Embedded newform 567.2.f.l.190.2

• Label: 567.2.f.l.190.2
• Level: $567$
• Weight: $2$
• Character: 567.190
• Analytic conductor: $4.528$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.52751779461\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.1156923.1
Defining polynomial:	\( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			190.2
Root			\(0.500000 + 2.43956i\) of defining polynomial
Character	\(\chi\)	\(=\)	567.190
Dual form			567.2.f.l.379.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.261988 - 0.453777i) q^{2} +(0.862724 - 1.49428i) q^{4} +(1.10074 - 1.90653i) q^{5} +(-0.500000 - 0.866025i) q^{7} -1.95205 q^{8} -1.15352 q^{10} +(-2.60074 - 4.50461i) q^{11} +(-1.57676 + 2.73103i) q^{13} +(-0.261988 + 0.453777i) q^{14} +(-1.21404 - 2.10277i) q^{16} -3.24943 q^{17} +7.45090 q^{19} +(-1.89926 - 3.28962i) q^{20} +(-1.36272 + 2.36031i) q^{22} +(-2.20147 + 3.81306i) q^{23} +(0.0767598 + 0.132952i) q^{25} +1.65237 q^{26} -1.72545 q^{28} +(0.576760 + 0.998977i) q^{29} +(-1.00000 + 1.73205i) q^{31} +(-2.58817 + 4.48285i) q^{32} +(0.851311 + 1.47451i) q^{34} -2.20147 q^{35} +5.00000 q^{37} +(-1.95205 - 3.38104i) q^{38} +(-2.14869 + 3.72164i) q^{40} +(5.72545 - 9.91677i) q^{41} +(-4.64869 - 8.05177i) q^{43} -8.97487 q^{44} +2.30704 q^{46} +(0.523976 + 0.907554i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(0.0402203 - 0.0696636i) q^{50} +(2.72062 + 4.71225i) q^{52} -0.249425 q^{53} -11.4509 q^{55} +(0.976024 + 1.69052i) q^{56} +(0.302209 - 0.523440i) q^{58} +(4.04795 - 7.01126i) q^{59} +(-4.30221 - 7.45164i) q^{61} +1.04795 q^{62} -2.14386 q^{64} +(3.47119 + 6.01228i) q^{65} +(3.80221 - 6.58562i) q^{67} +(-2.80336 + 4.85556i) q^{68} +(0.576760 + 0.998977i) q^{70} +9.60442 q^{71} +0.846480 q^{73} +(-1.30994 - 2.26888i) q^{74} +(6.42807 - 11.1337i) q^{76} +(-2.60074 + 4.50461i) q^{77} +(3.80221 + 6.58562i) q^{79} -5.34533 q^{80} -6.00000 q^{82} +(-5.72545 - 9.91677i) q^{83} +(-3.57676 + 6.19513i) q^{85} +(-2.43580 + 4.21894i) q^{86} +(5.07676 + 8.79321i) q^{88} +9.24943 q^{89} +3.15352 q^{91} +(3.79853 + 6.57924i) q^{92} +(0.274551 - 0.475537i) q^{94} +(8.20147 - 14.2054i) q^{95} +(1.72545 + 2.98856i) q^{97} +0.523976 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q - 6 * q^4 - 3 * q^5 - 3 * q^7 - 18 * q^8 + 6 * q^10 - 6 * q^11 - 3 * q^13 - 12 * q^16 + 6 * q^17 - 21 * q^20 + 3 * q^22 + 6 * q^23 - 6 * q^25 - 54 * q^26 + 12 * q^28 - 3 * q^29 - 6 * q^31 + 18 * q^32 + 21 * q^34 + 6 * q^35 + 30 * q^37 - 18 * q^38 + 3 * q^40 + 12 * q^41 - 12 * q^43 - 6 * q^44 - 12 * q^46 - 3 * q^49 - 27 * q^50 - 9 * q^52 + 24 * q^53 - 24 * q^55 + 9 * q^56 - 27 * q^58 + 18 * q^59 + 3 * q^61 + 6 * q^64 + 21 * q^65 - 6 * q^67 - 39 * q^68 - 3 * q^70 + 18 * q^73 + 48 * q^76 - 6 * q^77 - 6 * q^79 + 6 * q^80 - 36 * q^82 - 12 * q^83 - 15 * q^85 - 45 * q^86 + 24 * q^88 + 30 * q^89 + 6 * q^91 + 42 * q^92 + 24 * q^94 + 30 * q^95 - 12 * q^97

