Properties

 Label 6760.2.a.e Level $6760$ Weight $2$ Character orbit 6760.a Self dual yes Analytic conductor $53.979$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6760,2,Mod(1,6760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6760 = 2^{3} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6760.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: $$53.9788717664$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 520) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} + 3 q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 - q^5 + 3 * q^7 - 2 * q^9 $$q - q^{3} - q^{5} + 3 q^{7} - 2 q^{9} - 5 q^{11} + q^{15} + 5 q^{17} + q^{19} - 3 q^{21} - q^{23} + q^{25} + 5 q^{27} - 9 q^{29} + 4 q^{31} + 5 q^{33} - 3 q^{35} + 3 q^{37} + q^{41} - 3 q^{43} + 2 q^{45} + 8 q^{47} + 2 q^{49} - 5 q^{51} + 10 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} + 7 q^{61} - 6 q^{63} + 9 q^{67} + q^{69} - 7 q^{71} - 10 q^{73} - q^{75} - 15 q^{77} - 16 q^{79} + q^{81} - 12 q^{83} - 5 q^{85} + 9 q^{87} - 15 q^{89} - 4 q^{93} - q^{95} + 7 q^{97} + 10 q^{99}+O(q^{100})$$ q - q^3 - q^5 + 3 * q^7 - 2 * q^9 - 5 * q^11 + q^15 + 5 * q^17 + q^19 - 3 * q^21 - q^23 + q^25 + 5 * q^27 - 9 * q^29 + 4 * q^31 + 5 * q^33 - 3 * q^35 + 3 * q^37 + q^41 - 3 * q^43 + 2 * q^45 + 8 * q^47 + 2 * q^49 - 5 * q^51 + 10 * q^53 + 5 * q^55 - q^57 - 3 * q^59 + 7 * q^61 - 6 * q^63 + 9 * q^67 + q^69 - 7 * q^71 - 10 * q^73 - q^75 - 15 * q^77 - 16 * q^79 + q^81 - 12 * q^83 - 5 * q^85 + 9 * q^87 - 15 * q^89 - 4 * q^93 - q^95 + 7 * q^97 + 10 * 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
0 −1.00000 0 −1.00000 0 3.00000 0 −2.00000 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
$$2$$ $$+1$$
$$5$$ $$+1$$
$$13$$ $$+1$$

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 6760.2.a.e 1
13.b even 2 1 6760.2.a.f 1
13.e even 6 2 520.2.q.d 2
52.i odd 6 2 1040.2.q.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.q.d 2 13.e even 6 2
1040.2.q.g 2 52.i odd 6 2
6760.2.a.e 1 1.a even 1 1 trivial
6760.2.a.f 1 13.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6760))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7} - 3$$ T7 - 3 $$T_{11} + 5$$ T11 + 5

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T - 3$$
$11$ $$T + 5$$
$13$ $$T$$
$17$ $$T - 5$$
$19$ $$T - 1$$
$23$ $$T + 1$$
$29$ $$T + 9$$
$31$ $$T - 4$$
$37$ $$T - 3$$
$41$ $$T - 1$$
$43$ $$T + 3$$
$47$ $$T - 8$$
$53$ $$T - 10$$
$59$ $$T + 3$$
$61$ $$T - 7$$
$67$ $$T - 9$$
$71$ $$T + 7$$
$73$ $$T + 10$$
$79$ $$T + 16$$
$83$ $$T + 12$$
$89$ $$T + 15$$
$97$ $$T - 7$$