# Properties

 Label 208.4.i.b Level $208$ Weight $4$ Character orbit 208.i 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.i (of order $$3$$, 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: $$x^{2} - x + 1$$ 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{3} + 17 q^{5} + 20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9}+O(q^{10})$$ q + (-2*z + 2) * q^3 + 17 * q^5 + 20*z * q^7 + 23*z * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{3} + 17 q^{5} + 20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9} + (32 \zeta_{6} - 32) q^{11} + ( - 13 \zeta_{6} - 39) q^{13} + ( - 34 \zeta_{6} + 34) q^{15} + 13 \zeta_{6} q^{17} + 30 \zeta_{6} q^{19} + 40 q^{21} + ( - 78 \zeta_{6} + 78) q^{23} + 164 q^{25} + 100 q^{27} + (197 \zeta_{6} - 197) q^{29} + 74 q^{31} + 64 \zeta_{6} q^{33} + 340 \zeta_{6} q^{35} + ( - 227 \zeta_{6} + 227) q^{37} + (78 \zeta_{6} - 104) q^{39} + ( - 165 \zeta_{6} + 165) q^{41} - 156 \zeta_{6} q^{43} + 391 \zeta_{6} q^{45} + 162 q^{47} + (57 \zeta_{6} - 57) q^{49} + 26 q^{51} + 93 q^{53} + (544 \zeta_{6} - 544) q^{55} + 60 q^{57} - 864 \zeta_{6} q^{59} - 145 \zeta_{6} q^{61} + (460 \zeta_{6} - 460) q^{63} + ( - 221 \zeta_{6} - 663) q^{65} + ( - 862 \zeta_{6} + 862) q^{67} - 156 \zeta_{6} q^{69} + 654 \zeta_{6} q^{71} + 215 q^{73} + ( - 328 \zeta_{6} + 328) q^{75} - 640 q^{77} + 76 q^{79} + (421 \zeta_{6} - 421) q^{81} - 628 q^{83} + 221 \zeta_{6} q^{85} + 394 \zeta_{6} q^{87} + ( - 266 \zeta_{6} + 266) q^{89} + ( - 1040 \zeta_{6} + 260) q^{91} + ( - 148 \zeta_{6} + 148) q^{93} + 510 \zeta_{6} q^{95} - 238 \zeta_{6} q^{97} - 736 q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^3 + 17 * q^5 + 20*z * q^7 + 23*z * q^9 + (32*z - 32) * q^11 + (-13*z - 39) * q^13 + (-34*z + 34) * q^15 + 13*z * q^17 + 30*z * q^19 + 40 * q^21 + (-78*z + 78) * q^23 + 164 * q^25 + 100 * q^27 + (197*z - 197) * q^29 + 74 * q^31 + 64*z * q^33 + 340*z * q^35 + (-227*z + 227) * q^37 + (78*z - 104) * q^39 + (-165*z + 165) * q^41 - 156*z * q^43 + 391*z * q^45 + 162 * q^47 + (57*z - 57) * q^49 + 26 * q^51 + 93 * q^53 + (544*z - 544) * q^55 + 60 * q^57 - 864*z * q^59 - 145*z * q^61 + (460*z - 460) * q^63 + (-221*z - 663) * q^65 + (-862*z + 862) * q^67 - 156*z * q^69 + 654*z * q^71 + 215 * q^73 + (-328*z + 328) * q^75 - 640 * q^77 + 76 * q^79 + (421*z - 421) * q^81 - 628 * q^83 + 221*z * q^85 + 394*z * q^87 + (-266*z + 266) * q^89 + (-1040*z + 260) * q^91 + (-148*z + 148) * q^93 + 510*z * q^95 - 238*z * q^97 - 736 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 34 q^{5} + 20 q^{7} + 23 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 34 * q^5 + 20 * q^7 + 23 * q^9 $$2 q + 2 q^{3} + 34 q^{5} + 20 q^{7} + 23 q^{9} - 32 q^{11} - 91 q^{13} + 34 q^{15} + 13 q^{17} + 30 q^{19} + 80 q^{21} + 78 q^{23} + 328 q^{25} + 200 q^{27} - 197 q^{29} + 148 q^{31} + 64 q^{33} + 340 q^{35} + 