Embedded newform 168.3.x.b.61.13

• Label: 168.3.x.b.61.13
• 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			61.13
Character	\(\chi\)	\(=\)	168.61
Dual form			168.3.x.b.157.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.671855 + 1.88378i) q^{2} +(0.866025 + 1.50000i) q^{3} +(-3.09722 - 2.53125i) q^{4} +(2.65366 - 4.59628i) q^{5} +(-3.40751 + 0.623616i) q^{6} +(6.60250 - 2.32529i) q^{7} +(6.84919 - 4.13384i) q^{8} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 2 q^{2} - 2 q^{4} + 26 q^{7} + 32 q^{8} - 90 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.671855	+	1.88378i	−0.335927	+	0.941888i
							\(3\)	0.866025	+	1.50000i	0.288675	+	0.500000i
\(5\)	2.65366	−	4.59628i	0.530733	−	0.919256i								−0.468624	−	0.883398i	\(-0.655250\pi\)
							0.999357	−	0.0358582i	\(-0.0114165\pi\)
\(7\)	6.60250	−	2.32529i	0.943215	−	0.332184i
							\(11\)	6.46900	−	3.73488i	0.588091	−	0.339534i	−0.176251	−	0.984345i	\(-0.556397\pi\)
0.764342	+	0.644811i	\(0.223064\pi\)
\(13\)	10.3115			0.793195			0.396598	−	0.917993i	\(-0.370191\pi\)
							0.396598	+	0.917993i	\(0.370191\pi\)
\(17\)	−19.6827	+	11.3638i	−1.15780	+	0.668458i	−0.950777	−	0.309877i	\(-0.899712\pi\)
							−0.207027	+	0.978335i	\(0.566379\pi\)
\(19\)	11.8937	−	20.6004i	0.625983	−	1.08423i	−0.362367	−	0.932035i	\(-0.618031\pi\)
							0.988350	−	0.152199i	\(-0.0486353\pi\)
\(23\)	−8.19416	+	14.1927i	−0.356268	+	0.617074i	−0.987334	−	0.158655i	\(-0.949284\pi\)
							0.631066	+	0.775729i	\(0.282618\pi\)
\(29\)			5.55129i			0.191424i	0.995409	+	0.0957119i	\(0.0305128\pi\)
							−0.995409	+	0.0957119i	\(0.969487\pi\)
\(31\)	3.57088	−	2.06165i	0.115190	−	0.0665048i	−0.441298	−	0.897361i	\(-0.645482\pi\)
							0.556488	+	0.830856i	\(0.312149\pi\)
\(37\)	35.8054	+	20.6723i	0.967713	+	0.558709i	0.898538	−	0.438895i	\(-0.144630\pi\)
							0.0691749	+	0.997605i	\(0.477963\pi\)
\(41\)			0.0763421i			0.00186200i	1.00000		0.000931001i	\(0.000296347\pi\)
							−1.00000		0.000931001i	\(0.999704\pi\)
\(43\)		−	44.6499i		−	1.03837i	−0.854662	−	0.519184i	\(-0.826236\pi\)
							0.854662	−	0.519184i	\(-0.173764\pi\)
\(47\)	29.5657	+	17.0697i	0.629057	+	0.363186i	0.780387	−	0.625297i	\(-0.215022\pi\)
							−0.151330	+	0.988483i	\(0.548356\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.3.x.b.61.13	yes	60
4.3	odd	2		672.3.bf.b.145.5		60
7.3	odd	6	inner	168.3.x.b.157.8	yes	60
8.3	odd	2		672.3.bf.b.145.26		60
8.5	even	2	inner	168.3.x.b.61.8	✓	60
28.3	even	6		672.3.bf.b.241.26		60
56.3	even	6		672.3.bf.b.241.5		60
56.45	odd	6	inner	168.3.x.b.157.13	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.x.b.61.8	✓	60	8.5	even	2	inner
168.3.x.b.61.13	yes	60	1.1	even	1	trivial
168.3.x.b.157.8	yes	60	7.3	odd	6	inner
168.3.x.b.157.13	yes	60	56.45	odd	6	inner
672.3.bf.b.145.5		60	4.3	odd	2
672.3.bf.b.145.26		60	8.3	odd	2
672.3.bf.b.241.5		60	56.3	even	6
672.3.bf.b.241.26		60	28.3	even	6