# Properties

 Label 208.8.a.e Level $208$ Weight $8$ Character orbit 208.a Self dual yes Analytic conductor $64.976$ 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] = [208,8,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, 8, names="a")

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

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$64.9760853007$$ 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)

 $$f(q)$$ $$=$$ $$q + 87 q^{3} + 321 q^{5} + 181 q^{7} + 5382 q^{9}+O(q^{10})$$ q + 87 * q^3 + 321 * q^5 + 181 * q^7 + 5382 * q^9 $$q + 87 q^{3} + 321 q^{5} + 181 q^{7} + 5382 q^{9} - 7782 q^{11} + 2197 q^{13} + 27927 q^{15} + 9069 q^{17} + 37150 q^{19} + 15747 q^{21} - 19008 q^{23} + 24916 q^{25} + 277965 q^{27} + 174750 q^{29} - 29012 q^{31} - 677034 q^{33} + 58101 q^{35} + 323669 q^{37} + 191139 q^{39} + 795312 q^{41} + 314137 q^{43} + 1727622 q^{45} + 447441 q^{47} - 790782 q^{49} + 789003 q^{51} - 1469232 q^{53} - 2498022 q^{55} + 3232050 q^{57} - 1627770 q^{59} - 2399608 q^{61} + 974142 q^{63} + 705237 q^{65} + 64066 q^{67} - 1653696 q^{69} + 322383 q^{71} - 4454782 q^{73} + 2167692 q^{75} - 1408542 q^{77} - 753560 q^{79} + 12412521 q^{81} + 1219092 q^{83} + 2911149 q^{85} + 15203250 q^{87} + 3390330 q^{89} + 397657 q^{91} - 2524044 q^{93} + 11925150 q^{95} + 1628774 q^{97} - 41882724 q^{99}+O(q^{100})$$ q + 87 * q^3 + 321 * q^5 + 181 * q^7 + 5382 * q^9 - 7782 * q^11 + 2197 * q^13 + 27927 * q^15 + 9069 * q^17 + 37150 * q^19 + 15747 * q^21 - 19008 * q^23 + 24916 * q^25 + 277965 * q^27 + 174750 * q^29 - 29012 * q^31 - 677034 * q^33 + 58101 * q^35 + 323669 * q^37 + 191139 * q^39 + 795312 * q^41 + 314137 * q^43 + 1727622 * q^45 + 447441 * q^47 - 790782 * q^49 + 789003 * q^51 - 1469232 * q^53 - 2498022 * q^55 + 3232050 * q^57 - 1627770 * q^59 - 2399608 * q^61 + 974142 * q^63 + 705237 * q^65 + 64066 * q^67 - 1653696 * q^69 + 322383 * q^71 - 4454782 * q^73 + 2167692 * q^75 - 1408542 * q^77 - 753560 * q^79 + 12412521 * q^81 + 1219092 * q^83 + 2911149 * q^85 + 15203250 * q^87 + 3390330 * q^89 + 397657 * q^91 - 2524044 * q^93 + 11925150 * q^95 + 1628774 * q^97 - 41882724 * 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 87.0000 0 321.000 0 181.000 0 5382.00 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
$$2$$ $$-1$$
$$13$$ $$-1$$

## 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 208.8.a.e 1
4.b odd 2 1 26.8.a.b 1
12.b even 2 1 234.8.a.a 1
52.b odd 2 1 338.8.a.a 1
52.f even 4 2 338.8.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.b 1 4.b odd 2 1
208.8.a.e 1 1.a even 1 1 trivial
234.8.a.a 1 12.b even 2 1
338.8.a.a 1 52.b odd 2 1
338.8.b.a 2 52.f even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 87$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(208))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 87$$
$5$ $$T - 321$$
$7$ $$T - 181$$
$11$ $$T + 7782$$
$13$ $$T - 2197$$
$17$ $$T - 9069$$
$19$ $$T - 37150$$
$23$ $$T + 19008$$
$29$ $$T - 174750$$
$31$ $$T + 29012$$
$37$ $$T - 323669$$
$41$ $$T - 795312$$
$43$ $$T - 314137$$
$47$ $$T - 447441$$
$53$ $$T + 1469232$$
$59$ $$T + 1627770$$
$61$ $$T + 2399608$$
$67$ $$T - 64066$$
$71$ $$T - 322383$$
$73$ $$T + 4454782$$
$79$ $$T + 753560$$
$83$ $$T - 1219092$$
$89$ $$T - 3390330$$
$97$ $$T - 1628774$$