Embedded newform 285.2.bi.a.233.36

• Label: 285.2.bi.a.233.36
• 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			233.36
Character	\(\chi\)	\(=\)	285.233
Dual form			285.2.bi.a.137.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.42596 + 1.13124i) q^{2} +(-1.72611 + 0.143298i) q^{3} +(3.31998 + 3.95660i) q^{4} +(2.16915 + 0.542930i) q^{5} +(-4.34958 - 1.60502i) q^{6} +(-3.20982 - 0.860068i) q^{7} +(2.19268 + 8.18318i) q^{8} +(2.95893 - 0.494696i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 12 q^{3} - 24 q^{6} - 12 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.42596	+	1.13124i	1.71541	+	0.799909i	0.994243	+	0.107149i	\(0.0341721\pi\)
							0.721168	+	0.692760i	\(0.243606\pi\)
\(3\)	−1.72611	+	0.143298i	−0.996572	+	0.0827330i
							\(5\)	2.16915	+	0.542930i	0.970075	+	0.242806i
\(7\)	−3.20982	−	0.860068i	−1.21320	−	0.325075i								−0.405181	−	0.914236i	\(-0.632792\pi\)
							−0.808015	+	0.589161i	\(0.799458\pi\)
\(11\)	−0.365727	−	0.211152i	−0.110271	−	0.0636649i	0.443850	−	0.896101i	\(-0.353612\pi\)
							−0.554121	+	0.832436i	\(0.686946\pi\)
\(13\)	0.975842	−	1.39365i	0.270650	−	0.386528i	−0.660702	−	0.750648i	\(-0.729741\pi\)
							0.931352	+	0.364120i	\(0.118630\pi\)
\(17\)	−5.20217	−	2.42581i	−1.26171	−	0.588346i	−0.327428	−	0.944876i	\(-0.606182\pi\)
							−0.934284	+	0.356530i	\(0.883960\pi\)
\(19\)	4.19806	+	1.17315i	0.963102	+	0.269138i
							\(23\)	0.270148	−	3.08781i	0.0563298	−	0.643852i	−0.914580	−	0.404406i	\(-0.867478\pi\)
0.970910	−	0.239447i	\(-0.0769660\pi\)
\(29\)	1.65095	+	0.600896i	0.306573	+	0.111584i	0.490726	−	0.871314i	\(-0.336732\pi\)
							−0.184152	+	0.982898i	\(0.558954\pi\)
\(31\)	−3.00937	−	5.21237i	−0.540498	−	0.936170i	−0.998875	−	0.0474123i	\(-0.984903\pi\)
							0.458377	−	0.888758i	\(-0.348431\pi\)
\(37\)	7.07536	−	7.07536i	1.16318	−	1.16318i	0.179407	−	0.983775i	\(-0.442582\pi\)
							0.983775	−	0.179407i	\(-0.0574180\pi\)
\(41\)	−4.31645	−	0.761106i	−0.674116	−	0.118865i	−0.173897	−	0.984764i	\(-0.555636\pi\)
							−0.500219	+	0.865899i	\(0.666747\pi\)
\(43\)	0.239866	+	2.74168i	0.0365792	+	0.418102i	0.992303	+	0.123836i	\(0.0395196\pi\)
							−0.955724	+	0.294266i	\(0.904925\pi\)
\(47\)	−1.81973	−	3.90241i	−0.265434	−	0.569226i	0.727742	−	0.685851i	\(-0.240570\pi\)
							−0.993176	+	0.116626i	\(0.962792\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.bi.a.233.36	yes	432
3.2	odd	2	inner	285.2.bi.a.233.1	yes	432
5.2	odd	4	inner	285.2.bi.a.62.1	yes	432
15.2	even	4	inner	285.2.bi.a.62.36	yes	432
19.4	even	9	inner	285.2.bi.a.23.36	yes	432
57.23	odd	18	inner	285.2.bi.a.23.1	✓	432
95.42	odd	36	inner	285.2.bi.a.137.1	yes	432
285.137	even	36	inner	285.2.bi.a.137.36	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.bi.a.23.1	✓	432	57.23	odd	18	inner
285.2.bi.a.23.36	yes	432	19.4	even	9	inner
285.2.bi.a.62.1	yes	432	5.2	odd	4	inner
285.2.bi.a.62.36	yes	432	15.2	even	4	inner
285.2.bi.a.137.1	yes	432	95.42	odd	36	inner
285.2.bi.a.137.36	yes	432	285.137	even	36	inner
285.2.bi.a.233.1	yes	432	3.2	odd	2	inner
285.2.bi.a.233.36	yes	432	1.1	even	1	trivial