227 q^{37} - 130 q^{39} + 165 q^{41} - 156 q^{43} + 391 q^{45} + 324 q^{47} - 57 q^{49} + 52 q^{51} + 186 q^{53} - 544 q^{55} + 120 q^{57} - 864 q^{59} - 145 q^{61} - 460 q^{63} - 1547 q^{65} + 862 q^{67} - 156 q^{69} + 654 q^{71} + 430 q^{73} + 328 q^{75} - 1280 q^{77} + 152 q^{79} - 421 q^{81} - 1256 q^{83} + 221 q^{85} + 394 q^{87} + 266 q^{89} - 520 q^{91} + 148 q^{93} + 510 q^{95} - 238 q^{97} - 1472 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 34 * q^5 + 20 * q^7 + 23 * q^9 - 32 * q^11 - 91 * q^13 + 34 * q^15 + 13 * q^17 + 30 * q^19 + 80 * q^21 + 78 * q^23 + 328 * q^25 + 200 * q^27 - 197 * q^29 + 148 * q^31 + 64 * q^33 + 340 * q^35 + 227 * q^37 - 130 * q^39 + 165 * q^41 - 156 * q^43 + 391 * q^45 + 324 * q^47 - 57 * q^49 + 52 * q^51 + 186 * q^53 - 544 * q^55 + 120 * q^57 - 864 * q^59 - 145 * q^61 - 460 * q^63 - 1547 * q^65 + 862 * q^67 - 156 * q^69 + 654 * q^71 + 430 * q^73 + 328 * q^75 - 1280 * q^77 + 152 * q^79 - 421 * q^81 - 1256 * q^83 + 221 * q^85 + 394 * q^87 + 266 * q^89 - 520 * q^91 + 148 * q^93 + 510 * q^95 - 238 * q^97 - 1472 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.00000 + 1.73205i 0 17.0000 0 10.0000 17.3205i 0 11.5000 19.9186i 0
113.1 0 1.00000 1.73205i 0 17.0000 0 10.0000 + 17.3205i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.4.i.b 2
4.b odd 2 1 13.4.c.a 2
12.b even 2 1 117.4.g.c 2
13.c even 3 1 inner 208.4.i.b 2
52.b odd 2 1 169.4.c.d 2
52.f even 4 2 169.4.e.c 4
52.i odd 6 1 169.4.a.a 1
52.i odd 6 1 169.4.c.d 2
52.j odd 6 1 13.4.c.a 2
52.j odd 6 1 169.4.a.d 1
52.l even 12 2 169.4.b.c 2
52.l even 12 2 169.4.e.c 4
156.p even 6 1 117.4.g.c 2
156.p even 6 1 1521.4.a.b 1
156.r even 6 1 1521.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.a 2 4.b odd 2 1
13.4.c.a 2 52.j odd 6 1
117.4.g.c 2 12.b even 2 1
117.4.g.c 2 156.p even 6 1
169.4.a.a 1 52.i odd 6 1
169.4.a.d 1 52.j odd 6 1
169.4.b.c 2 52.l even 12 2
169.4.c.d 2 52.b odd 2 1
169.4.c.d 2 52.i odd 6 1
169.4.e.c 4 52.f even 4 2
169.4.e.c 4 52.l even 12 2
208.4.i.b 2 1.a even 1 1 trivial
208.4.i.b 2 13.c even 3 1 inner
1521.4.a.b 1 156.p even 6 1
1521.4.a.k 1 156.r even 6 1

## 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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$(T - 17)^{2}$$
$7$ $$T^{2} - 20T + 400$$
$11$ $$T^{2} + 32T + 1024$$
$13$ $$T^{2} + 91T + 2197$$
$17$ $$T^{2} - 13T + 169$$
$19$ $$T^{2} - 30T + 900$$
$23$ $$T^{2} - 78T + 6084$$
$29$ $$T^{2} + 197T + 38809$$
$31$ $$(T - 74)^{2}$$
$37$ $$T^{2} - 227T + 51529$$
$41$ $$T^{2} - 165T + 27225$$
$43$ $$T^{2} + 156T + 24336$$
$47$ $$(T - 162)^{2}$$
$53$ $$(T - 93)^{2}$$
$59$ $$T^{2} + 864T + 746496$$
$61$ $$T^{2} + 145T + 21025$$
$67$ $$T^{2} - 862T + 743044$$
$71$ $$T^{2} - 654T + 427716$$
$73$ $$(T - 215)^{2}$$
$79$ $$(T - 76)^{2}$$
$83$ $$(T + 628)^{2}$$
$89$ $$T^{2} - 266T + 70756$$
$97$ $$T^{2} + 238T + 56644$$