Properties

 Label 1859.1.c.b Level $1859$ Weight $1$ Character orbit 1859.c Self dual yes Analytic conductor $0.928$ Analytic rank $0$ Dimension $3$ Projective image $D_{7}$ CM discriminant -11 Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1859,1,Mod(846,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1859.846");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1859.c (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: yes Analytic conductor: $$0.927761858485$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.6424482779.1 Artin image: $D_{14}$ Artin field: Galois closure of 14.2.536561726709678316933.1

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1 - 1) q^{3} + q^{4} + \beta_1 q^{5} + ( - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b2 + b1 - 1) * q^3 + q^4 + b1 * q^5 + (-b1 + 1) * q^9 $$q + ( - \beta_{2} + \beta_1 - 1) q^{3} + q^{4} + \beta_1 q^{5} + ( - \beta_1 + 1) q^{9} - q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{12} + ( - \beta_1 + 1) q^{15} + q^{16} + \beta_1 q^{20} + \beta_{2} q^{23} + (\beta_{2} + 1) q^{25} + (\beta_1 - 1) q^{27} - \beta_{2} q^{31} + (\beta_{2} - \beta_1 + 1) q^{33} + ( - \beta_1 + 1) q^{36} + (\beta_{2} - \beta_1 + 1) q^{37} - q^{44} + ( - \beta_{2} + \beta_1 - 2) q^{45} - \beta_{2} q^{47} + ( - \beta_{2} + \beta_1 - 1) q^{48} + q^{49} - \beta_1 q^{53} - \beta_1 q^{55} + (\beta_{2} - \beta_1 + 1) q^{59} + ( - \beta_1 + 1) q^{60} + q^{64} - 2 q^{67} + (\beta_{2} - \beta_1) q^{69} + (\beta_{2} - \beta_1 + 1) q^{71} - q^{75} + \beta_1 q^{80} + (\beta_{2} - \beta_1 + 1) q^{81} - \beta_{2} q^{89} + \beta_{2} q^{92} + ( - \beta_{2} + \beta_1) q^{93} + \beta_1 q^{97} + (\beta_1 - 1) q^{99}+O(q^{100})$$ q + (-b2 + b1 - 1) * q^3 + q^4 + b1 * q^5 + (-b1 + 1) * q^9 - q^11 + (-b2 + b1 - 1) * q^12 + (-b1 + 1) * q^15 + q^16 + b1 * q^20 + b2 * q^23 + (b2 + 1) * q^25 + (b1 - 1) * q^27 - b2 * q^31 + (b2 - b1 + 1) * q^33 + (-b1 + 1) * q^36 + (b2 - b1 + 1) * q^37 - q^44 + (-b2 + b1 - 2) * q^45 - b2 * q^47 + (-b2 + b1 - 1) * q^48 + q^49 - b1 * q^53 - b1 * q^55 + (b2 - b1 + 1) * q^59 + (-b1 + 1) * q^60 + q^64 - 2 * q^67 + (b2 - b1) * q^69 + (b2 - b1 + 1) * q^71 - q^75 + b1 * q^80 + (b2 - b1 + 1) * q^81 - b2 * q^89 + b2 * q^92 + (-b2 + b1) * q^93 + b1 * q^97 + (b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} + 3 q^{4} + q^{5} + 2 q^{9}+O(q^{10})$$ 3 * q - q^3 + 3 * q^4 + q^5 + 2 * q^9 $$3 q - q^{3} + 3 q^{4} + q^{5} + 2 q^{9} - 3 q^{11} - q^{12} + 2 q^{15} + 3 q^{16} + q^{20} - q^{23} + 2 q^{25} - 2 q^{27} + q^{31} + q^{33} + 2 q^{36} + q^{37} - 3 q^{44} - 4 q^{45} + q^{47} - q^{48} + 3 q^{49} - q^{53} - q^{55} + q^{59} + 2 q^{60} + 3 q^{64} - 6 q^{67} - 2 q^{69} + q^{71} - 3 q^{75} + q^{80} + q^{81} + q^{89} - q^{92} + 2 q^{93} + q^{97} - 2 q^{99}+O(q^{100})$$ 3 * q - q^3 + 3 * q^4 + q^5 + 2 * q^9 - 3 * q^11 - q^12 + 2 * q^15 + 3 * q^16 + q^20 - q^23 + 2 * q^25 - 2 * q^27 + q^31 + q^33 + 2 * q^36 + q^37 - 3 * q^44 - 4 * q^45 + q^47 - q^48 + 3 * q^49 - q^53 - q^55 + q^59 + 2 * q^60 + 3 * q^64 - 6 * q^67 - 2 * q^69 + q^71 - 3 * q^75 + q^80 + q^81 + q^89 - q^92 + 2 * q^93 + q^97 - 2 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1859\mathbb{Z}\right)^\times$$.

 $$n$$ $$508$$ $$1354$$ $$\chi(n)$$ $$-1$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
846.1
 −1.24698 1.80194 0.445042
0 −1.80194 1.00000 −1.24698 0 0 0 2.24698 0
846.2 0 −0.445042 1.00000 1.80194 0 0 0 −0.801938 0
846.3 0 1.24698 1.00000 0.445042 0 0 0 0.554958 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1859.1.c.b yes 3
11.b odd 2 1 CM 1859.1.c.b yes 3
13.b even 2 1 1859.1.c.a 3
13.c even 3 2 1859.1.k.b 6
13.d odd 4 2 1859.1.d.a 6
13.e even 6 2 1859.1.k.a 6
13.f odd 12 4 1859.1.i.c 12
143.d odd 2 1 1859.1.c.a 3
143.g even 4 2 1859.1.d.a 6
143.i odd 6 2 1859.1.k.a 6
143.k odd 6 2 1859.1.k.b 6
143.o even 12 4 1859.1.i.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1859.1.c.a 3 13.b even 2 1
1859.1.c.a 3 143.d odd 2 1
1859.1.c.b yes 3 1.a even 1 1 trivial
1859.1.c.b yes 3 11.b odd 2 1 CM
1859.1.d.a 6 13.d odd 4 2
1859.1.d.a 6 143.g even 4 2
1859.1.i.c 12 13.f odd 12 4
1859.1.i.c 12 143.o even 12 4
1859.1.k.a 6 13.e even 6 2
1859.1.k.a 6 143.i odd 6 2
1859.1.k.b 6 13.c even 3 2
1859.1.k.b 6 143.k odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1859, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}^{3} - T_{5}^{2} - 2T_{5} + 1$$ T5^3 - T5^2 - 2*T5 + 1

Hecke characteristic polynomials

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