# Properties

 Label 1530.2.a.i Level $1530$ Weight $2$ Character orbit 1530.a Self dual yes Analytic conductor $12.217$ 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] = [1530,2,Mod(1,1530)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1530, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1530.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1530.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: $$12.2171115093$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} - 2 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - q^5 - 2 * q^7 + q^8 $$q + q^{2} + q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{10} - 4 q^{11} - 2 q^{14} + q^{16} - q^{17} + 4 q^{19} - q^{20} - 4 q^{22} - 4 q^{23} + q^{25} - 2 q^{28} - 6 q^{29} - 8 q^{31} + q^{32} - q^{34} + 2 q^{35} - 6 q^{37} + 4 q^{38} - q^{40} - 8 q^{41} + 2 q^{43} - 4 q^{44} - 4 q^{46} + 8 q^{47} - 3 q^{49} + q^{50} - 14 q^{53} + 4 q^{55} - 2 q^{56} - 6 q^{58} - 6 q^{59} + 2 q^{61} - 8 q^{62} + q^{64} + 2 q^{67} - q^{68} + 2 q^{70} + 10 q^{71} + 4 q^{73} - 6 q^{74} + 4 q^{76} + 8 q^{77} + 4 q^{79} - q^{80} - 8 q^{82} + 16 q^{83} + q^{85} + 2 q^{86} - 4 q^{88} - 6 q^{89} - 4 q^{92} + 8 q^{94} - 4 q^{95} - 8 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + q^4 - q^5 - 2 * q^7 + q^8 - q^10 - 4 * q^11 - 2 * q^14 + q^16 - q^17 + 4 * q^19 - q^20 - 4 * q^22 - 4 * q^23 + q^25 - 2 * q^28 - 6 * q^29 - 8 * q^31 + q^32 - q^34 + 2 * q^35 - 6 * q^37 + 4 * q^38 - q^40 - 8 * q^41 + 2 * q^43 - 4 * q^44 - 4 * q^46 + 8 * q^47 - 3 * q^49 + q^50 - 14 * q^53 + 4 * q^55 - 2 * q^56 - 6 * q^58 - 6 * q^59 + 2 * q^61 - 8 * q^62 + q^64 + 2 * q^67 - q^68 + 2 * q^70 + 10 * q^71 + 4 * q^73 - 6 * q^74 + 4 * q^76 + 8 * q^77 + 4 * q^79 - q^80 - 8 * q^82 + 16 * q^83 + q^85 + 2 * q^86 - 4 * q^88 - 6 * q^89 - 4 * q^92 + 8 * q^94 - 4 * q^95 - 8 * q^97 - 3 * q^98

## 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
1.00000 0 1.00000 −1.00000 0 −2.00000 1.00000 0 −1.00000
 $$n$$: 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 1530.2.a.i 1
3.b odd 2 1 510.2.a.b 1
5.b even 2 1 7650.2.a.y 1
12.b even 2 1 4080.2.a.n 1
15.d odd 2 1 2550.2.a.y 1
15.e even 4 2 2550.2.d.j 2
51.c odd 2 1 8670.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.b 1 3.b odd 2 1
1530.2.a.i 1 1.a even 1 1 trivial
2550.2.a.y 1 15.d odd 2 1
2550.2.d.j 2 15.e even 4 2
4080.2.a.n 1 12.b even 2 1
7650.2.a.y 1 5.b even 2 1
8670.2.a.c 1 51.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1530))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 4$$ T11 + 4 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 2$$
$11$ $$T + 4$$
$13$ $$T$$
$17$ $$T + 1$$
$19$ $$T - 4$$
$23$ $$T + 4$$
$29$ $$T + 6$$
$31$ $$T + 8$$
$37$ $$T + 6$$
$41$ $$T + 8$$
$43$ $$T - 2$$
$47$ $$T - 8$$
$53$ $$T + 14$$
$59$ $$T + 6$$
$61$ $$T - 2$$
$67$ $$T - 2$$
$71$ $$T - 10$$
$73$ $$T - 4$$
$79$ $$T - 4$$
$83$ $$T - 16$$
$89$ $$T + 6$$
$97$ $$T + 8$$