# Properties

 Label 855.2.k.g Level $855$ Weight $2$ Character orbit 855.k Analytic conductor $6.827$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.3518667.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49$$ x^6 - x^5 + 7*x^4 - 8*x^3 + 43*x^2 - 42*x + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 3) q^{4} + \beta_{3} q^{5} + ( - \beta_{4} + 1) q^{7} + (\beta_{4} - \beta_{2} + 2) q^{8}+O(q^{10})$$ q + b1 * q^2 + (-b5 + b4 - 2*b3 + b2 - 3) * q^4 + b3 * q^5 + (-b4 + 1) * q^7 + (b4 - b2 + 2) * q^8 $$q + \beta_1 q^{2} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 3) q^{4} + \beta_{3} q^{5} + ( - \beta_{4} + 1) q^{7} + (\beta_{4} - \beta_{2} + 2) q^{8} + (\beta_{2} - \beta_1) q^{10} + ( - \beta_{4} + 2) q^{11} + (5 \beta_{3} + 5) q^{13} + ( - 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{14} + (\beta_{5} + \beta_{3} + \beta_1) q^{16} + ( - 2 \beta_{5} - \beta_1) q^{17} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 - 1) q^{19} + ( - \beta_{4} - \beta_{2} + 3) q^{20} + ( - 2 \beta_{5} - \beta_{3} + 3 \beta_1) q^{22} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{23} + ( - \beta_{3} - 1) q^{25} + 5 \beta_{2} q^{26} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 4) q^{28} + ( - \beta_{5} + \beta_{4} + \beta_1 - 1) q^{29} - \beta_{2} q^{31} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{32} + ( - \beta_{5} + \beta_{4} + 10 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 9) q^{34} + (\beta_{5} - \beta_1) q^{35} + (5 \beta_{4} + 2 \beta_{2} - 3) q^{37} + (3 \beta_{5} + 5 \beta_{3} - 2 \beta_{2} - \beta_1 + 7) q^{38} + ( - \beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{40} + ( - \beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{41} + (\beta_{5} + 2 \beta_1) q^{43} + ( - 3 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} + 3 \beta_1 - 7) q^{44} + ( - \beta_{4} + 4 \beta_{2} - 9) q^{46} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{47} + ( - 2 \beta_{4} + \beta_{2} - 2) q^{49} - \beta_{2} q^{50} + ( - 5 \beta_{5} - 10 \beta_{3}) q^{52} + ( - 4 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{53} + (\beta_{5} + \beta_{3} - \beta_1) q^{55} + (\beta_{4} - \beta_{2} - 5) q^{56} + (2 \beta_{4} - 3) q^{58} + (\beta_{5} - 2 \beta_{3} + \beta_1) q^{59} + (2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 5) q^{61} + (\beta_{5} + 4 \beta_{3}) q^{62} + (5 \beta_{4} + 2 \beta_{2} - 5) q^{64} - 5 q^{65} + (2 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 8) q^{67} + (\beta_{4} + 7 \beta_{2} - 14) q^{68} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_1 + 3) q^{70} + (3 \beta_{5} + 8 \beta_{3} - 2 \beta_1) q^{71} + ( - 2 \beta_{5} - 7 \beta_{3} - \beta_1) q^{73} + (8 \beta_{5} - 