Embedded newform 567.2.e.g.487.6

• Label: 567.2.e.g.487.6
• Level: $567$
• Weight: $2$
• Character: 567.487
• Analytic conductor: $4.528$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.52751779461\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 4x^{14} + 14x^{12} - 39x^{10} + 77x^{8} - 156x^{6} + 224x^{4} - 256x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			487.6
Root			\(-0.776749 + 1.18180i\) of defining polynomial
Character	\(\chi\)	\(=\)	567.487
Dual form			567.2.e.g.163.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.635098 + 1.10002i) q^{2} +(0.193301 - 0.334806i) q^{4} +(-0.776749 - 1.34537i) q^{5} +(-1.48662 + 2.18860i) q^{7} +3.03145 q^{8} +(0.986623 - 1.70888i) q^{10} +(1.60500 - 2.77995i) q^{11} +4.78669 q^{13} +(-3.35166 - 0.245346i) q^{14} +(1.53867 + 2.66505i) q^{16} +(1.05918 - 1.83456i) q^{17} +(2.43201 + 4.21237i) q^{19} -0.600584 q^{20} +4.07734 q^{22} +(-1.85379 - 3.21086i) q^{23} +(1.29332 - 2.24010i) q^{25} +(3.04002 + 5.26547i) q^{26} +(0.445391 + 0.920788i) q^{28} +7.37944 q^{29} +(-2.75209 + 4.76676i) q^{31} +(1.07704 - 1.86549i) q^{32} +2.69074 q^{34} +(4.09920 + 0.300067i) q^{35} +(0.0932782 + 0.161563i) q^{37} +(-3.08914 + 5.35054i) q^{38} +(-2.35468 - 4.07842i) q^{40} -10.7972 q^{41} +4.86916 q^{43} +(-0.620496 - 1.07473i) q^{44} +(2.35468 - 4.07842i) q^{46} +(-0.885937 - 1.53449i) q^{47} +(-2.57990 - 6.50724i) q^{49} +3.28555 q^{50} +(0.925270 - 1.60261i) q^{52} +(-0.834432 + 1.44528i) q^{53} -4.98674 q^{55} +(-4.50663 + 6.63462i) q^{56} +(4.68667 + 8.11755i) q^{58} +(-2.91297 + 5.04541i) q^{59} +(-3.43865 - 5.95591i) q^{61} -6.99139 q^{62} +8.89078 q^{64} +(-3.71806 - 6.43986i) q^{65} +(-6.11868 + 10.5979i) q^{67} +(-0.409481 - 0.709241i) q^{68} +(2.27331 + 4.69978i) q^{70} -13.8101 q^{71} +(-5.93201 + 10.2745i) q^{73} +(-0.118482 + 0.205216i) q^{74} +1.88044 q^{76} +(3.69814 + 7.64544i) q^{77} +(0.654632 + 1.13386i) q^{79} +(2.39032 - 4.14015i) q^{80} +(-6.85728 - 11.8772i) q^{82} -0.346488 q^{83} -3.29087 q^{85} +(3.09239 + 5.35618i) q^{86} +(4.86549 - 8.42727i) q^{88} +(8.70319 + 15.0744i) q^{89} +(-7.11601 + 10.4761i) q^{91} -1.43335 q^{92} +(1.12531 - 1.94910i) q^{94} +(3.77813 - 6.54391i) q^{95} -10.5683 q^{97} +(5.51961 - 6.97068i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	16 * q - 6 * q^4 + 6 * q^7 - 14 * q^10 + 12 * q^13 - 6 * q^16 - 24 * q^19 + 4 * q^22 - 26 * q^28 - 20 * q^31 + 4 * q^37 - 36 * q^40 + 20 * q^43 + 36 * q^46 - 14 * q^49 - 34 * q^52 + 8 * q^55 + 22 * q^58 - 36 * q^61 + 76 * q^64 + 18 * q^67 - 58 * q^70 - 32 * q^73 + 116 * q^76 + 32 * q^79 + 2 * q^82 + 60 * q^85 + 72 * q^88 - 22 * q^91 - 54 * q^94 + 28 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.635098	+	1.10002i	0.449082	+	0.777833i	0.998327	−	0.0578286i	\(-0.0184177\pi\)
							−0.549244	+	0.835662i	\(0.685084\pi\)
\(3\)		0			0
							\(5\)	−0.776749	−	1.34537i	−0.347373	−	0.601667i	0.638409	−	0.769697i	\(-0.279593\pi\)
−0.985782	+	0.168030i	\(0.946259\pi\)
\(7\)	−1.48662	+	2.18860i	−0.561891	+	0.827211i
							\(11\)	1.60500	−	2.77995i	0.483927	−	0.838185i	−0.515903	−	0.856647i	\(-0.672544\pi\)
0.999830	+	0.0184616i	\(0.00587686\pi\)
\(13\)	4.78669			1.32759			0.663795	−	0.747915i	\(-0.268945\pi\)
							0.663795	+	0.747915i	\(0.268945\pi\)
\(17\)	1.05918	−	1.83456i	0.256889	−	0.444945i	−0.708518	−	0.705693i	\(-0.750636\pi\)
							0.965407	+	0.260748i	\(0.0839691\pi\)
