# Properties

 Label 2800.2.g.v Level $2800$ Weight $2$ Character orbit 2800.g Analytic conductor $22.358$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(449,2800)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.g (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1400) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + \beta_{2} q^{7} + (\beta_{3} - 2) q^{9}+O(q^{10})$$ q + b1 * q^3 + b2 * q^7 + (b3 - 2) * q^9 $$q + \beta_1 q^{3} + \beta_{2} q^{7} + (\beta_{3} - 2) q^{9} + (2 \beta_{3} + 1) q^{11} + 2 \beta_{2} q^{13} - \beta_1 q^{17} + (\beta_{3} + 1) q^{19} + ( - \beta_{3} + 1) q^{21} + ( - 3 \beta_{2} + \beta_1) q^{23} + (4 \beta_{2} + \beta_1) q^{27} + (\beta_{3} - 6) q^{29} + (2 \beta_{3} + 4) q^{31} + (8 \beta_{2} + \beta_1) q^{33} + ( - \beta_{2} - 5 \beta_1) q^{37} + ( - 2 \beta_{3} + 2) q^{39} + (\beta_{3} - 5) q^{41} + ( - 5 \beta_{2} - \beta_1) q^{43} + ( - 2 \beta_{2} - 4 \beta_1) q^{47} - q^{49} + ( - \beta_{3} + 5) q^{51} + (2 \beta_{2} - 2 \beta_1) q^{53} + (4 \beta_{2} + \beta_1) q^{57} + (6 \beta_{3} - 4) q^{59} + (2 \beta_{3} + 6) q^{61} + ( - \beta_{2} + \beta_1) q^{63} + (11 \beta_{2} - 2 \beta_1) q^{67} + (4 \beta_{3} - 8) q^{69} + ( - 3 \beta_{3} + 6) q^{71} + (8 \beta_{2} - \beta_1) q^{73} + (3 \beta_{2} + 2 \beta_1) q^{77} + ( - 3 \beta_{3} + 2) q^{79} - 7 q^{81} + (6 \beta_{2} - 3 \beta_1) q^{83} + (4 \beta_{2} - 6 \beta_1) q^{87} + ( - 5 \beta_{3} + 3) q^{89} - 2 q^{91} + (8 \beta_{2} + 4 \beta_1) q^{93} - 6 \beta_{2} q^{97} + ( - \beta_{3} + 6) q^{99}+O(q^{100})$$ q + b1 * q^3 + b2 * q^7 + (b3 - 2) * q^9 + (2*b3 + 1) * q^11 + 2*b2 * q^13 - b1 * q^17 + (b3 + 1) * q^19 + (-b3 + 1) * q^21 + (-3*b2 + b1) * q^23 + (4*b2 + b1) * q^27 + (b3 - 6) * q^29 + (2*b3 + 4) * q^31 + (8*b2 + b1) * q^33 + (-b2 - 5*b1) * q^37 + (-2*b3 + 2) * q^39 + (b3 - 5) * q^41 + (-5*b2 - b1) * q^43 + (-2*b2 - 4*b1) * q^47 - q^49 + (-b3 + 5) * q^51 + (2*b2 - 2*b1) * q^53 + (4*b2 + b1) * q^57 + (6*b3 - 4) * q^59 + (2*b3 + 6) * q^61 + (-b2 + b1) * q^63 + (11*b2 - 2*b1) * q^67 + (4*b3 - 8) * q^69 + (-3*b3 + 6) * q^71 + (8*b2 - b1) * q^73 + (3*b2 + 2*b1) * q^77 + (-3*b3 + 2) * q^79 - 7 * q^81 + (6*b2 - 3*b1) * q^83 + (4*b2 - 6*b1) * q^87 + (-5*b3 + 3) * q^89 - 2 * q^91 + (8*b2 + 4*b1) * q^93 - 6*b2 * q^97 + (-b3 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{9}+O(q^{10})$$ 4 * q - 6 * q^9 $$4 q - 6 q^{9} + 8 q^{11} + 6 q^{19} + 2 q^{21} - 22 q^{29} + 20 q^{31} + 4 q^{39} - 18 q^{41} - 4 q^{49} + 18 q^{51} - 4 q^{59} + 28 q^{61} - 24 q^{69} + 18 q^{71} + 2 q^{79} - 28 q^{81} + 2 q^{89} - 8 q^{91} + 22 q^{99}+O(q^{100})$$ 4 * q - 6 * q^9 + 8 * q^11 + 6 * q^19 + 2 * q^21 - 22 * q^29 + 20 * q^31 + 4 * q^39 - 18 * q^41 - 4 * q^49 + 18 * q^51 - 4 * q^59 + 28 * q^61 - 24 * q^69 + 18 * q^71 + 2 * q^79 - 28 * q^81 + 2 * q^89 - 8 * q^91 + 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{2} - 5\beta_1$$ 4*b2 - 5*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$801$$ $$2101$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 2.56155i − 1.56155i 1.56155i 2.56155i
0 2.56155i 0 0 0 1.00000i 0 −3.56155 0
449.2 0 1.56155i 0 0 0 1.00000i 0 0.561553 0
449.3 0 1.56155i 0 0 0 1.00000i 0 0.561553 0
449.4 0 2.56155i 0 0 0 1.00000i 0 −3.56155 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{4} + 9T_{3}^{2} + 16$$ T3^4 + 9*T3^2 + 16 $$T_{11}^{2} - 4T_{11} - 13$$ T11^2 - 4*T11 - 13 $$T_{13}^{2} + 4$$ T13^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 9T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T^{2} - 4 T - 13)^{2}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$T^{4} + 9T^{2} + 16$$
$19$ $$(T^{2} - 3 T - 2)^{2}$$
$23$ $$T^{4} + 33T^{2} + 64$$
$29$ $$(T^{2} + 11 T + 26)^{2}$$
$31$ $$(T^{2} - 10 T + 8)^{2}$$
$37$ $$T^{4} + 217 T^{2} + 10816$$
$41$ $$(T^{2} + 9 T + 16)^{2}$$
$43$ $$T^{4} + 49T^{2} + 256$$
$47$ $$(T^{2} + 68)^{2}$$
$53$ $$T^{4} + 52T^{2} + 64$$
$59$ $$(T^{2} + 2 T - 152)^{2}$$
$61$ $$(T^{2} - 14 T + 32)^{2}$$
$67$ $$T^{4} + 322 T^{2} + 16129$$
$71$ $$(T^{2} - 9 T - 18)^{2}$$
$73$ $$T^{4} + 153T^{2} + 4624$$
$79$ $$(T^{2} - T - 38)^{2}$$
$83$ $$T^{4} + 189T^{2} + 324$$
$89$ $$(T^{2} - T - 106)^{2}$$
$97$ $$(T^{2} + 36)^{2}$$