# Properties

 Label 8550.2.a.bn 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta - 3) q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b - 3) * q^7 - q^8 $$q - q^{2} + q^{4} + (\beta - 3) q^{7} - q^{8} - \beta q^{11} + (2 \beta - 3) q^{13} + ( - \beta + 3) q^{14} + q^{16} + q^{17} - q^{19} + \beta q^{22} + (3 \beta + 5) q^{23} + ( - 2 \beta + 3) q^{26} + (\beta - 3) q^{28} + (2 \beta + 3) q^{29} + ( - 3 \beta + 2) q^{31} - q^{32} - q^{34} - 6 \beta q^{37} + q^{38} - 3 \beta q^{41} + ( - 3 \beta - 6) q^{43} - \beta q^{44} + ( - 3 \beta - 5) q^{46} + ( - 6 \beta + 4) q^{49} + (2 \beta - 3) q^{52} + ( - 6 \beta - 3) q^{53} + ( - \beta + 3) q^{56} + ( - 2 \beta - 3) q^{58} + (7 \beta + 3) q^{59} + ( - 3 \beta + 10) q^{61} + (3 \beta - 2) q^{62} + q^{64} + ( - 3 \beta - 9) q^{67} + q^{68} + ( - \beta + 12) q^{71} + (6 \beta - 3) q^{73} + 6 \beta q^{74} - q^{76} + (3 \beta - 2) q^{77} + (6 \beta + 2) q^{79} + 3 \beta q^{82} + ( - 6 \beta + 6) q^{83} + (3 \beta + 6) q^{86} + \beta q^{88} + 5 \beta q^{89} + ( - 9 \beta + 13) q^{91} + (3 \beta + 5) q^{92} + (4 \beta + 6) q^{97} + (6 \beta - 4) q^{98} +O(q^{100})$$ q - q^2 + q^4 + (b - 3) * q^7 - q^8 - b * q^11 + (2*b - 3) * q^13 + (-b + 3) * q^14 + q^16 + q^17 - q^19 + b * q^22 + (3*b + 5) * q^23 + (-2*b + 3) * q^26 + (b - 3) * q^28 + (2*b + 3) * q^29 + (-3*b + 2) * q^31 - q^32 - q^34 - 6*b * q^37 + q^38 - 3*b * q^41 + (-3*b - 6) * q^43 - b * q^44 + (-3*b - 5) * q^46 + (-6*b + 4) * q^49 + (2*b - 3) * q^52 + (-6*b - 3) * q^53 + (-b + 3) * q^56 + (-2*b - 3) * q^58 + (7*b + 3) * q^59 + (-3*b + 10) * q^61 + (3*b - 2) * q^62 + q^64 + (-3*b - 9) * q^67 + q^68 + (-b + 12) * q^71 + (6*b - 3) * q^73 + 6*b * q^74 - q^76 + (3*b - 2) * q^77 + (6*b + 2) * q^79 + 3*b * q^82 + (-6*b + 6) * q^83 + (3*b + 6) * q^86 + b * q^88 + 5*b * q^89 + (-9*b + 13) * q^91 + (3*b + 5) * q^92 + (4*b + 6) * q^97 + (6*b - 4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8} - 6 q^{13} + 6 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 10 q^{23} + 6 q^{26} - 6 q^{28} + 6 q^{29} + 4 q^{31} - 2 q^{32} - 2 q^{34} + 2 q^{38} - 12 q^{43} - 10 q^{46} + 8 q^{49} - 6 q^{52} - 6 q^{53} + 6 q^{56} - 6 q^{58} + 6 q^{59} + 20 q^{61} - 4 q^{62} + 2 q^{64} - 18 q^{67} + 2 q^{68} + 24 q^{71} - 6 q^{73} - 2 q^{76} - 4 q^{77} + 4 q^{79} + 12 q^{83} + 12 q^{86} + 26 q^{91} + 10 q^{92} + 12 q^{97} - 8 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8 - 6 * q^13 + 6 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 + 10 * q^23 + 6 * q^26 - 6 * q^28 + 6 * q^29 + 4 * q^31 - 2 * q^32 - 2 * q^34 + 2 * q^38 - 12 * q^43 - 10 * q^46 + 8 * q^49 - 6 * q^52 - 6 * q^53 + 6 * q^56 - 6 * q^58 + 6 * q^59 + 20 * q^61 - 4 * q^62 + 2 * q^64 - 18 * q^67 + 2 * q^68 + 24 * q^71 - 6 * q^73 - 2 * q^76 - 4 * q^77 + 4 * q^79 + 12 * q^83 + 12 * q^86 + 26 * q^91 + 10 * q^92 + 12 * q^97 - 8 * 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
 −1.41421 1.41421
−1.00000 0 1.00000 0 0 −4.41421 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 −1.58579 −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.bn 2
3.b odd 2 1 950.2.a.g 2
5.b even 2 1 8550.2.a.cb 2
5.c odd 4 2 1710.2.d.c 4
12.b even 2 1 7600.2.a.bg 2
15.d odd 2 1 950.2.a.f 2
15.e even 4 2 190.2.b.a 4
60.h even 2 1 7600.2.a.v 2
60.l odd 4 2 1520.2.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.a 4 15.e even 4 2
950.2.a.f 2 15.d odd 2 1
950.2.a.g 2 3.b odd 2 1
1520.2.d.e 4 60.l odd 4 2
1710.2.d.c 4 5.c odd 4 2
7600.2.a.v 2 60.h even 2 1
7600.2.a.bg 2 12.b even 2 1
8550.2.a.bn 2 1.a even 1 1 trivial
8550.2.a.cb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7}^{2} + 6T_{7} + 7$$ T7^2 + 6*T7 + 7 $$T_{11}^{2} - 2$$ T11^2 - 2 $$T_{13}^{2} + 6T_{13} + 1$$ T13^2 + 6*T13 + 1 $$T_{17} - 1$$ T17 - 1 $$T_{23}^{2} - 10T_{23} + 7$$ T23^2 - 10*T23 + 7 $$T_{53}^{2} + 6T_{53} - 63$$ T53^2 + 6*T53 - 63

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6T + 7$$
$11$ $$T^{2} - 2$$
$13$ $$T^{2} + 6T + 1$$
$17$ $$(T - 1)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 10T + 7$$
$29$ $$T^{2} - 6T + 1$$
$31$ $$T^{2} - 4T - 14$$
$37$ $$T^{2} - 72$$
$41$ $$T^{2} - 18$$
$43$ $$T^{2} + 12T + 18$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T - 63$$
$59$ $$T^{2} - 6T - 89$$
$61$ $$T^{2} - 20T + 82$$
$67$ $$T^{2} + 18T + 63$$
$71$ $$T^{2} - 24T + 142$$
$73$ $$T^{2} + 6T - 63$$
$79$ $$T^{2} - 4T - 68$$
$83$ $$T^{2} - 12T - 36$$
$89$ $$T^{2} - 50$$
$97$ $$T^{2} - 12T + 4$$