# Properties

 Label 819.2.a.e Level $819$ Weight $2$ Character orbit 819.a Self dual yes Analytic conductor $6.540$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("819.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.a (trivial)

## 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: yes Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
2.00000 0 2.00000 1.00000 0 1.00000 0 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.a.e 1
3.b odd 2 1 273.2.a.a 1
7.b odd 2 1 5733.2.a.m 1
12.b even 2 1 4368.2.a.q 1
15.d odd 2 1 6825.2.a.l 1
21.c even 2 1 1911.2.a.a 1
39.d odd 2 1 3549.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.a 1 3.b odd 2 1
819.2.a.e 1 1.a even 1 1 trivial
1911.2.a.a 1 21.c even 2 1
3549.2.a.d 1 39.d odd 2 1
4368.2.a.q 1 12.b even 2 1
5733.2.a.m 1 7.b odd 2 1
6825.2.a.l 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(819))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{5} - 1$$ T5 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 1$$
$11$ $$T - 2$$
$13$ $$T - 1$$
$17$ $$T - 4$$
$19$ $$T - 3$$
$23$ $$T - 9$$
$29$ $$T - 1$$
$31$ $$T + 5$$
$37$ $$T + 8$$
$41$ $$T + 6$$
$43$ $$T + 9$$
$47$ $$T - 3$$
$53$ $$T + 3$$
$59$ $$T$$
$61$ $$T - 10$$
$67$ $$T + 2$$
$71$ $$T + 12$$
$73$ $$T - 5$$
$79$ $$T + 13$$
$83$ $$T - 11$$
$89$ $$T + 1$$
$97$ $$T - 1$$