Embedded newform 98.4.g.b.11.3

• Label: 98.4.g.b.11.3
• 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			11.3
Character	\(\chi\)	\(=\)	98.11
Dual form			98.4.g.b.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.46610 + 1.36035i) q^{2} +(-2.91630 + 0.439561i) q^{3} +(0.298920 - 3.98882i) q^{4} +(-5.72213 + 14.5797i) q^{5} +(3.67764 - 4.61161i) q^{6} +(18.2029 - 3.41371i) q^{7} +(4.98792 + 6.25465i) q^{8} +(-17.4889 + 5.39461i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 14 q^{2} + 2 q^{3} + 28 q^{4} - 9 q^{5} + 34 q^{6} + 28 q^{7} - 112 q^{8} - 175 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.91630	+	0.439561i	−0.561241	+	0.0845935i	−0.423535	−	0.905880i	\(-0.639211\pi\)
−0.137706	+	0.990473i	\(0.543973\pi\)
\(5\)	−5.72213	+	14.5797i	−0.511802	+	1.30405i	0.408470	+	0.912772i	\(0.366063\pi\)
							−0.920272	+	0.391279i	\(0.872033\pi\)
\(7\)	18.2029	−	3.41371i	0.982866	−	0.184323i
							\(11\)	−32.6840	−	10.0817i	−0.895873	−	0.276340i	−0.187572	−	0.982251i	\(-0.560062\pi\)
−0.708301	+	0.705910i	\(0.750538\pi\)
\(13\)	−20.2072	−	88.5337i	−0.431114	−	1.88883i	−0.457551	−	0.889184i	\(-0.651273\pi\)
							0.0264365	−	0.999650i	\(-0.491584\pi\)
\(17\)	−20.0604	+	13.6769i	−0.286198	+	0.195126i	−0.697906	−	0.716189i	\(-0.745885\pi\)
							0.411708	+	0.911316i	\(0.364932\pi\)
\(19\)	−8.71685	−	15.0980i	−0.105252	−	0.182301i	0.808589	−	0.588373i	\(-0.200231\pi\)
							−0.913841	+	0.406072i	\(0.866898\pi\)
\(23\)	−44.1959	−	30.1323i	−0.400673	−	0.273174i	0.346174	−	0.938170i	\(-0.387481\pi\)
							−0.746847	+	0.664996i	\(0.768433\pi\)
\(29\)	−182.497	−	87.8860i	−1.16858	−	0.562759i	−0.254018	−	0.967200i	\(-0.581752\pi\)
							−0.914564	+	0.404440i	\(0.867466\pi\)
\(31\)	6.37788	−	11.0468i	0.0369517	−	0.0640022i	−0.846958	−	0.531660i	\(-0.821569\pi\)
							0.883910	+	0.467657i	\(0.154902\pi\)
\(37\)	11.1371	+	148.615i	0.0494847	+	0.660327i	0.965703	+	0.259648i	\(0.0836067\pi\)
							−0.916219	+	0.400679i	\(0.868774\pi\)
\(41\)	147.761	+	185.287i	0.562840	+	0.705779i	0.979080	−	0.203476i	\(-0.0652238\pi\)
							−0.416240	+	0.909255i	\(0.636652\pi\)
\(43\)	−271.555	+	340.520i	−0.963065	+	1.20765i	0.0151149	+	0.999886i	\(0.495189\pi\)
							−0.978180	+	0.207760i	\(0.933383\pi\)
\(47\)	−293.059	+	271.919i	−0.909513	+	0.843905i	−0.988227	−	0.152992i	\(-0.951109\pi\)
							0.0787142	+	0.996897i	\(0.474919\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.g.b.11.3	yes	84
49.9	even	21	inner	98.4.g.b.9.3	✓	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.g.b.9.3	✓	84	49.9	even	21	inner
98.4.g.b.11.3	yes	84	1.1	even	1	trivial