# Properties

 Label 208.4.w.b Level $208$ Weight $4$ Character orbit 208.w Analytic conductor $12.272$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,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, 4, names="a")

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

chi := DirichletCharacter("208.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 1) q^{5} + ( - 8 \zeta_{6} + 16) q^{7} + 23 \zeta_{6} q^{9}+O(q^{10})$$ q + (-2*z + 2) * q^3 + (-2*z + 1) * q^5 + (-8*z + 16) * q^7 + 23*z * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 1) q^{5} + ( - 8 \zeta_{6} + 16) q^{7} + 23 \zeta_{6} q^{9} + ( - 8 \zeta_{6} - 8) q^{11} + (13 \zeta_{6} + 39) q^{13} + ( - 2 \zeta_{6} - 2) q^{15} - 117 \zeta_{6} q^{17} + ( - 66 \zeta_{6} + 132) q^{19} + ( - 32 \zeta_{6} + 16) q^{21} + (78 \zeta_{6} - 78) q^{23} + 122 q^{25} + 100 q^{27} + ( - 141 \zeta_{6} + 141) q^{29} + ( - 180 \zeta_{6} + 90) q^{31} + (16 \zeta_{6} - 32) q^{33} - 24 \zeta_{6} q^{35} + ( - 83 \zeta_{6} - 83) q^{37} + ( - 78 \zeta_{6} + 104) q^{39} + (157 \zeta_{6} + 157) q^{41} + 104 \zeta_{6} q^{43} + ( - 23 \zeta_{6} + 46) q^{45} + ( - 348 \zeta_{6} + 174) q^{47} + (151 \zeta_{6} - 151) q^{49} - 234 q^{51} + 93 q^{53} + (24 \zeta_{6} - 24) q^{55} + ( - 264 \zeta_{6} + 132) q^{57} + ( - 164 \zeta_{6} + 328) q^{59} - 145 \zeta_{6} q^{61} + (184 \zeta_{6} + 184) q^{63} + ( - 91 \zeta_{6} + 65) q^{65} + (454 \zeta_{6} + 454) q^{67} + 156 \zeta_{6} q^{69} + (610 \zeta_{6} - 1220) q^{71} + (530 \zeta_{6} - 265) q^{73} + ( - 244 \zeta_{6} + 244) q^{75} - 192 q^{77} - 1276 q^{79} + (421 \zeta_{6} - 421) q^{81} + (912 \zeta_{6} - 456) q^{83} + (117 \zeta_{6} - 234) q^{85} - 282 \zeta_{6} q^{87} + ( - 564 \zeta_{6} - 564) q^{89} + ( - 208 \zeta_{6} + 728) q^{91} + ( - 180 \zeta_{6} - 180) q^{93} - 198 \zeta_{6} q^{95} + ( - 116 \zeta_{6} + 232) q^{97} + ( - 368 \zeta_{6} + 184) q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^3 + (-2*z + 1) * q^5 + (-8*z + 16) * q^7 + 23*z * q^9 + (-8*z - 8) * q^11 + (13*z + 39) * q^13 + (-2*z - 2) * q^15 - 117*z * q^17 + (-66*z + 132) * q^19 + (-32*z + 16) * q^21 + (78*z - 78) * q^23 + 122 * q^25 + 100 * q^27 + (-141*z + 141) * q^29 + (-180*z + 90) * q^31 + (16*z - 32) * q^33 - 24*z * q^35 + (-83*z - 83) * q^37 + (-78*z + 104) * q^39 + (157*z + 157) * q^41 + 104*z * q^43 + (-23*z + 46) * q^45 + (-348*z + 174) * q^47 + (151*z - 151) * q^49 - 234 * q^51 + 93 * q^53 + (24*z - 24) * q^55 + (-264*z + 132) * q^57 + (-164*z + 328) * q^59 - 145*z * q^61 + (184*z + 184) * q^63 + (-91*z + 65) * q^65 + (454*z + 454) * q^67 + 156*z * q^69 + (610*z - 1220) * q^71 + (530*z - 265) * q^73 + (-244*z + 244) * q^75 - 192 * q^77 - 1276 * q^79 + (421*z - 421) * q^81 + (912*z - 456) * q^83 + (117*z - 234) * q^85 - 282*z * q^87 + (-564*z - 564) * q^89 + (-208*z + 728) * q^91 + (-180*z - 180) * q^93 - 198*z * q^95 + (-116*z + 232) * q^97 + (-368*z + 184) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 