# Properties

 Label 2760.2.a.r Level $2760$ Weight $2$ Character orbit 2760.a Self dual yes Analytic conductor $22.039$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(1,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, 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("2760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2760.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: $$22.0387109579$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + (\beta_{2} - 2) q^{7} + q^{9}+O(q^{10})$$ q + q^3 - q^5 + (b2 - 2) * q^7 + q^9 $$q + q^{3} - q^{5} + (\beta_{2} - 2) q^{7} + q^{9} + ( - \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{2} + 3 \beta_1 - 1) q^{13} - q^{15} + ( - 3 \beta_1 - 1) q^{17} + ( - \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{2} - 2) q^{21} + q^{23} + q^{25} + q^{27} + (2 \beta_{2} - 3 \beta_1 - 1) q^{29} + (3 \beta_{2} - 2) q^{31} + ( - \beta_{2} - \beta_1 + 1) q^{33} + ( - \beta_{2} + 2) q^{35} + (3 \beta_{2} - 2 \beta_1 - 2) q^{37} + ( - \beta_{2} + 3 \beta_1 - 1) q^{39} + ( - 2 \beta_{2} + \beta_1 - 5) q^{41} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{43} - q^{45} + ( - 3 \beta_{2} + 3 \beta_1 - 7) q^{47} + ( - 5 \beta_{2} + 2 \beta_1 + 1) q^{49} + ( - 3 \beta_1 - 1) q^{51} + (3 \beta_1 - 7) q^{53} + (\beta_{2} + \beta_1 - 1) q^{55} + ( - \beta_{2} + \beta_1 - 1) q^{57} + (2 \beta_{2} - 3 \beta_1 + 5) q^{59} + (\beta_{2} - \beta_1 - 5) q^{61} + (\beta_{2} - 2) q^{63} + (\beta_{2} - 3 \beta_1 + 1) q^{65} + (\beta_{2} - 2 \beta_1 - 4) q^{67} + q^{69} + ( - 6 \beta_{2} + 5 \beta_1 - 3) q^{71} + (3 \beta_{2} + 3 \beta_1 - 1) q^{73} + q^{75} + (3 \beta_{2} - \beta_1 - 7) q^{77} + q^{81} + (4 \beta_{2} - \beta_1 - 1) q^{83} + (3 \beta_1 + 1) q^{85} + (2 \beta_{2} - 3 \beta_1 - 1) q^{87} + (2 \beta_1 - 4) q^{89} + (5 \beta_{2} - 5 \beta_1 + 1) q^{91} + (3 \beta_{2} - 2) q^{93} + (\beta_{2} - \beta_1 + 1) q^{95} + (2 \beta_{2} - 4 \beta_1 + 2) q^{97} + ( - \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100})$$ q + q^3 - q^5 + (b2 - 2) * q^7 + q^9 + (-b2 - b1 + 1) * q^11 + (-b2 + 3*b1 - 1) * q^13 - q^15 + (-3*b1 - 1) * q^17 + (-b2 + b1 - 1) * q^19 + (b2 - 2) * q^21 + q^23 + q^25 + q^27 + (2*b2 - 3*b1 - 1) * q^29 + (3*b2 - 2) * q^31 + (-b2 - b1 + 1) * q^33 + (-b2 + 2) * q^35 + (3*b2 - 2*b1 - 2) * q^37 + (-b2 + 3*b1 - 1) * q^39 + (-2*b2 + b1 - 5) * q^41 + (-4*b2 + 2*b1 - 2) * q^43 - q^45 + (-3*b2 + 3*b1 - 7) * q^47 + (-5*b2 + 2*b1 + 1) * q^49 + (-3*b1 - 1) * q^51 + (3*b1 - 7) * q^53 + (b2 + b1 - 1) * q^55 + (-b2 + b1 - 1) * q^57 + (2*b2 - 3*b1 + 5) * q^59 + (b2 - b1 - 5) * q^61 + (b2 - 2) * q^63 + (b2 - 3*b1 + 1) * q^65 + (b2 - 2*b1 - 4) * q^67 + q^69 + (-6*b2 + 5*b1 - 3) * q^71 + (3*b2 + 3*b1 - 1) * q^73 + q^75 + (3*b2 - b1 - 7) * q^77 + q^81 + (4*b2 - b1 - 1) * q^83 + (3*b1 + 1) * q^85 + (2*b2 - 3*b1 - 1) * q^87 + (2*b1 - 4) * q^89 + (5*b2 - 5*b1 + 1) * q^91 + (3*b2 - 2) * q^93 + (b2 - b1 + 1) * q^95 + (2*b2 - 4*b1 + 2) * q^97 + (-b2 - b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 3 q^{5} - 6 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - 3 * q^5 - 6 * q^7 + 3 * q^9 $$3 q + 3 q^{3} - 3 q^{5} - 6 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{15} - 6 q^{17} - 2 q^{19} - 6 q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} - 6 q^{29} - 6 q^{31} + 2 q^{33} + 6 q^{35} - 8 q^{37} - 14 q^{41} - 4 q^{43} - 3 q^{45} - 18 q^{47} + 5 q^{49} - 6 q^{51} - 18 q^{53} - 2 q^{55} - 2 q^{57} + 12 q^{59} - 16 q^{61} - 6 q^{63} - 14 q^{67} + 3 q^{69} - 4 q^{71} + 3 q^{75} - 22 q^{77} + 3 q^{81} - 4 q^{83} + 6 q^{85} - 6 q^{87} - 10 q^{89} - 2 q^{91} - 6 q^{93} + 2 q^{95} + 2 q^{97} + 2 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - 3 * q^5 - 6 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^15 - 6 * q^17 - 2 * q^19 - 6 * q^21 + 3 * q^23 + 3 * q^25 + 3 * q^27 - 6 * q^29 - 6 * q^31 + 2 * q^33 + 6 * q^35 - 8 * q^37 - 14 * q^41 - 4 * q^43 - 3 * q^45 - 18 * q^47 + 5 * q^49 - 6 * q^51 - 18 * q^53 - 2 * q^55 - 2 * q^57 + 12 * q^59 - 16 * q^61 - 6 * q^63 - 14 * q^67 + 3 * q^69 - 4 * q^71 + 3 * q^75 - 22 * q^77 + 3 * q^81 - 4 * q^83 + 6 * q^85 - 6 * q^87 - 10 * q^89 - 2 * q^91 - 6 * q^93 + 2 * q^95 + 2 * q^97 + 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## 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
 0.470683 −1.81361 2.34292
0 1.00000 0 −1.00000 0 −4.77846 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −1.71083 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 0.489289 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$23$$ $$-1$$

