Embedded newform 98.4.g.a.11.6

• Label: 98.4.g.a.11.6
• Level: $98$
• Weight: $4$
• Character: 98.11
• Analytic conductor: $5.782$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	98.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.78218718056\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(7\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			11.6
Character	\(\chi\)	\(=\)	98.11
Dual form			98.4.g.a.9.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.46610 - 1.36035i) q^{2} +(6.59617 - 0.994213i) q^{3} +(0.298920 - 3.98882i) q^{4} +(1.69395 - 4.31612i) q^{5} +(8.31820 - 10.4307i) q^{6} +(-10.9576 - 14.9309i) q^{7} +(-4.98792 - 6.25465i) q^{8} +(16.7205 - 5.15760i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 14 q^{2} - 8 q^{3} + 28 q^{4} + 7 q^{5} + 10 q^{6} + 20 q^{7} + 112 q^{8} + 189 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.46610	−	1.36035i	0.518346	−	0.480955i
							\(3\)	6.59617	−	0.994213i	1.26943	−	0.191336i	0.520431	−	0.853904i	\(-0.325771\pi\)
0.749002	+	0.662567i	\(0.230533\pi\)
\(5\)	1.69395	−	4.31612i	0.151512	−	0.386046i	−0.834886	−	0.550423i	\(-0.814466\pi\)
							0.986397	+	0.164378i	\(0.0525616\pi\)
\(7\)	−10.9576	−	14.9309i	−0.591656	−	0.806191i
							\(11\)	24.4925	+	7.55493i	0.671342	+	0.207082i	0.611642	−	0.791135i	\(-0.290509\pi\)
0.0597001	+	0.998216i	\(0.480986\pi\)
\(13\)	12.5891	+	55.1566i	0.268584	+	1.17675i	0.911661	+	0.410943i	\(0.134801\pi\)
							−0.643077	+	0.765802i	\(0.722342\pi\)
\(17\)	12.7784	−	8.71215i	0.182307	−	0.124295i	−0.468728	−	0.883342i	\(-0.655288\pi\)
							0.651035	+	0.759048i	\(0.274335\pi\)
\(19\)	27.7543	+	48.0718i	0.335119	+	0.580444i	0.983508	−	0.180866i	\(-0.0578900\pi\)
							−0.648388	+	0.761310i	\(0.724557\pi\)
\(23\)	−35.8714	−	24.4567i	−0.325204	−	0.221720i	0.389702	−	0.920941i	\(-0.372578\pi\)
							−0.714906	+	0.699221i	\(0.753530\pi\)
\(29\)	−48.7666	−	23.4848i	−0.312267	−	0.150380i	0.271186	−	0.962527i	\(-0.412584\pi\)
							−0.583452	+	0.812147i	\(0.698299\pi\)
\(31\)	45.1126	−	78.1372i	0.261369	−	0.452705i	−0.705237	−	0.708972i	\(-0.749159\pi\)
							0.966606	+	0.256267i	\(0.0824926\pi\)
\(37\)	−11.5426	−	154.025i	−0.0512863	−	0.684368i	−0.962300	−	0.271991i	\(-0.912318\pi\)
							0.911013	−	0.412377i	\(-0.135301\pi\)
\(41\)	140.895	+	176.677i	0.536686	+	0.672983i	0.974058	−	0.226298i	\(-0.0726622\pi\)
							−0.437372	+	0.899281i	\(0.644091\pi\)
\(43\)	−250.238	+	313.788i	−0.887463	+	1.11284i	0.105500	+	0.994419i	\(0.466356\pi\)
							−0.992963	+	0.118424i	\(0.962216\pi\)
\(47\)	−80.5613	+	74.7500i	−0.250023	+	0.231987i	−0.795221	−	0.606320i	\(-0.792645\pi\)
							0.545198	+	0.838307i	\(0.316455\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.g.a.11.6	yes	84
49.9	even	21	inner	98.4.g.a.9.6	✓	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.g.a.9.6	✓	84	49.9	even	21	inner
98.4.g.a.11.6	yes	84	1.1	even	1	trivial