Embedded newform 930.2.bt.b.13.13

• Label: 930.2.bt.b.13.13
• 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			13.13
Character	\(\chi\)	\(=\)	930.13
Dual form			930.2.bt.b.787.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.156434 - 0.987688i) q^{2} +(0.358368 - 0.933580i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(-0.285666 + 2.21775i) q^{5} +(-0.866025 - 0.500000i) q^{6} +(2.60023 + 0.136272i) q^{7} +(-0.453990 + 0.891007i) q^{8} +(-0.743145 - 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 16 q^{7}+O(q^{10}) \)

Character values

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

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

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
\(5\)	−0.285666	+	2.21775i	−0.127754	+	0.991806i
							\(7\)	2.60023	+	0.136272i	0.982793	+	0.0515060i	0.536958	−	0.843609i	\(-0.319573\pi\)
0.445835	+	0.895115i	\(0.352907\pi\)
\(11\)	0.519015	+	2.44177i	0.156489	+	0.736222i	0.984482	+	0.175484i	\(0.0561491\pi\)
							−0.827993	+	0.560738i	\(0.810518\pi\)
\(13\)	−3.80097	−	4.69381i	−1.05420	−	1.30183i	−0.951678	−	0.307098i	\(-0.900642\pi\)
							−0.102522	−	0.994731i	\(-0.532691\pi\)
\(17\)	0.426091	+	0.656123i	0.103342	+	0.159133i	0.886559	−	0.462616i	\(-0.153089\pi\)
							−0.783217	+	0.621749i	\(0.786422\pi\)
\(19\)	5.92428	−	0.622667i	1.35912	−	0.142850i	0.603242	−	0.797558i	\(-0.293875\pi\)
							0.755881	+	0.654709i	\(0.227209\pi\)
\(23\)	8.27454	+	4.21609i	1.72536	+	0.879115i	0.976244	+	0.216676i	\(0.0695216\pi\)
							0.749116	+	0.662438i	\(0.230478\pi\)
\(29\)	7.57301	−	5.50211i	1.40627	−	1.02172i	0.412421	−	0.910993i	\(-0.364683\pi\)
							0.993851	−	0.110723i	\(-0.0353168\pi\)
\(31\)	5.50174	−	0.854896i	0.988142	−	0.153544i
							\(37\)	−2.85455	+	0.764874i	−0.469285	+	0.125744i	−0.485709	−	0.874121i	\(-0.661438\pi\)
0.0164241	+	0.999865i	\(0.494772\pi\)
\(41\)	7.86377	+	3.50118i	1.22811	+	0.546792i	0.915204	−	0.402991i	\(-0.132029\pi\)
							0.312910	+	0.949783i	\(0.398696\pi\)
\(43\)	3.79768	+	3.07530i	0.579142	+	0.468980i	0.873573	−	0.486693i	\(-0.161797\pi\)
							−0.294432	+	0.955673i	\(0.595130\pi\)
\(47\)	−2.61880	+	0.414777i	−0.381991	+	0.0605014i	−0.344477	−	0.938795i	\(-0.611944\pi\)
							−0.0375137	+	0.999296i	\(0.511944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.b.13.13	yes	256
5.2	odd	4		930.2.bt.a.757.15	yes	256
31.12	odd	30		930.2.bt.a.43.15	✓	256
155.12	even	60	inner	930.2.bt.b.787.13	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.43.15	✓	256	31.12	odd	30
930.2.bt.a.757.15	yes	256	5.2	odd	4
930.2.bt.b.13.13	yes	256	1.1	even	1	trivial
930.2.bt.b.787.13	yes	256	155.12	even	60	inner