# Properties

 Label 528.4.a.c 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 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 9 * q^9 $$q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 9 q^{9} - 11 q^{11} + 6 q^{13} + 18 q^{15} - 108 q^{17} + 98 q^{19} - 42 q^{21} + 32 q^{23} - 89 q^{25} - 27 q^{27} - 8 q^{29} + 40 q^{31} + 33 q^{33} - 84 q^{35} + 50 q^{37} - 18 q^{39} - 8 q^{41} + 486 q^{43} - 54 q^{45} - 40 q^{47} - 147 q^{49} + 324 q^{51} + 710 q^{53} + 66 q^{55} - 294 q^{57} + 604 q^{59} + 322 q^{61} + 126 q^{63} - 36 q^{65} + 476 q^{67} - 96 q^{69} - 216 q^{71} + 502 q^{73} + 267 q^{75} - 154 q^{77} + 862 q^{79} + 81 q^{81} - 592 q^{83} + 648 q^{85} + 24 q^{87} + 354 q^{89} + 84 q^{91} - 120 q^{93} - 588 q^{95} + 446 q^{97} - 99 q^{99}+O(q^{100})$$ q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 9 * q^9 - 11 * q^11 + 6 * q^13 + 18 * q^15 - 108 * q^17 + 98 * q^19 - 42 * q^21 + 32 * q^23 - 89 * q^25 - 27 * q^27 - 8 * q^29 + 40 * q^31 + 33 * q^33 - 84 * q^35 + 50 * q^37 - 18 * q^39 - 8 * q^41 + 486 * q^43 - 54 * q^45 - 40 * q^47 - 147 * q^49 + 324 * q^51 + 710 * q^53 + 66 * q^55 - 294 * q^57 + 604 * q^59 + 322 * q^61 + 126 * q^63 - 36 * q^65 + 476 * q^67 - 96 * q^69 - 216 * q^71 + 502 * q^73 + 267 * q^75 - 154 * q^77 + 862 * q^79 + 81 * q^81 - 592 * q^83 + 648 * q^85 + 24 * q^87 + 354 * q^89 + 84 * q^91 - 120 * q^93 - 588 * q^95 + 446 * 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 −6.00000 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.c 1
3.b odd 2 1 1584.4.a.n 1
4.b odd 2 1 264.4.a.c 1
8.b even 2 1 2112.4.a.w 1
8.d odd 2 1 2112.4.a.i 1
12.b even 2 1 792.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.4.a.c 1 4.b odd 2 1
528.4.a.c 1 1.a even 1 1 trivial
792.4.a.c 1 12.b even 2 1
1584.4.a.n 1 3.b odd 2 1
2112.4.a.i 1 8.d odd 2 1
2112.4.a.w 1 8.b even 2 1

## Hecke kernels

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

 $$T_{5} + 6$$ T5 + 6 $$T_{7} - 14$$ T7 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 6$$
$7$ $$T - 14$$
$11$ $$T + 11$$
$13$ $$T - 6$$
$17$ $$T + 108$$
$19$ $$T - 98$$
$23$ $$T - 32$$
$29$ $$T + 8$$
$31$ $$T - 40$$
$37$ $$T - 50$$
$41$ $$T + 8$$
$43$ $$T - 486$$
$47$ $$T + 40$$
$53$ $$T - 710$$
$59$ $$T - 604$$
$61$ $$T - 322$$
$67$ $$T - 476$$
$71$ $$T + 216$$
$73$ $$T - 502$$
$79$ $$T - 862$$
$83$ $$T + 592$$
$89$ $$T - 354$$
$97$ $$T - 446$$