Embedded newform 930.2.bt.a.13.6

• Label: 930.2.bt.a.13.6
• Level: $930$
• Weight: $2$
• Character: 930.13
• 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			13.6
Character	\(\chi\)	\(=\)	930.13
Dual form			930.2.bt.a.787.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.156434 + 0.987688i) q^{2} +(0.358368 - 0.933580i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(1.66702 + 1.49031i) q^{5} +(0.866025 + 0.500000i) q^{6} +(2.09913 + 0.110011i) q^{7} +(0.453990 - 0.891007i) q^{8} +(-0.743145 - 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 8 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(311\)	\(871\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(e\left(\frac{11}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.156434	+	0.987688i	−0.110616	+	0.698401i
							\(3\)	0.358368	−	0.933580i	0.206904	−	0.539003i
\(5\)	1.66702	+	1.49031i	0.745516	+	0.666488i
							\(7\)	2.09913	+	0.110011i	0.793395	+	0.0415801i	0.444727	−	0.895666i	\(-0.353301\pi\)
0.348668	+	0.937246i	\(0.386634\pi\)
\(11\)	0.553155	+	2.60239i	0.166783	+	0.784651i	0.979412	+	0.201870i	\(0.0647019\pi\)
							−0.812630	+	0.582780i	\(0.801965\pi\)
\(13\)	3.08657	+	3.81160i	0.856062	+	1.05715i	0.997737	+	0.0672351i	\(0.0214177\pi\)
							−0.141675	+	0.989913i	\(0.545249\pi\)
\(17\)	−2.31445	−	3.56394i	−0.561336	−	0.864381i	0.438067	−	0.898942i	\(-0.355663\pi\)
							−0.999403	+	0.0345610i	\(0.988997\pi\)
\(19\)	−0.541615	+	0.0569261i	−0.124255	+	0.0130597i	−0.166452	−	0.986050i	\(-0.553231\pi\)
							0.0421968	+	0.999109i	\(0.486564\pi\)
\(23\)	−3.22807	−	1.64478i	−0.673099	−	0.342961i	0.0838094	−	0.996482i	\(-0.473291\pi\)
							−0.756909	+	0.653521i	\(0.773291\pi\)
\(29\)	7.68684	−	5.58481i	1.42741	−	1.03707i	0.436918	−	0.899501i	\(-0.356070\pi\)
							0.990492	−	0.137572i	\(-0.0439300\pi\)
\(31\)	−1.70950	+	5.29883i	−0.307035	+	0.951698i
							\(37\)	−5.41734	+	1.45157i	−0.890606	+	0.238637i	−0.674977	−	0.737838i	\(-0.735847\pi\)
−0.215628	+	0.976475i	\(0.569180\pi\)
\(41\)	6.27290	+	2.79288i	0.979663	+	0.436174i	0.833157	−	0.553036i	\(-0.186531\pi\)
							0.146505	+	0.989210i	\(0.453197\pi\)
\(43\)	7.03772	+	5.69903i	1.07324	+	0.869094i	0.991828	−	0.127586i	\(-0.0407229\pi\)
							0.0814145	+	0.996680i	\(0.474056\pi\)
\(47\)	2.22339	−	0.352151i	0.324315	−	0.0513665i	0.00784599	−	0.999969i	\(-0.497503\pi\)
							0.316469	+	0.948603i	\(0.397503\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.a.13.6	✓	256
5.2	odd	4		930.2.bt.b.757.6	yes	256
31.12	odd	30		930.2.bt.b.43.6	yes	256
155.12	even	60	inner	930.2.bt.a.787.6	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.13.6	✓	256	1.1	even	1	trivial
930.2.bt.a.787.6	yes	256	155.12	even	60	inner
930.2.bt.b.43.6	yes	256	31.12	odd	30
930.2.bt.b.757.6	yes	256	5.2	odd	4