# Properties

 Label 819.2.a.k Level $819$ Weight $2$ Character orbit 819.a Self dual yes Analytic conductor $6.540$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.17428.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ x^4 - x^3 - 6*x^2 + 4*x + 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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)

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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{2} + 1) q^{5} + q^{7} + ( - \beta_{3} - 2 \beta_1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + 2) * q^4 + (b2 + 1) * q^5 + q^7 + (-b3 - 2*b1) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{2} + 1) q^{5} + q^{7} + ( - \beta_{3} - 2 \beta_1) q^{8} + ( - \beta_{3} - 3 \beta_1) q^{10} + (\beta_{3} + \beta_1) q^{11} + q^{13} - \beta_1 q^{14} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{16} + 2 \beta_1 q^{17} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{19} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 8) q^{20} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{22} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{23} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{25} - \beta_1 q^{26} + (\beta_{2} + 2) q^{28} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{29} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{31} + ( - 2 \beta_{2} - \beta_1 - 2) q^{32} + ( - 2 \beta_{2} - 8) q^{34} + (\beta_{2} + 1) q^{35} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{37} + ( - 2 \beta_{2} - 2) q^{38} + ( - \beta_{3} - 2 \beta_{2} - 7 \beta_1 - 2) q^{40} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 4) q^{41} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{43} + (\beta_{3} + 2 \beta_{2} + 5 \beta_1 + 2) q^{44} + (2 \beta_{3} + 4 \beta_1 - 6) q^{46} + ( - \beta_{2} - 2 \beta_1 - 1) q^{47} + q^{49} + ( - 2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 2) q^{50} + (\beta_{2} + 2) q^{52} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{53} + (2 \beta_{2} + 4 \beta_1 + 2) q^{55} + ( - \beta_{3} - 2 \beta_1) q^{56} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{58} + ( - \beta_{3} - 3 \beta_1 + 6) q^{59} + (4 \beta_1 + 2) q^{61} + (2 \beta_{2} + 14) q^{62} + ( - \beta_{2} + 4 \beta_1) q^{64} + (\beta_{2} + 1) q^{65} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{67} + (2 \beta_{3} + 8 \beta_1) q^{68} + ( - \beta_{3} - 3 \beta_1) q^{70} + (3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 2) q^{71} + ( - \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 1) q^{73} + (4 \beta_{2} - 6 \beta_1 + 4) q^{74} + (2 \beta_{2} + 4 \beta_1 - 2) q^{76} + (\beta_{3} + \beta_1) q^{77} + (\beta_{3} - \beta_{2} + 5 \beta_1 + 1) q^{79} + (\beta_{3} + 4 \beta_{2} + 5 \beta_1 + 10) q^{80} + (3 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{82} + ( - \beta_{2} + 2 \beta_1 - 1) q^{83} + (2 \beta_{3} + 6 \beta_1) q^{85} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{86} + ( - \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 