Embedded newform 196.2.p.a.59.21

• Label: 196.2.p.a.59.21
• Level: $196$
• Weight: $2$
• Character: 196.59
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	196.p (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.56506787962\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(26\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			59.21
Character	\(\chi\)	\(=\)	196.59
Dual form			196.2.p.a.103.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.976093 - 1.02335i) q^{2} +(0.268294 - 0.182920i) q^{3} +(-0.0944861 - 1.99777i) q^{4} +(3.60103 - 0.269860i) q^{5} +(0.0746891 - 0.453105i) q^{6} +(-1.91709 + 1.82340i) q^{7} +(-2.13664 - 1.85331i) q^{8} +(-1.05750 + 2.69447i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 13 q^{2} - 13 q^{4} - 22 q^{5} - 14 q^{6} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.976093	−	1.02335i	0.690202	−	0.723617i
							\(3\)	0.268294	−	0.182920i	0.154900	−	0.105609i	−0.483390	−	0.875405i	\(-0.660595\pi\)
0.638290	+	0.769796i	\(0.279642\pi\)
\(5\)	3.60103	−	0.269860i	1.61043	−	0.120685i	0.761130	−	0.648599i	\(-0.224645\pi\)
							0.849298	+	0.527914i	\(0.177026\pi\)
\(7\)	−1.91709	+	1.82340i	−0.724591	+	0.689179i
							\(11\)	0.946254	−	0.371377i	0.285306	−	0.111974i	−0.218372	−	0.975866i	\(-0.570075\pi\)
0.503679	+	0.863891i	\(0.331980\pi\)
\(13\)	−2.72262	−	2.17122i	−0.755119	−	0.602187i	0.168410	−	0.985717i	\(-0.446137\pi\)
							−0.923528	+	0.383530i	\(0.874708\pi\)
\(17\)	−1.83141	+	1.97379i	−0.444183	+	0.478715i	−0.914964	−	0.403535i	\(-0.867781\pi\)
							0.470782	+	0.882250i	\(0.343972\pi\)
\(19\)	−2.18150	−	3.77847i	−0.500471	−	0.866841i	−1.00000		0.000543834i	\(-0.999827\pi\)
							0.499529	−	0.866297i	\(-0.333506\pi\)
\(23\)	4.02678	+	4.33983i	0.839641	+	0.904918i	0.996575	−	0.0826915i	\(-0.0263516\pi\)
							−0.156934	+	0.987609i	\(0.550161\pi\)
\(29\)	1.26899	−	5.55980i	0.235645	−	1.03243i	−0.709225	−	0.704983i	\(-0.750955\pi\)
							0.944870	−	0.327446i	\(-0.106188\pi\)
\(31\)	−2.93635	+	5.08591i	−0.527384	+	0.913456i	0.472106	+	0.881542i	\(0.343494\pi\)
							−0.999491	+	0.0319145i	\(0.989840\pi\)
\(37\)	−2.33916	+	0.721535i	−0.384555	+	0.118620i	−0.481004	−	0.876718i	\(-0.659728\pi\)
							0.0964488	+	0.995338i	\(0.469252\pi\)
\(41\)	−2.01156	+	4.17704i	−0.314152	+	0.652344i	−0.996931	−	0.0782809i	\(-0.975057\pi\)
							0.682779	+	0.730625i	\(0.260771\pi\)
\(43\)	1.13347	+	2.35367i	0.172852	+	0.358931i	0.969340	−	0.245724i	\(-0.0790256\pi\)
							−0.796488	+	0.604654i	\(0.793311\pi\)
\(47\)	1.76704	+	0.266339i	0.257750	+	0.0388495i	0.276646	−	0.960972i	\(-0.410777\pi\)
							−0.0188961	+	0.999821i	\(0.506015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.p.a.59.21	yes	312
4.3	odd	2	inner	196.2.p.a.59.18	✓	312
49.5	odd	42	inner	196.2.p.a.103.18	yes	312
196.103	even	42	inner	196.2.p.a.103.21	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.p.a.59.18	✓	312	4.3	odd	2	inner
196.2.p.a.59.21	yes	312	1.1	even	1	trivial
196.2.p.a.103.18	yes	312	49.5	odd	42	inner
196.2.p.a.103.21	yes	312	196.103	even	42	inner