Embedded newform 98.4.e.b.15.7

• Label: 98.4.e.b.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.b.85.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.445042 - 1.94986i) q^{2} +(6.02089 - 7.54996i) q^{3} +(-3.60388 - 1.73553i) q^{4} +(-10.1068 + 12.6736i) q^{5} +(-12.0418 - 15.0999i) q^{6} +(-9.34591 - 15.9892i) q^{7} +(-4.98792 + 6.25465i) q^{8} +(-14.7427 - 64.5919i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 14 q^{2} - q^{3} - 28 q^{4} + 14 q^{5} + 2 q^{6} + 7 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

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
							\(3\)	6.02089	−	7.54996i	1.15872	−	1.45299i	0.290456	−	0.956888i	\(-0.406193\pi\)
0.868264	−	0.496102i	\(-0.165236\pi\)
\(5\)	−10.1068	+	12.6736i	−0.903982	+	1.13356i	0.0865465	+	0.996248i	\(0.472417\pi\)
							−0.990528	+	0.137309i	\(0.956155\pi\)
\(7\)	−9.34591	−	15.9892i	−0.504632	−	0.863335i
							\(11\)	11.9400	−	52.3125i	0.327276	−	1.43389i	−0.497023	−	0.867737i	\(-0.665574\pi\)
0.824299	−	0.566154i	\(-0.191569\pi\)
\(13\)	−8.74751	+	38.3254i	−0.186625	+	0.817657i	0.791755	+	0.610839i	\(0.209168\pi\)
							−0.978380	+	0.206818i	\(0.933689\pi\)
\(17\)	41.9196	−	20.1874i	0.598059	−	0.288010i	−0.110260	−	0.993903i	\(-0.535168\pi\)
							0.708319	+	0.705893i	\(0.249454\pi\)
\(19\)	79.0040			0.953935			0.476967	−	0.878921i	\(-0.341736\pi\)
							0.476967	+	0.878921i	\(0.341736\pi\)
\(23\)	112.412	+	54.1349i	1.01911	+	0.490778i	0.867382	−	0.497644i	\(-0.165801\pi\)
							0.151730	+	0.988422i	\(0.451516\pi\)
\(29\)	83.9435	−	40.4251i	0.537515	−	0.258853i	−0.145373	−	0.989377i	\(-0.546438\pi\)
							0.682887	+	0.730524i	\(0.260724\pi\)
\(31\)	142.772			0.827179			0.413590	−	0.910463i	\(-0.364275\pi\)
							0.413590	+	0.910463i	\(0.364275\pi\)
\(37\)	−166.185	+	80.0304i	−0.738395	+	0.355592i	−0.764980	−	0.644054i	\(-0.777251\pi\)
							0.0265849	+	0.999647i	\(0.491537\pi\)
\(41\)	−25.1657	+	31.5568i	−0.0958592	+	0.120204i	−0.827448	−	0.561543i	\(-0.810208\pi\)
							0.731588	+	0.681747i	\(0.238779\pi\)
\(43\)	80.6363	+	101.115i	0.285975	+	0.358601i	0.903981	−	0.427572i	\(-0.140631\pi\)
							−0.618007	+	0.786173i	\(0.712060\pi\)
\(47\)	29.7187	−	130.206i	0.0922324	−	0.404096i	−0.907645	−	0.419738i	\(-0.862122\pi\)
							0.999878	+	0.0156416i	\(0.00497908\pi\)

(See \(a_n\) instead)

Twists

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

    

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