# Properties

 Label 882.4.a.bg Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + 14 \beta q^{5} + 8 q^{8}+O(q^{10})$$ q + 2 * q^2 + 4 * q^4 + 14*b * q^5 + 8 * q^8 $$q + 2 q^{2} + 4 q^{4} + 14 \beta q^{5} + 8 q^{8} + 28 \beta q^{10} + 14 q^{11} + 36 \beta q^{13} + 16 q^{16} - \beta q^{17} - \beta q^{19} + 56 \beta q^{20} + 28 q^{22} - 140 q^{23} + 267 q^{25} + 72 \beta q^{26} + 286 q^{29} - 66 \beta q^{31} + 32 q^{32} - 2 \beta q^{34} - 38 q^{37} - 2 \beta q^{38} + 112 \beta q^{40} + 89 \beta q^{41} - 34 q^{43} + 56 q^{44} - 280 q^{46} - 370 \beta q^{47} + 534 q^{50} + 144 \beta q^{52} + 74 q^{53} + 196 \beta q^{55} + 572 q^{58} - 307 \beta q^{59} + 10 \beta q^{61} - 132 \beta q^{62} + 64 q^{64} + 1008 q^{65} + 684 q^{67} - 4 \beta q^{68} - 588 q^{71} - 191 \beta q^{73} - 76 q^{74} - 4 \beta q^{76} + 1220 q^{79} + 224 \beta q^{80} + 178 \beta q^{82} - 299 \beta q^{83} - 28 q^{85} - 68 q^{86} + 112 q^{88} - 437 \beta q^{89} - 560 q^{92} - 740 \beta q^{94} - 28 q^{95} + 1049 \beta q^{97} +O(q^{100})$$ q + 2 * q^2 + 4 * q^4 + 14*b * q^5 + 8 * q^8 + 28*b * q^10 + 14 * q^11 + 36*b * q^13 + 16 * q^16 - b * q^17 - b * q^19 + 56*b * q^20 + 28 * q^22 - 140 * q^23 + 267 * q^25 + 72*b * q^26 + 286 * q^29 - 66*b * q^31 + 32 * q^32 - 2*b * q^34 - 38 * q^37 - 2*b * q^38 + 112*b * q^40 + 89*b * q^41 - 34 * q^43 + 56 * q^44 - 280 * q^46 - 370*b * q^47 + 534 * q^50 + 144*b * q^52 + 74 * q^53 + 196*b * q^55 + 572 * q^58 - 307*b * q^59 + 10*b * q^61 - 132*b * q^62 + 64 * q^64 + 1008 * q^65 + 684 * q^67 - 4*b * q^68 - 588 * q^71 - 191*b * q^73 - 76 * q^74 - 4*b * q^76 + 1220 * q^79 + 224*b * q^80 + 178*b * q^82 - 299*b * q^83 - 28 * q^85 - 68 * q^86 + 112 * q^88 - 437*b * q^89 - 560 * q^92 - 740*b * q^94 - 28 * q^95 + 1049*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 8 q^{4} + 16 q^{8}+O(q^{10})$$ 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 $$2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 28 q^{11} + 32 q^{16} + 56 q^{22} - 280 q^{23} + 534 q^{25} + 572 q^{29} + 64 q^{32} - 76 q^{37} - 68 q^{43} + 112 q^{44} - 560 q^{46} + 1068 q^{50} + 148 q^{53} + 1144 q^{58} + 128 q^{64} + 2016 q^{65} + 1368 q^{67} - 1176 q^{71} - 152 q^{74} + 2440 q^{79} - 56 q^{85} - 136 q^{86} + 224 q^{88} - 1120 q^{92} - 56 q^{95}+O(q^{100})$$ 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 28 * q^11 + 32 * q^16 + 56 * q^22 - 280 * q^23 + 534 * q^25 + 572 * q^29 + 64 * q^32 - 76 * q^37 - 68 * q^43 + 112 * q^44 - 560 * q^46 + 1068 * q^50 + 148 * q^53 + 1144 * q^58 + 128 * q^64 + 2016 * q^65 + 1368 * q^67 - 1176 * q^71 - 152 * q^74 + 2440 * q^79 - 56 * q^85 - 136 * q^86 + 224 * q^88 - 1120 * q^92 - 56 * q^95

## 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
 −1.41421 1.41421
2.00000 0 4.00000 −19.7990 0 0 8.00000 0 −39.5980
1.2 2.00000 0 4.00000 19.7990 0 0 8.00000 0 39.5980
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.a.bg 2
3.b odd 2 1 98.4.a.g 2
7.b odd 2 1 inner 882.4.a.bg 2
7.c even 3 2 882.4.g.ba 4
7.d odd 6 2 882.4.g.ba 4
12.b even 2 1 784.4.a.y 2
15.d odd 2 1 2450.4.a.bx 2
21.c even 2 1 98.4.a.g 2
21.g even 6 2 98.4.c.h 4
21.h odd 6 2 98.4.c.h 4
84.h odd 2 1 784.4.a.y 2
105.g even 2 1 2450.4.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 3.b odd 2 1
98.4.a.g 2 21.c even 2 1
98.4.c.h 4 21.g even 6 2
98.4.c.h 4 21.h odd 6 2
784.4.a.y 2 12.b even 2 1
784.4.a.y 2 84.h odd 2 1
882.4.a.bg 2 1.a even 1 1 trivial
882.4.a.bg 2 7.b odd 2 1 inner
882.4.g.ba 4 7.c even 3 2
882.4.g.ba 4 7.d odd 6 2
2450.4.a.bx 2 15.d odd 2 1
2450.4.a.bx 2 105.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5}^{2} - 392$$ T5^2 - 392 $$T_{11} - 14$$ T11 - 14 $$T_{13}^{2} - 2592$$ T13^2 - 2592

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 392$$
$7$ $$T^{2}$$
$11$ $$(T - 14)^{2}$$
$13$ $$T^{2} - 2592$$
$17$ $$T^{2} - 2$$
$19$ $$T^{2} - 2$$
$23$ $$(T + 140)^{2}$$
$29$ $$(T - 286)^{2}$$
$31$ $$T^{2} - 8712$$
$37$ $$(T + 38)^{2}$$
$41$ $$T^{2} - 15842$$
$43$ $$(T + 34)^{2}$$
$47$ $$T^{2} - 273800$$
$53$ $$(T - 74)^{2}$$
$59$ $$T^{2} - 188498$$
$61$ $$T^{2} - 200$$
$67$ $$(T - 684)^{2}$$
$71$ $$(T + 588)^{2}$$
$73$ $$T^{2} - 72962$$
$79$ $$(T - 1220)^{2}$$
$83$ $$T^{2} - 178802$$
$89$ $$T^{2} - 381938$$
$97$ $$T^{2} - 2200802$$