# Properties

 Label 1840.2.a.s Level $1840$ Weight $2$ Character orbit 1840.a Self dual yes Analytic conductor $14.692$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1,1840)]

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([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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2597.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 9x + 8$$ x^3 - x^2 - 9*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + q^{5} + (\beta_1 - 1) q^{7} + (\beta_{2} + 3) q^{9}+O(q^{10})$$ q - b1 * q^3 + q^5 + (b1 - 1) * q^7 + (b2 + 3) * q^9 $$q - \beta_1 q^{3} + q^{5} + (\beta_1 - 1) q^{7} + (\beta_{2} + 3) q^{9} + ( - \beta_1 - 2) q^{11} - \beta_{2} q^{13} - \beta_1 q^{15} + (\beta_{2} + 3) q^{17} + (\beta_{2} + 4) q^{19} + ( - \beta_{2} + \beta_1 - 6) q^{21} - q^{23} + q^{25} + ( - \beta_{2} - 3 \beta_1 + 2) q^{27} + ( - \beta_{2} - \beta_1 + 5) q^{29} + ( - \beta_1 + 3) q^{31} + (\beta_{2} + 2 \beta_1 + 6) q^{33} + (\beta_1 - 1) q^{35} + ( - \beta_{2} - 3 \beta_1 + 3) q^{37} + (\beta_{2} + 3 \beta_1 - 2) q^{39} + ( - \beta_{2} + 3) q^{41} - 8 q^{43} + (\beta_{2} + 3) q^{45} - 2 \beta_{2} q^{47} + (\beta_{2} - 2 \beta_1) q^{49} + ( - \beta_{2} - 6 \beta_1 + 2) q^{51} + (\beta_{2} + 3 \beta_1 - 1) q^{53} + ( - \beta_1 - 2) q^{55} + ( - \beta_{2} - 7 \beta_1 + 2) q^{57} + (\beta_{2} + \beta_1 + 5) q^{59} + (2 \beta_{2} - \beta_1 + 4) q^{61} + (6 \beta_1 - 5) q^{63} - \beta_{2} q^{65} + (\beta_{2} + 3 \beta_1 - 3) q^{67} + \beta_1 q^{69} + ( - \beta_{2} + 2 \beta_1 + 7) q^{71} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{73} - \beta_1 q^{75} + ( - \beta_{2} - \beta_1 - 4) q^{77} + 4 \beta_1 q^{79} + (\beta_{2} + \beta_1 + 7) q^{81} + ( - \beta_{2} + \beta_1 + 5) q^{83} + (\beta_{2} + 3) q^{85} + (2 \beta_{2} - 2 \beta_1 + 4) q^{87} + (4 \beta_1 - 4) q^{89} + ( - 3 \beta_1 + 2) q^{91} + (\beta_{2} - 3 \beta_1 + 6) q^{93} + (\beta_{2} + 4) q^{95} + (3 \beta_1 - 2) q^{97} + ( - 3 \beta_{2} - 6 \beta_1 - 4) q^{99}+O(q^{100})$$ q - b1 * q^3 + q^5 + (b1 - 1) * q^7 + (b2 + 3) * q^9 + (-b1 - 2) * q^11 - b2 * q^13 - b1 * q^15 + (b2 + 3) * q^17 + (b2 + 4) * q^19 + (-b2 + b1 - 6) * q^21 - q^23 + q^25 + (-b2 - 3*b1 + 2) * q^27 + (-b2 - b1 + 5) * q^29 + (-b1 + 3) * q^31 + (b2 + 2*b1 + 6) * q^33 + (b1 - 1) * q^35 + (-b2 - 3*b1 + 3) * q^37 + (b2 + 3*b1 - 2) * q^39 + (-b2 + 3) * q^41 - 8 * q^43 + (b2 + 3) * q^45 - 2*b2 * q^47 + (b2 - 2*b1) * q^49 + (-b2 - 6*b1 + 2) * q^51 + (b2 + 3*b1 - 1) * q^53 + (-b1 - 2) * q^55 + (-b2 - 7*b1 + 2) * q^57 + (b2 + b1 + 5) * q^59 + (2*b2 - b1 + 4) * q^61 + (6*b1 - 5) * q^63 - b2 * q^65 + (b2 + 3*b1 - 3) * q^67 + b1 * q^69 + (-b2 + 2*b1 + 7) * q^71 + (-2*b2 - 4*b1 + 6) * q^73 - b1 * q^75 + (-b2 - b1 - 4) * q^77 + 4*b1 * q^79 + (b2 + b1 + 7) * q^81 + (-b2 + b1 + 5) * q^83 + (b2 + 3) * q^85 + (2*b2 - 2*b1 + 4) * q^87 + (4*b1 - 4) * q^89 + (-3*b1 + 2) * q^91 + (b2 - 3*b1 + 6) * q^93 + (b2 + 4) * q^95 + (3*b1 - 2) * q^97 + (-3*b2 - 6*b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} + 3 q^{5} - 2 q^{7} + 10 q^{9}+O(q^{10})$$ 3 * q - q^3 + 3 * q^5 - 2 * q^7 + 10 * q^9 $$3 q - q^{3} + 3 q^{5} - 2 q^{7} + 10 q^{9} - 7 q^{11} - q^{13} - q^{15} + 10 q^{17} + 13 q^{19} - 18 q^{21} - 3 q^{23} + 3 q^{25} + 2 q^{27} + 13 q^{29} + 8 q^{31} + 21 q^{33} - 2 q^{35} + 5 q^{37} - 2 q^{39} + 8 q^{41} - 24 q^{43} + 10 q^{45} - 2 q^{47} - q^{49} - q^{51} + q^{53} - 7 q^{55} - 2 q^{57} + 17 q^{59} + 13 q^{61} - 9 q^{63} - q^{65} - 5 q^{67} + q^{69} + 22 q^{71} + 12 q^{73} - q^{75} - 14 q^{77} + 4 q^{79} + 23 q^{81} + 15 q^{83} + 10 q^{85} + 12 q^{87} - 8 q^{89} + 3 q^{91} + 16 q^{93} + 13 q^{95} - 3 q^{97} - 21 q^{99}+O(q^{100})$$ 3 * q - q^3 + 3 * q^5 - 2 * q^7 + 10 * q^9 - 7 * q^11 - q^13 - q^15 + 10 * q^17 + 13 * q^19 - 18 * q^21 - 3 * q^23 + 3 * q^25 + 2 * q^27 + 13 * q^29 + 8 * q^31 + 21 * q^33 - 2 * q^35 + 5 * q^37 - 2 * q^39 + 8 * q^41 - 24 * q^43 + 10 * q^45 - 2 * q^47 - q^49 - q^51 + q^53 - 7 * q^55 - 2 * q^57 + 17 * q^59 + 13 * q^61 - 9 * q^63 - q^65 - 5 * q^67 + q^69 + 22 * q^71 + 12 * q^73 - q^75 - 14 * q^77 + 4 * q^79 + 23 * q^81 + 15 * q^83 + 10 * q^85 + 12 * q^87 - 8 * q^89 + 3 * q^91 + 16 * q^93 + 13 * q^95 - 3 * q^97 - 21 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 6$$ v^2 - 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 6$$ b2 + 6

