Embedded newform 930.2.bt.b.313.11

• Label: 930.2.bt.b.313.11
• Level: $930$
• Weight: $2$
• Character: 930.313
• 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			313.11
Character	\(\chi\)	\(=\)	930.313
Dual form			930.2.bt.b.517.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.156434 - 0.987688i) q^{2} +(0.629320 + 0.777146i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(-1.24001 - 1.86075i) q^{5} +(0.866025 - 0.500000i) q^{6} +(0.387221 - 0.596268i) q^{7} +(-0.453990 + 0.891007i) q^{8} +(-0.207912 + 0.978148i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 16 q^{7}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	0.629320	+	0.777146i	0.363338	+	0.448685i
\(5\)	−1.24001	−	1.86075i	−0.554548	−	0.832152i
							\(7\)	0.387221	−	0.596268i	0.146356	−	0.225368i	−0.757888	−	0.652385i	\(-0.773769\pi\)
0.904244	+	0.427016i	\(0.140435\pi\)
\(11\)	−4.62888	+	4.16786i	−1.39566	+	1.25666i	−0.467747	+	0.883862i	\(0.654934\pi\)
							−0.927913	+	0.372796i	\(0.878399\pi\)
\(13\)	−1.80724	+	4.70802i	−0.501238	+	1.30577i	0.416700	+	0.909044i	\(0.363187\pi\)
							−0.917938	+	0.396725i	\(0.870147\pi\)
\(17\)	0.128057	−	0.00671117i	0.0310583	−	0.00162770i	−0.0368005	−	0.999323i	\(-0.511717\pi\)
							0.0678588	+	0.997695i	\(0.478383\pi\)
\(19\)	−0.901033	+	2.02375i	−0.206711	+	0.464281i	−0.986914	−	0.161245i	\(-0.948449\pi\)
							0.780203	+	0.625526i	\(0.215116\pi\)
\(23\)	4.82979	+	2.46090i	1.00708	+	0.513133i	0.878081	−	0.478512i	\(-0.158824\pi\)
							0.128999	+	0.991645i	\(0.458824\pi\)
\(29\)	5.02991	−	3.65444i	0.934031	−	0.678613i	−0.0129456	−	0.999916i	\(-0.504121\pi\)
							0.946976	+	0.321303i	\(0.104121\pi\)
\(31\)	−3.66325	+	4.19292i	−0.657940	+	0.753071i
							\(37\)	−0.169457	+	0.632421i	−0.0278585	+	0.103969i	−0.978455	−	0.206459i	\(-0.933806\pi\)
0.950597	+	0.310429i	\(0.100473\pi\)
\(41\)	−0.833602	−	7.93120i	−0.130187	−	1.23864i	−0.843241	−	0.537536i	\(-0.819355\pi\)
							0.713054	−	0.701109i	\(-0.247311\pi\)
\(43\)	12.0181	−	4.61332i	1.83275	−	0.703525i	0.847884	−	0.530181i	\(-0.177876\pi\)
							0.984861	−	0.173344i	\(-0.0554572\pi\)
\(47\)	−11.6014	+	1.83749i	−1.69224	+	0.268025i	−0.926821	−	0.375504i	\(-0.877469\pi\)
							−0.765422	+	0.643529i	\(0.777469\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.b.313.11	yes	256
5.2	odd	4		930.2.bt.a.127.16	✓	256
31.21	odd	30		930.2.bt.a.703.16	yes	256
155.52	even	60	inner	930.2.bt.b.517.11	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.127.16	✓	256	5.2	odd	4
930.2.bt.a.703.16	yes	256	31.21	odd	30
930.2.bt.b.313.11	yes	256	1.1	even	1	trivial
930.2.bt.b.517.11	yes	256	155.52	even	60	inner