# Properties

 Label 528.4.a.l 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 132) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 22 q^{5} + 20 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 22 * q^5 + 20 * q^7 + 9 * q^9 $$q + 3 q^{3} + 22 q^{5} + 20 q^{7} + 9 q^{9} - 11 q^{11} + 22 q^{13} + 66 q^{15} + 110 q^{17} - 48 q^{19} + 60 q^{21} - 72 q^{23} + 359 q^{25} + 27 q^{27} - 142 q^{29} - 184 q^{31} - 33 q^{33} + 440 q^{35} - 194 q^{37} + 66 q^{39} - 482 q^{41} + 80 q^{43} + 198 q^{45} - 392 q^{47} + 57 q^{49} + 330 q^{51} - 34 q^{53} - 242 q^{55} - 144 q^{57} + 108 q^{59} + 382 q^{61} + 180 q^{63} + 484 q^{65} - 84 q^{67} - 216 q^{69} + 1040 q^{71} - 606 q^{73} + 1077 q^{75} - 220 q^{77} + 1292 q^{79} + 81 q^{81} - 356 q^{83} + 2420 q^{85} - 426 q^{87} - 406 q^{89} + 440 q^{91} - 552 q^{93} - 1056 q^{95} + 1090 q^{97} - 99 q^{99}+O(q^{100})$$ q + 3 * q^3 + 22 * q^5 + 20 * q^7 + 9 * q^9 - 11 * q^11 + 22 * q^13 + 66 * q^15 + 110 * q^17 - 48 * q^19 + 60 * q^21 - 72 * q^23 + 359 * q^25 + 27 * q^27 - 142 * q^29 - 184 * q^31 - 33 * q^33 + 440 * q^35 - 194 * q^37 + 66 * q^39 - 482 * q^41 + 80 * q^43 + 198 * q^45 - 392 * q^47 + 57 * q^49 + 330 * q^51 - 34 * q^53 - 242 * q^55 - 144 * q^57 + 108 * q^59 + 382 * q^61 + 180 * q^63 + 484 * q^65 - 84 * q^67 - 216 * q^69 + 1040 * q^71 - 606 * q^73 + 1077 * q^75 - 220 * q^77 + 1292 * q^79 + 81 * q^81 - 356 * q^83 + 2420 * q^85 - 426 * q^87 - 406 * q^89 + 440 * q^91 - 552 * q^93 - 1056 * q^95 + 1090 * 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 22.0000 0 20.0000 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.l 1
3.b odd 2 1 1584.4.a.a 1
4.b odd 2 1 132.4.a.c 1
8.b even 2 1 2112.4.a.a 1
8.d odd 2 1 2112.4.a.n 1
12.b even 2 1 396.4.a.a 1
44.c even 2 1 1452.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.a.c 1 4.b odd 2 1
396.4.a.a 1 12.b even 2 1
528.4.a.l 1 1.a even 1 1 trivial
1452.4.a.c 1 44.c even 2 1
1584.4.a.a 1 3.b odd 2 1
2112.4.a.a 1 8.b even 2 1
2112.4.a.n 1 8.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(528))$$:

 $$T_{5} - 22$$ T5 - 22 $$T_{7} - 20$$ T7 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 22$$
$7$ $$T - 20$$
$11$ $$T + 11$$
$13$ $$T - 22$$
$17$ $$T - 110$$
$19$ $$T + 48$$
$23$ $$T + 72$$
$29$ $$T + 142$$
$31$ $$T + 184$$
$37$ $$T + 194$$
$41$ $$T + 482$$
$43$ $$T - 80$$
$47$ $$T + 392$$
$53$ $$T + 34$$
$59$ $$T - 108$$
$61$ $$T - 382$$
$67$ $$T + 84$$
$71$ $$T - 1040$$
$73$ $$T + 606$$
$79$ $$T - 1292$$
$83$ $$T + 356$$
$89$ $$T + 406$$
$97$ $$T - 1090$$