# Properties

 Label 392.8.a.d Level $392$ Weight $8$ Character orbit 392.a Self dual yes Analytic conductor $122.455$ 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] = [392,8,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, 8, names="a")

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

chi := DirichletCharacter("392.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$122.454929990$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) 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)

 $$f(q)$$ $$=$$ $$q + 84 q^{3} + 82 q^{5} + 4869 q^{9}+O(q^{10})$$ q + 84 * q^3 + 82 * q^5 + 4869 * q^9 $$q + 84 q^{3} + 82 q^{5} + 4869 q^{9} - 2524 q^{11} + 10778 q^{13} + 6888 q^{15} + 11150 q^{17} - 4124 q^{19} + 81704 q^{23} - 71401 q^{25} + 225288 q^{27} + 99798 q^{29} + 40480 q^{31} - 212016 q^{33} - 419442 q^{37} + 905352 q^{39} - 141402 q^{41} - 690428 q^{43} + 399258 q^{45} + 682032 q^{47} + 936600 q^{51} + 1813118 q^{53} - 206968 q^{55} - 346416 q^{57} + 966028 q^{59} - 1887670 q^{61} + 883796 q^{65} + 2965868 q^{67} + 6863136 q^{69} - 2548232 q^{71} + 1680326 q^{73} - 5997684 q^{75} + 4038064 q^{79} + 8275689 q^{81} + 5385764 q^{83} + 914300 q^{85} + 8383032 q^{87} + 6473046 q^{89} + 3400320 q^{93} - 338168 q^{95} + 6065758 q^{97} - 12289356 q^{99}+O(q^{100})$$ q + 84 * q^3 + 82 * q^5 + 4869 * q^9 - 2524 * q^11 + 10778 * q^13 + 6888 * q^15 + 11150 * q^17 - 4124 * q^19 + 81704 * q^23 - 71401 * q^25 + 225288 * q^27 + 99798 * q^29 + 40480 * q^31 - 212016 * q^33 - 419442 * q^37 + 905352 * q^39 - 141402 * q^41 - 690428 * q^43 + 399258 * q^45 + 682032 * q^47 + 936600 * q^51 + 1813118 * q^53 - 206968 * q^55 - 346416 * q^57 + 966028 * q^59 - 1887670 * q^61 + 883796 * q^65 + 2965868 * q^67 + 6863136 * q^69 - 2548232 * q^71 + 1680326 * q^73 - 5997684 * q^75 + 4038064 * q^79 + 8275689 * q^81 + 5385764 * q^83 + 914300 * q^85 + 8383032 * q^87 + 6473046 * q^89 + 3400320 * q^93 - 338168 * q^95 + 6065758 * q^97 - 12289356 * 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 84.0000 0 82.0000 0 0 0 4869.00 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 392.8.a.d 1
7.b odd 2 1 8.8.a.a 1
21.c even 2 1 72.8.a.d 1
28.d even 2 1 16.8.a.c 1
35.c odd 2 1 200.8.a.i 1
35.f even 4 2 200.8.c.a 2
56.e even 2 1 64.8.a.a 1
56.h odd 2 1 64.8.a.g 1
84.h odd 2 1 144.8.a.g 1
112.j even 4 2 256.8.b.c 2
112.l odd 4 2 256.8.b.e 2
140.c even 2 1 400.8.a.b 1
140.j odd 4 2 400.8.c.b 2
168.e odd 2 1 576.8.a.k 1
168.i even 2 1 576.8.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 7.b odd 2 1
16.8.a.c 1 28.d even 2 1
64.8.a.a 1 56.e even 2 1
64.8.a.g 1 56.h odd 2 1
72.8.a.d 1 21.c even 2 1
144.8.a.g 1 84.h odd 2 1
200.8.a.i 1 35.c odd 2 1
200.8.c.a 2 35.f even 4 2
256.8.b.c 2 112.j even 4 2
256.8.b.e 2 112.l odd 4 2
392.8.a.d 1 1.a even 1 1 trivial
400.8.a.b 1 140.c even 2 1
400.8.c.b 2 140.j odd 4 2
576.8.a.j 1 168.i even 2 1
576.8.a.k 1 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 84$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(392))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 84$$
$5$ $$T - 82$$
$7$ $$T$$
$11$ $$T + 2524$$
$13$ $$T - 10778$$
$17$ $$T - 11150$$
$19$ $$T + 4124$$
$23$ $$T - 81704$$
$29$ $$T - 99798$$
$31$ $$T - 40480$$
$37$ $$T + 419442$$
$41$ $$T + 141402$$
$43$ $$T + 690428$$
$47$ $$T - 682032$$
$53$ $$T - 1813118$$
$59$ $$T - 966028$$
$61$ $$T + 1887670$$
$67$ $$T - 2965868$$
$71$ $$T + 2548232$$
$73$ $$T - 1680326$$
$79$ $$T - 4038064$$
$83$ $$T - 5385764$$
$89$ $$T - 6473046$$
$97$ $$T - 6065758$$