# Properties

 Label 528.6.a.i Level $528$ Weight $6$ Character orbit 528.a Self dual yes Analytic conductor $84.683$ 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] = [528,6,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, 6, names="a")

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

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$84.6826568613$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 9 q^{3} + 46 q^{5} - 148 q^{7} + 81 q^{9}+O(q^{10})$$ q + 9 * q^3 + 46 * q^5 - 148 * q^7 + 81 * q^9 $$q + 9 q^{3} + 46 q^{5} - 148 q^{7} + 81 q^{9} - 121 q^{11} + 574 q^{13} + 414 q^{15} - 722 q^{17} - 2160 q^{19} - 1332 q^{21} + 2536 q^{23} - 1009 q^{25} + 729 q^{27} + 4650 q^{29} - 5032 q^{31} - 1089 q^{33} - 6808 q^{35} + 8118 q^{37} + 5166 q^{39} - 5138 q^{41} - 8304 q^{43} + 3726 q^{45} - 24728 q^{47} + 5097 q^{49} - 6498 q^{51} - 28746 q^{53} - 5566 q^{55} - 19440 q^{57} + 5860 q^{59} - 53658 q^{61} - 11988 q^{63} + 26404 q^{65} - 30908 q^{67} + 22824 q^{69} + 69648 q^{71} - 18446 q^{73} - 9081 q^{75} + 17908 q^{77} + 25300 q^{79} + 6561 q^{81} + 17556 q^{83} - 33212 q^{85} + 41850 q^{87} + 132570 q^{89} - 84952 q^{91} - 45288 q^{93} - 99360 q^{95} + 70658 q^{97} - 9801 q^{99}+O(q^{100})$$ q + 9 * q^3 + 46 * q^5 - 148 * q^7 + 81 * q^9 - 121 * q^11 + 574 * q^13 + 414 * q^15 - 722 * q^17 - 2160 * q^19 - 1332 * q^21 + 2536 * q^23 - 1009 * q^25 + 729 * q^27 + 4650 * q^29 - 5032 * q^31 - 1089 * q^33 - 6808 * q^35 + 8118 * q^37 + 5166 * q^39 - 5138 * q^41 - 8304 * q^43 + 3726 * q^45 - 24728 * q^47 + 5097 * q^49 - 6498 * q^51 - 28746 * q^53 - 5566 * q^55 - 19440 * q^57 + 5860 * q^59 - 53658 * q^61 - 11988 * q^63 + 26404 * q^65 - 30908 * q^67 + 22824 * q^69 + 69648 * q^71 - 18446 * q^73 - 9081 * q^75 + 17908 * q^77 + 25300 * q^79 + 6561 * q^81 + 17556 * q^83 - 33212 * q^85 + 41850 * q^87 + 132570 * q^89 - 84952 * q^91 - 45288 * q^93 - 99360 * q^95 + 70658 * q^97 - 9801 * 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 9.00000 0 46.0000 0 −148.000 0 81.0000 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 528.6.a.i 1
4.b odd 2 1 33.6.a.a 1
12.b even 2 1 99.6.a.b 1
20.d odd 2 1 825.6.a.b 1
44.c even 2 1 363.6.a.c 1
132.d odd 2 1 1089.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.a 1 4.b odd 2 1
99.6.a.b 1 12.b even 2 1
363.6.a.c 1 44.c even 2 1
528.6.a.i 1 1.a even 1 1 trivial
825.6.a.b 1 20.d odd 2 1
1089.6.a.d 1 132.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 $$S_{6}^{\mathrm{new}}(\Gamma_0(528))$$:

 $$T_{5} - 46$$ T5 - 46 $$T_{7} + 148$$ T7 + 148

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 9$$
$5$ $$T - 46$$
$7$ $$T + 148$$
$11$ $$T + 121$$
$13$ $$T - 574$$
$17$ $$T + 722$$
$19$ $$T + 2160$$
$23$ $$T - 2536$$
$29$ $$T - 4650$$
$31$ $$T + 5032$$
$37$ $$T - 8118$$
$41$ $$T + 5138$$
$43$ $$T + 8304$$
$47$ $$T + 24728$$
$53$ $$T + 28746$$
$59$ $$T - 5860$$
$61$ $$T + 53658$$
$67$ $$T + 30908$$
$71$ $$T - 69648$$
$73$ $$T + 18446$$
$79$ $$T - 25300$$
$83$ $$T - 17556$$
$89$ $$T - 132570$$
$97$ $$T - 70658$$