Embedded newform 197.4.e.a.6.18

• Label: 197.4.e.a.6.18
• Level: $197$
• Weight: $4$
• Character: 197.6
• Analytic conductor: $11.623$
• Analytic rank: $0$
• Dimension: $288$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 197 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	197.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.6233762711\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(48\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			6.18
Character	\(\chi\)	\(=\)	197.6
Dual form			197.4.e.a.33.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77835 + 0.405896i) q^{2} +(3.55906 + 0.812332i) q^{3} +(-4.20998 + 2.02742i) q^{4} +(-2.71301 + 5.63362i) q^{5} -6.65897 q^{6} +(-6.79789 + 29.7835i) q^{7} +(18.0729 - 14.4126i) q^{8} +(-12.3191 - 5.93258i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 7 q^{2} - 7 q^{3} + 159 q^{4} - 7 q^{5} + 26 q^{6} - 53 q^{7} + 49 q^{8} + 391 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77835	+	0.405896i	−0.628741	+	0.143506i	−0.525005	−	0.851099i	\(-0.675937\pi\)
							−0.103736	+	0.994605i	\(0.533080\pi\)
\(3\)	3.55906	+	0.812332i	0.684941	+	0.156333i	0.550811	−	0.834630i	\(-0.314319\pi\)
							0.134131	+	0.990964i	\(0.457176\pi\)
\(5\)	−2.71301	+	5.63362i	−0.242659	+	0.503886i	−0.986355	−	0.164631i	\(-0.947357\pi\)
							0.743696	+	0.668518i	\(0.233071\pi\)
\(7\)	−6.79789	+	29.7835i	−0.367052	+	1.60816i	0.367782	+	0.929912i	\(0.380117\pi\)
							−0.734834	+	0.678247i	\(0.762740\pi\)
\(11\)	1.46669	−	0.334762i	0.0402021	−	0.00917587i	−0.202373	−	0.979309i	\(-0.564865\pi\)
							0.242575	+	0.970133i	\(0.422008\pi\)
\(13\)	4.27585	+	3.40988i	0.0912237	+	0.0727485i	0.668040	−	0.744126i	\(-0.267134\pi\)
							−0.576816	+	0.816874i	\(0.695705\pi\)
\(17\)	49.8853	−	103.588i	0.711704	−	1.47787i	−0.159630	−	0.987177i	\(-0.551030\pi\)
							0.871334	−	0.490691i	\(-0.163256\pi\)
\(19\)	−149.144			−1.80084			−0.900420	−	0.435022i	\(-0.856740\pi\)
							−0.900420	+	0.435022i	\(0.856740\pi\)
\(23\)	−5.84219	−	25.5963i	−0.0529644	−	0.232052i	0.941517	−	0.336966i	\(-0.109401\pi\)
							−0.994481	+	0.104913i	\(0.966543\pi\)
\(29\)	3.12361	+	13.6854i	0.0200014	+	0.0876318i	0.983944	−	0.178480i	\(-0.0571180\pi\)
							−0.963942	+	0.266112i	\(0.914261\pi\)
\(31\)	−90.6296	+	20.6856i	−0.525083	+	0.119847i	−0.476842	−	0.878989i	\(-0.658219\pi\)
							−0.0482407	+	0.998836i	\(0.515361\pi\)
\(37\)	28.4364	+	35.6581i	0.126349	+	0.158437i	0.840982	−	0.541063i	\(-0.181978\pi\)
							−0.714633	+	0.699499i	\(0.753406\pi\)
\(41\)	−153.112	−	73.7348i	−0.583221	−	0.280864i	0.118918	−	0.992904i	\(-0.462057\pi\)
							−0.702139	+	0.712040i	\(0.747772\pi\)
\(43\)	−123.951	−	543.067i	−0.439591	−	1.92597i	−0.371916	−	0.928266i	\(-0.621299\pi\)
							−0.0676746	−	0.997707i	\(-0.521558\pi\)
\(47\)	−36.0545	−	45.2109i	−0.111896	−	0.140313i	0.722730	−	0.691131i	\(-0.242887\pi\)
							−0.834625	+	0.550818i	\(0.814316\pi\)