Embedded newform 168.2.v.a.11.5

• Label: 168.2.v.a.11.5
• Level: $168$
• Weight: $2$
• Character: 168.11
• 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			11.5
Character	\(\chi\)	\(=\)	168.11
Dual form			168.2.v.a.107.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18883 - 0.765954i) q^{2} +(0.728136 - 1.57157i) q^{3} +(0.826627 + 1.82118i) q^{4} +(-1.00081 - 1.73346i) q^{5} +(-2.06938 + 1.31060i) q^{6} +(-2.50986 - 0.837013i) q^{7} +(0.412221 - 2.79823i) q^{8} +(-1.93964 - 2.28863i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(73\)	\(85\)	\(113\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18883	−	0.765954i	−0.840629	−	0.541612i
							\(3\)	0.728136	−	1.57157i	0.420390	−	0.907344i
\(5\)	−1.00081	−	1.73346i	−0.447577	−	0.775227i								0.550650	−	0.834736i	\(-0.314380\pi\)
							−0.998228	+	0.0595094i	\(0.981046\pi\)
\(7\)	−2.50986	−	0.837013i	−0.948639	−	0.316361i
							\(11\)	3.33711	+	1.92668i	1.00618	+	0.580916i	0.910070	−	0.414455i	\(-0.136028\pi\)
0.0961058	+	0.995371i	\(0.469361\pi\)
\(13\)		−	2.05924i		−	0.571129i	−0.958359	−	0.285565i	\(-0.907819\pi\)
							0.958359	−	0.285565i	\(-0.0921811\pi\)
\(17\)	−4.92484	−	2.84336i	−1.19445	−	0.689616i	−0.235137	−	0.971962i	\(-0.575554\pi\)
							−0.959313	+	0.282346i	\(0.908887\pi\)
\(19\)	−0.232846	−	0.403301i	−0.0534185	−	0.0925235i	0.838080	−	0.545548i	\(-0.183678\pi\)
							−0.891498	+	0.453024i	\(0.850345\pi\)
\(23\)	0.711115	+	1.23169i	0.148278	+	0.256825i	0.930591	−	0.366061i	\(-0.119294\pi\)
							−0.782313	+	0.622885i	\(0.785960\pi\)
\(29\)	9.25829			1.71922			0.859611	−	0.510949i	\(-0.170706\pi\)
							0.859611	+	0.510949i	\(0.170706\pi\)
\(31\)	4.99699	+	2.88502i	0.897486	+	0.518164i	0.876384	−	0.481613i	\(-0.159949\pi\)
							0.0211026	+	0.999777i	\(0.493282\pi\)
\(37\)	−0.569094	+	0.328567i	−0.0935586	+	0.0540161i	−0.546049	−	0.837753i	\(-0.683869\pi\)
							0.452491	+	0.891769i	\(0.350536\pi\)
\(41\)		−	8.42924i		−	1.31643i	−0.752832	−	0.658213i	\(-0.771313\pi\)
							0.752832	−	0.658213i	\(-0.228687\pi\)
\(43\)	6.77074			1.03253			0.516264	−	0.856430i	\(-0.327322\pi\)
							0.516264	+	0.856430i	\(0.327322\pi\)
\(47\)	1.79536	+	3.10966i	0.261881	+	0.453590i	0.966742	−	0.255755i	\(-0.0823240\pi\)
							−0.704861	+	0.709345i	\(0.748991\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.v.a.11.5	✓	56
3.2	odd	2	inner	168.2.v.a.11.24	yes	56
4.3	odd	2		672.2.bd.a.431.11		56
7.2	even	3	inner	168.2.v.a.107.14	yes	56
8.3	odd	2	inner	168.2.v.a.11.15	yes	56
8.5	even	2		672.2.bd.a.431.12		56
12.11	even	2		672.2.bd.a.431.10		56
21.2	odd	6	inner	168.2.v.a.107.15	yes	56
24.5	odd	2		672.2.bd.a.431.9		56
24.11	even	2	inner	168.2.v.a.11.14	yes	56
28.23	odd	6		672.2.bd.a.527.9		56
56.37	even	6		672.2.bd.a.527.10		56
56.51	odd	6	inner	168.2.v.a.107.24	yes	56
84.23	even	6		672.2.bd.a.527.12		56
168.107	even	6	inner	168.2.v.a.107.5	yes	56
168.149	odd	6		672.2.bd.a.527.11		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.5	✓	56	1.1	even	1	trivial
168.2.v.a.11.14	yes	56	24.11	even	2	inner
168.2.v.a.11.15	yes	56	8.3	odd	2	inner
168.2.v.a.11.24	yes	56	3.2	odd	2	inner
168.2.v.a.107.5	yes	56	168.107	even	6	inner
168.2.v.a.107.14	yes	56	7.2	even	3	inner
168.2.v.a.107.15	yes	56	21.2	odd	6	inner
168.2.v.a.107.24	yes	56	56.51	odd	6	inner
672.2.bd.a.431.9		56	24.5	odd	2
672.2.bd.a.431.10		56	12.11	even	2
672.2.bd.a.431.11		56	4.3	odd	2
672.2.bd.a.431.12		56	8.5	even	2
672.2.bd.a.527.9		56	28.23	odd	6
672.2.bd.a.527.10		56	56.37	even	6
672.2.bd.a.527.11		56	168.149	odd	6
672.2.bd.a.527.12		56	84.23	even	6