Embedded newform 196.2.j.a.111.19

• Label: 196.2.j.a.111.19
• Level: $196$
• Weight: $2$
• Character: 196.111
• 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			111.19
Character	\(\chi\)	\(=\)	196.111
Dual form			196.2.j.a.83.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.846372 - 1.13298i) q^{2} +(0.635719 + 0.797166i) q^{3} +(-0.567307 - 1.91785i) q^{4} +(-0.726306 + 0.579209i) q^{5} +(1.44123 - 0.0455600i) q^{6} +(2.60326 - 0.472259i) q^{7} +(-2.65305 - 0.980468i) q^{8} +(0.436227 - 1.91124i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.846372	−	1.13298i	0.598476	−	0.801141i
							\(3\)	0.635719	+	0.797166i	0.367033	+	0.460244i	0.930714	−	0.365748i	\(-0.119187\pi\)
−0.563681	+	0.825992i	\(0.690615\pi\)
\(5\)	−0.726306	+	0.579209i	−0.324814	+	0.259030i	−0.772297	−	0.635262i	\(-0.780892\pi\)
							0.447483	+	0.894293i	\(0.352321\pi\)
\(7\)	2.60326	−	0.472259i	0.983940	−	0.178497i
							\(11\)	5.04868	−	1.15233i	1.52224	−	0.347440i	0.622061	−	0.782969i	\(-0.286296\pi\)
0.900175	+	0.435529i	\(0.143439\pi\)
\(13\)	−2.71007	+	0.618555i	−0.751638	+	0.171556i	−0.581142	−	0.813802i	\(-0.697394\pi\)
							−0.170495	+	0.985358i	\(0.554537\pi\)
\(17\)	−2.67586	+	5.55648i	−0.648991	+	1.34764i	0.273593	+	0.961846i	\(0.411788\pi\)
							−0.922583	+	0.385798i	\(0.873926\pi\)
\(19\)	−5.31475			−1.21929			−0.609644	−	0.792675i	\(-0.708688\pi\)
							−0.609644	+	0.792675i	\(0.708688\pi\)
\(23\)	2.67158	+	5.54759i	0.557062	+	1.15675i	0.969347	+	0.245697i	\(0.0790167\pi\)
							−0.412284	+	0.911055i	\(0.635269\pi\)
\(29\)	−6.99898	−	3.37053i	−1.29968	−	0.625892i	−0.349307	−	0.937008i	\(-0.613583\pi\)
							−0.950372	+	0.311116i	\(0.899297\pi\)
\(31\)	−2.80797			−0.504326			−0.252163	−	0.967685i	\(-0.581142\pi\)
							−0.252163	+	0.967685i	\(0.581142\pi\)
\(37\)	−5.03268	−	2.42361i	−0.827367	−	0.398439i	−0.0282401	−	0.999601i	\(-0.508990\pi\)
							−0.799127	+	0.601162i	\(0.794705\pi\)
\(41\)	8.64941	−	6.89767i	1.35081	−	1.07724i	0.361351	−	0.932430i	\(-0.382316\pi\)
							0.989460	−	0.144805i	\(-0.0462556\pi\)
\(43\)	−1.76174	−	1.40494i	−0.268663	−	0.214252i	0.479886	−	0.877331i	\(-0.340678\pi\)
							−0.748549	+	0.663079i	\(0.769249\pi\)
\(47\)	1.31376	+	5.75595i	0.191631	+	0.839592i	0.975734	+	0.218960i	\(0.0702665\pi\)
							−0.784102	+	0.620631i	\(0.786876\pi\)