Embedded newform 930.2.bt.b.757.4

• Label: 930.2.bt.b.757.4
• Level: $930$
• Weight: $2$
• Character: 930.757
• 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			757.4
Character	\(\chi\)	\(=\)	930.757
Dual form			930.2.bt.b.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.987688 - 0.156434i) q^{2} +(-0.933580 - 0.358368i) q^{3} +(0.951057 + 0.309017i) q^{4} +(-0.437784 - 2.19279i) q^{5} +(0.866025 + 0.500000i) q^{6} +(0.0914165 - 1.74433i) q^{7} +(-0.891007 - 0.453990i) q^{8} +(0.743145 + 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 16 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.987688	−	0.156434i	−0.698401	−	0.110616i
							\(3\)	−0.933580	−	0.358368i	−0.539003	−	0.206904i
\(5\)	−0.437784	−	2.19279i	−0.195783	−	0.980647i
							\(7\)	0.0914165	−	1.74433i	0.0345522	−	0.659295i	−0.925376	−	0.379050i	\(-0.876251\pi\)
0.959928	−	0.280245i	\(-0.0904158\pi\)
\(11\)	−0.980098	−	4.61100i	−0.295511	−	1.39027i	−0.835913	−	0.548862i	\(-0.815061\pi\)
							0.540403	−	0.841407i	\(-0.318272\pi\)
\(13\)	4.28202	−	3.46751i	1.18762	−	0.961714i	0.187827	−	0.982202i	\(-0.439855\pi\)
							0.999790	+	0.0204885i	\(0.00652214\pi\)
\(17\)	−4.48068	+	2.90979i	−1.08673	+	0.705728i	−0.958235	−	0.285983i	\(-0.907680\pi\)
							−0.128491	+	0.991711i	\(0.541013\pi\)
\(19\)	6.00747	−	0.631411i	1.37821	−	0.144856i	0.613736	−	0.789512i	\(-0.289666\pi\)
							0.764473	+	0.644656i	\(0.222999\pi\)
\(23\)	−1.30791	+	2.56691i	−0.272717	+	0.535238i	−0.986225	−	0.165410i	\(-0.947105\pi\)
							0.713508	+	0.700647i	\(0.247105\pi\)
\(29\)	5.33298	−	3.87464i	0.990310	−	0.719503i	0.0303213	−	0.999540i	\(-0.490347\pi\)
							0.959989	+	0.280038i	\(0.0903470\pi\)
\(31\)	−5.54131	−	0.542073i	−0.995249	−	0.0973593i
							\(37\)	−2.06756	−	7.71624i	−0.339905	−	1.26854i	−0.898453	−	0.439069i	\(-0.855308\pi\)
0.558549	−	0.829472i	\(-0.311358\pi\)
\(41\)	8.01827	+	3.56996i	1.25224	+	0.557535i	0.922303	−	0.386467i	\(-0.126305\pi\)
							0.329940	+	0.944002i	\(0.392971\pi\)
\(43\)	−2.59046	+	3.19895i	−0.395042	+	0.487836i	−0.935396	−	0.353602i	\(-0.884957\pi\)
							0.540354	+	0.841438i	\(0.318290\pi\)
\(47\)	−0.0574468	−	0.362705i	−0.00837948	−	0.0529059i	0.983144	−	0.182831i	\(-0.0585262\pi\)
							−0.991524	+	0.129925i	\(0.958526\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.b.757.4	yes	256
5.3	odd	4		930.2.bt.a.13.1	✓	256
31.12	odd	30		930.2.bt.a.787.1	yes	256
155.43	even	60	inner	930.2.bt.b.43.4	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.13.1	✓	256	5.3	odd	4
930.2.bt.a.787.1	yes	256	31.12	odd	30
930.2.bt.b.43.4	yes	256	155.43	even	60	inner
930.2.bt.b.757.4	yes	256	1.1	even	1	trivial