Embedded newform 168.3.x.b.61.3

• Label: 168.3.x.b.61.3
• 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			61.3
Character	\(\chi\)	\(=\)	168.61
Dual form			168.3.x.b.157.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94006 - 0.485962i) q^{2} +(0.866025 + 1.50000i) q^{3} +(3.52768 + 1.88559i) q^{4} +(-2.81718 + 4.87950i) q^{5} +(-0.951200 - 3.33095i) q^{6} +(6.86898 + 1.34800i) q^{7} +(-5.92759 - 5.37249i) q^{8} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 2 q^{2} - 2 q^{4} + 26 q^{7} + 32 q^{8} - 90 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94006	−	0.485962i	−0.970031	−	0.242981i
							\(3\)	0.866025	+	1.50000i	0.288675	+	0.500000i
\(5\)	−2.81718	+	4.87950i	−0.563436	+	0.975900i								0.433757	+	0.901030i	\(0.357188\pi\)
							−0.997193	+	0.0748700i	\(0.976146\pi\)
\(7\)	6.86898	+	1.34800i	0.981283	+	0.192571i
							\(11\)	−6.32385	+	3.65108i	−0.574896	+	0.331916i	−0.759102	−	0.650971i	\(-0.774362\pi\)
0.184206	+	0.982888i	\(0.441028\pi\)
\(13\)	−14.1389			−1.08761			−0.543805	−	0.839212i	\(-0.683017\pi\)
							−0.543805	+	0.839212i	\(0.683017\pi\)
\(17\)	4.36523	−	2.52027i	0.256778	−	0.148251i	−0.366086	−	0.930581i	\(-0.619302\pi\)
							0.622864	+	0.782330i	\(0.285969\pi\)
\(19\)	1.44361	−	2.50041i	0.0759797	−	0.131601i	−0.825532	−	0.564355i	\(-0.809125\pi\)
							0.901512	+	0.432754i	\(0.142458\pi\)
\(23\)	−14.6264	+	25.3337i	−0.635931	+	1.10147i	0.350386	+	0.936605i	\(0.386050\pi\)
							−0.986317	+	0.164860i	\(0.947283\pi\)
\(29\)			35.4838i			1.22358i	0.791021	+	0.611789i	\(0.209550\pi\)
							−0.791021	+	0.611789i	\(0.790450\pi\)
\(31\)	−43.2072	+	24.9457i	−1.39378	+	0.804700i	−0.993731	−	0.111794i	\(-0.964340\pi\)
							−0.400050	+	0.916493i	\(0.631007\pi\)
\(37\)	52.4760	+	30.2970i	1.41827	+	0.818839i	0.996147	−	0.0877001i	\(-0.0279517\pi\)
							0.422123	+	0.906539i	\(0.361285\pi\)
\(41\)		−	72.2031i		−	1.76105i	−0.473997	−	0.880526i	\(-0.657189\pi\)
							0.473997	−	0.880526i	\(-0.342811\pi\)
\(43\)		−	30.7906i		−	0.716061i	−0.933710	−	0.358031i	\(-0.883448\pi\)
							0.933710	−	0.358031i	\(-0.116552\pi\)
\(47\)	10.8812	+	6.28228i	0.231515	+	0.133666i	0.611271	−	0.791421i	\(-0.290659\pi\)
							−0.379756	+	0.925087i	\(0.623992\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.3.x.b.61.3	✓	60
4.3	odd	2		672.3.bf.b.145.4		60
7.3	odd	6	inner	168.3.x.b.157.24	yes	60
8.3	odd	2		672.3.bf.b.145.27		60
8.5	even	2	inner	168.3.x.b.61.24	yes	60
28.3	even	6		672.3.bf.b.241.27		60
56.3	even	6		672.3.bf.b.241.4		60
56.45	odd	6	inner	168.3.x.b.157.3	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.x.b.61.3	✓	60	1.1	even	1	trivial
168.3.x.b.61.24	yes	60	8.5	even	2	inner
168.3.x.b.157.3	yes	60	56.45	odd	6	inner
168.3.x.b.157.24	yes	60	7.3	odd	6	inner
672.3.bf.b.145.4		60	4.3	odd	2
672.3.bf.b.145.27		60	8.3	odd	2
672.3.bf.b.241.4		60	56.3	even	6
672.3.bf.b.241.27		60	28.3	even	6