# Properties

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

# 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: 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 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} + (\beta_{2} + 3 \beta_1) q^{11} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{13} + \beta_{3} q^{14} + 5 \beta_1 q^{16} + (\beta_{3} - \beta_{2} + 6 \beta_1 - 6) q^{17} + (2 \beta_1 - 2) q^{19} + ( - \beta_{3} + \beta_{2}) q^{20} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{22} + ( - \beta_{2} + 3 \beta_1) q^{23} - 2 q^{25} + (2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 3) q^{26} - \beta_1 q^{28} - 3 \beta_1 q^{29} + ( - 3 \beta_{3} - 1) q^{31} + (3 \beta_{3} - 3 \beta_{2}) q^{32} + (6 \beta_{3} - 3) q^{34} + ( - \beta_{3} + \beta_{2}) q^{35} + 7 \beta_1 q^{37} + 2 \beta_{3} q^{38} - 3 q^{40} + 3 \beta_{2} q^{41} + ( - 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 5) q^{43} + ( - \beta_{3} - 3) q^{44} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{46} + (4 \beta_{3} - 6) q^{47} - \beta_1 q^{49} + 2 \beta_{2} q^{50} + (\beta_{3} - 2 \beta_{2} - 2) q^{52} + ( - 4 \beta_{3} + 3) q^{53} + (3 \beta_{2} + 3 \beta_1) q^{55} + (\beta_{3} - \beta_{2}) q^{56} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{58} + (\beta_{3} - \beta_{2} - 9 \beta_1 + 9) q^{59} + (3 \beta_{3} - 3 \beta_{2} - 10 \beta_1 + 10) q^{61} + (\beta_{2} + 9 \beta_1) q^{62} + q^{64} + (2 \beta_{2} + 3 \beta_1 - 6) q^{65} + ( - 3 \beta_{2} + \beta_1) q^{67} + (\beta_{2} - 6 \beta_1) q^{68} + 3 q^{70} + ( - 6 \beta_1 + 6) q^{71} + (3 \beta_{3} + 2) q^{73} + (7 \beta_{3} - 7 \beta_{2}) q^{74} - 2 \beta_1 q^{76} + ( - \beta_{3} - 3) q^{77} + ( - 3 \beta_{3} + 11) q^{79} + 5 \beta_{2} q^{80} + ( - 9 \beta_1 + 9) q^{82} + ( - 3 \beta_{3} - 3) q^{83} + ( - 6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 + 3) q^{85} + (5 \beta_{3} + 9) q^{86} + ( - 3 \beta_{2} - 3 \beta_1) q^{88} + ( - 4 \beta_{2} + 6 \beta_1) q^{89} + (\beta_{3} - 2 \beta_{2} - 2) q^{91} + (\beta_{3} - 3) q^{92} + (6 \beta_{2} - 12 \beta_1) q^{94} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{95} + (6 \beta_{3} - 6 \beta_{2} - 4 \beta_1 + 4) 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 + (b2 + 3*b1) * q^11 + (-2*b3 + b2 + 2*b1) * q^13 + b3 * q^14 + 5*b1 * q^16 + (b3 - b2 + 6*b1 - 6) * q^17 + (2*b1 - 2) * q^19 + (-b3 + b2) * q^20 + (3*b3 - 3*b2 - 3*b1 + 3) * q^22 + (-b2 + 3*b1) * q^23 - 2 * q^25 + (2*b3 - 2*b2 + 3*b1 + 3) * q^26 - b1 * q^28 - 3*b1 * q^29 + (-3*b3 - 1) * q^31 + (3*b3 - 3*b2) * q^32 + (6*b3 - 3) * q^34 + (-b3 + b2) * q^35 + 7*b1 * q^37 + 2*b3 * q^38 - 3 * q^40 + 3*b2 * q^41 + (-3*b3 + 3*b2 + 5*b1 - 5) * q^43 + (-b3 - 3) * q^44 + (3*b3 - 3*b2 + 3*b1 - 3) * q^46 + (4*b3 - 6) * q^47 - b1 * q^49 + 2*b2 * q^50 + (b3 - 2*b2 - 2) * q^52 + (-4*b3 + 3) * q^53 + (3*b2 + 3*b1) * q^55 + (b3 - b2) * q^56 + (-3*b3 + 3*b2) * q^58 + (b3 - b2 - 9*b1 + 9) * q^59 + (3*b3 - 3*b2 - 10*b1 + 10) * q^61 + (b2 + 9*b1) * q^62 + q^64 + (2*b2 + 3*b1 - 6) * q^65 + (-3*b2 + b1) * q^67 + (b2 - 6*b1) * q^68 + 3 * q^70 + (-6*b1 + 6) * q^71 + (3*b3 + 2) * q^73 + (7*b3 - 7*b2) * q^74 - 2*b1 * q^76 + (-b3 - 3) * q^77 + (-3*b3 + 11) * q^79 + 5*b2 * q^80 + (-9*b1 + 9) * q^82 + (-3*b3 - 3) * q^83 + (-6*b3 + 6*b2 - 3*b1 + 3) * q^85 + (5*b3 + 9) * q^86 + (-3*b2 - 3*b1) * q^88 + (-4*b2 + 6*b1) * q^89 + (b3 - 2*b2 - 2) * q^91 + (b3 - 3) * q^92 + (6*b2 - 12*b1) * q^94 + (-2*b3 + 2*b2) * q^95 + (6*b3 - 6*b2 - 4*b1 + 4) * 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 $$4 q - 2 q^{4} - 2 q^{7} - 6 q^{10} + 6 q^{11} + 4 q^{13} + 10 q^{16} - 12 q^{17} - 4 q^{19} + 6 q^{22} + 6 q^{23} - 8 q^{25} + 18 q^{26} - 2 q^{28} - 6 q^{29} - 4 q^{31} - 12 q^{34} + 14 q^{37} - 12 q^{40} - 10 q^{43} - 12 q^{44} - 6 q^{46} - 24 q^{47} - 2 q^{49} - 8 q^{52} + 12 q^{53} + 6 q^{55} + 18 q^{59} + 20 q^{61} + 18 q^{62} + 4 q^{64} - 18 q^{65} + 2 q^{67} - 12 q^{68} + 12 q^{70} + 12 q^{71} + 8 q^{73} - 4 q^{76} - 12 q^{77} + 44 q^{79} + 18 q^{82} - 12 q^{83} + 6 q^{85} + 36 q^{86} - 6 q^{88} + 12 q^{89} - 8 q^{91} - 12 q^{92} - 24 q^{94} + 8 q^{97}+O(q^{100})$$ 4 * q - 2 * q^4 - 2 * q^7 - 6 * q^10 + 6 * q^11 + 4 * q^13 + 10 * q^16 - 12 * q^17 - 4 * q^19 + 6 * q^22 + 6 * q^23 - 8 * q^25 + 18 * q^26 - 2 * q^28 - 6 * q^29 - 4 * q^31 - 12 * q^34 + 14 * q^37 - 12 * q^40 - 10 * q^43 - 12 * q^44 - 6 * q^46 - 24 * q^47 - 2 * q^49 - 8 * q^52 + 12 * q^53 + 6 * q^55 + 18 * q^59 + 20 * q^61 + 18 * q^62 + 4 * q^64 - 18 * q^65 + 2 * q^67 - 12 * q^68 + 12 * q^70 + 12 * q^71 + 8 * q^73 - 4 * q^76 - 12 * q^77 + 44 * q^79 + 18 * q^82 - 12 * q^83 + 6 * q^85 + 36 * q^86 - 6 * q^88 + 12 * q^89 - 8 * q^91 - 12 * q^92 - 24 * q^94 + 8 * 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
13.c 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.o.b 4
