# Properties

 Label 819.2.o.a Level $819$ Weight $2$ Character orbit 819.o Analytic conductor $6.540$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(568,819)]

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, 0, 4]))

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

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

chi := DirichletCharacter("819.568");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.o (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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
568.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 + 1.50000i 0 −0.500000 0.866025i 1.73205 0 −0.500000 0.866025i −1.73205 0 −1.50000 + 2.59808i
568.2 0.866025 1.50000i 0 −0.500000 0.866025i −1.73205 0 −0.500000 0.866025i 1.73205 0 −1.50000 + 2.59808i
757.1 −0.866025 1.50000i 0 −0.500000 + 0.866025i 1.73205 0 −0.500000 + 0.866025i −1.73205 0 −1.50000 2.59808i
757.2 0.866025 + 1.50000i 0 −0.500000 + 0.866025i −1.73205 0 −0.500000 + 0.866025i 1.73205 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
3.b odd 2 1 inner
13.c even 3 1 inner
39.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.o.a 4
3.b odd 2 1 inner 819.2.o.a 4
13.c even 3 1 inner 819.2.o.a 4
39.i odd 6 1 inner 819.2.o.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.o.a 4 1.a even 1 1 trivial
819.2.o.a 4 3.b odd 2 1 inner
819.2.o.a 4 13.c even 3 1 inner
819.2.o.a 4 39.i odd 6 1 inner

## Hecke kernels

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

 $$T_{2}^{4} + 3T_{2}^{2} + 9$$ T2^4 + 3*T2^2 + 9 $$T_{11}^{4} + 12T_{11}^{2} + 144$$ T11^4 + 12*T11^2 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 9$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$(T^{2} + T + 1)^{2}$$
$11$ $$T^{4} + 12T^{2} + 144$$
$13$ $$(T^{2} + 7 T + 13)^{2}$$
$17$ $$T^{4} + 3T^{2} + 9$$
$19$ $$(T^{2} + 2 T + 4)^{2}$$
$23$ $$T^{4} + 48T^{2} + 2304$$
$29$ $$T^{4} + 27T^{2} + 729$$
$31$ $$(T + 4)^{4}$$
$37$ $$(T^{2} + 5 T + 25)^{2}$$
$41$ $$T^{4} + 27T^{2} + 729$$
$43$ $$(T^{2} - 10 T + 100)^{2}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} - 75)^{2}$$
$59$ $$T^{4} + 12T^{2} + 144$$
$61$ $$(T^{2} + 11 T + 121)^{2}$$
$67$ $$(T^{2} - 4 T + 16)^{2}$$
$71$ $$T^{4} + 108 T^{2} + 11664$$
$73$ $$(T - 11)^{4}$$
$79$ $$(T + 10)^{4}$$
$83$ $$(T^{2} - 108)^{2}$$
$89$ $$T^{4} + 192 T^{2} + 36864$$
$97$ $$(T^{2} - 10 T + 100)^{2}$$