# Properties

 Label 208.4.i.e Level $208$ Weight $4$ Character orbit 208.i Analytic conductor $12.272$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 3 \beta_1) q^{3} + ( - 5 \beta_{3} - 10) q^{5} + ( - \beta_{3} - 7 \beta_{2} - \beta_1 + 7) q^{7} + (15 \beta_{3} + 10 \beta_{2} + 15 \beta_1 - 10) q^{9}+O(q^{10})$$ q + (b2 + 3*b1) * q^3 + (-5*b3 - 10) * q^5 + (-b3 - 7*b2 - b1 + 7) * q^7 + (15*b3 + 10*b2 + 15*b1 - 10) * q^9 $$q + (\beta_{2} + 3 \beta_1) q^{3} + ( - 5 \beta_{3} - 10) q^{5} + ( - \beta_{3} - 7 \beta_{2} - \beta_1 + 7) q^{7} + (15 \beta_{3} + 10 \beta_{2} + 15 \beta_1 - 10) q^{9} + (\beta_{2} + 15 \beta_1) q^{11} + ( - 7 \beta_{3} + 40 \beta_{2} - 5 \beta_1 + 9) q^{13} + (50 \beta_{2} - 10 \beta_1) q^{15} + ( - 16 \beta_{3} + 43 \beta_{2} - 16 \beta_1 - 43) q^{17} + ( - 5 \beta_{3} + 73 \beta_{2} - 5 \beta_1 - 73) q^{19} + ( - 25 \beta_{3} + 19) q^{21} + (89 \beta_{2} - 33 \beta_1) q^{23} + (75 \beta_{3} + 75) q^{25} + (9 \beta_{3} - 163) q^{27} + (13 \beta_{2} - 60 \beta_1) q^{29} + (100 \beta_{3} + 120) q^{31} + (63 \beta_{3} + 181 \beta_{2} + 63 \beta_1 - 181) q^{33} + ( - 30 \beta_{3} + 50 \beta_{2} - 30 \beta_1 - 50) q^{35} + (73 \beta_{2} + 44 \beta_1) q^{37} + (100 \beta_{3} + 73 \beta_{2} + 155 \beta_1 + 20) q^{39} + ( - 259 \beta_{2} - 20 \beta_1) q^{41} + ( - 97 \beta_{3} - 179 \beta_{2} - 97 \beta_1 + 179) q^{43} + ( - 25 \beta_{3} + 200 \beta_{2} - 25 \beta_1 - 200) q^{45} + ( - 140 \beta_{3} - 100) q^{47} + (290 \beta_{2} - 15 \beta_1) q^{49} + (65 \beta_{3} + 149) q^{51} + ( - 165 \beta_{3} + 190) q^{53} + (290 \beta_{2} - 70 \beta_1) q^{55} + (199 \beta_{3} - 13) q^{57} + (55 \beta_{3} + 377 \beta_{2} + 55 \beta_1 - 377) q^{59} + ( - 200 \beta_{3} + 351 \beta_{2} - 200 \beta_1 - 351) q^{61} + (130 \beta_{2} + 130 \beta_1) q^{63} + ( - 10 \beta_{3} - 500 \beta_{2} + 225 \beta_1 + 50) q^{65} + ( - 283 \beta_{2} + 91 \beta_1) q^{67} + (135 \beta_{3} - 307 \beta_{2} + 135 \beta_1 + 307) q^{69} + ( - 105 \beta_{3} - 11 \beta_{2} - 105 \beta_1 + 11) q^{71} + ( - 85 \beta_{3} + 250) q^{73} + ( - 825 \beta_{2} - 75 \beta_1) q^{75} + ( - 121 \beta_{3} + 67) q^{77} + ( - 40 \beta_{3} - 140) q^{79} + ( - \beta_{2} - 120 \beta_1) q^{81} + ( - 100 \beta_{3} - 180) q^{83} + (295 \beta_{3} - 750 \beta_{2} + 295 \beta_1 + 750) q^{85} + ( - 201 \beta_{3} - 707 \beta_{2} - 201 \beta_1 + 707) q^{87} + ( - 523 \beta_{2} + 125 \beta_1) q^{89} + ( - 65 \beta_{3} - 91 \beta_{2} - 65 \beta_1 + 351) q^{91} + ( - 1080 \beta_{2} - 40 \beta_1) q^{93} + (390 \beta_{3} - 830 \beta_{2} + 390 \beta_1 + 830) q^{95} + ( - 469 \beta_{3} + 27 \beta_{2} - 469 \beta_1 - 27) q^{97} + (390 \beta_{3} - 910) q^{99}+O(q^{100})$$ q + (b2 + 3*b1) * q^3 + (-5*b3 - 10) * q^5 + (-b3 - 7*b2 - b1 + 7) * q^7 + (15*b3 + 10*b2 + 15*b1 - 10) * q^9 + (b2 + 