Embedded newform 567.2.g.l.109.5

• Label: 567.2.g.l.109.5
• 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			109.5
Root			\(-1.04779 - 0.949812i\) of defining polynomial
Character	\(\chi\)	\(=\)	567.109
Dual form			567.2.g.l.541.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.298668 - 0.517308i) q^{2} +(0.821595 + 1.42304i) q^{4} +2.09557 q^{5} +(-1.51053 + 2.17216i) q^{7} +2.17621 q^{8} +(0.625881 - 1.08406i) q^{10} +1.65141 q^{11} +(0.213022 - 0.368965i) q^{13} +(0.672530 + 1.43017i) q^{14} +(-0.993225 + 1.72032i) q^{16} +(-3.03819 + 5.26230i) q^{17} +(-2.70625 - 4.68736i) q^{19} +(1.72171 + 2.98209i) q^{20} +(0.493225 - 0.854291i) q^{22} +7.63457 q^{23} -0.608573 q^{25} +(-0.127246 - 0.220396i) q^{26} +(-4.33213 - 0.364918i) q^{28} +(-1.82688 - 3.16426i) q^{29} +(2.65372 + 4.59638i) q^{31} +(2.76950 + 4.79691i) q^{32} +(1.81482 + 3.14336i) q^{34} +(-3.16543 + 4.55193i) q^{35} +(2.33890 + 4.05110i) q^{37} -3.23308 q^{38} +4.56041 q^{40} +(0.742827 - 1.28661i) q^{41} +(-4.24499 - 7.35253i) q^{43} +(1.35679 + 2.35004i) q^{44} +(2.28020 - 3.94943i) q^{46} +(5.66624 - 9.81422i) q^{47} +(-2.43658 - 6.56225i) q^{49} +(-0.181761 + 0.314820i) q^{50} +0.700071 q^{52} +(2.74496 - 4.75441i) q^{53} +3.46066 q^{55} +(-3.28724 + 4.72708i) q^{56} -2.18253 q^{58} +(-0.779098 - 1.34944i) q^{59} +(-2.52408 + 4.37184i) q^{61} +3.17033 q^{62} -0.664256 q^{64} +(0.446403 - 0.773193i) q^{65} +(2.61498 + 4.52928i) q^{67} -9.98464 q^{68} +(1.40933 + 2.99702i) q^{70} +12.5604 q^{71} +(-0.793753 + 1.37482i) q^{73} +2.79422 q^{74} +(4.44688 - 7.70222i) q^{76} +(-2.49452 + 3.58714i) q^{77} +(3.81482 - 6.60746i) q^{79} +(-2.08138 + 3.60505i) q^{80} +(-0.443717 - 0.768541i) q^{82} +(2.62806 + 4.55193i) q^{83} +(-6.36674 + 11.0275i) q^{85} -5.07137 q^{86} +3.59382 q^{88} +(-9.27808 - 16.0701i) q^{89} +(0.479675 + 1.02005i) q^{91} +(6.27252 + 10.8643i) q^{92} +(-3.38465 - 5.86239i) q^{94} +(-5.67114 - 9.82270i) q^{95} +(-6.87728 - 11.9118i) q^{97} +(-4.12243 - 0.699472i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.298668	−	0.517308i	0.211190	−	0.365792i	−0.740897	−	0.671619i	\(-0.765599\pi\)
							0.952087	+	0.305826i	\(0.0989327\pi\)
\(3\)		0			0
							\(5\)	2.09557			0.937169			0.468584	−	0.883419i	\(-0.344764\pi\)
0.468584	+	0.883419i	\(0.344764\pi\)
\(7\)	−1.51053	+	2.17216i	−0.570928	+	0.821000i
							\(11\)	1.65141			0.497920			0.248960	−	0.968514i	\(-0.419911\pi\)
0.248960	+	0.968514i	\(0.419911\pi\)
\(13\)	0.213022	−	0.368965i	0.0590817	−	0.102333i	−0.834972	−	0.550293i	\(-0.814516\pi\)
							0.894053	+	0.447960i	\(0.147849\pi\)
\(17\)	−3.03819	+	5.26230i	−0.736869	+	1.27629i	0.217030	+	0.976165i	\(0.430363\pi\)
							−0.953899	+	0.300129i	\(0.902970\pi\)
\(19\)	−2.70625	−	4.68736i	−0.620856	−	1.07535i	−0.989327	−	0.145714i	\(-0.953452\pi\)
							0.368471	−	0.929639i	\(-0.379881\pi\)
\(23\)	7.63457			1.59192			0.795959	−	0.605351i	\(-0.206967\pi\)
							0.795959	+	0.605351i	\(0.206967\pi\)
\(29\)	−1.82688	−	3.16426i	−0.339244	−	0.587588i	0.645047	−	0.764143i	\(-0.276838\pi\)
							−0.984291	+	0.176555i	\(0.943505\pi\)
