Embedded newform 567.2.s.g.26.14

• Label: 567.2.s.g.26.14
• 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			26.14
Character	\(\chi\)	\(=\)	567.26
Dual form			567.2.s.g.458.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.88023 - 1.08555i) q^{2} +(1.35685 - 2.35014i) q^{4} +1.23750 q^{5} +(0.654161 + 2.56361i) q^{7} -1.54953i q^{8} +(2.32678 - 1.34337i) q^{10} -1.11788i q^{11} +(4.67580 - 2.69958i) q^{13} +(4.01291 + 4.11005i) q^{14} +(1.03161 + 1.78680i) q^{16} +(-0.913955 - 1.58302i) q^{17} +(-3.40985 - 1.96868i) q^{19} +(1.67910 - 2.90829i) q^{20} +(-1.21352 - 2.10188i) q^{22} +0.193095i q^{23} -3.46860 q^{25} +(5.86107 - 10.1517i) q^{26} +(6.91242 + 1.94106i) q^{28} +(-7.34939 - 4.24317i) q^{29} +(4.27901 + 2.47049i) q^{31} +(6.56319 + 3.78926i) q^{32} +(-3.43690 - 1.98429i) q^{34} +(0.809523 + 3.17245i) q^{35} +(-5.81453 + 10.0711i) q^{37} -8.54841 q^{38} -1.91753i q^{40} +(-5.69201 - 9.85884i) q^{41} +(-1.71352 + 2.96791i) q^{43} +(-2.62718 - 1.51680i) q^{44} +(0.209614 + 0.363063i) q^{46} +(4.03316 + 6.98563i) q^{47} +(-6.14415 + 3.35402i) q^{49} +(-6.52178 + 3.76535i) q^{50} -14.6517i q^{52} +(2.54245 - 1.46788i) q^{53} -1.38338i q^{55} +(3.97237 - 1.01364i) q^{56} -18.4248 q^{58} +(6.10123 - 10.5676i) q^{59} +(-6.61605 + 3.81978i) q^{61} +10.7274 q^{62} +12.3273 q^{64} +(5.78629 - 3.34072i) q^{65} +(0.587063 - 1.01682i) q^{67} -4.96041 q^{68} +(4.96596 + 5.08617i) q^{70} +11.7500i q^{71} +(0.662676 - 0.382596i) q^{73} +25.2479i q^{74} +(-9.25331 + 5.34240i) q^{76} +(2.86581 - 0.731275i) q^{77} +(0.436035 + 0.755235i) q^{79} +(1.27661 + 2.21116i) q^{80} +(-21.4046 - 12.3580i) q^{82} +(5.30068 - 9.18104i) q^{83} +(-1.13102 - 1.95898i) q^{85} +7.44047i q^{86} -1.73219 q^{88} +(0.705529 - 1.22201i) q^{89} +(9.97938 + 10.2210i) q^{91} +(0.453799 + 0.262001i) q^{92} +(15.1666 + 8.75641i) q^{94} +(-4.21967 - 2.43623i) q^{95} +(4.26169 + 2.46049i) q^{97} +(-7.91146 + 12.9761i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.88023	−	1.08555i	1.32953	−	0.767602i	0.344300	−	0.938860i	\(-0.388116\pi\)
							0.985226	+	0.171258i	\(0.0547831\pi\)
\(3\)		0			0
							\(5\)	1.23750			0.553425			0.276713	−	0.960953i	\(-0.410755\pi\)
0.276713	+	0.960953i	\(0.410755\pi\)
\(7\)	0.654161	+	2.56361i	0.247250	+	0.968952i
							\(11\)		−	1.11788i		−	0.337054i	−0.985697	−	0.168527i	\(-0.946099\pi\)
0.985697	−	0.168527i	\(-0.0539011\pi\)
\(13\)	4.67580	−	2.69958i	1.29683	−	0.748728i	0.316978	−	0.948433i	\(-0.397332\pi\)
							0.979856	+	0.199705i	\(0.0639984\pi\)
\(17\)	−0.913955	−	1.58302i	−0.221667	−	0.383938i	0.733648	−	0.679530i	\(-0.237816\pi\)
							−0.955314	+	0.295592i	\(0.904483\pi\)
\(19\)	−3.40985	−	1.96868i	−0.782272	−	0.451645i	0.0549627	−	0.998488i	\(-0.482496\pi\)
							−0.837235	+	0.546843i	\(0.815829\pi\)
\(23\)			0.193095i			0.0402630i	0.999797	+	0.0201315i	\(0.00640849\pi\)
							−0.999797	+	0.0201315i	\(0.993592\pi\)
\(29\)	−7.34939	−	4.24317i	−1.36475	−	0.787938i	−0.374496	−	0.927228i	\(-0.622184\pi\)
