# Properties

 Label 855.2.k.d Level $855$ Weight $2$ Character orbit 855.k Analytic conductor $6.827$ Analytic rank $0$ 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: $$0$$ 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 285) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} - 2 q^{7} +O(q^{10})$$ q + (-2*z + 2) * q^2 - 2*z * q^4 + (z - 1) * q^5 - 2 * q^7 $$q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} - 2 q^{7} + 2 \zeta_{6} q^{10} + 3 q^{11} - 6 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{14} + ( - 4 \zeta_{6} + 4) q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + ( - 3 \zeta_{6} - 2) q^{19} + 2 q^{20} + ( - 6 \zeta_{6} + 6) q^{22} - 8 \zeta_{6} q^{23} - \zeta_{6} q^{25} - 12 q^{26} + 4 \zeta_{6} q^{28} + 7 \zeta_{6} q^{29} + 9 q^{31} - 8 \zeta_{6} q^{32} - 12 \zeta_{6} q^{34} + ( - 2 \zeta_{6} + 2) q^{35} - 2 q^{37} + (4 \zeta_{6} - 10) q^{38} + ( - 6 \zeta_{6} + 6) q^{41} + (10 \zeta_{6} - 10) q^{43} - 6 \zeta_{6} q^{44} - 16 q^{46} + 4 \zeta_{6} q^{47} - 3 q^{49} - 2 q^{50} + (12 \zeta_{6} - 12) q^{52} + 14 \zeta_{6} q^{53} + (3 \zeta_{6} - 3) q^{55} + 14 q^{58} + ( - 3 \zeta_{6} + 3) q^{59} + 7 \zeta_{6} q^{61} + ( - 18 \zeta_{6} + 18) q^{62} - 8 q^{64} + 6 q^{65} - 4 \zeta_{6} q^{67} - 12 q^{68} - 4 \zeta_{6} q^{70} + ( - 7 \zeta_{6} + 7) q^{71} + (2 \zeta_{6} - 2) q^{73} + (4 \zeta_{6} - 4) q^{74} + (10 \zeta_{6} - 6) q^{76} - 6 q^{77} + (5 \zeta_{6} - 5) q^{79} + 4 \zeta_{6} q^{80} - 12 \zeta_{6} q^{82} + 6 q^{83} + 6 \zeta_{6} q^{85} + 20 \zeta_{6} q^{86} + 3 \zeta_{6} q^{89} + 12 \zeta_{6} q^{91} + (16 \zeta_{6} - 16) q^{92} + 8 q^{94} + ( - 2 \zeta_{6} + 5) q^{95} + ( - 12 \zeta_{6} + 12) q^{97} + (6 \zeta_{6} - 6) q^{98} +O(q^{100})$$ q + (-2*z + 2) * q^2 - 2*z * q^4 + (z - 1) * q^5 - 2 * q^7 + 2*z * q^10 + 3 * q^11 - 6*z * q^13 + (4*z - 4) * q^14 + (-4*z + 4) * q^16 + (-6*z + 6) * q^17 + (-3*z - 2) * q^19 + 2 * q^20 + (-6*z + 6) * q^22 - 8*z * q^23 - z * q^25 - 12 * q^26 + 4*z * q^28 + 7*z * q^29 + 9 * q^31 - 8*z * q^32 - 12*z * q^34 + (-2*z + 2) * q^35 - 2 * q^37 + (4*z - 10) * q^38 + (-6*z + 6) * q^41 + (10*z - 10) * q^43 - 6*z * q^44 - 16 * q^46 + 4*z * q^47 - 3 * q^49 - 2 * q^50 + (12*z - 12) * q^52 + 14*z * q^53 + (3*z - 3) * q^55 + 14 * q^58 + (-3*z + 3) * q^59 + 7*z * q^61 + (-18*z + 18) * q^62 - 8 * q^64 + 6 * q^65 - 4*z * q^67 - 12 * q^68 - 4*z * q^70 + (-7*z + 7) * q^71 + (2*z - 2) * q^73 + (4*z - 4) * q^74 + (10*z - 6) * q^76 - 6 * q^77 + (5*z - 5) * q^79 + 4*z * q^80 - 12*z * q^82 + 6 * q^83 + 6*z * q^85 + 20*z * q^86 + 3*z * q^89 + 12*z * q^91 + (16*z - 16) * q^92 + 8 * q^94 + (-2*z + 5) * q^95 + (-12*z + 12) * q^97 + (6*z - 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{4} - q^{5} - 4 q^{7}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^4 - q^5 - 4 * q^7 $$2 q + 2 q^{2} - 2 q^{4} - q^{5} - 4 q^{7} + 2 q^{10} + 6 q^{11} - 6 q^{13} - 4 q^{14} + 4 q^{16} + 6 q^{17} - 7 q^{19} + 4 q^{20} + 6 q^{22} - 8 q^{23} - q^{25} - 24 q^{26} + 4 q^{28} + 7 q^{29} + 18 q^{31} - 8 q^{32} - 12 q^{34} + 2 q^{35} - 4 q^{37} - 16 q^{38} + 6 q^{41} - 10 q^{43} - 6 q^{44} - 32 q^{46} + 4 q^{47} - 6 q^{49} - 4 q^{50} - 12 q^{52} + 14 q^{53} - 3 q^{55} + 28 q^{58} + 3 q^{59} + 7 q^{61} + 18 q^{62} - 16 q^{64} + 12 q^{65} - 4 q^{67} - 24 q^{68} - 4 q^{70} + 7 q^{71} - 2 q^{73} - 4 q^{74} - 2 q^{76} - 12 q^{77} - 5 q^{79} + 4 q^{80} - 12 q^{82} + 12 q^{83} + 6 q^{85} + 20 q^{86} + 3 q^{89} + 12 q^{91} - 16 q^{92} + 16 q^{94} + 8 q^{95} + 12 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^4 - q^5 - 4 * q^7 + 2 * q^10 + 6 * q^11 - 6 * q^13 - 4 * q^14 + 4 * q^16 + 6 * q^17 - 7 * q^19 + 4 * q^20 + 6 * q^22 - 8 * q^23 - q^25 - 24 * q^26 + 4 * q^28 + 7 * q^29 + 18 * q^31 - 8 * q^32 - 12 * q^34 + 2 * q^35 - 4 * q^37 - 16 * q^38 + 6 * q^41 - 10 * q^43 - 6 * q^44 - 32 * q^46 + 4 * q^47 - 6 * q^49 - 4 * q^50 - 12 * q^52 + 14 * q^53 - 3 * q^55 + 28 * q^58 + 3 * q^59 + 7 * q^61 + 18 * q^62 - 16 * q^64 + 12 * q^65 - 4 * q^67 - 24 * q^68 - 4 * q^70 + 7 * q^71 - 2 * q^73 - 4 * q^74 - 2 * q^76 - 12 * q^77 - 5 * q^79 + 4 * q^80 - 12 * q^82 + 12 * q^83 + 6 * q^85 + 20 * q^86 + 3 * q^89 + 12 * q^91 - 16 * q^92 + 16 * q^94 + 8 * q^95 + 12 * q^97 - 6 * q^98

## 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
1.00000 + 1.73205i 0 −1.00000 + 1.73205i −0.500000 0.866025i 0 −2.00000 0 0 1.00000 1.73205i
676.1 1.00000 1.73205i 0 −1.00000 1.73205i −0.500000 + 0.866025i 0 −2.00000 0 0 1.00000 + 1.73205i
 $$n$$: 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.d 2
3.b odd 2 1 285.2.i.a 2
19.c even 3 1 inner 855.2.k.d 2
57.f even 6 1 5415.2.a.a 1
57.h odd 6 1 285.2.i.a 2
57.h odd 6 1 5415.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.a 2 3.b odd 2 1
285.2.i.a 2 57.h odd 6 1
855.2.k.d 2 1.a even 1 1 trivial
855.2.k.d 2 19.c even 3 1 inner
5415.2.a.a 1 57.f even 6 1
5415.2.a.k 1 57.h odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 2T_{2} + 4$$ T2^2 - 2*T2 + 4 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$(T + 2)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 6T + 36$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 7T + 19$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$T^{2} - 7T + 49$$
$31$ $$(T - 9)^{2}$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} + 10T + 100$$
$47$ $$T^{2} - 4T + 16$$
$53$ $$T^{2} - 14T + 196$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2} - 7T + 49$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} + 5T + 25$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 3T + 9$$
$97$ $$T^{2} - 12T + 144$$