# Properties

 Label 8550.2.a.br Level $8550$ Weight $2$ Character orbit 8550.a Self dual yes Analytic conductor $68.272$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + b * q^7 - q^8 $$q - q^{2} + q^{4} + \beta q^{7} - q^{8} - 4 q^{11} + ( - 3 \beta + 2) q^{13} - \beta q^{14} + q^{16} + ( - \beta + 6) q^{17} - q^{19} + 4 q^{22} + 3 \beta q^{23} + (3 \beta - 2) q^{26} + \beta q^{28} + (3 \beta - 2) q^{29} - 2 \beta q^{31} - q^{32} + (\beta - 6) q^{34} + 6 q^{37} + q^{38} + ( - 4 \beta - 2) q^{41} + ( - 2 \beta + 8) q^{43} - 4 q^{44} - 3 \beta q^{46} + (4 \beta - 4) q^{47} + (\beta - 3) q^{49} + ( - 3 \beta + 2) q^{52} + ( - \beta - 2) q^{53} - \beta q^{56} + ( - 3 \beta + 2) q^{58} - \beta q^{59} + (2 \beta + 6) q^{61} + 2 \beta q^{62} + q^{64} + \beta q^{67} + ( - \beta + 6) q^{68} - 4 \beta q^{71} + (3 \beta - 6) q^{73} - 6 q^{74} - q^{76} - 4 \beta q^{77} - 2 \beta q^{79} + (4 \beta + 2) q^{82} + ( - 2 \beta + 8) q^{83} + (2 \beta - 8) q^{86} + 4 q^{88} - 2 q^{89} + ( - \beta - 12) q^{91} + 3 \beta q^{92} + ( - 4 \beta + 4) q^{94} - 6 q^{97} + ( - \beta + 3) q^{98} +O(q^{100})$$ q - q^2 + q^4 + b * q^7 - q^8 - 4 * q^11 + (-3*b + 2) * q^13 - b * q^14 + q^16 + (-b + 6) * q^17 - q^19 + 4 * q^22 + 3*b * q^23 + (3*b - 2) * q^26 + b * q^28 + (3*b - 2) * q^29 - 2*b * q^31 - q^32 + (b - 6) * q^34 + 6 * q^37 + q^38 + (-4*b - 2) * q^41 + (-2*b + 8) * q^43 - 4 * q^44 - 3*b * q^46 + (4*b - 4) * q^47 + (b - 3) * q^49 + (-3*b + 2) * q^52 + (-b - 2) * q^53 - b * q^56 + (-3*b + 2) * q^58 - b * q^59 + (2*b + 6) * q^61 + 2*b * q^62 + q^64 + b * q^67 + (-b + 6) * q^68 - 4*b * q^71 + (3*b - 6) * q^73 - 6 * q^74 - q^76 - 4*b * q^77 - 2*b * q^79 + (4*b + 2) * q^82 + (-2*b + 8) * q^83 + (2*b - 8) * q^86 + 4 * q^88 - 2 * q^89 + (-b - 12) * q^91 + 3*b * q^92 + (-4*b + 4) * q^94 - 6 * q^97 + (-b + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8} - 8 q^{11} + q^{13} - q^{14} + 2 q^{16} + 11 q^{17} - 2 q^{19} + 8 q^{22} + 3 q^{23} - q^{26} + q^{28} - q^{29} - 2 q^{31} - 2 q^{32} - 11 q^{34} + 12 q^{37} + 2 q^{38} - 8 q^{41} + 14 q^{43} - 8 q^{44} - 3 q^{46} - 4 q^{47} - 5 q^{49} + q^{52} - 5 q^{53} - q^{56} + q^{58} - q^{59} + 14 q^{61} + 2 q^{62} + 2 q^{64} + q^{67} + 11 q^{68} - 4 q^{71} - 9 q^{73} - 12 q^{74} - 2 q^{76} - 4 q^{77} - 2 q^{79} + 8 q^{82} + 14 q^{83} - 14 q^{86} + 8 q^{88} - 4 q^{89} - 25 q^{91} + 3 q^{92} + 4 q^{94} - 12 q^{97} + 5 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8 - 8 * q^11 + q^13 - q^14 + 2 * q^16 + 11 * q^17 - 2 * q^19 + 8 * q^22 + 3 * q^23 - q^26 + q^28 - q^29 - 2 * q^31 - 2 * q^32 - 11 * q^34 + 12 * q^37 + 2 * q^38 - 8 * q^41 + 14 * q^43 - 8 * q^44 - 3 * q^46 - 4 * q^47 - 5 * q^49 + q^52 - 5 * q^53 - q^56 + q^58 - q^59 + 14 * q^61 + 2 * q^62 + 2 * q^64 + q^67 + 11 * q^68 - 4 * q^71 - 9 * q^73 - 12 * q^74 - 2 * q^76 - 4 * q^77 - 2 * q^79 + 8 * q^82 + 14 * q^83 - 14 * q^86 + 8 * q^88 - 4 * q^89 - 25 * q^91 + 3 * q^92 + 4 * q^94 - 12 * q^97 + 5 * 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
 −1.56155 2.56155
−1.00000 0 1.00000 0 0 −1.56155 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 2.56155 −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.br 2
3.b odd 2 1 950.2.a.h 2
5.b even 2 1 1710.2.a.w 2
12.b even 2 1 7600.2.a.y 2
15.d odd 2 1 190.2.a.d 2
15.e even 4 2 950.2.b.f 4
60.h even 2 1 1520.2.a.n 2
105.g even 2 1 9310.2.a.bc 2
120.i odd 2 1 6080.2.a.bh 2
120.m even 2 1 6080.2.a.bb 2
285.b even 2 1 3610.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 15.d odd 2 1
950.2.a.h 2 3.b odd 2 1
950.2.b.f 4 15.e even 4 2
1520.2.a.n 2 60.h even 2 1
1710.2.a.w 2 5.b even 2 1
3610.2.a.t 2 285.b even 2 1
6080.2.a.bb 2 120.m even 2 1
6080.2.a.bh 2 120.i odd 2 1
7600.2.a.y 2 12.b even 2 1
8550.2.a.br 2 1.a even 1 1 trivial
9310.2.a.bc 2 105.g even 2 1

## Hecke kernels

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

 $$T_{7}^{2} - T_{7} - 4$$ T7^2 - T7 - 4 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} - T_{13} - 38$$ T13^2 - T13 - 38 $$T_{17}^{2} - 11T_{17} + 26$$ T17^2 - 11*T17 + 26 $$T_{23}^{2} - 3T_{23} - 36$$ T23^2 - 3*T23 - 36 $$T_{53}^{2} + 5T_{53} + 2$$ T53^2 + 5*T53 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T - 4$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} - T - 38$$
$17$ $$T^{2} - 11T + 26$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 3T - 36$$
$29$ $$T^{2} + T - 38$$
$31$ $$T^{2} + 2T - 16$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} + 8T - 52$$
$43$ $$T^{2} - 14T + 32$$
$47$ $$T^{2} + 4T - 64$$
$53$ $$T^{2} + 5T + 2$$
$59$ $$T^{2} + T - 4$$
$61$ $$T^{2} - 14T + 32$$
$67$ $$T^{2} - T - 4$$
$71$ $$T^{2} + 4T - 64$$
$73$ $$T^{2} + 9T - 18$$
$79$ $$T^{2} + 2T - 16$$
$83$ $$T^{2} - 14T + 32$$
$89$ $$(T + 2)^{2}$$
$97$ $$(T + 6)^{2}$$