Embedded newform 98.4.g.a.65.2

• Label: 98.4.g.a.65.2
• Level: $98$
• Weight: $4$
• Character: 98.65
• 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			65.2
Character	\(\chi\)	\(=\)	98.65
Dual form			98.4.g.a.95.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.730682 - 1.86175i) q^{2} +(-0.524442 - 6.99820i) q^{3} +(-2.93221 + 2.72069i) q^{4} +(-5.50602 + 3.75394i) q^{5} +(-12.6457 + 6.08984i) q^{6} +(-17.1709 + 6.93965i) q^{7} +(7.20775 + 3.47107i) q^{8} +(-22.0013 + 3.31616i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 14 q^{2} - 8 q^{3} + 28 q^{4} + 7 q^{5} + 10 q^{6} + 20 q^{7} + 112 q^{8} + 189 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{10}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.730682	−	1.86175i	−0.258335	−	0.658227i
							\(3\)	−0.524442	−	6.99820i	−0.100929	−	1.34680i	−0.785954	−	0.618285i	\(-0.787828\pi\)
0.685025	−	0.728519i	\(-0.259791\pi\)
\(5\)	−5.50602	+	3.75394i	−0.492474	+	0.335763i	−0.783959	−	0.620812i	\(-0.786803\pi\)
							0.291486	+	0.956575i	\(0.405850\pi\)
\(7\)	−17.1709	+	6.93965i	−0.927144	+	0.374706i
							\(11\)	−9.03637	−	1.36201i	−0.247688	−	0.0373330i	0.0240247	−	0.999711i	\(-0.492352\pi\)
−0.271713	+	0.962378i	\(0.587590\pi\)
\(13\)	−4.48594	+	5.62519i	−0.0957058	+	0.120011i	−0.827378	−	0.561645i	\(-0.810169\pi\)
							0.731672	+	0.681656i	\(0.238740\pi\)
\(17\)	35.8924	−	11.0713i	0.512070	−	0.157953i	−0.0279429	−	0.999610i	\(-0.508896\pi\)
							0.540013	+	0.841657i	\(0.318419\pi\)
\(19\)	−27.4793	+	47.5956i	−0.331799	+	0.574693i	−0.982865	−	0.184328i	\(-0.940989\pi\)
							0.651065	+	0.759022i	\(0.274322\pi\)
\(23\)	−154.500	−	47.6570i	−1.40068	−	0.432051i	−0.499804	−	0.866138i	\(-0.666595\pi\)
							−0.900871	+	0.434087i	\(0.857071\pi\)
\(29\)	−39.3655	+	172.472i	−0.252069	+	1.10439i	0.677438	+	0.735580i	\(0.263090\pi\)
							−0.929507	+	0.368805i	\(0.879767\pi\)
\(31\)	−111.148	−	192.514i	−0.643961	−	1.11537i	−0.984540	−	0.175158i	\(-0.943956\pi\)
							0.340579	−	0.940216i	\(-0.389377\pi\)
\(37\)	−268.527	−	249.157i	−1.19312	−	1.10706i	−0.991835	−	0.127526i	\(-0.959296\pi\)
							−0.201289	−	0.979532i	\(-0.564513\pi\)
\(41\)	143.773	+	69.2376i	0.547649	+	0.263734i	0.687184	−	0.726483i	\(-0.258847\pi\)
							−0.139535	+	0.990217i	\(0.544561\pi\)
\(43\)	−0.691600	+	0.333057i	−0.00245275	+	0.00118118i	−0.435110	−	0.900377i	\(-0.643290\pi\)
							0.432657	+	0.901559i	\(0.357576\pi\)
\(47\)	−83.5498	−	212.882i	−0.259298	−	0.660680i	0.740639	−	0.671903i	\(-0.234523\pi\)
							−0.999937	+	0.0112229i	\(0.996428\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.g.a.65.2	✓	84
49.46	even	21	inner	98.4.g.a.95.2	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.g.a.65.2	✓	84	1.1	even	1	trivial
98.4.g.a.95.2	yes	84	49.46	even	21	inner