# Properties

 Label 528.4.a.j Level $528$ Weight $4$ Character orbit 528.a Self dual yes Analytic conductor $31.153$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 10 q^{5} - 16 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 10 * q^5 - 16 * q^7 + 9 * q^9 $$q + 3 q^{3} + 10 q^{5} - 16 q^{7} + 9 q^{9} - 11 q^{11} + 10 q^{13} + 30 q^{15} - 10 q^{17} + 144 q^{19} - 48 q^{21} + 84 q^{23} - 25 q^{25} + 27 q^{27} + 218 q^{29} + 176 q^{31} - 33 q^{33} - 160 q^{35} + 46 q^{37} + 30 q^{39} - 26 q^{41} + 488 q^{43} + 90 q^{45} - 404 q^{47} - 87 q^{49} - 30 q^{51} + 194 q^{53} - 110 q^{55} + 432 q^{57} - 444 q^{59} + 202 q^{61} - 144 q^{63} + 100 q^{65} + 84 q^{67} + 252 q^{69} + 764 q^{71} + 354 q^{73} - 75 q^{75} + 176 q^{77} - 1312 q^{79} + 81 q^{81} + 1252 q^{83} - 100 q^{85} + 654 q^{87} - 1222 q^{89} - 160 q^{91} + 528 q^{93} + 1440 q^{95} - 1358 q^{97} - 99 q^{99}+O(q^{100})$$ q + 3 * q^3 + 10 * q^5 - 16 * q^7 + 9 * q^9 - 11 * q^11 + 10 * q^13 + 30 * q^15 - 10 * q^17 + 144 * q^19 - 48 * q^21 + 84 * q^23 - 25 * q^25 + 27 * q^27 + 218 * q^29 + 176 * q^31 - 33 * q^33 - 160 * q^35 + 46 * q^37 + 30 * q^39 - 26 * q^41 + 488 * q^43 + 90 * q^45 - 404 * q^47 - 87 * q^49 - 30 * q^51 + 194 * q^53 - 110 * q^55 + 432 * q^57 - 444 * q^59 + 202 * q^61 - 144 * q^63 + 100 * q^65 + 84 * q^67 + 252 * q^69 + 764 * q^71 + 354 * q^73 - 75 * q^75 + 176 * q^77 - 1312 * q^79 + 81 * q^81 + 1252 * q^83 - 100 * q^85 + 654 * q^87 - 1222 * q^89 - 160 * q^91 + 528 * q^93 + 1440 * q^95 - 1358 * 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 10.0000 0 −16.0000 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.j 1
3.b odd 2 1 1584.4.a.e 1
4.b odd 2 1 66.4.a.b 1
8.b even 2 1 2112.4.a.d 1
8.d odd 2 1 2112.4.a.r 1
12.b even 2 1 198.4.a.a 1
20.d odd 2 1 1650.4.a.e 1
20.e even 4 2 1650.4.c.e 2
44.c even 2 1 726.4.a.b 1
132.d odd 2 1 2178.4.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.a.b 1 4.b odd 2 1
198.4.a.a 1 12.b even 2 1
528.4.a.j 1 1.a even 1 1 trivial
726.4.a.b 1 44.c even 2 1
1584.4.a.e 1 3.b odd 2 1
1650.4.a.e 1 20.d odd 2 1
1650.4.c.e 2 20.e even 4 2
2112.4.a.d 1 8.b even 2 1
2112.4.a.r 1 8.d odd 2 1
2178.4.a.m 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(528))$$:

 $$T_{5} - 10$$ T5 - 10 $$T_{7} + 16$$ T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 10$$
$7$ $$T + 16$$
$11$ $$T + 11$$
$13$ $$T - 10$$
$17$ $$T + 10$$
$19$ $$T - 144$$
$23$ $$T - 84$$
$29$ $$T - 218$$
$31$ $$T - 176$$
$37$ $$T - 46$$
$41$ $$T + 26$$
$43$ $$T - 488$$
$47$ $$T + 404$$
$53$ $$T - 194$$
$59$ $$T + 444$$
$61$ $$T - 202$$
$67$ $$T - 84$$
$71$ $$T - 764$$
$73$ $$T - 354$$
$79$ $$T + 1312$$
$83$ $$T - 1252$$
$89$ $$T + 1222$$
$97$ $$T + 1358$$