3.b odd 2 1 91.2.f.b 4
12.b even 2 1 1456.2.s.o 4
13.c even 3 1 inner 819.2.o.b 4
21.c even 2 1 637.2.f.d 4
21.g even 6 1 637.2.g.d 4
21.g even 6 1 637.2.h.e 4
21.h odd 6 1 637.2.g.e 4
21.h odd 6 1 637.2.h.d 4
39.h odd 6 1 1183.2.a.e 2
39.i odd 6 1 91.2.f.b 4
39.i odd 6 1 1183.2.a.f 2
39.k even 12 2 1183.2.c.e 4
156.p even 6 1 1456.2.s.o 4
273.r even 6 1 637.2.g.d 4
273.s odd 6 1 637.2.g.e 4
273.u even 6 1 8281.2.a.t 2
273.bf even 6 1 637.2.h.e 4
273.bm odd 6 1 637.2.h.d 4
273.bn even 6 1 637.2.f.d 4
273.bn even 6 1 8281.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b 4 3.b odd 2 1
91.2.f.b 4 39.i odd 6 1
637.2.f.d 4 21.c even 2 1
637.2.f.d 4 273.bn even 6 1
637.2.g.d 4 21.g even 6 1
637.2.g.d 4 273.r even 6 1
637.2.g.e 4 21.h odd 6 1
637.2.g.e 4 273.s odd 6 1
637.2.h.d 4 21.h odd 6 1
637.2.h.d 4 273.bm odd 6 1
637.2.h.e 4 21.g even 6 1
637.2.h.e 4 273.bf even 6 1
819.2.o.b 4 1.a even 1 1 trivial
819.2.o.b 4 13.c even 3 1 inner
1183.2.a.e 2 39.h odd 6 1
1183.2.a.f 2 39.i odd 6 1
1183.2.c.e 4 39.k even 12 2
1456.2.s.o 4 12.b even 2 1
1456.2.s.o 4 156.p even 6 1
8281.2.a.r 2 273.bn even 6 1
8281.2.a.t 2 273.u even 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 3T_{2}^{2} + 9$$ T2^4 + 3*T2^2 + 9 $$T_{11}^{4} - 6T_{11}^{3} + 30T_{11}^{2} - 36T_{11} + 36$$ T11^4 - 6*T11^3 + 30*T11^2 - 36*T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 9$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$(T^{2} + T + 1)^{2}$$
$11$ $$T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36$$
$13$ $$T^{4} - 4 T^{3} + 3 T^{2} - 52 T + 169$$
$17$ $$T^{4} + 12 T^{3} + 111 T^{2} + \cdots + 1089$$
$19$ $$(T^{2} + 2 T + 4)^{2}$$
$23$ $$T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36$$
$29$ $$(T^{2} + 3 T + 9)^{2}$$
$31$ $$(T^{2} + 2 T - 26)^{2}$$
$37$ $$(T^{2} - 7 T + 49)^{2}$$
$41$ $$T^{4} + 27T^{2} + 729$$
$43$ $$T^{4} + 10 T^{3} + 102 T^{2} - 20 T + 4$$
$47$ $$(T^{2} + 12 T - 12)^{2}$$
$53$ $$(T^{2} - 6 T - 39)^{2}$$
$59$ $$T^{4} - 18 T^{3} + 246 T^{2} + \cdots + 6084$$
$61$ $$T^{4} - 20 T^{3} + 327 T^{2} + \cdots + 5329$$
$67$ $$T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676$$
$71$ $$(T^{2} - 6 T + 36)^{2}$$
$73$ $$(T^{2} - 4 T - 23)^{2}$$
$79$ $$(T^{2} - 22 T + 94)^{2}$$
$83$ $$(T^{2} + 6 T - 18)^{2}$$
$89$ $$T^{4} - 12 T^{3} + 156 T^{2} + \cdots + 144$$
$97$ $$T^{4} - 8 T^{3} + 156 T^{2} + \cdots + 8464$$