# Properties

 Label 855.2.k.f Level $855$ Weight $2$ Character orbit 855.k Analytic conductor $6.827$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + ( - \beta_{2} - 1) q^{5} + (2 \beta_{3} - 1) q^{7} + (\beta_{3} - 3) q^{8}+O(q^{10})$$ q + (b2 + b1 + 1) * q^2 + (2*b3 + b2 + 2*b1) * q^4 + (-b2 - 1) * q^5 + (2*b3 - 1) * q^7 + (b3 - 3) * q^8 $$q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + ( - \beta_{2} - 1) q^{5} + (2 \beta_{3} - 1) q^{7} + (\beta_{3} - 3) q^{8} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{10} + 2 \beta_{3} q^{11} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{13} + ( - 5 \beta_{2} - 3 \beta_1 - 5) q^{14} + ( - 3 \beta_{2} - 3) q^{16} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{17} + (2 \beta_{2} + 5) q^{19} + ( - 2 \beta_{3} + 1) q^{20} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{22} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{23} + \beta_{2} q^{25} + (3 \beta_{3} - 5) q^{26} + ( - 4 \beta_{3} - 9 \beta_{2} - 4 \beta_1) q^{28} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{29} - 5 q^{31} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{32} + ( - 6 \beta_{3} - 8 \beta_{2} - 6 \beta_1) q^{34} + (\beta_{2} + 2 \beta_1 + 1) q^{35} + (2 \beta_{3} - 5) q^{37} + (2 \beta_{3} + 5 \beta_{2} + 5 \beta_1 + 3) q^{38} + (3 \beta_{2} + \beta_1 + 3) q^{40} - 2 \beta_1 q^{41} + ( - 5 \beta_{2} + 2 \beta_1 - 5) q^{43} + ( - 2 \beta_{3} - 8 \beta_{2} - 2 \beta_1) q^{44} + ( - 4 \beta_{3} + 6) q^{46} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{47} + ( - 4 \beta_{3} + 2) q^{49} + (\beta_{3} - 1) q^{50} + ( - 9 \beta_{2} - 4 \beta_1 - 9) q^{52} + 2 \beta_{2} q^{53} + 2 \beta_1 q^{55} + ( - 7 \beta_{3} + 7) q^{56} - 4 q^{58} + (8 \beta_{3} + 3 \beta_{2} + 8 \beta_1) q^{61} + ( - 5 \beta_{2} - 5 \beta_1 - 5) q^{62} + (2 \beta_{3} - 7) q^{64} + ( - 2 \beta_{3} + 1) q^{65} + ( - 6 \beta_{3} - 3 \beta_{2} - 6 \beta_1) q^{67} + ( - 10 \beta_{3} + 12) q^{68} + (3 \beta_{3} + 5 \beta_{2} + 3 \beta_1) q^{70} + (10 \beta_{2} + 10) q^{71} + ( - \beta_{2} + 6 \beta_1 - 1) q^{73} + ( - 9 \beta_{2} - 7 \beta_1 - 9) q^{74} + (10 \beta_{3} + 3 \beta_{2} + 6 \beta_1 - 2) q^{76} + ( - 2 \beta_{3} + 8) q^{77} + (9 \beta_{2} + 4 \beta_1 + 9) q^{79} + 3 \beta_{2} q^{80} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{82} + 8 q^{83} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{85} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{86} + ( - 6 \beta_{3} + 4) q^{88} + (6 \beta_{3} - 4 \beta_{2} + 6 \beta_1) q^{89} + ( - 4 \beta_{3} - 9 \beta_{2} - 4 \beta_1) q^{91} + (10 \beta_{2} + 6 \beta_1 + 10) q^{92} + (8 \beta_{3} - 10) q^{94} + ( - 5 \beta_{2} - 3) q^{95} + (6 \beta_{2} + 6) q^{97} + (10 \beta_{2} + 6 \beta_1 + 10) q^{98}+O(q^{100})$$ q + (b2 + b1 + 1) * q^2 + (2*b3 + b2 + 2*b1) * q^4 + (-b2 - 1) * q^5 + (2*b3 - 1) * q^7 + (b3 - 3) * q^8 + (-b3 - b2 - b1) * q^10 + 2*b3 * q^11 + (2*b3 + b2 + 2*b1) * q^13 + (-5*b2 - 3*b1 - 5) * q^14 + (-3*b2 - 3) * q^16 + (-4*b2 - 2*b1 - 4) * q^17 + (2*b2 + 5) * q^19 + (-2*b3 + 1) * q^20 + (-4*b2 - 2*b1 - 4) * q^22 + (-2*b3 - 2*b2 - 2*b1) * q^23 + b2 * q^25 + (3*b3 - 5) * q^26 + (-4*b3 - 9*b2 - 4*b1) * q^28 + (4*b3 - 4*b2 + 4*b1) * q^29 - 5 * q^31 + (-b3 + 3*b2 - b1) * q^32 + (-6*b3 - 8*b2 - 6*b1) * q^34 + (b2 + 2*b1 + 1) * q^35 + (2*b3 - 5) * q^37 + (2*b3 + 5*b2 + 5*b1 + 3) * q^38 + (3*b2 + b1 + 3) * q^40 - 2*b1 * q^41 + (-5*b2 + 2*b1 - 5) * q^43 + (-2*b3 - 8*b2 - 2*b1) * q^44 + (-4*b3 + 6) * q^46 + (2*b3 + 6*b2 + 2*b1) * q^47 + (-4*b3 + 2) * q^49 + (b3 - 1) * q^50 + (-9*b2 - 4*b1 - 9) * q^52 + 2*b2 * q^53 + 2*b1 * q^55 + (-7*b3 + 7) * q^56 - 4 * q^58 + (8*b3 + 3*b2 + 8*b1) * q^61 + (-5*b2 - 5*b1 - 5) * q^62 + (2*b3 - 7) * q^64 + (-2*b3 + 1) * q^65 + (-6*b3 - 3*b2 - 6*b1) * q^67 + (-10*b3 + 12) * q^68 + (3*b3 + 5*b2 + 3*b1) * q^70 + (10*b2 + 10) * q^71 + (-b2 + 6*b1 - 1) * q^73 + (-9*b2 - 7*b1 - 9) * q^74 + (10*b3 + 3*b2 + 6*b1 - 2) * q^76 + (-2*b3 + 8) * q^77 + (9*b2 + 4*b1 + 9) * q^79 + 3*b2 * q^80 + (-2*b3 - 4*b2 - 2*b1) * q^82 + 8 * q^83 + (2*b3 + 4*b2 + 2*b1) * q^85 + (-3*b3 - b2 - 3*b1) * q^86 + (-6*b3 + 4) * q^88 + (6*b3 - 4*b2 + 6*b1) * q^89 + (-4*b3 - 9*b2 - 4*b1) * q^91 + (10*b2 + 6*b1 + 10) * q^92 + (8*b3 - 10) * q^94 + (-5*b2 - 3) * q^95 + (6*b2 + 6) * q^97 + (10*b2 + 6*b1 + 10) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8 $$4 q + 2 q^{2} - 2 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8} + 2 q^{10} - 2 q^{13} - 10 q^{14} - 6 q^{16} - 8 q^{17} + 16 q^{19} + 4 q^{20} - 8 q^{22} + 4 q^{23} - 2 q^{25} - 20 q^{26} + 18 q^{28} + 8 q^{29} - 20 q^{31} - 6 q^{32} + 16 q^{34} + 2 q^{35} - 20 q^{37} + 2 q^{38} + 6 q^{40} - 10 q^{43} + 16 q^{44} + 24 q^{46} - 12 q^{47} + 8 q^{49} - 4 q^{50} - 18 q^{52} - 4 q^{53} + 28 q^{56} - 16 q^{58} - 6 q^{61} - 10 q^{62} - 28 q^{64} + 4 q^{65} + 6 q^{67} + 48 q^{68} - 10 q^{70} + 20 q^{71} - 2 q^{73} - 18 q^{74} - 14 q^{76} + 32 q^{77} + 18 q^{79} - 6 q^{80} + 8 q^{82} + 32 q^{83} - 8 q^{85} + 2 q^{86} + 16 q^{88} + 8 q^{89} + 18 q^{91} + 20 q^{92} - 40 q^{94} - 2 q^{95} + 12 q^{97} + 20 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8 + 2 * q^10 - 2 * q^13 - 10 * q^14 - 6 * q^16 - 8 * q^17 + 16 * q^19 + 4 * q^20 - 8 * q^22 + 4 * q^23 - 2 * q^25 - 20 * q^26 + 18 * q^28 + 8 * q^29 - 20 * q^31 - 6 * q^32 + 16 * q^34 + 2 * q^35 - 20 * q^37 + 2 * q^38 + 6 * q^40 - 10 * q^43 + 16 * q^44 + 24 * q^46 - 12 * q^47 + 8 * q^49 - 4 * q^50 - 18 * q^52 - 4 * q^53 + 28 * q^56 - 16 * q^58 - 6 * q^61 - 10 * q^62 - 28 * q^64 + 4 * q^65 + 6 * q^67 + 48 * q^68 - 10 * q^70 + 20 * q^71 - 2 * q^73 - 18 * q^74 - 14 * q^76 + 32 * q^77 + 18 * q^79 - 6 * q^80 + 8 * q^82 + 32 * q^83 - 8 * q^85 + 2 * q^86 + 16 * q^88 + 8 * q^89 + 18 * q^91 + 20 * q^92 - 40 * q^94 - 2 * q^95 + 12 * q^97 + 20 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$172$$ $$191$$ $$496$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.207107 0.358719i 0 0.914214 1.58346i −0.500000 0.866025i 0 1.82843 −1.58579 0 −0.207107 + 0.358719i
406.2 1.20711 + 2.09077i 0 −1.91421 + 3.31552i −0.500000 0.866025i 0 −3.82843 −4.41421 0 1.20711 2.09077i
676.1 −0.207107 + 0.358719i 0 0.914214 + 1.58346i −0.500000 + 0.866025i 0 1.82843 −1.58579 0 −0.207107 0.358719i
676.2 1.20711 2.09077i 0 −1.91421 3.31552i −0.500000 + 0.866025i 0 −3.82843 −4.41421 0 1.20711 + 2.09077i
 $$n$$: 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.f 4
3.b odd 2 1 285.2.i.d 4
19.c even 3 1 inner 855.2.k.f 4
57.f even 6 1 5415.2.a.o 2
57.h odd 6 1 285.2.i.d 4
57.h odd 6 1 5415.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.d 4 3.b odd 2 1
285.2.i.d 4 57.h odd 6 1
855.2.k.f 4 1.a even 1 1 trivial
855.2.k.f 4 19.c even 3 1 inner
5415.2.a.o 2 57.f even 6 1
5415.2.a.u 2 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 $$S_{2}^{\mathrm{new}}(855, [\chi])$$:

 $$T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1$$ T2^4 - 2*T2^3 + 5*T2^2 + 2*T2 + 1 $$T_{7}^{2} + 2T_{7} - 7$$ T7^2 + 2*T7 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$(T^{2} + 2 T - 7)^{2}$$
$11$ $$(T^{2} - 8)^{2}$$
$13$ $$T^{4} + 2 T^{3} + 11 T^{2} - 14 T + 49$$
$17$ $$T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64$$
$19$ $$(T^{2} - 8 T + 19)^{2}$$
$23$ $$T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16$$
$29$ $$T^{4} - 8 T^{3} + 80 T^{2} + 128 T + 256$$
$31$ $$(T + 5)^{4}$$
$37$ $$(T^{2} + 10 T + 17)^{2}$$
$41$ $$T^{4} + 8T^{2} + 64$$
$43$ $$T^{4} + 10 T^{3} + 83 T^{2} + \cdots + 289$$
$47$ $$T^{4} + 12 T^{3} + 116 T^{2} + \cdots + 784$$
$53$ $$(T^{2} + 2 T + 4)^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 6 T^{3} + 155 T^{2} + \cdots + 14161$$
$67$ $$T^{4} - 6 T^{3} + 99 T^{2} + \cdots + 3969$$
$71$ $$(T^{2} - 10 T + 100)^{2}$$
$73$ $$T^{4} + 2 T^{3} + 75 T^{2} + \cdots + 5041$$
$79$ $$T^{4} - 18 T^{3} + 275 T^{2} + \cdots + 2401$$
$83$ $$(T - 8)^{4}$$
$89$ $$T^{4} - 8 T^{3} + 120 T^{2} + \cdots + 3136$$
$97$ $$(T^{2} - 6 T + 36)^{2}$$