# Properties

 Label 528.4.a.a 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 33) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 14 q^{5} + 32 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 14 * q^5 + 32 * q^7 + 9 * q^9 $$q - 3 q^{3} - 14 q^{5} + 32 q^{7} + 9 q^{9} + 11 q^{11} - 38 q^{13} + 42 q^{15} - 2 q^{17} - 72 q^{19} - 96 q^{21} - 68 q^{23} + 71 q^{25} - 27 q^{27} - 54 q^{29} + 152 q^{31} - 33 q^{33} - 448 q^{35} + 174 q^{37} + 114 q^{39} + 94 q^{41} + 528 q^{43} - 126 q^{45} + 340 q^{47} + 681 q^{49} + 6 q^{51} - 438 q^{53} - 154 q^{55} + 216 q^{57} - 20 q^{59} + 570 q^{61} + 288 q^{63} + 532 q^{65} + 460 q^{67} + 204 q^{69} + 1092 q^{71} + 562 q^{73} - 213 q^{75} + 352 q^{77} + 16 q^{79} + 81 q^{81} - 372 q^{83} + 28 q^{85} + 162 q^{87} - 966 q^{89} - 1216 q^{91} - 456 q^{93} + 1008 q^{95} - 526 q^{97} + 99 q^{99}+O(q^{100})$$ q - 3 * q^3 - 14 * q^5 + 32 * q^7 + 9 * q^9 + 11 * q^11 - 38 * q^13 + 42 * q^15 - 2 * q^17 - 72 * q^19 - 96 * q^21 - 68 * q^23 + 71 * q^25 - 27 * q^27 - 54 * q^29 + 152 * q^31 - 33 * q^33 - 448 * q^35 + 174 * q^37 + 114 * q^39 + 94 * q^41 + 528 * q^43 - 126 * q^45 + 340 * q^47 + 681 * q^49 + 6 * q^51 - 438 * q^53 - 154 * q^55 + 216 * q^57 - 20 * q^59 + 570 * q^61 + 288 * q^63 + 532 * q^65 + 460 * q^67 + 204 * q^69 + 1092 * q^71 + 562 * q^73 - 213 * q^75 + 352 * q^77 + 16 * q^79 + 81 * q^81 - 372 * q^83 + 28 * q^85 + 162 * q^87 - 966 * q^89 - 1216 * q^91 - 456 * q^93 + 1008 * q^95 - 526 * 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 −14.0000 0 32.0000 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.a 1
3.b odd 2 1 1584.4.a.t 1
4.b odd 2 1 33.4.a.a 1
8.b even 2 1 2112.4.a.y 1
8.d odd 2 1 2112.4.a.l 1
12.b even 2 1 99.4.a.b 1
20.d odd 2 1 825.4.a.i 1
20.e even 4 2 825.4.c.a 2
28.d even 2 1 1617.4.a.a 1
44.c even 2 1 363.4.a.h 1
60.h even 2 1 2475.4.a.b 1
132.d odd 2 1 1089.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.a 1 4.b odd 2 1
99.4.a.b 1 12.b even 2 1
363.4.a.h 1 44.c even 2 1
528.4.a.a 1 1.a even 1 1 trivial
825.4.a.i 1 20.d odd 2 1
825.4.c.a 2 20.e even 4 2
1089.4.a.a 1 132.d odd 2 1
1584.4.a.t 1 3.b odd 2 1
1617.4.a.a 1 28.d even 2 1
2112.4.a.l 1 8.d odd 2 1
2112.4.a.y 1 8.b even 2 1
2475.4.a.b 1 60.h even 2 1

## Hecke kernels

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

 $$T_{5} + 14$$ T5 + 14 $$T_{7} - 32$$ T7 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 14$$
$7$ $$T - 32$$
$11$ $$T - 11$$
$13$ $$T + 38$$
$17$ $$T + 2$$
$19$ $$T + 72$$
$23$ $$T + 68$$
$29$ $$T + 54$$
$31$ $$T - 152$$
$37$ $$T - 174$$
$41$ $$T - 94$$
$43$ $$T - 528$$
$47$ $$T - 340$$
$53$ $$T + 438$$
$59$ $$T + 20$$
$61$ $$T - 570$$
$67$ $$T - 460$$
$71$ $$T - 1092$$
$73$ $$T - 562$$
$79$ $$T - 16$$
$83$ $$T + 372$$
$89$ $$T + 966$$
$97$ $$T + 526$$