# Properties

 Label 882.4.a.c Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ 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] = [882,4,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, 4, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$52.0396846251$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} - 9 q^{5} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 - 9 * q^5 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} - 9 q^{5} - 8 q^{8} + 18 q^{10} + 57 q^{11} + 70 q^{13} + 16 q^{16} + 51 q^{17} - 5 q^{19} - 36 q^{20} - 114 q^{22} - 69 q^{23} - 44 q^{25} - 140 q^{26} - 114 q^{29} - 23 q^{31} - 32 q^{32} - 102 q^{34} - 253 q^{37} + 10 q^{38} + 72 q^{40} - 42 q^{41} - 124 q^{43} + 228 q^{44} + 138 q^{46} + 201 q^{47} + 88 q^{50} + 280 q^{52} + 393 q^{53} - 513 q^{55} + 228 q^{58} + 219 q^{59} + 709 q^{61} + 46 q^{62} + 64 q^{64} - 630 q^{65} + 419 q^{67} + 204 q^{68} + 96 q^{71} + 313 q^{73} + 506 q^{74} - 20 q^{76} + 461 q^{79} - 144 q^{80} + 84 q^{82} - 588 q^{83} - 459 q^{85} + 248 q^{86} - 456 q^{88} - 1017 q^{89} - 276 q^{92} - 402 q^{94} + 45 q^{95} + 1834 q^{97}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 - 9 * q^5 - 8 * q^8 + 18 * q^10 + 57 * q^11 + 70 * q^13 + 16 * q^16 + 51 * q^17 - 5 * q^19 - 36 * q^20 - 114 * q^22 - 69 * q^23 - 44 * q^25 - 140 * q^26 - 114 * q^29 - 23 * q^31 - 32 * q^32 - 102 * q^34 - 253 * q^37 + 10 * q^38 + 72 * q^40 - 42 * q^41 - 124 * q^43 + 228 * q^44 + 138 * q^46 + 201 * q^47 + 88 * q^50 + 280 * q^52 + 393 * q^53 - 513 * q^55 + 228 * q^58 + 219 * q^59 + 709 * q^61 + 46 * q^62 + 64 * q^64 - 630 * q^65 + 419 * q^67 + 204 * q^68 + 96 * q^71 + 313 * q^73 + 506 * q^74 - 20 * q^76 + 461 * q^79 - 144 * q^80 + 84 * q^82 - 588 * q^83 - 459 * q^85 + 248 * q^86 - 456 * q^88 - 1017 * q^89 - 276 * q^92 - 402 * q^94 + 45 * q^95 + 1834 * 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
−2.00000 0 4.00000 −9.00000 0 0 −8.00000 0 18.0000
 $$n$$: 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 882.4.a.c 1
3.b odd 2 1 98.4.a.f 1
7.b odd 2 1 882.4.a.f 1
7.c even 3 2 882.4.g.u 2
7.d odd 6 2 126.4.g.d 2
12.b even 2 1 784.4.a.c 1
15.d odd 2 1 2450.4.a.d 1
21.c even 2 1 98.4.a.d 1
21.g even 6 2 14.4.c.a 2
21.h odd 6 2 98.4.c.a 2
84.h odd 2 1 784.4.a.p 1
84.j odd 6 2 112.4.i.a 2
105.g even 2 1 2450.4.a.q 1
105.p even 6 2 350.4.e.e 2
105.w odd 12 4 350.4.j.b 4
168.ba even 6 2 448.4.i.b 2
168.be odd 6 2 448.4.i.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 21.g even 6 2
98.4.a.d 1 21.c even 2 1
98.4.a.f 1 3.b odd 2 1
98.4.c.a 2 21.h odd 6 2
112.4.i.a 2 84.j odd 6 2
126.4.g.d 2 7.d odd 6 2
350.4.e.e 2 105.p even 6 2
350.4.j.b 4 105.w odd 12 4
448.4.i.b 2 168.ba even 6 2
448.4.i.e 2 168.be odd 6 2
784.4.a.c 1 12.b even 2 1
784.4.a.p 1 84.h odd 2 1
882.4.a.c 1 1.a even 1 1 trivial
882.4.a.f 1 7.b odd 2 1
882.4.g.u 2 7.c even 3 2
2450.4.a.d 1 15.d odd 2 1
2450.4.a.q 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5} + 9$$ T5 + 9 $$T_{11} - 57$$ T11 - 57 $$T_{13} - 70$$ T13 - 70

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T + 9$$
$7$ $$T$$
$11$ $$T - 57$$
$13$ $$T - 70$$
$17$ $$T - 51$$
$19$ $$T + 5$$
$23$ $$T + 69$$
$29$ $$T + 114$$
$31$ $$T + 23$$
$37$ $$T + 253$$
$41$ $$T + 42$$
$43$ $$T + 124$$
$47$ $$T - 201$$
$53$ $$T - 393$$
$59$ $$T - 219$$
$61$ $$T - 709$$
$67$ $$T - 419$$
$71$ $$T - 96$$
$73$ $$T - 313$$
$79$ $$T - 461$$
$83$ $$T + 588$$
$89$ $$T + 1017$$
$97$ $$T - 1834$$