# Properties

 Label 882.6.a.i Level $882$ Weight $6$ Character orbit 882.a Self dual yes Analytic conductor $141.459$ 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] = [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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 16 q^{4} + 24 q^{5} - 64 q^{8}+O(q^{10})$$ q - 4 * q^2 + 16 * q^4 + 24 * q^5 - 64 * q^8 $$q - 4 q^{2} + 16 q^{4} + 24 q^{5} - 64 q^{8} - 96 q^{10} - 66 q^{11} - 98 q^{13} + 256 q^{16} - 216 q^{17} + 340 q^{19} + 384 q^{20} + 264 q^{22} + 1038 q^{23} - 2549 q^{25} + 392 q^{26} + 2490 q^{29} + 7048 q^{31} - 1024 q^{32} + 864 q^{34} - 12238 q^{37} - 1360 q^{38} - 1536 q^{40} + 6468 q^{41} - 15412 q^{43} - 1056 q^{44} - 4152 q^{46} + 20604 q^{47} + 10196 q^{50} - 1568 q^{52} - 32490 q^{53} - 1584 q^{55} - 9960 q^{58} + 34224 q^{59} - 35654 q^{61} - 28192 q^{62} + 4096 q^{64} - 2352 q^{65} + 12680 q^{67} - 3456 q^{68} + 42642 q^{71} - 33734 q^{73} + 48952 q^{74} + 5440 q^{76} - 85108 q^{79} + 6144 q^{80} - 25872 q^{82} - 106764 q^{83} - 5184 q^{85} + 61648 q^{86} + 4224 q^{88} + 34884 q^{89} + 16608 q^{92} - 82416 q^{94} + 8160 q^{95} - 18662 q^{97}+O(q^{100})$$ q - 4 * q^2 + 16 * q^4 + 24 * q^5 - 64 * q^8 - 96 * q^10 - 66 * q^11 - 98 * q^13 + 256 * q^16 - 216 * q^17 + 340 * q^19 + 384 * q^20 + 264 * q^22 + 1038 * q^23 - 2549 * q^25 + 392 * q^26 + 2490 * q^29 + 7048 * q^31 - 1024 * q^32 + 864 * q^34 - 12238 * q^37 - 1360 * q^38 - 1536 * q^40 + 6468 * q^41 - 15412 * q^43 - 1056 * q^44 - 4152 * q^46 + 20604 * q^47 + 10196 * q^50 - 1568 * q^52 - 32490 * q^53 - 1584 * q^55 - 9960 * q^58 + 34224 * q^59 - 35654 * q^61 - 28192 * q^62 + 4096 * q^64 - 2352 * q^65 + 12680 * q^67 - 3456 * q^68 + 42642 * q^71 - 33734 * q^73 + 48952 * q^74 + 5440 * q^76 - 85108 * q^79 + 6144 * q^80 - 25872 * q^82 - 106764 * q^83 - 5184 * q^85 + 61648 * q^86 + 4224 * q^88 + 34884 * q^89 + 16608 * q^92 - 82416 * q^94 + 8160 * q^95 - 18662 * 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 24.0000 0 0 −64.0000 0 −96.0000
 $$n$$: 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.i 1
3.b odd 2 1 294.6.a.i 1
7.b odd 2 1 126.6.a.b 1
21.c even 2 1 42.6.a.f 1
21.g even 6 2 294.6.e.b 2
21.h odd 6 2 294.6.e.f 2
28.d even 2 1 1008.6.a.k 1
84.h odd 2 1 336.6.a.g 1
105.g even 2 1 1050.6.a.a 1
105.k odd 4 2 1050.6.g.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.f 1 21.c even 2 1
126.6.a.b 1 7.b odd 2 1
294.6.a.i 1 3.b odd 2 1
294.6.e.b 2 21.g even 6 2
294.6.e.f 2 21.h odd 6 2
336.6.a.g 1 84.h odd 2 1
882.6.a.i 1 1.a even 1 1 trivial
1008.6.a.k 1 28.d even 2 1
1050.6.a.a 1 105.g even 2 1
1050.6.g.m 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5} - 24$$ T5 - 24 $$T_{11} + 66$$ T11 + 66

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T$$
$5$ $$T - 24$$
$7$ $$T$$
$11$ $$T + 66$$
$13$ $$T + 98$$
$17$ $$T + 216$$
$19$ $$T - 340$$
$23$ $$T - 1038$$
$29$ $$T - 2490$$
$31$ $$T - 7048$$
$37$ $$T + 12238$$
$41$ $$T - 6468$$
$43$ $$T + 15412$$
$47$ $$T - 20604$$
$53$ $$T + 32490$$
$59$ $$T - 34224$$
$61$ $$T + 35654$$
$67$ $$T - 12680$$
$71$ $$T - 42642$$
$73$ $$T + 33734$$
$79$ $$T + 85108$$
$83$ $$T + 106764$$
$89$ $$T - 34884$$
$97$ $$T + 18662$$