Embedded newform 196.2.j.a.27.9

• Label: 196.2.j.a.27.9
• 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			27.9
Character	\(\chi\)	\(=\)	196.27
Dual form			196.2.j.a.167.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.894470 + 1.09541i) q^{2} +(0.0687858 - 0.301370i) q^{3} +(-0.399847 - 1.95962i) q^{4} +(-0.644334 - 0.147065i) q^{5} +(0.268597 + 0.344915i) q^{6} +(-1.15895 + 2.37841i) q^{7} +(2.50424 + 1.31483i) q^{8} +(2.61681 + 1.26019i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.894470	+	1.09541i	−0.632486	+	0.774572i
							\(3\)	0.0687858	−	0.301370i	0.0397135	−	0.173996i	−0.951182	−	0.308631i	\(-0.900129\pi\)
0.990895	+	0.134635i	\(0.0429862\pi\)
\(5\)	−0.644334	−	0.147065i	−0.288155	−	0.0657695i	0.0760000	−	0.997108i	\(-0.475785\pi\)
							−0.364155	+	0.931338i	\(0.618642\pi\)
\(7\)	−1.15895	+	2.37841i	−0.438043	+	0.898954i
							\(11\)	1.52133	+	3.15908i	0.458700	+	0.952500i	0.994158	+	0.107938i	\(0.0344247\pi\)
−0.535458	+	0.844562i	\(0.679861\pi\)
\(13\)	2.04222	+	4.24072i	0.566411	+	1.17616i	0.965777	+	0.259373i	\(0.0835159\pi\)
							−0.399366	+	0.916791i	\(0.630770\pi\)
\(17\)	5.62829	−	4.48841i	1.36506	−	1.08860i	0.378429	−	0.925630i	\(-0.376464\pi\)
							0.986631	−	0.162969i	\(-0.0521071\pi\)
\(19\)	−3.99801			−0.917206			−0.458603	−	0.888641i	\(-0.651650\pi\)
							−0.458603	+	0.888641i	\(0.651650\pi\)
\(23\)	1.01511	+	0.809522i	0.211665	+	0.168797i	0.723584	−	0.690237i	\(-0.242494\pi\)
							−0.511919	+	0.859034i	\(0.671065\pi\)
\(29\)	3.68573	+	4.62176i	0.684423	+	0.858240i	0.995753	−	0.0920621i	\(-0.0293458\pi\)
							−0.311330	+	0.950302i	\(0.600774\pi\)
\(31\)	−8.06065			−1.44773			−0.723867	−	0.689939i	\(-0.757637\pi\)
							−0.723867	+	0.689939i	\(0.757637\pi\)
\(37\)	−2.22032	−	2.78419i	−0.365018	−	0.457718i	0.565077	−	0.825038i	\(-0.308847\pi\)
							−0.930094	+	0.367321i	\(0.880275\pi\)
\(41\)	−4.89556	−	1.11738i	−0.764558	−	0.174505i	−0.177578	−	0.984107i	\(-0.556826\pi\)
							−0.586980	+	0.809601i	\(0.699683\pi\)
\(43\)	11.4763	−	2.61939i	1.75012	−	0.399454i	0.776955	−	0.629556i	\(-0.216763\pi\)
							0.973167	+	0.230102i	\(0.0739059\pi\)
\(47\)	9.75455	−	4.69755i	1.42285	−	0.685207i	0.445195	−	0.895434i	\(-0.353134\pi\)
							0.977653	+	0.210226i	\(0.0674200\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.j.a.27.9	yes	156
4.3	odd	2	inner	196.2.j.a.27.1	✓	156
49.20	odd	14	inner	196.2.j.a.167.1	yes	156
196.167	even	14	inner	196.2.j.a.167.9	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.j.a.27.1	✓	156	4.3	odd	2	inner
196.2.j.a.27.9	yes	156	1.1	even	1	trivial
196.2.j.a.167.1	yes	156	49.20	odd	14	inner
196.2.j.a.167.9	yes	156	196.167	even	14	inner