# Properties

 Label 882.6.a.s Level $882$ Weight $6$ Character orbit 882.a Self dual yes Analytic conductor $141.459$ 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] = [882,6,Mod(1,882)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 882.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: $$141.458529075$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 16 q^{4} + 26 q^{5} + 64 q^{8}+O(q^{10})$$ q + 4 * q^2 + 16 * q^4 + 26 * q^5 + 64 * q^8 $$q + 4 q^{2} + 16 q^{4} + 26 q^{5} + 64 q^{8} + 104 q^{10} - 664 q^{11} - 318 q^{13} + 256 q^{16} + 1582 q^{17} - 236 q^{19} + 416 q^{20} - 2656 q^{22} - 2212 q^{23} - 2449 q^{25} - 1272 q^{26} + 4954 q^{29} + 7128 q^{31} + 1024 q^{32} + 6328 q^{34} + 4358 q^{37} - 944 q^{38} + 1664 q^{40} + 10542 q^{41} - 8452 q^{43} - 10624 q^{44} - 8848 q^{46} + 5352 q^{47} - 9796 q^{50} - 5088 q^{52} + 33354 q^{53} - 17264 q^{55} + 19816 q^{58} - 15436 q^{59} + 36762 q^{61} + 28512 q^{62} + 4096 q^{64} - 8268 q^{65} + 40972 q^{67} + 25312 q^{68} + 9092 q^{71} + 73454 q^{73} + 17432 q^{74} - 3776 q^{76} + 89400 q^{79} + 6656 q^{80} + 42168 q^{82} - 6428 q^{83} + 41132 q^{85} - 33808 q^{86} - 42496 q^{88} - 122658 q^{89} - 35392 q^{92} + 21408 q^{94} - 6136 q^{95} - 21370 q^{97}+O(q^{100})$$ q + 4 * q^2 + 16 * q^4 + 26 * q^5 + 64 * q^8 + 104 * q^10 - 664 * q^11 - 318 * q^13 + 256 * q^16 + 1582 * q^17 - 236 * q^19 + 416 * q^20 - 2656 * q^22 - 2212 * q^23 - 2449 * q^25 - 1272 * q^26 + 4954 * q^29 + 7128 * q^31 + 1024 * q^32 + 6328 * q^34 + 4358 * q^37 - 944 * q^38 + 1664 * q^40 + 10542 * q^41 - 8452 * q^43 - 10624 * q^44 - 8848 * q^46 + 5352 * q^47 - 9796 * q^50 - 5088 * q^52 + 33354 * q^53 - 17264 * q^55 + 19816 * q^58 - 15436 * q^59 + 36762 * q^61 + 28512 * q^62 + 4096 * q^64 - 8268 * q^65 + 40972 * q^67 + 25312 * q^68 + 9092 * q^71 + 73454 * q^73 + 17432 * q^74 - 3776 * q^76 + 89400 * q^79 + 6656 * q^80 + 42168 * q^82 - 6428 * q^83 + 41132 * q^85 - 33808 * q^86 - 42496 * q^88 - 122658 * q^89 - 35392 * q^92 + 21408 * q^94 - 6136 * q^95 - 21370 * q^97

## 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
4.00000 0 16.0000 26.0000 0 0 64.0000 0 104.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 882.6.a.s 1
3.b odd 2 1 294.6.a.b 1
7.b odd 2 1 126.6.a.i 1
21.c even 2 1 42.6.a.d 1
21.g even 6 2 294.6.e.i 2
21.h odd 6 2 294.6.e.p 2
28.d even 2 1 1008.6.a.j 1
84.h odd 2 1 336.6.a.h 1
105.g even 2 1 1050.6.a.k 1
105.k odd 4 2 1050.6.g.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.d 1 21.c even 2 1
126.6.a.i 1 7.b odd 2 1
294.6.a.b 1 3.b odd 2 1
294.6.e.i 2 21.g even 6 2
294.6.e.p 2 21.h odd 6 2
336.6.a.h 1 84.h odd 2 1
882.6.a.s 1 1.a even 1 1 trivial
1008.6.a.j 1 28.d even 2 1
1050.6.a.k 1 105.g even 2 1
1050.6.g.i 2 105.k odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5} - 26$$ T5 - 26 $$T_{11} + 664$$ T11 + 664

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T$$
$5$ $$T - 26$$
$7$ $$T$$
$11$ $$T + 664$$
$13$ $$T + 318$$
$17$ $$T - 1582$$
$19$ $$T + 236$$
$23$ $$T + 2212$$
$29$ $$T - 4954$$
$31$ $$T - 7128$$
$37$ $$T - 4358$$
$41$ $$T - 10542$$
$43$ $$T + 8452$$
$47$ $$T - 5352$$
$53$ $$T - 33354$$
$59$ $$T + 15436$$
$61$ $$T - 36762$$
$67$ $$T - 40972$$
$71$ $$T - 9092$$
$73$ $$T - 73454$$
$79$ $$T - 89400$$
$83$ $$T + 6428$$
$89$ $$T + 122658$$
$97$ $$T + 21370$$