Embedded newform 98.4.e.b.15.6

• Label: 98.4.e.b.15.6
• 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			15.6
Character	\(\chi\)	\(=\)	98.15
Dual form			98.4.e.b.85.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.445042 - 1.94986i) q^{2} +(4.72767 - 5.92832i) q^{3} +(-3.60388 - 1.73553i) q^{4} +(8.54463 - 10.7146i) q^{5} +(-9.45535 - 11.8566i) q^{6} +(13.8633 + 12.2804i) q^{7} +(-4.98792 + 6.25465i) q^{8} +(-6.78596 - 29.7312i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 14 q^{2} - q^{3} - 28 q^{4} + 14 q^{5} + 2 q^{6} + 7 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

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
							\(3\)	4.72767	−	5.92832i	0.909841	−	1.14091i	−0.0797235	−	0.996817i	\(-0.525404\pi\)
0.989565	−	0.144088i	\(-0.0460248\pi\)
\(5\)	8.54463	−	10.7146i	0.764255	−	0.958345i	−0.235654	−	0.971837i	\(-0.575723\pi\)
							0.999909	+	0.0134918i	\(0.00429469\pi\)
\(7\)	13.8633	+	12.2804i	0.748548	+	0.663080i
							\(11\)	−0.842354	+	3.69060i	−0.0230890	+	0.101160i	−0.985159	−	0.171641i	\(-0.945093\pi\)
0.962070	+	0.272801i	\(0.0879501\pi\)
\(13\)	−10.5980	+	46.4330i	−0.226105	+	0.990630i	0.726678	+	0.686978i	\(0.241063\pi\)
							−0.952783	+	0.303652i	\(0.901794\pi\)
\(17\)	−100.202	+	48.2545i	−1.42956	+	0.688438i	−0.978915	−	0.204268i	\(-0.934519\pi\)
							−0.450641	+	0.892705i	\(0.648804\pi\)
\(19\)	119.356			1.44116			0.720580	−	0.693372i	\(-0.243876\pi\)
							0.720580	+	0.693372i	\(0.243876\pi\)
\(23\)	−141.621	−	68.2013i	−1.28392	−	0.618302i	−0.337524	−	0.941317i	\(-0.609589\pi\)
							−0.946394	+	0.323015i	\(0.895304\pi\)
\(29\)	−19.9929	+	9.62806i	−0.128020	+	0.0616513i	−0.496797	−	0.867867i	\(-0.665491\pi\)
							0.368777	+	0.929518i	\(0.379777\pi\)
\(31\)	−93.6758			−0.542731			−0.271366	−	0.962476i	\(-0.587475\pi\)
							−0.271366	+	0.962476i	\(0.587475\pi\)
\(37\)	302.287	−	145.574i	1.34313	−	0.646815i	0.382318	−	0.924031i	\(-0.375126\pi\)
							0.960808	+	0.277215i	\(0.0894115\pi\)
\(41\)	158.384	−	198.607i	0.603303	−	0.756518i	−0.382586	−	0.923920i	\(-0.624966\pi\)
							0.985889	+	0.167402i	\(0.0535377\pi\)
\(43\)	109.997	+	137.932i	0.390102	+	0.489172i	0.937640	−	0.347609i	\(-0.113006\pi\)
							−0.547538	+	0.836781i	\(0.684435\pi\)
\(47\)	−127.008	+	556.458i	−0.394170	+	1.72697i	0.255546	+	0.966797i	\(0.417745\pi\)
							−0.649717	+	0.760176i	\(0.725112\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.e.b.15.6	✓	42
49.36	even	7	inner	98.4.e.b.85.6	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.e.b.15.6	✓	42	1.1	even	1	trivial
98.4.e.b.85.6	yes	42	49.36	even	7	inner