) q^{47} + q^{49} + (\beta + 4) q^{51} + ( - 2 \beta - 2) q^{53} + ( - \beta + 4) q^{57} + (6 \beta - 2) q^{59} + ( - 2 \beta + 8) q^{61} + (\beta + 1) q^{63} + (2 \beta + 11) q^{67} + (4 \beta + 4) q^{69} + (3 \beta + 3) q^{71} + ( - \beta - 8) q^{73} + ( - 2 \beta + 3) q^{77} + ( - 3 \beta + 1) q^{79} - 7 q^{81} + ( - 3 \beta - 6) q^{83} + (6 \beta + 4) q^{87} + ( - 5 \beta + 2) q^{89} - 2 q^{91} + (4 \beta - 8) q^{93} - 6 q^{97} + ( - \beta - 5) q^{99} +O(q^{100})$$ q + b * q^3 + q^7 + (b + 1) * q^9 + (-2*b + 3) * q^11 - 2 * q^13 + b * q^17 + (b - 2) * q^19 + b * q^21 + (b + 3) * q^23 + (-b + 4) * q^27 + (b + 5) * q^29 + (-2*b + 6) * q^31 + (b - 8) * q^33 + (5*b - 1) * q^37 - 2*b * q^39 + (-b - 4) * q^41 + (-b + 5) * q^43 + (4*b - 2) * q^47 + q^49 + (b + 4) * q^51 + (-2*b - 2) * q^53 + (-b + 4) * q^57 + (6*b - 2) * q^59 + (-2*b + 8) * q^61 + (b + 1) * q^63 + (2*b + 11) * q^67 + (4*b + 4) * q^69 + (3*b + 3) * q^71 + (-b - 8) * q^73 + (-2*b + 3) * q^77 + (-3*b + 1) * q^79 - 7 * q^81 + (-3*b - 6) * q^83 + (6*b + 4) * q^87 + (-5*b + 2) * q^89 - 2 * q^91 + (4*b - 8) * q^93 - 6 * q^97 + (-b - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^7 + 3 * q^9 $$2 q + q^{3} + 2 q^{7} + 3 q^{9} + 4 q^{11} - 4 q^{13} + q^{17} - 3 q^{19} + q^{21} + 7 q^{23} + 7 q^{27} + 11 q^{29} + 10 q^{31} - 15 q^{33} + 3 q^{37} - 2 q^{39} - 9 q^{41} + 9 q^{43} + 2 q^{49} + 9 q^{51} - 6 q^{53} + 7 q^{57} + 2 q^{59} + 14 q^{61} + 3 q^{63} + 24 q^{67} + 12 q^{69} + 9 q^{71} - 17 q^{73} + 4 q^{77} - q^{79} - 14 q^{81} - 15 q^{83} + 14 q^{87} - q^{89} - 4 q^{91} - 12 q^{93} - 12 q^{97} - 11 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^7 + 3 * q^9 + 4 * q^11 - 4 * q^13 + q^17 - 3 * q^19 + q^21 + 7 * q^23 + 7 * q^27 + 11 * q^29 + 10 * q^31 - 15 * q^33 + 3 * q^37 - 2 * q^39 - 9 * q^41 + 9 * q^43 + 2 * q^49 + 9 * q^51 - 6 * q^53 + 7 * q^57 + 2 * q^59 + 14 * q^61 + 3 * q^63 + 24 * q^67 + 12 * q^69 + 9 * q^71 - 17 * q^73 + 4 * q^77 - q^79 - 14 * q^81 - 15 * q^83 + 14 * q^87 - q^89 - 4 * q^91 - 12 * q^93 - 12 * q^97 - 11 * 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 1.00000 0 −0.561553 0
1.2 0 2.56155 0 0 0 1.00000 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
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 2800.2.a.bo 2
4.b odd 2 1 1400.2.a.o 2
5.b even 2 1 2800.2.a.bj 2
5.c odd 4 2 2800.2.g.v 4
20.d odd 2 1 1400.2.a.q yes 2
20.e even 4 2 1400.2.g.j 4
28.d even 2 1 9800.2.a.bx 2
140.c even 2 1 9800.2.a.bt 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.o 2 4.b odd 2 1
1400.2.a.q yes 2 20.d odd 2 1
1400.2.g.j 4 20.e even 4 2
2800.2.a.bj 2 5.b even 2 1
2800.2.a.bo 2 1.a even 1 1 trivial
2800.2.g.v 4 5.c odd 4 2
9800.2.a.bt 2 140.c even 2 1
9800.2.a.bx 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2800))$$:

 $$T_{3}^{2} - T_{3} - 4$$ T3^2 - T3 - 4 $$T_{11}^{2} - 4T_{11} - 13$$ T11^2 - 4*T11 - 13 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 4$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 4T - 13$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - T - 4$$
$19$ $$T^{2} + 3T - 2$$
$23$ $$T^{2} - 7T + 8$$
$29$ $$T^{2} - 11T + 26$$
$31$ $$T^{2} - 10T + 8$$
$37$ $$T^{2} - 3T - 104$$
$41$ $$T^{2} + 9T + 16$$
$43$ $$T^{2} - 9T + 16$$
$47$ $$T^{2} - 68$$
$53$ $$T^{2} + 6T - 8$$
$59$ $$T^{2} - 2T - 152$$
$61$ $$T^{2} - 14T + 32$$
$67$ $$T^{2} - 24T + 127$$
$71$ $$T^{2} - 9T - 18$$
$73$ $$T^{2} + 17T + 68$$
$79$ $$T^{2} + T - 38$$
$83$ $$T^{2} + 15T + 18$$
$89$ $$T^{2} + T - 106$$
$97$ $$(T + 6)^{2}$$