Embedded newform 930.2.bt.b.817.12

• Label: 930.2.bt.b.817.12
• 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			817.12
Character	\(\chi\)	\(=\)	930.817
Dual form			930.2.bt.b.823.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.453990 - 0.891007i) q^{2} +(-0.0523360 + 0.998630i) q^{3} +(-0.587785 - 0.809017i) q^{4} +(-0.971799 - 2.01385i) q^{5} +(0.866025 + 0.500000i) q^{6} +(2.64533 - 3.26672i) q^{7} +(-0.987688 + 0.156434i) q^{8} +(-0.994522 - 0.104528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 16 q^{7}+O(q^{10}) \)

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
\(5\)	−0.971799	−	2.01385i	−0.434602	−	0.900623i
							\(7\)	2.64533	−	3.26672i	0.999842	−	1.23470i	0.0278689	−	0.999612i	\(-0.491128\pi\)
0.971973	−	0.235091i	\(-0.0755388\pi\)
\(11\)	0.903840	−	2.03006i	0.272518	−	0.612085i	−0.724498	−	0.689276i	\(-0.757929\pi\)
							0.997016	+	0.0771913i	\(0.0245952\pi\)
\(13\)	−5.44433	−	3.53559i	−1.50999	−	0.980596i	−0.992142	−	0.125121i	\(-0.960068\pi\)
							−0.517844	−	0.855475i	\(-0.673265\pi\)
\(17\)	−1.47747	+	3.84895i	−0.358340	+	0.933508i	0.629355	+	0.777118i	\(0.283319\pi\)
							−0.987695	+	0.156390i	\(0.950014\pi\)
\(19\)	−0.913728	+	4.29875i	−0.209624	+	0.986202i	0.739951	+	0.672661i	\(0.234849\pi\)
							−0.949575	+	0.313541i	\(0.898485\pi\)
\(23\)	1.25823	+	7.94414i	0.262359	+	1.65647i	0.669285	+	0.743006i	\(0.266601\pi\)
							−0.406926	+	0.913461i	\(0.633399\pi\)
\(29\)	−2.43671	−	7.49943i	−0.452486	−	1.39261i	−0.874061	−	0.485816i	\(-0.838522\pi\)
							0.421575	−	0.906794i	\(-0.361478\pi\)
\(31\)	1.12282	−	5.45337i	0.201665	−	0.979455i
							\(37\)	−2.59834	−	9.69715i	−0.427165	−	1.59420i	−0.759149	−	0.650917i	\(-0.774385\pi\)
0.331984	−	0.943285i	\(-0.392282\pi\)
\(41\)	−2.42663	+	2.69505i	−0.378976	+	0.420895i	−0.902212	−	0.431292i	\(-0.858058\pi\)
							0.523236	+	0.852188i	\(0.324724\pi\)
\(43\)	−3.53816	−	5.44829i	−0.539565	−	0.830857i	0.458769	−	0.888555i	\(-0.348290\pi\)
							−0.998334	+	0.0576984i	\(0.981624\pi\)
\(47\)	9.67717	−	4.93076i	1.41156	−	0.719226i	0.428677	−	0.903458i	\(-0.358980\pi\)
							0.982883	+	0.184232i	\(0.0589797\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.b.817.12	yes	256
5.3	odd	4		930.2.bt.a.73.3	✓	256
31.17	odd	30		930.2.bt.a.637.3	yes	256
155.48	even	60	inner	930.2.bt.b.823.12	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.73.3	✓	256	5.3	odd	4
930.2.bt.a.637.3	yes	256	31.17	odd	30
930.2.bt.b.817.12	yes	256	1.1	even	1	trivial
930.2.bt.b.823.12	yes	256	155.48	even	60	inner