Embedded newform 98.4.e.b.85.6

• Label: 98.4.e.b.85.6
• Level: $98$
• Weight: $4$
• Character: 98.85
• 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			85.6
Character	\(\chi\)	\(=\)	98.85
Dual form			98.4.e.b.15.6

$q$-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 \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.e.b.15.6	✓	42	49.15	even	7	inner
98.4.e.b.85.6	yes	42	1.1	even	1	trivial