Embedded newform 216.2.v.b.11.32

• Label: 216.2.v.b.11.32
• Level: $216$
• Weight: $2$
• Character: 216.11
• 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			11.32
Character	\(\chi\)	\(=\)	216.11
Dual form			216.2.v.b.59.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41380 + 0.0343398i) q^{2} +(1.40807 - 1.00863i) q^{3} +(1.99764 + 0.0970990i) q^{4} +(-2.25586 - 0.821067i) q^{5} +(2.02536 - 1.37765i) q^{6} +(-1.95474 - 0.344674i) q^{7} +(2.82092 + 0.205877i) q^{8} +(0.965315 - 2.84045i) q^{9} +(-3.16114 - 1.23829i) q^{10} +(0.553700 + 1.52128i) q^{11} +(2.91075 - 1.87817i) q^{12} +(3.57715 + 4.26309i) q^{13} +(-2.75177 - 0.554424i) q^{14} +(-4.00457 + 1.11922i) q^{15} +(3.98114 + 0.387938i) q^{16} +(-6.40600 + 3.69851i) q^{17} +(1.46230 - 3.98267i) q^{18} +(-1.39557 + 2.41719i) q^{19} +(-4.42668 - 1.85924i) q^{20} +(-3.10006 + 1.48630i) q^{21} +(0.730579 + 2.16979i) q^{22} +(-0.465506 - 2.64002i) q^{23} +(4.17971 - 2.55539i) q^{24} +(0.584547 + 0.490493i) q^{25} +(4.91098 + 6.14998i) q^{26} +(-1.50575 - 4.97320i) q^{27} +(-3.87141 - 0.878338i) q^{28} +(-0.138481 - 0.116199i) q^{29} +(-5.70008 + 1.44484i) q^{30} +(1.76926 - 0.311969i) q^{31} +(5.61521 + 0.685177i) q^{32} +(2.31406 + 1.58358i) q^{33} +(-9.18379 + 5.00895i) q^{34} +(4.12663 + 2.38251i) q^{35} +(2.20416 - 5.58047i) q^{36} +(-3.55469 + 2.05230i) q^{37} +(-2.05606 + 3.36950i) q^{38} +(9.33677 + 2.39468i) q^{39} +(-6.19458 - 2.78060i) q^{40} +(-2.85334 - 3.40048i) q^{41} +(-4.43390 + 1.99486i) q^{42} +(11.1139 - 4.04511i) q^{43} +(0.958380 + 3.09273i) q^{44} +(-4.50982 + 5.61508i) q^{45} +(-0.567474 - 3.74843i) q^{46} +(2.08588 - 11.8296i) q^{47} +(5.99701 - 3.46927i) q^{48} +(-2.87563 - 1.04664i) q^{49} +(0.809587 + 0.713530i) q^{50} +(-5.28965 + 11.6691i) q^{51} +(6.73193 + 8.86346i) q^{52} -10.1087 q^{53} +(-1.95804 - 7.08280i) q^{54} -3.88642i q^{55} +(-5.44322 - 1.37474i) q^{56} +(0.473009 + 4.81119i) q^{57} +(-0.191794 - 0.169038i) q^{58} +(-1.96718 + 5.40478i) q^{59} +(-8.10836 + 1.84696i) q^{60} +(7.60780 + 1.34146i) q^{61} +(2.51209 - 0.380305i) q^{62} +(-2.86597 + 5.21963i) q^{63} +(7.91523 + 1.16153i) q^{64} +(-4.56929 - 12.5540i) q^{65} +(3.21723 + 2.31833i) q^{66} +(1.38817 - 1.16481i) q^{67} +(-13.1560 + 6.76627i) q^{68} +(-3.31828 - 3.24780i) q^{69} +(5.75240 + 3.51010i) q^{70} +(-1.25119 - 2.16713i) q^{71} +(3.30786 - 7.81396i) q^{72} +(-3.79038 + 6.56514i) q^{73} +(-5.09608 + 2.77947i) q^{74} +(1.31781 + 0.101054i) q^{75} +(-3.02255 + 4.69318i) q^{76} +(-0.557996 - 3.16455i) q^{77} +(13.1181 + 3.70621i) q^{78} +(6.77140 - 8.06984i) q^{79} +(-8.66239 - 4.14392i) q^{80} +(-7.13633 - 5.48386i) q^{81} +(-3.91727 - 4.90557i) q^{82} +(0.718131 - 0.855836i) q^{83} +(-6.33713 + 2.66807i) q^{84} +(17.4878 - 3.08357i) q^{85} +(15.8516 - 5.33732i) q^{86} +(-0.312194 - 0.0239400i) q^{87} +(1.24875 + 4.40541i) q^{88} +(9.87804 + 5.70309i) q^{89} +(-6.56879 + 7.78372i) q^{90} +(-5.52304 - 9.56619i) q^{91} +(-0.673572 - 5.31901i) q^{92} +(2.17658 - 2.22381i) q^{93} +(3.35523 - 16.6530i) q^{94} +(5.13289 - 4.30701i) q^{95} +(8.59769 - 4.69891i) q^{96} +(-5.25587 + 1.91298i) q^{97} +(-4.02961 - 1.57849i) q^{98} +(4.85561 - 0.104246i) 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 \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{13}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41380	+	0.0343398i	0.999705	+	0.0242819i
