# Properties

 Label 528.4.a.d 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 66) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 14 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 14 * q^7 + 9 * q^9 $$q - 3 q^{3} - 14 q^{7} + 9 q^{9} - 11 q^{11} + 80 q^{13} + 30 q^{17} - 56 q^{19} + 42 q^{21} + 126 q^{23} - 125 q^{25} - 27 q^{27} - 222 q^{29} + 16 q^{31} + 33 q^{33} - 106 q^{37} - 240 q^{39} + 114 q^{41} + 52 q^{43} - 246 q^{47} - 147 q^{49} - 90 q^{51} - 264 q^{53} + 168 q^{57} - 264 q^{59} + 92 q^{61} - 126 q^{63} + 796 q^{67} - 378 q^{69} - 426 q^{71} - 1174 q^{73} + 375 q^{75} + 154 q^{77} - 842 q^{79} + 81 q^{81} - 852 q^{83} + 666 q^{87} - 1062 q^{89} - 1120 q^{91} - 48 q^{93} - 1282 q^{97} - 99 q^{99}+O(q^{100})$$ q - 3 * q^3 - 14 * q^7 + 9 * q^9 - 11 * q^11 + 80 * q^13 + 30 * q^17 - 56 * q^19 + 42 * q^21 + 126 * q^23 - 125 * q^25 - 27 * q^27 - 222 * q^29 + 16 * q^31 + 33 * q^33 - 106 * q^37 - 240 * q^39 + 114 * q^41 + 52 * q^43 - 246 * q^47 - 147 * q^49 - 90 * q^51 - 264 * q^53 + 168 * q^57 - 264 * q^59 + 92 * q^61 - 126 * q^63 + 796 * q^67 - 378 * q^69 - 426 * q^71 - 1174 * q^73 + 375 * q^75 + 154 * q^77 - 842 * q^79 + 81 * q^81 - 852 * q^83 + 666 * q^87 - 1062 * q^89 - 1120 * q^91 - 48 * q^93 - 1282 * 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 0 0 −14.0000 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.d 1
3.b odd 2 1 1584.4.a.i 1
4.b odd 2 1 66.4.a.a 1
8.b even 2 1 2112.4.a.s 1
8.d odd 2 1 2112.4.a.g 1
12.b even 2 1 198.4.a.f 1
20.d odd 2 1 1650.4.a.h 1
20.e even 4 2 1650.4.c.g 2
44.c even 2 1 726.4.a.h 1
132.d odd 2 1 2178.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.a.a 1 4.b odd 2 1
198.4.a.f 1 12.b even 2 1
528.4.a.d 1 1.a even 1 1 trivial
726.4.a.h 1 44.c even 2 1
1584.4.a.i 1 3.b odd 2 1
1650.4.a.h 1 20.d odd 2 1
1650.4.c.g 2 20.e even 4 2
2112.4.a.g 1 8.d odd 2 1
2112.4.a.s 1 8.b even 2 1
2178.4.a.g 1 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(528))$$:

 $$T_{5}$$ T5 $$T_{7} + 14$$ T7 + 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T$$
$7$ $$T + 14$$
$11$ $$T + 11$$
$13$ $$T - 80$$
$17$ $$T - 30$$
$19$ $$T + 56$$
$23$ $$T - 126$$
$29$ $$T + 222$$
$31$ $$T - 16$$
$37$ $$T + 106$$
$41$ $$T - 114$$
$43$ $$T - 52$$
$47$ $$T + 246$$
$53$ $$T + 264$$
$59$ $$T + 264$$
$61$ $$T - 92$$
$67$ $$T - 796$$
$71$ $$T + 426$$
$73$ $$T + 1174$$
$79$ $$T + 842$$
$83$ $$T + 852$$
$89$ $$T + 1062$$
$97$ $$T + 1282$$