# Properties

 Label 819.2.n.c Level $819$ Weight $2$ Character orbit 819.n 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(100,819)]

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, 4]))

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

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

chi := DirichletCharacter("819.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.n (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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + ( - 3 \zeta_{6} + 3) q^{5} + (\zeta_{6} - 3) q^{7} + 3 q^{8}+O(q^{10})$$ q + z * q^2 + (-z + 1) * q^4 + (-3*z + 3) * q^5 + (z - 3) * q^7 + 3 * q^8 $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + ( - 3 \zeta_{6} + 3) q^{5} + (\zeta_{6} - 3) q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} + ( - 4 \zeta_{6} + 1) q^{13} + ( - 2 \zeta_{6} - 1) q^{14} + \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} - q^{19} - 3 \zeta_{6} q^{20} + 3 \zeta_{6} q^{22} - 4 \zeta_{6} q^{25} + ( - 3 \zeta_{6} + 4) q^{26} + (3 \zeta_{6} - 2) q^{28} + ( - 7 \zeta_{6} + 7) q^{29} - 3 \zeta_{6} q^{31} + ( - 5 \zeta_{6} + 5) q^{32} - 2 q^{34} + (9 \zeta_{6} - 6) q^{35} - 2 \zeta_{6} q^{37} - \zeta_{6} q^{38} + ( - 9 \zeta_{6} + 9) q^{40} + ( - 3 \zeta_{6} + 3) q^{41} + 7 \zeta_{6} q^{43} + ( - 3 \zeta_{6} + 3) q^{44} + ( - \zeta_{6} + 1) q^{47} + ( - 5 \zeta_{6} + 8) q^{49} + ( - 4 \zeta_{6} + 4) q^{50} + ( - \zeta_{6} - 3) q^{52} + 3 \zeta_{6} q^{53} + ( - 9 \zeta_{6} + 9) q^{55} + (3 \zeta_{6} - 9) q^{56} + 7 q^{58} + (4 \zeta_{6} - 4) q^{59} - 13 q^{61} + ( - 3 \zeta_{6} + 3) q^{62} + 7 q^{64} + ( - 3 \zeta_{6} - 9) q^{65} - 3 q^{67} + 2 \zeta_{6} q^{68} + (3 \zeta_{6} - 9) q^{70} + 13 \zeta_{6} q^{71} + 13 \zeta_{6} q^{73} + ( - 2 \zeta_{6} + 2) q^{74} + (\zeta_{6} - 1) q^{76} + (3 \zeta_{6} - 9) q^{77} + ( - 3 \zeta_{6} + 3) q^{79} + 3 q^{80} + 3 q^{82} + 6 \zeta_{6} q^{85} + (7 \zeta_{6} - 7) q^{86} + 9 q^{88} + 6 \zeta_{6} q^{89} + (9 \zeta_{6} + 1) q^{91} + q^{94} + (3 \zeta_{6} - 3) q^{95} + 5 \zeta_{6} q^{97} + (3 \zeta_{6} + 5) q^{98} +O(q^{100})$$ q + z * q^2 + (-z + 1) * q^4 + (-3*z + 3) * q^5 + (z - 3) * q^7 + 3 * q^8 + 3 * q^10 + 3 * q^11 + (-4*z + 1) * q^13 + (-2*z - 1) * q^14 + z * q^16 + (2*z - 2) * q^17 - q^19 - 3*z * q^20 + 3*z * q^22 - 4*z * q^25 + (-3*z + 4) * q^26 + (3*z - 2) * q^28 + (-7*z + 7) * q^29 - 3*z * q^31 + (-5*z + 5) * q^32 - 2 * q^34 + (9*z - 6) * q^35 - 2*z * q^37 - z * q^38 + (-9*z + 9) * q^40 + (-3*z + 3) * q^41 + 7*z * q^43 + (-3*z + 3) * q^44 + (-z + 1) * q^47 + (-5*z + 8) * q^49 + (-4*z + 4) * q^50 + (-z - 3) * q^52 + 3*z * q^53 + (-9*z + 9) * q^55 + (3*z - 9) * q^56 + 7 * q^58 + (4*z - 4) * q^59 - 13 * q^61 + (-3*z + 3) * q^62 + 7 * q^64 + (-3*z - 9) * q^65 - 3 * q^67 + 2*z * q^68 + (3*z - 9) * q^70 + 13*z * q^71 + 13*z * q^73 + (-2*z + 2) * q^74 + (z - 1) * q^76 + (3*z - 9) * q^77 + (-3*z + 3) * q^79 + 3 * q^80 + 3 * q^82 + 6*z * q^85 + (7*z - 7) * q^86 + 9 * q^88 + 6*z * q^89 + (9*z + 1) * q^91 + q^94 + (3*z - 3) * q^95 + 5*z * q^97 + (3*z + 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 $$2 q + q^{2} + q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} + 6 q^{10} + 6 q^{11} - 2 q^{13} - 4 q^{14} + q^{16} - 2 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 4 q^{25} + 5 q^{26} - q^{28} + 7 q^{29} - 3 q^{31} + 5 q^{32} - 4 q^{34} - 3 q^{35} - 2 q^{37} - q^{38} + 9 q^{40} + 3 q^{41} + 7 q^{43} + 3 q^{44} + q^{47} + 11 q^{49} + 4 q^{50} - 7 q^{52} + 3 q^{53} + 9 q^{55} - 15 q^{56} + 14 q^{58} - 4 q^{59} - 26 q^{61} + 3 q^{62} + 14 q^{64} - 21 q^{65} - 6 q^{67} + 2 q^{68} - 15 q^{70} + 13 q^{71} + 13 q^{73} + 2 q^{74} - q^{76} - 15 q^{77} + 3 q^{79} + 6 q^{80} + 6 q^{82} + 6 q^{85} - 7 q^{86} + 18 q^{88} + 6 q^{89} + 11 q^{91} + 2 q^{94} - 3 q^{95} + 5 q^{97} + 13 q^{98}+O(q^{100})$$ 2 * q + q^2 + q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 + 6 * q^10 + 6 * q^11 - 2 * q^13 - 4 * q^14 + q^16 - 2 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 4 * q^25 + 5 * q^26 - q^28 + 7 * q^29 - 3 * q^31 + 5 * q^32 - 4 * q^34 - 3 * q^35 - 2 * q^37 - q^38 + 9 * q^40 + 3 * q^41 + 7 * q^43 + 3 * q^44 + q^47 + 11 * q^49 + 4 * q^50 - 7 * q^52 + 3 * q^53 + 9 * q^55 - 15 * q^56 + 14 * q^58 - 4 * q^59 - 26 * q^61 + 3 * q^62 + 14 * q^64 - 21 * q^65 - 6 * q^67 + 2 * q^68 - 15 * q^70 + 13 * q^71 + 13 * q^73 + 2 * q^74 - q^76 - 15 * q^77 + 3 * q^79 + 6 * q^80 + 6 * q^82 + 6 * q^85 - 7 * q^86 + 18 * q^88 + 6 * q^89 + 11 * q^91 + 2 * q^94 - 3 * q^95 + 5 * q^97 + 13 * q^98