3 \beta_{3} - 8 \beta_1) q^{74} + (2 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 1) q^{76} + ( - 3 \beta_{4} + \beta_{2} + 6) q^{77} + (2 \beta_{5} - 8 \beta_{3} + 2 \beta_1) q^{79} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{80} + ( - 4 \beta_{5} + 4 \beta_{4} - 9 \beta_{3} + 4 \beta_{2} - 13) q^{82} + ( - 3 \beta_{2} + 2) q^{83} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} + \beta_1 + 2) q^{85} + ( - \beta_{5} + \beta_{4} - 11 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 12) q^{86} + (2 \beta_{4} - 2 \beta_{2} - 3) q^{88} + (4 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} - 4 \beta_{2} - 2) q^{89} + (5 \beta_{5} - 5 \beta_{4} - 5 \beta_1 + 5) q^{91} + ( - 2 \beta_{5} - 15 \beta_{3} - 6 \beta_1) q^{92} + 7 q^{94} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{95} + ( - 3 \beta_{5} + 3 \beta_{3} - \beta_1) q^{97} + ( - 5 \beta_{5} - 6 \beta_{3}) q^{98}+O(q^{100})$$ q + b1 * q^2 + (-b5 + b4 - 2*b3 + b2 - 3) * q^4 + b3 * q^5 + (-b4 + 1) * q^7 + (b4 - b2 + 2) * q^8 + (b2 - b1) * q^10 + (-b4 + 2) * q^11 + (5*b3 + 5) * q^13 + (-2*b5 - b3 + 2*b1) * q^14 + (b5 + b3 + b1) * q^16 + (-2*b5 - b1) * q^17 + (-b5 + 2*b4 + b2 - b1 - 1) * q^19 + (-b4 - b2 + 3) * q^20 + (-2*b5 - b3 + 3*b1) * q^22 + (2*b5 - 2*b4 + b3 - 3*b2 + b1 + 3) * q^23 + (-b3 - 1) * q^25 + 5*b2 * q^26 + (-2*b5 + 2*b4 - 2*b3 - b2 + 3*b1 - 4) * q^28 + (-b5 + b4 + b1 - 1) * q^29 - b2 * q^31 + (-2*b5 + 2*b4 - b3 + b2 + b1 - 3) * q^32 + (-b5 + b4 + 10*b3 - 3*b2 + 4*b1 + 9) * q^34 + (b5 - b1) * q^35 + (5*b4 + 2*b2 - 3) * q^37 + (3*b5 + 5*b3 - 2*b2 - b1 + 7) * q^38 + (-b5 + 3*b3 + 2*b1) * q^40 + (-b5 + 2*b3 + 3*b1) * q^41 + (b5 + 2*b1) * q^43 + (-3*b5 + 3*b4 - 4*b3 + 3*b1 - 7) * q^44 + (-b4 + 4*b2 - 9) * q^46 + (-b5 + b4 + 2*b3 + 2*b2 - b1 + 1) * q^47 + (-2*b4 + b2 - 2) * q^49 - b2 * q^50 + (-5*b5 - 10*b3) * q^52 + (-4*b5 + 4*b4 + 6*b3 + b2 + 3*b1 + 2) * q^53 + (b5 + b3 - b1) * q^55 + (b4 - b2 - 5) * q^56 + (2*b4 - 3) * q^58 + (b5 - 2*b3 + b1) * q^59 + (2*b5 - 2*b4 + 3*b3 - 4*b2 + 2*b1 + 5) * q^61 + (b5 + 4*b3) * q^62 + (5*b4 + 2*b2 - 5) * q^64 - 5 * q^65 + (2*b5 - 2*b4 + 6*b3 - 2*b2 + 8) * q^67 + (b4 + 7*b2 - 14) * q^68 + (2*b5 - 2*b4 + b3 - 2*b1 + 3) * q^70 + (3*b5 + 8*b3 - 2*b1) * q^71 + (-2*b5 - 7*b3 - b1) * q^73 + (8*b5 - 3*b3 - 8*b1) * q^74 + (2*b5 - 2*b4 + 5*b3 + 5*b2 - 2*b1 - 1) * q^76 + (-3*b4 + b2 + 6) * q^77 + (2*b5 - 8*b3 + 2*b1) * q^79 + (-b5 + b4 - b3 + 2*b2 - b1 - 2) * q^80 + (-4*b5 + 4*b4 - 9*b3 + 4*b2 - 13) * q^82 + (-3*b2 + 2) * q^83 + (2*b5 - 2*b4 - 3*b2 + b1 + 2) * q^85 + (-b5 + b4 - 11*b3 + 3*b2 - 2*b1 - 12) * q^86 + (2*b4 - 2*b2 - 3) * q^88 + (4*b5 - 4*b4 - 6*b3 - 4*b2 - 2) * q^89 + (5*b5 - 5*b4 - 5*b1 + 5) * q^91 + (-2*b5 - 15*b3 - 