# Properties

 Label 2760.2.a.w Level $2760$ Weight $2$ Character orbit 2760.a Self dual yes Analytic conductor $22.039$ Analytic rank $0$ Dimension $5$ 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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.20087896.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4$$ x^5 - x^4 - 21*x^3 + 5*x^2 + 84*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + (\beta_1 - 1) q^{7} + q^{9}+O(q^{10})$$ q - q^3 + q^5 + (b1 - 1) * q^7 + q^9 $$q - q^{3} + q^{5} + (\beta_1 - 1) q^{7} + q^{9} + (\beta_{2} - 1) q^{11} + ( - \beta_{4} + 1) q^{13} - q^{15} + (\beta_{3} + \beta_1 + 2) q^{17} + ( - \beta_{3} - 1) q^{19} + ( - \beta_1 + 1) q^{21} - q^{23} + q^{25} - q^{27} + (\beta_{2} - \beta_1 + 2) q^{29} + (\beta_{4} - \beta_{2} + \beta_1 + 1) q^{31} + ( - \beta_{2} + 1) q^{33} + (\beta_1 - 1) q^{35} + (\beta_1 + 1) q^{37} + (\beta_{4} - 1) q^{39} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{41} + ( - \beta_{4} + \beta_{3}) q^{43} + q^{45} + (\beta_{3} + 1) q^{47} + (\beta_{4} - \beta_{2} - \beta_1 + 4) q^{49} + ( - \beta_{3} - \beta_1 - 2) q^{51} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{53} + (\beta_{2} - 1) q^{55} + (\beta_{3} + 1) q^{57} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{59} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 3) q^{61} + (\beta_1 - 1) q^{63} + ( - \beta_{4} + 1) q^{65} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{67} + q^{69} + ( - \beta_{2} - \beta_1 + 2) q^{71} + ( - \beta_{3} - 2 \beta_1 + 3) q^{73} - q^{75} + (2 \beta_{4} - \beta_{3} - 2 \beta_1 + 3) q^{77} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 4) q^{79} + q^{81} + ( - \beta_{3} - \beta_1 - 4) q^{83} + (\beta_{3} + \beta_1 + 2) q^{85} + ( - \beta_{2} + \beta_1 - 2) q^{87} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} + 4) q^{89} + (3 \beta_{3} - 2 \beta_{2} + 5) q^{91} + ( - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{93} + ( - \beta_{3} - 1) q^{95} + (\beta_{4} - \beta_{3} + 2 \beta_{2}) q^{97} + (\beta_{2} - 1) q^{99}+O(q^{100})$$ q - q^3 + q^5 + (b1 - 1) * q^7 + q^9 + (b2 - 1) * q^11 + (-b4 + 1) * q^13 - q^15 + (b3 + b1 + 2) * q^17 + (-b3 - 1) * q^19 + (-b1 + 1) * q^21 - q^23 + q^25 - q^27 + (b2 - b1 + 2) * q^29 + (b4 - b2 + b1 + 1) * q^31 + (-b2 + 1) * q^33 + (b1 - 1) * q^35 + (b1 + 1) * q^37 + (b4 - 1) * q^39 + (b4 - b3 - b2 + b1 + 2) * q^41 + (-b4 + b3) * q^43 + q^45 + (b3 + 1) * q^47 + (b4 - b2 - b1 + 4) * q^49 + (-b3 - b1 - 2) * q^51 + (b4 - b3 - b2 - b1) * q^53 + (b2 - 1) * q^55 + (b3 + 1) * q^57 + (-b4 + b3 + b2 - 3*b1 + 2) * q^59 + (-b4 - b3 + b2 + 3) * q^61 + (b1 - 1) * q^63 + (-b4 + 1) * q^65 + (-b3 - b2 + b1 - 1) * q^67 + q^69 + (-b2 - b1 + 2) * q^71 + (-b3 - 2*b1 + 3) * q^73 - q^75 + (2*b4 - b3 - 2*b1 + 3) * q^77 + (-b4 + 2*b3 + b2 + 4) * q^79 + q^81 + (-b3 - b1 - 4) * q^83 + (b3 + b1 + 2) * q^85 + (-b2 + b1 - 2) * q^87 + (-2*b4 + b3 + b2 + 4) * q^89 + (3*b3 - 2*b2 + 5) * q^91 + (-b4 + b2 - b1 - 1) * q^93 + (-b3 - 1) * q^95 + (b4 - b3 + 2*b2) * q^97 + (b2 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{3} + 5 q^{5} - 4 q^{7} + 5 q^{9}+O(q^{10})$$ 5 * q - 5 * q^3 + 5 * q^5 - 4 * q^7 + 5 * q^9 $$5 q - 5 q^{3} + 5 q^{5} - 4 q^{7} + 5 q^{9} - 4 q^{11} + 4 q^{13} - 5 q^{15} + 10 q^{17} - 4 q^{19} + 4 q^{21} - 5 q^{23} + 5 q^{25} - 5 q^{27} + 10 q^{29} + 6 q^{31} + 4 q^{33} - 4 q^{35} + 6 q^{37} - 4 q^{39} + 12 q^{41} - 2 q^{43} + 5 q^{45} + 4 q^{47} + 19 q^{49} - 10 q^{51} - 4 q^{55} + 4 q^{57} + 6 q^{59} + 16 q^{61} - 4 q^{63} + 4 q^{65} - 4 q^{67} + 5 q^{69} + 8 q^{71} + 14 q^{73} - 5 q^{75} + 16 q^{77} + 18 q^{79} + 5 q^{81} - 20 q^{83} + 10 q^{85} - 10 q^{87} + 18 q^{89} + 20 q^{91} - 6 q^{93} - 4 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100})$$ 5 * q - 5 * q^3 + 5 * q^5 - 4 * q^7 + 5 * q^9 - 4 * q^11 + 4 * q^13 - 5 * q^15 + 10 * q^17 - 4 * q^19 + 4 * q^21 - 5 * q^23 + 5 * q^25 - 5 * q^27 + 10 * q^29 + 6 * q^31 + 4 * q^33 - 4 * q^35 + 6 * q^37 - 4 * q^39 + 12 * q^41 - 2 * q^43 + 5 * q^45 + 4 * q^47 + 19 * q^49 - 10 * q^51 - 4 * q^55 + 4 * q^57 + 6 * q^59 + 16 * q^61 - 4 * q^63 + 4 * q^65 - 4 * q^67 + 5 * q^69 + 8 * q^71 + 14 * q^73 - 5 * q^75 + 16 * q^77 + 18 * q^79 + 5 * q^81 - 20 * q^83 + 10 * q^85 - 10 * q^87 + 18 * q^89 + 20 * q^91 - 6 * q^93 - 4 * q^95 + 4 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} - \nu - 8 ) / 2$$ (v^2 - v - 8) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{4} + 4\nu^{3} + 13\nu^{2} - 36\nu - 20 ) / 8$$ (-v^4 + 4*v^3 + 13*v^2 - 36*v - 20) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 21\nu^{2} - 16\nu + 60 ) / 8$$ (v^4 - 21*v^2 - 16*v + 60) / 8 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - 21\nu^{2} + 60 ) / 8$$ (v^4 - 21*v^2 + 60) / 8