## 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 2760.2.a.r 3
3.b odd 2 1 8280.2.a.bm 3
4.b odd 2 1 5520.2.a.bx 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.r 3 1.a even 1 1 trivial
5520.2.a.bx 3 4.b odd 2 1
8280.2.a.bm 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2760))$$:

 $$T_{7}^{3} + 6T_{7}^{2} + 5T_{7} - 4$$ T7^3 + 6*T7^2 + 5*T7 - 4 $$T_{11}^{3} - 2T_{11}^{2} - 14T_{11} + 32$$ T11^3 - 2*T11^2 - 14*T11 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 6 T^{2} + 5 T - 4$$
$11$ $$T^{3} - 2 T^{2} - 14 T + 32$$
$13$ $$T^{3} - 34T + 76$$
$17$ $$T^{3} + 6 T^{2} - 27 T - 86$$
$19$ $$T^{3} + 2 T^{2} - 6 T - 8$$
$23$ $$(T - 1)^{3}$$
$29$ $$T^{3} + 6 T^{2} - 31 T - 122$$
$31$ $$T^{3} + 6 T^{2} - 51 T - 64$$
$37$ $$T^{3} + 8 T^{2} - 35 T + 22$$
$41$ $$T^{3} + 14 T^{2} + 41 T - 58$$
$43$ $$T^{3} + 4 T^{2} - 92 T - 496$$
$47$ $$T^{3} + 18 T^{2} + 42 T - 272$$
$53$ $$T^{3} + 18 T^{2} + 69 T - 2$$
$59$ $$T^{3} - 12 T^{2} + 5 T + 64$$
$61$ $$T^{3} + 16 T^{2} + 78 T + 116$$
$67$ $$T^{3} + 14 T^{2} + 49 T + 4$$
$71$ $$T^{3} + 4 T^{2} - 235 T - 1376$$
$73$ $$T^{3} - 138T - 596$$
$79$ $$T^{3}$$
$83$ $$T^{3} + 4 T^{2} - 95 T + 164$$
$89$ $$T^{3} + 10 T^{2} + 16 T - 16$$
$97$ $$T^{3} - 2 T^{2} - 64 T - 128$$