Embedded newform 930.2.bt.b.817.9

• Label: 930.2.bt.b.817.9
• Level: $930$
• Weight: $2$
• Character: 930.817
• 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			817.9
Character	\(\chi\)	\(=\)	930.817
Dual form			930.2.bt.b.823.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.453990 - 0.891007i) q^{2} +(-0.0523360 + 0.998630i) q^{3} +(-0.587785 - 0.809017i) q^{4} +(-2.21080 + 0.335214i) q^{5} +(0.866025 + 0.500000i) q^{6} +(2.00224 - 2.47256i) q^{7} +(-0.987688 + 0.156434i) q^{8} +(-0.994522 - 0.104528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 16 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(311\)	\(871\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(e\left(\frac{23}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.453990	−	0.891007i	0.321020	−	0.630037i
							\(3\)	−0.0523360	+	0.998630i	−0.0302162	+	0.576559i
\(5\)	−2.21080	+	0.335214i	−0.988699	+	0.149912i
							\(7\)	2.00224	−	2.47256i	0.756777	−	0.934541i	−0.242590	−	0.970129i	\(-0.577997\pi\)
0.999367	+	0.0355876i	\(0.0113303\pi\)
\(11\)	−1.04457	+	2.34613i	−0.314948	+	0.707386i	−0.999773	−	0.0213226i	\(-0.993212\pi\)
							0.684824	+	0.728708i	\(0.259879\pi\)
\(13\)	3.49111	+	2.26716i	0.968261	+	0.628796i	0.928861	−	0.370428i	\(-0.120789\pi\)
							0.0393995	+	0.999224i	\(0.487455\pi\)
\(17\)	1.39805	−	3.64205i	0.339077	−	0.883326i	−0.652962	−	0.757391i	\(-0.726474\pi\)
							0.992039	−	0.125935i	\(-0.0401930\pi\)
\(19\)	1.08327	−	5.09639i	0.248519	−	1.16919i	−0.659973	−	0.751290i	\(-0.729432\pi\)
							0.908492	−	0.417902i	\(-0.137234\pi\)
\(23\)	−0.472733	−	2.98472i	−0.0985716	−	0.622356i	−0.986674	−	0.162710i	\(-0.947976\pi\)
							0.888102	−	0.459646i	\(-0.152024\pi\)
\(29\)	−2.83067	−	8.71190i	−0.525642	−	1.61776i	−0.763044	−	0.646347i	\(-0.776296\pi\)
							0.237402	−	0.971411i	\(-0.423704\pi\)
\(31\)	0.900608	−	5.49444i	0.161754	−	0.986831i
							\(37\)	0.684257	+	2.55368i	0.112491	+	0.419823i	0.999087	−	0.0427221i	\(-0.0136030\pi\)
−0.886596	+	0.462545i	\(0.846936\pi\)
\(41\)	1.67458	−	1.85981i	0.261526	−	0.290454i	−0.598054	−	0.801456i	\(-0.704059\pi\)
							0.859579	+	0.511002i	\(0.170726\pi\)
\(43\)	3.51168	+	5.40751i	0.535526	+	0.824637i	0.998079	−	0.0619543i	\(-0.0197333\pi\)
							−0.462553	+	0.886591i	\(0.653067\pi\)
\(47\)	−6.46940	+	3.29633i	−0.943660	+	0.480819i	−0.856941	−	0.515414i	\(-0.827638\pi\)
							−0.0867185	+	0.996233i	\(0.527638\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.b.817.9	yes	256
5.3	odd	4		930.2.bt.a.73.6	✓	256
31.17	odd	30		930.2.bt.a.637.6	yes	256
155.48	even	60	inner	930.2.bt.b.823.9	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.73.6	✓	256	5.3	odd	4
930.2.bt.a.637.6	yes	256	31.17	odd	30
930.2.bt.b.817.9	yes	256	1.1	even	1	trivial
930.2.bt.b.823.9	yes	256	155.48	even	60	inner