Embedded newform 162.4.c.g.55.1

• Label: 162.4.c.g.55.1
• Level: $162$
• Weight: $4$
• Character: 162.55
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.55830942093\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	162.55
Dual form			162.4.c.g.109.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(1.50000 + 2.59808i) q^{5} +(-14.5000 + 25.1147i) q^{7} -8.00000 q^{8} +6.00000 q^{10} +(-28.5000 + 49.3634i) q^{11} +(-10.0000 - 17.3205i) q^{13} +(29.0000 + 50.2295i) q^{14} +(-8.00000 + 13.8564i) q^{16} +72.0000 q^{17} -106.000 q^{19} +(6.00000 - 10.3923i) q^{20} +(57.0000 + 98.7269i) q^{22} +(87.0000 + 150.688i) q^{23} +(58.0000 - 100.459i) q^{25} -40.0000 q^{26} +116.000 q^{28} +(-105.000 + 181.865i) q^{29} +(-23.5000 - 40.7032i) q^{31} +(16.0000 + 27.7128i) q^{32} +(72.0000 - 124.708i) q^{34} -87.0000 q^{35} +2.00000 q^{37} +(-106.000 + 183.597i) q^{38} +(-12.0000 - 20.7846i) q^{40} +(-3.00000 - 5.19615i) q^{41} +(-109.000 + 188.794i) q^{43} +228.000 q^{44} +348.000 q^{46} +(237.000 - 410.496i) q^{47} +(-249.000 - 431.281i) q^{49} +(-116.000 - 200.918i) q^{50} +(-40.0000 + 69.2820i) q^{52} -81.0000 q^{53} -171.000 q^{55} +(116.000 - 200.918i) q^{56} +(210.000 + 363.731i) q^{58} +(42.0000 + 72.7461i) q^{59} +(-28.0000 + 48.4974i) q^{61} -94.0000 q^{62} +64.0000 q^{64} +(30.0000 - 51.9615i) q^{65} +(71.0000 + 122.976i) q^{67} +(-144.000 - 249.415i) q^{68} +(-87.0000 + 150.688i) q^{70} -360.000 q^{71} -1159.00 q^{73} +(2.00000 - 3.46410i) q^{74} +(212.000 + 367.195i) q^{76} +(-826.500 - 1431.54i) q^{77} +(80.0000 - 138.564i) q^{79} -48.0000 q^{80} -12.0000 q^{82} +(367.500 - 636.529i) q^{83} +(108.000 + 187.061i) q^{85} +(218.000 + 377.587i) q^{86} +(228.000 - 394.908i) q^{88} +954.000 q^{89} +580.000 q^{91} +(348.000 - 602.754i) q^{92} +(-474.000 - 820.992i) q^{94} +(-159.000 - 275.396i) q^{95} +(-95.5000 + 165.411i) q^{97} -996.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 - 4 * q^4 + 3 * q^5 - 29 * q^7 - 16 * q^8 + 12 * q^10 - 57 * q^11 - 20 * q^13 + 58 * q^14 - 16 * q^16 + 144 * q^17 - 212 * q^19 + 12 * q^20 + 114 * q^22 + 174 * q^23 + 116 * q^25 - 80 * q^26 + 232 * q^28 - 210 * q^29 - 47 * q^31 + 32 * q^32 + 144 * q^34 - 174 * q^35 + 4 * q^37 - 212 * q^38 - 24 * q^40 - 6 * q^41 - 218 * q^43 + 456 * q^44 + 696 * q^46 + 474 * q^47 - 498 * q^49 - 232 * q^50 - 80 * q^52 - 162 * q^53 - 342 * q^55 + 232 * q^56 + 420 * q^58 + 84 * q^59 - 56 * q^61 - 188 * q^62 + 128 * q^64 + 60 * q^65 + 142 * q^67 - 288 * q^68 - 174 * q^70 - 720 * q^71 - 2318 * q^73 + 4 * q^74 + 424 * q^76 - 1653 * q^77 + 160 * q^79 - 96 * q^80 - 24 * q^82 + 735 * q^83 + 216 * q^85 + 436 * q^86 + 456 * q^88 + 1908 * q^89 + 1160 * q^91 + 696 * q^92 - 948 * q^94 - 318 * q^95 - 191 * q^97 - 1992 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	−	1.73205i	0.353553	−	0.612372i
							\(3\)		0			0
\(5\)	1.50000	+	2.59808i	0.134164	+	0.232379i								0.925278	−	0.379290i	\(-0.123832\pi\)
							−0.791114	+	0.611669i	\(0.790498\pi\)
\(7\)	−14.5000	+	25.1147i	−0.782926	+	1.35607i	0.147304	+	0.989091i	\(0.452941\pi\)
							−0.930230	+	0.366977i	\(0.880393\pi\)
\(11\)	−28.5000	+	49.3634i	−0.781188	+	1.35306i	0.150061	+	0.988677i	\(0.452053\pi\)
							−0.931250	+	0.364381i	\(0.881280\pi\)
\(13\)	−10.0000	−	17.3205i	−0.213346	−	0.369527i	0.739413	−	0.673252i	\(-0.235103\pi\)
							−0.952760	+	0.303725i	\(0.901770\pi\)
\(17\)	72.0000			1.02721			0.513605	−	0.858027i	\(-0.328310\pi\)
							0.513605	+	0.858027i	\(0.328310\pi\)
