Embedded newform 930.2.bt.b.313.16

• Label: 930.2.bt.b.313.16
• 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			313.16
Character	\(\chi\)	\(=\)	930.313
Dual form			930.2.bt.b.517.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.156434 - 0.987688i) q^{2} +(0.629320 + 0.777146i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(2.07778 - 0.826327i) q^{5} +(0.866025 - 0.500000i) q^{6} +(-2.52374 + 3.88622i) q^{7} +(-0.453990 + 0.891007i) q^{8} +(-0.207912 + 0.978148i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 16 q^{7}+O(q^{10}) \)

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
\(5\)	2.07778	−	0.826327i	0.929213	−	0.369545i
							\(7\)	−2.52374	+	3.88622i	−0.953885	+	1.46885i	−0.0719911	+	0.997405i	\(0.522935\pi\)
−0.881894	+	0.471448i	\(0.843731\pi\)
\(11\)	−3.85079	+	3.46727i	−1.16106	+	1.04542i	−0.162786	+	0.986661i	\(0.552048\pi\)
							−0.998272	+	0.0587597i	\(0.981285\pi\)
\(13\)	0.457930	−	1.19295i	0.127007	−	0.330864i	−0.855303	−	0.518129i	\(-0.826629\pi\)
							0.982310	+	0.187265i	\(0.0599621\pi\)
\(17\)	−4.50257	+	0.235970i	−1.09203	+	0.0572310i	−0.589837	−	0.807522i	\(-0.700808\pi\)
							−0.502197	+	0.864753i	\(0.667475\pi\)
\(19\)	−1.79369	+	4.02870i	−0.411501	+	0.924247i	0.582290	+	0.812981i	\(0.302157\pi\)
							−0.993791	+	0.111265i	\(0.964510\pi\)
\(23\)	−3.34341	−	1.70355i	−0.697149	−	0.355215i	0.0692327	−	0.997601i	\(-0.477945\pi\)
							−0.766382	+	0.642385i	\(0.777945\pi\)
\(29\)	−6.38417	+	4.63837i	−1.18551	+	0.861323i	−0.992782	−	0.119929i	\(-0.961733\pi\)
							−0.192727	+	0.981252i	\(0.561733\pi\)
\(31\)	5.40155	+	1.35027i	0.970148	+	0.242515i
							\(37\)	−1.06272	+	3.96611i	−0.174710	+	0.652025i	0.821891	+	0.569644i	\(0.192919\pi\)
−0.996601	+	0.0823807i	\(0.973748\pi\)
\(41\)	−0.915268	−	8.70819i	−0.142941	−	1.35999i	−0.797197	−	0.603719i	\(-0.793685\pi\)
							0.654257	−	0.756273i	\(-0.272982\pi\)
\(43\)	3.26593	−	1.25367i	0.498050	−	0.191183i	−0.0963586	−	0.995347i	\(-0.530720\pi\)
							0.594409	+	0.804163i	\(0.297386\pi\)
\(47\)	8.53175	−	1.35130i	1.24448	−	0.197107i	0.500756	−	0.865589i	\(-0.333056\pi\)
							0.743729	+	0.668482i	\(0.233056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.b.313.16	yes	256
5.2	odd	4		930.2.bt.a.127.11	✓	256
31.21	odd	30		930.2.bt.a.703.11	yes	256
155.52	even	60	inner	930.2.bt.b.517.16	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.127.11	✓	256	5.2	odd	4
930.2.bt.a.703.11	yes	256	31.21	odd	30
930.2.bt.b.313.16	yes	256	1.1	even	1	trivial
930.2.bt.b.517.16	yes	256	155.52	even	60	inner