15*b1) * q^11 + (-7*b3 + 40*b2 - 5*b1 + 9) * q^13 + (50*b2 - 10*b1) * q^15 + (-16*b3 + 43*b2 - 16*b1 - 43) * q^17 + (-5*b3 + 73*b2 - 5*b1 - 73) * q^19 + (-25*b3 + 19) * q^21 + (89*b2 - 33*b1) * q^23 + (75*b3 + 75) * q^25 + (9*b3 - 163) * q^27 + (13*b2 - 60*b1) * q^29 + (100*b3 + 120) * q^31 + (63*b3 + 181*b2 + 63*b1 - 181) * q^33 + (-30*b3 + 50*b2 - 30*b1 - 50) * q^35 + (73*b2 + 44*b1) * q^37 + (100*b3 + 73*b2 + 155*b1 + 20) * q^39 + (-259*b2 - 20*b1) * q^41 + (-97*b3 - 179*b2 - 97*b1 + 179) * q^43 + (-25*b3 + 200*b2 - 25*b1 - 200) * q^45 + (-140*b3 - 100) * q^47 + (290*b2 - 15*b1) * q^49 + (65*b3 + 149) * q^51 + (-165*b3 + 190) * q^53 + (290*b2 - 70*b1) * q^55 + (199*b3 - 13) * q^57 + (55*b3 + 377*b2 + 55*b1 - 377) * q^59 + (-200*b3 + 351*b2 - 200*b1 - 351) * q^61 + (130*b2 + 130*b1) * q^63 + (-10*b3 - 500*b2 + 225*b1 + 50) * q^65 + (-283*b2 + 91*b1) * q^67 + (135*b3 - 307*b2 + 135*b1 + 307) * q^69 + (-105*b3 - 11*b2 - 105*b1 + 11) * q^71 + (-85*b3 + 250) * q^73 + (-825*b2 - 75*b1) * q^75 + (-121*b3 + 67) * q^77 + (-40*b3 - 140) * q^79 + (-b2 - 120*b1) * q^81 + (-100*b3 - 180) * q^83 + (295*b3 - 750*b2 + 295*b1 + 750) * q^85 + (-201*b3 - 707*b2 - 201*b1 + 707) * q^87 + (-523*b2 + 125*b1) * q^89 + (-65*b3 - 91*b2 - 65*b1 + 351) * q^91 + (-1080*b2 - 40*b1) * q^93 + (390*b3 - 830*b2 + 390*b1 + 830) * q^95 + (-469*b3 + 27*b2 - 469*b1 - 27) * q^97 + (390*b3 - 910) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9}+O(q^{10})$$ 4 * q + 5 * q^3 - 30 * q^5 + 15 * q^7 - 35 * q^9 $$4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9} + 17 q^{11} + 125 q^{13} + 90 q^{15} - 70 q^{17} - 141 q^{19} + 126 q^{21} + 145 q^{23} + 150 q^{25} - 670 q^{27} - 34 q^{29} + 280 q^{31} - 425 q^{33} - 70 q^{35} + 190 q^{37} + 181 q^{39} - 538 q^{41} + 455 q^{43} - 375 q^{45} - 120 q^{47} + 565 q^{49} + 466 q^{51} + 1090 q^{53} + 510 q^{55} - 450 q^{57} - 809 q^{59} - 502 q^{61} + 390 q^{63} - 555 q^{65} - 475 q^{67} + 479 q^{69} + 127 q^{71} + 1170 q^{73} - 1725 q^{75} + 510 q^{77} - 480 q^{79} - 122 q^{81} - 520 q^{83} + 1205 q^{85} + 1615 q^{87} - 921 q^{89} + 1287 q^{91} - 2200 q^{93} + 1270 q^{95} + 415 q^{97} - 4420 q^{99}+O(q^{100})$$ 4 * q + 5 * q^3 - 30 * q^5 + 15 * q^7 - 35 * q^9 + 17 * q^11 + 125 * q^13 + 90 * q^15 - 70 * q^17 - 141 * q^19 + 126 * q^21 + 145 * q^23 + 150 * q^25 - 670 * q^27 - 34 * q^29 + 280 * q^31 - 425 * q^33 - 70 * q^35 + 190 * q^37 + 181 * q^39 - 538 * q^41 + 455 * q^43 - 375 * q^45 - 120 * q^47 + 565 * q^49 + 466 * q^51 + 1090 * q^53 + 510 * q^55 - 450 * q^57 - 809 * q^59 - 502 * q^61 + 390 * q^63 - 555 * q^65 - 475 * q^67 + 479 * q^69 + 127 * q^71 + 1170 * q^73 - 1725 * q^75 + 510 * q^77 - 480 * q^79 - 122 * q^81 - 520 * q^83 + 1205 * q^85 + 1615 * q^87 - 921 * q^89 + 1287 * q^91 - 2200 * q^93 + 1270 * q^95 + 415 * q^97 - 4420 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

