Properties

 Label 882.2.a.i Level $882$ Weight $2$ Character orbit 882.a Self dual yes Analytic conductor $7.043$ 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] = [882,2,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, 2, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$7.04280545828$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^8 $$q + q^{2} + q^{4} + q^{8} + 4 q^{13} + q^{16} + 6 q^{17} - 2 q^{19} - 5 q^{25} + 4 q^{26} + 6 q^{29} + 4 q^{31} + q^{32} + 6 q^{34} + 2 q^{37} - 2 q^{38} + 6 q^{41} + 8 q^{43} - 12 q^{47} - 5 q^{50} + 4 q^{52} - 6 q^{53} + 6 q^{58} - 6 q^{59} - 8 q^{61} + 4 q^{62} + q^{64} - 4 q^{67} + 6 q^{68} - 2 q^{73} + 2 q^{74} - 2 q^{76} + 8 q^{79} + 6 q^{82} - 6 q^{83} + 8 q^{86} - 6 q^{89} - 12 q^{94} + 10 q^{97}+O(q^{100})$$ q + q^2 + q^4 + q^8 + 4 * q^13 + q^16 + 6 * q^17 - 2 * q^19 - 5 * q^25 + 4 * q^26 + 6 * q^29 + 4 * q^31 + q^32 + 6 * q^34 + 2 * q^37 - 2 * q^38 + 6 * q^41 + 8 * q^43 - 12 * q^47 - 5 * q^50 + 4 * q^52 - 6 * q^53 + 6 * q^58 - 6 * q^59 - 8 * q^61 + 4 * q^62 + q^64 - 4 * q^67 + 6 * q^68 - 2 * q^73 + 2 * q^74 - 2 * q^76 + 8 * q^79 + 6 * q^82 - 6 * q^83 + 8 * q^86 - 6 * q^89 - 12 * q^94 + 10 * 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
1.00000 0 1.00000 0 0 0 1.00000 0 0
 $$n$$: 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 882.2.a.i 1
3.b odd 2 1 98.2.a.a 1
4.b odd 2 1 7056.2.a.bd 1
7.b odd 2 1 126.2.a.b 1
7.c even 3 2 882.2.g.d 2
7.d odd 6 2 882.2.g.c 2
12.b even 2 1 784.2.a.b 1
15.d odd 2 1 2450.2.a.t 1
15.e even 4 2 2450.2.c.c 2
21.c even 2 1 14.2.a.a 1
21.g even 6 2 98.2.c.b 2
21.h odd 6 2 98.2.c.a 2
24.f even 2 1 3136.2.a.z 1
24.h odd 2 1 3136.2.a.e 1
28.d even 2 1 1008.2.a.h 1
35.c odd 2 1 3150.2.a.i 1
35.f even 4 2 3150.2.g.j 2
56.e even 2 1 4032.2.a.r 1
56.h odd 2 1 4032.2.a.w 1
63.l odd 6 2 1134.2.f.f 2
63.o even 6 2 1134.2.f.l 2
84.h odd 2 1 112.2.a.c 1
84.j odd 6 2 784.2.i.c 2
84.n even 6 2 784.2.i.i 2
105.g even 2 1 350.2.a.f 1
105.k odd 4 2 350.2.c.d 2
168.e odd 2 1 448.2.a.a 1
168.i even 2 1 448.2.a.g 1
231.h odd 2 1 1694.2.a.e 1
273.g even 2 1 2366.2.a.j 1
273.o odd 4 2 2366.2.d.b 2
336.v odd 4 2 1792.2.b.g 2
336.y even 4 2 1792.2.b.c 2
357.c even 2 1 4046.2.a.f 1
399.h odd 2 1 5054.2.a.c 1
420.o odd 2 1 2800.2.a.g 1
420.w even 4 2 2800.2.g.h 2
483.c odd 2 1 7406.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 21.c even 2 1
98.2.a.a 1 3.b odd 2 1
98.2.c.a 2 21.h odd 6 2
98.2.c.b 2 21.g even 6 2
112.2.a.c 1 84.h odd 2 1
126.2.a.b 1 7.b odd 2 1
350.2.a.f 1 105.g even 2 1
350.2.c.d 2 105.k odd 4 2
448.2.a.a 1 168.e odd 2 1
448.2.a.g 1 168.i even 2 1
784.2.a.b 1 12.b even 2 1
784.2.i.c 2 84.j odd 6 2
784.2.i.i 2 84.n even 6 2
882.2.a.i 1 1.a even 1 1 trivial
882.2.g.c 2 7.d odd 6 2
882.2.g.d 2 7.c even 3 2
1008.2.a.h 1 28.d even 2 1
1134.2.f.f 2 63.l odd 6 2
1134.2.f.l 2 63.o even 6 2
1694.2.a.e 1 231.h odd 2 1
1792.2.b.c 2 336.y even 4 2
1792.2.b.g 2 336.v odd 4 2
2366.2.a.j 1 273.g even 2 1
2366.2.d.b 2 273.o odd 4 2
2450.2.a.t 1 15.d odd 2 1
2450.2.c.c 2 15.e even 4 2
2800.2.a.g 1 420.o odd 2 1
2800.2.g.h 2 420.w even 4 2
3136.2.a.e 1 24.h odd 2 1
3136.2.a.z 1 24.f even 2 1
3150.2.a.i 1 35.c odd 2 1
3150.2.g.j 2 35.f even 4 2
4032.2.a.r 1 56.e even 2 1
4032.2.a.w 1 56.h odd 2 1
4046.2.a.f 1 357.c even 2 1
5054.2.a.c 1 399.h odd 2 1
7056.2.a.bd 1 4.b odd 2 1
7406.2.a.a 1 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5}$$ T5 $$T_{11}$$ T11 $$T_{13} - 4$$ T13 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 4$$
$17$ $$T - 6$$
$19$ $$T + 2$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T - 4$$
$37$ $$T - 2$$
$41$ $$T - 6$$
$43$ $$T - 8$$
$47$ $$T + 12$$
$53$ $$T + 6$$
$59$ $$T + 6$$
$61$ $$T + 8$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T + 6$$
$89$ $$T + 6$$
$97$ $$T - 10$$