24 q^{7} + 23 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 24 * q^7 + 23 * q^9 $$2 q + 2 q^{3} + 24 q^{7} + 23 q^{9} - 24 q^{11} + 91 q^{13} - 6 q^{15} - 117 q^{17} + 198 q^{19} - 78 q^{23} + 244 q^{25} + 200 q^{27} + 141 q^{29} - 48 q^{33} - 24 q^{35} - 249 q^{37} + 130 q^{39} + 471 q^{41} + 104 q^{43} + 69 q^{45} - 151 q^{49} - 468 q^{51} + 186 q^{53} - 24 q^{55} + 492 q^{59} - 145 q^{61} + 552 q^{63} + 39 q^{65} + 1362 q^{67} + 156 q^{69} - 1830 q^{71} + 244 q^{75} - 384 q^{77} - 2552 q^{79} - 421 q^{81} - 351 q^{85} - 282 q^{87} - 1692 q^{89} + 1248 q^{91} - 540 q^{93} - 198 q^{95} + 348 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 24 * q^7 + 23 * q^9 - 24 * q^11 + 91 * q^13 - 6 * q^15 - 117 * q^17 + 198 * q^19 - 78 * q^23 + 244 * q^25 + 200 * q^27 + 141 * q^29 - 48 * q^33 - 24 * q^35 - 249 * q^37 + 130 * q^39 + 471 * q^41 + 104 * q^43 + 69 * q^45 - 151 * q^49 - 468 * q^51 + 186 * q^53 - 24 * q^55 + 492 * q^59 - 145 * q^61 + 552 * q^63 + 39 * q^65 + 1362 * q^67 + 156 * q^69 - 1830 * q^71 + 244 * q^75 - 384 * q^77 - 2552 * q^79 - 421 * q^81 - 351 * q^85 - 282 * q^87 - 1692 * q^89 + 1248 * q^91 - 540 * q^93 - 198 * q^95 + 348 * 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 12.0000 6.92820i 0 11.5000 + 19.9186i 0
49.1 0 1.00000 + 1.73205i 0 1.73205i 0 12.0000 + 6.92820i 0 11.5000 19.9186i 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 208.4.w.b 2
4.b odd 2 1 13.4.e.b 2
12.b even 2 1 117.4.q.a 2
13.e even 6 1 inner 208.4.w.b 2
52.b odd 2 1 169.4.e.a 2
52.f even 4 2 169.4.c.h 4
52.i odd 6 1 13.4.e.b 2
52.i odd 6 1 169.4.b.d 2
52.j odd 6 1 169.4.b.d 2
52.j odd 6 1 169.4.e.a 2
52.l even 12 2 169.4.a.i 2
52.l even 12 2 169.4.c.h 4
156.r even 6 1 117.4.q.a 2
156.v odd 12 2 1521.4.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 4.b odd 2 1
13.4.e.b 2 52.i odd 6 1
117.4.q.a 2 12.b even 2 1
117.4.q.a 2 156.r even 6 1
169.4.a.i 2 52.l even 12 2
169.4.b.d 2 52.i odd 6 1
169.4.b.d 2 52.j odd 6 1
169.4.c.h 4 52.f even 4 2
169.4.c.h 4 52.l even 12 2
169.4.e.a 2 52.b odd 2 1
169.4.e.a 2 52.j odd 6 1
208.4.w.b 2 1.a even 1 1 trivial
208.4.w.b 2 13.e even 6 1 inner
1521.4.a.o 2 156.v 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 $$S_{4}^{\mathrm{new}}(208, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2} - 24T + 192$$
$11$ $$T^{2} + 24T + 192$$
$13$ $$T^{2} - 91T + 2197$$
$17$ $$T^{2} + 117T + 13689$$
$19$ $$T^{2} - 198T + 13068$$
$23$ $$T^{2} + 78T + 6084$$
$29$ $$T^{2} - 141T + 19881$$
$31$ $$T^{2} + 24300$$
$37$ $$T^{2} + 249T + 20667$$
$41$ $$T^{2} - 471T + 73947$$
$43$ $$T^{2} - 104T + 10816$$
$47$ $$T^{2} + 90828$$
$53$ $$(T - 93)^{2}$$
$59$ $$T^{2} - 492T + 80688$$
$61$ $$T^{2} + 145T + 21025$$
$67$ $$T^{2} - 1362 T + 618348$$
$71$ $$T^{2} + 1830 T + 1116300$$
$73$ $$T^{2} + 210675$$
$79$ $$(T + 1276)^{2}$$
$83$ $$T^{2} + 623808$$
$89$ $$T^{2} + 1692 T + 954288$$
$97$ $$T^{2} - 348T + 40368$$