Embedded newform 930.2.bt.a.13.11

• Label: 930.2.bt.a.13.11
• 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			13.11
Character	\(\chi\)	\(=\)	930.13
Dual form			930.2.bt.a.787.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.156434 - 0.987688i) q^{2} +(-0.358368 + 0.933580i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(-1.16716 - 1.90728i) q^{5} +(0.866025 + 0.500000i) q^{6} +(2.78313 + 0.145858i) q^{7} +(-0.453990 + 0.891007i) q^{8} +(-0.743145 - 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 8 q^{7}+O(q^{10}) \)

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
\(5\)	−1.16716	−	1.90728i	−0.521971	−	0.852963i
							\(7\)	2.78313	+	0.145858i	1.05193	+	0.0551291i	0.570452	−	0.821331i	\(-0.306768\pi\)
0.481474	+	0.876460i	\(0.340101\pi\)
\(11\)	1.13244	+	5.32773i	0.341445	+	1.60637i	0.728993	+	0.684521i	\(0.239989\pi\)
							−0.387548	+	0.921849i	\(0.626678\pi\)
\(13\)	1.67102	+	2.06354i	0.463457	+	0.572322i	0.954271	−	0.298942i	\(-0.0966339\pi\)
							−0.490814	+	0.871264i	\(0.663301\pi\)
\(17\)	−1.77409	−	2.73186i	−0.430280	−	0.662573i	0.555326	−	0.831633i	\(-0.312594\pi\)
							−0.985606	+	0.169060i	\(0.945927\pi\)
\(19\)	−1.17727	+	0.123736i	−0.270084	+	0.0283869i	−0.238602	−	0.971117i	\(-0.576689\pi\)
							−0.0314815	+	0.999504i	\(0.510023\pi\)
\(23\)	2.54109	+	1.29475i	0.529854	+	0.269974i	0.698385	−	0.715722i	\(-0.253902\pi\)
							−0.168531	+	0.985696i	\(0.553902\pi\)
\(29\)	5.97170	−	4.33869i	1.10892	−	0.805675i	0.126424	−	0.991976i	\(-0.459650\pi\)
							0.982493	+	0.186302i	\(0.0596501\pi\)
\(31\)	5.56006	−	0.292731i	0.998617	−	0.0525761i
							\(37\)	7.79414	−	2.08843i	1.28135	−	0.343337i	0.446981	−	0.894544i	\(-0.352499\pi\)
0.834369	+	0.551207i	\(0.185833\pi\)
\(41\)	−2.68926	−	1.19734i	−0.419993	−	0.186993i	0.185849	−	0.982578i	\(-0.440497\pi\)
							−0.605841	+	0.795586i	\(0.707163\pi\)
\(43\)	8.71095	+	7.05398i	1.32841	+	1.07572i	0.990929	+	0.134389i	\(0.0429072\pi\)
							0.337478	+	0.941333i	\(0.390426\pi\)
\(47\)	−4.05994	+	0.643031i	−0.592202	+	0.0937957i	−0.445340	−	0.895361i	\(-0.646917\pi\)
							−0.146862	+	0.989157i	\(0.546917\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.a.13.11	✓	256
5.2	odd	4		930.2.bt.b.757.11	yes	256
31.12	odd	30		930.2.bt.b.43.11	yes	256
155.12	even	60	inner	930.2.bt.a.787.11	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.13.11	✓	256	1.1	even	1	trivial
930.2.bt.a.787.11	yes	256	155.12	even	60	inner
930.2.bt.b.43.11	yes	256	31.12	odd	30
930.2.bt.b.757.11	yes	256	5.2	odd	4