 $$\nu$$ $$=$$ $$( \beta_{4} - \beta_{3} ) / 2$$ (b4 - b3) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{4} - \beta_{3} + 4\beta _1 + 16 ) / 2$$ (b4 - b3 + 4*b1 + 16) / 2 $$\nu^{3}$$ $$=$$ $$( 15\beta_{4} - 11\beta_{3} + 4\beta_{2} + 8\beta _1 + 12 ) / 2$$ (15*b4 - 11*b3 + 4*b2 + 8*b1 + 12) / 2 $$\nu^{4}$$ $$=$$ $$( 37\beta_{4} - 21\beta_{3} + 84\beta _1 + 216 ) / 2$$ (37*b4 - 21*b3 + 84*b1 + 216) / 2

## 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
 0.0475116 2.27399 −2.39144 −3.43588 4.50582
0 −1.00000 0 1.00000 0 −5.02263 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −3.55148 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 −0.944775 0 1.00000 0
1.4 0 −1.00000 0 1.00000 0 2.62057 0 1.00000 0
1.5 0 −1.00000 0 1.00000 0 2.89831 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.w 5
3.b odd 2 1 8280.2.a.br 5
4.b odd 2 1 5520.2.a.cc 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.w 5 1.a even 1 1 trivial
5520.2.a.cc 5 4.b odd 2 1
8280.2.a.br 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2760))$$:

 $$T_{7}^{5} + 4T_{7}^{4} - 19T_{7}^{3} - 54T_{7}^{2} + 104T_{7} + 128$$ T7^5 + 4*T7^4 - 19*T7^3 - 54*T7^2 + 104*T7 + 128 $$T_{11}^{5} + 4T_{11}^{4} - 46T_{11}^{3} - 160T_{11}^{2} + 408T_{11} + 1312$$ T11^5 + 4*T11^4 - 46*T11^3 - 160*T11^2 + 408*T11 + 1312

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$(T + 1)^{5}$$
$5$ $$(T - 1)^{5}$$
$7$ $$T^{5} + 4 T^{4} - 19 T^{3} - 54 T^{2} + \cdots + 128$$
$11$ $$T^{5} + 4 T^{4} - 46 T^{3} + \cdots + 1312$$
$13$ $$T^{5} - 4 T^{4} - 66 T^{3} + \cdots - 3584$$
$17$ $$T^{5} - 10 T^{4} - 25 T^{3} + \cdots + 2416$$
$19$ $$T^{5} + 4 T^{4} - 54 T^{3} - 120 T^{2} + \cdots + 512$$
$23$ $$(T + 1)^{5}$$
$29$ $$T^{5} - 10 T^{4} - 25 T^{3} + \cdots + 1424$$
$31$ $$T^{5} - 6 T^{4} - 79 T^{3} + \cdots - 4096$$
$37$ $$T^{5} - 6 T^{4} - 11 T^{3} + 76 T^{2} + \cdots - 112$$
$41$ $$T^{5} - 12 T^{4} - 53 T^{3} + \cdots + 1912$$
$43$ $$T^{5} + 2 T^{4} - 84 T^{3} - 40 T^{2} + \cdots + 128$$
$47$ $$T^{5} - 4 T^{4} - 54 T^{3} + 120 T^{2} + \cdots - 512$$
$53$ $$T^{5} - 141 T^{3} + 202 T^{2} + \cdots - 2344$$
$59$ $$T^{5} - 6 T^{4} - 269 T^{3} + \cdots - 53248$$
$61$ $$T^{5} - 16 T^{4} - 126 T^{3} + \cdots + 25856$$
$67$ $$T^{5} + 4 T^{4} - 123 T^{3} + \cdots - 6176$$
$71$ $$T^{5} - 8 T^{4} - 65 T^{3} + \cdots - 6464$$
$73$ $$T^{5} - 14 T^{4} - 42 T^{3} + \cdots - 656$$
$79$ $$T^{5} - 18 T^{4} - 88 T^{3} + \cdots - 32768$$
$83$ $$T^{5} + 20 T^{4} + 95 T^{3} + \cdots - 9056$$
$89$ $$T^{5} - 18 T^{4} - 120 T^{3} + \cdots - 3328$$
$97$ $$T^{5} - 4 T^{4} - 364 T^{3} + \cdots - 13952$$