# Properties

 Label 891.2.g.a Level $891$ Weight $2$ Character orbit 891.g Analytic conductor $7.115$ Analytic rank $0$ Dimension $4$ CM discriminant -11 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,2,Mod(296,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("891.296");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$891 = 3^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 891.g (of order $$6$$, degree $$2$$, not 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: $$7.11467082010$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{2} + 2) q^{4} + \beta_1 q^{5}+O(q^{10})$$ q + (-2*b2 + 2) * q^4 + b1 * q^5 $$q + ( - 2 \beta_{2} + 2) q^{4} + \beta_1 q^{5} + \beta_{3} q^{11} - 4 \beta_{2} q^{16} + 2 \beta_{3} q^{20} + \beta_1 q^{23} + 6 \beta_{2} q^{25} + (5 \beta_{2} - 5) q^{31} - 7 q^{37} + (2 \beta_{3} - 2 \beta_1) q^{44} + 2 \beta_{3} q^{47} + (7 \beta_{2} - 7) q^{49} + (4 \beta_{3} - 4 \beta_1) q^{53} + 11 q^{55} + \beta_1 q^{59} - 8 q^{64} + ( - 13 \beta_{2} + 13) q^{67} + ( - 5 \beta_{3} + 5 \beta_1) q^{71} + (4 \beta_{3} - 4 \beta_1) q^{80} + ( - 5 \beta_{3} + 5 \beta_1) q^{89} + 2 \beta_{3} q^{92} - 17 \beta_{2} q^{97}+O(q^{100})$$ q + (-2*b2 + 2) * q^4 + b1 * q^5 + b3 * q^11 - 4*b2 * q^16 + 2*b3 * q^20 + b1 * q^23 + 6*b2 * q^25 + (5*b2 - 5) * q^31 - 7 * q^37 + (2*b3 - 2*b1) * q^44 + 2*b3 * q^47 + (7*b2 - 7) * q^49 + (4*b3 - 4*b1) * q^53 + 11 * q^55 + b1 * q^59 - 8 * q^64 + (-13*b2 + 13) * q^67 + (-5*b3 + 5*b1) * q^71 + (4*b3 - 4*b1) * q^80 + (-5*b3 + 5*b1) * q^89 + 2*b3 * q^92 - 17*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4}+O(q^{10})$$ 4 * q + 4 * q^4 $$4 q + 4 q^{4} - 8 q^{16} + 12 q^{25} - 10 q^{31} - 28 q^{37} - 14 q^{49} + 44 q^{55} - 32 q^{64} + 26 q^{67} - 34 q^{97}+O(q^{100})$$ 4 * q + 4 * q^4 - 8 * q^16 + 12 * q^25 - 10 * q^31 - 28 * q^37 - 14 * q^49 + 44 * q^55 - 32 * q^64 + 26 * q^67 - 34 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + 10\nu - 9 ) / 6$$ (v^3 + 2*v^2 + 10*v - 9) / 6 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{3}$$ $$=$$ $$( -5\nu^{3} + 2\nu^{2} + 10\nu + 15 ) / 6$$ (-5*v^3 + 2*v^2 + 10*v + 15) / 6
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta _1 + 1 ) / 2$$ (-b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 5\beta_{2} ) / 2$$ (b3 + 5*b2) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta _1 + 4$$ -b3 + b1 + 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/891\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$650$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
296.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
0 0 1.00000 1.73205i −2.87228 1.65831i 0 0 0 0 0
296.2 0 0 1.00000 1.73205i 2.87228 + 1.65831i 0 0 0 0 0
593.1 0 0 1.00000 + 1.73205i −2.87228 + 1.65831i 0 0 0 0 0
593.2 0 0 1.00000 + 1.73205i 2.87228 1.65831i 0 0 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
