Embedded newform 196.2.j.a.27.4

• Label: 196.2.j.a.27.4
• 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			27.4
Character	\(\chi\)	\(=\)	196.27
Dual form			196.2.j.a.167.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26947 - 0.623262i) q^{2} +(-0.743013 + 3.25535i) q^{3} +(1.22309 + 1.58242i) q^{4} +(2.48779 + 0.567822i) q^{5} +(2.97216 - 3.66947i) q^{6} +(2.27356 + 1.35311i) q^{7} +(-0.566410 - 2.77113i) q^{8} +(-7.34233 - 3.53588i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26947	−	0.623262i	−0.897648	−	0.440713i
							\(3\)	−0.743013	+	3.25535i	−0.428979	+	1.87948i	0.0450784	+	0.998983i	\(0.485646\pi\)
−0.474057	+	0.880494i	\(0.657211\pi\)
\(5\)	2.48779	+	0.567822i	1.11257	+	0.253938i	0.739028	−	0.673674i	\(-0.235285\pi\)
							0.373545	+	0.927612i	\(0.378142\pi\)
\(7\)	2.27356	+	1.35311i	0.859326	+	0.511428i
							\(11\)	0.384733	+	0.798906i	0.116001	+	0.240879i	0.950883	−	0.309550i	\(-0.100178\pi\)
−0.834882	+	0.550429i	\(0.814464\pi\)
\(13\)	0.912607	+	1.89505i	0.253112	+	0.525592i	0.988347	−	0.152221i	\(-0.0486424\pi\)
							−0.735235	+	0.677812i	\(0.762928\pi\)
\(17\)	3.00169	−	2.39377i	0.728016	−	0.580573i	−0.187783	−	0.982211i	\(-0.560130\pi\)
							0.915799	+	0.401637i	\(0.131559\pi\)
\(19\)	−1.86887			−0.428748			−0.214374	−	0.976752i	\(-0.568771\pi\)
							−0.214374	+	0.976752i	\(0.568771\pi\)
\(23\)	−4.77020	−	3.80411i	−0.994656	−	0.793212i	−0.0162424	−	0.999868i	\(-0.505170\pi\)
							−0.978414	+	0.206656i	\(0.933742\pi\)
\(29\)	−0.186951	−	0.234429i	−0.0347159	−	0.0435324i	0.764170	−	0.645014i	\(-0.223149\pi\)
							−0.798886	+	0.601482i	\(0.794577\pi\)
\(31\)	−4.86707			−0.874151			−0.437076	−	0.899425i	\(-0.643986\pi\)
							−0.437076	+	0.899425i	\(0.643986\pi\)
\(37\)	4.10523	+	5.14779i	0.674895	+	0.846292i	0.994873	−	0.101133i	\(-0.0322468\pi\)
							−0.319977	+	0.947425i	\(0.603675\pi\)
\(41\)	−2.27651	−	0.519598i	−0.355531	−	0.0811476i	0.0410251	−	0.999158i	\(-0.486938\pi\)
							−0.396556	+	0.918010i	\(0.629795\pi\)
\(43\)	2.38875	−	0.545217i	0.364281	−	0.0831448i	−0.0364625	−	0.999335i	\(-0.511609\pi\)
							0.400744	+	0.916190i	\(0.368752\pi\)
\(47\)	8.30887	−	4.00134i	1.21197	−	0.583656i	0.284908	−	0.958555i	\(-0.408037\pi\)
							0.927066	+	0.374899i	\(0.122323\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.j.a.27.4	✓	156
4.3	odd	2	inner	196.2.j.a.27.12	yes	156
49.20	odd	14	inner	196.2.j.a.167.12	yes	156
196.167	even	14	inner	196.2.j.a.167.4	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.j.a.27.4	✓	156	1.1	even	1	trivial
196.2.j.a.27.12	yes	156	4.3	odd	2	inner
196.2.j.a.167.4	yes	156	196.167	even	14	inner
196.2.j.a.167.12	yes	156	49.20	odd	14	inner