# Properties

 Label 208.4.w.a 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: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (7 \zeta_{6} - 7) q^{3} + (16 \zeta_{6} - 8) q^{5} + (13 \zeta_{6} - 26) q^{7} - 22 \zeta_{6} q^{9} +O(q^{10})$$ q + (7*z - 7) * q^3 + (16*z - 8) * q^5 + (13*z - 26) * q^7 - 22*z * q^9 $$q + (7 \zeta_{6} - 7) q^{3} + (16 \zeta_{6} - 8) q^{5} + (13 \zeta_{6} - 26) q^{7} - 22 \zeta_{6} q^{9} + (13 \zeta_{6} + 13) q^{11} + (52 \zeta_{6} - 39) q^{13} + ( - 56 \zeta_{6} - 56) q^{15} - 27 \zeta_{6} q^{17} + ( - 51 \zeta_{6} + 102) q^{19} + ( - 182 \zeta_{6} + 91) q^{21} + ( - 57 \zeta_{6} + 57) q^{23} - 67 q^{25} - 35 q^{27} + ( - 69 \zeta_{6} + 69) q^{29} + (84 \zeta_{6} - 42) q^{31} + (91 \zeta_{6} - 182) q^{33} - 312 \zeta_{6} q^{35} + ( - 23 \zeta_{6} - 23) q^{37} + ( - 273 \zeta_{6} - 91) q^{39} + ( - 227 \zeta_{6} - 227) q^{41} - 85 \zeta_{6} q^{43} + ( - 176 \zeta_{6} + 352) q^{45} + (396 \zeta_{6} - 198) q^{47} + ( - 164 \zeta_{6} + 164) q^{49} + 189 q^{51} + 426 q^{53} + (312 \zeta_{6} - 312) q^{55} + (714 \zeta_{6} - 357) q^{57} + ( - 11 \zeta_{6} + 22) q^{59} + 17 \zeta_{6} q^{61} + (286 \zeta_{6} + 286) q^{63} + ( - 208 \zeta_{6} - 520) q^{65} + ( - 95 \zeta_{6} - 95) q^{67} + 399 \zeta_{6} q^{69} + (337 \zeta_{6} - 674) q^{71} + (1160 \zeta_{6} - 580) q^{73} + ( - 469 \zeta_{6} + 469) q^{75} - 507 q^{77} + 1244 q^{79} + ( - 839 \zeta_{6} + 839) q^{81} + (492 \zeta_{6} - 246) q^{83} + ( - 216 \zeta_{6} + 432) q^{85} + 483 \zeta_{6} q^{87} + (177 \zeta_{6} + 177) q^{89} + ( - 1183 \zeta_{6} + 338) q^{91} + ( - 294 \zeta_{6} - 294) q^{93} + 1224 \zeta_{6} q^{95} + ( - 713 \zeta_{6} + 1426) q^{97} + ( - 572 \zeta_{6} + 286) q^{99} +O(q^{100})$$ q + (7*z - 7) * q^3 + (16*z - 8) * q^5 + (13*z - 26) * q^7 - 22*z * q^9 + (13*z + 13) * q^11 + (52*z - 39) * q^13 + (-56*z - 56) * q^15 - 27*z * q^17 + (-51*z + 102) * q^19 + (-182*z + 91) * q^21 + (-57*z + 57) * q^23 - 67 * q^25 - 35 * q^27 + (-69*z + 69) * q^29 + (84*z - 42) * q^31 + (91*z - 182) * q^33 - 312*z * q^35 + (-23*z - 23) * q^37 + (-273*z - 91) * q^39 + (-227*z - 227) * q^41 - 85*z * q^43 + (-176*z + 352) * q^45 + (396*z - 198) * q^47 + (-164*z + 164) * q^49 + 189 * q^51 + 426 * q^53 + (312*z - 312) * q^55 + (714*z - 357) * q^57 + (-11*z + 22) * q^59 + 17*z * q^61 + (286*z + 286) * q^63 + (-208*z - 520) * q^65 + (-95*z - 95) * q^67 + 399*z * q^69 + (337*z - 674) * q^71 + (1160*z - 580) * q^73 + (-469*z + 469) * q^75 - 507 * q^77 + 1244 * q^79 + (-839*z + 839) * q^81 + (492*z - 246) * q^83 + (-216*z + 432) * q^85 + 483*z * q^87 + (177*z + 177) * q^89 + (-1183*z + 338) * q^91 + (-294*z - 294) * q^93 + 1224*z * q^95 + (-713*z + 