Embedded newform 567.2.g.l.109.6

• Label: 567.2.g.l.109.6
• Level: $567$
• Weight: $2$
• Character: 567.109
• 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.g (of order \(3\), degree \(2\), not 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			109.6
Root			\(-0.776749 - 1.18180i\) of defining polynomial
Character	\(\chi\)	\(=\)	567.109
Dual form			567.2.g.l.541.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.635098 - 1.10002i) q^{2} +(0.193301 + 0.334806i) q^{4} +1.55350 q^{5} +(2.63869 + 0.193156i) q^{7} +3.03145 q^{8} +(0.986623 - 1.70888i) q^{10} -3.21001 q^{11} +(-2.39335 + 4.14540i) q^{13} +(1.88830 - 2.77995i) q^{14} +(1.53867 - 2.66505i) q^{16} +(1.05918 - 1.83456i) q^{17} +(2.43201 + 4.21237i) q^{19} +(0.300292 + 0.520121i) q^{20} +(-2.03867 + 3.53108i) q^{22} +3.70758 q^{23} -2.58665 q^{25} +(3.04002 + 5.26547i) q^{26} +(0.445391 + 0.920788i) q^{28} +(-3.68972 - 6.39078i) q^{29} +(-2.75209 - 4.76676i) q^{31} +(1.07704 + 1.86549i) q^{32} +(-1.34537 - 2.33025i) q^{34} +(4.09920 + 0.300067i) q^{35} +(0.0932782 + 0.161563i) q^{37} +6.17827 q^{38} +4.70935 q^{40} +(5.39860 - 9.35065i) q^{41} +(-2.43458 - 4.21681i) q^{43} +(-0.620496 - 1.07473i) q^{44} +(2.35468 - 4.07842i) q^{46} +(-0.885937 + 1.53449i) q^{47} +(6.92538 + 1.01936i) q^{49} +(-1.64277 + 2.84537i) q^{50} -1.85054 q^{52} +(-0.834432 + 1.44528i) q^{53} -4.98674 q^{55} +(7.99907 + 0.585543i) q^{56} -9.37334 q^{58} +(-2.91297 - 5.04541i) q^{59} +(-3.43865 + 5.95591i) q^{61} -6.99139 q^{62} +8.89078 q^{64} +(-3.71806 + 6.43986i) q^{65} +(-6.11868 - 10.5979i) q^{67} +0.818961 q^{68} +(2.93347 - 4.31864i) q^{70} -13.8101 q^{71} +(-5.93201 + 10.2745i) q^{73} +0.236963 q^{74} +(-0.940219 + 1.62851i) q^{76} +(-8.47021 - 0.620031i) q^{77} +(0.654632 - 1.13386i) q^{79} +(2.39032 - 4.14015i) q^{80} +(-6.85728 - 11.8772i) q^{82} +(0.173244 + 0.300067i) q^{83} +(1.64544 - 2.84998i) q^{85} -6.18478 q^{86} -9.73098 q^{88} +(8.70319 + 15.0744i) q^{89} +(-7.11601 + 10.4761i) q^{91} +(0.716677 + 1.24132i) q^{92} +(1.12531 + 1.94910i) q^{94} +(3.77813 + 6.54391i) q^{95} +(5.28413 + 9.15238i) q^{97} +(5.51961 - 6.97068i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	16 * q - 6 * q^4 - 14 * q^10 - 6 * q^13 - 6 * q^16 - 24 * q^19 - 2 * q^22 - 26 * q^28 - 20 * q^31 + 4 * q^37 + 72 * q^40 - 10 * q^43 + 36 * q^46 + 4 * q^49 + 68 * q^52 + 8 * q^55 - 44 * q^58 - 36 * q^61 + 76 * q^64 + 18 * q^67 + 104 * q^70 - 32 * q^73 - 58 * q^76 + 32 * q^79 + 2 * q^82 - 30 * q^85 - 144 * q^88 - 22 * q^91 - 54 * q^94 - 14 * q^97

