# Properties

 Label 8550.2.a.bu Level $8550$ Weight $2$ Character orbit 8550.a Self dual yes Analytic conductor $68.272$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta + 2) q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b + 2) * q^7 - q^8 $$q - q^{2} + q^{4} + (\beta + 2) q^{7} - q^{8} + ( - \beta - 1) q^{11} - \beta q^{13} + ( - \beta - 2) q^{14} + q^{16} + ( - \beta - 2) q^{17} + q^{19} + (\beta + 1) q^{22} + q^{23} + \beta q^{26} + (\beta + 2) q^{28} + (3 \beta - 3) q^{29} - 3 q^{31} - q^{32} + (\beta + 2) q^{34} + ( - 4 \beta + 2) q^{37} - q^{38} + (2 \beta - 4) q^{41} - \beta q^{43} + ( - \beta - 1) q^{44} - q^{46} + (4 \beta - 2) q^{47} + (4 \beta + 3) q^{49} - \beta q^{52} + (\beta + 5) q^{53} + ( - \beta - 2) q^{56} + ( - 3 \beta + 3) q^{58} + ( - \beta - 4) q^{59} + ( - \beta - 7) q^{61} + 3 q^{62} + q^{64} + (5 \beta + 3) q^{67} + ( - \beta - 2) q^{68} + ( - \beta + 4) q^{71} - q^{73} + (4 \beta - 2) q^{74} + q^{76} + ( - 3 \beta - 8) q^{77} - 5 q^{79} + ( - 2 \beta + 4) q^{82} + ( - 3 \beta - 1) q^{83} + \beta q^{86} + (\beta + 1) q^{88} + (2 \beta - 7) q^{89} + ( - 2 \beta - 6) q^{91} + q^{92} + ( - 4 \beta + 2) q^{94} + ( - \beta + 4) q^{97} + ( - 4 \beta - 3) q^{98} +O(q^{100})$$ q - q^2 + q^4 + (b + 2) * q^7 - q^8 + (-b - 1) * q^11 - b * q^13 + (-b - 2) * q^14 + q^16 + (-b - 2) * q^17 + q^19 + (b + 1) * q^22 + q^23 + b * q^26 + (b + 2) * q^28 + (3*b - 3) * q^29 - 3 * q^31 - q^32 + (b + 2) * q^34 + (-4*b + 2) * q^37 - q^38 + (2*b - 4) * q^41 - b * q^43 + (-b - 1) * q^44 - q^46 + (4*b - 2) * q^47 + (4*b + 3) * q^49 - b * q^52 + (b + 5) * q^53 + (-b - 2) * q^56 + (-3*b + 3) * q^58 + (-b - 4) * q^59 + (-b - 7) * q^61 + 3 * q^62 + q^64 + (5*b + 3) * q^67 + (-b - 2) * q^68 + (-b + 4) * q^71 - q^73 + (4*b - 2) * q^74 + q^76 + (-3*b - 8) * q^77 - 5 * q^79 + (-2*b + 4) * q^82 + (-3*b - 1) * q^83 + b * q^86 + (b + 1) * q^88 + (2*b - 7) * q^89 + (-2*b - 6) * q^91 + q^92 + (-4*b + 2) * q^94 + (-b + 4) * q^97 + (-4*b - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8} - 2 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{19} + 2 q^{22} + 2 q^{23} + 4 q^{28} - 6 q^{29} - 6 q^{31} - 2 q^{32} + 4 q^{34} + 4 q^{37} - 2 q^{38} - 8 q^{41} - 2 q^{44} - 2 q^{46} - 4 q^{47} + 6 q^{49} + 10 q^{53} - 4 q^{56} + 6 q^{58} - 8 q^{59} - 14 q^{61} + 6 q^{62} + 2 q^{64} + 6 q^{67} - 4 q^{68} + 8 q^{71} - 2 q^{73} - 4 q^{74} + 2 q^{76} - 16 q^{77} - 10 q^{79} + 8 q^{82} - 2 q^{83} + 2 q^{88} - 14 q^{89} - 12 q^{91} + 2 q^{92} + 4 q^{94} + 8 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 2 * q^8 - 2 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^19 + 2 * q^22 + 2 * q^23 + 4 * q^28 - 6 * q^29 - 6 * q^31 - 2 * q^32 + 4 * q^34 + 4 * q^37 - 2 * q^38 - 8 * q^41 - 2 * q^44 - 2 * q^46 - 4 * q^47 + 6 * q^49 + 10 * q^53 - 4 * q^56 + 6 * q^58 - 8 * q^59 - 14 * q^61 + 6 * q^62 + 2 * q^64 + 6 * q^67 - 4 * q^68 + 8 * q^71 - 2 * q^73 - 4 * q^74 + 2 * q^76 - 16 * q^77 - 10 * q^79 + 8 * q^82 - 2 * q^83 + 2 * q^88 - 14 * q^89 - 12 * q^91 + 2 * q^92 + 4 * q^94 + 8 * 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
 −2.44949 2.44949
−1.00000 0 1.00000 0 0 −0.449490 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 4.44949 −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.bu 2
3.b odd 2 1 2850.2.a.bj yes 2
5.b even 2 1 8550.2.a.bv 2
15.d odd 2 1 2850.2.a.bc 2
15.e even 4 2 2850.2.d.w 4

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

 $$T_{7}^{2} - 4T_{7} - 2$$ T7^2 - 4*T7 - 2 $$T_{11}^{2} + 2T_{11} - 5$$ T11^2 + 2*T11 - 5 $$T_{13}^{2} - 6$$ T13^2 - 6 $$T_{17}^{2} + 4T_{17} - 2$$ T17^2 + 4*T17 - 2 $$T_{23} - 1$$ T23 - 1 $$T_{53}^{2} - 10T_{53} + 19$$ T53^2 - 10*T53 + 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T - 2$$
$11$ $$T^{2} + 2T - 5$$
$13$ $$T^{2} - 6$$
$17$ $$T^{2} + 4T - 2$$
$19$ $$(T - 1)^{2}$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 6T - 45$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} - 4T - 92$$
$41$ $$T^{2} + 8T - 8$$
$43$ $$T^{2} - 6$$
$47$ $$T^{2} + 4T - 92$$
$53$ $$T^{2} - 10T + 19$$
$59$ $$T^{2} + 8T + 10$$
$61$ $$T^{2} + 14T + 43$$
$67$ $$T^{2} - 6T - 141$$
$71$ $$T^{2} - 8T + 10$$
$73$ $$(T + 1)^{2}$$
$79$ $$(T + 5)^{2}$$
$83$ $$T^{2} + 2T - 53$$
$89$ $$T^{2} + 14T + 25$$
$97$ $$T^{2} - 8T + 10$$