Embedded newform 168.3.x.b.61.2

• Label: 168.3.x.b.61.2
• Level: $168$
• Weight: $3$
• Character: 168.61
• Analytic conductor: $4.578$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	168.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.57766844125\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			61.2
Character	\(\chi\)	\(=\)	168.61
Dual form			168.3.x.b.157.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.96845 - 0.353828i) q^{2} +(0.866025 + 1.50000i) q^{3} +(3.74961 + 1.39299i) q^{4} +(3.07593 - 5.32766i) q^{5} +(-1.17399 - 3.25910i) q^{6} +(-5.56076 - 4.25182i) q^{7} +(-6.88805 - 4.06875i) q^{8} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 2 q^{2} - 2 q^{4} + 26 q^{7} + 32 q^{8} - 90 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\)	\(73\)	\(85\)	\(113\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.96845	−	0.353828i	−0.984226	−	0.176914i
							\(3\)	0.866025	+	1.50000i	0.288675	+	0.500000i
\(5\)	3.07593	−	5.32766i	0.615186	−	1.06553i								−0.375166	−	0.926958i	\(-0.622414\pi\)
							0.990352	−	0.138575i	\(-0.0442523\pi\)
\(7\)	−5.56076	−	4.25182i	−0.794394	−	0.607403i
							\(11\)	15.3495	−	8.86207i	1.39541	−	0.805642i	0.401506	−	0.915857i	\(-0.368487\pi\)
0.993908	+	0.110214i	\(0.0351537\pi\)
\(13\)	−21.5235			−1.65565			−0.827825	−	0.560986i	\(-0.810422\pi\)
							−0.827825	+	0.560986i	\(0.810422\pi\)
\(17\)	0.757646	−	0.437427i	0.0445674	−	0.0257310i	−0.477551	−	0.878604i	\(-0.658475\pi\)
							0.522118	+	0.852873i	\(0.325142\pi\)
\(19\)	12.0479	−	20.8676i	0.634102	−	1.09830i	−0.352603	−	0.935773i	\(-0.614703\pi\)
							0.986705	−	0.162524i	\(-0.0519634\pi\)
\(23\)	5.94767	−	10.3017i	0.258594	−	0.447898i	−0.707271	−	0.706942i	\(-0.750074\pi\)
							0.965866	+	0.259044i	\(0.0834074\pi\)
\(29\)		−	37.8644i		−	1.30567i	−0.757501	−	0.652834i	\(-0.773580\pi\)
							0.757501	−	0.652834i	\(-0.226420\pi\)
\(31\)	23.6226	−	13.6385i	0.762021	−	0.439953i	−0.0680000	−	0.997685i	\(-0.521662\pi\)
							0.830021	+	0.557732i	\(0.188328\pi\)
\(37\)	−0.460874	−	0.266086i	−0.0124561	−	0.00719151i	0.493759	−	0.869599i	\(-0.335622\pi\)
							−0.506215	+	0.862407i	\(0.668956\pi\)
\(41\)			23.0486i			0.562160i	0.959684	+	0.281080i	\(0.0906927\pi\)
							−0.959684	+	0.281080i	\(0.909307\pi\)
\(43\)			51.9635i			1.20845i	0.796812	+	0.604227i	\(0.206518\pi\)
							−0.796812	+	0.604227i	\(0.793482\pi\)
\(47\)	55.5732	+	32.0852i	1.18241	+	0.682664i	0.956571	−	0.291500i	\(-0.0941544\pi\)
							0.225839	+	0.974165i	\(0.427488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.3.x.b.61.2	✓	60
4.3	odd	2		672.3.bf.b.145.3		60
7.3	odd	6	inner	168.3.x.b.157.23	yes	60
8.3	odd	2		672.3.bf.b.145.28		60
8.5	even	2	inner	168.3.x.b.61.23	yes	60
28.3	even	6		672.3.bf.b.241.28		60
56.3	even	6		672.3.bf.b.241.3		60
56.45	odd	6	inner	168.3.x.b.157.2	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.x.b.61.2	✓	60	1.1	even	1	trivial
168.3.x.b.61.23	yes	60	8.5	even	2	inner
168.3.x.b.157.2	yes	60	56.45	odd	6	inner
168.3.x.b.157.23	yes	60	7.3	odd	6	inner
672.3.bf.b.145.3		60	4.3	odd	2
672.3.bf.b.145.28		60	8.3	odd	2
672.3.bf.b.241.3		60	56.3	even	6
672.3.bf.b.241.28		60	28.3	even	6