Embedded newform 216.2.v.b.59.2

• Label: 216.2.v.b.59.2
• Level: $216$
• Weight: $2$
• Character: 216.59
• Analytic conductor: $1.725$
• Analytic rank: $0$
• Dimension: $192$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	216.v (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.72476868366\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			59.2
Character	\(\chi\)	\(=\)	216.59
Dual form			216.2.v.b.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40972 - 0.112665i) q^{2} +(0.00307742 + 1.73205i) q^{3} +(1.97461 + 0.317652i) q^{4} +(2.47648 - 0.901367i) q^{5} +(0.190803 - 2.44205i) q^{6} +(2.55949 - 0.451307i) q^{7} +(-2.74786 - 0.670270i) q^{8} +(-2.99998 + 0.0106605i) q^{9} +(-3.59270 + 0.991660i) q^{10} +(0.556608 - 1.52927i) q^{11} +(-0.544112 + 3.42110i) q^{12} +(1.88389 - 2.24514i) q^{13} +(-3.65901 + 0.347851i) q^{14} +(1.56883 + 4.28662i) q^{15} +(3.79819 + 1.25448i) q^{16} +(-3.28845 - 1.89859i) q^{17} +(4.23033 + 0.322965i) q^{18} +(4.30904 + 7.46347i) q^{19} +(5.17642 - 0.993190i) q^{20} +(0.789562 + 4.43177i) q^{21} +(-0.956956 + 2.09313i) q^{22} +(-1.07009 + 6.06879i) q^{23} +(1.15248 - 4.76149i) q^{24} +(1.49029 - 1.25050i) q^{25} +(-2.90871 + 2.95276i) q^{26} +(-0.0276967 - 5.19608i) q^{27} +(5.19736 - 0.0781292i) q^{28} +(3.88174 - 3.25716i) q^{29} +(-1.72866 - 6.21968i) q^{30} +(-3.57300 - 0.630016i) q^{31} +(-5.21305 - 2.19639i) q^{32} +(2.65048 + 0.959365i) q^{33} +(4.42189 + 3.04697i) q^{34} +(5.93174 - 3.42469i) q^{35} +(-5.92719 - 0.931900i) q^{36} +(-6.40179 - 3.69607i) q^{37} +(-5.23366 - 11.0069i) q^{38} +(3.89448 + 3.25608i) q^{39} +(-7.40919 + 0.816916i) q^{40} +(-4.43232 + 5.28224i) q^{41} +(-0.613754 - 6.33650i) q^{42} +(-3.77811 - 1.37512i) q^{43} +(1.58486 - 2.84290i) q^{44} +(-7.41980 + 2.73048i) q^{45} +(2.19227 - 8.43472i) q^{46} +(0.253807 + 1.43941i) q^{47} +(-2.16113 + 6.58252i) q^{48} +(-0.230541 + 0.0839100i) q^{49} +(-2.24178 + 1.59495i) q^{50} +(3.27833 - 5.70160i) q^{51} +(4.43313 - 3.83485i) q^{52} +0.180986 q^{53} +(-0.546372 + 7.32813i) q^{54} -4.28892i q^{55} +(-7.33562 - 0.475421i) q^{56} +(-12.9138 + 7.48643i) q^{57} +(-5.83912 + 4.15435i) q^{58} +(0.253090 + 0.695359i) q^{59} +(1.73618 + 8.96275i) q^{60} +(11.1190 - 1.96057i) q^{61} +(4.96594 + 1.29070i) q^{62} +(-7.67361 + 1.38120i) q^{63} +(7.10148 + 3.68362i) q^{64} +(2.64174 - 7.25812i) q^{65} +(-3.62834 - 1.65105i) q^{66} +(-10.5159 - 8.82387i) q^{67} +(-5.89033 - 4.79357i) q^{68} +(-10.5147 - 1.83477i) q^{69} +(-8.74793 + 4.15955i) q^{70} +(2.57253 - 4.45576i) q^{71} +(8.25068 + 1.98150i) q^{72} +(-2.62831 - 4.55236i) q^{73} +(8.60830 + 5.93168i) q^{74} +(2.17052 + 2.57741i) q^{75} +(6.13789 + 16.1062i) q^{76} +(0.734463 - 4.16534i) q^{77} +(-5.12328 - 5.02893i) q^{78} +(-6.72831 - 8.01849i) q^{79} +(10.5369 - 0.316864i) q^{80} +(8.99977 - 