Embedded newform 93.2.p.b.86.2

• Label: 93.2.p.b.86.2
• Level: $93$
• Weight: $2$
• Character: 93.86
• Analytic conductor: $0.743$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	93.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.742608738798\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(8\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			86.2
Character	\(\chi\)	\(=\)	93.86
Dual form			93.2.p.b.53.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77867 + 0.577925i) q^{2} +(0.476813 - 1.66513i) q^{3} +(1.21163 - 0.880304i) q^{4} +(-1.69224 - 0.977017i) q^{5} +(0.114225 + 3.23727i) q^{6} +(-2.38117 - 1.06017i) q^{7} +(0.552209 - 0.760050i) q^{8} +(-2.54530 - 1.58791i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77867	+	0.577925i	−1.25771	+	0.408655i	−0.860678	−	0.509150i	\(-0.829960\pi\)
							−0.397032	+	0.917805i	\(0.629960\pi\)
\(3\)	0.476813	−	1.66513i	0.275288	−	0.961362i
							\(5\)	−1.69224	−	0.977017i	−0.756794	−	0.436935i	0.0713496	−	0.997451i	\(-0.477269\pi\)
−0.828143	+	0.560516i	\(0.810603\pi\)
\(7\)	−2.38117	−	1.06017i	−0.899998	−	0.400705i	−0.0960307	−	0.995378i	\(-0.530615\pi\)
							−0.803967	+	0.594673i	\(0.797281\pi\)
\(11\)	−0.175252	−	1.66741i	−0.0528403	−	0.502742i	−0.988650	−	0.150237i	\(-0.951996\pi\)
							0.935810	−	0.352506i	\(-0.114670\pi\)
\(13\)	3.38587	−	3.04865i	0.939070	−	0.845543i	−0.0491170	−	0.998793i	\(-0.515641\pi\)
							0.988187	+	0.153250i	\(0.0489740\pi\)
\(17\)	0.161055	−	1.53234i	0.0390617	−	0.371647i	−0.957477	−	0.288511i	\(-0.906840\pi\)
							0.996538	−	0.0831360i	\(-0.0264936\pi\)
\(19\)	−2.97075	+	3.29935i	−0.681537	+	0.756924i	−0.980324	−	0.197396i	\(-0.936751\pi\)
							0.298787	+	0.954320i	\(0.403418\pi\)
\(23\)	5.64314	+	4.09998i	1.17668	+	0.854906i	0.991793	−	0.127855i	\(-0.0408091\pi\)
							0.184884	+	0.982760i	\(0.440809\pi\)
\(29\)	1.31099	+	4.03481i	0.243445	+	0.749246i	0.995888	+	0.0905893i	\(0.0288751\pi\)
							−0.752444	+	0.658657i	\(0.771125\pi\)
\(31\)	4.51205	−	3.26212i	0.810387	−	0.585895i
							\(37\)	8.07914	−	4.66449i	1.32820	−	0.766838i	0.343181	−	0.939269i	\(-0.388496\pi\)
0.985022	+	0.172431i	\(0.0551623\pi\)
\(41\)	0.405705	−	1.90869i	0.0633605	−	0.298088i	−0.935045	−	0.354528i	\(-0.884642\pi\)
							0.998406	+	0.0564400i	\(0.0179750\pi\)
\(43\)	−2.27585	−	2.04919i	−0.347064	−	0.312498i	0.477100	−	0.878849i	\(-0.341688\pi\)
							−0.824164	+	0.566351i	\(0.808355\pi\)
\(47\)	−10.2349	−	3.32552i	−1.49292	−	0.485078i	−0.554973	−	0.831869i	\(-0.687271\pi\)
							−0.937942	+	0.346791i	\(0.887271\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.b.86.2	yes	64
3.2	odd	2	inner	93.2.p.b.86.7	yes	64
31.22	odd	30	inner	93.2.p.b.53.7	yes	64
93.53	even	30	inner	93.2.p.b.53.2	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.53.2	✓	64	93.53	even	30	inner
93.2.p.b.53.7	yes	64	31.22	odd	30	inner
93.2.p.b.86.2	yes	64	1.1	even	1	trivial
93.2.p.b.86.7	yes	64	3.2	odd	2	inner