Embedded newform 168.2.v.a.11.15

• Label: 168.2.v.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	168.11
Dual form			168.2.v.a.107.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0689217 + 1.41253i) q^{2} +(0.728136 - 1.57157i) q^{3} +(-1.99050 + 0.194708i) q^{4} +(1.00081 + 1.73346i) q^{5} +(2.27007 + 0.920201i) q^{6} +(2.50986 + 0.837013i) q^{7} +(-0.412221 - 2.79823i) q^{8} +(-1.93964 - 2.28863i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0689217	+	1.41253i	0.0487350	+	0.998812i
							\(3\)	0.728136	−	1.57157i	0.420390	−	0.907344i
\(5\)	1.00081	+	1.73346i	0.447577	+	0.775227i								0.998228	−	0.0595094i	\(-0.0189536\pi\)
							−0.550650	+	0.834736i	\(0.685620\pi\)
\(7\)	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	\(0.0921811\pi\)
							−0.958359	+	0.285565i	\(0.907819\pi\)
\(17\)	−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	0.782313	−	0.622885i	\(-0.214040\pi\)
							−0.930591	+	0.366061i	\(0.880706\pi\)
\(29\)	−9.25829			−1.71922			−0.859611	−	0.510949i	\(-0.829294\pi\)
							−0.859611	+	0.510949i	\(0.829294\pi\)
\(31\)	−4.99699	−	2.88502i	−0.897486	−	0.518164i	−0.0211026	−	0.999777i	\(-0.506718\pi\)
							−0.876384	+	0.481613i	\(0.840051\pi\)
\(37\)	0.569094	−	0.328567i	0.0935586	−	0.0540161i	−0.452491	−	0.891769i	\(-0.649464\pi\)
							0.546049	+	0.837753i	\(0.316131\pi\)
\(41\)		−	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	0.704861	−	0.709345i	\(-0.251009\pi\)
							−0.966742	+	0.255755i	\(0.917676\pi\)

(See \(a_n\) instead)

Twists

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

    

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