Embedded newform 567.2.ba.a.143.15

• Label: 567.2.ba.a.143.15
• Level: $567$
• Weight: $2$
• Character: 567.143
• Analytic conductor: $4.528$
• Analytic rank: $0$
• Dimension: $132$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.52751779461\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{18})\)
Twist minimal:	no (minimal twist has level 189)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			143.15
Character	\(\chi\)	\(=\)	567.143
Dual form			567.2.ba.a.341.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.582560 - 0.694268i) q^{2} +(0.204665 + 1.16071i) q^{4} +(0.931176 - 0.781349i) q^{5} +(-2.06750 - 1.65089i) q^{7} +(2.49483 + 1.44039i) q^{8} -1.10167i q^{10} +(2.30628 - 2.74851i) q^{11} +(0.206055 - 0.566132i) q^{13} +(-2.35060 + 0.473658i) q^{14} +(0.238336 - 0.0867472i) q^{16} +7.94939 q^{17} -1.41477i q^{19} +(1.09750 + 0.920911i) q^{20} +(-0.564660 - 3.20235i) q^{22} +(0.349041 - 0.958983i) q^{23} +(-0.611659 + 3.46889i) q^{25} +(-0.273008 - 0.472864i) q^{26} +(1.49306 - 2.73765i) q^{28} +(0.413325 + 1.13560i) q^{29} +(8.17053 - 1.44069i) q^{31} +(-1.89196 + 5.19810i) q^{32} +(4.63100 - 5.51901i) q^{34} +(-3.21513 + 0.0781722i) q^{35} +(-4.22521 + 7.31828i) q^{37} +(-0.982231 - 0.824190i) q^{38} +(3.44858 - 0.608078i) q^{40} +(-6.73814 - 2.45248i) q^{41} +(1.41438 - 8.02135i) q^{43} +(3.66224 + 2.11440i) q^{44} +(-0.462454 - 0.800993i) q^{46} +(0.0548081 - 0.310832i) q^{47} +(1.54913 + 6.82643i) q^{49} +(2.05201 + 2.44549i) q^{50} +(0.699288 + 0.123303i) q^{52} +(-5.74259 - 3.31549i) q^{53} -4.36136i q^{55} +(-2.78014 - 7.09671i) q^{56} +(1.02920 + 0.374598i) q^{58} +(0.936973 + 0.341030i) q^{59} +(-12.0683 - 2.12796i) q^{61} +(3.75960 - 6.51182i) q^{62} +(2.76033 + 4.78103i) q^{64} +(-0.250473 - 0.688170i) q^{65} +(-7.90859 + 6.63609i) q^{67} +(1.62696 + 9.22694i) q^{68} +(-1.81873 + 2.27770i) q^{70} +(0.952881 - 0.550146i) q^{71} +(-0.808112 + 0.466563i) q^{73} +(2.61941 + 7.19676i) q^{74} +(1.64214 - 0.289554i) q^{76} +(-9.30573 + 1.87515i) q^{77} +(-10.3048 - 8.64674i) q^{79} +(0.154153 - 0.267000i) q^{80} +(-5.62805 + 3.24936i) q^{82} +(5.66119 - 2.06050i) q^{83} +(7.40228 - 6.21125i) q^{85} +(-4.74500 - 5.65488i) q^{86} +(9.71272 - 3.53514i) q^{88} +5.92334 q^{89} +(-1.36064 + 0.830305i) q^{91} +(1.18454 + 0.208866i) q^{92} +(-0.183872 - 0.219130i) q^{94} +(-1.10543 - 1.31740i) q^{95} +(-8.29293 - 1.46227i) q^{97} +(5.64183 + 2.90130i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	132 * q + 3 * q^2 - 3 * q^4 + 9 * q^5 - 6 * q^7 + 18 * q^8 + 9 * q^11 - 3 * q^14 + 3 * q^16 + 18 * q^17 - 18 * q^20 - 12 * q^22 + 6 * q^23 - 3 * q^25 - 12 * q^28 - 6 * q^29 - 9 * q^31 - 3 * q^32 - 18 * q^34 - 18 * q^35 + 3 * q^37 + 99 * q^38 - 54 * q^40 - 12 * q^43 + 9 * q^44 + 3 * q^46 - 45 * q^47 - 24 * q^49 + 9 * q^50 - 9 * q^52 + 45 * q^53 - 3 * q^56 - 3 * q^58 - 36 * q^59 - 9 * q^61 + 99 * q^62 + 18 * q^64 - 69 * q^65 - 3 * q^67 - 36 * q^68 + 66 * q^70 - 18 * q^71 - 9 * q^73 - 75 * q^74 + 36 * q^76 - 15 * q^77 - 21 * q^79 - 72 * q^80 - 18 * q^82 + 90 * q^83 + 9 * q^85 + 105 * q^86 - 63 * q^88 + 18 * q^89 + 6 * q^91 - 150 * q^92 - 9 * q^94 - 45 * q^95 - 27 * q^98

