Properties

 Label 208.2.a.d Level $208$ Weight $2$ Character orbit 208.a Self dual yes Analytic conductor $1.661$ 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] = [208,2,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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - q^{5} - q^{7} + 6 q^{9}+O(q^{10})$$ q + 3 * q^3 - q^5 - q^7 + 6 * q^9 $$q + 3 q^{3} - q^{5} - q^{7} + 6 q^{9} + 2 q^{11} - q^{13} - 3 q^{15} - 3 q^{17} - 6 q^{19} - 3 q^{21} + 4 q^{23} - 4 q^{25} + 9 q^{27} + 2 q^{29} - 4 q^{31} + 6 q^{33} + q^{35} + 3 q^{37} - 3 q^{39} + 5 q^{43} - 6 q^{45} - 13 q^{47} - 6 q^{49} - 9 q^{51} + 12 q^{53} - 2 q^{55} - 18 q^{57} + 10 q^{59} - 8 q^{61} - 6 q^{63} + q^{65} + 2 q^{67} + 12 q^{69} + 5 q^{71} - 10 q^{73} - 12 q^{75} - 2 q^{77} + 4 q^{79} + 9 q^{81} + 3 q^{85} + 6 q^{87} + 6 q^{89} + q^{91} - 12 q^{93} + 6 q^{95} + 14 q^{97} + 12 q^{99}+O(q^{100})$$ q + 3 * q^3 - q^5 - q^7 + 6 * q^9 + 2 * q^11 - q^13 - 3 * q^15 - 3 * q^17 - 6 * q^19 - 3 * q^21 + 4 * q^23 - 4 * q^25 + 9 * q^27 + 2 * q^29 - 4 * q^31 + 6 * q^33 + q^35 + 3 * q^37 - 3 * q^39 + 5 * q^43 - 6 * q^45 - 13 * q^47 - 6 * q^49 - 9 * q^51 + 12 * q^53 - 2 * q^55 - 18 * q^57 + 10 * q^59 - 8 * q^61 - 6 * q^63 + q^65 + 2 * q^67 + 12 * q^69 + 5 * q^71 - 10 * q^73 - 12 * q^75 - 2 * q^77 + 4 * q^79 + 9 * q^81 + 3 * q^85 + 6 * q^87 + 6 * q^89 + q^91 - 12 * q^93 + 6 * q^95 + 14 * 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 3.00000 0 −1.00000 0 −1.00000 0 6.00000 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.d 1
3.b odd 2 1 1872.2.a.m 1
4.b odd 2 1 26.2.a.b 1
5.b even 2 1 5200.2.a.c 1
8.b even 2 1 832.2.a.a 1
8.d odd 2 1 832.2.a.j 1
12.b even 2 1 234.2.a.b 1
13.b even 2 1 2704.2.a.n 1
13.d odd 4 2 2704.2.f.j 2
16.e even 4 2 3328.2.b.k 2
16.f odd 4 2 3328.2.b.g 2
20.d odd 2 1 650.2.a.g 1
20.e even 4 2 650.2.b.a 2
24.f even 2 1 7488.2.a.w 1
24.h odd 2 1 7488.2.a.v 1
28.d even 2 1 1274.2.a.o 1
28.f even 6 2 1274.2.f.a 2
28.g odd 6 2 1274.2.f.l 2
36.f odd 6 2 2106.2.e.h 2
36.h even 6 2 2106.2.e.t 2
44.c even 2 1 3146.2.a.a 1
52.b odd 2 1 338.2.a.a 1
52.f even 4 2 338.2.b.a 2
52.i odd 6 2 338.2.c.g 2
52.j odd 6 2 338.2.c.c 2
52.l even 12 4 338.2.e.d 4
60.h even 2 1 5850.2.a.bn 1
60.l odd 4 2 5850.2.e.v 2
68.d odd 2 1 7514.2.a.i 1
76.d even 2 1 9386.2.a.f 1
156.h even 2 1 3042.2.a.l 1
156.l odd 4 2 3042.2.b.f 2
260.g odd 2 1 8450.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 4.b odd 2 1
208.2.a.d 1 1.a even 1 1 trivial
234.2.a.b 1 12.b even 2 1
338.2.a.a 1 52.b odd 2 1
338.2.b.a 2 52.f even 4 2
338.2.c.c 2 52.j odd 6 2
338.2.c.g 2 52.i odd 6 2
338.2.e.d 4 52.l even 12 4
650.2.a.g 1 20.d odd 2 1
650.2.b.a 2 20.e even 4 2
832.2.a.a 1 8.b even 2 1
832.2.a.j 1 8.d odd 2 1
1274.2.a.o 1 28.d even 2 1
1274.2.f.a 2 28.f even 6 2
1274.2.f.l 2 28.g odd 6 2
1872.2.a.m 1 3.b odd 2 1
2106.2.e.h 2 36.f odd 6 2
2106.2.e.t 2 36.h even 6 2
2704.2.a.n 1 13.b even 2 1
2704.2.f.j 2 13.d odd 4 2
3042.2.a.l 1 156.h even 2 1
3042.2.b.f 2 156.l odd 4 2
3146.2.a.a 1 44.c even 2 1
3328.2.b.g 2 16.f odd 4 2
3328.2.b.k 2 16.e even 4 2
5200.2.a.c 1 5.b even 2 1
5850.2.a.bn 1 60.h even 2 1
5850.2.e.v 2 60.l odd 4 2
7488.2.a.v 1 24.h odd 2 1
7488.2.a.w 1 24.f even 2 1
7514.2.a.i 1 68.d odd 2 1
8450.2.a.y 1 260.g odd 2 1
9386.2.a.f 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(208))$$:

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

Hecke characteristic polynomials

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