# Properties

 Label 208.2.w.b Level $208$ Weight $2$ Character orbit 208.w Analytic conductor $1.661$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,2,Mod(17,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("208.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 208.w (of order $$6$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$1.66088836204$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 1) q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^3 + (-2*z + 1) * q^5 - z * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 1) q^{5} - \zeta_{6} q^{9} + ( - 3 \zeta_{6} - 1) q^{13} + ( - 2 \zeta_{6} - 2) q^{15} + 3 \zeta_{6} q^{17} + ( - 2 \zeta_{6} + 4) q^{19} + (6 \zeta_{6} - 6) q^{23} + 2 q^{25} + 4 q^{27} + (3 \zeta_{6} - 3) q^{29} + ( - 4 \zeta_{6} + 2) q^{31} + (5 \zeta_{6} + 5) q^{37} + (2 \zeta_{6} - 8) q^{39} + ( - 3 \zeta_{6} - 3) q^{41} + 8 \zeta_{6} q^{43} + (\zeta_{6} - 2) q^{45} + (4 \zeta_{6} - 2) q^{47} + (7 \zeta_{6} - 7) q^{49} + 6 q^{51} - 3 q^{53} + ( - 8 \zeta_{6} + 4) q^{57} + (4 \zeta_{6} - 8) q^{59} - \zeta_{6} q^{61} + (5 \zeta_{6} - 7) q^{65} + ( - 2 \zeta_{6} - 2) q^{67} + 12 \zeta_{6} q^{69} + (2 \zeta_{6} - 4) q^{71} + (2 \zeta_{6} - 1) q^{73} + ( - 4 \zeta_{6} + 4) q^{75} - 4 q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + ( - 16 \zeta_{6} + 8) q^{83} + ( - 3 \zeta_{6} + 6) q^{85} + 6 \zeta_{6} q^{87} + ( - 4 \zeta_{6} - 4) q^{89} + ( - 4 \zeta_{6} - 4) q^{93} - 6 \zeta_{6} q^{95} + ( - 4 \zeta_{6} + 8) q^{97} +O(q^{100})$$ q + (-2*z + 2) * q^3 + (-2*z + 1) * q^5 - z * q^9 + (-3*z - 1) * q^13 + (-2*z - 2) * q^15 + 3*z * q^17 + (-2*z + 4) * q^19 + (6*z - 6) * q^23 + 2 * q^25 + 4 * q^27 + (3*z - 3) * q^29 + (-4*z + 2) * q^31 + (5*z + 5) * q^37 + (2*z - 8) * q^39 + (-3*z - 3) * q^41 + 8*z * q^43 + (z - 2) * q^45 + (4*z - 2) * q^47 + (7*z - 7) * q^49 + 6 * q^51 - 3 * q^53 + (-8*z + 4) * q^57 + (4*z - 8) * q^59 - z * q^61 + (5*z - 7) * q^65 + (-2*z - 2) * q^67 + 12*z * q^69 + (2*z - 4) * q^71 + (2*z - 1) * q^73 + (-4*z + 4) * q^75 - 4 * q^79 + (-11*z + 11) * q^81 + (-16*z + 8) * q^83 + (-3*z + 6) * q^85 + 6*z * q^87 + (-4*z - 4) * q^89 + (-4*z - 4) * q^93 - 6*z * q^95 + (-4*z + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - q^9 $$2 q + 2 q^{3} - q^{9} - 5 q^{13} - 6 q^{15} + 3 q^{17} + 6 q^{19} - 6 q^{23} + 4 q^{25} + 8 q^{27} - 3 q^{29} + 15 q^{37} - 14 q^{39} - 9 q^{41} + 8 q^{43} - 3 q^{45} - 7 q^{49} + 12 q^{51} - 6 q^{53} - 12 q^{59} - q^{61} - 9 q^{65} - 6 q^{67} + 12 q^{69} - 6 q^{71} + 4 q^{75} - 8 q^{79} + 11 q^{81} + 9 q^{85} + 6 q^{87} - 12 q^{89} - 12 q^{93} - 6 q^{95} + 12 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 - q^9 - 5 * q^13 - 6 * q^15 + 3 * q^17 + 6 * q^19 - 6 * q^23 + 4 * q^25 + 8 * q^27 - 3 * q^29 + 15 * q^37 - 14 * q^39 - 9 * q^41 + 8 * q^43 - 3 * q^45 - 7 * q^49 + 12 * q^51 - 6 * q^53 - 12 * q^59 - q^61 - 9 * q^65 - 6 * q^67 + 12 * q^69 - 6 * q^71 + 4 * q^75 - 8 * q^79 + 11 * q^81 + 9 * q^85 + 6 * q^87 - 12 * q^89 - 12 * q^93 - 6 * q^95 + 12 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.00000 1.73205i 0 1.73205i 0 0 0 −0.500000 0.866025i 0
