Embedded newform 285.2.v.a.83.16

• Label: 285.2.v.a.83.16
• Level: $285$
• Weight: $2$
• Character: 285.83
• Analytic conductor: $2.276$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.27573645761\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			83.16
Character	\(\chi\)	\(=\)	285.83
Dual form			285.2.v.a.182.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.136660 + 0.510023i) q^{2} +(-1.72937 + 0.0963563i) q^{3} +(1.49060 + 0.860600i) q^{4} +(2.14256 - 0.639882i) q^{5} +(0.187192 - 0.895186i) q^{6} +(-2.68311 + 2.68311i) q^{7} +(-1.38936 + 1.38936i) q^{8} +(2.98143 - 0.333271i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 2 q^{3} - 4 q^{6} - 16 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

\(n\)	\(172\)	\(191\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.136660	+	0.510023i	−0.0966334	+	0.360641i	−0.997262	−	0.0739487i	\(-0.976440\pi\)
							0.900629	+	0.434590i	\(0.143107\pi\)
\(3\)	−1.72937	+	0.0963563i	−0.998451	+	0.0556313i
							\(5\)	2.14256	−	0.639882i	0.958181	−	0.286164i
\(7\)	−2.68311	+	2.68311i	−1.01412	+	1.01412i								−0.0142199	+	0.999899i	\(0.504526\pi\)
							−0.999899	+	0.0142199i	\(0.995474\pi\)
\(11\)		−	0.858267i		−	0.258777i	−0.991594	−	0.129389i	\(-0.958699\pi\)
							0.991594	−	0.129389i	\(-0.0413014\pi\)
\(13\)	4.40001	−	1.17898i	1.22034	−	0.326990i	0.409531	−	0.912296i	\(-0.365692\pi\)
							0.810813	+	0.585306i	\(0.199026\pi\)
\(17\)	−1.55915	+	5.81883i	−0.378150	+	1.41127i	0.470538	+	0.882380i	\(0.344060\pi\)
							−0.848687	+	0.528895i	\(0.822607\pi\)
\(19\)	−4.09105	+	1.50444i	−0.938551	+	0.345142i
							\(23\)	1.15613	+	4.31473i	0.241069	+	0.899683i	0.975318	+	0.220803i	\(0.0708677\pi\)
−0.734249	+	0.678880i	\(0.762466\pi\)
\(29\)	−0.442208	+	0.765926i	−0.0821159	+	0.142229i	−0.904159	−	0.427197i	\(-0.859501\pi\)
							0.822043	+	0.569426i	\(0.192834\pi\)
\(31\)	3.36718			0.604763			0.302381	−	0.953187i	\(-0.402218\pi\)
							0.302381	+	0.953187i	\(0.402218\pi\)
\(37\)	1.87723	−	1.87723i	0.308614	−	0.308614i	−0.535758	−	0.844372i	\(-0.679974\pi\)
							0.844372	+	0.535758i	\(0.179974\pi\)
\(41\)	−3.33846	+	1.92746i	−0.521380	+	0.301019i	−0.737499	−	0.675348i	\(-0.763993\pi\)
							0.216119	+	0.976367i	\(0.430660\pi\)
\(43\)	2.87472	−	10.7286i	0.438391	−	1.63610i	−0.294426	−	0.955674i	\(-0.595129\pi\)
							0.732818	−	0.680425i	\(-0.238205\pi\)
\(47\)	10.6580	−	2.85580i	1.55463	−	0.416561i	0.623670	−	0.781688i	\(-0.285641\pi\)
							0.930958	+	0.365126i	\(0.118974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.v.a.83.16	yes	144
3.2	odd	2	inner	285.2.v.a.83.21	yes	144
5.2	odd	4	inner	285.2.v.a.197.16	yes	144
15.2	even	4	inner	285.2.v.a.197.21	yes	144
19.11	even	3	inner	285.2.v.a.68.21	yes	144
57.11	odd	6	inner	285.2.v.a.68.16	✓	144
95.87	odd	12	inner	285.2.v.a.182.21	yes	144
285.182	even	12	inner	285.2.v.a.182.16	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.v.a.68.16	✓	144	57.11	odd	6	inner
285.2.v.a.68.21	yes	144	19.11	even	3	inner
285.2.v.a.83.16	yes	144	1.1	even	1	trivial
285.2.v.a.83.21	yes	144	3.2	odd	2	inner
285.2.v.a.182.16	yes	144	285.182	even	12	inner
285.2.v.a.182.21	yes	144	95.87	odd	12	inner
285.2.v.a.197.16	yes	144	5.2	odd	4	inner
285.2.v.a.197.21	yes	144	15.2	even	4	inner