## 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
 3.07912 0.878468 −2.95759
0 −3.07912 0 1.00000 0 2.07912 0 6.48097 0
1.2 0 −0.878468 0 1.00000 0 −0.121532 0 −2.22829 0
1.3 0 2.95759 0 1.00000 0 −3.95759 0 5.74732 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
$$2$$ $$1$$
$$5$$ $$-1$$
$$23$$ $$1$$

## 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 1840.2.a.s 3
4.b odd 2 1 920.2.a.h 3
5.b even 2 1 9200.2.a.ce 3
8.b even 2 1 7360.2.a.cc 3
8.d odd 2 1 7360.2.a.by 3
12.b even 2 1 8280.2.a.bj 3
20.d odd 2 1 4600.2.a.x 3
20.e even 4 2 4600.2.e.p 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.h 3 4.b odd 2 1
1840.2.a.s 3 1.a even 1 1 trivial
4600.2.a.x 3 20.d odd 2 1
4600.2.e.p 6 20.e even 4 2
7360.2.a.by 3 8.d odd 2 1
7360.2.a.cc 3 8.b even 2 1
8280.2.a.bj 3 12.b even 2 1
9200.2.a.ce 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{3} + T_{3}^{2} - 9T_{3} - 8$$ T3^3 + T3^2 - 9*T3 - 8 $$T_{7}^{3} + 2T_{7}^{2} - 8T_{7} - 1$$ T7^3 + 2*T7^2 - 8*T7 - 1 $$T_{11}^{3} + 7T_{11}^{2} + 7T_{11} - 14$$ T11^3 + 7*T11^2 + 7*T11 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 9T - 8$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + 2 T^{2} + \cdots - 1$$
$11$ $$T^{3} + 7 T^{2} + \cdots - 14$$
$13$ $$T^{3} + T^{2} + \cdots - 50$$
$17$ $$T^{3} - 10 T^{2} + \cdots + 83$$
$19$ $$T^{3} - 13 T^{2} + \cdots + 62$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} - 13 T^{2} + \cdots + 76$$
$31$ $$T^{3} - 8 T^{2} + \cdots + 1$$
$37$ $$T^{3} - 5 T^{2} + \cdots + 496$$
$41$ $$T^{3} - 8 T^{2} + \cdots + 1$$
$43$ $$(T + 8)^{3}$$
$47$ $$T^{3} + 2 T^{2} + \cdots - 400$$
$53$ $$T^{3} - T^{2} + \cdots - 300$$
$59$ $$T^{3} - 17 T^{2} + \cdots - 36$$
$61$ $$T^{3} - 13 T^{2} + \cdots + 720$$
$67$ $$T^{3} + 5 T^{2} + \cdots - 496$$
$71$ $$T^{3} - 22 T^{2} + \cdots + 225$$
$73$ $$T^{3} - 12 T^{2} + \cdots + 2120$$
$79$ $$T^{3} - 4 T^{2} + \cdots + 512$$
$83$ $$T^{3} - 15 T^{2} + \cdots + 36$$
$89$ $$T^{3} + 8 T^{2} + \cdots - 64$$
$97$ $$T^{3} + 3 T^{2} + \cdots + 50$$