14) q^{88} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 3) q^{89} + q^{91} + ( - 4 \beta_{2} + 2 \beta_1 - 10) q^{92} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 + 8) q^{94} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{95} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 - 5) q^{97} - \beta_1 q^{98}+O(q^{100})$$ q - b1 * q^2 + (b2 + 2) * q^4 + (b2 + 1) * q^5 + q^7 + (-b3 - 2*b1) * q^8 + (-b3 - 3*b1) * q^10 + (b3 + b1) * q^11 + q^13 - b1 * q^14 + (b3 + b2 + b1 + 2) * q^16 + 2*b1 * q^17 + (b3 - b2 + b1 + 1) * q^19 + (b3 + 2*b2 + b1 + 8) * q^20 + (-b3 - 2*b2 - b1 - 2) * q^22 + (-b3 - b2 + b1 - 1) * q^23 + (b3 + b2 + b1 + 2) * q^25 - b1 * q^26 + (b2 + 2) * q^28 + (b3 - b2 + b1 - 1) * q^29 + (b3 - b2 - 3*b1 + 1) * q^31 + (-2*b2 - b1 - 2) * q^32 + (-2*b2 - 8) * q^34 + (b2 + 1) * q^35 + (-2*b3 + 2*b2 - 2*b1 + 4) * q^37 + (-2*b2 - 2) * q^38 + (-b3 - 2*b2 - 7*b1 - 2) * q^40 + (-b3 - 2*b2 - b1 + 4) * q^41 + (-b3 - b2 - b1 + 1) * q^43 + (b3 + 2*b2 + 5*b1 + 2) * q^44 + (2*b3 + 4*b1 - 6) * q^46 + (-b2 - 2*b1 - 1) * q^47 + q^49 + (-2*b3 - 2*b2 - 5*b1 - 2) * q^50 + (b2 + 2) * q^52 + (-b3 - b2 - b1 - 1) * q^53 + (2*b2 + 4*b1 + 2) * q^55 + (-b3 - 2*b1) * q^56 + (-2*b2 + 2*b1 - 2) * q^58 + (-b3 - 3*b1 + 6) * q^59 + (4*b1 + 2) * q^61 + (2*b2 + 14) * q^62 + (-b2 + 4*b1) * q^64 + (b2 + 1) * q^65 + (-2*b3 - 2*b2 + 2*b1 - 6) * q^67 + (2*b3 + 8*b1) * q^68 + (-b3 - 3*b1) * q^70 + (3*b3 - 2*b2 + 3*b1 - 2) * q^71 + (-b3 + 3*b2 - 5*b1 - 1) * q^73 + (4*b2 - 6*b1 + 4) * q^74 + (2*b2 + 4*b1 - 2) * q^76 + (b3 + b1) * q^77 + (b3 - b2 + 5*b1 + 1) * q^79 + (b3 + 4*b2 + 5*b1 + 10) * q^80 + (3*b3 + 2*b2 + b1 + 2) * q^82 + (-b2 + 2*b1 - 1) * q^83 + (2*b3 + 6*b1) * q^85 + (2*b3 + 2*b2 + 2*b1 + 2) * q^86 + (-b3 - 2*b2 - 5*b1 - 14) * q^88 + (-2*b3 + 3*b2 - 2*b1 + 3) * q^89 + q^91 + (-4*b2 + 2*b1 - 10) * q^92 + (b3 + 2*b2 + 3*b1 + 8) * q^94 + (-b3 + 3*b2 + 3*b1 - 3) * q^95 + (-b3 - b2 + 3*b1 - 5) * q^97 - b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10})$$ 4 * q - q^2 + 7 * q^4 + 3 * q^5 + 4 * q^7 - 3 * q^8 $$4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8} - 4 q^{10} + 2 q^{11} + 4 q^{13} - q^{14} + 9 q^{16} + 2 q^{17} + 7 q^{19} + 32 q^{20} - 8 q^{22} - 3 q^{23} + 9 q^{25} - q^{26} + 7 q^{28} - q^{29} + 3 q^{31} - 7 q^{32} - 30 q^{34} + 3 q^{35} + 10 q^{37} - 6 q^{38} - 14 q^{40} + 16 q^{41} + 3 q^{43} + 12 q^{44} - 18 q^{46} - 5 q^{47} + 4 q^{49} - 13 q^{50} + 7 q^{52} - 5 q^{53} + 10 q^{55} - 3 q^{56} - 4 q^{58} + 20 q^{59} + 12 q^{61} + 54 q^{62} + 5 q^{64} + 3 q^{65} - 22 q^{67} + 10 q^{68} - 4 q^{70} - 13 q^{73} + 6 q^{74} - 6 q^{76} + 2 q^{77} + 11 q^{79} + 42 q^{80} + 10 q^{82} - q^{83} + 8 q^{85} + 10 q^{86} - 60 q^{88} + 5 q^{89} + 4 q^{91} - 34 q^{92} + 34 q^{94} - 13 q^{95} - 17 q^{97} - q^{98}+O(q^{100})$$ 4 * q - q^2 + 7 * q^4 + 3 * q^5 + 4 * q^7 - 3 * q^8 - 4 * q^10 + 2 * q^11 + 4 * q^13 - q^14 + 9 * q^16 + 2 * q^17 + 7 * q^19 + 32 * q^20 - 8 * q^22 - 3 * q^23 + 9 * q^25 - q^26 + 7 * q^28 - q^29 + 3 * q^31 - 7 * q^32 - 30 * q^34 + 3 * q^35 + 10 * q^37 - 6 * q^38 - 14 * q^40 + 16 * q^41 + 3 * q^43 + 12 * q^44 - 18 * q^46 - 5 * q^47 + 4 * q^49 - 13 * q^50 + 7 * q^52 - 5 * q^53 + 10 * q^55 - 3 * q^56 - 4 * q^58 + 20 * q^59 + 12 * q^61 + 54 * q^62 + 5 * q^64 + 3 * q^65 - 22 * q^67 + 10 * q^68 - 4 * q^70 - 13 * q^73 + 6 * q^74 - 6 * q^76 + 2 * q^77 + 11 * q^79 + 42 * q^80 + 10 * q^82 - q^83 + 8 * q^85 + 10 * q^86 - 60 * q^88 + 5 * q^89 + 4 * q^91 - 34 * q^92 + 34 * q^94 - 13 * q^95 - 17 * q^97 - q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1 $$\beta_{3}$$ $$=$$ $$-\nu^{2} + 2\nu + 3$$ -v^2 + 2*v + 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 3$$ b1 + 3 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + \beta_{2} + 2\beta _1 + 1$$ 2*b3 + b2 + 2*b1 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.36865 −2.10710 1.52616 −0.787711
−2.61050 0 4.81471 3.81471 0 1.00000 −7.34780 0 −9.95830
1.2 −1.43986 0 0.0731828 −0.926817 0 1.00000 2.77434 0 1.33448
1.3 0.670843 0 −1.54997 −2.54997 0 1.00000 −2.38147 0 −1.71063
1.4 2.37951 0 3.66208 2.66208 0 1.00000 3.95493 0 6.33445
 $$n$$: 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.k 4
3.b odd 2 1 273.2.a.e 4
7.b odd 2 1 5733.2.a.bf 4
12.b even 2 1 4368.2.a.br 4
15.d odd 2 1 6825.2.a.bg 4
21.c even 2 1 1911.2.a.s 4
39.d odd 2 1 3549.2.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.e 4 3.b odd 2 1
819.2.a.k 4 1.a even 1 1 trivial
1911.2.a.s 4 21.c even 2 1
3549.2.a.w 4 39.d odd 2 1
4368.2.a.br 4 12.b even 2 1
5733.2.a.bf 4 7.b odd 2 1
6825.2.a.bg 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(819))$$:

 $$T_{2}^{4} + T_{2}^{3} - 7T_{2}^{2} - 5T_{2} + 6$$ T2^4 + T2^3 - 7*T2^2 - 5*T2 + 6 $$T_{5}^{4} - 3T_{5}^{3} - 10T_{5}^{2} + 20T_{5} + 24$$ T5^4 - 3*T5^3 - 10*T5^2 + 20*T5 + 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} - 7 T^{2} - 5 T + 6$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 3 T^{3} - 10 T^{2} + 20 T + 24$$
$7$ $$(T - 1)^{4}$$
$11$ $$T^{4} - 2 T^{3} - 24 T^{2} + 32 T + 96$$
$13$ $$(T - 1)^{4}$$
$17$ $$T^{4} - 2 T^{3} - 28 T^{2} + 40 T + 96$$
$19$ $$T^{4} - 7 T^{3} - 12 T^{2} + 48 T + 64$$
$23$ $$T^{4} + 3 T^{3} - 52 T^{2} - 256 T - 288$$
$29$ $$T^{4} + T^{3} - 30 T^{2} - 52 T + 72$$
$31$ $$T^{4} - 3 T^{3} - 128 T^{2} + \cdots + 3968$$
$37$ $$T^{4} - 10 T^{3} - 84 T^{2} + \cdots - 128$$
$41$ $$T^{4} - 16 T^{3} + 688 T - 1392$$
$43$ $$T^{4} - 3 T^{3} - 44 T^{2} + 112 T - 64$$
$47$ $$T^{4} + 5 T^{3} - 40 T^{2} - 16 T + 144$$
$53$ $$T^{4} + 5 T^{3} - 38 T^{2} - 68 T - 24$$
$59$ $$T^{4} - 20 T^{3} + 80 T^{2} + \cdots - 1536$$
$61$ $$T^{4} - 12 T^{3} - 64 T^{2} + \cdots + 496$$
$67$ $$T^{4} + 22 T^{3} - 40 T^{2} + \cdots - 15488$$
$71$ $$T^{4} - 232 T^{2} - 304 T + 10176$$
$73$ $$T^{4} + 13 T^{3} - 166 T^{2} + \cdots - 11672$$
$79$ $$T^{4} - 11 T^{3} - 120 T^{2} + \cdots - 3456$$
$83$ $$T^{4} + T^{3} - 36 T^{2} + 80 T - 48$$
$89$ $$T^{4} - 5 T^{3} - 162 T^{2} + \cdots - 1704$$
$97$ $$T^{4} + 17 T^{3} - 14 T^{2} + \cdots - 1528$$