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 1.50000 2.59808i 0 −2.50000 + 0.866025i 3.00000 0 3.00000
172.1 0.500000 0.866025i 0 0.500000 + 0.866025i 1.50000 + 2.59808i 0 −2.50000 0.866025i 3.00000 0 3.00000
 $$n$$: 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.g 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.n.c 2
3.b odd 2 1 91.2.g.a 2
7.c even 3 1 819.2.s.a 2
13.c even 3 1 819.2.s.a 2
21.c even 2 1 637.2.g.a 2
21.g even 6 1 637.2.f.a 2
21.g even 6 1 637.2.h.a 2
21.h odd 6 1 91.2.h.a yes 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.h.a yes 2
39.i odd 6 1 1183.2.e.a 2
91.g even 3 1 inner 819.2.n.c 2
273.r even 6 1 637.2.f.a 2
273.s odd 6 1 637.2.f.b 2
273.s odd 6 1 1183.2.e.a 2
273.x odd 6 1 8281.2.a.c 1
273.y even 6 1 8281.2.a.g 1
273.bf even 6 1 637.2.g.a 2
273.bf even 6 1 8281.2.a.j 1
273.bm odd 6 1 91.2.g.a 2
273.bm odd 6 1 8281.2.a.i 1
273.bn even 6 1 637.2.h.a 2
273.bp odd 6 1 1183.2.e.c 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 7T + 49$$
$31$ $$T^{2} + 3T + 9$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - 7T + 49$$
$47$ $$T^{2} - T + 1$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$(T + 13)^{2}$$
$67$ $$(T + 3)^{2}$$
$71$ $$T^{2} - 13T + 169$$
$73$ $$T^{2} - 13T + 169$$
$79$ $$T^{2} - 3T + 9$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$T^{2} - 5T + 25$$