# Properties

 Label 6760.2.a.b 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10})$$ q - 3 * q^3 + q^5 - 3 * q^7 + 6 * q^9 $$q - 3 q^{3} + q^{5} - 3 q^{7} + 6 q^{9} - 5 q^{11} - 3 q^{15} + 3 q^{17} + 5 q^{19} + 9 q^{21} - 3 q^{23} + q^{25} - 9 q^{27} - 5 q^{29} - 8 q^{31} + 15 q^{33} - 3 q^{35} + 9 q^{37} - 3 q^{41} - q^{43} + 6 q^{45} + 12 q^{47} + 2 q^{49} - 9 q^{51} + 2 q^{53} - 5 q^{55} - 15 q^{57} + 5 q^{59} - q^{61} - 18 q^{63} + 15 q^{67} + 9 q^{69} + q^{71} + 10 q^{73} - 3 q^{75} + 15 q^{77} - 4 q^{79} + 9 q^{81} - 12 q^{83} + 3 q^{85} + 15 q^{87} + q^{89} + 24 q^{93} + 5 q^{95} - 7 q^{97} - 30 q^{99}+O(q^{100})$$ q - 3 * q^3 + q^5 - 3 * q^7 + 6 * q^9 - 5 * q^11 - 3 * q^15 + 3 * q^17 + 5 * q^19 + 9 * q^21 - 3 * q^23 + q^25 - 9 * q^27 - 5 * q^29 - 8 * q^31 + 15 * q^33 - 3 * q^35 + 9 * q^37 - 3 * q^41 - q^43 + 6 * q^45 + 12 * q^47 + 2 * q^49 - 9 * q^51 + 2 * q^53 - 5 * q^55 - 15 * q^57 + 5 * q^59 - q^61 - 18 * q^63 + 15 * q^67 + 9 * q^69 + q^71 + 10 * q^73 - 3 * q^75 + 15 * q^77 - 4 * q^79 + 9 * q^81 - 12 * q^83 + 3 * q^85 + 15 * q^87 + q^89 + 24 * q^93 + 5 * q^95 - 7 * q^97 - 30 * 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 −3.00000 0 1.00000 0 −3.00000 0 6.00000 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.b 1
13.b even 2 1 6760.2.a.a 1
13.e even 6 2 520.2.q.e 2
52.i odd 6 2 1040.2.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.q.e 2 13.e even 6 2
1040.2.q.a 2 52.i odd 6 2
6760.2.a.a 1 13.b even 2 1
6760.2.a.b 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6760))$$:

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

## Hecke characteristic polynomials

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