Embedded newform 285.2.bi.a.233.13

• Label: 285.2.bi.a.233.13
• Level: $285$
• Weight: $2$
• Character: 285.233
• Analytic conductor: $2.276$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.bi (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.27573645761\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(36\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			233.13
Character	\(\chi\)	\(=\)	285.233
Dual form			285.2.bi.a.137.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.911447 - 0.425015i) q^{2} +(1.70258 + 0.318156i) q^{3} +(-0.635477 - 0.757331i) q^{4} +(2.21511 + 0.305406i) q^{5} +(-1.41659 - 1.01360i) q^{6} +(0.0637212 + 0.0170740i) q^{7} +(0.777900 + 2.90316i) q^{8} +(2.79755 + 1.08337i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 12 q^{3} - 24 q^{6} - 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(172\)	\(191\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.911447	−	0.425015i	−0.644491	−	0.300531i	0.0727566	−	0.997350i	\(-0.476820\pi\)
							−0.717247	+	0.696819i	\(0.754598\pi\)
\(3\)	1.70258	+	0.318156i	0.982985	+	0.183687i
							\(5\)	2.21511	+	0.305406i	0.990629	+	0.136582i
\(7\)	0.0637212	+	0.0170740i	0.0240844	+	0.00645338i								0.270841	−	0.962624i	\(-0.412698\pi\)
							−0.246757	+	0.969077i	\(0.579365\pi\)
\(11\)	0.508093	+	0.293348i	0.153196	+	0.0884476i	0.574638	−	0.818407i	\(-0.305143\pi\)
							−0.421443	+	0.906855i	\(0.638476\pi\)
\(13\)	0.566140	−	0.808532i	0.157019	−	0.224246i	−0.732933	−	0.680301i	\(-0.761849\pi\)
							0.889952	+	0.456055i	\(0.150738\pi\)
\(17\)	−2.69807	−	1.25813i	−0.654378	−	0.305141i	0.0669282	−	0.997758i	\(-0.478680\pi\)
							−0.721306	+	0.692616i	\(0.756458\pi\)
\(19\)	1.84387	−	3.94970i	0.423013	−	0.906124i
							\(23\)	−0.319702	+	3.65422i	−0.0666626	+	0.761957i	0.887136	+	0.461509i	\(0.152692\pi\)
−0.953798	+	0.300448i	\(0.902864\pi\)
\(29\)	−2.64543	−	0.962857i	−0.491243	−	0.178798i	0.0845080	−	0.996423i	\(-0.473068\pi\)
							−0.575751	+	0.817625i	\(0.695290\pi\)
\(31\)	−1.34144	−	2.32345i	−0.240931	−	0.417304i	0.720049	−	0.693923i	\(-0.244119\pi\)
							−0.960980	+	0.276619i	\(0.910786\pi\)
\(37\)	4.97782	−	4.97782i	0.818349	−	0.818349i	−0.167520	−	0.985869i	\(-0.553576\pi\)
							0.985869	+	0.167520i	\(0.0535757\pi\)
\(41\)	−9.04391	−	1.59469i	−1.41242	−	0.249048i	−0.585184	−	0.810900i	\(-0.698978\pi\)
							−0.827238	+	0.561852i	\(0.810089\pi\)
\(43\)	0.786528	+	8.99006i	0.119944	+	1.37097i	0.782823	+	0.622244i	\(0.213779\pi\)
							−0.662879	+	0.748727i	\(0.730666\pi\)
\(47\)	1.34443	+	2.88314i	0.196105	+	0.420550i	0.979630	−	0.200811i	\(-0.0643578\pi\)
							−0.783524	+	0.621361i	\(0.786580\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.bi.a.233.13	yes	432
3.2	odd	2	inner	285.2.bi.a.233.24	yes	432
5.2	odd	4	inner	285.2.bi.a.62.24	yes	432
15.2	even	4	inner	285.2.bi.a.62.13	yes	432
19.4	even	9	inner	285.2.bi.a.23.13	✓	432
57.23	odd	18	inner	285.2.bi.a.23.24	yes	432
95.42	odd	36	inner	285.2.bi.a.137.24	yes	432
285.137	even	36	inner	285.2.bi.a.137.13	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.bi.a.23.13	✓	432	19.4	even	9	inner
285.2.bi.a.23.24	yes	432	57.23	odd	18	inner
285.2.bi.a.62.13	yes	432	15.2	even	4	inner
285.2.bi.a.62.24	yes	432	5.2	odd	4	inner
285.2.bi.a.137.13	yes	432	285.137	even	36	inner
285.2.bi.a.137.24	yes	432	95.42	odd	36	inner
285.2.bi.a.233.13	yes	432	1.1	even	1	trivial
285.2.bi.a.233.24	yes	432	3.2	odd	2	inner