# Properties

 Label 2280.2.a.p Level $2280$ Weight $2$ Character orbit 2280.a Self dual yes Analytic conductor $18.206$ 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] = [2280,2,Mod(1,2280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2280.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: $$18.2058916609$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + (\beta + 3) q^{7} + q^{9}+O(q^{10})$$ q + q^3 - q^5 + (b + 3) * q^7 + q^9 $$q + q^{3} - q^{5} + (\beta + 3) q^{7} + q^{9} + ( - \beta + 1) q^{11} + ( - \beta + 3) q^{13} - q^{15} + 2 \beta q^{17} + q^{19} + (\beta + 3) q^{21} + (2 \beta - 2) q^{23} + q^{25} + q^{27} + ( - \beta + 3) q^{29} - 4 \beta q^{31} + ( - \beta + 1) q^{33} + ( - \beta - 3) q^{35} + (\beta + 1) q^{37} + ( - \beta + 3) q^{39} + ( - 3 \beta - 3) q^{41} + (\beta + 3) q^{43} - q^{45} + (2 \beta - 2) q^{47} + (6 \beta + 7) q^{49} + 2 \beta q^{51} + ( - 2 \beta + 4) q^{53} + (\beta - 1) q^{55} + q^{57} + ( - 2 \beta - 6) q^{59} + 2 \beta q^{61} + (\beta + 3) q^{63} + (\beta - 3) q^{65} + ( - 2 \beta + 6) q^{67} + (2 \beta - 2) q^{69} + (2 \beta + 8) q^{73} + q^{75} + ( - 2 \beta - 2) q^{77} + q^{81} + 4 q^{83} - 2 \beta q^{85} + ( - \beta + 3) q^{87} + ( - 3 \beta + 5) q^{89} + 4 q^{91} - 4 \beta q^{93} - q^{95} + (\beta - 7) q^{97} + ( - \beta + 1) q^{99} +O(q^{100})$$ q + q^3 - q^5 + (b + 3) * q^7 + q^9 + (-b + 1) * q^11 + (-b + 3) * q^13 - q^15 + 2*b * q^17 + q^19 + (b + 3) * q^21 + (2*b - 2) * q^23 + q^25 + q^27 + (-b + 3) * q^29 - 4*b * q^31 + (-b + 1) * q^33 + (-b - 3) * q^35 + (b + 1) * q^37 + (-b + 3) * q^39 + (-3*b - 3) * q^41 + (b + 3) * q^43 - q^45 + (2*b - 2) * q^47 + (6*b + 7) * q^49 + 2*b * q^51 + (-2*b + 4) * q^53 + (b - 1) * q^55 + q^57 + (-2*b - 6) * q^59 + 2*b * q^61 + (b + 3) * q^63 + (b - 3) * q^65 + (-2*b + 6) * q^67 + (2*b - 2) * q^69 + (2*b + 8) * q^73 + q^75 + (-2*b - 2) * q^77 + q^81 + 4 * q^83 - 2*b * q^85 + (-b + 3) * q^87 + (-3*b + 5) * q^89 + 4 * q^91 - 4*b * q^93 - q^95 + (b - 7) * q^97 + (-b + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 + 6 * q^7 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{19} + 6 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{33} - 6 q^{35} + 2 q^{37} + 6 q^{39} - 6 q^{41} + 6 q^{43} - 2 q^{45} - 4 q^{47} + 14 q^{49} + 8 q^{53} - 2 q^{55} + 2 q^{57} - 12 q^{59} + 6 q^{63} - 6 q^{65} + 12 q^{67} - 4 q^{69} + 16 q^{73} + 2 q^{75} - 4 q^{77} + 2 q^{81} + 8 q^{83} + 6 q^{87} + 10 q^{89} + 8 q^{91} - 2 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 + 6 * q^7 + 2 * q^9 + 2 * q^11 + 6 * q^13 - 2 * q^15 + 2 * q^19 + 6 * q^21 - 4 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 + 2 * q^33 - 6 * q^35 + 2 * q^37 + 6 * q^39 - 6 * q^41 + 6 * q^43 - 2 * q^45 - 4 * q^47 + 14 * q^49 + 8 * q^53 - 2 * q^55 + 2 * q^57 - 12 * q^59 + 6 * q^63 - 6 * q^65 + 12 * q^67 - 4 * q^69 + 16 * q^73 + 2 * q^75 - 4 * q^77 + 2 * q^81 + 8 * q^83 + 6 * q^87 + 10 * q^89 + 8 * q^91 - 2 * q^95 - 14 * q^97 + 2 * q^99

## 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
 −0.618034 1.61803
0 1.00000 0 −1.00000 0 0.763932 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 5.23607 0 1.00000 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 2280.2.a.p 2
3.b odd 2 1 6840.2.a.bd 2
4.b odd 2 1 4560.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.p 2 1.a even 1 1 trivial
4560.2.a.be 2 4.b odd 2 1
6840.2.a.bd 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2280))$$:

 $$T_{7}^{2} - 6T_{7} + 4$$ T7^2 - 6*T7 + 4 $$T_{11}^{2} - 2T_{11} - 4$$ T11^2 - 2*T11 - 4 $$T_{13}^{2} - 6T_{13} + 4$$ T13^2 - 6*T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 6T + 4$$
$11$ $$T^{2} - 2T - 4$$
$13$ $$T^{2} - 6T + 4$$
$17$ $$T^{2} - 20$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 4T - 16$$
$29$ $$T^{2} - 6T + 4$$
$31$ $$T^{2} - 80$$
$37$ $$T^{2} - 2T - 4$$
$41$ $$T^{2} + 6T - 36$$
$43$ $$T^{2} - 6T + 4$$
$47$ $$T^{2} + 4T - 16$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} + 12T + 16$$
$61$ $$T^{2} - 20$$
$67$ $$T^{2} - 12T + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 16T + 44$$
$79$ $$T^{2}$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} - 10T - 20$$
$97$ $$T^{2} + 14T + 44$$