Embedded newform 168.2.v.a.107.8

• Label: 168.2.v.a.107.8
• Level: $168$
• Weight: $2$
• Character: 168.107
• Analytic conductor: $1.341$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34148675396\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			107.8
Character	\(\chi\)	\(=\)	168.107
Dual form			168.2.v.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.935706 + 1.06040i) q^{2} +(-1.45135 - 0.945295i) q^{3} +(-0.248908 - 1.98445i) q^{4} +(-0.187787 + 0.325257i) q^{5} +(2.36043 - 0.654497i) q^{6} +(0.198565 + 2.63829i) q^{7} +(2.33722 + 1.59292i) q^{8} +(1.21283 + 2.74391i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(73\)	\(85\)	\(113\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.935706	+	1.06040i	−0.661644	+	0.749818i
							\(3\)	−1.45135	−	0.945295i	−0.837937	−	0.545766i
\(5\)	−0.187787	+	0.325257i	−0.0839810	+	0.145459i								−0.904957	−	0.425504i	\(-0.860097\pi\)
							0.820976	+	0.570963i	\(0.193430\pi\)
\(7\)	0.198565	+	2.63829i	0.0750504	+	0.997180i
							\(11\)	−4.18624	+	2.41693i	−1.26220	+	0.728731i	−0.973500	−	0.228688i	\(-0.926556\pi\)
−0.288700	+	0.957420i	\(0.593223\pi\)
\(13\)			4.40701i			1.22228i	0.791521	+	0.611142i	\(0.209290\pi\)
							−0.791521	+	0.611142i	\(0.790710\pi\)
\(17\)	2.39410	−	1.38223i	0.580653	−	0.335240i	−0.180740	−	0.983531i	\(-0.557849\pi\)
							0.761393	+	0.648291i	\(0.224516\pi\)
\(19\)	1.83569	−	3.17951i	0.421137	−	0.729431i	−0.574914	−	0.818214i	\(-0.694965\pi\)
							0.996051	+	0.0887833i	\(0.0282979\pi\)
\(23\)	−3.60943	+	6.25172i	−0.752618	+	1.30357i	0.193932	+	0.981015i	\(0.437876\pi\)
							−0.946550	+	0.322558i	\(0.895457\pi\)
\(29\)	−0.968468			−0.179840			−0.0899200	−	0.995949i	\(-0.528661\pi\)
							−0.0899200	+	0.995949i	\(0.528661\pi\)
\(31\)	−2.78588	+	1.60843i	−0.500359	+	0.288883i	−0.728862	−	0.684661i	\(-0.759950\pi\)
							0.228503	+	0.973543i	\(0.426617\pi\)
\(37\)	0.459519	+	0.265303i	0.0755445	+	0.0436156i	0.537296	−	0.843393i	\(-0.319446\pi\)
							−0.461752	+	0.887009i	\(0.652779\pi\)
\(41\)		−	3.88604i		−	0.606897i	−0.952848	−	0.303449i	\(-0.901862\pi\)
							0.952848	−	0.303449i	\(-0.0981381\pi\)
\(43\)	0.747189			0.113945			0.0569726	−	0.998376i	\(-0.481855\pi\)
							0.0569726	+	0.998376i	\(0.481855\pi\)
\(47\)	5.20322	−	9.01223i	0.758967	−	1.31457i	−0.184411	−	0.982849i	\(-0.559038\pi\)
							0.943378	−	0.331720i	\(-0.107629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.v.a.107.8	yes	56
3.2	odd	2	inner	168.2.v.a.107.21	yes	56
4.3	odd	2		672.2.bd.a.527.23		56
7.4	even	3	inner	168.2.v.a.11.11	yes	56
8.3	odd	2	inner	168.2.v.a.107.18	yes	56
8.5	even	2		672.2.bd.a.527.24		56
12.11	even	2		672.2.bd.a.527.14		56
21.11	odd	6	inner	168.2.v.a.11.18	yes	56
24.5	odd	2		672.2.bd.a.527.13		56
24.11	even	2	inner	168.2.v.a.107.11	yes	56
28.11	odd	6		672.2.bd.a.431.13		56
56.11	odd	6	inner	168.2.v.a.11.21	yes	56
56.53	even	6		672.2.bd.a.431.14		56
84.11	even	6		672.2.bd.a.431.24		56
168.11	even	6	inner	168.2.v.a.11.8	✓	56
168.53	odd	6		672.2.bd.a.431.23		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.8	✓	56	168.11	even	6	inner
168.2.v.a.11.11	yes	56	7.4	even	3	inner
168.2.v.a.11.18	yes	56	21.11	odd	6	inner
168.2.v.a.11.21	yes	56	56.11	odd	6	inner
168.2.v.a.107.8	yes	56	1.1	even	1	trivial
168.2.v.a.107.11	yes	56	24.11	even	2	inner
168.2.v.a.107.18	yes	56	8.3	odd	2	inner
168.2.v.a.107.21	yes	56	3.2	odd	2	inner
672.2.bd.a.431.13		56	28.11	odd	6
672.2.bd.a.431.14		56	56.53	even	6
672.2.bd.a.431.23		56	168.53	odd	6
672.2.bd.a.431.24		56	84.11	even	6
672.2.bd.a.527.13		56	24.5	odd	2
672.2.bd.a.527.14		56	12.11	even	2
672.2.bd.a.527.23		56	4.3	odd	2
672.2.bd.a.527.24		56	8.5	even	2