# Properties

 Label 392.2.a.c Level $392$ Weight $2$ Character orbit 392.a Self dual yes Analytic conductor $3.130$ 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] = [392,2,Mod(1,392)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("392.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 392.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: $$3.13013575923$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} - 2 q^{9}+O(q^{10})$$ q - q^3 - q^5 - 2 * q^9 $$q - q^{3} - q^{5} - 2 q^{9} + 3 q^{11} - 6 q^{13} + q^{15} - 5 q^{17} + q^{19} - 7 q^{23} - 4 q^{25} + 5 q^{27} + 2 q^{29} - 5 q^{31} - 3 q^{33} + 3 q^{37} + 6 q^{39} - 2 q^{41} - 4 q^{43} + 2 q^{45} + 5 q^{47} + 5 q^{51} - q^{53} - 3 q^{55} - q^{57} + 15 q^{59} - 5 q^{61} + 6 q^{65} - 9 q^{67} + 7 q^{69} + 7 q^{73} + 4 q^{75} + q^{79} + q^{81} + 12 q^{83} + 5 q^{85} - 2 q^{87} + 7 q^{89} + 5 q^{93} - q^{95} - 2 q^{97} - 6 q^{99}+O(q^{100})$$ q - q^3 - q^5 - 2 * q^9 + 3 * q^11 - 6 * q^13 + q^15 - 5 * q^17 + q^19 - 7 * q^23 - 4 * q^25 + 5 * q^27 + 2 * q^29 - 5 * q^31 - 3 * q^33 + 3 * q^37 + 6 * q^39 - 2 * q^41 - 4 * q^43 + 2 * q^45 + 5 * q^47 + 5 * q^51 - q^53 - 3 * q^55 - q^57 + 15 * q^59 - 5 * q^61 + 6 * q^65 - 9 * q^67 + 7 * q^69 + 7 * q^73 + 4 * q^75 + q^79 + q^81 + 12 * q^83 + 5 * q^85 - 2 * q^87 + 7 * q^89 + 5 * q^93 - q^95 - 2 * q^97 - 6 * q^99

## 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
0 −1.00000 0 −1.00000 0 0 0 −2.00000 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
$$2$$ $$1$$
$$7$$ $$1$$

## 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 392.2.a.c 1
3.b odd 2 1 3528.2.a.p 1
4.b odd 2 1 784.2.a.h 1
5.b even 2 1 9800.2.a.be 1
7.b odd 2 1 392.2.a.e 1
7.c even 3 2 56.2.i.b 2
7.d odd 6 2 392.2.i.b 2
8.b even 2 1 3136.2.a.u 1
8.d odd 2 1 3136.2.a.j 1
12.b even 2 1 7056.2.a.bj 1
21.c even 2 1 3528.2.a.j 1
21.g even 6 2 3528.2.s.q 2
21.h odd 6 2 504.2.s.c 2
28.d even 2 1 784.2.a.c 1
28.f even 6 2 784.2.i.h 2
28.g odd 6 2 112.2.i.a 2
35.c odd 2 1 9800.2.a.s 1
35.j even 6 2 1400.2.q.d 2
35.l odd 12 4 1400.2.bh.a 4
56.e even 2 1 3136.2.a.t 1
56.h odd 2 1 3136.2.a.i 1
56.k odd 6 2 448.2.i.d 2
56.p even 6 2 448.2.i.b 2
84.h odd 2 1 7056.2.a.u 1
84.n even 6 2 1008.2.s.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.b 2 7.c even 3 2
112.2.i.a 2 28.g odd 6 2
392.2.a.c 1 1.a even 1 1 trivial
392.2.a.e 1 7.b odd 2 1
392.2.i.b 2 7.d odd 6 2
448.2.i.b 2 56.p even 6 2
448.2.i.d 2 56.k odd 6 2
504.2.s.c 2 21.h odd 6 2
784.2.a.c 1 28.d even 2 1
784.2.a.h 1 4.b odd 2 1
784.2.i.h 2 28.f even 6 2
1008.2.s.g 2 84.n even 6 2
1400.2.q.d 2 35.j even 6 2
1400.2.bh.a 4 35.l odd 12 4
3136.2.a.i 1 56.h odd 2 1
3136.2.a.j 1 8.d odd 2 1
3136.2.a.t 1 56.e even 2 1
3136.2.a.u 1 8.b even 2 1
3528.2.a.j 1 21.c even 2 1
3528.2.a.p 1 3.b odd 2 1
3528.2.s.q 2 21.g even 6 2
7056.2.a.u 1 84.h odd 2 1
7056.2.a.bj 1 12.b even 2 1
9800.2.a.s 1 35.c odd 2 1
9800.2.a.be 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(392))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T - 3$$
$13$ $$T + 6$$
$17$ $$T + 5$$
$19$ $$T - 1$$
$23$ $$T + 7$$
$29$ $$T - 2$$
$31$ $$T + 5$$
$37$ $$T - 3$$
$41$ $$T + 2$$
$43$ $$T + 4$$
$47$ $$T - 5$$
$53$ $$T + 1$$
$59$ $$T - 15$$
$61$ $$T + 5$$
$67$ $$T + 9$$
$71$ $$T$$
$73$ $$T - 7$$
$79$ $$T - 1$$
$83$ $$T - 12$$
$89$ $$T - 7$$
$97$ $$T + 2$$