Embedded newform 98.4.g.b.11.6

• Label: 98.4.g.b.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.b.9.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.46610 + 1.36035i) q^{2} +(6.60186 - 0.995070i) q^{3} +(0.298920 - 3.98882i) q^{4} +(-1.48581 + 3.78578i) q^{5} +(-8.32537 + 10.4397i) q^{6} +(2.76568 + 18.3126i) q^{7} +(4.98792 + 6.25465i) q^{8} +(16.7939 - 5.18025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 14 q^{2} + 2 q^{3} + 28 q^{4} - 9 q^{5} + 34 q^{6} + 28 q^{7} - 112 q^{8} - 175 q^{9}+O(q^{10}) \)

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
							\(3\)	6.60186	−	0.995070i	1.27053	−	0.191501i	0.521048	−	0.853527i	\(-0.325541\pi\)
0.749481	+	0.662026i	\(0.230303\pi\)
\(5\)	−1.48581	+	3.78578i	−0.132895	+	0.338611i	−0.981795	−	0.189942i	\(-0.939170\pi\)
							0.848900	+	0.528553i	\(0.177265\pi\)
\(7\)	2.76568	+	18.3126i	0.149333	+	0.988787i
							\(11\)	57.4838	+	17.7314i	1.57564	+	0.486020i	0.954633	−	0.297786i	\(-0.0962483\pi\)
0.621005	+	0.783806i	\(0.286725\pi\)
\(13\)	−6.81702	−	29.8673i	−0.145439	−	0.637208i	−0.994118	−	0.108301i	\(-0.965459\pi\)
							0.848680	−	0.528907i	\(-0.177398\pi\)
\(17\)	29.9419	−	20.4140i	0.427175	−	0.291243i	−0.330585	−	0.943776i	\(-0.607246\pi\)
							0.757760	+	0.652533i	\(0.226294\pi\)
\(19\)	52.1124	+	90.2614i	0.629232	+	1.08986i	0.987706	+	0.156322i	\(0.0499639\pi\)
							−0.358474	+	0.933540i	\(0.616703\pi\)
\(23\)	−55.0603	−	37.5395i	−0.499168	−	0.340327i	0.287420	−	0.957805i	\(-0.407202\pi\)
							−0.786589	+	0.617477i	\(0.788155\pi\)
\(29\)	−161.316	−	77.6855i	−1.03295	−	0.497443i	−0.160958	−	0.986961i	\(-0.551458\pi\)
							−0.871993	+	0.489518i	\(0.837173\pi\)
\(31\)	14.6202	−	25.3229i	0.0847053	−	0.146714i	−0.820560	−	0.571560i	\(-0.806338\pi\)
							0.905266	+	0.424846i	\(0.139672\pi\)
\(37\)	−20.6367	−	275.377i	−0.0916931	−	1.22356i	−0.833337	−	0.552766i	\(-0.813573\pi\)
							0.741644	−	0.670794i	\(-0.234047\pi\)
\(41\)	−191.936	−	240.681i	−0.731108	−	0.916781i	0.267801	−	0.963474i	\(-0.413703\pi\)
							−0.998909	+	0.0466934i	\(0.985132\pi\)
\(43\)	−0.461403	+	0.578581i	−0.00163636	+	0.00205192i	−0.782649	−	0.622463i	\(-0.786132\pi\)
							0.781013	+	0.624515i	\(0.214703\pi\)
\(47\)	17.0564	−	15.8260i	0.0529348	−	0.0491163i	−0.653262	−	0.757132i	\(-0.726600\pi\)
							0.706197	+	0.708016i	\(0.250409\pi\)

(See \(a_n\) instead)

Twists

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

    

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