Embedded newform 930.2.bt.a.13.16

• Label: 930.2.bt.a.13.16
• Level: $930$
• Weight: $2$
• Character: 930.13
• Analytic conductor: $7.426$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.bt (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.42608738798\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			13.16
Character	\(\chi\)	\(=\)	930.13
Dual form			930.2.bt.a.787.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.156434 - 0.987688i) q^{2} +(-0.358368 + 0.933580i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(1.76830 + 1.36862i) q^{5} +(0.866025 + 0.500000i) q^{6} +(-0.704223 - 0.0369068i) q^{7} +(-0.453990 + 0.891007i) q^{8} +(-0.743145 - 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(311\)	\(871\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(e\left(\frac{11}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.156434	−	0.987688i	0.110616	−	0.698401i
							\(3\)	−0.358368	+	0.933580i	−0.206904	+	0.539003i
\(5\)	1.76830	+	1.36862i	0.790806	+	0.612067i
							\(7\)	−0.704223	−	0.0369068i	−0.266171	−	0.0139494i	−0.0812165	−	0.996696i	\(-0.525881\pi\)
−0.184955	+	0.982747i	\(0.559214\pi\)
\(11\)	0.725740	+	3.41434i	0.218819	+	1.02946i	0.941169	+	0.337935i	\(0.109728\pi\)
							−0.722351	+	0.691527i	\(0.756938\pi\)
\(13\)	−1.21519	−	1.50063i	−0.337032	−	0.416200i	0.580250	−	0.814439i	\(-0.302955\pi\)
							−0.917282	+	0.398238i	\(0.869622\pi\)
\(17\)	0.571059	+	0.879354i	0.138502	+	0.213275i	0.901115	−	0.433581i	\(-0.142750\pi\)
							−0.762612	+	0.646856i	\(0.776084\pi\)
\(19\)	1.81729	−	0.191004i	0.416914	−	0.0438194i	0.106250	−	0.994339i	\(-0.466115\pi\)
							0.310663	+	0.950520i	\(0.399449\pi\)
\(23\)	3.89468	+	1.98444i	0.812096	+	0.413784i	0.810156	−	0.586214i	\(-0.199382\pi\)
							0.00194039	+	0.999998i	\(0.499382\pi\)
\(29\)	−7.51245	+	5.45811i	−1.39503	+	1.01355i	−0.399734	+	0.916631i	\(0.630897\pi\)
							−0.995293	+	0.0969147i	\(0.969103\pi\)
\(31\)	−2.95938	+	4.71615i	−0.531521	+	0.847045i
							\(37\)	−2.60942	+	0.699192i	−0.428986	+	0.114946i	−0.466850	−	0.884337i	\(-0.654611\pi\)
0.0378643	+	0.999283i	\(0.487945\pi\)
\(41\)	2.49276	+	1.10985i	0.389304	+	0.173329i	0.592048	−	0.805902i	\(-0.298319\pi\)
							−0.202745	+	0.979232i	\(0.564986\pi\)
\(43\)	5.78611	+	4.68550i	0.882374	+	0.714532i	0.959265	−	0.282507i	\(-0.0911661\pi\)
							−0.0768913	+	0.997039i	\(0.524499\pi\)
\(47\)	2.25685	−	0.357450i	0.329195	−	0.0521394i	0.0103508	−	0.999946i	\(-0.496705\pi\)
							0.318845	+	0.947807i	\(0.396705\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.a.13.16	✓	256
5.2	odd	4		930.2.bt.b.757.12	yes	256
31.12	odd	30		930.2.bt.b.43.12	yes	256
155.12	even	60	inner	930.2.bt.a.787.16	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.13.16	✓	256	1.1	even	1	trivial
930.2.bt.a.787.16	yes	256	155.12	even	60	inner
930.2.bt.b.43.12	yes	256	31.12	odd	30
930.2.bt.b.757.12	yes	256	5.2	odd	4