Embedded newform 197.4.g.a.37.3

• Label: 197.4.g.a.37.3
• Level: $197$
• Weight: $4$
• Character: 197.37
• Analytic conductor: $11.623$
• Analytic rank: $0$
• Dimension: $2058$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 197 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	197.g (of order \(49\), degree \(42\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.6233762711\)
Analytic rank:	\(0\)
Dimension:	\(2058\)
Relative dimension:	\(49\) over \(\Q(\zeta_{49})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{49}]$

Embedding invariants

Embedding label			37.3
Character	\(\chi\)	\(=\)	197.37
Dual form			197.4.g.a.16.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.92908 + 0.316457i) q^{2} +(4.70953 + 6.75146i) q^{3} +(16.2613 - 2.09667i) q^{4} +(6.63060 + 4.31604i) q^{5} +(-25.3502 - 31.7881i) q^{6} +(4.33945 + 0.278602i) q^{7} +(-40.7048 + 7.92726i) q^{8} +(-14.0777 + 38.2538i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2058 q - 42 q^{2} - 42 q^{3} - 84 q^{4} - 42 q^{5} - 119 q^{6} - 84 q^{7} - 98 q^{8} - 42 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/197\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{48}{49}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.92908	+	0.316457i	−1.74269	+	0.111885i	−0.902578	−	0.430526i	\(-0.858328\pi\)
							−0.840114	+	0.542410i	\(0.817512\pi\)
\(3\)	4.70953	+	6.75146i	0.906349	+	1.29932i	0.953799	+	0.300446i	\(0.0971355\pi\)
							−0.0474497	+	0.998874i	\(0.515109\pi\)
\(5\)	6.63060	+	4.31604i	0.593059	+	0.386038i	0.806776	−	0.590857i	\(-0.201210\pi\)
							−0.213718	+	0.976895i	\(0.568557\pi\)
\(7\)	4.33945	+	0.278602i	0.234308	+	0.0150431i	0.180781	−	0.983523i	\(-0.442137\pi\)
							0.0535268	+	0.998566i	\(0.482954\pi\)
\(11\)	8.49702	+	28.6293i	0.232904	+	0.784732i	0.991148	+	0.132761i	\(0.0423843\pi\)
							−0.758244	+	0.651971i	\(0.773942\pi\)
\(13\)	0.892088	+	27.8186i	0.0190324	+	0.593499i	0.963934	+	0.266143i	\(0.0857494\pi\)
							−0.944901	+	0.327356i	\(0.893842\pi\)
\(17\)	20.1806	−	19.5438i	0.287912	−	0.278828i	−0.537614	−	0.843191i	\(-0.680674\pi\)
							0.825526	+	0.564363i	\(0.190878\pi\)
\(19\)	80.5898	−	38.8100i	0.973083	−	0.468612i	0.121362	−	0.992608i	\(-0.461274\pi\)
							0.851721	+	0.523996i	\(0.175559\pi\)
\(23\)	−14.0692	+	87.0228i	−0.127549	+	0.788935i	0.841169	+	0.540772i	\(0.181868\pi\)
							−0.968718	+	0.248163i	\(0.920173\pi\)
\(29\)	150.601	+	165.828i	0.964342	+	1.06185i	0.998050	+	0.0624156i	\(0.0198804\pi\)
							−0.0337087	+	0.999432i	\(0.510732\pi\)
\(31\)	−68.3505	−	27.6721i	−0.396004	−	0.160324i	0.168603	−	0.985684i	\(-0.446074\pi\)
							−0.564607	+	0.825360i	\(0.690972\pi\)
\(37\)	27.2570	+	7.14762i	0.121109	+	0.0317584i	0.313711	−	0.949518i	\(-0.398428\pi\)
							−0.192603	+	0.981277i	\(0.561693\pi\)
\(41\)	−295.899	+	286.562i	−1.12711	+	1.09155i	−0.132159	+	0.991229i	\(0.542191\pi\)
							−0.994955	+	0.100320i	\(0.968013\pi\)
\(43\)	−16.3071	−	54.9439i	−0.0578326	−	0.194857i	0.924788	−	0.380483i	\(-0.124242\pi\)
							−0.982621	+	0.185626i	\(0.940569\pi\)
\(47\)	−1.17965	+	2.26117i	−0.00366105	+	0.00701755i	−0.888423	−	0.459026i	\(-0.848199\pi\)
							0.884762	+	0.466043i	\(0.154321\pi\)