6*b1) * q^92 + 7 * q^94 + (-b5 - b4 + b3 - 2*b2 + 2*b1 + 1) * q^95 + (-3*b5 + 3*b3 - b1) * q^97 + (-5*b5 - 6*b3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{2} - 7 q^{4} - 3 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10})$$ 6 * q + q^2 - 7 * q^4 - 3 * q^5 + 4 * q^7 + 12 * q^8 $$6 q + q^{2} - 7 q^{4} - 3 q^{5} + 4 q^{7} + 12 q^{8} + q^{10} + 10 q^{11} + 15 q^{13} + 7 q^{14} - 3 q^{16} + q^{17} + 14 q^{20} + 8 q^{22} + 4 q^{23} - 3 q^{25} + 10 q^{26} - 11 q^{28} - 2 q^{29} - 2 q^{31} - 6 q^{32} + 25 q^{34} - 2 q^{35} - 4 q^{37} + 19 q^{38} - 6 q^{40} - 2 q^{41} + q^{43} - 18 q^{44} - 48 q^{46} + 6 q^{47} - 14 q^{49} - 2 q^{50} + 35 q^{52} + 11 q^{53} - 5 q^{55} - 30 q^{56} - 14 q^{58} + 6 q^{59} + 9 q^{61} - 13 q^{62} - 16 q^{64} - 30 q^{65} + 20 q^{67} - 68 q^{68} + 7 q^{70} - 29 q^{71} + 22 q^{73} - 7 q^{74} - 19 q^{76} + 32 q^{77} + 24 q^{79} - 3 q^{80} - 31 q^{82} + 6 q^{83} + q^{85} - 32 q^{86} - 18 q^{88} - 14 q^{89} + 10 q^{91} + 41 q^{92} + 42 q^{94} - 7 q^{97} + 23 q^{98}+O(q^{100})$$ 6 * q + q^2 - 7 * q^4 - 3 * q^5 + 4 * q^7 + 12 * q^8 + q^10 + 10 * q^11 + 15 * q^13 + 7 * q^14 - 3 * q^16 + q^17 + 14 * q^20 + 8 * q^22 + 4 * q^23 - 3 * q^25 + 10 * q^26 - 11 * q^28 - 2 * q^29 - 2 * q^31 - 6 * q^32 + 25 * q^34 - 2 * q^35 - 4 * q^37 + 19 * q^38 - 6 * q^40 - 2 * q^41 + q^43 - 18 * q^44 - 48 * q^46 + 6 * q^47 - 14 * q^49 - 2 * q^50 + 35 * q^52 + 11 * q^53 - 5 * q^55 - 30 * q^56 - 14 * q^58 + 6 * q^59 + 9 * q^61 - 13 * q^62 - 16 * q^64 - 30 * q^65 + 20 * q^67 - 68 * q^68 + 7 * q^70 - 29 * q^71 + 22 * q^73 - 7 * q^74 - 19 * q^76 + 32 * q^77 + 24 * q^79 - 3 * q^80 - 31 * q^82 + 6 * q^83 + q^85 - 32 * q^86 - 18 * q^88 - 14 * q^89 + 10 * q^91 + 41 * q^92 + 42 * q^94 - 7 * q^97 + 23 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 7\nu^{4} - 49\nu^{3} + 43\nu^{2} - 42\nu + 294 ) / 259$$ (-v^5 + 7*v^4 - 49*v^3 + 43*v^2 - 42*v + 294) / 259 $$\beta_{3}$$ $$=$$ $$( -6\nu^{5} + 5\nu^{4} - 35\nu^{3} - \nu^{2} - 215\nu - 49 ) / 259$$ (-6*v^5 + 5*v^4 - 35*v^3 - v^2 - 215*v - 49) / 259 $$\beta_{4}$$ $$=$$ $$( -5\nu^{5} + 35\nu^{4} + 14\nu^{3} + 215\nu^{2} - 210\nu + 952 ) / 259$$ (-5*v^5 + 35*v^4 + 14*v^3 + 215*v^2 - 210*v + 952) / 259 $$\beta_{5}$$ $$=$$ $$( 18\nu^{5} + 22\nu^{4} + 105\nu^{3} + 3\nu^{2} + 608\nu + 147 ) / 259$$ (18*v^5 + 22*v^4 + 105*v^3 + 3*v^2 + 608*v + 147) / 259
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} - 5$$ -b5 + b4 - 4*b3 + b2 - 5 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 5\beta_{2} + 2$$ b4 - 5*b2 + 2 $$\nu^{4}$$ $$=$$ $$7\beta_{5} + 21\beta_{3} + \beta_1$$ 7*b5 + 21*b3 + b1 $$\nu^{5}$$ $$=$$ $$6\beta_{5} - 6\beta_{4} - 25\beta_{3} + 29\beta_{2} - 35\beta _1 - 19$$ 6*b5 - 6*b4 - 25*b3 + 29*b2 - 35*b1 - 19

