Embedded newform 930.2.bt.b.43.3

• Label: 930.2.bt.b.43.3
• Level: $930$
• Weight: $2$
• Character: 930.43
• 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			43.3
Character	\(\chi\)	\(=\)	930.43
Dual form			930.2.bt.b.757.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.987688 + 0.156434i) q^{2} +(-0.933580 + 0.358368i) q^{3} +(0.951057 - 0.309017i) q^{4} +(-0.873557 - 2.05837i) q^{5} +(0.866025 - 0.500000i) q^{6} +(0.239568 + 4.57122i) q^{7} +(-0.891007 + 0.453990i) q^{8} +(0.743145 - 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 16 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(311\)	\(871\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(e\left(\frac{19}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.987688	+	0.156434i	−0.698401	+	0.110616i
							\(3\)	−0.933580	+	0.358368i	−0.539003	+	0.206904i
\(5\)	−0.873557	−	2.05837i	−0.390667	−	0.920532i
							\(7\)	0.239568	+	4.57122i	0.0905480	+	1.72776i	0.546246	+	0.837625i	\(0.316056\pi\)
−0.455698	+	0.890134i	\(0.650610\pi\)
\(11\)	0.789597	−	3.71476i	0.238073	−	1.12004i	−0.682929	−	0.730485i	\(-0.739294\pi\)
							0.921002	−	0.389559i	\(-0.127372\pi\)
\(13\)	−0.488812	−	0.395832i	−0.135572	−	0.109784i	0.559136	−	0.829076i	\(-0.311133\pi\)
							−0.694708	+	0.719292i	\(0.744466\pi\)
\(17\)	−6.25507	−	4.06209i	−1.51708	−	0.985201i	−0.990867	−	0.134846i	\(-0.956946\pi\)
							−0.526210	−	0.850355i	\(-0.676387\pi\)
\(19\)	3.53798	+	0.371857i	0.811669	+	0.0853098i	0.501264	−	0.865295i	\(-0.332869\pi\)
							0.310405	+	0.950604i	\(0.399535\pi\)
\(23\)	3.22916	+	6.33758i	0.673326	+	1.32148i	0.934423	+	0.356165i	\(0.115916\pi\)
							−0.261097	+	0.965313i	\(0.584084\pi\)
\(29\)	4.55554	+	3.30979i	0.845942	+	0.614613i	0.924024	−	0.382334i	\(-0.124880\pi\)
							−0.0780821	+	0.996947i	\(0.524880\pi\)
\(31\)	−2.21195	−	5.10953i	−0.397278	−	0.917698i
							\(37\)	2.22388	−	8.29963i	0.365603	−	1.36445i	−0.500998	−	0.865449i	\(-0.667034\pi\)
0.866601	−	0.499001i	\(-0.166300\pi\)
\(41\)	−0.958157	+	0.426599i	−0.149639	+	0.0666236i	−0.480190	−	0.877165i	\(-0.659432\pi\)
							0.330551	+	0.943788i	\(0.392766\pi\)
\(43\)	0.00204784	+	0.00252888i	0.000312293	+	0.000385650i	0.777302	−	0.629128i	\(-0.216588\pi\)
							−0.776990	+	0.629513i	\(0.783254\pi\)
\(47\)	1.95492	−	12.3429i	0.285155	−	1.80040i	−0.263834	−	0.964568i	\(-0.584987\pi\)
							0.548989	−	0.835830i	\(-0.315013\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.b.43.3	yes	256
5.2	odd	4		930.2.bt.a.787.8	yes	256
31.13	odd	30		930.2.bt.a.13.8	✓	256
155.137	even	60	inner	930.2.bt.b.757.3	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.13.8	✓	256	31.13	odd	30
930.2.bt.a.787.8	yes	256	5.2	odd	4
930.2.bt.b.43.3	yes	256	1.1	even	1	trivial
930.2.bt.b.757.3	yes	256	155.137	even	60	inner