Embedded newform 285.2.bf.c.89.2

• Label: 285.2.bf.c.89.2
• Level: $285$
• Weight: $2$
• Character: 285.89
• Analytic conductor: $2.276$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.bf (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.27573645761\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			89.2
Character	\(\chi\)	\(=\)	285.89
Dual form			285.2.bf.c.269.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65736 - 1.97516i) q^{2} +(1.36405 - 1.06741i) q^{3} +(-0.807134 + 4.57748i) q^{4} +(-1.55003 + 1.61165i) q^{5} +(-4.36903 - 0.925132i) q^{6} +(-2.53293 - 1.46239i) q^{7} +(5.91308 - 3.41392i) q^{8} +(0.721258 - 2.91201i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{4} - 30 q^{6} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(172\)	\(191\)	\(211\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65736	−	1.97516i	−1.17193	−	1.39665i	−0.900871	−	0.434087i	\(-0.857071\pi\)
							−0.271058	−	0.962563i	\(-0.587374\pi\)
\(3\)	1.36405	−	1.06741i	0.787534	−	0.616271i
							\(5\)	−1.55003	+	1.61165i	−0.693193	+	0.720752i
\(7\)	−2.53293	−	1.46239i	−0.957356	−	0.552730i								−0.0619979	−	0.998076i	\(-0.519747\pi\)
							−0.895358	+	0.445346i	\(0.853081\pi\)
\(11\)	−4.74881	+	2.74173i	−1.43182	+	0.826662i	−0.997260	−	0.0739802i	\(-0.976430\pi\)
							−0.434561	+	0.900642i	\(0.643097\pi\)
\(13\)	−2.14863	+	0.782039i	−0.595924	+	0.216899i	−0.622333	−	0.782753i	\(-0.713815\pi\)
							0.0264090	+	0.999651i	\(0.491593\pi\)
\(17\)	−3.16143	+	2.65276i	−0.766761	+	0.643389i	−0.939877	−	0.341513i	\(-0.889061\pi\)
							0.173117	+	0.984901i	\(0.444616\pi\)
\(19\)	1.91290	+	3.91674i	0.438849	+	0.898561i
							\(23\)	0.343923	−	1.95048i	0.0717128	−	0.406704i	−0.927728	−	0.373258i	\(-0.878241\pi\)
0.999441	−	0.0334459i	\(-0.0106482\pi\)
\(29\)	0.831239	+	0.697492i	0.154357	+	0.129521i	0.716696	−	0.697386i	\(-0.245654\pi\)
							−0.562338	+	0.826907i	\(0.690098\pi\)
\(31\)	−4.42004	−	2.55191i	−0.793863	−	0.458337i	0.0474576	−	0.998873i	\(-0.484888\pi\)
							−0.841321	+	0.540536i	\(0.818221\pi\)
\(37\)	−2.34480			−0.385483			−0.192741	−	0.981250i	\(-0.561738\pi\)
							−0.192741	+	0.981250i	\(0.561738\pi\)
\(41\)	6.39168	+	2.32638i	0.998213	+	0.363320i	0.788895	−	0.614528i	\(-0.210653\pi\)
							0.209318	+	0.977848i	\(0.432876\pi\)
\(43\)	−1.32322	+	0.233319i	−0.201789	+	0.0355808i	−0.273629	−	0.961835i	\(-0.588224\pi\)
							0.0718402	+	0.997416i	\(0.477113\pi\)
\(47\)	−5.65959	−	4.74896i	−0.825536	−	0.692707i	0.128726	−	0.991680i	\(-0.458911\pi\)
							−0.954261	+	0.298974i	\(0.903356\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.bf.c.89.2	yes	192
3.2	odd	2	inner	285.2.bf.c.89.32	yes	192
5.4	even	2	inner	285.2.bf.c.89.31	yes	192
15.14	odd	2	inner	285.2.bf.c.89.1	✓	192
19.3	odd	18	inner	285.2.bf.c.269.1	yes	192
57.41	even	18	inner	285.2.bf.c.269.31	yes	192
95.79	odd	18	inner	285.2.bf.c.269.32	yes	192
285.269	even	18	inner	285.2.bf.c.269.2	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.bf.c.89.1	✓	192	15.14	odd	2	inner
285.2.bf.c.89.2	yes	192	1.1	even	1	trivial
285.2.bf.c.89.31	yes	192	5.4	even	2	inner
285.2.bf.c.89.32	yes	192	3.2	odd	2	inner
285.2.bf.c.269.1	yes	192	19.3	odd	18	inner
285.2.bf.c.269.2	yes	192	285.269	even	18	inner
285.2.bf.c.269.31	yes	192	57.41	even	18	inner
285.2.bf.c.269.32	yes	192	95.79	odd	18	inner