1426) * q^97 + (-572*z + 286) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 7 q^{3} - 39 q^{7} - 22 q^{9}+O(q^{10})$$ 2 * q - 7 * q^3 - 39 * q^7 - 22 * q^9 $$2 q - 7 q^{3} - 39 q^{7} - 22 q^{9} + 39 q^{11} - 26 q^{13} - 168 q^{15} - 27 q^{17} + 153 q^{19} + 57 q^{23} - 134 q^{25} - 70 q^{27} + 69 q^{29} - 273 q^{33} - 312 q^{35} - 69 q^{37} - 455 q^{39} - 681 q^{41} - 85 q^{43} + 528 q^{45} + 164 q^{49} + 378 q^{51} + 852 q^{53} - 312 q^{55} + 33 q^{59} + 17 q^{61} + 858 q^{63} - 1248 q^{65} - 285 q^{67} + 399 q^{69} - 1011 q^{71} + 469 q^{75} - 1014 q^{77} + 2488 q^{79} + 839 q^{81} + 648 q^{85} + 483 q^{87} + 531 q^{89} - 507 q^{91} - 882 q^{93} + 1224 q^{95} + 2139 q^{97}+O(q^{100})$$ 2 * q - 7 * q^3 - 39 * q^7 - 22 * q^9 + 39 * q^11 - 26 * q^13 - 168 * q^15 - 27 * q^17 + 153 * q^19 + 57 * q^23 - 134 * q^25 - 70 * q^27 + 69 * q^29 - 273 * q^33 - 312 * q^35 - 69 * q^37 - 455 * q^39 - 681 * q^41 - 85 * q^43 + 528 * q^45 + 164 * q^49 + 378 * q^51 + 852 * q^53 - 312 * q^55 + 33 * q^59 + 17 * q^61 + 858 * q^63 - 1248 * q^65 - 285 * q^67 + 399 * q^69 - 1011 * q^71 + 469 * q^75 - 1014 * q^77 + 2488 * q^79 + 839 * q^81 + 648 * q^85 + 483 * q^87 + 531 * q^89 - 507 * q^91 - 882 * q^93 + 1224 * q^95 + 2139 * 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.

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 −3.50000 + 6.06218i 0 13.8564i 0 −19.5000 + 11.2583i 0 −11.0000 19.0526i 0
49.1 0 −3.50000 6.06218i 0 13.8564i 0 −19.5000 11.2583i 0 −11.0000 + 19.0526i 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.a 2
4.b odd 2 1 13.4.e.a 2
12.b even 2 1 117.4.q.c 2
13.e even 6 1 inner 208.4.w.a 2
52.b odd 2 1 169.4.e.b 2
52.f even 4 2 169.4.c.i 4
52.i odd 6 1 13.4.e.a 2
52.i odd 6 1 169.4.b.b 2
52.j odd 6 1 169.4.b.b 2
52.j odd 6 1 169.4.e.b 2
52.l even 12 2 169.4.a.h 2
52.l even 12 2 169.4.c.i 4
156.r even 6 1 117.4.q.c 2
156.v odd 12 2 1521.4.a.q 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 7T_{3} + 49$$ acting on $$S_{4}^{\mathrm{new}}(208, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 7T + 49$$
$5$ $$T^{2} + 192$$
$7$ $$T^{2} + 39T + 507$$
$11$ $$T^{2} - 39T + 507$$
$13$ $$T^{2} + 26T + 2197$$
$17$ $$T^{2} + 27T + 729$$
$19$ $$T^{2} - 153T + 7803$$
$23$ $$T^{2} - 57T + 3249$$
$29$ $$T^{2} - 69T + 4761$$
$31$ $$T^{2} + 5292$$
$37$ $$T^{2} + 69T + 1587$$
$41$ $$T^{2} + 681T + 154587$$
$43$ $$T^{2} + 85T + 7225$$
$47$ $$T^{2} + 117612$$
$53$ $$(T - 426)^{2}$$
$59$ $$T^{2} - 33T + 363$$
$61$ $$T^{2} - 17T + 289$$
$67$ $$T^{2} + 285T + 27075$$
$71$ $$T^{2} + 1011 T + 340707$$
$73$ $$T^{2} + 1009200$$
$79$ $$(T - 1244)^{2}$$
$83$ $$T^{2} + 181548$$
$89$ $$T^{2} - 531T + 93987$$
$97$ $$T^{2} - 2139 T + 1525107$$