# Properties

 Label 208.4.a.e Level $208$ Weight $4$ Character orbit 208.a Self dual yes Analytic conductor $12.272$ 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] = [208,4,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, 4, names="a")

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

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$12.2723972812$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + 17 q^{5} + 35 q^{7} - 26 q^{9}+O(q^{10})$$ q + q^3 + 17 * q^5 + 35 * q^7 - 26 * q^9 $$q + q^{3} + 17 q^{5} + 35 q^{7} - 26 q^{9} - 2 q^{11} + 13 q^{13} + 17 q^{15} - 19 q^{17} - 94 q^{19} + 35 q^{21} + 72 q^{23} + 164 q^{25} - 53 q^{27} + 246 q^{29} + 100 q^{31} - 2 q^{33} + 595 q^{35} - 11 q^{37} + 13 q^{39} - 280 q^{41} - 241 q^{43} - 442 q^{45} - 137 q^{47} + 882 q^{49} - 19 q^{51} - 232 q^{53} - 34 q^{55} - 94 q^{57} + 386 q^{59} + 64 q^{61} - 910 q^{63} + 221 q^{65} + 670 q^{67} + 72 q^{69} - 55 q^{71} - 838 q^{73} + 164 q^{75} - 70 q^{77} - 1016 q^{79} + 649 q^{81} - 420 q^{83} - 323 q^{85} + 246 q^{87} - 934 q^{89} + 455 q^{91} + 100 q^{93} - 1598 q^{95} - 1154 q^{97} + 52 q^{99}+O(q^{100})$$ q + q^3 + 17 * q^5 + 35 * q^7 - 26 * q^9 - 2 * q^11 + 13 * q^13 + 17 * q^15 - 19 * q^17 - 94 * q^19 + 35 * q^21 + 72 * q^23 + 164 * q^25 - 53 * q^27 + 246 * q^29 + 100 * q^31 - 2 * q^33 + 595 * q^35 - 11 * q^37 + 13 * q^39 - 280 * q^41 - 241 * q^43 - 442 * q^45 - 137 * q^47 + 882 * q^49 - 19 * q^51 - 232 * q^53 - 34 * q^55 - 94 * q^57 + 386 * q^59 + 64 * q^61 - 910 * q^63 + 221 * q^65 + 670 * q^67 + 72 * q^69 - 55 * q^71 - 838 * q^73 + 164 * q^75 - 70 * q^77 - 1016 * q^79 + 649 * q^81 - 420 * q^83 - 323 * q^85 + 246 * q^87 - 934 * q^89 + 455 * q^91 + 100 * q^93 - 1598 * q^95 - 1154 * q^97 + 52 * 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 17.0000 0 35.0000 0 −26.0000 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 208.4.a.e 1
3.b odd 2 1 1872.4.a.b 1
4.b odd 2 1 26.4.a.b 1
8.b even 2 1 832.4.a.g 1
8.d odd 2 1 832.4.a.j 1
12.b even 2 1 234.4.a.a 1
20.d odd 2 1 650.4.a.c 1
20.e even 4 2 650.4.b.d 2
28.d even 2 1 1274.4.a.f 1
52.b odd 2 1 338.4.a.b 1
52.f even 4 2 338.4.b.b 2
52.i odd 6 2 338.4.c.g 2
52.j odd 6 2 338.4.c.c 2
52.l even 12 4 338.4.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.b 1 4.b odd 2 1
208.4.a.e 1 1.a even 1 1 trivial
234.4.a.a 1 12.b even 2 1
338.4.a.b 1 52.b odd 2 1
338.4.b.b 2 52.f even 4 2
338.4.c.c 2 52.j odd 6 2
338.4.c.g 2 52.i odd 6 2
338.4.e.c 4 52.l even 12 4
650.4.a.c 1 20.d odd 2 1
650.4.b.d 2 20.e even 4 2
832.4.a.g 1 8.b even 2 1
832.4.a.j 1 8.d odd 2 1
1274.4.a.f 1 28.d even 2 1
1872.4.a.b 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(208))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 17$$
$7$ $$T - 35$$
$11$ $$T + 2$$
$13$ $$T - 13$$
$17$ $$T + 19$$
$19$ $$T + 94$$
$23$ $$T - 72$$
$29$ $$T - 246$$
$31$ $$T - 100$$
$37$ $$T + 11$$
$41$ $$T + 280$$
$43$ $$T + 241$$
$47$ $$T + 137$$
$53$ $$T + 232$$
$59$ $$T - 386$$
$61$ $$T - 64$$
$67$ $$T - 670$$
$71$ $$T + 55$$
$73$ $$T + 838$$
$79$ $$T + 1016$$
$83$ $$T + 420$$
$89$ $$T + 934$$
$97$ $$T + 1154$$