# Properties

 Label 528.4.a.b Level $528$ Weight $4$ Character orbit 528.a Self dual yes Analytic conductor $31.153$ 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] = [528,4,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 528.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: $$31.1530084830$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 264) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 6 q^{5} + 8 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 6 * q^5 + 8 * q^7 + 9 * q^9 $$q - 3 q^{3} - 6 q^{5} + 8 q^{7} + 9 q^{9} + 11 q^{11} - 30 q^{13} + 18 q^{15} - 18 q^{17} + 56 q^{19} - 24 q^{21} + 100 q^{23} - 89 q^{25} - 27 q^{27} + 26 q^{29} + 136 q^{31} - 33 q^{33} - 48 q^{35} - 178 q^{37} + 90 q^{39} + 110 q^{41} - 288 q^{43} - 54 q^{45} - 116 q^{47} - 279 q^{49} + 54 q^{51} - 398 q^{53} - 66 q^{55} - 168 q^{57} - 196 q^{59} - 782 q^{61} + 72 q^{63} + 180 q^{65} - 292 q^{67} - 300 q^{69} - 180 q^{71} - 398 q^{73} + 267 q^{75} + 88 q^{77} - 56 q^{79} + 81 q^{81} - 548 q^{83} + 108 q^{85} - 78 q^{87} + 282 q^{89} - 240 q^{91} - 408 q^{93} - 336 q^{95} - 142 q^{97} + 99 q^{99}+O(q^{100})$$ q - 3 * q^3 - 6 * q^5 + 8 * q^7 + 9 * q^9 + 11 * q^11 - 30 * q^13 + 18 * q^15 - 18 * q^17 + 56 * q^19 - 24 * q^21 + 100 * q^23 - 89 * q^25 - 27 * q^27 + 26 * q^29 + 136 * q^31 - 33 * q^33 - 48 * q^35 - 178 * q^37 + 90 * q^39 + 110 * q^41 - 288 * q^43 - 54 * q^45 - 116 * q^47 - 279 * q^49 + 54 * q^51 - 398 * q^53 - 66 * q^55 - 168 * q^57 - 196 * q^59 - 782 * q^61 + 72 * q^63 + 180 * q^65 - 292 * q^67 - 300 * q^69 - 180 * q^71 - 398 * q^73 + 267 * q^75 + 88 * q^77 - 56 * q^79 + 81 * q^81 - 548 * q^83 + 108 * q^85 - 78 * q^87 + 282 * q^89 - 240 * q^91 - 408 * q^93 - 336 * q^95 - 142 * q^97 + 99 * 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 −6.00000 0 8.00000 0 9.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$11$$ $$-1$$

## 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 528.4.a.b 1
3.b odd 2 1 1584.4.a.m 1
4.b odd 2 1 264.4.a.d 1
8.b even 2 1 2112.4.a.v 1
8.d odd 2 1 2112.4.a.j 1
12.b even 2 1 792.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.4.a.d 1 4.b odd 2 1
528.4.a.b 1 1.a even 1 1 trivial
792.4.a.d 1 12.b even 2 1
1584.4.a.m 1 3.b odd 2 1
2112.4.a.j 1 8.d odd 2 1
2112.4.a.v 1 8.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(528))$$:

 $$T_{5} + 6$$ T5 + 6 $$T_{7} - 8$$ T7 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 6$$
$7$ $$T - 8$$
$11$ $$T - 11$$
$13$ $$T + 30$$
$17$ $$T + 18$$
$19$ $$T - 56$$
$23$ $$T - 100$$
$29$ $$T - 26$$
$31$ $$T - 136$$
$37$ $$T + 178$$
$41$ $$T - 110$$
$43$ $$T + 288$$
$47$ $$T + 116$$
$53$ $$T + 398$$
$59$ $$T + 196$$
$61$ $$T + 782$$
$67$ $$T + 292$$
$71$ $$T + 180$$
$73$ $$T + 398$$
$79$ $$T + 56$$
$83$ $$T + 548$$
$89$ $$T - 282$$
$97$ $$T + 142$$