Character values

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

\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{18}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.582560	−	0.694268i	0.411932	−	0.490922i	−0.519688	−	0.854356i	\(-0.673952\pi\)
							0.931620	+	0.363435i	\(0.118396\pi\)
\(3\)		0			0
							\(5\)	0.931176	−	0.781349i	0.416434	−	0.349430i	−0.410370	−	0.911919i	\(-0.634601\pi\)
0.826805	+	0.562489i	\(0.190156\pi\)
\(7\)	−2.06750	−	1.65089i	−0.781442	−	0.623978i
							\(11\)	2.30628	−	2.74851i	0.695369	−	0.828708i	−0.296625	−	0.954994i	\(-0.595861\pi\)
0.991994	+	0.126286i	\(0.0403056\pi\)
\(13\)	0.206055	−	0.566132i	0.0571495	−	0.157017i	−0.907833	−	0.419333i	\(-0.862264\pi\)
							0.964982	+	0.262316i	\(0.0844863\pi\)
\(17\)	7.94939			1.92801			0.964005	−	0.265883i	\(-0.0856635\pi\)
							0.964005	+	0.265883i	\(0.0856635\pi\)
\(19\)		−	1.41477i		−	0.324571i	−0.986744	−	0.162286i	\(-0.948113\pi\)
							0.986744	−	0.162286i	\(-0.0518866\pi\)
\(23\)	0.349041	−	0.958983i	0.0727802	−	0.199962i	−0.897969	−	0.440060i	\(-0.854957\pi\)
							0.970749	+	0.240098i	\(0.0771795\pi\)
\(29\)	0.413325	+	1.13560i	0.0767526	+	0.210876i	0.972135	−	0.234422i	\(-0.0753199\pi\)
							−0.895382	+	0.445298i	\(0.853098\pi\)
\(31\)	8.17053	−	1.44069i	1.46747	−	0.258755i	0.617912	−	0.786247i	\(-0.287979\pi\)
							0.849560	+	0.527493i	\(0.176868\pi\)
\(37\)	−4.22521	+	7.31828i	−0.694620	+	1.20312i	0.275689	+	0.961247i	\(0.411094\pi\)
							−0.970309	+	0.241870i	\(0.922239\pi\)
\(41\)	−6.73814	−	2.45248i	−1.05232	−	0.383013i	−0.242782	−	0.970081i	\(-0.578060\pi\)
							−0.809538	+	0.587067i	\(0.800282\pi\)
\(43\)	1.41438	−	8.02135i	0.215691	−	1.22324i	−0.664012	−	0.747722i	\(-0.731148\pi\)
							0.879703	−	0.475523i	\(-0.157741\pi\)
\(47\)	0.0548081	−	0.310832i	0.00799458	−	0.0453395i	−0.980550	−	0.196272i	\(-0.937116\pi\)
							0.988544	+	0.150932i	\(0.0482276\pi\)
\(53\)	−5.74259	−	3.31549i	−0.788806	−	0.455417i	0.0507361	−	0.998712i	\(-0.483843\pi\)
							−0.839542	+	0.543295i	\(0.817177\pi\)
\(59\)	0.936973	+	0.341030i	0.121983	+	0.0443983i	0.402291	−	0.915512i	\(-0.368214\pi\)
							−0.280307	+	0.959910i	\(0.590436\pi\)
\(61\)	−12.0683	−	2.12796i	−1.54518	−	0.272458i	−0.664911	−	0.746923i	\(-0.731530\pi\)
							−0.880274	+	0.474465i	\(0.842641\pi\)
\(67\)	−7.90859	+	6.63609i	−0.966188	+	0.810728i	−0.981949	−	0.189148i	\(-0.939427\pi\)
							0.0157609	+	0.999876i	\(0.494983\pi\)
\(71\)	0.952881	−	0.550146i	0.113086	−	0.0652904i	−0.442390	−	0.896823i	\(-0.645869\pi\)
							0.555476	+	0.831532i	\(0.312536\pi\)
\(73\)	−0.808112	+	0.466563i	−0.0945823	+	0.0546071i	−0.546545	−	0.837430i	\(-0.684057\pi\)
							0.451963	+	0.892037i	\(0.350724\pi\)
\(79\)	−10.3048	−	8.64674i	−1.15938	−	0.972834i	−0.159482	−	0.987201i	\(-0.550982\pi\)
							−0.999897	+	0.0143668i	\(0.995427\pi\)
\(83\)	5.66119	−	2.06050i	0.621396	−	0.226170i	−0.0120862	−	0.999927i	\(-0.503847\pi\)
							0.633482	+	0.773757i	\(0.281625\pi\)
\(89\)	5.92334			0.627873			0.313937	−	0.949444i	\(-0.398352\pi\)
							0.313937	+	0.949444i	\(0.398352\pi\)
\(97\)	−8.29293	−	1.46227i	−0.842019	−	0.148471i	−0.264029	−	0.964515i	\(-0.585052\pi\)
							−0.577990	+	0.816044i	\(0.696163\pi\)

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.ba.a.143.15		132
3.2	odd	2		189.2.ba.a.101.8	✓	132
7.5	odd	6		567.2.bd.a.467.8		132
21.5	even	6		189.2.bd.a.47.15	yes	132
27.4	even	9		189.2.bd.a.185.15	yes	132
27.23	odd	18		567.2.bd.a.17.8		132
189.131	even	18	inner	567.2.ba.a.341.15		132
189.166	odd	18		189.2.ba.a.131.8	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.ba.a.101.8	✓	132	3.2	odd	2
189.2.ba.a.131.8	yes	132	189.166	odd	18
189.2.bd.a.47.15	yes	132	21.5	even	6
189.2.bd.a.185.15	yes	132	27.4	even	9
567.2.ba.a.143.15		132	1.1	even	1	trivial
567.2.ba.a.341.15		132	189.131	even	18	inner
567.2.bd.a.17.8		132	27.23	odd	18
567.2.bd.a.467.8		132	7.5	odd	6