# Properties

 Label 528.2.a.d Level $528$ Weight $2$ Character orbit 528.a Self dual yes Analytic conductor $4.216$ 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] = [528,2,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, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$4.21610122672$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 2 * q^7 + q^9 $$q - q^{3} - 2 q^{7} + q^{9} + q^{11} - 4 q^{13} - 6 q^{17} + 4 q^{19} + 2 q^{21} - 6 q^{23} - 5 q^{25} - q^{27} + 6 q^{29} - 8 q^{31} - q^{33} - 10 q^{37} + 4 q^{39} + 6 q^{41} - 8 q^{43} + 6 q^{47} - 3 q^{49} + 6 q^{51} - 4 q^{57} + 8 q^{61} - 2 q^{63} + 4 q^{67} + 6 q^{69} - 6 q^{71} + 2 q^{73} + 5 q^{75} - 2 q^{77} - 14 q^{79} + q^{81} + 12 q^{83} - 6 q^{87} - 6 q^{89} + 8 q^{91} + 8 q^{93} + 14 q^{97} + q^{99}+O(q^{100})$$ q - q^3 - 2 * q^7 + q^9 + q^11 - 4 * q^13 - 6 * q^17 + 4 * q^19 + 2 * q^21 - 6 * q^23 - 5 * q^25 - q^27 + 6 * q^29 - 8 * q^31 - q^33 - 10 * q^37 + 4 * q^39 + 6 * q^41 - 8 * q^43 + 6 * q^47 - 3 * q^49 + 6 * q^51 - 4 * q^57 + 8 * q^61 - 2 * q^63 + 4 * q^67 + 6 * q^69 - 6 * q^71 + 2 * q^73 + 5 * q^75 - 2 * q^77 - 14 * q^79 + q^81 + 12 * q^83 - 6 * q^87 - 6 * q^89 + 8 * q^91 + 8 * q^93 + 14 * q^97 + 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 −1.00000 0 0 0 −2.00000 0 1.00000 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 528.2.a.d 1
3.b odd 2 1 1584.2.a.h 1
4.b odd 2 1 66.2.a.a 1
8.b even 2 1 2112.2.a.v 1
8.d odd 2 1 2112.2.a.i 1
11.b odd 2 1 5808.2.a.l 1
12.b even 2 1 198.2.a.e 1
20.d odd 2 1 1650.2.a.m 1
20.e even 4 2 1650.2.c.d 2
24.f even 2 1 6336.2.a.bj 1
24.h odd 2 1 6336.2.a.bf 1
28.d even 2 1 3234.2.a.d 1
36.f odd 6 2 1782.2.e.s 2
36.h even 6 2 1782.2.e.f 2
44.c even 2 1 726.2.a.i 1
44.g even 10 4 726.2.e.b 4
44.h odd 10 4 726.2.e.k 4
60.h even 2 1 4950.2.a.g 1
60.l odd 4 2 4950.2.c.r 2
84.h odd 2 1 9702.2.a.bu 1
132.d odd 2 1 2178.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 4.b odd 2 1
198.2.a.e 1 12.b even 2 1
528.2.a.d 1 1.a even 1 1 trivial
726.2.a.i 1 44.c even 2 1
726.2.e.b 4 44.g even 10 4
726.2.e.k 4 44.h odd 10 4
1584.2.a.h 1 3.b odd 2 1
1650.2.a.m 1 20.d odd 2 1
1650.2.c.d 2 20.e even 4 2
1782.2.e.f 2 36.h even 6 2
1782.2.e.s 2 36.f odd 6 2
2112.2.a.i 1 8.d odd 2 1
2112.2.a.v 1 8.b even 2 1
2178.2.a.b 1 132.d odd 2 1
3234.2.a.d 1 28.d even 2 1
4950.2.a.g 1 60.h even 2 1
4950.2.c.r 2 60.l odd 4 2
5808.2.a.l 1 11.b odd 2 1
6336.2.a.bf 1 24.h odd 2 1
6336.2.a.bj 1 24.f even 2 1
9702.2.a.bu 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(528))$$:

 $$T_{5}$$ T5 $$T_{7} + 2$$ T7 + 2 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T + 6$$
$19$ $$T - 4$$
$23$ $$T + 6$$
$29$ $$T - 6$$
$31$ $$T + 8$$
$37$ $$T + 10$$
$41$ $$T - 6$$
$43$ $$T + 8$$
$47$ $$T - 6$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 8$$
$67$ $$T - 4$$
$71$ $$T + 6$$
$73$ $$T - 2$$
$79$ $$T + 14$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T - 14$$