# Properties

 Label 882.3.b.a Level $882$ Weight $3$ Character orbit 882.b Analytic conductor $24.033$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(197,882)]

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([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 882.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N # Warning: the index may be different

gp: f = lf \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$24.0327593166$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 2 q^{4} + 3 \beta q^{5} - 2 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 2 * q^4 + 3*b * q^5 - 2*b * q^8 $$q + \beta q^{2} - 2 q^{4} + 3 \beta q^{5} - 2 \beta q^{8} - 6 q^{10} + 12 \beta q^{11} - 8 q^{13} + 4 q^{16} + 9 \beta q^{17} + 16 q^{19} - 6 \beta q^{20} - 24 q^{22} - 12 \beta q^{23} + 7 q^{25} - 8 \beta q^{26} + 3 \beta q^{29} - 44 q^{31} + 4 \beta q^{32} - 18 q^{34} - 34 q^{37} + 16 \beta q^{38} + 12 q^{40} - 33 \beta q^{41} - 40 q^{43} - 24 \beta q^{44} + 24 q^{46} + 60 \beta q^{47} + 7 \beta q^{50} + 16 q^{52} + 27 \beta q^{53} - 72 q^{55} - 6 q^{58} - 24 \beta q^{59} - 50 q^{61} - 44 \beta q^{62} - 8 q^{64} - 24 \beta q^{65} + 8 q^{67} - 18 \beta q^{68} - 36 \beta q^{71} + 16 q^{73} - 34 \beta q^{74} - 32 q^{76} - 76 q^{79} + 12 \beta q^{80} + 66 q^{82} - 84 \beta q^{83} - 54 q^{85} - 40 \beta q^{86} + 48 q^{88} - 9 \beta q^{89} + 24 \beta q^{92} - 120 q^{94} + 48 \beta q^{95} - 176 q^{97} +O(q^{100})$$ q + b * q^2 - 2 * q^4 + 3*b * q^5 - 2*b * q^8 - 6 * q^10 + 12*b * q^11 - 8 * q^13 + 4 * q^16 + 9*b * q^17 + 16 * q^19 - 6*b * q^20 - 24 * q^22 - 12*b * q^23 + 7 * q^25 - 8*b * q^26 + 3*b * q^29 - 44 * q^31 + 4*b * q^32 - 18 * q^34 - 34 * q^37 + 16*b * q^38 + 12 * q^40 - 33*b * q^41 - 40 * q^43 - 24*b * q^44 + 24 * q^46 + 60*b * q^47 + 7*b * q^50 + 16 * q^52 + 27*b * q^53 - 72 * q^55 - 6 * q^58 - 24*b * q^59 - 50 * q^61 - 44*b * q^62 - 8 * q^64 - 24*b * q^65 + 8 * q^67 - 18*b * q^68 - 36*b * q^71 + 16 * q^73 - 34*b * q^74 - 32 * q^76 - 76 * q^79 + 12*b * q^80 + 66 * q^82 - 84*b * q^83 - 54 * q^85 - 40*b * q^86 + 48 * q^88 - 9*b * q^89 + 24*b * q^92 - 120 * q^94 + 48*b * q^95 - 176 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4}+O(q^{10})$$ 2 * q - 4 * q^4 $$2 q - 4 q^{4} - 12 q^{10} - 16 q^{13} + 8 q^{16} + 32 q^{19} - 48 q^{22} + 14 q^{25} - 88 q^{31} - 36 q^{34} - 68 q^{37} + 24 q^{40} - 80 q^{43} + 48 q^{46} + 32 q^{52} - 144 q^{55} - 12 q^{58} - 100 q^{61} - 16 q^{64} + 16 q^{67} + 32 q^{73} - 64 q^{76} - 152 q^{79} + 132 q^{82} - 108 q^{85} + 96 q^{88} - 240 q^{94} - 352 q^{97}+O(q^{100})$$ 2 * q - 4 * q^4 - 12 * q^10 - 16 * q^13 + 8 * q^16 + 32 * q^19 - 48 * q^22 + 14 * q^25 - 88 * q^31 - 36 * q^34 - 68 * q^37 + 24 * q^40 - 80 * q^43 + 48 * q^46 + 32 * q^52 - 144 * q^55 - 12 * q^58 - 100 * q^61 - 16 * q^64 + 16 * q^67 + 32 * q^73 - 64 * q^76 - 152 * q^79 + 132 * q^82 - 108 * q^85 + 96 * q^88 - 240 * q^94 - 352 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 − 1.41421i 1.41421i
