# Properties

 Label 1840.2.m.b Level $1840$ Weight $2$ Character orbit 1840.m Analytic conductor $14.692$ Analytic rank $0$ Dimension $4$ CM discriminant -115 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1839,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))

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

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

chi := DirichletCharacter("1840.1839");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.m (of order $$2$$, degree $$1$$, 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: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{5}, \sqrt{-23})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 49$$ x^4 + 9*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} - \beta_{2} q^{7} - 3 q^{9}+O(q^{10})$$ q + b1 * q^5 - b2 * q^7 - 3 * q^9 $$q + \beta_1 q^{5} - \beta_{2} q^{7} - 3 q^{9} + 3 \beta_1 q^{17} + \beta_{2} q^{23} + 5 q^{25} - q^{29} + \beta_{3} q^{31} + \beta_{3} q^{35} - 5 \beta_1 q^{37} + 7 q^{41} - 2 \beta_{2} q^{43} - 3 \beta_1 q^{45} - 16 q^{49} - \beta_1 q^{53} + \beta_{3} q^{59} + 3 \beta_{2} q^{63} - \beta_{2} q^{67} + \beta_{3} q^{71} + 9 q^{81} + 3 \beta_{2} q^{83} + 15 q^{85} - 2 \beta_1 q^{97}+O(q^{100})$$ q + b1 * q^5 - b2 * q^7 - 3 * q^9 + 3*b1 * q^17 + b2 * q^23 + 5 * q^25 - q^29 + b3 * q^31 + b3 * q^35 - 5*b1 * q^37 + 7 * q^41 - 2*b2 * q^43 - 3*b1 * q^45 - 16 * q^49 - b1 * q^53 + b3 * q^59 + 3*b2 * q^63 - b2 * q^67 + b3 * q^71 + 9 * q^81 + 3*b2 * q^83 + 15 * q^85 - 2*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^9 $$4 q - 12 q^{9} + 20 q^{25} - 4 q^{29} + 28 q^{41} - 64 q^{49} + 36 q^{81} + 60 q^{85}+O(q^{100})$$ 4 * q - 12 * q^9 + 20 * q^25 - 4 * q^29 + 28 * q^41 - 64 * q^49 + 36 * q^81 + 60 * q^85

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu ) / 7$$ (v^3 + 2*v) / 7 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 16\nu ) / 7$$ (v^3 + 16*v) / 7 $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 9$$ 2*v^2 + 9
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 9 ) / 2$$ (b3 - 9) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 8\beta_1$$ -b2 + 8*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1839.1
 1.11803 + 2.39792i 1.11803 − 2.39792i −1.11803 + 2.39792i −1.11803 − 2.39792i
0 0 0 −2.23607 0 4.79583i 0 −3.00000 0
1839.2 0 0 0 −2.23607 0 4.79583i 0 −3.00000 0
1839.3 0 0 0 2.23607 0 4.79583i 0 −3.00000 0
1839.4 0 0 0 2.23607 0 4.79583i 0 −3.00000 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
115.c odd 2 1 CM by $$\Q(\sqrt{-115})$$
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner
460.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.m.b 4
4.b odd 2 1 inner 1840.2.m.b 4
5.b even 2 1 inner 1840.2.m.b 4
20.d odd 2 1 inner 1840.2.m.b 4
23.b odd 2 1 inner 1840.2.m.b 4
92.b even 2 1 inner 1840.2.m.b 4
115.c odd 2 1 CM 1840.2.m.b 4
460.g even 2 1 inner 1840.2.m.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.b 4 1.a even 1 1 trivial
1840.2.m.b 4 4.b odd 2 1 inner
1840.2.m.b 4 5.b even 2 1 inner
1840.2.m.b 4 20.d odd 2 1 inner
1840.2.m.b 4 23.b odd 2 1 inner
1840.2.m.b 4 92.b even 2 1 inner
1840.2.m.b 4 115.c odd 2 1 CM
1840.2.m.b 4 460.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1840, [\chi])$$:

 $$T_{3}$$ T3 $$T_{7}^{2} + 23$$ T7^2 + 23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 5)^{2}$$
$7$ $$(T^{2} + 23)^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 45)^{2}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 23)^{2}$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T^{2} + 115)^{2}$$
$37$ $$(T^{2} - 125)^{2}$$
$41$ $$(T - 7)^{4}$$
$43$ $$(T^{2} + 92)^{2}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} - 5)^{2}$$
$59$ $$(T^{2} + 115)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 23)^{2}$$
$71$ $$(T^{2} + 115)^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 207)^{2}$$
$89$ $$T^{4}$$
$97$ $$(T^{2} - 20)^{2}$$