\(19\)	−106.000			−1.27990			−0.639949	−	0.768417i	\(-0.721045\pi\)
							−0.639949	+	0.768417i	\(0.721045\pi\)
\(23\)	87.0000	+	150.688i	0.788728	+	1.36612i	0.926746	+	0.375688i	\(0.122594\pi\)
							−0.138018	+	0.990430i	\(0.544073\pi\)
\(29\)	−105.000	+	181.865i	−0.672345	+	1.16454i	0.304892	+	0.952387i	\(0.401380\pi\)
							−0.977237	+	0.212149i	\(0.931954\pi\)
\(31\)	−23.5000	−	40.7032i	−0.136152	−	0.235823i	0.789885	−	0.613255i	\(-0.210140\pi\)
							−0.926037	+	0.377433i	\(0.876807\pi\)
\(37\)	2.00000			0.00888643			0.00444322	−	0.999990i	\(-0.498586\pi\)
							0.00444322	+	0.999990i	\(0.498586\pi\)
\(41\)	−3.00000	−	5.19615i	−0.0114273	−	0.0197927i	0.860255	−	0.509864i	\(-0.170304\pi\)
							−0.871683	+	0.490071i	\(0.836971\pi\)
\(43\)	−109.000	+	188.794i	−0.386566	+	0.669552i	−0.991985	−	0.126355i	\(-0.959672\pi\)
							0.605419	+	0.795907i	\(0.293006\pi\)
\(47\)	237.000	−	410.496i	0.735532	−	1.27398i	−0.218958	−	0.975734i	\(-0.570266\pi\)
							0.954490	−	0.298244i	\(-0.0964010\pi\)
\(53\)	−81.0000			−0.209928			−0.104964	−	0.994476i	\(-0.533473\pi\)
							−0.104964	+	0.994476i	\(0.533473\pi\)
\(59\)	42.0000	+	72.7461i	0.0926769	+	0.160521i	0.908637	−	0.417588i	\(-0.137124\pi\)
							−0.815960	+	0.578109i	\(0.803791\pi\)
\(61\)	−28.0000	+	48.4974i	−0.0587710	+	0.101794i	−0.893914	−	0.448239i	\(-0.852052\pi\)
							0.835143	+	0.550033i	\(0.185385\pi\)
\(67\)	71.0000	+	122.976i	0.129463	+	0.224237i	0.923469	−	0.383674i	\(-0.125341\pi\)
							−0.794006	+	0.607910i	\(0.792008\pi\)
\(71\)	−360.000			−0.601748			−0.300874	−	0.953664i	\(-0.597278\pi\)
							−0.300874	+	0.953664i	\(0.597278\pi\)
\(73\)	−1159.00			−1.85823			−0.929114	−	0.369793i	\(-0.879429\pi\)
							−0.929114	+	0.369793i	\(0.879429\pi\)
\(79\)	80.0000	−	138.564i	0.113933	−	0.197338i	−0.803420	−	0.595413i	\(-0.796988\pi\)
							0.917353	+	0.398075i	\(0.130322\pi\)
\(83\)	367.500	−	636.529i	0.486004	−	0.841784i	−0.513866	−	0.857870i	\(-0.671787\pi\)
							0.999871	+	0.0160860i	\(0.00512056\pi\)
\(89\)	954.000			1.13622			0.568111	−	0.822952i	\(-0.307674\pi\)
							0.568111	+	0.822952i	\(0.307674\pi\)
\(97\)	−95.5000	+	165.411i	−0.0999645	+	0.173144i	−0.911670	−	0.410924i	\(-0.865206\pi\)
							0.811705	+	0.584067i	\(0.198540\pi\)

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	162.4.c.g.55.1		2
3.2	odd	2		162.4.c.b.55.1		2
9.2	odd	6		54.4.a.c.1.1	yes	1
9.4	even	3	inner	162.4.c.g.109.1		2
9.5	odd	6		162.4.c.b.109.1		2
9.7	even	3		54.4.a.b.1.1	✓	1
36.7	odd	6		432.4.a.e.1.1		1
36.11	even	6		432.4.a.j.1.1		1
45.2	even	12		1350.4.c.b.649.2		2
45.7	odd	12		1350.4.c.s.649.1		2
45.29	odd	6		1350.4.a.a.1.1		1
45.34	even	6		1350.4.a.o.1.1		1
45.38	even	12		1350.4.c.b.649.1		2
45.43	odd	12		1350.4.c.s.649.2		2
72.11	even	6		1728.4.a.k.1.1		1
72.29	odd	6		1728.4.a.l.1.1		1
72.43	odd	6		1728.4.a.u.1.1		1
72.61	even	6		1728.4.a.v.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.4.a.b.1.1	✓	1	9.7	even	3
54.4.a.c.1.1	yes	1	9.2	odd	6
162.4.c.b.55.1		2	3.2	odd	2
162.4.c.b.109.1		2	9.5	odd	6
162.4.c.g.55.1		2	1.1	even	1	trivial
162.4.c.g.109.1		2	9.4	even	3	inner
432.4.a.e.1.1		1	36.7	odd	6
432.4.a.j.1.1		1	36.11	even	6
1350.4.a.a.1.1		1	45.29	odd	6
1350.4.a.o.1.1		1	45.34	even	6
1350.4.c.b.649.1		2	45.38	even	12
1350.4.c.b.649.2		2	45.2	even	12
1350.4.c.s.649.1		2	45.7	odd	12
1350.4.c.s.649.2		2	45.43	odd	12
1728.4.a.k.1.1		1	72.11	even	6
1728.4.a.l.1.1		1	72.29	odd	6
1728.4.a.u.1.1		1	72.43	odd	6
1728.4.a.v.1.1		1	72.61	even	6