Embedded newform 567.2.s.g.26.13

• Label: 567.2.s.g.26.13
• Level: $567$
• Weight: $2$
• Character: 567.26
• Analytic conductor: $4.528$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.52751779461\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			26.13
Character	\(\chi\)	\(=\)	567.26
Dual form			567.2.s.g.458.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41253 - 0.815523i) q^{2} +(0.330156 - 0.571847i) q^{4} +2.00581 q^{5} +(2.28410 - 1.33524i) q^{7} +2.18509i q^{8} +(2.83327 - 1.63579i) q^{10} -1.31085i q^{11} +(-2.65208 + 1.53118i) q^{13} +(2.13744 - 3.74881i) q^{14} +(2.44231 + 4.23020i) q^{16} +(-0.683325 - 1.18355i) q^{17} +(4.92262 + 2.84207i) q^{19} +(0.662231 - 1.14702i) q^{20} +(-1.06902 - 1.85161i) q^{22} -6.58539i q^{23} -0.976718 q^{25} +(-2.49742 + 4.32566i) q^{26} +(-0.00944366 - 1.74700i) q^{28} +(5.41377 + 3.12564i) q^{29} +(-6.20195 - 3.58070i) q^{31} +(3.11496 + 1.79842i) q^{32} +(-1.93043 - 1.11453i) q^{34} +(4.58148 - 2.67824i) q^{35} +(-2.17992 + 3.77573i) q^{37} +9.27111 q^{38} +4.38288i q^{40} +(-2.79967 - 4.84917i) q^{41} +(-1.56902 + 2.71763i) q^{43} +(-0.749603 - 0.432784i) q^{44} +(-5.37054 - 9.30205i) q^{46} +(-3.86407 - 6.69277i) q^{47} +(3.43426 - 6.09966i) q^{49} +(-1.37964 + 0.796536i) q^{50} +2.02211i q^{52} +(8.54669 - 4.93443i) q^{53} -2.62931i q^{55} +(2.91763 + 4.99098i) q^{56} +10.1961 q^{58} +(-5.78441 + 10.0189i) q^{59} +(-12.5188 + 7.22776i) q^{61} -11.6806 q^{62} -3.90260 q^{64} +(-5.31957 + 3.07126i) q^{65} +(-5.21234 + 9.02803i) q^{67} -0.902416 q^{68} +(4.28730 - 7.51940i) q^{70} -9.13734i q^{71} +(1.16332 - 0.671641i) q^{73} +7.11110i q^{74} +(3.25047 - 1.87666i) q^{76} +(-1.75030 - 2.99411i) q^{77} +(6.78595 + 11.7536i) q^{79} +(4.89881 + 8.48498i) q^{80} +(-7.90922 - 4.56639i) q^{82} +(-0.206808 + 0.358202i) q^{83} +(-1.37062 - 2.37398i) q^{85} +5.11830i q^{86} +2.86432 q^{88} +(-2.39824 + 4.15388i) q^{89} +(-4.01313 + 7.03854i) q^{91} +(-3.76584 - 2.17421i) q^{92} +(-10.9162 - 6.30248i) q^{94} +(9.87384 + 5.70067i) q^{95} +(-15.6273 - 9.02240i) q^{97} +(-0.123433 - 11.4167i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	32 * q + 16 * q^4 + 4 * q^7 - 12 * q^13 - 28 * q^16 - 12 * q^22 + 32 * q^25 - 16 * q^28 + 48 * q^31 - 4 * q^37 - 28 * q^43 + 12 * q^46 - 16 * q^49 - 72 * q^58 - 12 * q^61 - 80 * q^64 - 20 * q^67 - 60 * q^70 - 36 * q^73 + 108 * q^76 + 4 * q^79 + 12 * q^85 - 24 * q^88 + 36 * q^91 - 72 * q^94 - 12 * q^97

