Embedded newform 196.2.j.a.27.10

• Label: 196.2.j.a.27.10
• Level: $196$
• Weight: $2$
• Character: 196.27
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $156$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	196.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.56506787962\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(26\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			27.10
Character	\(\chi\)	\(=\)	196.27
Dual form			196.2.j.a.167.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.530578 - 1.31091i) q^{2} +(-0.325950 + 1.42808i) q^{3} +(-1.43697 + 1.39108i) q^{4} +(0.906872 + 0.206988i) q^{5} +(2.04503 - 0.330416i) q^{6} +(-2.40631 - 1.09985i) q^{7} +(2.58601 + 1.14567i) q^{8} +(0.769737 + 0.370686i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.530578	−	1.31091i	−0.375175	−	0.926954i
							\(3\)	−0.325950	+	1.42808i	−0.188187	+	0.824503i	0.789384	+	0.613899i	\(0.210400\pi\)
−0.977572	+	0.210603i	\(0.932457\pi\)
\(5\)	0.906872	+	0.206988i	0.405565	+	0.0925677i	0.420437	−	0.907322i	\(-0.361877\pi\)
							−0.0148715	+	0.999889i	\(0.504734\pi\)
\(7\)	−2.40631	−	1.09985i	−0.909500	−	0.415705i
							\(11\)	2.41212	+	5.00882i	0.727282	+	1.51022i	0.855125	+	0.518422i	\(0.173480\pi\)
−0.127843	+	0.991794i	\(0.540805\pi\)
\(13\)	2.37739	+	4.93670i	0.659369	+	1.36919i	0.915408	+	0.402527i	\(0.131868\pi\)
							−0.256039	+	0.966666i	\(0.582418\pi\)
\(17\)	−0.542474	+	0.432609i	−0.131569	+	0.104923i	−0.687063	−	0.726597i	\(-0.741101\pi\)
							0.555494	+	0.831520i	\(0.312529\pi\)
\(19\)	3.02950			0.695014			0.347507	−	0.937677i	\(-0.387028\pi\)
							0.347507	+	0.937677i	\(0.387028\pi\)
\(23\)	−2.77073	−	2.20958i	−0.577737	−	0.460730i	0.290503	−	0.956874i	\(-0.406177\pi\)
							−0.868240	+	0.496144i	\(0.834749\pi\)
\(29\)	−6.02277	−	7.55231i	−1.11840	−	1.40243i	−0.904972	−	0.425470i	\(-0.860109\pi\)
							−0.213427	−	0.976959i	\(-0.568463\pi\)
\(31\)	8.30033			1.49078			0.745391	−	0.666627i	\(-0.232263\pi\)
							0.745391	+	0.666627i	\(0.232263\pi\)
\(37\)	−1.60313	−	2.01027i	−0.263554	−	0.330486i	0.632393	−	0.774648i	\(-0.282073\pi\)
							−0.895946	+	0.444162i	\(0.853501\pi\)
\(41\)	4.23097	+	0.965691i	0.660766	+	0.150816i	0.539737	−	0.841833i	\(-0.318524\pi\)
							0.121029	+	0.992649i	\(0.461381\pi\)
\(43\)	6.95620	−	1.58771i	1.06081	−	0.242123i	0.343698	−	0.939080i	\(-0.388320\pi\)
							0.717113	+	0.696957i	\(0.245463\pi\)
\(47\)	−4.87707	+	2.34867i	−0.711393	+	0.342589i	−0.754336	−	0.656489i	\(-0.772041\pi\)
							0.0429426	+	0.999078i	\(0.486327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.j.a.27.10	✓	156
4.3	odd	2	inner	196.2.j.a.27.19	yes	156
49.20	odd	14	inner	196.2.j.a.167.19	yes	156
196.167	even	14	inner	196.2.j.a.167.10	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.j.a.27.10	✓	156	1.1	even	1	trivial
196.2.j.a.27.19	yes	156	4.3	odd	2	inner
196.2.j.a.167.10	yes	156	196.167	even	14	inner
196.2.j.a.167.19	yes	156	49.20	odd	14	inner