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

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \beta_{2}$$

## 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
 −0.780776 − 1.35234i 1.28078 + 2.21837i −0.780776 + 1.35234i 1.28078 − 2.21837i
0 −1.84233 3.19101i 0 −17.8078 0 2.71922 4.70983i 0 6.71165 11.6249i 0
81.2 0 4.34233 + 7.52113i 0 2.80776 0 4.78078 8.28055i 0 −24.2116 + 41.9358i 0
113.1 0 −1.84233 + 3.19101i 0 −17.8078 0 2.71922 + 4.70983i 0 6.71165 + 11.6249i 0
113.2 0 4.34233 7.52113i 0 2.80776 0 4.78078 + 8.28055i 0 −24.2116 41.9358i 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.e 4
4.b odd 2 1 13.4.c.b 4
12.b even 2 1 117.4.g.d 4
13.c even 3 1 inner 208.4.i.e 4
52.b odd 2 1 169.4.c.f 4
52.f even 4 2 169.4.e.g 8
52.i odd 6 1 169.4.a.j 2
52.i odd 6 1 169.4.c.f 4
52.j odd 6 1 13.4.c.b 4
52.j odd 6 1 169.4.a.f 2
52.l even 12 2 169.4.b.e 4
52.l even 12 2 169.4.e.g 8
156.p even 6 1 117.4.g.d 4
156.p even 6 1 1521.4.a.t 2
156.r even 6 1 1521.4.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 4.b odd 2 1
13.4.c.b 4 52.j odd 6 1
117.4.g.d 4 12.b even 2 1
117.4.g.d 4 156.p even 6 1
169.4.a.f 2 52.j odd 6 1
169.4.a.j 2 52.i odd 6 1
169.4.b.e 4 52.l even 12 2
169.4.c.f 4 52.b odd 2 1
169.4.c.f 4 52.i odd 6 1
169.4.e.g 8 52.f even 4 2
169.4.e.g 8 52.l even 12 2
208.4.i.e 4 1.a even 1 1 trivial
208.4.i.e 4 13.c even 3 1 inner
1521.4.a.l 2 156.r even 6 1
1521.4.a.t 2 156.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 5T_{3}^{3} + 57T_{3}^{2} + 160T_{3} + 1024$$ acting on $$S_{4}^{\mathrm{new}}(208, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 5 T^{3} + 57 T^{2} + \cdots + 1024$$
$5$ $$(T^{2} + 15 T - 50)^{2}$$
$7$ $$T^{4} - 15 T^{3} + 173 T^{2} + \cdots + 2704$$
$11$ $$T^{4} - 17 T^{3} + 1173 T^{2} + \cdots + 781456$$
$13$ $$T^{4} - 125 T^{3} + 7956 T^{2} + \cdots + 4826809$$
$17$ $$T^{4} + 70 T^{3} + 4763 T^{2} + \cdots + 18769$$
$19$ $$T^{4} + 141 T^{3} + \cdots + 23658496$$
$23$ $$T^{4} - 145 T^{3} + 20397 T^{2} + \cdots + 394384$$
$29$ $$T^{4} + 34 T^{3} + \cdots + 225330121$$
$31$ $$(T^{2} - 140 T - 37600)^{2}$$
$37$ $$T^{4} - 190 T^{3} + 35303 T^{2} + \cdots + 635209$$
$41$ $$T^{4} + 538 T^{3} + \cdots + 4992976921$$
$43$ $$T^{4} - 455 T^{3} + \cdots + 138485824$$
$47$ $$(T^{2} + 60 T - 82400)^{2}$$
$53$ $$(T^{2} - 545 T - 41450)^{2}$$
$59$ $$T^{4} + 809 T^{3} + \cdots + 22729783696$$
$61$ $$T^{4} + 502 T^{3} + \cdots + 11448786001$$
$67$ $$T^{4} + 475 T^{3} + \cdots + 449948944$$
$71$ $$T^{4} - 127 T^{3} + \cdots + 1833894976$$
$73$ $$(T^{2} - 585 T + 54850)^{2}$$
$79$ $$(T^{2} + 240 T + 7600)^{2}$$
$83$ $$(T^{2} + 260 T - 25600)^{2}$$
$89$ $$T^{4} + 921 T^{3} + \cdots + 21215087716$$
$97$ $$T^{4} - 415 T^{3} + \cdots + 795268001284$$