# Properties

 Label 208.2.a.a Level $208$ Weight $2$ Character orbit 208.a Self dual yes Analytic conductor $1.661$ Analytic rank $1$ 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] = [208,2,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, 2, names="a")

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

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$1.66088836204$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 3 q^{5} + q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 - 3 * q^5 + q^7 - 2 * q^9 $$q - q^{3} - 3 q^{5} + q^{7} - 2 q^{9} - 6 q^{11} + q^{13} + 3 q^{15} - 3 q^{17} - 2 q^{19} - q^{21} + 4 q^{25} + 5 q^{27} + 6 q^{29} + 4 q^{31} + 6 q^{33} - 3 q^{35} - 7 q^{37} - q^{39} + q^{43} + 6 q^{45} - 3 q^{47} - 6 q^{49} + 3 q^{51} + 18 q^{55} + 2 q^{57} + 6 q^{59} + 8 q^{61} - 2 q^{63} - 3 q^{65} - 14 q^{67} + 3 q^{71} + 2 q^{73} - 4 q^{75} - 6 q^{77} - 8 q^{79} + q^{81} - 12 q^{83} + 9 q^{85} - 6 q^{87} - 6 q^{89} + q^{91} - 4 q^{93} + 6 q^{95} - 10 q^{97} + 12 q^{99}+O(q^{100})$$ q - q^3 - 3 * q^5 + q^7 - 2 * q^9 - 6 * q^11 + q^13 + 3 * q^15 - 3 * q^17 - 2 * q^19 - q^21 + 4 * q^25 + 5 * q^27 + 6 * q^29 + 4 * q^31 + 6 * q^33 - 3 * q^35 - 7 * q^37 - q^39 + q^43 + 6 * q^45 - 3 * q^47 - 6 * q^49 + 3 * q^51 + 18 * q^55 + 2 * q^57 + 6 * q^59 + 8 * q^61 - 2 * q^63 - 3 * q^65 - 14 * q^67 + 3 * q^71 + 2 * q^73 - 4 * q^75 - 6 * q^77 - 8 * q^79 + q^81 - 12 * q^83 + 9 * q^85 - 6 * q^87 - 6 * q^89 + q^91 - 4 * q^93 + 6 * q^95 - 10 * q^97 + 12 * 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 −3.00000 0 1.00000 0 −2.00000 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
$$2$$ $$-1$$
$$13$$ $$-1$$

## 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 208.2.a.a 1
3.b odd 2 1 1872.2.a.q 1
4.b odd 2 1 26.2.a.a 1
5.b even 2 1 5200.2.a.x 1
8.b even 2 1 832.2.a.i 1
8.d odd 2 1 832.2.a.d 1
12.b even 2 1 234.2.a.e 1
13.b even 2 1 2704.2.a.f 1
13.d odd 4 2 2704.2.f.d 2
16.e even 4 2 3328.2.b.j 2
16.f odd 4 2 3328.2.b.m 2
20.d odd 2 1 650.2.a.j 1
20.e even 4 2 650.2.b.d 2
24.f even 2 1 7488.2.a.g 1
24.h odd 2 1 7488.2.a.h 1
28.d even 2 1 1274.2.a.d 1
28.f even 6 2 1274.2.f.r 2
28.g odd 6 2 1274.2.f.p 2
36.f odd 6 2 2106.2.e.ba 2
36.h even 6 2 2106.2.e.b 2
44.c even 2 1 3146.2.a.n 1
52.b odd 2 1 338.2.a.f 1
52.f even 4 2 338.2.b.c 2
52.i odd 6 2 338.2.c.a 2
52.j odd 6 2 338.2.c.d 2
52.l even 12 4 338.2.e.a 4
60.h even 2 1 5850.2.a.p 1
60.l odd 4 2 5850.2.e.a 2
68.d odd 2 1 7514.2.a.c 1
76.d even 2 1 9386.2.a.j 1
156.h even 2 1 3042.2.a.a 1
156.l odd 4 2 3042.2.b.a 2
260.g odd 2 1 8450.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 4.b odd 2 1
208.2.a.a 1 1.a even 1 1 trivial
234.2.a.e 1 12.b even 2 1
338.2.a.f 1 52.b odd 2 1
338.2.b.c 2 52.f even 4 2
338.2.c.a 2 52.i odd 6 2
338.2.c.d 2 52.j odd 6 2
338.2.e.a 4 52.l even 12 4
650.2.a.j 1 20.d odd 2 1
650.2.b.d 2 20.e even 4 2
832.2.a.d 1 8.d odd 2 1
832.2.a.i 1 8.b even 2 1
1274.2.a.d 1 28.d even 2 1
1274.2.f.p 2 28.g odd 6 2
1274.2.f.r 2 28.f even 6 2
1872.2.a.q 1 3.b odd 2 1
2106.2.e.b 2 36.h even 6 2
2106.2.e.ba 2 36.f odd 6 2
2704.2.a.f 1 13.b even 2 1
2704.2.f.d 2 13.d odd 4 2
3042.2.a.a 1 156.h even 2 1
3042.2.b.a 2 156.l odd 4 2
3146.2.a.n 1 44.c even 2 1
3328.2.b.j 2 16.e even 4 2
3328.2.b.m 2 16.f odd 4 2
5200.2.a.x 1 5.b even 2 1
5850.2.a.p 1 60.h even 2 1
5850.2.e.a 2 60.l odd 4 2
7488.2.a.g 1 24.f even 2 1
7488.2.a.h 1 24.h odd 2 1
7514.2.a.c 1 68.d odd 2 1
8450.2.a.c 1 260.g odd 2 1
9386.2.a.j 1 76.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(208))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{5} + 3$$ T5 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 3$$
$7$ $$T - 1$$
$11$ $$T + 6$$
$13$ $$T - 1$$
$17$ $$T + 3$$
$19$ $$T + 2$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T - 4$$
$37$ $$T + 7$$
$41$ $$T$$
$43$ $$T - 1$$
$47$ $$T + 3$$
$53$ $$T$$
$59$ $$T - 6$$
$61$ $$T - 8$$
$67$ $$T + 14$$
$71$ $$T - 3$$
$73$ $$T - 2$$
$79$ $$T + 8$$
$83$ $$T + 12$$
$89$ $$T + 6$$
$97$ $$T + 10$$