## Character values

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

 $$n$$ $$172$$ $$191$$ $$496$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{3}$$

## 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
 −1.25351 − 2.17114i 0.610938 + 1.05818i 1.14257 + 1.97899i −1.25351 + 2.17114i 0.610938 − 1.05818i 1.14257 − 1.97899i
−1.25351 2.17114i 0 −2.14257 + 3.71104i −0.500000 0.866025i 0 −0.221876 5.72889 0 −1.25351 + 2.17114i
406.2 0.610938 + 1.05818i 0 0.253509 0.439091i −0.500000 0.866025i 0 −1.28514 3.06327 0 0.610938 1.05818i
406.3 1.14257 + 1.97899i 0 −1.61094 + 2.79023i −0.500000 0.866025i 0 3.50702 −2.79216 0 1.14257 1.97899i
676.1 −1.25351 + 2.17114i 0 −2.14257 3.71104i −0.500000 + 0.866025i 0 −0.221876 5.72889 0 −1.25351 2.17114i
676.2 0.610938 1.05818i 0 0.253509 + 0.439091i −0.500000 + 0.866025i 0 −1.28514 3.06327 0 0.610938 + 1.05818i
676.3 1.14257 1.97899i 0 −1.61094 2.79023i −0.500000 + 0.866025i 0 3.50702 −2.79216 0 1.14257 + 1.97899i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 676.3 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.g 6
3.b odd 2 1 95.2.e.b 6
12.b even 2 1 1520.2.q.j 6
15.d odd 2 1 475.2.e.d 6
15.e even 4 2 475.2.j.b 12
19.c even 3 1 inner 855.2.k.g 6
57.f even 6 1 1805.2.a.g 3
57.h odd 6 1 95.2.e.b 6
57.h odd 6 1 1805.2.a.h 3
228.m even 6 1 1520.2.q.j 6
285.n odd 6 1 475.2.e.d 6
285.n odd 6 1 9025.2.a.z 3
285.q even 6 1 9025.2.a.ba 3
285.v even 12 2 475.2.j.b 12

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

## Hecke kernels

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

 $$T_{2}^{6} - T_{2}^{5} + 7T_{2}^{4} - 8T_{2}^{3} + 43T_{2}^{2} - 42T_{2} + 49$$ T2^6 - T2^5 + 7*T2^4 - 8*T2^3 + 43*T2^2 - 42*T2 + 49 $$T_{7}^{3} - 2T_{7}^{2} - 5T_{7} - 1$$ T7^3 - 2*T7^2 - 5*T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{5} + 7 T^{4} - 8 T^{3} + \cdots + 49$$
$3$ $$T^{6}$$
$5$ $$(T^{2} + T + 1)^{3}$$
$7$ $$(T^{3} - 2 T^{2} - 5 T - 1)^{2}$$
$11$ $$(T^{3} - 5 T^{2} + 2 T + 1)^{2}$$
$13$ $$(T^{2} - 5 T + 25)^{3}$$
$17$ $$T^{6} - T^{5} + 45 T^{4} + 30 T^{3} + \cdots + 49$$
$19$ $$T^{6} + 133T^{3} + 6859$$
$23$ $$T^{6} - 4 T^{5} + 55 T^{4} + \cdots + 2401$$
$29$ $$T^{6} + 2 T^{5} + 9 T^{4} - 12 T^{3} + \cdots + 1$$
$31$ $$(T^{3} + T^{2} - 6 T - 7)^{2}$$
$37$ $$(T^{3} + 2 T^{2} - 119 T - 227)^{2}$$
$41$ $$T^{6} + 2 T^{5} + 47 T^{4} + \cdots + 1369$$
$43$ $$T^{6} - T^{5} + 45 T^{4} - 198 T^{3} + \cdots + 14641$$
$47$ $$T^{6} - 6 T^{5} + 43 T^{4} + \cdots + 2401$$
$53$ $$T^{6} - 11 T^{5} + 163 T^{4} + \cdots + 96721$$
$59$ $$T^{6} - 6 T^{5} + 43 T^{4} + \cdots + 2401$$
$61$ $$T^{6} - 9 T^{5} + 130 T^{4} + \cdots + 2401$$
$67$ $$T^{6} - 20 T^{5} + 292 T^{4} + \cdots + 7744$$
$71$ $$T^{6} + 29 T^{5} + 605 T^{4} + \cdots + 218089$$
$73$ $$T^{6} - 22 T^{5} + 367 T^{4} + \cdots + 5929$$
$79$ $$T^{6} - 24 T^{5} + 460 T^{4} + \cdots + 61504$$
$83$ $$(T^{3} - 3 T^{2} - 54 T - 77)^{2}$$
$89$ $$T^{6} + 14 T^{5} + 232 T^{4} + \cdots + 3136$$
$97$ $$T^{6} + 7 T^{5} + 115 T^{4} + \cdots + 14641$$