Properties

 Label 208.6.a.b Level $208$ Weight $6$ Character orbit 208.a Self dual yes Analytic conductor $33.360$ 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] = [208,6,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 208.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: $$33.3598345211$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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)

 $$f(q)$$ $$=$$ $$q - 14 q^{5} + 170 q^{7} - 243 q^{9}+O(q^{10})$$ q - 14 * q^5 + 170 * q^7 - 243 * q^9 $$q - 14 q^{5} + 170 q^{7} - 243 q^{9} + 250 q^{11} - 169 q^{13} + 1062 q^{17} + 78 q^{19} - 1576 q^{23} - 2929 q^{25} + 2578 q^{29} + 8654 q^{31} - 2380 q^{35} + 10986 q^{37} + 1050 q^{41} + 5900 q^{43} + 3402 q^{45} + 5962 q^{47} + 12093 q^{49} + 29046 q^{53} - 3500 q^{55} + 13922 q^{59} - 32882 q^{61} - 41310 q^{63} + 2366 q^{65} + 69566 q^{67} + 50542 q^{71} - 46750 q^{73} + 42500 q^{77} + 19348 q^{79} + 59049 q^{81} + 87438 q^{83} - 14868 q^{85} + 94170 q^{89} - 28730 q^{91} - 1092 q^{95} + 182786 q^{97} - 60750 q^{99}+O(q^{100})$$ q - 14 * q^5 + 170 * q^7 - 243 * q^9 + 250 * q^11 - 169 * q^13 + 1062 * q^17 + 78 * q^19 - 1576 * q^23 - 2929 * q^25 + 2578 * q^29 + 8654 * q^31 - 2380 * q^35 + 10986 * q^37 + 1050 * q^41 + 5900 * q^43 + 3402 * q^45 + 5962 * q^47 + 12093 * q^49 + 29046 * q^53 - 3500 * q^55 + 13922 * q^59 - 32882 * q^61 - 41310 * q^63 + 2366 * q^65 + 69566 * q^67 + 50542 * q^71 - 46750 * q^73 + 42500 * q^77 + 19348 * q^79 + 59049 * q^81 + 87438 * q^83 - 14868 * q^85 + 94170 * q^89 - 28730 * q^91 - 1092 * q^95 + 182786 * q^97 - 60750 * 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 0 0 −14.0000 0 170.000 0 −243.000 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
$$2$$ $$-1$$
$$13$$ $$+1$$

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 208.6.a.b 1
4.b odd 2 1 26.6.a.a 1
8.b even 2 1 832.6.a.e 1
8.d odd 2 1 832.6.a.d 1
12.b even 2 1 234.6.a.g 1
20.d odd 2 1 650.6.a.a 1
20.e even 4 2 650.6.b.a 2
52.b odd 2 1 338.6.a.d 1
52.f even 4 2 338.6.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.a.a 1 4.b odd 2 1
208.6.a.b 1 1.a even 1 1 trivial
234.6.a.g 1 12.b even 2 1
338.6.a.d 1 52.b odd 2 1
338.6.b.a 2 52.f even 4 2
650.6.a.a 1 20.d odd 2 1
650.6.b.a 2 20.e even 4 2
832.6.a.d 1 8.d odd 2 1
832.6.a.e 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(208))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 14$$
$7$ $$T - 170$$
$11$ $$T - 250$$
$13$ $$T + 169$$
$17$ $$T - 1062$$
$19$ $$T - 78$$
$23$ $$T + 1576$$
$29$ $$T - 2578$$
$31$ $$T - 8654$$
$37$ $$T - 10986$$
$41$ $$T - 1050$$
$43$ $$T - 5900$$
$47$ $$T - 5962$$
$53$ $$T - 29046$$
$59$ $$T - 13922$$
$61$ $$T + 32882$$
$67$ $$T - 69566$$
$71$ $$T - 50542$$
$73$ $$T + 46750$$
$79$ $$T - 19348$$
$83$ $$T - 87438$$
$89$ $$T - 94170$$
$97$ $$T - 182786$$