Embedded newform 196.2.j.a.27.2

• Label: 196.2.j.a.27.2
• 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			27.2
Character	\(\chi\)	\(=\)	196.27
Dual form			196.2.j.a.167.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41194 + 0.0802101i) q^{2} +(0.320649 - 1.40485i) q^{3} +(1.98713 - 0.226503i) q^{4} +(3.61677 + 0.825505i) q^{5} +(-0.340052 + 2.00928i) q^{6} +(-1.15779 + 2.37897i) q^{7} +(-2.78754 + 0.479196i) q^{8} +(0.832108 + 0.400722i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41194	+	0.0802101i	−0.998390	+	0.0567171i
							\(3\)	0.320649	−	1.40485i	0.185127	−	0.811093i	−0.794012	−	0.607901i	\(-0.792012\pi\)
0.979139	−	0.203191i	\(-0.0651313\pi\)
\(5\)	3.61677	+	0.825505i	1.61747	+	0.369177i	0.933003	−	0.359868i	\(-0.117178\pi\)
							0.684466	+	0.729045i	\(0.260035\pi\)
\(7\)	−1.15779	+	2.37897i	−0.437604	+	0.899168i
							\(11\)	0.0133800	+	0.0277839i	0.00403423	+	0.00837716i	0.902976	−	0.429690i	\(-0.141377\pi\)
−0.898942	+	0.438068i	\(0.855663\pi\)
\(13\)	−0.604201	−	1.25464i	−0.167575	−	0.347974i	0.800222	−	0.599704i	\(-0.204715\pi\)
							−0.967798	+	0.251730i	\(0.919001\pi\)
\(17\)	−2.27351	+	1.81307i	−0.551408	+	0.439733i	−0.859140	−	0.511740i	\(-0.829001\pi\)
							0.307733	+	0.951473i	\(0.400430\pi\)
\(19\)	5.55319			1.27399			0.636995	−	0.770868i	\(-0.280177\pi\)
							0.636995	+	0.770868i	\(0.280177\pi\)
\(23\)	−4.05564	−	3.23426i	−0.845659	−	0.674390i	0.101612	−	0.994824i	\(-0.467600\pi\)
							−0.947271	+	0.320434i	\(0.896171\pi\)
\(29\)	−6.40291	−	8.02899i	−1.18899	−	1.49095i	−0.830143	−	0.557550i	\(-0.811741\pi\)
							−0.358847	−	0.933396i	\(-0.616830\pi\)
\(31\)	−2.17084			−0.389894			−0.194947	−	0.980814i	\(-0.562453\pi\)
							−0.194947	+	0.980814i	\(0.562453\pi\)
\(37\)	1.02460	+	1.28481i	0.168443	+	0.211221i	0.858887	−	0.512165i	\(-0.171156\pi\)
							−0.690444	+	0.723386i	\(0.742585\pi\)
\(41\)	−2.46823	−	0.563358i	−0.385473	−	0.0879817i	0.0253927	−	0.999678i	\(-0.491916\pi\)
							−0.410866	+	0.911696i	\(0.634774\pi\)
\(43\)	3.48404	−	0.795210i	0.531312	−	0.121268i	0.0515552	−	0.998670i	\(-0.483582\pi\)
							0.479756	+	0.877402i	\(0.340725\pi\)
\(47\)	−8.77574	+	4.22617i	−1.28007	+	0.616451i	−0.945409	−	0.325886i	\(-0.894337\pi\)
							−0.334665	+	0.942337i	\(0.608623\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.j.a.27.2	✓	156
4.3	odd	2	inner	196.2.j.a.27.8	yes	156
49.20	odd	14	inner	196.2.j.a.167.8	yes	156
196.167	even	14	inner	196.2.j.a.167.2	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.j.a.27.2	✓	156	1.1	even	1	trivial
196.2.j.a.27.8	yes	156	4.3	odd	2	inner
196.2.j.a.167.2	yes	156	196.167	even	14	inner
196.2.j.a.167.8	yes	156	49.20	odd	14	inner