Embedded newform 196.2.p.a.103.18

• Label: 196.2.p.a.103.18
• 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			103.18
Character	\(\chi\)	\(=\)	196.103
Dual form			196.2.p.a.59.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.631090 - 1.26559i) q^{2} +(-0.268294 - 0.182920i) q^{3} +(-1.20345 - 1.59741i) q^{4} +(3.60103 + 0.269860i) q^{5} +(-0.400820 + 0.224112i) q^{6} +(1.91709 + 1.82340i) q^{7} +(-2.78115 + 0.514970i) q^{8} +(-1.05750 - 2.69447i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 13 q^{2} - 13 q^{4} - 22 q^{5} - 14 q^{6} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.631090	−	1.26559i	0.446248	−	0.894909i
							\(3\)	−0.268294	−	0.182920i	−0.154900	−	0.105609i	0.483390	−	0.875405i	\(-0.339405\pi\)
−0.638290	+	0.769796i	\(0.720358\pi\)
\(5\)	3.60103	+	0.269860i	1.61043	+	0.120685i	0.849298	−	0.527914i	\(-0.177026\pi\)
							0.761130	+	0.648599i	\(0.224645\pi\)
\(7\)	1.91709	+	1.82340i	0.724591	+	0.689179i
							\(11\)	−0.946254	−	0.371377i	−0.285306	−	0.111974i	0.218372	−	0.975866i	\(-0.429925\pi\)
−0.503679	+	0.863891i	\(0.668020\pi\)
\(13\)	−2.72262	+	2.17122i	−0.755119	+	0.602187i	−0.923528	−	0.383530i	\(-0.874708\pi\)
							0.168410	+	0.985717i	\(0.446137\pi\)
\(17\)	−1.83141	−	1.97379i	−0.444183	−	0.478715i	0.470782	−	0.882250i	\(-0.343972\pi\)
							−0.914964	+	0.403535i	\(0.867781\pi\)
\(19\)	2.18150	−	3.77847i	0.500471	−	0.866841i	−0.499529	−	0.866297i	\(-0.666494\pi\)
							1.00000		0.000543834i	\(-0.000173108\pi\)
\(23\)	−4.02678	+	4.33983i	−0.839641	+	0.904918i	−0.996575	−	0.0826915i	\(-0.973648\pi\)
							0.156934	+	0.987609i	\(0.449839\pi\)
\(29\)	1.26899	+	5.55980i	0.235645	+	1.03243i	0.944870	+	0.327446i	\(0.106188\pi\)
							−0.709225	+	0.704983i	\(0.750955\pi\)
\(31\)	2.93635	+	5.08591i	0.527384	+	0.913456i	0.999491	+	0.0319145i	\(0.0101604\pi\)
							−0.472106	+	0.881542i	\(0.656506\pi\)
\(37\)	−2.33916	−	0.721535i	−0.384555	−	0.118620i	0.0964488	−	0.995338i	\(-0.469252\pi\)
							−0.481004	+	0.876718i	\(0.659728\pi\)
\(41\)	−2.01156	−	4.17704i	−0.314152	−	0.652344i	0.682779	−	0.730625i	\(-0.260771\pi\)
							−0.996931	+	0.0782809i	\(0.975057\pi\)
\(43\)	−1.13347	+	2.35367i	−0.172852	+	0.358931i	−0.969340	−	0.245724i	\(-0.920974\pi\)
							0.796488	+	0.604654i	\(0.206689\pi\)
\(47\)	−1.76704	+	0.266339i	−0.257750	+	0.0388495i	−0.276646	−	0.960972i	\(-0.589223\pi\)
							0.0188961	+	0.999821i	\(0.493985\pi\)