Embedded newform 930.2.bt.a.313.6

• Label: 930.2.bt.a.313.6
• 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			313.6
Character	\(\chi\)	\(=\)	930.313
Dual form			930.2.bt.a.517.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.156434 + 0.987688i) q^{2} +(0.629320 + 0.777146i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(1.28645 - 1.82895i) q^{5} +(-0.866025 + 0.500000i) q^{6} +(-2.32513 + 3.58039i) q^{7} +(0.453990 - 0.891007i) q^{8} +(-0.207912 + 0.978148i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 8 q^{7}+O(q^{10}) \)

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
\(5\)	1.28645	−	1.82895i	0.575319	−	0.817929i
							\(7\)	−2.32513	+	3.58039i	−0.878817	+	1.35326i	0.0561123	+	0.998424i	\(0.482130\pi\)
−0.934929	+	0.354835i	\(0.884537\pi\)
\(11\)	1.57517	−	1.41829i	0.474930	−	0.427629i	−0.396585	−	0.917998i	\(-0.629805\pi\)
							0.871515	+	0.490369i	\(0.163138\pi\)
\(13\)	−2.19626	+	5.72145i	−0.609133	+	1.58685i	0.187423	+	0.982279i	\(0.439987\pi\)
							−0.796555	+	0.604566i	\(0.793347\pi\)
\(17\)	−0.781272	+	0.0409447i	−0.189486	+	0.00993056i	−0.146843	−	0.989160i	\(-0.546911\pi\)
							−0.0426433	+	0.999090i	\(0.513578\pi\)
\(19\)	−0.0837507	+	0.188107i	−0.0192137	+	0.0431548i	−0.922897	−	0.385046i	\(-0.874186\pi\)
							0.903684	+	0.428201i	\(0.140852\pi\)
\(23\)	3.26985	+	1.66607i	0.681810	+	0.347399i	0.760352	−	0.649511i	\(-0.225026\pi\)
							−0.0785421	+	0.996911i	\(0.525026\pi\)
\(29\)	−0.170416	+	0.123815i	−0.0316455	+	0.0229918i	−0.603496	−	0.797366i	\(-0.706226\pi\)
							0.571850	+	0.820358i	\(0.306226\pi\)
\(31\)	−5.35667	+	1.51859i	−0.962086	+	0.272747i
							\(37\)	−2.89930	+	10.8203i	−0.476642	+	1.77885i	0.138419	+	0.990374i	\(0.455798\pi\)
−0.615061	+	0.788480i	\(0.710869\pi\)
\(41\)	0.811082	+	7.71693i	0.126670	+	1.20518i	0.854510	+	0.519435i	\(0.173858\pi\)
							−0.727840	+	0.685747i	\(0.759476\pi\)
\(43\)	9.45930	−	3.63108i	1.44253	−	0.553735i	0.493730	−	0.869615i	\(-0.335633\pi\)
							0.948799	+	0.315880i	\(0.102300\pi\)
\(47\)	−11.9027	+	1.88520i	−1.73618	+	0.274984i	−0.942708	−	0.333619i	\(-0.891730\pi\)
							−0.793475	+	0.608603i	\(0.791730\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.a.313.6	✓	256
5.2	odd	4		930.2.bt.b.127.5	yes	256
31.21	odd	30		930.2.bt.b.703.5	yes	256
155.52	even	60	inner	930.2.bt.a.517.6	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.313.6	✓	256	1.1	even	1	trivial
930.2.bt.a.517.6	yes	256	155.52	even	60	inner
930.2.bt.b.127.5	yes	256	5.2	odd	4
930.2.bt.b.703.5	yes	256	31.21	odd	30