# Properties

 Label 392.2.a.b Level $392$ Weight $2$ Character orbit 392.a Self dual yes Analytic conductor $3.130$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + 4 q^{5} + q^{9}+O(q^{10})$$ q - 2 * q^3 + 4 * q^5 + q^9 $$q - 2 q^{3} + 4 q^{5} + q^{9} - 8 q^{15} + 2 q^{17} + 2 q^{19} + 8 q^{23} + 11 q^{25} + 4 q^{27} + 2 q^{29} - 4 q^{31} - 6 q^{37} + 2 q^{41} + 8 q^{43} + 4 q^{45} + 4 q^{47} - 4 q^{51} - 10 q^{53} - 4 q^{57} - 6 q^{59} - 4 q^{61} - 12 q^{67} - 16 q^{69} + 14 q^{73} - 22 q^{75} - 8 q^{79} - 11 q^{81} - 6 q^{83} + 8 q^{85} - 4 q^{87} - 10 q^{89} + 8 q^{93} + 8 q^{95} + 2 q^{97}+O(q^{100})$$ q - 2 * q^3 + 4 * q^5 + q^9 - 8 * q^15 + 2 * q^17 + 2 * q^19 + 8 * q^23 + 11 * q^25 + 4 * q^27 + 2 * q^29 - 4 * q^31 - 6 * q^37 + 2 * q^41 + 8 * q^43 + 4 * q^45 + 4 * q^47 - 4 * q^51 - 10 * q^53 - 4 * q^57 - 6 * q^59 - 4 * q^61 - 12 * q^67 - 16 * q^69 + 14 * q^73 - 22 * q^75 - 8 * q^79 - 11 * q^81 - 6 * q^83 + 8 * q^85 - 4 * q^87 - 10 * q^89 + 8 * q^93 + 8 * q^95 + 2 * 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
0 −2.00000 0 4.00000 0 0 0 1.00000 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.b 1
3.b odd 2 1 3528.2.a.b 1
4.b odd 2 1 784.2.a.i 1
5.b even 2 1 9800.2.a.bj 1
7.b odd 2 1 56.2.a.b 1
7.c even 3 2 392.2.i.e 2
7.d odd 6 2 392.2.i.a 2
8.b even 2 1 3136.2.a.w 1
8.d odd 2 1 3136.2.a.c 1
12.b even 2 1 7056.2.a.c 1
21.c even 2 1 504.2.a.h 1
21.g even 6 2 3528.2.s.a 2
21.h odd 6 2 3528.2.s.ba 2
28.d even 2 1 112.2.a.a 1
28.f even 6 2 784.2.i.j 2
28.g odd 6 2 784.2.i.b 2
35.c odd 2 1 1400.2.a.a 1
35.f even 4 2 1400.2.g.b 2
56.e even 2 1 448.2.a.h 1
56.h odd 2 1 448.2.a.c 1
77.b even 2 1 6776.2.a.h 1
84.h odd 2 1 1008.2.a.m 1
91.b odd 2 1 9464.2.a.h 1
112.j even 4 2 1792.2.b.h 2
112.l odd 4 2 1792.2.b.a 2
140.c even 2 1 2800.2.a.bd 1
140.j odd 4 2 2800.2.g.g 2
168.e odd 2 1 4032.2.a.a 1
168.i even 2 1 4032.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 7.b odd 2 1
112.2.a.a 1 28.d even 2 1
392.2.a.b 1 1.a even 1 1 trivial
392.2.i.a 2 7.d odd 6 2
392.2.i.e 2 7.c even 3 2
448.2.a.c 1 56.h odd 2 1
448.2.a.h 1 56.e even 2 1
504.2.a.h 1 21.c even 2 1
784.2.a.i 1 4.b odd 2 1
784.2.i.b 2 28.g odd 6 2
784.2.i.j 2 28.f even 6 2
1008.2.a.m 1 84.h odd 2 1
1400.2.a.a 1 35.c odd 2 1
1400.2.g.b 2 35.f even 4 2
1792.2.b.a 2 112.l odd 4 2
1792.2.b.h 2 112.j even 4 2
2800.2.a.bd 1 140.c even 2 1
2800.2.g.g 2 140.j odd 4 2
3136.2.a.c 1 8.d odd 2 1
3136.2.a.w 1 8.b even 2 1
3528.2.a.b 1 3.b odd 2 1
3528.2.s.a 2 21.g even 6 2
3528.2.s.ba 2 21.h odd 6 2
4032.2.a.a 1 168.e odd 2 1
4032.2.a.d 1 168.i even 2 1
6776.2.a.h 1 77.b even 2 1
7056.2.a.c 1 12.b even 2 1
9464.2.a.h 1 91.b odd 2 1
9800.2.a.bj 1 5.b even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T - 4$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T - 2$$
$23$ $$T - 8$$
$29$ $$T - 2$$
$31$ $$T + 4$$
$37$ $$T + 6$$
$41$ $$T - 2$$
$43$ $$T - 8$$
$47$ $$T - 4$$
$53$ $$T + 10$$
$59$ $$T + 6$$
$61$ $$T + 4$$
$67$ $$T + 12$$
$71$ $$T$$
$73$ $$T - 14$$
$79$ $$T + 8$$
$83$ $$T + 6$$
$89$ $$T + 10$$
$97$ $$T - 2$$