# Properties

 Label 8550.2.a.bq 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: $$\Q(\sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 10$$ x^2 - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2850) 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 $$\beta = \sqrt{10}$$. 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 $$q - q^{2} + q^{4} + \beta q^{7} - q^{8} + ( - \beta - 1) q^{11} + (\beta + 2) q^{13} - \beta q^{14} + q^{16} + (\beta - 4) q^{17} - q^{19} + (\beta + 1) q^{22} + (2 \beta - 1) q^{23} + ( - \beta - 2) q^{26} + \beta q^{28} + ( - \beta + 7) q^{29} + (2 \beta - 1) q^{31} - q^{32} + ( - \beta + 4) q^{34} + 10 q^{37} + q^{38} + 2 \beta q^{41} + ( - \beta - 6) q^{43} + ( - \beta - 1) q^{44} + ( - 2 \beta + 1) q^{46} - 6 q^{47} + 3 q^{49} + (\beta + 2) q^{52} + (\beta - 5) q^{53} - \beta q^{56} + (\beta - 7) q^{58} + (3 \beta - 2) q^{59} + ( - \beta - 1) q^{61} + ( - 2 \beta + 1) q^{62} + q^{64} + ( - 3 \beta + 3) q^{67} + (\beta - 4) q^{68} + ( - \beta + 2) q^{71} - q^{73} - 10 q^{74} - q^{76} + ( - \beta - 10) q^{77} + ( - 2 \beta + 1) q^{79} - 2 \beta q^{82} + ( - 3 \beta + 3) q^{83} + (\beta + 6) q^{86} + (\beta + 1) q^{88} + (2 \beta + 1) q^{89} + (2 \beta + 10) q^{91} + (2 \beta - 1) q^{92} + 6 q^{94} + ( - 3 \beta + 10) q^{97} - 3 q^{98} +O(q^{100})$$ q - q^2 + q^4 + b * q^7 - q^8 + (-b - 1) * q^11 + (b + 2) * q^13 - b * q^14 + q^16 + (b - 4) * q^17 - q^19 + (b + 1) * q^22 + (2*b - 1) * q^23 + (-b - 2) * q^26 + b * q^28 + (-b + 7) * q^29 + (2*b - 1) * q^31 - q^32 + (-b + 4) * q^34 + 10 * q^37 + q^38 + 2*b * q^41 + (-b - 6) * q^43 + (-b - 1) * q^44 + (-2*b + 1) * q^46 - 6 * q^47 + 3 * q^49 + (b + 2) * q^52 + (b - 5) * q^53 - b * q^56 + (b - 7) * q^58 + (3*b - 2) * q^59 + (-b - 1) * q^61 + (-2*b + 1) * q^62 + q^64 + (-3*b + 3) * q^67 + (b - 4) * q^68 + (-b + 2) * q^71 - q^73 - 10 * q^74 - q^76 + (-b - 10) * q^77 + (-2*b + 1) * q^79 - 2*b * q^82 + (-3*b + 3) * q^83 + (b + 6) * q^86 + (b + 1) * q^88 + (2*b + 1) * q^89 + (2*b + 10) * q^91 + (2*b - 1) * q^92 + 6 * q^94 + (-3*b + 10) * q^97 - 3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{11} + 4 q^{13} + 2 q^{16} - 8 q^{17} - 2 q^{19} + 2 q^{22} - 2 q^{23} - 4 q^{26} + 14 q^{29} - 2 q^{31} - 2 q^{32} + 8 q^{34} + 20 q^{37} + 2 q^{38} - 12 q^{43} - 2 q^{44} + 2 q^{46} - 12 q^{47} + 6 q^{49} + 4 q^{52} - 10 q^{53} - 14 q^{58} - 4 q^{59} - 2 q^{61} + 2 q^{62} + 2 q^{64} + 6 q^{67} - 8 q^{68} + 4 q^{71} - 2 q^{73} - 20 q^{74} - 2 q^{76} - 20 q^{77} + 2 q^{79} + 6 q^{83} + 12 q^{86} + 2 q^{88} + 2 q^{89} + 20 q^{91} - 2 q^{92} + 12 q^{94} + 20 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^11 + 4 * q^13 + 2 * q^16 - 8 * q^17 - 2 * q^19 + 2 * q^22 - 2 * q^23 - 4 * q^26 + 14 * q^29 - 2 * q^31 - 2 * q^32 + 8 * q^34 + 20 * q^37 + 2 * q^38 - 12 * q^43 - 2 * q^44 + 2 * q^46 - 12 * q^47 + 6 * q^49 + 4 * q^52 - 10 * q^53 - 14 * q^58 - 4 * q^59 - 2 * q^61 + 2 * q^62 + 2 * q^64 + 6 * q^67 - 8 * q^68 + 4 * q^71 - 2 * q^73 - 20 * q^74 - 2 * q^76 - 20 * q^77 + 2 * q^79 + 6 * q^83 + 12 * q^86 + 2 * q^88 + 2 * q^89 + 20 * q^91 - 2 * q^92 + 12 * q^94 + 20 * q^97 - 6 * 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
 −3.16228 3.16228
−1.00000 0 1.00000 0 0 −3.16228 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 3.16228 −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.bq 2
3.b odd 2 1 2850.2.a.bg yes 2
5.b even 2 1 8550.2.a.ca 2
15.d odd 2 1 2850.2.a.bf 2
15.e even 4 2 2850.2.d.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bf 2 15.d odd 2 1
2850.2.a.bg yes 2 3.b odd 2 1
2850.2.d.v 4 15.e even 4 2
8550.2.a.bq 2 1.a even 1 1 trivial
8550.2.a.ca 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7}^{2} - 10$$ T7^2 - 10 $$T_{11}^{2} + 2T_{11} - 9$$ T11^2 + 2*T11 - 9 $$T_{13}^{2} - 4T_{13} - 6$$ T13^2 - 4*T13 - 6 $$T_{17}^{2} + 8T_{17} + 6$$ T17^2 + 8*T17 + 6 $$T_{23}^{2} + 2T_{23} - 39$$ T23^2 + 2*T23 - 39 $$T_{53}^{2} + 10T_{53} + 15$$ T53^2 + 10*T53 + 15

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 10$$
$11$ $$T^{2} + 2T - 9$$
$13$ $$T^{2} - 4T - 6$$
$17$ $$T^{2} + 8T + 6$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 2T - 39$$
$29$ $$T^{2} - 14T + 39$$
$31$ $$T^{2} + 2T - 39$$
$37$ $$(T - 10)^{2}$$
$41$ $$T^{2} - 40$$
$43$ $$T^{2} + 12T + 26$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} + 10T + 15$$
$59$ $$T^{2} + 4T - 86$$
$61$ $$T^{2} + 2T - 9$$
$67$ $$T^{2} - 6T - 81$$
$71$ $$T^{2} - 4T - 6$$
$73$ $$(T + 1)^{2}$$
$79$ $$T^{2} - 2T - 39$$
$83$ $$T^{2} - 6T - 81$$
$89$ $$T^{2} - 2T - 39$$
$97$ $$T^{2} - 20T + 10$$