Character values

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

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

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.261988	−	0.453777i	−0.185254	−	0.320869i	0.758408	−	0.651780i	\(-0.225977\pi\)
							−0.943662	+	0.330911i	\(0.892644\pi\)
\(3\)		0			0
							\(5\)	1.10074	−	1.90653i	0.492264	−	0.852627i	−0.507696	−	0.861536i	\(-0.669503\pi\)
0.999960	+	0.00890964i	\(0.00283606\pi\)
\(7\)	−0.500000	−	0.866025i	−0.188982	−	0.327327i
							\(11\)	−2.60074	−	4.50461i	−0.784151	−	1.35819i	−0.929505	−	0.368810i	\(-0.879765\pi\)
0.145353	−	0.989380i	\(-0.453568\pi\)
\(13\)	−1.57676	+	2.73103i	−0.437314	+	0.757451i	−0.997481	−	0.0709289i	\(-0.977404\pi\)
							0.560167	+	0.828380i	\(0.310737\pi\)
\(17\)	−3.24943			−0.788101			−0.394051	−	0.919089i	\(-0.628927\pi\)
							−0.394051	+	0.919089i	\(0.628927\pi\)
\(19\)	7.45090			1.70935			0.854677	−	0.519161i	\(-0.173755\pi\)
							0.854677	+	0.519161i	\(0.173755\pi\)
\(23\)	−2.20147	+	3.81306i	−0.459039	+	0.795078i	−0.998910	−	0.0466689i	\(-0.985139\pi\)
							0.539872	+	0.841747i	\(0.318473\pi\)
\(29\)	0.576760	+	0.998977i	0.107102	+	0.185505i	0.914595	−	0.404371i	\(-0.132510\pi\)
							−0.807493	+	0.589877i	\(0.799176\pi\)
\(31\)	−1.00000	+	1.73205i	−0.179605	+	0.311086i	−0.941745	−	0.336327i	\(-0.890815\pi\)
							0.762140	+	0.647412i	\(0.224149\pi\)
\(37\)	5.00000			0.821995			0.410997	−	0.911636i	\(-0.365181\pi\)
							0.410997	+	0.911636i	\(0.365181\pi\)
\(41\)	5.72545	−	9.91677i	0.894165	−	1.54874i	0.0593301	−	0.998238i	\(-0.481104\pi\)
							0.834835	−	0.550501i	\(-0.185563\pi\)
\(43\)	−4.64869	−	8.05177i	−0.708918	−	1.22788i	−0.965259	−	0.261296i	\(-0.915850\pi\)
							0.256340	−	0.966587i	\(-0.417483\pi\)
\(47\)	0.523976	+	0.907554i	0.0764298	+	0.132380i	0.901707	−	0.432347i	\(-0.142315\pi\)
							−0.825277	+	0.564728i	\(0.808981\pi\)
\(53\)	−0.249425			−0.0342612			−0.0171306	−	0.999853i	\(-0.505453\pi\)
							−0.0171306	+	0.999853i	\(0.505453\pi\)
\(59\)	4.04795	−	7.01126i	0.526999	−	0.912788i	−0.472506	−	0.881327i	\(-0.656651\pi\)
							0.999505	−	0.0314611i	\(-0.0100160\pi\)
\(61\)	−4.30221	−	7.45164i	−0.550841	−	0.954085i	−0.998214	−	0.0597376i	\(-0.980974\pi\)
							0.447373	−	0.894348i	\(-0.352360\pi\)
\(67\)	3.80221	−	6.58562i	0.464514	−	0.804561i	−0.534666	−	0.845064i	\(-0.679562\pi\)
							0.999179	+	0.0405023i	\(0.0128958\pi\)
\(71\)	9.60442			1.13983			0.569917	−	0.821702i	\(-0.306975\pi\)
							0.569917	+	0.821702i	\(0.306975\pi\)
\(73\)	0.846480			0.0990730			0.0495365	−	0.998772i	\(-0.484226\pi\)
							0.0495365	+	0.998772i	\(0.484226\pi\)
\(79\)	3.80221	+	6.58562i	0.427782	+	0.740940i	0.996676	−	0.0814710i	\(-0.0259618\pi\)
							−0.568894	+	0.822411i	\(0.692628\pi\)
\(83\)	−5.72545	−	9.91677i	−0.628450	−	1.08851i	−0.987863	−	0.155328i	\(-0.950356\pi\)
							0.359413	−	0.933178i	\(-0.382977\pi\)
\(89\)	9.24943			0.980437			0.490219	−	0.871600i	\(-0.336917\pi\)
							0.490219	+	0.871600i	\(0.336917\pi\)
\(97\)	1.72545	+	2.98856i	0.175193	+	0.303443i	0.940228	−	0.340546i	\(-0.110612\pi\)
							−0.765035	+	0.643988i	\(0.777278\pi\)

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.f.l.190.2		6
3.2	odd	2		567.2.f.m.190.2		6
9.2	odd	6		567.2.f.m.379.2		6
9.4	even	3		567.2.a.f.1.2	yes	3
9.5	odd	6		567.2.a.e.1.2	✓	3
9.7	even	3	inner	567.2.f.l.379.2		6
36.23	even	6		9072.2.a.bu.1.3		3
36.31	odd	6		9072.2.a.cb.1.1		3
63.13	odd	6		3969.2.a.n.1.2		3
63.41	even	6		3969.2.a.o.1.2		3

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.a.e.1.2	✓	3	9.5	odd	6
567.2.a.f.1.2	yes	3	9.4	even	3
567.2.f.l.190.2		6	1.1	even	1	trivial
567.2.f.l.379.2		6	9.7	even	3	inner
567.2.f.m.190.2		6	3.2	odd	2
567.2.f.m.379.2		6	9.2	odd	6
3969.2.a.n.1.2		3	63.13	odd	6
3969.2.a.o.1.2		3	63.41	even	6
9072.2.a.bu.1.3		3	36.23	even	6
9072.2.a.cb.1.1		3	36.31	odd	6