Embedded newform 216.2.v.b.59.24

• Label: 216.2.v.b.59.24
• 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			59.24
Character	\(\chi\)	\(=\)	216.59
Dual form			216.2.v.b.11.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.860856 + 1.12202i) q^{2} +(1.59734 - 0.669704i) q^{3} +(-0.517853 + 1.93179i) q^{4} +(3.00936 - 1.09532i) q^{5} +(2.12650 + 1.21573i) q^{6} +(-4.58614 + 0.808660i) q^{7} +(-2.61331 + 1.08196i) q^{8} +(2.10299 - 2.13949i) q^{9} +(3.81960 + 2.43365i) q^{10} +(-0.303411 + 0.833616i) q^{11} +(0.466542 + 3.43254i) q^{12} +(1.22056 - 1.45461i) q^{13} +(-4.85534 - 4.44959i) q^{14} +(4.07344 - 3.76498i) q^{15} +(-3.46366 - 2.00077i) q^{16} +(-4.03247 - 2.32815i) q^{17} +(4.21092 + 0.517807i) q^{18} +(-0.171350 - 0.296787i) q^{19} +(0.557523 + 6.38069i) q^{20} +(-6.78406 + 4.36306i) q^{21} +(-1.19653 + 0.377190i) q^{22} +(-1.00156 + 5.68012i) q^{23} +(-3.44975 + 3.47839i) q^{24} +(4.02633 - 3.37849i) q^{25} +(2.68283 + 0.117284i) q^{26} +(1.92637 - 4.82588i) q^{27} +(0.812780 - 9.27824i) q^{28} +(-4.16935 + 3.49850i) q^{29} +(7.73102 + 1.32937i) q^{30} +(7.80815 + 1.37679i) q^{31} +(-0.736809 - 5.60866i) q^{32} +(0.0736241 + 1.53476i) q^{33} +(-0.859151 - 6.52870i) q^{34} +(-12.9156 + 7.45683i) q^{35} +(3.04401 + 5.17049i) q^{36} +(-2.31812 - 1.33837i) q^{37} +(0.185493 - 0.447749i) q^{38} +(0.975496 - 3.14092i) q^{39} +(-6.67930 + 6.11841i) q^{40} +(0.0346849 - 0.0413359i) q^{41} +(-10.7355 - 3.85588i) q^{42} +(-2.85915 - 1.04064i) q^{43} +(-1.45325 - 1.01782i) q^{44} +(3.98525 - 8.74195i) q^{45} +(-7.23540 + 3.76600i) q^{46} +(-1.90179 - 10.7856i) q^{47} +(-6.87256 - 0.876288i) q^{48} +(13.8009 - 5.02311i) q^{49} +(7.25682 + 1.60922i) q^{50} +(-8.00039 - 1.01828i) q^{51} +(2.17793 + 3.11115i) q^{52} +1.27415 q^{53} +(7.07306 - 1.99296i) q^{54} +2.84099i q^{55} +(11.1100 - 7.07528i) q^{56} +(-0.472464 - 0.359316i) q^{57} +(-7.51460 - 1.66638i) q^{58} +(2.43399 + 6.68734i) q^{59} +(5.16372 + 9.81875i) q^{60} +(8.19039 - 1.44419i) q^{61} +(5.17691 + 9.94610i) q^{62} +(-7.91450 + 11.5126i) q^{63} +(5.65874 - 5.65497i) q^{64} +(2.07985 - 5.71435i) q^{65} +(-1.65865 + 1.40382i) q^{66} +(-6.31205 - 5.29644i) q^{67} +(6.58572 - 6.58426i) q^{68} +(2.20417 + 9.74384i) q^{69} +(-19.4852 - 8.07230i) q^{70} +(-0.186788 + 0.323526i) q^{71} +(-3.18093 + 7.86649i) q^{72} +(6.29550 + 10.9041i) q^{73} +(-0.493895 - 3.75312i) q^{74} +(4.16883 - 8.09304i) q^{75} +(0.662065 - 0.177321i) q^{76} +(0.717374 - 4.06843i) q^{77} +(4.36393 - 1.60936i) q^{78} +(-2.54509 - 3.03312i) q^{79} +(-12.6149 - 2.22724i) q^{80} +(-0.154829 - 8.99867i) q^{81} +(0.0762384 + 0.00333288i) q^{82} +(8.90996 + 10.6185i) q^{83} +(-4.91538 - 15.3648i) q^{84} +(-14.6852 - 2.58940i) q^{85} +(-1.29369 - 4.10386i) q^{86} +(-4.31692 + 8.38053i) q^{87} +(-0.109029 - 2.50677i) q^{88} +(-6.35012 + 3.66624i) q^{89} +(13.2394 - 3.05404i) q^{90} +(-4.42138 + 7.65805i) q^{91} +(-10.4542 - 4.87627i) q^{92} +(13.3943 - 3.02994i) q^{93} +(10.4645 - 11.4187i) q^{94} +(-0.840731 - 0.705457i) q^{95} +(-4.93308 - 8.46550i) q^{96} +(-0.833364 - 0.303320i) q^{97} +(17.5166 + 11.1607i) q^{98} +(1.14544 + 2.40223i) 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 \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.860856	+	1.12202i	0.608717	+	0.793387i
