# Properties

 Label 882.4.a.n Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ 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] = [882,4,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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 8 q^{8}+O(q^{10})$$ q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 8 * q^8 $$q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 8 q^{8} + 12 q^{10} - 12 q^{11} - 38 q^{13} + 16 q^{16} - 126 q^{17} - 20 q^{19} + 24 q^{20} - 24 q^{22} - 168 q^{23} - 89 q^{25} - 76 q^{26} - 30 q^{29} + 88 q^{31} + 32 q^{32} - 252 q^{34} + 254 q^{37} - 40 q^{38} + 48 q^{40} + 42 q^{41} - 52 q^{43} - 48 q^{44} - 336 q^{46} - 96 q^{47} - 178 q^{50} - 152 q^{52} - 198 q^{53} - 72 q^{55} - 60 q^{58} - 660 q^{59} + 538 q^{61} + 176 q^{62} + 64 q^{64} - 228 q^{65} + 884 q^{67} - 504 q^{68} - 792 q^{71} - 218 q^{73} + 508 q^{74} - 80 q^{76} - 520 q^{79} + 96 q^{80} + 84 q^{82} - 492 q^{83} - 756 q^{85} - 104 q^{86} - 96 q^{88} + 810 q^{89} - 672 q^{92} - 192 q^{94} - 120 q^{95} - 1154 q^{97}+O(q^{100})$$ q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 8 * q^8 + 12 * q^10 - 12 * q^11 - 38 * q^13 + 16 * q^16 - 126 * q^17 - 20 * q^19 + 24 * q^20 - 24 * q^22 - 168 * q^23 - 89 * q^25 - 76 * q^26 - 30 * q^29 + 88 * q^31 + 32 * q^32 - 252 * q^34 + 254 * q^37 - 40 * q^38 + 48 * q^40 + 42 * q^41 - 52 * q^43 - 48 * q^44 - 336 * q^46 - 96 * q^47 - 178 * q^50 - 152 * q^52 - 198 * q^53 - 72 * q^55 - 60 * q^58 - 660 * q^59 + 538 * q^61 + 176 * q^62 + 64 * q^64 - 228 * q^65 + 884 * q^67 - 504 * q^68 - 792 * q^71 - 218 * q^73 + 508 * q^74 - 80 * q^76 - 520 * q^79 + 96 * q^80 + 84 * q^82 - 492 * q^83 - 756 * q^85 - 104 * q^86 - 96 * q^88 + 810 * q^89 - 672 * q^92 - 192 * q^94 - 120 * q^95 - 1154 * 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 6.00000 0 0 8.00000 0 12.0000
 $$n$$: 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.n 1
3.b odd 2 1 294.4.a.e 1
7.b odd 2 1 18.4.a.a 1
7.c even 3 2 882.4.g.f 2
7.d odd 6 2 882.4.g.i 2
12.b even 2 1 2352.4.a.e 1
21.c even 2 1 6.4.a.a 1
21.g even 6 2 294.4.e.h 2
21.h odd 6 2 294.4.e.g 2
28.d even 2 1 144.4.a.c 1
35.c odd 2 1 450.4.a.h 1
35.f even 4 2 450.4.c.e 2
56.e even 2 1 576.4.a.r 1
56.h odd 2 1 576.4.a.q 1
63.l odd 6 2 162.4.c.c 2
63.o even 6 2 162.4.c.f 2
77.b even 2 1 2178.4.a.e 1
84.h odd 2 1 48.4.a.c 1
105.g even 2 1 150.4.a.i 1
105.k odd 4 2 150.4.c.d 2
168.e odd 2 1 192.4.a.c 1
168.i even 2 1 192.4.a.i 1
231.h odd 2 1 726.4.a.f 1
273.g even 2 1 1014.4.a.g 1
273.o odd 4 2 1014.4.b.d 2
336.v odd 4 2 768.4.d.c 2
336.y even 4 2 768.4.d.n 2
357.c even 2 1 1734.4.a.d 1
399.h odd 2 1 2166.4.a.i 1
420.o odd 2 1 1200.4.a.b 1
420.w even 4 2 1200.4.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 21.c even 2 1
18.4.a.a 1 7.b odd 2 1
48.4.a.c 1 84.h odd 2 1
144.4.a.c 1 28.d even 2 1
150.4.a.i 1 105.g even 2 1
150.4.c.d 2 105.k odd 4 2
162.4.c.c 2 63.l odd 6 2
162.4.c.f 2 63.o even 6 2
192.4.a.c 1 168.e odd 2 1
192.4.a.i 1 168.i even 2 1
294.4.a.e 1 3.b odd 2 1
294.4.e.g 2 21.h odd 6 2
294.4.e.h 2 21.g even 6 2
450.4.a.h 1 35.c odd 2 1
450.4.c.e 2 35.f even 4 2
576.4.a.q 1 56.h odd 2 1
576.4.a.r 1 56.e even 2 1
726.4.a.f 1 231.h odd 2 1
768.4.d.c 2 336.v odd 4 2
768.4.d.n 2 336.y even 4 2
882.4.a.n 1 1.a even 1 1 trivial
882.4.g.f 2 7.c even 3 2
882.4.g.i 2 7.d odd 6 2
1014.4.a.g 1 273.g even 2 1
1014.4.b.d 2 273.o odd 4 2
1200.4.a.b 1 420.o odd 2 1
1200.4.f.j 2 420.w even 4 2
1734.4.a.d 1 357.c even 2 1
2166.4.a.i 1 399.h odd 2 1
2178.4.a.e 1 77.b even 2 1
2352.4.a.e 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5} - 6$$ T5 - 6 $$T_{11} + 12$$ T11 + 12 $$T_{13} + 38$$ T13 + 38

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T - 6$$
$7$ $$T$$
$11$ $$T + 12$$
$13$ $$T + 38$$
$17$ $$T + 126$$
$19$ $$T + 20$$
$23$ $$T + 168$$
$29$ $$T + 30$$
$31$ $$T - 88$$
$37$ $$T - 254$$
$41$ $$T - 42$$
$43$ $$T + 52$$
$47$ $$T + 96$$
$53$ $$T + 198$$
$59$ $$T + 660$$
$61$ $$T - 538$$
$67$ $$T - 884$$
$71$ $$T + 792$$
$73$ $$T + 218$$
$79$ $$T + 520$$
$83$ $$T + 492$$
$89$ $$T - 810$$
$97$ $$T + 1154$$