# Properties

 Label 8550.2.a.cj Level $8550$ Weight $2$ Character orbit 8550.a Self dual yes Analytic conductor $68.272$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.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: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 3$$ x^3 - x^2 - 4*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 950) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (2 \beta_{2} - \beta_1 + 1) q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (2*b2 - b1 + 1) * q^7 - q^8 $$q - q^{2} + q^{4} + (2 \beta_{2} - \beta_1 + 1) q^{7} - q^{8} + (2 \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{2} - 2 \beta_1) q^{13} + ( - 2 \beta_{2} + \beta_1 - 1) q^{14} + q^{16} + (\beta_{2} - 2 \beta_1 + 2) q^{17} - q^{19} + ( - 2 \beta_{2} + 2 \beta_1) q^{22} + ( - \beta_{2} + \beta_1 - 5) q^{23} + (\beta_{2} + 2 \beta_1) q^{26} + (2 \beta_{2} - \beta_1 + 1) q^{28} + ( - \beta_{2} - 2 \beta_1 - 4) q^{29} + (2 \beta_{2} + 2 \beta_1 - 2) q^{31} - q^{32} + ( - \beta_{2} + 2 \beta_1 - 2) q^{34} + (2 \beta_{2} - 2 \beta_1 - 5) q^{37} + q^{38} + ( - 4 \beta_1 + 4) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + (2 \beta_{2} - 2 \beta_1) q^{44} + (\beta_{2} - \beta_1 + 5) q^{46} + (2 \beta_{2} - 4 \beta_1 - 3) q^{47} + ( - 3 \beta_{2} - 2 \beta_1 + 9) q^{49} + ( - \beta_{2} - 2 \beta_1) q^{52} + ( - 2 \beta_{2} + \beta_1 + 3) q^{53} + ( - 2 \beta_{2} + \beta_1 - 1) q^{56} + (\beta_{2} + 2 \beta_1 + 4) q^{58} + ( - \beta_{2} + 3 \beta_1 + 1) q^{59} + (4 \beta_{2} - 2 \beta_1 + 8) q^{61} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{62} + q^{64} + (4 \beta_{2} + 3 \beta_1 - 1) q^{67} + (\beta_{2} - 2 \beta_1 + 2) q^{68} + 2 \beta_1 q^{71} + (3 \beta_{2} - 3 \beta_1 - 3) q^{73} + ( - 2 \beta_{2} + 2 \beta_1 + 5) q^{74} - q^{76} + ( - 6 \beta_{2} - 4 \beta_1 + 18) q^{77} + (4 \beta_{2} + 8) q^{79} + (4 \beta_1 - 4) q^{82} + (4 \beta_{2} - 6 \beta_1 - 2) q^{83} + ( - 2 \beta_{2} - 2 \beta_1) q^{86} + ( - 2 \beta_{2} + 2 \beta_1) q^{88} + 10 q^{89} - 7 \beta_1 q^{91} + ( - \beta_{2} + \beta_1 - 5) q^{92} + ( - 2 \beta_{2} + 4 \beta_1 + 3) q^{94} + 6 \beta_{2} q^{97} + (3 \beta_{2} + 2 \beta_1 - 9) q^{98}+O(q^{100})$$ q - q^2 + q^4 + (2*b2 - b1 + 1) * q^7 - q^8 + (2*b2 - 2*b1) * q^11 + (-b2 - 2*b1) * q^13 + (-2*b2 + b1 - 1) * q^14 + q^16 + (b2 - 2*b1 + 2) * q^17 - q^19 + (-2*b2 + 2*b1) * q^22 + (-b2 + b1 - 5) * q^23 + (b2 + 2*b1) * q^26 + (2*b2 - b1 + 1) * q^28 + (-b2 - 2*b1 - 4) * q^29 + (2*b2 + 2*b1 - 2) * q^31 - q^32 + (-b2 + 2*b1 - 2) * q^34 + (2*b2 - 2*b1 - 5) * q^37 + q^38 + (-4*b1 + 4) * q^41 + (2*b2 + 2*b1) * q^43 + (2*b2 - 2*b1) * q^44 + (b2 - b1 + 5) * q^46 + (2*b2 - 4*b1 - 3) * q^47 + (-3*b2 - 2*b1 + 9) * q^49 + (-b2 - 2*b1) * q^52 + (-2*b2 + b1 + 3) * q^53 + (-2*b2 + b1 - 1) * q^56 + (b2 + 2*b1 + 4) * q^58 + (-b2 + 3*b1 + 1) * q^59 + (4*b2 - 2*b1 + 8) * q^61 + (-2*b2 - 2*b1 + 2) * q^62 + q^64 + (4*b2 + 3*b1 - 1) * q^67 + (b2 - 2*b1 + 2) * q^68 + 2*b1 * q^71 + (3*b2 - 3*b1 - 3) * q^73 + (-2*b2 + 2*b1 + 5) * q^74 - q^76 + (-6*b2 - 4*b1 + 18) * q^77 + (4*b2 + 8) * q^79 + (4*b1 - 4) * q^82 + (4*b2 - 6*b1 - 2) * q^83 + (-2*b2 - 2*b1) * q^86 + (-2*b2 + 2*b1) * q^88 + 10 * q^89 - 7*b1 * q^91 + (-b2 + b1 - 5) * q^92 + (-2*b2 + 4*b1 + 3) * q^94 + 6*b2 * q^97 + (3*b2 + 2*b1 - 9) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 + 2 * q^7 - 3 * q^8 $$3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8} - 2 q^{11} - 2 q^{13} - 2 q^{14} + 3 q^{16} + 4 q^{17} - 3 q^{19} + 2 q^{22} - 14 q^{23} + 2 q^{26} + 2 q^{28} - 14 q^{29} - 4 q^{31} - 3 q^{32} - 4 q^{34} - 17 q^{37} + 3 q^{38} + 8 q^{41} + 2 q^{43} - 2 q^{44} + 14 q^{46} - 13 q^{47} + 25 q^{49} - 2 q^{52} + 10 q^{53} - 2 q^{56} + 14 q^{58} + 6 q^{59} + 22 q^{61} + 4 q^{62} + 3 q^{64} + 4 q^{68} + 2 q^{71} - 12 q^{73} + 17 q^{74} - 3 q^{76} + 50 q^{77} + 24 q^{79} - 8 q^{82} - 12 q^{83} - 2 q^{86} + 2 q^{88} + 30 q^{89} - 7 q^{91} - 14 q^{92} + 13 q^{94} - 25 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 + 2 * q^7 - 3 * q^8 - 2 * q^11 - 2 * q^13 - 2 * q^14 + 3 * q^16 + 4 * q^17 - 3 * q^19 + 2 * q^22 - 14 * q^23 + 2 * q^26 + 2 * q^28 - 14 * q^29 - 4 * q^31 - 3 * q^32 - 4 * q^34 - 17 * q^37 + 3 * q^38 + 8 * q^41 + 2 * q^43 - 2 * q^44 + 14 * q^46 - 13 * q^47 + 25 * q^49 - 2 * q^52 + 10 * q^53 - 2 * q^56 + 14 * q^58 + 6 * q^59 + 22 * q^61 + 4 * q^62 + 3 * q^64 + 4 * q^68 + 2 * q^71 - 12 * q^73 + 17 * q^74 - 3 * q^76 + 50 * q^77 + 24 * q^79 - 8 * q^82 - 12 * q^83 - 2 * q^86 + 2 * q^88 + 30 * q^89 - 7 * q^91 - 14 * q^92 + 13 * q^94 - 25 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## 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
 0.713538 2.19869 −1.91223
−1.00000 0 1.00000 0 0 −4.69527 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 2.46980 −1.00000 0 0
1.3 −1.00000 0 1.00000 0 0 4.22547 −1.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$19$$ $$+1$$

## 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 8550.2.a.cj 3
3.b odd 2 1 950.2.a.m yes 3
5.b even 2 1 8550.2.a.co 3
12.b even 2 1 7600.2.a.bm 3
15.d odd 2 1 950.2.a.k 3
15.e even 4 2 950.2.b.g 6
60.h even 2 1 7600.2.a.cb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.k 3 15.d odd 2 1
950.2.a.m yes 3 3.b odd 2 1
950.2.b.g 6 15.e even 4 2
7600.2.a.bm 3 12.b even 2 1
7600.2.a.cb 3 60.h even 2 1
8550.2.a.cj 3 1.a even 1 1 trivial
8550.2.a.co 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7}^{3} - 2T_{7}^{2} - 21T_{7} + 49$$ T7^3 - 2*T7^2 - 21*T7 + 49 $$T_{11}^{3} + 2T_{11}^{2} - 32T_{11} - 24$$ T11^3 + 2*T11^2 - 32*T11 - 24 $$T_{13}^{3} + 2T_{13}^{2} - 23T_{13} + 21$$ T13^3 + 2*T13^2 - 23*T13 + 21 $$T_{17}^{3} - 4T_{17}^{2} - 15T_{17} - 7$$ T17^3 - 4*T17^2 - 15*T17 - 7 $$T_{23}^{3} + 14T_{23}^{2} + 57T_{23} + 63$$ T23^3 + 14*T23^2 + 57*T23 + 63 $$T_{53}^{3} - 10T_{53}^{2} + 11T_{53} + 3$$ T53^3 - 10*T53^2 + 11*T53 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 2 T^{2} + \cdots + 49$$
$11$ $$T^{3} + 2 T^{2} + \cdots - 24$$
$13$ $$T^{3} + 2 T^{2} + \cdots + 21$$
$17$ $$T^{3} - 4 T^{2} + \cdots - 7$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} + 14 T^{2} + \cdots + 63$$
$29$ $$T^{3} + 14 T^{2} + \cdots + 25$$
$31$ $$T^{3} + 4 T^{2} + \cdots - 152$$
$37$ $$T^{3} + 17 T^{2} + \cdots - 9$$
$41$ $$T^{3} - 8 T^{2} + \cdots + 64$$
$43$ $$T^{3} - 2 T^{2} + \cdots - 72$$
$47$ $$T^{3} + 13 T^{2} + \cdots - 525$$
$53$ $$T^{3} - 10 T^{2} + \cdots + 3$$
$59$ $$T^{3} - 6 T^{2} + \cdots + 175$$
$61$ $$T^{3} - 22 T^{2} + \cdots + 536$$
$67$ $$T^{3} - 131T - 469$$
$71$ $$T^{3} - 2 T^{2} + \cdots + 24$$
$73$ $$T^{3} + 12 T^{2} + \cdots - 243$$
$79$ $$T^{3} - 24 T^{2} + \cdots + 320$$
$83$ $$T^{3} + 12 T^{2} + \cdots - 1544$$
$89$ $$(T - 10)^{3}$$
$97$ $$T^{3} - 180T + 648$$