Embedded newform 93.2.k.a.89.8

• Label: 93.2.k.a.89.8
• Level: $93$
• Weight: $2$
• Character: 93.89
• Analytic conductor: $0.743$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	93.k (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.742608738798\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			89.8
Character	\(\chi\)	\(=\)	93.89
Dual form			93.2.k.a.23.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.99894 - 0.649494i) q^{2} +(-1.51648 - 0.836828i) q^{3} +(1.95587 - 1.42103i) q^{4} +0.445040i q^{5} +(-3.57487 - 0.687819i) q^{6} +(1.36818 - 0.994042i) q^{7} +(0.515899 - 0.710074i) q^{8} +(1.59944 + 2.53807i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 5 q^{3} - 6 q^{4} + 8 q^{7} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.99894	−	0.649494i	1.41346	−	0.459262i	0.499943	−	0.866058i	\(-0.333354\pi\)
							0.913519	+	0.406796i	\(0.133354\pi\)
\(3\)	−1.51648	−	0.836828i	−0.875542	−	0.483143i
							\(5\)			0.445040i			0.199028i	0.995036	+	0.0995140i	\(0.0317288\pi\)
−0.995036	+	0.0995140i	\(0.968271\pi\)
\(7\)	1.36818	−	0.994042i	0.517124	−	0.375712i	−0.298396	−	0.954442i	\(-0.596451\pi\)
							0.815519	+	0.578730i	\(0.196451\pi\)
\(11\)	−4.29256	+	3.11873i	−1.29426	+	0.940332i	−0.999882	−	0.0153587i	\(-0.995111\pi\)
							−0.294374	+	0.955690i	\(0.595111\pi\)
\(13\)	−0.520922	−	0.169258i	−0.144478	−	0.0469437i	0.235885	−	0.971781i	\(-0.424201\pi\)
							−0.380363	+	0.924837i	\(0.624201\pi\)
\(17\)	1.54386	+	1.12168i	0.374440	+	0.272047i	0.759050	−	0.651033i	\(-0.225664\pi\)
							−0.384609	+	0.923079i	\(0.625664\pi\)
\(19\)	−1.89011	−	5.81717i	−0.433621	−	1.33455i	−0.894493	−	0.447082i	\(-0.852463\pi\)
							0.460871	−	0.887467i	\(-0.347537\pi\)
\(23\)	1.07911	+	0.784020i	0.225010	+	0.163479i	0.694579	−	0.719416i	\(-0.255591\pi\)
							−0.469569	+	0.882896i	\(0.655591\pi\)
\(29\)	−2.68876	−	8.27515i	−0.499290	−	1.53666i	−0.810163	−	0.586205i	\(-0.800621\pi\)
							0.310873	−	0.950451i	\(-0.399379\pi\)
\(31\)	1.92591	−	5.22407i	0.345904	−	0.938270i
							\(37\)			2.92752i			0.481282i	0.970614	+	0.240641i	\(0.0773576\pi\)
−0.970614	+	0.240641i	\(0.922642\pi\)
\(41\)	−4.60047	+	1.49478i	−0.718473	+	0.233446i	−0.645361	−	0.763878i	\(-0.723293\pi\)
							−0.0731122	+	0.997324i	\(0.523293\pi\)
\(43\)	−3.10699	+	1.00952i	−0.473812	+	0.153951i	−0.536182	−	0.844102i	\(-0.680134\pi\)
							0.0623705	+	0.998053i	\(0.480134\pi\)
\(47\)	−9.40980	−	3.05743i	−1.37256	−	0.445972i	−0.472343	−	0.881415i	\(-0.656592\pi\)
							−0.900216	+	0.435443i	\(0.856592\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.k.a.89.8	yes	32
3.2	odd	2	inner	93.2.k.a.89.1	yes	32
31.23	odd	10	inner	93.2.k.a.23.1	✓	32
93.23	even	10	inner	93.2.k.a.23.8	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.k.a.23.1	✓	32	31.23	odd	10	inner
93.2.k.a.23.8	yes	32	93.23	even	10	inner
93.2.k.a.89.1	yes	32	3.2	odd	2	inner
93.2.k.a.89.8	yes	32	1.1	even	1	trivial