Embedded newform 95.11.n.a.86.9

• Label: 95.11.n.a.86.9
• Level: $95$
• Weight: $11$
• Character: 95.86
• Analytic conductor: $60.359$
• Analytic rank: $0$
• Dimension: $396$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	95.n (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(60.3589390040\)
Analytic rank:	\(0\)
Dimension:	\(396\)
Relative dimension:	\(66\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			86.9
Character	\(\chi\)	\(=\)	95.86
Dual form			95.11.n.a.21.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-49.0540 + 8.64954i) q^{2} +(84.6169 + 100.843i) q^{3} +(1369.23 - 498.360i) q^{4} +(-1313.26 - 477.988i) q^{5} +(-5023.04 - 4214.83i) q^{6} +(-12628.7 + 21873.6i) q^{7} +(-18683.1 + 10786.7i) q^{8} +(7244.56 - 41085.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 396 q + 132 q^{3} - 1830 q^{4} + 3066 q^{6} - 95436 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(77\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−49.0540	+	8.64954i	−1.53294	+	0.270298i	−0.875502	−	0.483214i	\(-0.839469\pi\)
							−0.657434	+	0.753512i	\(0.728358\pi\)
\(3\)	84.6169	+	100.843i	0.348218	+	0.414990i	0.911516	−	0.411264i	\(-0.134912\pi\)
							−0.563299	+	0.826253i	\(0.690468\pi\)
\(5\)	−1313.26	−	477.988i	−0.420243	−	0.152956i
							\(7\)	−12628.7	+	21873.6i	−0.751397	+	1.30146i	0.195749	+	0.980654i	\(0.437286\pi\)
−0.947146	+	0.320804i	\(0.896047\pi\)
\(11\)	−86332.3	−	149532.i	−0.536056	−	0.928476i	−0.999111	−	0.0421466i	\(-0.986580\pi\)
							0.463056	−	0.886329i	\(-0.346753\pi\)
\(13\)	−33306.7	+	39693.4i	−0.0897046	+	0.106906i	−0.809030	−	0.587768i	\(-0.800007\pi\)
							0.719325	+	0.694674i	\(0.244451\pi\)
\(17\)	−32858.2	−	186348.i	−0.0231419	−	0.131244i	0.971048	−	0.238884i	\(-0.0767817\pi\)
							−0.994190	+	0.107640i	\(0.965671\pi\)
\(19\)	−1.75422e6	+	1.74751e6i	−0.708461	+	0.705750i
							\(23\)	4.21627e6	−	1.53460e6i	0.655072	−	0.238427i	0.00696463	−	0.999976i	\(-0.497783\pi\)
0.648107	+	0.761549i	\(0.275561\pi\)
\(29\)	−2.64089e7	−	4.65660e6i	−1.28754	−	0.227028i	−0.512360	−	0.858770i	\(-0.671229\pi\)
							−0.775178	+	0.631743i	\(0.782340\pi\)
\(31\)	1.09563e7	+	6.32562e6i	0.382697	+	0.220950i	0.678991	−	0.734146i	\(-0.262417\pi\)
							−0.296294	+	0.955097i	\(0.595751\pi\)
\(37\)		−	6.98449e7i		−	1.00722i	−0.863930	−	0.503612i	\(-0.832004\pi\)
							0.863930	−	0.503612i	\(-0.167996\pi\)
\(41\)	−6.39878e7	−	7.62577e7i	−0.552304	−	0.658210i	0.415595	−	0.909550i	\(-0.363573\pi\)
							−0.967899	+	0.251340i	\(0.919129\pi\)
\(43\)	7.00414e6	+	2.54930e6i	0.0476445	+	0.0173412i	0.365732	−	0.930720i	\(-0.380819\pi\)
							−0.318088	+	0.948061i	\(0.603041\pi\)
\(47\)	4.61942e7	−	2.61981e8i	0.201418	−	1.14230i	−0.701559	−	0.712611i	\(-0.747512\pi\)
							0.902977	−	0.429688i	\(-0.141376\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.11.n.a.86.9	yes	396
19.2	odd	18	inner	95.11.n.a.21.9	✓	396

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.11.n.a.21.9	✓	396	19.2	odd	18	inner
95.11.n.a.86.9	yes	396	1.1	even	1	trivial