# 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

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$12.2723972812$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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

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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} + ( 1 - 2 \zeta_{6} ) q^{5} + ( 16 - 8 \zeta_{6} ) q^{7} + 23 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} + ( 1 - 2 \zeta_{6} ) q^{5} + ( 16 - 8 \zeta_{6} ) q^{7} + 23 \zeta_{6} q^{9} + ( -8 - 8 \zeta_{6} ) q^{11} + ( 39 + 13 \zeta_{6} ) q^{13} + ( -2 - 2 \zeta_{6} ) q^{15} -117 \zeta_{6} q^{17} + ( 132 - 66 \zeta_{6} ) q^{19} + ( 16 - 32 \zeta_{6} ) q^{21} + ( -78 + 78 \zeta_{6} ) q^{23} + 122 q^{25} + 100 q^{27} + ( 141 - 141 \zeta_{6} ) q^{29} + ( 90 - 180 \zeta_{6} ) q^{31} + ( -32 + 16 \zeta_{6} ) q^{33} -24 \zeta_{6} q^{35} + ( -83 - 83 \zeta_{6} ) q^{37} + ( 104 - 78 \zeta_{6} ) q^{39} + ( 157 + 157 \zeta_{6} ) q^{41} + 104 \zeta_{6} q^{43} + ( 46 - 23 \zeta_{6} ) q^{45} + ( 174 - 348 \zeta_{6} ) q^{47} + ( -151 + 151 \zeta_{6} ) q^{49} -234 q^{51} + 93 q^{53} + ( -24 + 24 \zeta_{6} ) q^{55} + ( 132 - 264 \zeta_{6} ) q^{57} + ( 328 - 164 \zeta_{6} ) q^{59} -145 \zeta_{6} q^{61} + ( 184 + 184 \zeta_{6} ) q^{63} + ( 65 - 91 \zeta_{6} ) q^{65} + ( 454 + 454 \zeta_{6} ) q^{67} + 156 \zeta_{6} q^{69} + ( -1220 + 610 \zeta_{6} ) q^{71} + ( -265 + 530 \zeta_{6} ) q^{73} + ( 244 - 244 \zeta_{6} ) q^{75} -192 q^{77} -1276 q^{79} + ( -421 + 421 \zeta_{6} ) q^{81} + ( -456 + 912 \zeta_{6} ) q^{83} + ( -234 + 117 \zeta_{6} ) q^{85} -282 \zeta_{6} q^{87} + ( -564 - 564 \zeta_{6} ) q^{89} + ( 728 - 208 \zeta_{6} ) q^{91} + ( -180 - 180 \zeta_{6} ) q^{93} -198 \zeta_{6} q^{95} + ( 232 - 116 \zeta_{6} ) q^{97} + ( 184 - 368 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 24q^{7} + 23q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 24q^{7} + 23q^{9} - 24q^{11} + 91q^{13} - 6q^{15} - 117q^{17} + 198q^{19} - 78q^{23} + 244q^{25} + 200q^{27} + 141q^{29} - 48q^{33} - 24q^{35} - 249q^{37} + 130q^{39} + 471q^{41} + 104q^{43} + 69q^{45} - 151q^{49} - 468q^{51} + 186q^{53} - 24q^{55} + 492q^{59} - 145q^{61} + 552q^{63} + 39q^{65} + 1362q^{67} + 156q^{69} - 1830q^{71} + 244q^{75} - 384q^{77} - 2552q^{79} - 421q^{81} - 351q^{85} - 282q^{87} - 1692q^{89} + 1248q^{91} - 540q^{93} - 198q^{95} + 348q^{97} + O(q^{100})$$

## 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.

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 $$T_{3}^{2} - 2 T_{3} + 4$$ acting on $$S_{4}^{\mathrm{new}}(208, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T - 23 T^{2} - 54 T^{3} + 729 T^{4}$$
$5$ $$1 - 247 T^{2} + 15625 T^{4}$$
$7$ $$1 - 24 T + 535 T^{2} - 8232 T^{3} + 117649 T^{4}$$
$11$ $$1 + 24 T + 1523 T^{2} + 31944 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 91 T + 2197 T^{2}$$
$17$ $$1 + 117 T + 8776 T^{2} + 574821 T^{3} + 24137569 T^{4}$$
$19$ $$1 - 198 T + 19927 T^{2} - 1358082 T^{3} + 47045881 T^{4}$$
$23$ $$1 + 78 T - 6083 T^{2} + 949026 T^{3} + 148035889 T^{4}$$
$29$ $$1 - 141 T - 4508 T^{2} - 3438849 T^{3} + 594823321 T^{4}$$
$31$ $$( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} )$$
$37$ $$1 + 249 T + 71320 T^{2} + 12612597 T^{3} + 2565726409 T^{4}$$
$41$ $$1 - 471 T + 142868 T^{2} - 32461791 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 104 T - 68691 T^{2} - 8268728 T^{3} + 6321363049 T^{4}$$
$47$ $$1 - 116818 T^{2} + 10779215329 T^{4}$$
$53$ $$( 1 - 93 T + 148877 T^{2} )^{2}$$
$59$ $$1 - 492 T + 286067 T^{2} - 101046468 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4}$$
$67$ $$1 - 1362 T + 919111 T^{2} - 409639206 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 1830 T + 1474211 T^{2} + 654977130 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 567359 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 + 1276 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 519766 T^{2} + 326940373369 T^{4}$$
$89$ $$1 + 1692 T + 1659257 T^{2} + 1192807548 T^{3} + 496981290961 T^{4}$$
$97$ $$1 - 348 T + 953041 T^{2} - 317610204 T^{3} + 832972004929 T^{4}$$