# Properties

 Label 855.2.k.b Level $855$ Weight $2$ Character orbit 855.k Analytic conductor $6.827$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(406,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("855.406");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$855 = 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 855.k (of order $$3$$, degree $$2$$, 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: $$6.82720937282$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} - 4 q^{7}+O(q^{10})$$ q + 2*z * q^4 + (z - 1) * q^5 - 4 * q^7 $$q + 2 \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} - 4 q^{7} - 3 q^{11} - 2 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + ( - 3 \zeta_{6} - 2) q^{19} - 2 q^{20} - \zeta_{6} q^{25} - 8 \zeta_{6} q^{28} - 3 \zeta_{6} q^{29} - 7 q^{31} + ( - 4 \zeta_{6} + 4) q^{35} + 8 q^{37} + (6 \zeta_{6} - 6) q^{41} + ( - 4 \zeta_{6} + 4) q^{43} - 6 \zeta_{6} q^{44} + 6 \zeta_{6} q^{47} + 9 q^{49} + ( - 4 \zeta_{6} + 4) q^{52} - 6 \zeta_{6} q^{53} + ( - 3 \zeta_{6} + 3) q^{55} + (15 \zeta_{6} - 15) q^{59} - 5 \zeta_{6} q^{61} - 8 q^{64} + 2 q^{65} - 2 \zeta_{6} q^{67} + 12 q^{68} + (3 \zeta_{6} - 3) q^{71} + (8 \zeta_{6} - 8) q^{73} + ( - 10 \zeta_{6} + 6) q^{76} + 12 q^{77} + (5 \zeta_{6} - 5) q^{79} - 4 \zeta_{6} q^{80} - 12 q^{83} + 6 \zeta_{6} q^{85} - 15 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + ( - 2 \zeta_{6} + 5) q^{95} + (8 \zeta_{6} - 8) q^{97} +O(q^{100})$$ q + 2*z * q^4 + (z - 1) * q^5 - 4 * q^7 - 3 * q^11 - 2*z * q^13 + (4*z - 4) * q^16 + (-6*z + 6) * q^17 + (-3*z - 2) * q^19 - 2 * q^20 - z * q^25 - 8*z * q^28 - 3*z * q^29 - 7 * q^31 + (-4*z + 4) * q^35 + 8 * q^37 + (6*z - 6) * q^41 + (-4*z + 4) * q^43 - 6*z * q^44 + 6*z * q^47 + 9 * q^49 + (-4*z + 4) * q^52 - 6*z * q^53 + (-3*z + 3) * q^55 + (15*z - 15) * q^59 - 5*z * q^61 - 8 * q^64 + 2 * q^65 - 2*z * q^67 + 12 * q^68 + (3*z - 3) * q^71 + (8*z - 8) * q^73 + (-10*z + 6) * q^76 + 12 * q^77 + (5*z - 5) * q^79 - 4*z * q^80 - 12 * q^83 + 6*z * q^85 - 15*z * q^89 + 8*z * q^91 + (-2*z + 5) * q^95 + (8*z - 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - q^{5} - 8 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 - q^5 - 8 * q^7 $$2 q + 2 q^{4} - q^{5} - 8 q^{7} - 6 q^{11} - 2 q^{13} - 4 q^{16} + 6 q^{17} - 7 q^{19} - 4 q^{20} - q^{25} - 8 q^{28} - 3 q^{29} - 14 q^{31} + 4 q^{35} + 16 q^{37} - 6 q^{41} + 4 q^{43} - 6 q^{44} + 6 q^{47} + 18 q^{49} + 4 q^{52} - 6 q^{53} + 3 q^{55} - 15 q^{59} - 5 q^{61} - 16 q^{64} + 4 q^{65} - 2 q^{67} + 24 q^{68} - 3 q^{71} - 8 q^{73} + 2 q^{76} + 24 q^{77} - 5 q^{79} - 4 q^{80} - 24 q^{83} + 6 q^{85} - 15 q^{89} + 8 q^{91} + 8 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 - q^5 - 8 * q^7 - 6 * q^11 - 2 * q^13 - 4 * q^16 + 6 * q^17 - 7 * q^19 - 4 * q^20 - q^25 - 8 * q^28 - 3 * q^29 - 14 * q^31 + 4 * q^35 + 16 * q^37 - 6 * q^41 + 4 * q^43 - 6 * q^44 + 6 * q^47 + 18 * q^49 + 4 * q^52 - 6 * q^53 + 3 * q^55 - 15 * q^59 - 5 * q^61 - 16 * q^64 + 4 * q^65 - 2 * q^67 + 24 * q^68 - 3 * q^71 - 8 * q^73 + 2 * q^76 + 24 * q^77 - 5 * q^79 - 4 * q^80 - 24 * q^83 + 6 * q^85 - 15 * q^89 + 8 * q^91 + 8 * q^95 - 8 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$$.

 $$n$$ $$172$$ $$191$$ $$496$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
406.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 1.00000 1.73205i −0.500000 0.866025i 0 −4.00000 0 0 0
676.1 0 0 1.00000 + 1.73205i −0.500000 + 0.866025i 0 −4.00000 0 0 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.k.b 2
3.b odd 2 1 95.2.e.a 2
12.b even 2 1 1520.2.q.c 2
15.d odd 2 1 475.2.e.b 2
15.e even 4 2 475.2.j.a 4
19.c even 3 1 inner 855.2.k.b 2
57.f even 6 1 1805.2.a.b 1
57.h odd 6 1 95.2.e.a 2
57.h odd 6 1 1805.2.a.a 1
228.m even 6 1 1520.2.q.c 2
285.n odd 6 1 475.2.e.b 2
285.n odd 6 1 9025.2.a.g 1
285.q even 6 1 9025.2.a.e 1
285.v even 12 2 475.2.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.a 2 3.b odd 2 1
95.2.e.a 2 57.h odd 6 1
475.2.e.b 2 15.d odd 2 1
475.2.e.b 2 285.n odd 6 1
475.2.j.a 4 15.e even 4 2
475.2.j.a 4 285.v even 12 2
855.2.k.b 2 1.a even 1 1 trivial
855.2.k.b 2 19.c even 3 1 inner
1520.2.q.c 2 12.b even 2 1
1520.2.q.c 2 228.m even 6 1
1805.2.a.a 1 57.h odd 6 1
1805.2.a.b 1 57.f even 6 1
9025.2.a.e 1 285.q even 6 1
9025.2.a.g 1 285.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(855, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7} + 4$$ T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$(T + 4)^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 7T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$(T + 7)^{2}$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} + 15T + 225$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$T^{2} + 3T + 9$$
$73$ $$T^{2} + 8T + 64$$
$79$ $$T^{2} + 5T + 25$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 15T + 225$$
$97$ $$T^{2} + 8T + 64$$