Embedded newform 196.2.j.a.111.21

• Label: 196.2.j.a.111.21
• 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			111.21
Character	\(\chi\)	\(=\)	196.111
Dual form			196.2.j.a.83.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.04119 - 0.957034i) q^{2} +(1.57092 + 1.96987i) q^{3} +(0.168171 - 1.99292i) q^{4} +(1.55435 - 1.23956i) q^{5} +(3.52087 + 0.547594i) q^{6} +(-2.03422 + 1.69173i) q^{7} +(-1.73219 - 2.23596i) q^{8} +(-0.745040 + 3.26424i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.04119	−	0.957034i	0.736236	−	0.676725i
							\(3\)	1.57092	+	1.96987i	0.906971	+	1.13731i	0.990043	+	0.140762i	\(0.0449553\pi\)
−0.0830725	+	0.996544i	\(0.526473\pi\)
\(5\)	1.55435	−	1.23956i	0.695128	−	0.554346i	−0.210929	−	0.977501i	\(-0.567649\pi\)
							0.906057	+	0.423155i	\(0.139078\pi\)
\(7\)	−2.03422	+	1.69173i	−0.768861	+	0.639415i
							\(11\)	−3.09970	+	0.707487i	−0.934596	+	0.213315i	−0.662592	−	0.748981i	\(-0.730544\pi\)
−0.272004	+	0.962296i	\(0.587686\pi\)
\(13\)	−5.38387	+	1.22883i	−1.49322	+	0.340817i	−0.889705	−	0.456537i	\(-0.849090\pi\)
							−0.603512	+	0.797354i	\(0.706233\pi\)
\(17\)	2.48020	−	5.15019i	0.601537	−	1.24910i	−0.348601	−	0.937271i	\(-0.613343\pi\)
							0.950137	−	0.311832i	\(-0.100943\pi\)
\(19\)	7.59495			1.74240			0.871201	−	0.490927i	\(-0.163342\pi\)
							0.871201	+	0.490927i	\(0.163342\pi\)
\(23\)	0.751225	+	1.55993i	0.156641	+	0.325269i	0.964489	−	0.264122i	\(-0.0850821\pi\)
							−0.807848	+	0.589391i	\(0.799368\pi\)
\(29\)	0.626525	+	0.301719i	0.116343	+	0.0560277i	0.491150	−	0.871075i	\(-0.336577\pi\)
							−0.374807	+	0.927103i	\(0.622291\pi\)
\(31\)	−8.05654			−1.44700			−0.723499	−	0.690326i	\(-0.757467\pi\)
							−0.723499	+	0.690326i	\(0.757467\pi\)
\(37\)	4.24933	+	2.04637i	0.698586	+	0.336421i	0.749243	−	0.662295i	\(-0.230417\pi\)
							−0.0506571	+	0.998716i	\(0.516132\pi\)
\(41\)	−0.0970188	+	0.0773699i	−0.0151518	+	0.0120832i	−0.631036	−	0.775753i	\(-0.717370\pi\)
							0.615885	+	0.787836i	\(0.288799\pi\)
\(43\)	−3.53279	−	2.81730i	−0.538745	−	0.429635i	0.315942	−	0.948779i	\(-0.397680\pi\)
							−0.854687	+	0.519144i	\(0.826251\pi\)
\(47\)	0.299542	+	1.31238i	0.0436927	+	0.191430i	0.992064	−	0.125731i	\(-0.0401276\pi\)
							−0.948372	+	0.317161i	\(0.897270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.j.a.111.21	yes	156
4.3	odd	2	inner	196.2.j.a.111.3	yes	156
49.34	odd	14	inner	196.2.j.a.83.3	✓	156
196.83	even	14	inner	196.2.j.a.83.21	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.j.a.83.3	✓	156	49.34	odd	14	inner
196.2.j.a.83.21	yes	156	196.83	even	14	inner
196.2.j.a.111.3	yes	156	4.3	odd	2	inner
196.2.j.a.111.21	yes	156	1.1	even	1	trivial