Embedded newform 98.4.e.a.15.7

• Label: 98.4.e.a.15.7
• Level: $98$
• Weight: $4$
• Character: 98.15
• Analytic conductor: $5.782$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	98.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.78218718056\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			15.7
Character	\(\chi\)	\(=\)	98.15
Dual form			98.4.e.a.85.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.445042 + 1.94986i) q^{2} +(5.26476 - 6.60180i) q^{3} +(-3.60388 - 1.73553i) q^{4} +(5.40383 - 6.77619i) q^{5} +(10.5295 + 13.2036i) q^{6} +(-1.27168 + 18.4765i) q^{7} +(4.98792 - 6.25465i) q^{8} +(-9.85802 - 43.1908i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 14 q^{2} - 5 q^{3} - 28 q^{4} + 12 q^{5} - 10 q^{6} - 7 q^{7} - 56 q^{8} - 98 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.445042	+	1.94986i	−0.157346	+	0.689378i
							\(3\)	5.26476	−	6.60180i	1.01320	−	1.27052i	0.0508535	−	0.998706i	\(-0.483806\pi\)
0.962350	−	0.271812i	\(-0.0876227\pi\)
\(5\)	5.40383	−	6.77619i	0.483333	−	0.606081i	−0.479046	−	0.877790i	\(-0.659017\pi\)
							0.962379	+	0.271709i	\(0.0875888\pi\)
\(7\)	−1.27168	+	18.4765i	−0.0686640	+	0.997640i
							\(11\)	13.1260	−	57.5087i	0.359785	−	1.57632i	−0.393943	−	0.919135i	\(-0.628889\pi\)
0.753728	−	0.657187i	\(-0.228254\pi\)
\(13\)	1.36896	−	5.99782i	0.0292063	−	0.127961i	−0.958223	−	0.286022i	\(-0.907667\pi\)
							0.987429	+	0.158061i	\(0.0505241\pi\)
\(17\)	58.1926	−	28.0241i	0.830222	−	0.399814i	0.0300234	−	0.999549i	\(-0.490442\pi\)
							0.800198	+	0.599735i	\(0.204728\pi\)
\(19\)	−119.215			−1.43946			−0.719732	−	0.694252i	\(-0.755735\pi\)
							−0.719732	+	0.694252i	\(0.755735\pi\)
\(23\)	160.401	+	77.2450i	1.45417	+	0.700291i	0.983314	−	0.181918i	\(-0.0582307\pi\)
							0.470856	+	0.882210i	\(0.343945\pi\)
\(29\)	−179.714	+	86.5455i	−1.15076	+	0.554176i	−0.909261	−	0.416226i	\(-0.863353\pi\)
							−0.241497	+	0.970402i	\(0.577638\pi\)
\(31\)	88.4324			0.512353			0.256176	−	0.966630i	\(-0.417537\pi\)
							0.256176	+	0.966630i	\(0.417537\pi\)
\(37\)	244.337	−	117.666i	1.08564	−	0.522817i	0.196524	−	0.980499i	\(-0.437035\pi\)
							0.889116	+	0.457682i	\(0.151320\pi\)
\(41\)	−295.757	+	370.868i	−1.12657	+	1.41268i	−0.228107	+	0.973636i	\(0.573254\pi\)
							−0.898466	+	0.439043i	\(0.855318\pi\)
\(43\)	79.6360	+	99.8604i	0.282427	+	0.354153i	0.902728	−	0.430211i	\(-0.141561\pi\)
							−0.620301	+	0.784364i	\(0.712989\pi\)
\(47\)	−75.7905	+	332.060i	−0.235217	+	1.03055i	0.710024	+	0.704177i	\(0.248684\pi\)
							−0.945241	+	0.326374i	\(0.894173\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.e.a.15.7	✓	42
49.36	even	7	inner	98.4.e.a.85.7	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.e.a.15.7	✓	42	1.1	even	1	trivial
98.4.e.a.85.7	yes	42	49.36	even	7	inner