0.0639625i) q^{81} +(6.84345 - 6.94710i) q^{82} +(7.39656 + 8.81488i) q^{83} +(0.151318 + 9.00184i) q^{84} +(-9.85513 - 1.73773i) q^{85} +(5.17114 + 2.36419i) q^{86} +(5.65351 + 6.71333i) q^{87} +(-2.55450 + 3.82914i) q^{88} +(-0.211259 + 0.121971i) q^{89} +(10.7675 - 3.01326i) q^{90} +(3.80856 - 6.59662i) q^{91} +(-4.04078 + 11.6436i) q^{92} +(1.08022 - 6.19054i) q^{93} +(-0.195625 - 2.05776i) q^{94} +(17.3986 + 14.5992i) q^{95} +(3.78821 - 9.03601i) q^{96} +(1.95441 + 0.711348i) q^{97} +(0.334451 - 0.0923156i) q^{98} +(-1.65351 + 4.59371i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	192 * q - 6 * q^2 - 12 * q^3 - 6 * q^4 - 6 * q^6 - 9 * q^8 - 12 * q^9 - 3 * q^10 - 30 * q^11 - 3 * q^12 + 9 * q^14 - 6 * q^16 - 18 * q^17 + 63 * q^18 - 6 * q^19 - 27 * q^20 - 18 * q^22 - 30 * q^24 - 12 * q^25 + 18 * q^27 - 12 * q^28 - 21 * q^30 - 36 * q^32 - 30 * q^33 + 12 * q^34 - 18 * q^35 - 36 * q^36 - 102 * q^38 + 9 * q^40 + 18 * q^41 - 6 * q^42 - 42 * q^43 - 81 * q^44 - 3 * q^46 - 81 * q^48 - 12 * q^49 + 57 * q^50 - 18 * q^51 + 21 * q^52 + 78 * q^54 - 69 * q^56 - 36 * q^57 - 33 * q^58 - 84 * q^59 - 54 * q^60 + 90 * q^62 - 51 * q^64 - 12 * q^65 + 87 * q^66 + 30 * q^67 + 63 * q^68 - 33 * q^70 + 12 * q^72 - 6 * q^73 + 51 * q^74 - 96 * q^75 + 30 * q^76 + 90 * q^78 - 12 * q^81 - 12 * q^82 - 72 * q^83 - 48 * q^84 + 42 * q^86 - 78 * q^88 + 144 * q^89 + 120 * q^90 - 6 * q^91 - 3 * q^92 - 33 * q^94 - 78 * q^96 - 42 * q^97 + 162 * q^98 - 12 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{18}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40972	−	0.112665i	−0.996822	−	0.0796663i
							\(3\)	0.00307742	+	1.73205i	0.00177675	+	0.999998i
\(5\)	2.47648	−	0.901367i	1.10752	−	0.403103i								0.277435	−	0.960744i	\(-0.410516\pi\)
							0.830082	+	0.557641i	\(0.188293\pi\)
\(7\)	2.55949	−	0.451307i	0.967396	−	0.170578i	0.332438	−	0.943125i	\(-0.392129\pi\)
							0.634958	+	0.772547i	\(0.281018\pi\)
\(11\)	0.556608	−	1.52927i	0.167824	−	0.461091i	−0.827061	−	0.562113i	\(-0.809989\pi\)
							0.994884	+	0.101021i	\(0.0322110\pi\)
\(13\)	1.88389	−	2.24514i	0.522498	−	0.622689i	−0.438672	−	0.898647i	\(-0.644551\pi\)
							0.961169	+	0.275959i	\(0.0889953\pi\)
\(17\)	−3.28845	−	1.89859i	−0.797567	−	0.460476i	0.0450525	−	0.998985i	\(-0.485654\pi\)
							−0.842620	+	0.538509i	\(0.818988\pi\)
\(19\)	4.30904	+	7.46347i	0.988561	+	1.71224i	0.624895	+	0.780708i	\(0.285142\pi\)
							0.363666	+	0.931530i	\(0.381525\pi\)
\(23\)	−1.07009	+	6.06879i	−0.223129	+	1.26543i	0.643100	+	0.765783i	\(0.277648\pi\)
							−0.866229	+	0.499647i	\(0.833463\pi\)
\(29\)	3.88174	−	3.25716i	0.720820	−	0.604840i	−0.206792	−	0.978385i	\(-0.566302\pi\)
							0.927612	+	0.373545i	\(0.121858\pi\)
\(31\)	−3.57300	−	0.630016i	−0.641729	−	0.113154i	−0.156692	−	0.987648i	\(-0.550083\pi\)
							−0.485037	+	0.874493i	\(0.661194\pi\)
