Embedded newform 196.2.j.a.111.24

• Label: 196.2.j.a.111.24
• 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			111.24
Character	\(\chi\)	\(=\)	196.111
Dual form			196.2.j.a.83.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32010 + 0.507284i) q^{2} +(-1.45168 - 1.82035i) q^{3} +(1.48533 + 1.33933i) q^{4} +(1.13412 - 0.904431i) q^{5} +(-0.992928 - 3.13945i) q^{6} +(-0.147062 - 2.64166i) q^{7} +(1.28136 + 2.52153i) q^{8} +(-0.538730 + 2.36033i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32010	+	0.507284i	0.933451	+	0.358704i
							\(3\)	−1.45168	−	1.82035i	−0.838127	−	1.05098i	−0.997961	−	0.0638337i	\(-0.979667\pi\)
0.159834	−	0.987144i	\(-0.448904\pi\)
\(5\)	1.13412	−	0.904431i	0.507194	−	0.404474i	−0.336182	−	0.941797i	\(-0.609136\pi\)
							0.843377	+	0.537323i	\(0.180564\pi\)
\(7\)	−0.147062	−	2.64166i	−0.0555841	−	0.998454i
							\(11\)	−0.655703	+	0.149660i	−0.197702	+	0.0451242i	−0.320225	−	0.947341i	\(-0.603759\pi\)
0.122523	+	0.992466i	\(0.460901\pi\)
\(13\)	1.04204	−	0.237840i	0.289011	−	0.0659648i	−0.0755576	−	0.997141i	\(-0.524074\pi\)
							0.364568	+	0.931177i	\(0.381217\pi\)
\(17\)	−0.527427	+	1.09521i	−0.127920	+	0.265628i	−0.955086	−	0.296330i	\(-0.904237\pi\)
							0.827166	+	0.561958i	\(0.189952\pi\)
\(19\)	3.97680			0.912341			0.456171	−	0.889892i	\(-0.349221\pi\)
							0.456171	+	0.889892i	\(0.349221\pi\)
\(23\)	2.80013	+	5.81454i	0.583868	+	1.21242i	0.958460	+	0.285226i	\(0.0920685\pi\)
							−0.374592	+	0.927190i	\(0.622217\pi\)
\(29\)	−7.01088	−	3.37626i	−1.30189	−	0.626956i	−0.350967	−	0.936388i	\(-0.614147\pi\)
							−0.950922	+	0.309431i	\(0.899861\pi\)
\(31\)	−9.48293			−1.70318			−0.851592	−	0.524205i	\(-0.824363\pi\)
							−0.851592	+	0.524205i	\(0.824363\pi\)
\(37\)	3.82889	+	1.84390i	0.629465	+	0.303135i	0.721282	−	0.692641i	\(-0.243553\pi\)
							−0.0918168	+	0.995776i	\(0.529267\pi\)
\(41\)	−0.0397106	+	0.0316681i	−0.00620175	+	0.00494573i	−0.626586	−	0.779352i	\(-0.715548\pi\)
							0.620384	+	0.784298i	\(0.286977\pi\)
\(43\)	1.03558	+	0.825848i	0.157925	+	0.125941i	0.699264	−	0.714863i	\(-0.253511\pi\)
							−0.541339	+	0.840804i	\(0.682083\pi\)
\(47\)	2.66423	+	11.6727i	0.388617	+	1.70264i	0.669425	+	0.742880i	\(0.266541\pi\)
							−0.280808	+	0.959764i	\(0.590602\pi\)