Embedded newform 930.2.bt.b.13.11

• Label: 930.2.bt.b.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.b.787.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.156434 - 0.987688i) q^{2} +(0.358368 - 0.933580i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(-0.657500 - 2.13722i) q^{5} +(-0.866025 - 0.500000i) q^{6} +(-3.81046 - 0.199698i) q^{7} +(-0.453990 + 0.891007i) q^{8} +(-0.743145 - 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 16 q^{7}+O(q^{10}) \)

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
\(5\)	−0.657500	−	2.13722i	−0.294043	−	0.955792i
							\(7\)	−3.81046	−	0.199698i	−1.44022	−	0.0754787i	−0.683850	−	0.729623i	\(-0.739696\pi\)
−0.756370	+	0.654144i	\(0.773029\pi\)
\(11\)	−0.436103	−	2.05170i	−0.131490	−	0.618611i	−0.993701	−	0.112062i	\(-0.964254\pi\)
							0.862211	−	0.506549i	\(-0.169079\pi\)
\(13\)	−2.66724	−	3.29377i	−0.739759	−	0.913526i	0.258868	−	0.965913i	\(-0.416651\pi\)
							−0.998627	+	0.0523864i	\(0.983317\pi\)
\(17\)	4.22574	+	6.50708i	1.02489	+	1.57820i	0.796509	+	0.604627i	\(0.206678\pi\)
							0.228385	+	0.973571i	\(0.426656\pi\)
\(19\)	4.36812	−	0.459107i	1.00211	−	0.105326i	0.410764	−	0.911742i	\(-0.365262\pi\)
							0.591350	+	0.806415i	\(0.298595\pi\)
\(23\)	0.582963	+	0.297035i	0.121556	+	0.0619360i	0.513711	−	0.857963i	\(-0.328270\pi\)
							−0.392155	+	0.919899i	\(0.628270\pi\)
\(29\)	−6.08559	+	4.42144i	−1.13007	+	0.821041i	−0.985704	−	0.168484i	\(-0.946113\pi\)
							−0.144362	+	0.989525i	\(0.546113\pi\)
\(31\)	4.45579	+	3.33855i	0.800284	+	0.599621i
							\(37\)	−0.852002	+	0.228293i	−0.140068	+	0.0375312i	−0.328172	−	0.944618i	\(-0.606433\pi\)
0.188104	+	0.982149i	\(0.439766\pi\)
\(41\)	−7.64904	−	3.40557i	−1.19458	−	0.531861i	−0.289531	−	0.957169i	\(-0.593499\pi\)
							−0.905049	+	0.425308i	\(0.860166\pi\)
\(43\)	−1.09318	−	0.885236i	−0.166708	−	0.134997i	0.542336	−	0.840162i	\(-0.317540\pi\)
							−0.709044	+	0.705164i	\(0.750873\pi\)
\(47\)	−11.7873	+	1.86692i	−1.71935	+	0.272319i	−0.936698	−	0.350139i	\(-0.886134\pi\)
							−0.782654	+	0.622457i	\(0.786134\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.b.13.11	yes	256
5.2	odd	4		930.2.bt.a.757.12	yes	256
31.12	odd	30		930.2.bt.a.43.12	✓	256
155.12	even	60	inner	930.2.bt.b.787.11	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.43.12	✓	256	31.12	odd	30
930.2.bt.a.757.12	yes	256	5.2	odd	4
930.2.bt.b.13.11	yes	256	1.1	even	1	trivial
930.2.bt.b.787.11	yes	256	155.12	even	60	inner