1.41421i 0 −2.00000 4.24264i 0 0 2.82843i 0 −6.00000
197.2 1.41421i 0 −2.00000 4.24264i 0 0 2.82843i 0 −6.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.b.a 2
3.b odd 2 1 inner 882.3.b.a 2
7.b odd 2 1 18.3.b.a 2
7.c even 3 2 882.3.s.d 4
7.d odd 6 2 882.3.s.b 4
21.c even 2 1 18.3.b.a 2
21.g even 6 2 882.3.s.b 4
21.h odd 6 2 882.3.s.d 4
28.d even 2 1 144.3.e.b 2
35.c odd 2 1 450.3.d.f 2
35.f even 4 2 450.3.b.b 4
56.e even 2 1 576.3.e.f 2
56.h odd 2 1 576.3.e.c 2
63.l odd 6 2 162.3.d.b 4
63.o even 6 2 162.3.d.b 4
77.b even 2 1 2178.3.c.d 2
84.h odd 2 1 144.3.e.b 2
91.b odd 2 1 3042.3.c.e 2
91.i even 4 2 3042.3.d.a 4
105.g even 2 1 450.3.d.f 2
105.k odd 4 2 450.3.b.b 4
112.j even 4 2 2304.3.h.c 4
112.l odd 4 2 2304.3.h.f 4
140.c even 2 1 3600.3.l.d 2
140.j odd 4 2 3600.3.c.b 4
168.e odd 2 1 576.3.e.f 2
168.i even 2 1 576.3.e.c 2
231.h odd 2 1 2178.3.c.d 2
252.s odd 6 2 1296.3.q.f 4
252.bi even 6 2 1296.3.q.f 4
273.g even 2 1 3042.3.c.e 2
273.o odd 4 2 3042.3.d.a 4
336.v odd 4 2 2304.3.h.c 4
336.y even 4 2 2304.3.h.f 4
420.o odd 2 1 3600.3.l.d 2
420.w even 4 2 3600.3.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 7.b odd 2 1
18.3.b.a 2 21.c even 2 1
144.3.e.b 2 28.d even 2 1
144.3.e.b 2 84.h odd 2 1
162.3.d.b 4 63.l odd 6 2
162.3.d.b 4 63.o even 6 2
450.3.b.b 4 35.f even 4 2
450.3.b.b 4 105.k odd 4 2
450.3.d.f 2 35.c odd 2 1
450.3.d.f 2 105.g even 2 1
576.3.e.c 2 56.h odd 2 1
576.3.e.c 2 168.i even 2 1
576.3.e.f 2 56.e even 2 1
576.3.e.f 2 168.e odd 2 1
882.3.b.a 2 1.a even 1 1 trivial
882.3.b.a 2 3.b odd 2 1 inner
882.3.s.b 4 7.d odd 6 2
882.3.s.b 4 21.g even 6 2
882.3.s.d 4 7.c even 3 2
882.3.s.d 4 21.h odd 6 2
1296.3.q.f 4 252.s odd 6 2
1296.3.q.f 4 252.bi even 6 2
2178.3.c.d 2 77.b even 2 1
2178.3.c.d 2 231.h odd 2 1
2304.3.h.c 4 112.j even 4 2
2304.3.h.c 4 336.v odd 4 2
2304.3.h.f 4 112.l odd 4 2
2304.3.h.f 4 336.y even 4 2
3042.3.c.e 2 91.b odd 2 1
3042.3.c.e 2 273.g even 2 1
3042.3.d.a 4 91.i even 4 2
3042.3.d.a 4 273.o odd 4 2
3600.3.c.b 4 140.j odd 4 2
3600.3.c.b 4 420.w even 4 2
3600.3.l.d 2 140.c even 2 1
3600.3.l.d 2 420.o odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} + 18$$ T5^2 + 18 $$T_{11}^{2} + 288$$ T11^2 + 288 $$T_{13} + 8$$ T13 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 18$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 288$$
$13$ $$(T + 8)^{2}$$
$17$ $$T^{2} + 162$$
$19$ $$(T - 16)^{2}$$
$23$ $$T^{2} + 288$$
$29$ $$T^{2} + 18$$
$31$ $$(T + 44)^{2}$$
$37$ $$(T + 34)^{2}$$
$41$ $$T^{2} + 2178$$
$43$ $$(T + 40)^{2}$$
$47$ $$T^{2} + 7200$$
$53$ $$T^{2} + 1458$$
$59$ $$T^{2} + 1152$$
$61$ $$(T + 50)^{2}$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} + 2592$$
$73$ $$(T - 16)^{2}$$
$79$ $$(T + 76)^{2}$$
$83$ $$T^{2} + 14112$$
$89$ $$T^{2} + 162$$
$97$ $$(T + 176)^{2}$$