Embedded newform 162.2.c.b.109.1

• Label: 162.2.c.b.109.1
• Level: $162$
• Weight: $2$
• Character: 162.109
• Analytic conductor: $1.294$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.29357651274\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	162.109
Dual form			162.2.c.b.55.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.50000 - 2.59808i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} -3.00000 q^{10} +(-1.50000 - 2.59808i) q^{11} +(2.00000 - 3.46410i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +2.00000 q^{19} +(1.50000 + 2.59808i) q^{20} +(-1.50000 + 2.59808i) q^{22} +(-3.00000 + 5.19615i) q^{23} +(-2.00000 - 3.46410i) q^{25} -4.00000 q^{26} -1.00000 q^{28} +(3.00000 + 5.19615i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(-0.500000 + 0.866025i) q^{32} +3.00000 q^{35} +2.00000 q^{37} +(-1.00000 - 1.73205i) q^{38} +(1.50000 - 2.59808i) q^{40} +(-3.00000 + 5.19615i) q^{41} +(5.00000 + 8.66025i) q^{43} +3.00000 q^{44} +6.00000 q^{46} +(3.00000 + 5.19615i) q^{47} +(3.00000 - 5.19615i) q^{49} +(-2.00000 + 3.46410i) q^{50} +(2.00000 + 3.46410i) q^{52} -9.00000 q^{53} -9.00000 q^{55} +(0.500000 + 0.866025i) q^{56} +(3.00000 - 5.19615i) q^{58} +(6.00000 - 10.3923i) q^{59} +(-4.00000 - 6.92820i) q^{61} +5.00000 q^{62} +1.00000 q^{64} +(-6.00000 - 10.3923i) q^{65} +(-7.00000 + 12.1244i) q^{67} +(-1.50000 - 2.59808i) q^{70} -7.00000 q^{73} +(-1.00000 - 1.73205i) q^{74} +(-1.00000 + 1.73205i) q^{76} +(1.50000 - 2.59808i) q^{77} +(-4.00000 - 6.92820i) q^{79} -3.00000 q^{80} +6.00000 q^{82} +(-1.50000 - 2.59808i) q^{83} +(5.00000 - 8.66025i) q^{86} +(-1.50000 - 2.59808i) q^{88} +18.0000 q^{89} +4.00000 q^{91} +(-3.00000 - 5.19615i) q^{92} +(3.00000 - 5.19615i) q^{94} +(3.00000 - 5.19615i) q^{95} +(0.500000 + 0.866025i) q^{97} -6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 - q^4 + 3 * q^5 + q^7 + 2 * q^8 - 6 * q^10 - 3 * q^11 + 4 * q^13 + q^14 - q^16 + 4 * q^19 + 3 * q^20 - 3 * q^22 - 6 * q^23 - 4 * q^25 - 8 * q^26 - 2 * q^28 + 6 * q^29 - 5 * q^31 - q^32 + 6 * q^35 + 4 * q^37 - 2 * q^38 + 3 * q^40 - 6 * q^41 + 10 * q^43 + 6 * q^44 + 12 * q^46 + 6 * q^47 + 6 * q^49 - 4 * q^50 + 4 * q^52 - 18 * q^53 - 18 * q^55 + q^56 + 6 * q^58 + 12 * q^59 - 8 * q^61 + 10 * q^62 + 2 * q^64 - 12 * q^65 - 14 * q^67 - 3 * q^70 - 14 * q^73 - 2 * q^74 - 2 * q^76 + 3 * q^77 - 8 * q^79 - 6 * q^80 + 12 * q^82 - 3 * q^83 + 10 * q^86 - 3 * q^88 + 36 * q^89 + 8 * q^91 - 6 * q^92 + 6 * q^94 + 6 * q^95 + q^97 - 12 * q^98

