Embedded newform 98.4.e.b.15.4

• Label: 98.4.e.b.15.4
• 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			15.4
Character	\(\chi\)	\(=\)	98.15
Dual form			98.4.e.b.85.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.445042 - 1.94986i) q^{2} +(0.220311 - 0.276261i) q^{3} +(-3.60388 - 1.73553i) q^{4} +(-7.39079 + 9.26776i) q^{5} +(-0.440622 - 0.552522i) q^{6} +(-9.68982 + 15.7831i) q^{7} +(-4.98792 + 6.25465i) q^{8} +(5.98028 + 26.2013i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 14 q^{2} - q^{3} - 28 q^{4} + 14 q^{5} + 2 q^{6} + 7 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

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
							\(3\)	0.220311	−	0.276261i	0.0423988	−	0.0531665i	−0.760180	−	0.649713i	\(-0.774889\pi\)
0.802579	+	0.596546i	\(0.203461\pi\)
\(5\)	−7.39079	+	9.26776i	−0.661052	+	0.828934i	−0.993457	−	0.114203i	\(-0.963569\pi\)
							0.332405	+	0.943137i	\(0.392140\pi\)
\(7\)	−9.68982	+	15.7831i	−0.523201	+	0.852209i
							\(11\)	0.956397	−	4.19025i	0.0262149	−	0.114855i	−0.960128	−	0.279562i	\(-0.909811\pi\)
0.986343	+	0.164707i	\(0.0526678\pi\)
\(13\)	3.27973	−	14.3694i	0.0699718	−	0.306567i	−0.927817	−	0.373037i	\(-0.878317\pi\)
							0.997788	+	0.0664701i	\(0.0211737\pi\)
\(17\)	−76.0132	+	36.6060i	−1.08447	+	0.522251i	−0.888742	−	0.458408i	\(-0.848420\pi\)
							−0.195724	+	0.980659i	\(0.562706\pi\)
\(19\)	8.66723			0.104653			0.0523263	−	0.998630i	\(-0.483336\pi\)
							0.0523263	+	0.998630i	\(0.483336\pi\)
\(23\)	76.1381	+	36.6662i	0.690256	+	0.332410i	0.745916	−	0.666040i	\(-0.232012\pi\)
							−0.0556601	+	0.998450i	\(0.517726\pi\)
\(29\)	−36.2967	+	17.4796i	−0.232418	+	0.111927i	−0.546469	−	0.837480i	\(-0.684028\pi\)
							0.314050	+	0.949406i	\(0.398314\pi\)
\(31\)	−255.090			−1.47792			−0.738961	−	0.673748i	\(-0.764683\pi\)
							−0.738961	+	0.673748i	\(0.764683\pi\)
\(37\)	174.711	−	84.1363i	0.776278	−	0.373836i	−0.00341805	−	0.999994i	\(-0.501088\pi\)
							0.779696	+	0.626158i	\(0.215374\pi\)
\(41\)	−124.511	+	156.132i	−0.474277	+	0.594725i	−0.960213	−	0.279270i	\(-0.909908\pi\)
							0.485935	+	0.873995i	\(0.338479\pi\)
\(43\)	33.0911	+	41.4949i	0.117357	+	0.147161i	0.837040	−	0.547142i	\(-0.184284\pi\)
							−0.719683	+	0.694303i	\(0.755713\pi\)
\(47\)	65.1749	−	285.550i	0.202271	−	0.886207i	−0.767279	−	0.641313i	\(-0.778390\pi\)
							0.969550	−	0.244893i	\(-0.0787529\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.e.b.15.4	✓	42
49.36	even	7	inner	98.4.e.b.85.4	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.e.b.15.4	✓	42	1.1	even	1	trivial
98.4.e.b.85.4	yes	42	49.36	even	7	inner