Character values

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

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

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	−0.549244	−	0.835662i	\(-0.685084\pi\)
							0.998327	+	0.0578286i	\(0.0184177\pi\)
\(3\)		0			0
							\(5\)	1.55350			0.694745			0.347373	−	0.937727i	\(-0.387074\pi\)
0.347373	+	0.937727i	\(0.387074\pi\)
\(7\)	2.63869	+	0.193156i	0.997331	+	0.0730060i
							\(11\)	−3.21001			−0.967853			−0.483927	−	0.875109i	\(-0.660790\pi\)
−0.483927	+	0.875109i	\(0.660790\pi\)
\(13\)	−2.39335	+	4.14540i	−0.663795	+	1.14973i	0.315816	+	0.948820i	\(0.397722\pi\)
							−0.979611	+	0.200905i	\(0.935612\pi\)
\(17\)	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	\(0.644924\pi\)
\(23\)	3.70758			0.773084			0.386542	−	0.922272i	\(-0.373670\pi\)
							0.386542	+	0.922272i	\(0.373670\pi\)
\(29\)	−3.68972	−	6.39078i	−0.685164	−	1.18674i	−0.973385	−	0.229175i	\(-0.926397\pi\)
							0.288222	−	0.957564i	\(-0.406936\pi\)
\(31\)	−2.75209	−	4.76676i	−0.494290	−	0.856135i	0.505688	−	0.862716i	\(-0.331239\pi\)
							−0.999978	+	0.00658088i	\(0.997905\pi\)
\(37\)	0.0932782	+	0.161563i	0.0153348	+	0.0265607i	0.873591	−	0.486661i	\(-0.161785\pi\)
							−0.858256	+	0.513222i	\(0.828452\pi\)
\(41\)	5.39860	−	9.35065i	0.843120	−	1.46033i	−0.0441242	−	0.999026i	\(-0.514050\pi\)
							0.887244	−	0.461300i	\(-0.152617\pi\)
\(43\)	−2.43458	−	4.21681i	−0.371270	−	0.643058i	0.618491	−	0.785792i	\(-0.287744\pi\)
							−0.989761	+	0.142734i	\(0.954411\pi\)
\(47\)	−0.885937	+	1.53449i	−0.129227	+	0.223828i	−0.923377	−	0.383893i	\(-0.874583\pi\)
							0.794150	+	0.607722i	\(0.207916\pi\)
\(53\)	−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	0.611715	−	0.791078i	\(-0.290480\pi\)
							−0.990951	+	0.134221i	\(0.957147\pi\)
\(61\)	−3.43865	+	5.95591i	−0.440274	+	0.762576i	−0.997710	−	0.0676438i	\(-0.978452\pi\)
							0.557436	+	0.830220i	\(0.311785\pi\)
\(67\)	−6.11868	−	10.5979i	−0.747516	−	1.29474i	−0.949010	−	0.315246i	\(-0.897913\pi\)
							0.201494	−	0.979490i	\(-0.435420\pi\)
\(71\)	−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	−0.826847	−	0.562426i	\(-0.809868\pi\)
							0.900499	+	0.434858i	\(0.143201\pi\)
\(83\)	0.173244	+	0.300067i	0.0190160	+	0.0329366i	0.875377	−	0.483441i	\(-0.160613\pi\)
							−0.856361	+	0.516378i	\(0.827280\pi\)
\(89\)	8.70319	+	15.0744i	0.922537	+	1.59788i	0.795476	+	0.605985i	\(0.207221\pi\)
							0.127061	+	0.991895i	\(0.459446\pi\)
\(97\)	5.28413	+	9.15238i	0.536522	+	0.929283i	0.999088	+	0.0426982i	\(0.0135954\pi\)
							−0.462566	+	0.886585i	\(0.653071\pi\)

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.g.l.109.6		16
3.2	odd	2	inner	567.2.g.l.109.3		16
7.2	even	3		567.2.h.l.352.3		16
9.2	odd	6		567.2.h.l.298.6		16
9.4	even	3		567.2.e.g.487.6	yes	16
9.5	odd	6		567.2.e.g.487.3	yes	16
9.7	even	3		567.2.h.l.298.3		16
21.2	odd	6		567.2.h.l.352.6		16
63.2	odd	6	inner	567.2.g.l.541.3		16
63.4	even	3		3969.2.a.bg.1.3		8
63.16	even	3	inner	567.2.g.l.541.6		16
63.23	odd	6		567.2.e.g.163.3	✓	16
63.31	odd	6		3969.2.a.bf.1.3		8
63.32	odd	6		3969.2.a.bg.1.6		8
63.58	even	3		567.2.e.g.163.6	yes	16
63.59	even	6		3969.2.a.bf.1.6		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.e.g.163.3	✓	16	63.23	odd	6
567.2.e.g.163.6	yes	16	63.58	even	3
567.2.e.g.487.3	yes	16	9.5	odd	6
567.2.e.g.487.6	yes	16	9.4	even	3
567.2.g.l.109.3		16	3.2	odd	2	inner
567.2.g.l.109.6		16	1.1	even	1	trivial
567.2.g.l.541.3		16	63.2	odd	6	inner
567.2.g.l.541.6		16	63.16	even	3	inner
567.2.h.l.298.3		16	9.7	even	3
567.2.h.l.298.6		16	9.2	odd	6
567.2.h.l.352.3		16	7.2	even	3
567.2.h.l.352.6		16	21.2	odd	6
3969.2.a.bf.1.3		8	63.31	odd	6
3969.2.a.bf.1.6		8	63.59	even	6
3969.2.a.bg.1.3		8	63.4	even	3
3969.2.a.bg.1.6		8	63.32	odd	6