# Properties

 Label 1840.2.a.r Level $1840$ Weight $2$ Character orbit 1840.a Self dual yes Analytic conductor $14.692$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 9x + 12$$ x^3 - x^2 - 9*x + 12 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - q^{5} + (\beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 4) q^{9}+O(q^{10})$$ q - b1 * q^3 - q^5 + (b2 + b1 - 1) * q^7 + (b2 - b1 + 4) * q^9 $$q - \beta_1 q^{3} - q^{5} + (\beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 4) q^{9} + ( - 2 \beta_{2} + \beta_1 - 2) q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{13} + \beta_1 q^{15} + ( - \beta_1 - 2) q^{17} + (\beta_{2} + \beta_1 - 1) q^{19} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{21} + q^{23} + q^{25} + ( - \beta_{2} - 2 \beta_1 + 5) q^{27} + (2 \beta_1 - 2) q^{29} + ( - \beta_{2} + \beta_1 + 1) q^{31} + (3 \beta_{2} + 3 \beta_1 - 3) q^{33} + ( - \beta_{2} - \beta_1 + 1) q^{35} + 2 \beta_{2} q^{37} + (\beta_{2} + 2 \beta_1 - 5) q^{39} + ( - 2 \beta_{2} - \beta_1) q^{41} - 8 q^{43} + ( - \beta_{2} + \beta_1 - 4) q^{45} + (2 \beta_{2} - 2 \beta_1 + 6) q^{47} + (2 \beta_{2} - \beta_1 + 11) q^{49} + (\beta_{2} + \beta_1 + 7) q^{51} - 6 q^{53} + (2 \beta_{2} - \beta_1 + 2) q^{55} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{57} + ( - 2 \beta_1 - 4) q^{59} + (4 \beta_{2} - \beta_1 + 2) q^{61} + (\beta_{2} + 8 \beta_1 - 5) q^{63} + (\beta_{2} - \beta_1 + 1) q^{65} + ( - 4 \beta_{2} - 4) q^{67} - \beta_1 q^{69} + (\beta_1 - 4) q^{71} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{73} - \beta_1 q^{75} + (\beta_{2} - 8 \beta_1 - 5) q^{77} + (4 \beta_{2} - 4 \beta_1 + 4) q^{79} + (\beta_{2} - 4 \beta_1 + 4) q^{81} + (2 \beta_{2} - 2) q^{83} + (\beta_1 + 2) q^{85} + ( - 2 \beta_{2} + 4 \beta_1 - 14) q^{87} + ( - 2 \beta_{2} + 4 \beta_1 + 4) q^{89} + (2 \beta_{2} - 5 \beta_1 + 2) q^{91} + (\beta_{2} - 5) q^{93} + ( - \beta_{2} - \beta_1 + 1) q^{95} + ( - 3 \beta_1 - 10) q^{97} + ( - 3 \beta_{2} + 3 \beta_1 - 21) q^{99}+O(q^{100})$$ q - b1 * q^3 - q^5 + (b2 + b1 - 1) * q^7 + (b2 - b1 + 4) * q^9 + (-2*b2 + b1 - 2) * q^11 + (-b2 + b1 - 1) * q^13 + b1 * q^15 + (-b1 - 2) * q^17 + (b2 + b1 - 1) * q^19 + (-3*b2 + 2*b1 - 9) * q^21 + q^23 + q^25 + (-b2 - 2*b1 + 5) * q^27 + (2*b1 - 2) * q^29 + (-b2 + b1 + 1) * q^31 + (3*b2 + 3*b1 - 3) * q^33 + (-b2 - b1 + 1) * q^35 + 2*b2 * q^37 + (b2 + 2*b1 - 5) * q^39 + (-2*b2 - b1) * q^41 - 8 * q^43 + (-b2 + b1 - 4) * q^45 + (2*b2 - 2*b1 + 6) * q^47 + (2*b2 - b1 + 11) * q^49 + (b2 + b1 + 7) * q^51 - 6 * q^53 + (2*b2 - b1 + 2) * q^55 + (-3*b2 + 2*b1 - 9) * q^57 + (-2*b1 - 4) * q^59 + (4*b2 - b1 + 2) * q^61 + (b2 + 8*b1 - 5) * q^63 + (b2 - b1 + 1) * q^65 + (-4*b2 - 4) * q^67 - b1 * q^69 + (b1 - 4) * q^71 + (-2*b2 + 2*b1 - 4) * q^73 - b1 * q^75 + (b2 - 8*b1 - 5) * q^77 + (4*b2 - 4*b1 + 4) * q^79 + (b2 - 4*b1 + 4) * q^81 + (2*b2 - 2) * q^83 + (b1 + 2) * q^85 + (-2*b2 + 4*b1 - 14) * q^87 + (-2*b2 + 4*b1 + 4) * q^89 + (2*b2 - 5*b1 + 2) * q^91 + (b2 - 5) * q^93 + (-b2 - b1 + 1) * q^95 + (-3*b1 - 10) * q^97 + (-3*b2 + 3*b1 - 21) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{5} - 3 q^{7} + 10 q^{9}+O(q^{10})$$ 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 10 * q^9 $$3 q - q^{3} - 3 q^{5} - 3 q^{7} + 10 q^{9} - 3 q^{11} - q^{13} + q^{15} - 7 q^{17} - 3 q^{19} - 22 q^{21} + 3 q^{23} + 3 q^{25} + 14 q^{27} - 4 q^{29} + 5 q^{31} - 9 q^{33} + 3 q^{35} - 2 q^{37} - 14 q^{39} + q^{41} - 24 q^{43} - 10 q^{45} + 14 q^{47} + 30 q^{49} + 21 q^{51} - 18 q^{53} + 3 q^{55} - 22 q^{57} - 14 q^{59} + q^{61} - 8 q^{63} + q^{65} - 8 q^{67} - q^{69} - 11 q^{71} - 8 q^{73} - q^{75} - 24 q^{77} + 4 q^{79} + 7 q^{81} - 8 q^{83} + 7 q^{85} - 36 q^{87} + 18 q^{89} - q^{91} - 16 q^{93} + 3 q^{95} - 33 q^{97} - 57 q^{99}+O(q^{100})$$ 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 10 * q^9 - 3 * q^11 - q^13 + q^15 - 7 * q^17 - 3 * q^19 - 22 * q^21 + 3 * q^23 + 3 * q^25 + 14 * q^27 - 4 * q^29 + 5 * q^31 - 9 * q^33 + 3 * q^35 - 2 * q^37 - 14 * q^39 + q^41 - 24 * q^43 - 10 * q^45 + 14 * q^47 + 30 * q^49 + 21 * q^51 - 18 * q^53 + 3 * q^55 - 22 * q^57 - 14 * q^59 + q^61 - 8 * q^63 + q^65 - 8 * q^67 - q^69 - 11 * q^71 - 8 * q^73 - q^75 - 24 * q^77 + 4 * q^79 + 7 * q^81 - 8 * q^83 + 7 * q^85 - 36 * q^87 + 18 * q^89 - q^91 - 16 * q^93 + 3 * q^95 - 33 * q^97 - 57 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9x + 12$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$ v^2 + v - 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 7$$ b2 - b1 + 7

## 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
 2.68740 1.43163 −3.11903
0 −2.68740 0 −1.00000 0 4.59692 0 4.22212 0
1.2 0 −1.43163 0 −1.00000 0 −3.08719 0 −0.950444 0
1.3 0 3.11903 0 −1.00000 0 −4.50973 0 6.72833 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.r 3
4.b odd 2 1 230.2.a.d 3
5.b even 2 1 9200.2.a.cf 3
8.b even 2 1 7360.2.a.ce 3
8.d odd 2 1 7360.2.a.bz 3
12.b even 2 1 2070.2.a.z 3
20.d odd 2 1 1150.2.a.q 3
20.e even 4 2 1150.2.b.j 6
92.b even 2 1 5290.2.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.d 3 4.b odd 2 1
1150.2.a.q 3 20.d odd 2 1
1150.2.b.j 6 20.e even 4 2
1840.2.a.r 3 1.a even 1 1 trivial
2070.2.a.z 3 12.b even 2 1
5290.2.a.r 3 92.b even 2 1
7360.2.a.bz 3 8.d odd 2 1
7360.2.a.ce 3 8.b even 2 1
9200.2.a.cf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{3} + T_{3}^{2} - 9T_{3} - 12$$ T3^3 + T3^2 - 9*T3 - 12 $$T_{7}^{3} + 3T_{7}^{2} - 21T_{7} - 64$$ T7^3 + 3*T7^2 - 21*T7 - 64 $$T_{11}^{3} + 3T_{11}^{2} - 39T_{11} - 144$$ T11^3 + 3*T11^2 - 39*T11 - 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 9T - 12$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 3 T^{2} + \cdots - 64$$
$11$ $$T^{3} + 3 T^{2} + \cdots - 144$$
$13$ $$T^{3} + T^{2} + \cdots - 18$$
$17$ $$T^{3} + 7 T^{2} + \cdots - 18$$
$19$ $$T^{3} + 3 T^{2} + \cdots - 64$$
$23$ $$(T - 1)^{3}$$
$29$ $$T^{3} + 4 T^{2} + \cdots + 24$$
$31$ $$T^{3} - 5 T^{2} + \cdots + 8$$
$37$ $$T^{3} + 2 T^{2} + \cdots - 32$$
$41$ $$T^{3} - T^{2} + \cdots + 186$$
$43$ $$(T + 8)^{3}$$
$47$ $$T^{3} - 14 T^{2} + \cdots + 288$$
$53$ $$(T + 6)^{3}$$
$59$ $$T^{3} + 14 T^{2} + \cdots - 144$$
$61$ $$T^{3} - T^{2} + \cdots + 526$$
$67$ $$T^{3} + 8 T^{2} + \cdots - 384$$
$71$ $$T^{3} + 11 T^{2} + \cdots + 24$$
$73$ $$T^{3} + 8 T^{2} + \cdots - 248$$
$79$ $$T^{3} - 4 T^{2} + \cdots + 1152$$
$83$ $$T^{3} + 8 T^{2} + \cdots - 96$$
$89$ $$T^{3} - 18 T^{2} + \cdots + 1152$$
$97$ $$T^{3} + 33 T^{2} + \cdots + 166$$