Embedded newform 196.2.p.a.103.4

• Label: 196.2.p.a.103.4
• Level: $196$
• Weight: $2$
• Character: 196.103
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $312$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	196.p (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.56506787962\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(26\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			103.4
Character	\(\chi\)	\(=\)	196.103
Dual form			196.2.p.a.59.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34826 + 0.426855i) q^{2} +(1.15179 + 0.785275i) q^{3} +(1.63559 - 1.15102i) q^{4} +(2.72463 + 0.204183i) q^{5} +(-1.88810 - 0.567106i) q^{6} +(1.69335 - 2.03287i) q^{7} +(-1.71388 + 2.25003i) q^{8} +(-0.386067 - 0.983684i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 13 q^{2} - 13 q^{4} - 22 q^{5} - 14 q^{6} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34826	+	0.426855i	−0.953361	+	0.301832i
							\(3\)	1.15179	+	0.785275i	0.664984	+	0.453379i	0.848187	−	0.529697i	\(-0.177694\pi\)
−0.183203	+	0.983075i	\(0.558647\pi\)
\(5\)	2.72463	+	0.204183i	1.21849	+	0.0913134i	0.668408	−	0.743795i	\(-0.266976\pi\)
							0.550085	+	0.835109i	\(0.314595\pi\)
\(7\)	1.69335	−	2.03287i	0.640027	−	0.768353i
							\(11\)	−4.00653	−	1.57245i	−1.20801	−	0.474110i	−0.326016	−	0.945364i	\(-0.605706\pi\)
−0.881998	+	0.471254i	\(0.843802\pi\)
\(13\)	2.14265	−	1.70870i	0.594264	−	0.473909i	−0.279576	−	0.960124i	\(-0.590194\pi\)
							0.873840	+	0.486214i	\(0.161622\pi\)
\(17\)	2.62031	+	2.82402i	0.635518	+	0.684926i	0.965898	−	0.258922i	\(-0.0833674\pi\)
							−0.330380	+	0.943848i	\(0.607177\pi\)
\(19\)	−3.33091	+	5.76930i	−0.764162	+	1.32357i	0.176526	+	0.984296i	\(0.443514\pi\)
							−0.940688	+	0.339272i	\(0.889819\pi\)
\(23\)	−3.71540	+	4.00425i	−0.774714	+	0.834943i	−0.989920	−	0.141625i	\(-0.954767\pi\)
							0.215206	+	0.976569i	\(0.430958\pi\)
\(29\)	0.688317	+	3.01572i	0.127817	+	0.560004i	0.997763	+	0.0668560i	\(0.0212968\pi\)
							−0.869945	+	0.493148i	\(0.835846\pi\)
\(31\)	1.69507	+	2.93595i	0.304444	+	0.527312i	0.977137	−	0.212609i	\(-0.0681962\pi\)
							−0.672694	+	0.739921i	\(0.734863\pi\)
\(37\)	−7.82638	−	2.41412i	−1.28665	−	0.396879i	−0.425374	−	0.905018i	\(-0.639857\pi\)
							−0.861275	+	0.508139i	\(0.830334\pi\)
\(41\)	3.34988	+	6.95609i	0.523163	+	1.08636i	0.980401	+	0.197015i	\(0.0631247\pi\)
							−0.457237	+	0.889345i	\(0.651161\pi\)
\(43\)	1.60718	−	3.33735i	0.245093	−	0.508940i	−0.741739	−	0.670688i	\(-0.765999\pi\)
							0.986832	+	0.161748i	\(0.0517131\pi\)
\(47\)	−8.45195	+	1.27393i	−1.23284	+	0.185821i	−0.732960	−	0.680271i	\(-0.761862\pi\)
							−0.499883	+	0.866093i	\(0.666624\pi\)