Embedded newform 285.2.v.a.68.10

• Label: 285.2.v.a.68.10
• Level: $285$
• Weight: $2$
• Character: 285.68
• 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			68.10
Character	\(\chi\)	\(=\)	285.68
Dual form			285.2.v.a.197.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.51172 + 0.405064i) q^{2} +(-1.64008 + 0.556903i) q^{3} +(0.389164 - 0.224684i) q^{4} +(0.718854 + 2.11737i) q^{5} +(2.25376 - 1.50622i) q^{6} +(2.61601 - 2.61601i) q^{7} +(1.71601 - 1.71601i) q^{8} +(2.37972 - 1.82673i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 2 q^{3} - 4 q^{6} - 16 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(172\)	\(191\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.51172	+	0.405064i	−1.06895	+	0.286423i	−0.750060	−	0.661370i	\(-0.769975\pi\)
							−0.318886	+	0.947793i	\(0.603309\pi\)
\(3\)	−1.64008	+	0.556903i	−0.946900	+	0.321528i
							\(5\)	0.718854	+	2.11737i	0.321481	+	0.946916i
\(7\)	2.61601	−	2.61601i	0.988759	−	0.988759i								−0.0111785	−	0.999938i	\(-0.503558\pi\)
							0.999938	+	0.0111785i	\(0.00355831\pi\)
\(11\)			5.27961i			1.59186i	0.605388	+	0.795931i	\(0.293018\pi\)
							−0.605388	+	0.795931i	\(0.706982\pi\)
\(13\)	0.467989	−	1.74656i	0.129797	−	0.484408i	−0.870168	−	0.492754i	\(-0.835990\pi\)
							0.999965	+	0.00834636i	\(0.00265676\pi\)
\(17\)	−2.61818	+	0.701538i	−0.635001	+	0.170148i	−0.561938	−	0.827179i	\(-0.689944\pi\)
							−0.0730629	+	0.997327i	\(0.523277\pi\)
\(19\)	−0.232539	+	4.35269i	−0.0533482	+	0.998576i
							\(23\)	4.20489	+	1.12670i	0.876781	+	0.234933i	0.669018	−	0.743247i	\(-0.266715\pi\)
0.207763	+	0.978179i	\(0.433382\pi\)
\(29\)	−0.173588	−	0.300664i	−0.0322345	−	0.0558319i	0.849458	−	0.527656i	\(-0.176929\pi\)
							−0.881693	+	0.471824i	\(0.843596\pi\)
\(31\)	7.90446			1.41968			0.709842	−	0.704361i	\(-0.248767\pi\)
							0.709842	+	0.704361i	\(0.248767\pi\)
\(37\)	−5.09871	+	5.09871i	−0.838223	+	0.838223i	−0.988625	−	0.150402i	\(-0.951943\pi\)
							0.150402	+	0.988625i	\(0.451943\pi\)
\(41\)	−8.19995	−	4.73424i	−1.28062	−	0.739365i	−0.303656	−	0.952782i	\(-0.598207\pi\)
							−0.976961	+	0.213417i	\(0.931541\pi\)
\(43\)	1.11704	−	0.299311i	0.170348	−	0.0456445i	−0.172637	−	0.984986i	\(-0.555229\pi\)
							0.342985	+	0.939341i	\(0.388562\pi\)
\(47\)	−2.00325	+	7.47623i	−0.292204	+	1.09052i	0.651208	+	0.758899i	\(0.274262\pi\)
							−0.943412	+	0.331622i	\(0.892404\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.v.a.68.10	✓	144
3.2	odd	2	inner	285.2.v.a.68.27	yes	144
5.2	odd	4	inner	285.2.v.a.182.10	yes	144
15.2	even	4	inner	285.2.v.a.182.27	yes	144
19.7	even	3	inner	285.2.v.a.83.27	yes	144
57.26	odd	6	inner	285.2.v.a.83.10	yes	144
95.7	odd	12	inner	285.2.v.a.197.27	yes	144
285.197	even	12	inner	285.2.v.a.197.10	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.v.a.68.10	✓	144	1.1	even	1	trivial
285.2.v.a.68.27	yes	144	3.2	odd	2	inner
285.2.v.a.83.10	yes	144	57.26	odd	6	inner
285.2.v.a.83.27	yes	144	19.7	even	3	inner
285.2.v.a.182.10	yes	144	5.2	odd	4	inner
285.2.v.a.182.27	yes	144	15.2	even	4	inner
285.2.v.a.197.10	yes	144	285.197	even	12	inner
285.2.v.a.197.27	yes	144	95.7	odd	12	inner