\(31\)	2.65372	+	4.59638i	0.476623	+	0.825535i	0.999641	−	0.0267866i	\(-0.00852747\pi\)
							−0.523018	+	0.852321i	\(0.675194\pi\)
\(37\)	2.33890	+	4.05110i	0.384513	+	0.665997i	0.991702	−	0.128561i	\(-0.0410360\pi\)
							−0.607188	+	0.794558i	\(0.707703\pi\)
\(41\)	0.742827	−	1.28661i	0.116010	−	0.200935i	−0.802173	−	0.597092i	\(-0.796323\pi\)
							0.918183	+	0.396156i	\(0.129656\pi\)
\(43\)	−4.24499	−	7.35253i	−0.647354	−	1.12125i	−0.983752	−	0.179531i	\(-0.942542\pi\)
							0.336398	−	0.941720i	\(-0.390791\pi\)
\(47\)	5.66624	−	9.81422i	0.826506	−	1.43155i	−0.0742560	−	0.997239i	\(-0.523658\pi\)
							0.900763	−	0.434312i	\(-0.143008\pi\)
\(53\)	2.74496	−	4.75441i	0.377050	−	0.653069i	−0.613582	−	0.789631i	\(-0.710272\pi\)
							0.990632	+	0.136562i	\(0.0436053\pi\)
\(59\)	−0.779098	−	1.34944i	−0.101430	−	0.175682i	0.810844	−	0.585262i	\(-0.199008\pi\)
							−0.912274	+	0.409581i	\(0.865675\pi\)
\(61\)	−2.52408	+	4.37184i	−0.323176	+	0.559757i	−0.981141	−	0.193291i	\(-0.938084\pi\)
							0.657966	+	0.753048i	\(0.271417\pi\)
\(67\)	2.61498	+	4.52928i	0.319471	+	0.553340i	0.980378	−	0.197128i	\(-0.0631615\pi\)
							−0.660907	+	0.750468i	\(0.729828\pi\)
\(71\)	12.5604			1.49065			0.745324	−	0.666703i	\(-0.232295\pi\)
							0.745324	+	0.666703i	\(0.232295\pi\)
\(73\)	−0.793753	+	1.37482i	−0.0929017	+	0.160911i	−0.908731	−	0.417382i	\(-0.862948\pi\)
							0.815829	+	0.578293i	\(0.196281\pi\)
\(79\)	3.81482	−	6.60746i	0.429201	−	0.743398i	−0.567602	−	0.823303i	\(-0.692129\pi\)
							0.996802	+	0.0799058i	\(0.0254620\pi\)
\(83\)	2.62806	+	4.55193i	0.288467	+	0.499639i	0.973444	−	0.228926i	\(-0.0735213\pi\)
							−0.684977	+	0.728564i	\(0.740188\pi\)
\(89\)	−9.27808	−	16.0701i	−0.983474	−	1.70343i	−0.648528	−	0.761191i	\(-0.724615\pi\)
							−0.334946	−	0.942237i	\(-0.608718\pi\)
\(97\)	−6.87728	−	11.9118i	−0.698282	−	1.20946i	−0.969062	−	0.246818i	\(-0.920615\pi\)
							0.270780	−	0.962641i	\(-0.412718\pi\)

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.109.5		16
3.2	odd	2	inner	567.2.g.l.109.4		16
7.2	even	3		567.2.h.l.352.4		16
9.2	odd	6		567.2.h.l.298.5		16
9.4	even	3		567.2.e.g.487.5	yes	16
9.5	odd	6		567.2.e.g.487.4	yes	16
9.7	even	3		567.2.h.l.298.4		16
21.2	odd	6		567.2.h.l.352.5		16
63.2	odd	6	inner	567.2.g.l.541.4		16
63.4	even	3		3969.2.a.bg.1.4		8
63.16	even	3	inner	567.2.g.l.541.5		16
63.23	odd	6		567.2.e.g.163.4	✓	16
63.31	odd	6		3969.2.a.bf.1.4		8
63.32	odd	6		3969.2.a.bg.1.5		8
63.58	even	3		567.2.e.g.163.5	yes	16
63.59	even	6		3969.2.a.bf.1.5		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.e.g.163.4	✓	16	63.23	odd	6
567.2.e.g.163.5	yes	16	63.58	even	3
567.2.e.g.487.4	yes	16	9.5	odd	6
567.2.e.g.487.5	yes	16	9.4	even	3
567.2.g.l.109.4		16	3.2	odd	2	inner
567.2.g.l.109.5		16	1.1	even	1	trivial
567.2.g.l.541.4		16	63.2	odd	6	inner
567.2.g.l.541.5		16	63.16	even	3	inner
567.2.h.l.298.4		16	9.7	even	3
567.2.h.l.298.5		16	9.2	odd	6
567.2.h.l.352.4		16	7.2	even	3
567.2.h.l.352.5		16	21.2	odd	6
3969.2.a.bf.1.4		8	63.31	odd	6
3969.2.a.bf.1.5		8	63.59	even	6
3969.2.a.bg.1.4		8	63.4	even	3
3969.2.a.bg.1.5		8	63.32	odd	6