# Properties

 Label 819.2.s.a Level $819$ Weight $2$ Character orbit 819.s Analytic conductor $6.540$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(289,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("819.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.s (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: $$6.53974792554$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 0 −1.00000 1.50000 2.59808i 0 2.00000 + 1.73205i 3.00000 0 −1.50000 + 2.59808i
802.1 −1.00000 0 −1.00000 1.50000 + 2.59808i 0 2.00000 1.73205i 3.00000 0 −1.50000 2.59808i
 $$n$$: 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
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.s.a 2
3.b odd 2 1 91.2.h.a yes 2
7.c even 3 1 819.2.n.c 2
13.c even 3 1 819.2.n.c 2
21.c even 2 1 637.2.h.a 2
21.g even 6 1 637.2.f.a 2
21.g even 6 1 637.2.g.a 2
21.h odd 6 1 91.2.g.a 2
21.h odd 6 1 637.2.f.b 2
39.h odd 6 1 1183.2.e.c 2
39.i odd 6 1 91.2.g.a 2
39.i odd 6 1 1183.2.e.a 2
91.h even 3 1 inner 819.2.s.a 2
273.r even 6 1 637.2.h.a 2
273.r even 6 1 8281.2.a.j 1
273.s odd 6 1 91.2.h.a yes 2
273.s odd 6 1 8281.2.a.i 1
273.x odd 6 1 1183.2.e.c 2
273.bf even 6 1 637.2.f.a 2
273.bm odd 6 1 637.2.f.b 2
273.bm odd 6 1 1183.2.e.a 2
273.bn even 6 1 637.2.g.a 2
273.bp odd 6 1 8281.2.a.c 1
273.br even 6 1 8281.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 21.h odd 6 1
91.2.g.a 2 39.i odd 6 1
91.2.h.a yes 2 3.b odd 2 1
91.2.h.a yes 2 273.s odd 6 1
637.2.f.a 2 21.g even 6 1
637.2.f.a 2 273.bf even 6 1
637.2.f.b 2 21.h odd 6 1
637.2.f.b 2 273.bm odd 6 1
637.2.g.a 2 21.g even 6 1
637.2.g.a 2 273.bn even 6 1
637.2.h.a 2 21.c even 2 1
637.2.h.a 2 273.r even 6 1
819.2.n.c 2 7.c even 3 1
819.2.n.c 2 13.c even 3 1
819.2.s.a 2 1.a even 1 1 trivial
819.2.s.a 2 91.h even 3 1 inner
1183.2.e.a 2 39.i odd 6 1
1183.2.e.a 2 273.bm odd 6 1
1183.2.e.c 2 39.h odd 6 1
1183.2.e.c 2 273.x odd 6 1
8281.2.a.c 1 273.bp odd 6 1
8281.2.a.g 1 273.br even 6 1
8281.2.a.i 1 273.s odd 6 1
8281.2.a.j 1 273.r even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 7T + 49$$
$31$ $$T^{2} + 3T + 9$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - 7T + 49$$
$47$ $$T^{2} - T + 1$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} - 3T + 9$$
$71$ $$T^{2} - 13T + 169$$
$73$ $$T^{2} - 13T + 169$$
$79$ $$T^{2} - 3T + 9$$
$83$ $$T^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} - 5T + 25$$