# Properties

 Label 4080.2.a.bq Level $4080$ Weight $2$ Character orbit 4080.a Self dual yes Analytic conductor $32.579$ 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] = [4080,2,Mod(1,4080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4080, 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("4080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4080.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: $$32.5789640247$$ Analytic rank: $$0$$ 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: $$2$$ Twist minimal: no (minimal twist has level 510) 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 $$\beta = 2\sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + \beta q^{7} + q^{9}+O(q^{10})$$ q + q^3 + q^5 + b * q^7 + q^9 $$q + q^{3} + q^{5} + \beta q^{7} + q^{9} + (\beta + 2) q^{13} + q^{15} - q^{17} - 4 q^{19} + \beta q^{21} + 4 q^{23} + q^{25} + q^{27} + 6 q^{29} - 4 q^{31} + \beta q^{35} + 6 q^{37} + (\beta + 2) q^{39} + ( - \beta + 2) q^{41} + ( - \beta - 4) q^{43} + q^{45} - 2 \beta q^{47} + 17 q^{49} - q^{51} + ( - 2 \beta + 2) q^{53} - 4 q^{57} - \beta q^{59} + (2 \beta + 2) q^{61} + \beta q^{63} + (\beta + 2) q^{65} + ( - \beta + 4) q^{67} + 4 q^{69} + (\beta + 4) q^{71} + ( - \beta - 6) q^{73} + q^{75} + (2 \beta - 4) q^{79} + q^{81} + ( - 2 \beta - 4) q^{83} - q^{85} + 6 q^{87} + ( - 2 \beta + 2) q^{89} + (2 \beta + 24) q^{91} - 4 q^{93} - 4 q^{95} + ( - 3 \beta + 2) q^{97} +O(q^{100})$$ q + q^3 + q^5 + b * q^7 + q^9 + (b + 2) * q^13 + q^15 - q^17 - 4 * q^19 + b * q^21 + 4 * q^23 + q^25 + q^27 + 6 * q^29 - 4 * q^31 + b * q^35 + 6 * q^37 + (b + 2) * q^39 + (-b + 2) * q^41 + (-b - 4) * q^43 + q^45 - 2*b * q^47 + 17 * q^49 - q^51 + (-2*b + 2) * q^53 - 4 * q^57 - b * q^59 + (2*b + 2) * q^61 + b * q^63 + (b + 2) * q^65 + (-b + 4) * q^67 + 4 * q^69 + (b + 4) * q^71 + (-b - 6) * q^73 + q^75 + (2*b - 4) * q^79 + q^81 + (-2*b - 4) * q^83 - q^85 + 6 * q^87 + (-2*b + 2) * q^89 + (2*b + 24) * q^91 - 4 * q^93 - 4 * q^95 + (-3*b + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 2 q^{17} - 8 q^{19} + 8 q^{23} + 2 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{31} + 12 q^{37} + 4 q^{39} + 4 q^{41} - 8 q^{43} + 2 q^{45} + 34 q^{49} - 2 q^{51} + 4 q^{53} - 8 q^{57} + 4 q^{61} + 4 q^{65} + 8 q^{67} + 8 q^{69} + 8 q^{71} - 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} - 8 q^{83} - 2 q^{85} + 12 q^{87} + 4 q^{89} + 48 q^{91} - 8 q^{93} - 8 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 2 * q^17 - 8 * q^19 + 8 * q^23 + 2 * q^25 + 2 * q^27 + 12 * q^29 - 8 * q^31 + 12 * q^37 + 4 * q^39 + 4 * q^41 - 8 * q^43 + 2 * q^45 + 34 * q^49 - 2 * q^51 + 4 * q^53 - 8 * q^57 + 4 * q^61 + 4 * q^65 + 8 * q^67 + 8 * q^69 + 8 * q^71 - 12 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 - 8 * q^83 - 2 * q^85 + 12 * q^87 + 4 * q^89 + 48 * q^91 - 8 * q^93 - 8 * q^95 + 4 * q^97

## 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
0 1.00000 0 1.00000 0 −4.89898 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 4.89898 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$17$$ $$1$$

## 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 4080.2.a.bq 2
4.b odd 2 1 510.2.a.h 2
12.b even 2 1 1530.2.a.s 2
20.d odd 2 1 2550.2.a.bl 2
20.e even 4 2 2550.2.d.u 4
60.h even 2 1 7650.2.a.cu 2
68.d odd 2 1 8670.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.h 2 4.b odd 2 1
1530.2.a.s 2 12.b even 2 1
2550.2.a.bl 2 20.d odd 2 1
2550.2.d.u 4 20.e even 4 2
4080.2.a.bq 2 1.a even 1 1 trivial
7650.2.a.cu 2 60.h even 2 1
8670.2.a.be 2 68.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4080))$$:

 $$T_{7}^{2} - 24$$ T7^2 - 24 $$T_{11}$$ T11 $$T_{13}^{2} - 4T_{13} - 20$$ T13^2 - 4*T13 - 20 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 24$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T - 20$$
$17$ $$(T + 1)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$(T - 4)^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} - 4T - 20$$
$43$ $$T^{2} + 8T - 8$$
$47$ $$T^{2} - 96$$
$53$ $$T^{2} - 4T - 92$$
$59$ $$T^{2} - 24$$
$61$ $$T^{2} - 4T - 92$$
$67$ $$T^{2} - 8T - 8$$
$71$ $$T^{2} - 8T - 8$$
$73$ $$T^{2} + 12T + 12$$
$79$ $$T^{2} + 8T - 80$$
$83$ $$T^{2} + 8T - 80$$
$89$ $$T^{2} - 4T - 92$$
$97$ $$T^{2} - 4T - 212$$