# Properties

 Label 528.4.a.k 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 12 q^{5} - 22 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 12 * q^5 - 22 * q^7 + 9 * q^9 $$q + 3 q^{3} + 12 q^{5} - 22 q^{7} + 9 q^{9} - 11 q^{11} - 48 q^{13} + 36 q^{15} - 54 q^{17} - 100 q^{19} - 66 q^{21} - 58 q^{23} + 19 q^{25} + 27 q^{27} + 262 q^{29} - 248 q^{31} - 33 q^{33} - 264 q^{35} - 130 q^{37} - 144 q^{39} - 26 q^{41} - 216 q^{43} + 108 q^{45} - 22 q^{47} + 141 q^{49} - 162 q^{51} + 620 q^{53} - 132 q^{55} - 300 q^{57} + 424 q^{59} + 340 q^{61} - 198 q^{63} - 576 q^{65} + 620 q^{67} - 174 q^{69} - 810 q^{71} - 1118 q^{73} + 57 q^{75} + 242 q^{77} + 214 q^{79} + 81 q^{81} - 988 q^{83} - 648 q^{85} + 786 q^{87} - 6 q^{89} + 1056 q^{91} - 744 q^{93} - 1200 q^{95} + 590 q^{97} - 99 q^{99}+O(q^{100})$$ q + 3 * q^3 + 12 * q^5 - 22 * q^7 + 9 * q^9 - 11 * q^11 - 48 * q^13 + 36 * q^15 - 54 * q^17 - 100 * q^19 - 66 * q^21 - 58 * q^23 + 19 * q^25 + 27 * q^27 + 262 * q^29 - 248 * q^31 - 33 * q^33 - 264 * q^35 - 130 * q^37 - 144 * q^39 - 26 * q^41 - 216 * q^43 + 108 * q^45 - 22 * q^47 + 141 * q^49 - 162 * q^51 + 620 * q^53 - 132 * q^55 - 300 * q^57 + 424 * q^59 + 340 * q^61 - 198 * q^63 - 576 * q^65 + 620 * q^67 - 174 * q^69 - 810 * q^71 - 1118 * q^73 + 57 * q^75 + 242 * q^77 + 214 * q^79 + 81 * q^81 - 988 * q^83 - 648 * q^85 + 786 * q^87 - 6 * q^89 + 1056 * q^91 - 744 * q^93 - 1200 * q^95 + 590 * 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 12.0000 0 −22.0000 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.k 1
3.b odd 2 1 1584.4.a.d 1
4.b odd 2 1 264.4.a.b 1
8.b even 2 1 2112.4.a.c 1
8.d odd 2 1 2112.4.a.p 1
12.b even 2 1 792.4.a.a 1

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

 $$T_{5} - 12$$ T5 - 12 $$T_{7} + 22$$ T7 + 22

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 12$$
$7$ $$T + 22$$
$11$ $$T + 11$$
$13$ $$T + 48$$
$17$ $$T + 54$$
$19$ $$T + 100$$
$23$ $$T + 58$$
$29$ $$T - 262$$
$31$ $$T + 248$$
$37$ $$T + 130$$
$41$ $$T + 26$$
$43$ $$T + 216$$
$47$ $$T + 22$$
$53$ $$T - 620$$
$59$ $$T - 424$$
$61$ $$T - 340$$
$67$ $$T - 620$$
$71$ $$T + 810$$
$73$ $$T + 1118$$
$79$ $$T - 214$$
$83$ $$T + 988$$
$89$ $$T + 6$$
$97$ $$T - 590$$