# Properties

 Label 208.6.f.b Level $208$ Weight $6$ Character orbit 208.f Analytic conductor $33.360$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,6,Mod(129,208)]

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, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("208.129");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 208.f (of order $$2$$, degree $$1$$, 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: $$33.3598345211$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 3i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 13 q^{3} + 17 \beta q^{5} - 35 \beta q^{7} - 74 q^{9} +O(q^{10})$$ q + 13 * q^3 + 17*b * q^5 - 35*b * q^7 - 74 * q^9 $$q + 13 q^{3} + 17 \beta q^{5} - 35 \beta q^{7} - 74 q^{9} - 40 \beta q^{11} + (39 \beta - 598) q^{13} + 221 \beta q^{15} - 1101 q^{17} + 390 \beta q^{19} - 455 \beta q^{21} - 1050 q^{23} + 524 q^{25} - 4121 q^{27} - 4104 q^{29} - 3208 \beta q^{31} - 520 \beta q^{33} + 5355 q^{35} + 2903 \beta q^{37} + (507 \beta - 7774) q^{39} - 3160 \beta q^{41} - 9995 q^{43} - 1258 \beta q^{45} + 981 \beta q^{47} + 5782 q^{49} - 14313 q^{51} - 750 q^{53} + 6120 q^{55} + 5070 \beta q^{57} + 13646 \beta q^{59} - 57920 q^{61} + 2590 \beta q^{63} + ( - 10166 \beta - 5967) q^{65} - 7604 \beta q^{67} - 13650 q^{69} - 21247 \beta q^{71} + 19622 \beta q^{73} + 6812 q^{75} - 12600 q^{77} - 63202 q^{79} - 35591 q^{81} - 18486 \beta q^{83} - 18717 \beta q^{85} - 53352 q^{87} - 34926 \beta q^{89} + (20930 \beta + 12285) q^{91} - 41704 \beta q^{93} - 59670 q^{95} + 53484 \beta q^{97} + 2960 \beta q^{99} +O(q^{100})$$ q + 13 * q^3 + 17*b * q^5 - 35*b * q^7 - 74 * q^9 - 40*b * q^11 + (39*b - 598) * q^13 + 221*b * q^15 - 1101 * q^17 + 390*b * q^19 - 455*b * q^21 - 1050 * q^23 + 524 * q^25 - 4121 * q^27 - 4104 * q^29 - 3208*b * q^31 - 520*b * q^33 + 5355 * q^35 + 2903*b * q^37 + (507*b - 7774) * q^39 - 3160*b * q^41 - 9995 * q^43 - 1258*b * q^45 + 981*b * q^47 + 5782 * q^49 - 14313 * q^51 - 750 * q^53 + 6120 * q^55 + 5070*b * q^57 + 13646*b * q^59 - 57920 * q^61 + 2590*b * q^63 + (-10166*b - 5967) * q^65 - 7604*b * q^67 - 13650 * q^69 - 21247*b * q^71 + 19622*b * q^73 + 6812 * q^75 - 12600 * q^77 - 63202 * q^79 - 35591 * q^81 - 18486*b * q^83 - 18717*b * q^85 - 53352 * q^87 - 34926*b * q^89 + (20930*b + 12285) * q^91 - 41704*b * q^93 - 59670 * q^95 + 53484*b * q^97 + 2960*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 26 q^{3} - 148 q^{9}+O(q^{10})$$ 2 * q + 26 * q^3 - 148 * q^9 $$2 q + 26 q^{3} - 148 q^{9} - 1196 q^{13} - 2202 q^{17} - 2100 q^{23} + 1048 q^{25} - 8242 q^{27} - 8208 q^{29} + 10710 q^{35} - 15548 q^{39} - 19990 q^{43} + 11564 q^{49} - 28626 q^{51} - 1500 q^{53} + 12240 q^{55} - 115840 q^{61} - 11934 q^{65} - 27300 q^{69} + 13624 q^{75} - 25200 q^{77} - 126404 q^{79} - 71182 q^{81} - 106704 q^{87} + 24570 q^{91} - 119340 q^{95}+O(q^{100})$$ 2 * q + 26 * q^3 - 148 * q^9 - 1196 * q^13 - 2202 * q^17 - 2100 * q^23 + 1048 * q^25 - 8242 * q^27 - 8208 * q^29 + 10710 * q^35 - 15548 * q^39 - 19990 * q^43 + 11564 * q^49 - 28626 * q^51 - 1500 * q^53 + 12240 * q^55 - 115840 * q^61 - 11934 * q^65 - 27300 * q^69 + 13624 * q^75 - 25200 * q^77 - 126404 * q^79 - 71182 * q^81 - 106704 * q^87 + 24570 * q^91 - 119340 * q^95

## Character values

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

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 1.00000i 1.00000i
0 13.0000 0 51.0000i 0 105.000i 0 −74.0000 0
129.2 0 13.0000 0 51.0000i 0 105.000i 0 −74.0000 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.6.f.b 2
4.b odd 2 1 26.6.b.a 2
12.b even 2 1 234.6.b.b 2
13.b even 2 1 inner 208.6.f.b 2
52.b odd 2 1 26.6.b.a 2
52.f even 4 1 338.6.a.a 1
52.f even 4 1 338.6.a.c 1
156.h even 2 1 234.6.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.b.a 2 4.b odd 2 1
26.6.b.a 2 52.b odd 2 1
208.6.f.b 2 1.a even 1 1 trivial
208.6.f.b 2 13.b even 2 1 inner
234.6.b.b 2 12.b even 2 1
234.6.b.b 2 156.h even 2 1
338.6.a.a 1 52.f even 4 1
338.6.a.c 1 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 13$$ acting on $$S_{6}^{\mathrm{new}}(208, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 13)^{2}$$
$5$ $$T^{2} + 2601$$
$7$ $$T^{2} + 11025$$
$11$ $$T^{2} + 14400$$
$13$ $$T^{2} + 1196 T + 371293$$
$17$ $$(T + 1101)^{2}$$
$19$ $$T^{2} + 1368900$$
$23$ $$(T + 1050)^{2}$$
$29$ $$(T + 4104)^{2}$$
$31$ $$T^{2} + 92621376$$
$37$ $$T^{2} + 75846681$$
$41$ $$T^{2} + 89870400$$
$43$ $$(T + 9995)^{2}$$
$47$ $$T^{2} + 8661249$$
$53$ $$(T + 750)^{2}$$
$59$ $$T^{2} + 1675919844$$
$61$ $$(T + 57920)^{2}$$
$67$ $$T^{2} + 520387344$$
$71$ $$T^{2} + 4062915081$$
$73$ $$T^{2} + 3465205956$$
$79$ $$(T + 63202)^{2}$$
$83$ $$T^{2} + 3075589764$$
$89$ $$T^{2} + 10978429284$$
$97$ $$T^{2} + 25744844304$$