49.1 0 1.00000 + 1.73205i 0 1.73205i 0 0 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.2.w.b 2
3.b odd 2 1 1872.2.by.d 2
4.b odd 2 1 13.2.e.a 2
8.b even 2 1 832.2.w.a 2
8.d odd 2 1 832.2.w.d 2
12.b even 2 1 117.2.q.c 2
13.c even 3 1 2704.2.f.b 2
13.e even 6 1 inner 208.2.w.b 2
13.e even 6 1 2704.2.f.b 2
13.f odd 12 2 2704.2.a.o 2
20.d odd 2 1 325.2.n.a 2
20.e even 4 2 325.2.m.a 4
28.d even 2 1 637.2.q.a 2
28.f even 6 1 637.2.k.c 2
28.f even 6 1 637.2.u.b 2
28.g odd 6 1 637.2.k.a 2
28.g odd 6 1 637.2.u.c 2
39.h odd 6 1 1872.2.by.d 2
52.b odd 2 1 169.2.e.a 2
52.f even 4 2 169.2.c.a 4
52.i odd 6 1 13.2.e.a 2
52.i odd 6 1 169.2.b.a 2
52.j odd 6 1 169.2.b.a 2
52.j odd 6 1 169.2.e.a 2
52.l even 12 2 169.2.a.a 2
52.l even 12 2 169.2.c.a 4
104.p odd 6 1 832.2.w.d 2
104.s even 6 1 832.2.w.a 2
156.p even 6 1 1521.2.b.a 2
156.r even 6 1 117.2.q.c 2
156.r even 6 1 1521.2.b.a 2
156.v odd 12 2 1521.2.a.k 2
260.w odd 6 1 325.2.n.a 2
260.bc even 12 2 4225.2.a.v 2
260.bg even 12 2 325.2.m.a 4
364.s odd 6 1 637.2.k.a 2
364.w even 6 1 637.2.u.b 2
364.bc even 6 1 637.2.q.a 2
364.bk odd 6 1 637.2.u.c 2
364.bp even 6 1 637.2.k.c 2
364.bv odd 12 2 8281.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 4.b odd 2 1
13.2.e.a 2 52.i odd 6 1
117.2.q.c 2 12.b even 2 1
117.2.q.c 2 156.r even 6 1
169.2.a.a 2 52.l even 12 2
169.2.b.a 2 52.i odd 6 1
169.2.b.a 2 52.j odd 6 1
169.2.c.a 4 52.f even 4 2
169.2.c.a 4 52.l even 12 2
169.2.e.a 2 52.b odd 2 1
169.2.e.a 2 52.j odd 6 1
208.2.w.b 2 1.a even 1 1 trivial
208.2.w.b 2 13.e even 6 1 inner
325.2.m.a 4 20.e even 4 2
325.2.m.a 4 260.bg even 12 2
325.2.n.a 2 20.d odd 2 1
325.2.n.a 2 260.w odd 6 1
637.2.k.a 2 28.g odd 6 1
637.2.k.a 2 364.s odd 6 1
637.2.k.c 2 28.f even 6 1
637.2.k.c 2 364.bp even 6 1
637.2.q.a 2 28.d even 2 1
637.2.q.a 2 364.bc even 6 1
637.2.u.b 2 28.f even 6 1
637.2.u.b 2 364.w even 6 1
637.2.u.c 2 28.g odd 6 1
637.2.u.c 2 364.bk odd 6 1
832.2.w.a 2 8.b even 2 1
832.2.w.a 2 104.s even 6 1
832.2.w.d 2 8.d odd 2 1
832.2.w.d 2 104.p odd 6 1
1521.2.a.k 2 156.v odd 12 2
1521.2.b.a 2 156.p even 6 1
1521.2.b.a 2 156.r even 6 1
1872.2.by.d 2 3.b odd 2 1
1872.2.by.d 2 39.h odd 6 1
2704.2.a.o 2 13.f odd 12 2
2704.2.f.b 2 13.c even 3 1
2704.2.f.b 2 13.e even 6 1
4225.2.a.v 2 260.bc even 12 2
8281.2.a.q 2 364.bv odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(208, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 5T + 13$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} - 6T + 12$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} - 15T + 75$$
$41$ $$T^{2} + 9T + 27$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2} + 12T + 48$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 6T + 12$$
$71$ $$T^{2} + 6T + 12$$
$73$ $$T^{2} + 3$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} + 12T + 48$$
$97$ $$T^{2} - 12T + 48$$