33.d even 2 1 inner
99.g even 6 1 inner
99.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.g.a 4
3.b odd 2 1 inner 891.2.g.a 4
9.c even 3 1 33.2.d.a 2
9.c even 3 1 inner 891.2.g.a 4
9.d odd 6 1 33.2.d.a 2
9.d odd 6 1 inner 891.2.g.a 4
11.b odd 2 1 CM 891.2.g.a 4
33.d even 2 1 inner 891.2.g.a 4
36.f odd 6 1 528.2.b.a 2
36.h even 6 1 528.2.b.a 2
45.h odd 6 1 825.2.f.a 2
45.j even 6 1 825.2.f.a 2
45.k odd 12 2 825.2.d.a 4
45.l even 12 2 825.2.d.a 4
72.j odd 6 1 2112.2.b.e 2
72.l even 6 1 2112.2.b.f 2
72.n even 6 1 2112.2.b.e 2
72.p odd 6 1 2112.2.b.f 2
99.g even 6 1 33.2.d.a 2
99.g even 6 1 inner 891.2.g.a 4
99.h odd 6 1 33.2.d.a 2
99.h odd 6 1 inner 891.2.g.a 4
99.m even 15 4 363.2.f.c 8
99.n odd 30 4 363.2.f.c 8
99.o odd 30 4 363.2.f.c 8
99.p even 30 4 363.2.f.c 8
396.k even 6 1 528.2.b.a 2
396.o odd 6 1 528.2.b.a 2
495.o odd 6 1 825.2.f.a 2
495.r even 6 1 825.2.f.a 2
495.bd odd 12 2 825.2.d.a 4
495.bf even 12 2 825.2.d.a 4
792.s odd 6 1 2112.2.b.f 2
792.w even 6 1 2112.2.b.e 2
792.z even 6 1 2112.2.b.f 2
792.be odd 6 1 2112.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 9.c even 3 1
33.2.d.a 2 9.d odd 6 1
33.2.d.a 2 99.g even 6 1
33.2.d.a 2 99.h odd 6 1
363.2.f.c 8 99.m even 15 4
363.2.f.c 8 99.n odd 30 4
363.2.f.c 8 99.o odd 30 4
363.2.f.c 8 99.p even 30 4
528.2.b.a 2 36.f odd 6 1
528.2.b.a 2 36.h even 6 1
528.2.b.a 2 396.k even 6 1
528.2.b.a 2 396.o odd 6 1
825.2.d.a 4 45.k odd 12 2
825.2.d.a 4 45.l even 12 2
825.2.d.a 4 495.bd odd 12 2
825.2.d.a 4 495.bf even 12 2
825.2.f.a 2 45.h odd 6 1
825.2.f.a 2 45.j even 6 1
825.2.f.a 2 495.o odd 6 1
825.2.f.a 2 495.r even 6 1
891.2.g.a 4 1.a even 1 1 trivial
891.2.g.a 4 3.b odd 2 1 inner
891.2.g.a 4 9.c even 3 1 inner
891.2.g.a 4 9.d odd 6 1 inner
891.2.g.a 4 11.b odd 2 1 CM
891.2.g.a 4 33.d even 2 1 inner
891.2.g.a 4 99.g even 6 1 inner
891.2.g.a 4 99.h odd 6 1 inner
2112.2.b.e 2 72.j odd 6 1
2112.2.b.e 2 72.n even 6 1
2112.2.b.e 2 792.w even 6 1
2112.2.b.e 2 792.be odd 6 1
2112.2.b.f 2 72.l even 6 1
2112.2.b.f 2 72.p odd 6 1
2112.2.b.f 2 792.s odd 6 1
2112.2.b.f 2 792.z even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(891, [\chi])$$:

 $$T_{2}$$ T2 $$T_{23}^{4} - 11T_{23}^{2} + 121$$ T23^4 - 11*T23^2 + 121

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 11T^{2} + 121$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 11T^{2} + 121$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} - 11T^{2} + 121$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 5 T + 25)^{2}$$
$37$ $$(T + 7)^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4} - 44T^{2} + 1936$$
$53$ $$(T^{2} + 176)^{2}$$
$59$ $$T^{4} - 11T^{2} + 121$$
$61$ $$T^{4}$$
$67$ $$(T^{2} - 13 T + 169)^{2}$$
$71$ $$(T^{2} + 275)^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} + 275)^{2}$$
$97$ $$(T^{2} + 17 T + 289)^{2}$$