# Properties

 Label 882.6.a.g 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 14) 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 16 q^{4} + 10 q^{5} - 64 q^{8}+O(q^{10})$$ q - 4 * q^2 + 16 * q^4 + 10 * q^5 - 64 * q^8 $$q - 4 q^{2} + 16 q^{4} + 10 q^{5} - 64 q^{8} - 40 q^{10} + 340 q^{11} + 294 q^{13} + 256 q^{16} + 1226 q^{17} - 2432 q^{19} + 160 q^{20} - 1360 q^{22} - 2000 q^{23} - 3025 q^{25} - 1176 q^{26} + 6746 q^{29} - 8856 q^{31} - 1024 q^{32} - 4904 q^{34} + 9182 q^{37} + 9728 q^{38} - 640 q^{40} - 14574 q^{41} + 8108 q^{43} + 5440 q^{44} + 8000 q^{46} - 312 q^{47} + 12100 q^{50} + 4704 q^{52} + 14634 q^{53} + 3400 q^{55} - 26984 q^{58} - 27656 q^{59} - 34338 q^{61} + 35424 q^{62} + 4096 q^{64} + 2940 q^{65} + 12316 q^{67} + 19616 q^{68} - 36920 q^{71} + 61718 q^{73} - 36728 q^{74} - 38912 q^{76} - 64752 q^{79} + 2560 q^{80} + 58296 q^{82} - 77056 q^{83} + 12260 q^{85} - 32432 q^{86} - 21760 q^{88} - 8166 q^{89} - 32000 q^{92} + 1248 q^{94} - 24320 q^{95} - 20650 q^{97}+O(q^{100})$$ q - 4 * q^2 + 16 * q^4 + 10 * q^5 - 64 * q^8 - 40 * q^10 + 340 * q^11 + 294 * q^13 + 256 * q^16 + 1226 * q^17 - 2432 * q^19 + 160 * q^20 - 1360 * q^22 - 2000 * q^23 - 3025 * q^25 - 1176 * q^26 + 6746 * q^29 - 8856 * q^31 - 1024 * q^32 - 4904 * q^34 + 9182 * q^37 + 9728 * q^38 - 640 * q^40 - 14574 * q^41 + 8108 * q^43 + 5440 * q^44 + 8000 * q^46 - 312 * q^47 + 12100 * q^50 + 4704 * q^52 + 14634 * q^53 + 3400 * q^55 - 26984 * q^58 - 27656 * q^59 - 34338 * q^61 + 35424 * q^62 + 4096 * q^64 + 2940 * q^65 + 12316 * q^67 + 19616 * q^68 - 36920 * q^71 + 61718 * q^73 - 36728 * q^74 - 38912 * q^76 - 64752 * q^79 + 2560 * q^80 + 58296 * q^82 - 77056 * q^83 + 12260 * q^85 - 32432 * q^86 - 21760 * q^88 - 8166 * q^89 - 32000 * q^92 + 1248 * q^94 - 24320 * q^95 - 20650 * 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 10.0000 0 0 −64.0000 0 −40.0000
 $$n$$: 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.g 1
3.b odd 2 1 98.6.a.b 1
7.b odd 2 1 126.6.a.c 1
12.b even 2 1 784.6.a.h 1
21.c even 2 1 14.6.a.b 1
21.g even 6 2 98.6.c.a 2
21.h odd 6 2 98.6.c.b 2
28.d even 2 1 1008.6.a.n 1
84.h odd 2 1 112.6.a.d 1
105.g even 2 1 350.6.a.b 1
105.k odd 4 2 350.6.c.f 2
168.e odd 2 1 448.6.a.k 1
168.i even 2 1 448.6.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 21.c even 2 1
98.6.a.b 1 3.b odd 2 1
98.6.c.a 2 21.g even 6 2
98.6.c.b 2 21.h odd 6 2
112.6.a.d 1 84.h odd 2 1
126.6.a.c 1 7.b odd 2 1
350.6.a.b 1 105.g even 2 1
350.6.c.f 2 105.k odd 4 2
448.6.a.f 1 168.i even 2 1
448.6.a.k 1 168.e odd 2 1
784.6.a.h 1 12.b even 2 1
882.6.a.g 1 1.a even 1 1 trivial
1008.6.a.n 1 28.d even 2 1

## Hecke kernels

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

 $$T_{5} - 10$$ T5 - 10 $$T_{11} - 340$$ T11 - 340

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T$$
$5$ $$T - 10$$
$7$ $$T$$
$11$ $$T - 340$$
$13$ $$T - 294$$
$17$ $$T - 1226$$
$19$ $$T + 2432$$
$23$ $$T + 2000$$
$29$ $$T - 6746$$
$31$ $$T + 8856$$
$37$ $$T - 9182$$
$41$ $$T + 14574$$
$43$ $$T - 8108$$
$47$ $$T + 312$$
$53$ $$T - 14634$$
$59$ $$T + 27656$$
$61$ $$T + 34338$$
$67$ $$T - 12316$$
$71$ $$T + 36920$$
$73$ $$T - 61718$$
$79$ $$T + 64752$$
$83$ $$T + 77056$$
$89$ $$T + 8166$$
$97$ $$T + 20650$$