\(37\)	−6.40179	−	3.69607i	−1.05245	−	0.607631i	−0.129114	−	0.991630i	\(-0.541213\pi\)
							−0.923333	+	0.383999i	\(0.874547\pi\)
\(41\)	−4.43232	+	5.28224i	−0.692212	+	0.824947i	−0.991621	−	0.129178i	\(-0.958766\pi\)
							0.299409	+	0.954125i	\(0.403211\pi\)
\(43\)	−3.77811	−	1.37512i	−0.576156	−	0.209704i	0.0374734	−	0.999298i	\(-0.488069\pi\)
							−0.613630	+	0.789594i	\(0.710291\pi\)
\(47\)	0.253807	+	1.43941i	0.0370215	+	0.209960i	0.997707	−	0.0676790i	\(-0.0215594\pi\)
							−0.960686	+	0.277639i	\(0.910448\pi\)
\(53\)	0.180986			0.0248604			0.0124302	−	0.999923i	\(-0.496043\pi\)
							0.0124302	+	0.999923i	\(0.496043\pi\)
\(59\)	0.253090	+	0.695359i	0.0329495	+	0.0905280i	0.955077	−	0.296359i	\(-0.0957725\pi\)
							−0.922127	+	0.386887i	\(0.873550\pi\)
\(61\)	11.1190	−	1.96057i	1.42364	−	0.251026i	0.591817	−	0.806072i	\(-0.298411\pi\)
							0.831819	+	0.555047i	\(0.187300\pi\)
\(67\)	−10.5159	−	8.82387i	−1.28472	−	1.07801i	−0.992575	−	0.121633i	\(-0.961187\pi\)
							−0.292144	−	0.956374i	\(-0.594369\pi\)
\(71\)	2.57253	−	4.45576i	0.305303	−	0.528801i	−0.672025	−	0.740528i	\(-0.734575\pi\)
							0.977329	+	0.211727i	\(0.0679088\pi\)
\(73\)	−2.62831	−	4.55236i	−0.307620	−	0.532814i	0.670221	−	0.742161i	\(-0.266199\pi\)
							−0.977841	+	0.209348i	\(0.932866\pi\)
\(79\)	−6.72831	−	8.01849i	−0.756994	−	0.902151i	0.240660	−	0.970610i	\(-0.422636\pi\)
							−0.997654	+	0.0684588i	\(0.978192\pi\)
\(83\)	7.39656	+	8.81488i	0.811878	+	0.967559i	0.999893	−	0.0146072i	\(-0.00464979\pi\)
							−0.188015	+	0.982166i	\(0.560205\pi\)
\(89\)	−0.211259	+	0.121971i	−0.0223934	+	0.0129289i	−0.511155	−	0.859489i	\(-0.670782\pi\)
							0.488761	+	0.872417i	\(0.337449\pi\)
\(97\)	1.95441	+	0.711348i	0.198441	+	0.0722264i	0.439329	−	0.898326i	\(-0.355216\pi\)
							−0.240888	+	0.970553i	\(0.577439\pi\)

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	216.2.v.b.59.2	yes	192
3.2	odd	2		648.2.v.b.611.31		192
4.3	odd	2		864.2.bh.b.815.18		192
8.3	odd	2	inner	216.2.v.b.59.17	yes	192
8.5	even	2		864.2.bh.b.815.17		192
24.11	even	2		648.2.v.b.611.16		192
27.11	odd	18	inner	216.2.v.b.11.17	yes	192
27.16	even	9		648.2.v.b.35.16		192
108.11	even	18		864.2.bh.b.335.17		192
216.11	even	18	inner	216.2.v.b.11.2	✓	192
216.43	odd	18		648.2.v.b.35.31		192
216.173	odd	18		864.2.bh.b.335.18		192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.b.11.2	✓	192	216.11	even	18	inner
216.2.v.b.11.17	yes	192	27.11	odd	18	inner
216.2.v.b.59.2	yes	192	1.1	even	1	trivial
216.2.v.b.59.17	yes	192	8.3	odd	2	inner
648.2.v.b.35.16		192	27.16	even	9
648.2.v.b.35.31		192	216.43	odd	18
648.2.v.b.611.16		192	24.11	even	2
648.2.v.b.611.31		192	3.2	odd	2
864.2.bh.b.335.17		192	108.11	even	18
864.2.bh.b.335.18		192	216.173	odd	18
864.2.bh.b.815.17		192	8.5	even	2
864.2.bh.b.815.18		192	4.3	odd	2