\(19\)	2.43201	+	4.21237i	0.557942	+	0.966384i	0.997668	+	0.0682523i	\(0.0217423\pi\)
							−0.439726	+	0.898132i	\(0.644924\pi\)
\(23\)	−1.85379	−	3.21086i	−0.386542	−	0.669510i	0.605440	−	0.795891i	\(-0.292997\pi\)
							−0.991982	+	0.126381i	\(0.959664\pi\)
\(29\)	7.37944			1.37033			0.685164	−	0.728389i	\(-0.259731\pi\)
							0.685164	+	0.728389i	\(0.259731\pi\)
\(31\)	−2.75209	+	4.76676i	−0.494290	+	0.856135i	−0.999978	−	0.00658088i	\(-0.997905\pi\)
							0.505688	+	0.862716i	\(0.331239\pi\)
\(37\)	0.0932782	+	0.161563i	0.0153348	+	0.0265607i	0.873591	−	0.486661i	\(-0.161785\pi\)
							−0.858256	+	0.513222i	\(0.828452\pi\)
\(41\)	−10.7972			−1.68624			−0.843120	−	0.537726i	\(-0.819284\pi\)
							−0.843120	+	0.537726i	\(0.819284\pi\)
\(43\)	4.86916			0.742539			0.371270	−	0.928525i	\(-0.378923\pi\)
							0.371270	+	0.928525i	\(0.378923\pi\)
\(47\)	−0.885937	−	1.53449i	−0.129227	−	0.223828i	0.794150	−	0.607722i	\(-0.207916\pi\)
							−0.923377	+	0.383893i	\(0.874583\pi\)
\(53\)	−0.834432	+	1.44528i	−0.114618	+	0.198524i	−0.917627	−	0.397443i	\(-0.869898\pi\)
							0.803009	+	0.595967i	\(0.203231\pi\)
\(59\)	−2.91297	+	5.04541i	−0.379236	+	0.656857i	−0.990951	−	0.134221i	\(-0.957147\pi\)
							0.611715	+	0.791078i	\(0.290480\pi\)
\(61\)	−3.43865	−	5.95591i	−0.440274	−	0.762576i	0.557436	−	0.830220i	\(-0.311785\pi\)
							−0.997710	+	0.0676438i	\(0.978452\pi\)
\(67\)	−6.11868	+	10.5979i	−0.747516	+	1.29474i	0.201494	+	0.979490i	\(0.435420\pi\)
							−0.949010	+	0.315246i	\(0.897913\pi\)
\(71\)	−13.8101			−1.63895			−0.819477	−	0.573112i	\(-0.805736\pi\)
							−0.819477	+	0.573112i	\(0.805736\pi\)
\(73\)	−5.93201	+	10.2745i	−0.694290	+	1.20255i	0.276130	+	0.961120i	\(0.410948\pi\)
							−0.970420	+	0.241425i	\(0.922385\pi\)
\(79\)	0.654632	+	1.13386i	0.0736518	+	0.127569i	0.900499	−	0.434858i	\(-0.143201\pi\)
							−0.826847	+	0.562426i	\(0.809868\pi\)
\(83\)	−0.346488			−0.0380320			−0.0190160	−	0.999819i	\(-0.506053\pi\)
							−0.0190160	+	0.999819i	\(0.506053\pi\)
\(89\)	8.70319	+	15.0744i	0.922537	+	1.59788i	0.795476	+	0.605985i	\(0.207221\pi\)
							0.127061	+	0.991895i	\(0.459446\pi\)
\(97\)	−10.5683			−1.07304			−0.536522	−	0.843886i	\(-0.680262\pi\)
							−0.536522	+	0.843886i	\(0.680262\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.2.e.g.487.6	yes	16
3.2	odd	2	inner	567.2.e.g.487.3	yes	16
7.2	even	3	inner	567.2.e.g.163.6	yes	16
7.3	odd	6		3969.2.a.bf.1.3		8
7.4	even	3		3969.2.a.bg.1.3		8
9.2	odd	6		567.2.g.l.109.3		16
9.4	even	3		567.2.h.l.298.3		16
9.5	odd	6		567.2.h.l.298.6		16
9.7	even	3		567.2.g.l.109.6		16
21.2	odd	6	inner	567.2.e.g.163.3	✓	16
21.11	odd	6		3969.2.a.bg.1.6		8
21.17	even	6		3969.2.a.bf.1.6		8
63.2	odd	6		567.2.h.l.352.6		16
63.16	even	3		567.2.h.l.352.3		16
63.23	odd	6		567.2.g.l.541.3		16
63.58	even	3		567.2.g.l.541.6		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.e.g.163.3	✓	16	21.2	odd	6	inner
567.2.e.g.163.6	yes	16	7.2	even	3	inner
567.2.e.g.487.3	yes	16	3.2	odd	2	inner
567.2.e.g.487.6	yes	16	1.1	even	1	trivial
567.2.g.l.109.3		16	9.2	odd	6
567.2.g.l.109.6		16	9.7	even	3
567.2.g.l.541.3		16	63.23	odd	6
567.2.g.l.541.6		16	63.58	even	3
567.2.h.l.298.3		16	9.4	even	3
567.2.h.l.298.6		16	9.5	odd	6
567.2.h.l.352.3		16	63.16	even	3
567.2.h.l.352.6		16	63.2	odd	6
3969.2.a.bf.1.3		8	7.3	odd	6
3969.2.a.bf.1.6		8	21.17	even	6
3969.2.a.bg.1.3		8	7.4	even	3
3969.2.a.bg.1.6		8	21.11	odd	6