Character values

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

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.500000	−	0.866025i	−0.353553	−	0.612372i
							\(3\)		0			0
\(5\)	1.50000	−	2.59808i	0.670820	−	1.16190i								−0.306851	−	0.951757i	\(-0.599275\pi\)
							0.977672	−	0.210138i	\(-0.0673912\pi\)
\(7\)	0.500000	+	0.866025i	0.188982	+	0.327327i	0.944911	−	0.327327i	\(-0.106148\pi\)
							−0.755929	+	0.654654i	\(0.772814\pi\)
\(11\)	−1.50000	−	2.59808i	−0.452267	−	0.783349i	0.546259	−	0.837616i	\(-0.316051\pi\)
							−0.998526	+	0.0542666i	\(0.982718\pi\)
\(13\)	2.00000	−	3.46410i	0.554700	−	0.960769i	−0.443227	−	0.896410i	\(-0.646166\pi\)
							0.997927	−	0.0643593i	\(-0.0205004\pi\)
\(17\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(19\)	2.00000			0.458831			0.229416	−	0.973329i	\(-0.426318\pi\)
							0.229416	+	0.973329i	\(0.426318\pi\)
\(23\)	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	\(0.381789\pi\)
							−0.988436	+	0.151642i	\(0.951544\pi\)
\(29\)	3.00000	+	5.19615i	0.557086	+	0.964901i	0.997738	+	0.0672232i	\(0.0214140\pi\)
							−0.440652	+	0.897678i	\(0.645253\pi\)
\(31\)	−2.50000	+	4.33013i	−0.449013	+	0.777714i	−0.998322	−	0.0579057i	\(-0.981558\pi\)
							0.549309	+	0.835619i	\(0.314891\pi\)
\(37\)	2.00000			0.328798			0.164399	−	0.986394i	\(-0.447432\pi\)
							0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	−3.00000	+	5.19615i	−0.468521	+	0.811503i	−0.999353	−	0.0359748i	\(-0.988546\pi\)
							0.530831	+	0.847477i	\(0.321880\pi\)
\(43\)	5.00000	+	8.66025i	0.762493	+	1.32068i	0.941562	+	0.336840i	\(0.109358\pi\)
							−0.179069	+	0.983836i	\(0.557309\pi\)
\(47\)	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	\(-0.0224970\pi\)
							−0.559908	+	0.828554i	\(0.689164\pi\)
\(53\)	−9.00000			−1.23625			−0.618123	−	0.786082i	\(-0.712106\pi\)
							−0.618123	+	0.786082i	\(0.712106\pi\)
\(59\)	6.00000	−	10.3923i	0.781133	−	1.35296i	−0.150148	−	0.988663i	\(-0.547975\pi\)
							0.931282	−	0.364299i	\(-0.118692\pi\)
\(61\)	−4.00000	−	6.92820i	−0.512148	−	0.887066i	−0.999901	−	0.0140840i	\(-0.995517\pi\)
							0.487753	−	0.872982i	\(-0.337817\pi\)
\(67\)	−7.00000	+	12.1244i	−0.855186	+	1.48123i	0.0212861	+	0.999773i	\(0.493224\pi\)
							−0.876472	+	0.481452i	\(0.840109\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)	−7.00000			−0.819288			−0.409644	−	0.912245i	\(-0.634347\pi\)
							−0.409644	+	0.912245i	\(0.634347\pi\)
\(79\)	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	\(-0.315255\pi\)
							−0.998388	+	0.0567635i	\(0.981922\pi\)
\(83\)	−1.50000	−	2.59808i	−0.164646	−	0.285176i	0.771883	−	0.635764i	\(-0.219315\pi\)
							−0.936530	+	0.350588i	\(0.885982\pi\)
\(89\)	18.0000			1.90800			0.953998	−	0.299813i	\(-0.0969242\pi\)
							0.953998	+	0.299813i	\(0.0969242\pi\)
\(97\)	0.500000	+	0.866025i	0.0507673	+	0.0879316i	0.890292	−	0.455389i	\(-0.150500\pi\)
							−0.839525	+	0.543321i	\(0.817167\pi\)

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	162.2.c.b.109.1		2
3.2	odd	2		162.2.c.c.109.1		2
4.3	odd	2		1296.2.i.o.433.1		2
9.2	odd	6		162.2.c.c.55.1		2
9.4	even	3		54.2.a.b.1.1	yes	1
9.5	odd	6		54.2.a.a.1.1	✓	1
9.7	even	3	inner	162.2.c.b.55.1		2
12.11	even	2		1296.2.i.c.433.1		2
36.7	odd	6		1296.2.i.o.865.1		2
36.11	even	6		1296.2.i.c.865.1		2
36.23	even	6		432.2.a.g.1.1		1
36.31	odd	6		432.2.a.b.1.1		1
45.4	even	6		1350.2.a.h.1.1		1
45.13	odd	12		1350.2.c.k.649.1		2
45.14	odd	6		1350.2.a.r.1.1		1
45.22	odd	12		1350.2.c.k.649.2		2
45.23	even	12		1350.2.c.b.649.2		2
45.32	even	12		1350.2.c.b.649.1		2
63.13	odd	6		2646.2.a.bd.1.1		1
63.41	even	6		2646.2.a.a.1.1		1
72.5	odd	6		1728.2.a.c.1.1		1
72.13	even	6		1728.2.a.y.1.1		1
72.59	even	6		1728.2.a.d.1.1		1
72.67	odd	6		1728.2.a.z.1.1		1
99.32	even	6		6534.2.a.bc.1.1		1
99.76	odd	6		6534.2.a.b.1.1		1
117.77	odd	6		9126.2.a.u.1.1		1
117.103	even	6		9126.2.a.r.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.2.a.a.1.1	✓	1	9.5	odd	6
54.2.a.b.1.1	yes	1	9.4	even	3
162.2.c.b.55.1		2	9.7	even	3	inner
162.2.c.b.109.1		2	1.1	even	1	trivial
162.2.c.c.55.1		2	9.2	odd	6
162.2.c.c.109.1		2	3.2	odd	2
432.2.a.b.1.1		1	36.31	odd	6
432.2.a.g.1.1		1	36.23	even	6
1296.2.i.c.433.1		2	12.11	even	2
1296.2.i.c.865.1		2	36.11	even	6
1296.2.i.o.433.1		2	4.3	odd	2
1296.2.i.o.865.1		2	36.7	odd	6
1350.2.a.h.1.1		1	45.4	even	6
1350.2.a.r.1.1		1	45.14	odd	6
1350.2.c.b.649.1		2	45.32	even	12
1350.2.c.b.649.2		2	45.23	even	12
1350.2.c.k.649.1		2	45.13	odd	12
1350.2.c.k.649.2		2	45.22	odd	12
1728.2.a.c.1.1		1	72.5	odd	6
1728.2.a.d.1.1		1	72.59	even	6
1728.2.a.y.1.1		1	72.13	even	6
1728.2.a.z.1.1		1	72.67	odd	6
2646.2.a.a.1.1		1	63.41	even	6
2646.2.a.bd.1.1		1	63.13	odd	6
6534.2.a.b.1.1		1	99.76	odd	6
6534.2.a.bc.1.1		1	99.32	even	6
9126.2.a.r.1.1		1	117.103	even	6
9126.2.a.u.1.1		1	117.77	odd	6