# Properties

 Label 882.4.a.r 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 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + 15 q^{5} + 8 q^{8}+O(q^{10})$$ q + 2 * q^2 + 4 * q^4 + 15 * q^5 + 8 * q^8 $$q + 2 q^{2} + 4 q^{4} + 15 q^{5} + 8 q^{8} + 30 q^{10} + 9 q^{11} - 88 q^{13} + 16 q^{16} + 84 q^{17} + 104 q^{19} + 60 q^{20} + 18 q^{22} + 84 q^{23} + 100 q^{25} - 176 q^{26} - 51 q^{29} + 185 q^{31} + 32 q^{32} + 168 q^{34} + 44 q^{37} + 208 q^{38} + 120 q^{40} + 168 q^{41} + 326 q^{43} + 36 q^{44} + 168 q^{46} + 138 q^{47} + 200 q^{50} - 352 q^{52} - 639 q^{53} + 135 q^{55} - 102 q^{58} - 159 q^{59} + 722 q^{61} + 370 q^{62} + 64 q^{64} - 1320 q^{65} - 166 q^{67} + 336 q^{68} - 1086 q^{71} + 218 q^{73} + 88 q^{74} + 416 q^{76} - 583 q^{79} + 240 q^{80} + 336 q^{82} + 597 q^{83} + 1260 q^{85} + 652 q^{86} + 72 q^{88} + 1038 q^{89} + 336 q^{92} + 276 q^{94} + 1560 q^{95} - 169 q^{97}+O(q^{100})$$ q + 2 * q^2 + 4 * q^4 + 15 * q^5 + 8 * q^8 + 30 * q^10 + 9 * q^11 - 88 * q^13 + 16 * q^16 + 84 * q^17 + 104 * q^19 + 60 * q^20 + 18 * q^22 + 84 * q^23 + 100 * q^25 - 176 * q^26 - 51 * q^29 + 185 * q^31 + 32 * q^32 + 168 * q^34 + 44 * q^37 + 208 * q^38 + 120 * q^40 + 168 * q^41 + 326 * q^43 + 36 * q^44 + 168 * q^46 + 138 * q^47 + 200 * q^50 - 352 * q^52 - 639 * q^53 + 135 * q^55 - 102 * q^58 - 159 * q^59 + 722 * q^61 + 370 * q^62 + 64 * q^64 - 1320 * q^65 - 166 * q^67 + 336 * q^68 - 1086 * q^71 + 218 * q^73 + 88 * q^74 + 416 * q^76 - 583 * q^79 + 240 * q^80 + 336 * q^82 + 597 * q^83 + 1260 * q^85 + 652 * q^86 + 72 * q^88 + 1038 * q^89 + 336 * q^92 + 276 * q^94 + 1560 * q^95 - 169 * 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 15.0000 0 0 8.00000 0 30.0000
 $$n$$: 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.r 1
3.b odd 2 1 294.4.a.a 1
7.b odd 2 1 882.4.a.h 1
7.c even 3 2 126.4.g.a 2
7.d odd 6 2 882.4.g.l 2
12.b even 2 1 2352.4.a.u 1
21.c even 2 1 294.4.a.g 1
21.g even 6 2 294.4.e.e 2
21.h odd 6 2 42.4.e.b 2
84.h odd 2 1 2352.4.a.q 1
84.n even 6 2 336.4.q.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 21.h odd 6 2
126.4.g.a 2 7.c even 3 2
294.4.a.a 1 3.b odd 2 1
294.4.a.g 1 21.c even 2 1
294.4.e.e 2 21.g even 6 2
336.4.q.d 2 84.n even 6 2
882.4.a.h 1 7.b odd 2 1
882.4.a.r 1 1.a even 1 1 trivial
882.4.g.l 2 7.d odd 6 2
2352.4.a.q 1 84.h odd 2 1
2352.4.a.u 1 12.b even 2 1

## Hecke kernels

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

 $$T_{5} - 15$$ T5 - 15 $$T_{11} - 9$$ T11 - 9 $$T_{13} + 88$$ T13 + 88

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T - 15$$
$7$ $$T$$
$11$ $$T - 9$$
$13$ $$T + 88$$
$17$ $$T - 84$$
$19$ $$T - 104$$
$23$ $$T - 84$$
$29$ $$T + 51$$
$31$ $$T - 185$$
$37$ $$T - 44$$
$41$ $$T - 168$$
$43$ $$T - 326$$
$47$ $$T - 138$$
$53$ $$T + 639$$
$59$ $$T + 159$$
$61$ $$T - 722$$
$67$ $$T + 166$$
$71$ $$T + 1086$$
$73$ $$T - 218$$
$79$ $$T + 583$$
$83$ $$T - 597$$
$89$ $$T - 1038$$
$97$ $$T + 169$$