							−0.990252	+	0.139291i	\(0.955518\pi\)
\(31\)	4.27901	+	2.47049i	0.768533	+	0.443713i	0.832351	−	0.554249i	\(-0.186994\pi\)
							−0.0638179	+	0.997962i	\(0.520328\pi\)
\(37\)	−5.81453	+	10.0711i	−0.955903	+	1.65567i	−0.223614	+	0.974678i	\(0.571785\pi\)
							−0.732289	+	0.680994i	\(0.761548\pi\)
\(41\)	−5.69201	−	9.85884i	−0.888942	−	1.53969i	−0.841128	−	0.540836i	\(-0.818108\pi\)
							−0.0478142	−	0.998856i	\(-0.515226\pi\)
\(43\)	−1.71352	+	2.96791i	−0.261310	+	0.452601i	−0.966590	−	0.256327i	\(-0.917488\pi\)
							0.705281	+	0.708928i	\(0.250821\pi\)
\(47\)	4.03316	+	6.98563i	0.588296	+	1.01896i	0.994456	+	0.105157i	\(0.0335345\pi\)
							−0.406159	+	0.913802i	\(0.633132\pi\)
\(53\)	2.54245	−	1.46788i	0.349232	−	0.201629i	−0.315115	−	0.949053i	\(-0.602043\pi\)
							0.664347	+	0.747424i	\(0.268710\pi\)
\(59\)	6.10123	−	10.5676i	0.794313	−	1.37579i	−0.128962	−	0.991650i	\(-0.541165\pi\)
							0.923275	−	0.384140i	\(-0.125502\pi\)
\(61\)	−6.61605	+	3.81978i	−0.847098	+	0.489072i	−0.859671	−	0.510848i	\(-0.829331\pi\)
							0.0125724	+	0.999921i	\(0.495998\pi\)
\(67\)	0.587063	−	1.01682i	0.0717212	−	0.124225i	−0.827935	−	0.560825i	\(-0.810484\pi\)
							0.899656	+	0.436600i	\(0.143817\pi\)
\(71\)			11.7500i			1.39447i	0.716843	+	0.697234i	\(0.245586\pi\)
							−0.716843	+	0.697234i	\(0.754414\pi\)
\(73\)	0.662676	−	0.382596i	0.0775603	−	0.0447795i	−0.460718	−	0.887546i	\(-0.652408\pi\)
							0.538279	+	0.842767i	\(0.319075\pi\)
\(79\)	0.436035	+	0.755235i	0.0490578	+	0.0849706i	0.889512	−	0.456913i	\(-0.151045\pi\)
							−0.840454	+	0.541883i	\(0.817711\pi\)
\(83\)	5.30068	−	9.18104i	0.581825	−	1.00775i	−0.413438	−	0.910532i	\(-0.635672\pi\)
							0.995263	−	0.0972179i	\(-0.0309944\pi\)
\(89\)	0.705529	−	1.22201i	0.0747859	−	0.129533i	−0.826207	−	0.563366i	\(-0.809506\pi\)
							0.900993	+	0.433833i	\(0.142839\pi\)
\(97\)	4.26169	+	2.46049i	0.432709	+	0.249825i	0.700500	−	0.713652i	\(-0.252960\pi\)
							−0.267791	+	0.963477i	\(0.586294\pi\)

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.26.14		32
3.2	odd	2	inner	567.2.s.g.26.3		32
7.3	odd	6		567.2.i.g.269.14		32
9.2	odd	6		567.2.p.e.404.14	yes	32
9.4	even	3		567.2.i.g.215.14		32
9.5	odd	6		567.2.i.g.215.3		32
9.7	even	3		567.2.p.e.404.3	yes	32
21.17	even	6		567.2.i.g.269.3		32
63.31	odd	6	inner	567.2.s.g.458.3		32
63.38	even	6		567.2.p.e.80.3	✓	32
63.52	odd	6		567.2.p.e.80.14	yes	32
63.59	even	6	inner	567.2.s.g.458.14		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.i.g.215.3		32	9.5	odd	6
567.2.i.g.215.14		32	9.4	even	3
567.2.i.g.269.3		32	21.17	even	6
567.2.i.g.269.14		32	7.3	odd	6
567.2.p.e.80.3	✓	32	63.38	even	6
567.2.p.e.80.14	yes	32	63.52	odd	6
567.2.p.e.404.3	yes	32	9.7	even	3
567.2.p.e.404.14	yes	32	9.2	odd	6
567.2.s.g.26.3		32	3.2	odd	2	inner
567.2.s.g.26.14		32	1.1	even	1	trivial
567.2.s.g.458.3		32	63.31	odd	6	inner
567.2.s.g.458.14		32	63.59	even	6	inner