							\(3\)	1.59734	−	0.669704i	0.922225	−	0.386654i
\(5\)	3.00936	−	1.09532i	1.34583	−	0.489842i								0.434185	−	0.900824i	\(-0.357036\pi\)
							0.911643	+	0.410982i	\(0.134814\pi\)
\(7\)	−4.58614	+	0.808660i	−1.73340	+	0.305645i	−0.949155	−	0.314810i	\(-0.898059\pi\)
							−0.784242	+	0.620455i	\(0.786948\pi\)
\(11\)	−0.303411	+	0.833616i	−0.0914819	+	0.251345i	−0.976992	−	0.213275i	\(-0.931587\pi\)
							0.885510	+	0.464620i	\(0.153809\pi\)
\(13\)	1.22056	−	1.45461i	0.338523	−	0.403436i	−0.569747	−	0.821820i	\(-0.692959\pi\)
							0.908270	+	0.418384i	\(0.137403\pi\)
\(17\)	−4.03247	−	2.32815i	−0.978017	−	0.564658i	−0.0763460	−	0.997081i	\(-0.524325\pi\)
							−0.901671	+	0.432423i	\(0.857659\pi\)
\(19\)	−0.171350	−	0.296787i	−0.0393104	−	0.0680876i	0.845701	−	0.533657i	\(-0.179183\pi\)
							−0.885011	+	0.465570i	\(0.845849\pi\)
\(23\)	−1.00156	+	5.68012i	−0.208839	+	1.18439i	0.682443	+	0.730939i	\(0.260918\pi\)
							−0.891282	+	0.453449i	\(0.850194\pi\)
\(29\)	−4.16935	+	3.49850i	−0.774229	+	0.649655i	−0.941788	−	0.336207i	\(-0.890856\pi\)
							0.167559	+	0.985862i	\(0.446411\pi\)
\(31\)	7.80815	+	1.37679i	1.40238	+	0.247278i	0.823123	−	0.567864i	\(-0.192230\pi\)
							0.579262	+	0.815142i	\(0.303341\pi\)
\(37\)	−2.31812	−	1.33837i	−0.381097	−	0.220026i	0.297199	−	0.954816i	\(-0.403948\pi\)
							−0.678295	+	0.734789i	\(0.737281\pi\)
\(41\)	0.0346849	−	0.0413359i	0.00541687	−	0.00645558i	−0.763329	−	0.646010i	\(-0.776437\pi\)
							0.768746	+	0.639554i	\(0.220881\pi\)
\(43\)	−2.85915	−	1.04064i	−0.436016	−	0.158697i	0.114681	−	0.993402i	\(-0.463416\pi\)
							−0.550696	+	0.834706i	\(0.685638\pi\)
\(47\)	−1.90179	−	10.7856i	−0.277404	−	1.57324i	−0.731219	−	0.682143i	\(-0.761048\pi\)
							0.453814	−	0.891096i	\(-0.350063\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.2.v.b.59.24	yes	192
3.2	odd	2		648.2.v.b.611.9		192
4.3	odd	2		864.2.bh.b.815.4		192
8.3	odd	2	inner	216.2.v.b.59.25	yes	192
8.5	even	2		864.2.bh.b.815.3		192
24.11	even	2		648.2.v.b.611.8		192
27.11	odd	18	inner	216.2.v.b.11.25	yes	192
27.16	even	9		648.2.v.b.35.8		192
108.11	even	18		864.2.bh.b.335.3		192
216.11	even	18	inner	216.2.v.b.11.24	✓	192
216.43	odd	18		648.2.v.b.35.9		192
216.173	odd	18		864.2.bh.b.335.4		192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.b.11.24	✓	192	216.11	even	18	inner
216.2.v.b.11.25	yes	192	27.11	odd	18	inner
216.2.v.b.59.24	yes	192	1.1	even	1	trivial
216.2.v.b.59.25	yes	192	8.3	odd	2	inner
648.2.v.b.35.8		192	27.16	even	9
648.2.v.b.35.9		192	216.43	odd	18
648.2.v.b.611.8		192	24.11	even	2
648.2.v.b.611.9		192	3.2	odd	2
864.2.bh.b.335.3		192	108.11	even	18
864.2.bh.b.335.4		192	216.173	odd	18
864.2.bh.b.815.3		192	8.5	even	2
864.2.bh.b.815.4		192	4.3	odd	2