# Properties

 Label 1260.2.a.d Level $1260$ Weight $2$ Character orbit 1260.a Self dual yes Analytic conductor $10.061$ 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] = [1260,2,Mod(1,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, 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("1260.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1260.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: $$10.0611506547$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{5} + q^{7}+O(q^{10})$$ q - q^5 + q^7 $$q - q^{5} + q^{7} - 4 q^{13} - 6 q^{17} + 2 q^{19} - 6 q^{23} + q^{25} + 2 q^{31} - q^{35} + 2 q^{37} - 6 q^{41} - 4 q^{43} + q^{49} + 6 q^{53} - 12 q^{59} - 10 q^{61} + 4 q^{65} - 4 q^{67} - 12 q^{71} - 4 q^{73} + 8 q^{79} - 12 q^{83} + 6 q^{85} - 6 q^{89} - 4 q^{91} - 2 q^{95} + 8 q^{97}+O(q^{100})$$ q - q^5 + q^7 - 4 * q^13 - 6 * q^17 + 2 * q^19 - 6 * q^23 + q^25 + 2 * q^31 - q^35 + 2 * q^37 - 6 * q^41 - 4 * q^43 + q^49 + 6 * q^53 - 12 * q^59 - 10 * q^61 + 4 * q^65 - 4 * q^67 - 12 * q^71 - 4 * q^73 + 8 * q^79 - 12 * q^83 + 6 * q^85 - 6 * q^89 - 4 * q^91 - 2 * q^95 + 8 * 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
 0
0 0 0 −1.00000 0 1.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 1260.2.a.d 1
3.b odd 2 1 1260.2.a.j yes 1
4.b odd 2 1 5040.2.a.e 1
5.b even 2 1 6300.2.a.i 1
5.c odd 4 2 6300.2.k.i 2
7.b odd 2 1 8820.2.a.u 1
12.b even 2 1 5040.2.a.w 1
15.d odd 2 1 6300.2.a.h 1
15.e even 4 2 6300.2.k.j 2
21.c even 2 1 8820.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.a.d 1 1.a even 1 1 trivial
1260.2.a.j yes 1 3.b odd 2 1
5040.2.a.e 1 4.b odd 2 1
5040.2.a.w 1 12.b even 2 1
6300.2.a.h 1 15.d odd 2 1
6300.2.a.i 1 5.b even 2 1
6300.2.k.i 2 5.c odd 4 2
6300.2.k.j 2 15.e even 4 2
8820.2.a.h 1 21.c even 2 1
8820.2.a.u 1 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1260))$$:

 $$T_{11}$$ T11 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T + 6$$
$19$ $$T - 2$$
$23$ $$T + 6$$
$29$ $$T$$
$31$ $$T - 2$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T + 10$$
$67$ $$T + 4$$
$71$ $$T + 12$$
$73$ $$T + 4$$
$79$ $$T - 8$$
$83$ $$T + 12$$
$89$ $$T + 6$$
$97$ $$T - 8$$