							\(3\)	1.40807	−	1.00863i	0.812949	−	0.582335i
\(5\)	−2.25586	−	0.821067i	−1.00885	−	0.367192i								−0.215861	−	0.976424i	\(-0.569256\pi\)
							−0.792992	+	0.609232i	\(0.791478\pi\)
\(7\)	−1.95474	−	0.344674i	−0.738823	−	0.130274i	−0.208443	−	0.978034i	\(-0.566840\pi\)
							−0.530380	+	0.847760i	\(0.677951\pi\)
\(11\)	0.553700	+	1.52128i	0.166947	+	0.458683i	0.994750	−	0.102336i	\(-0.0326317\pi\)
							−0.827803	+	0.561019i	\(0.810410\pi\)
\(13\)	3.57715	+	4.26309i	0.992124	+	1.18237i	0.983223	+	0.182407i	\(0.0583889\pi\)
							0.00890108	+	0.999960i	\(0.497167\pi\)
\(17\)	−6.40600	+	3.69851i	−1.55368	+	0.897020i	−0.555846	+	0.831285i	\(0.687606\pi\)
							−0.997837	+	0.0657345i	\(0.979061\pi\)
\(19\)	−1.39557	+	2.41719i	−0.320165	+	0.554543i	−0.980522	−	0.196410i	\(-0.937072\pi\)
							0.660357	+	0.750952i	\(0.270405\pi\)
\(23\)	−0.465506	−	2.64002i	−0.0970648	−	0.550482i	−0.994095	−	0.108513i	\(-0.965391\pi\)
							0.897030	−	0.441969i	\(-0.145720\pi\)
\(29\)	−0.138481	−	0.116199i	−0.0257153	−	0.0215777i	0.629839	−	0.776725i	\(-0.283121\pi\)
							−0.655555	+	0.755148i	\(0.727565\pi\)
\(31\)	1.76926	−	0.311969i	0.317769	−	0.0560313i	−0.0124887	−	0.999922i	\(-0.503975\pi\)
							0.330258	+	0.943891i	\(0.392864\pi\)
\(37\)	−3.55469	+	2.05230i	−0.584387	+	0.337396i	−0.762875	−	0.646546i	\(-0.776213\pi\)
							0.178488	+	0.983942i	\(0.442880\pi\)
\(41\)	−2.85334	−	3.40048i	−0.445617	−	0.531066i	0.495743	−	0.868469i	\(-0.334896\pi\)
							−0.941360	+	0.337404i	\(0.890451\pi\)
\(43\)	11.1139	−	4.04511i	1.69485	−	0.616874i	0.699625	−	0.714510i	\(-0.253350\pi\)
							0.995222	+	0.0976364i	\(0.0311282\pi\)
\(47\)	2.08588	−	11.8296i	0.304257	−	1.72552i	−0.322727	−	0.946492i	\(-0.604599\pi\)
							0.626983	−	0.779033i	\(-0.284289\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.2.v.b.11.32	yes	192
3.2	odd	2		648.2.v.b.35.1		192
4.3	odd	2		864.2.bh.b.335.5		192
8.3	odd	2	inner	216.2.v.b.11.16	✓	192
8.5	even	2		864.2.bh.b.335.6		192
24.11	even	2		648.2.v.b.35.17		192
27.5	odd	18	inner	216.2.v.b.59.16	yes	192
27.22	even	9		648.2.v.b.611.17		192
108.59	even	18		864.2.bh.b.815.6		192
216.5	odd	18		864.2.bh.b.815.5		192
216.59	even	18	inner	216.2.v.b.59.32	yes	192
216.211	odd	18		648.2.v.b.611.1		192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.b.11.16	✓	192	8.3	odd	2	inner
216.2.v.b.11.32	yes	192	1.1	even	1	trivial
216.2.v.b.59.16	yes	192	27.5	odd	18	inner
216.2.v.b.59.32	yes	192	216.59	even	18	inner
648.2.v.b.35.1		192	3.2	odd	2
648.2.v.b.35.17		192	24.11	even	2
648.2.v.b.611.1		192	216.211	odd	18
648.2.v.b.611.17		192	27.22	even	9
864.2.bh.b.335.5		192	4.3	odd	2
864.2.bh.b.335.6		192	8.5	even	2
864.2.bh.b.815.5		192	216.5	odd	18
864.2.bh.b.815.6		192	108.59	even	18