# Properties

 Label 1530.2.a.f 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + q^5 - q^8 $$q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 4 q^{11} + 2 q^{13} + q^{16} - q^{17} - 4 q^{19} + q^{20} + 4 q^{22} - 4 q^{23} + q^{25} - 2 q^{26} - 2 q^{29} - 4 q^{31} - q^{32} + q^{34} - 6 q^{37} + 4 q^{38} - q^{40} + 10 q^{41} - 8 q^{43} - 4 q^{44} + 4 q^{46} - 7 q^{49} - q^{50} + 2 q^{52} - 6 q^{53} - 4 q^{55} + 2 q^{58} + 8 q^{59} + 10 q^{61} + 4 q^{62} + q^{64} + 2 q^{65} - 8 q^{67} - q^{68} - 8 q^{71} - 2 q^{73} + 6 q^{74} - 4 q^{76} + 4 q^{79} + q^{80} - 10 q^{82} - 4 q^{83} - q^{85} + 8 q^{86} + 4 q^{88} + 14 q^{89} - 4 q^{92} - 4 q^{95} - 10 q^{97} + 7 q^{98}+O(q^{100})$$ q - q^2 + q^4 + q^5 - q^8 - q^10 - 4 * q^11 + 2 * q^13 + q^16 - q^17 - 4 * q^19 + q^20 + 4 * q^22 - 4 * q^23 + q^25 - 2 * q^26 - 2 * q^29 - 4 * q^31 - q^32 + q^34 - 6 * q^37 + 4 * q^38 - q^40 + 10 * q^41 - 8 * q^43 - 4 * q^44 + 4 * q^46 - 7 * q^49 - q^50 + 2 * q^52 - 6 * q^53 - 4 * q^55 + 2 * q^58 + 8 * q^59 + 10 * q^61 + 4 * q^62 + q^64 + 2 * q^65 - 8 * q^67 - q^68 - 8 * q^71 - 2 * q^73 + 6 * q^74 - 4 * q^76 + 4 * q^79 + q^80 - 10 * q^82 - 4 * q^83 - q^85 + 8 * q^86 + 4 * q^88 + 14 * q^89 - 4 * q^92 - 4 * q^95 - 10 * q^97 + 7 * 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 0 −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.f 1
3.b odd 2 1 510.2.a.f 1
5.b even 2 1 7650.2.a.bw 1
12.b even 2 1 4080.2.a.c 1
15.d odd 2 1 2550.2.a.d 1
15.e even 4 2 2550.2.d.s 2
51.c odd 2 1 8670.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.f 1 3.b odd 2 1
1530.2.a.f 1 1.a even 1 1 trivial
2550.2.a.d 1 15.d odd 2 1
2550.2.d.s 2 15.e even 4 2
4080.2.a.c 1 12.b even 2 1
7650.2.a.bw 1 5.b even 2 1
8670.2.a.r 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1530))$$:

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

## Hecke characteristic polynomials

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