Embedded newform 168.2.v.a.107.15

• Label: 168.2.v.a.107.15
• 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			107.15
Character	\(\chi\)	\(=\)	168.107
Dual form			168.2.v.a.11.15

$q$-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 \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0689217	−	1.41253i	0.0487350	−	0.998812i
							\(3\)	0.728136	+	1.57157i	0.420390	+	0.907344i
\(5\)	1.00081	−	1.73346i	0.447577	−	0.775227i								−0.550650	−	0.834736i	\(-0.685620\pi\)
							0.998228	+	0.0595094i	\(0.0189536\pi\)
\(7\)	2.50986	−	0.837013i	0.948639	−	0.316361i
							\(11\)	3.33711	−	1.92668i	1.00618	−	0.580916i	0.0961058	−	0.995371i	\(-0.469361\pi\)
0.910070	+	0.414455i	\(0.136028\pi\)
\(13\)		−	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	−0.959313	−	0.282346i	\(-0.908887\pi\)
							−0.235137	+	0.971962i	\(0.575554\pi\)
\(19\)	−0.232846	+	0.403301i	−0.0534185	+	0.0925235i	−0.891498	−	0.453024i	\(-0.850345\pi\)
							0.838080	+	0.545548i	\(0.183678\pi\)
\(23\)	−0.711115	+	1.23169i	−0.148278	+	0.256825i	−0.930591	−	0.366061i	\(-0.880706\pi\)
							0.782313	+	0.622885i	\(0.214040\pi\)
\(29\)	−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	−0.876384	−	0.481613i	\(-0.840051\pi\)
							−0.0211026	+	0.999777i	\(0.506718\pi\)
\(37\)	0.569094	+	0.328567i	0.0935586	+	0.0540161i	0.546049	−	0.837753i	\(-0.316131\pi\)
							−0.452491	+	0.891769i	\(0.649464\pi\)
\(41\)			8.42924i			1.31643i	0.752832	+	0.658213i	\(0.228687\pi\)
							−0.752832	+	0.658213i	\(0.771313\pi\)
\(43\)	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	\(-0.917676\pi\)
							0.704861	+	0.709345i	\(0.251009\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.v.a.107.15	yes	56
3.2	odd	2	inner	168.2.v.a.107.14	yes	56
4.3	odd	2		672.2.bd.a.527.12		56
7.4	even	3	inner	168.2.v.a.11.24	yes	56
8.3	odd	2	inner	168.2.v.a.107.5	yes	56
8.5	even	2		672.2.bd.a.527.11		56
12.11	even	2		672.2.bd.a.527.9		56
21.11	odd	6	inner	168.2.v.a.11.5	✓	56
24.5	odd	2		672.2.bd.a.527.10		56
24.11	even	2	inner	168.2.v.a.107.24	yes	56
28.11	odd	6		672.2.bd.a.431.10		56
56.11	odd	6	inner	168.2.v.a.11.14	yes	56
56.53	even	6		672.2.bd.a.431.9		56
84.11	even	6		672.2.bd.a.431.11		56
168.11	even	6	inner	168.2.v.a.11.15	yes	56
168.53	odd	6		672.2.bd.a.431.12		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.5	✓	56	21.11	odd	6	inner
168.2.v.a.11.14	yes	56	56.11	odd	6	inner
168.2.v.a.11.15	yes	56	168.11	even	6	inner
168.2.v.a.11.24	yes	56	7.4	even	3	inner
168.2.v.a.107.5	yes	56	8.3	odd	2	inner
168.2.v.a.107.14	yes	56	3.2	odd	2	inner
168.2.v.a.107.15	yes	56	1.1	even	1	trivial
168.2.v.a.107.24	yes	56	24.11	even	2	inner
672.2.bd.a.431.9		56	56.53	even	6
672.2.bd.a.431.10		56	28.11	odd	6
672.2.bd.a.431.11		56	84.11	even	6
672.2.bd.a.431.12		56	168.53	odd	6
672.2.bd.a.527.9		56	12.11	even	2
672.2.bd.a.527.10		56	24.5	odd	2
672.2.bd.a.527.11		56	8.5	even	2
672.2.bd.a.527.12		56	4.3	odd	2