Character values

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

\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41253	−	0.815523i	0.998808	−	0.576662i	0.0909125	−	0.995859i	\(-0.471022\pi\)
							0.907895	+	0.419197i	\(0.137688\pi\)
\(3\)		0			0
							\(5\)	2.00581			0.897026			0.448513	−	0.893776i	\(-0.351954\pi\)
0.448513	+	0.893776i	\(0.351954\pi\)
\(7\)	2.28410	−	1.33524i	0.863310	−	0.504674i
							\(11\)		−	1.31085i		−	0.395235i	−0.980279	−	0.197617i	\(-0.936680\pi\)
0.980279	−	0.197617i	\(-0.0633203\pi\)
\(13\)	−2.65208	+	1.53118i	−0.735554	+	0.424673i	−0.820451	−	0.571717i	\(-0.806277\pi\)
							0.0848963	+	0.996390i	\(0.472944\pi\)
\(17\)	−0.683325	−	1.18355i	−0.165731	−	0.287054i	0.771184	−	0.636612i	\(-0.219665\pi\)
							−0.936914	+	0.349559i	\(0.886332\pi\)
\(19\)	4.92262	+	2.84207i	1.12933	+	0.652016i	0.943765	−	0.330616i	\(-0.107257\pi\)
							0.185560	+	0.982633i	\(0.440590\pi\)
\(23\)		−	6.58539i		−	1.37315i	−0.727060	−	0.686574i	\(-0.759114\pi\)
							0.727060	−	0.686574i	\(-0.240886\pi\)
\(29\)	5.41377	+	3.12564i	1.00531	+	0.580417i	0.909816	−	0.415013i	\(-0.136223\pi\)
							0.0954960	+	0.995430i	\(0.469556\pi\)
\(31\)	−6.20195	−	3.58070i	−1.11390	−	0.643112i	−0.174065	−	0.984734i	\(-0.555690\pi\)
							−0.939837	+	0.341622i	\(0.889024\pi\)
\(37\)	−2.17992	+	3.77573i	−0.358377	+	0.620727i	−0.987690	−	0.156425i	\(-0.950003\pi\)
							0.629313	+	0.777152i	\(0.283336\pi\)
\(41\)	−2.79967	−	4.84917i	−0.437235	−	0.757313i	0.560240	−	0.828330i	\(-0.310709\pi\)
							−0.997475	+	0.0710171i	\(0.977376\pi\)
\(43\)	−1.56902	+	2.71763i	−0.239274	+	0.414435i	−0.960506	−	0.278259i	\(-0.910243\pi\)
							0.721232	+	0.692693i	\(0.243576\pi\)
\(47\)	−3.86407	−	6.69277i	−0.563633	−	0.976241i	−0.997175	−	0.0751077i	\(-0.976070\pi\)
							0.433543	−	0.901133i	\(-0.357263\pi\)
\(53\)	8.54669	−	4.93443i	1.17398	−	0.677796i	0.219364	−	0.975643i	\(-0.429602\pi\)
							0.954614	+	0.297847i	\(0.0962684\pi\)
\(59\)	−5.78441	+	10.0189i	−0.753066	+	1.30435i	0.193264	+	0.981147i	\(0.438093\pi\)
							−0.946330	+	0.323202i	\(0.895241\pi\)
\(61\)	−12.5188	+	7.22776i	−1.60287	+	0.925419i	−0.611964	+	0.790885i	\(0.709620\pi\)
							−0.990909	+	0.134534i	\(0.957046\pi\)
\(67\)	−5.21234	+	9.02803i	−0.636788	+	1.10295i	0.349345	+	0.936994i	\(0.386404\pi\)
							−0.986133	+	0.165955i	\(0.946929\pi\)
\(71\)		−	9.13734i		−	1.08440i	−0.840249	−	0.542201i	\(-0.817591\pi\)
							0.840249	−	0.542201i	\(-0.182409\pi\)
\(73\)	1.16332	−	0.671641i	0.136156	−	0.0786097i	−0.430375	−	0.902650i	\(-0.641619\pi\)
							0.566531	+	0.824041i	\(0.308285\pi\)
\(79\)	6.78595	+	11.7536i	0.763479	+	1.32238i	0.941047	+	0.338275i	\(0.109844\pi\)
							−0.177569	+	0.984108i	\(0.556823\pi\)
\(83\)	−0.206808	+	0.358202i	−0.0227001	+	0.0393177i	−0.877152	−	0.480212i	\(-0.840560\pi\)
							0.854452	+	0.519530i	\(0.173893\pi\)
\(89\)	−2.39824	+	4.15388i	−0.254213	+	0.440310i	−0.964681	−	0.263419i	\(-0.915150\pi\)
							0.710468	+	0.703729i	\(0.248483\pi\)
\(97\)	−15.6273	−	9.02240i	−1.58671	−	0.916086i	−0.993845	−	0.110782i	\(-0.964664\pi\)
							−0.592863	−	0.805304i	\(-0.702002\pi\)

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.s.g.26.13		32
3.2	odd	2	inner	567.2.s.g.26.4		32
7.3	odd	6		567.2.i.g.269.13		32
9.2	odd	6		567.2.p.e.404.13	yes	32
9.4	even	3		567.2.i.g.215.13		32
9.5	odd	6		567.2.i.g.215.4		32
9.7	even	3		567.2.p.e.404.4	yes	32
21.17	even	6		567.2.i.g.269.4		32
63.31	odd	6	inner	567.2.s.g.458.4		32
63.38	even	6		567.2.p.e.80.4	✓	32
63.52	odd	6		567.2.p.e.80.13	yes	32
63.59	even	6	inner	567.2.s.g.458.13		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.i.g.215.4		32	9.5	odd	6
567.2.i.g.215.13		32	9.4	even	3
567.2.i.g.269.4		32	21.17	even	6
567.2.i.g.269.13		32	7.3	odd	6
567.2.p.e.80.4	✓	32	63.38	even	6
567.2.p.e.80.13	yes	32	63.52	odd	6
567.2.p.e.404.4	yes	32	9.7	even	3
567.2.p.e.404.13	yes	32	9.2	odd	6
567.2.s.g.26.4		32	3.2	odd	2	inner
567.2.s.g.26.13		32	1.1	even	1	trivial
567.2.s.g.458.4		32	63.31	odd	6	inner
567.2.s.g.458.13		32	63.59	even	6	inner