Embedded newform 85.4.r.a.12.20

• Label: 85.4.r.a.12.20
• Level: $85$
• Weight: $4$
• Character: 85.12
• Analytic conductor: $5.015$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	85.r (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.01516235049\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(25\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			12.20
Character	\(\chi\)	\(=\)	85.12
Dual form			85.4.r.a.78.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.16376 - 2.80956i) q^{2} +(-0.780312 + 3.92289i) q^{3} +(-0.882452 - 0.882452i) q^{4} +(2.38753 - 10.9224i) q^{5} +(10.1135 + 6.75764i) q^{6} +(-1.26824 - 0.847409i) q^{7} +(18.9702 - 7.85773i) q^{8} +(10.1645 + 4.21029i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 72 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.16376	−	2.80956i	0.411451	−	0.993330i	−0.573298	−	0.819347i	\(-0.694336\pi\)
							0.984749	−	0.173983i	\(-0.0556638\pi\)
\(3\)	−0.780312	+	3.92289i	−0.150171	+	0.754961i	0.830148	+	0.557543i	\(0.188256\pi\)
							−0.980319	+	0.197419i	\(0.936744\pi\)
\(5\)	2.38753	−	10.9224i	0.213547	−	0.976933i
							\(7\)	−1.26824	−	0.847409i	−0.0684783	−	0.0457558i	0.520860	−	0.853642i	\(-0.325611\pi\)
−0.589338	+	0.807886i	\(0.700611\pi\)
\(11\)	32.8718	−	49.1961i	0.901019	−	1.34847i	−0.0360587	−	0.999350i	\(-0.511480\pi\)
							0.937078	−	0.349121i	\(-0.113520\pi\)
\(13\)	12.2498			0.261345			0.130672	−	0.991426i	\(-0.458286\pi\)
							0.130672	+	0.991426i	\(0.458286\pi\)
\(17\)	−7.47483	+	69.6931i	−0.106642	+	0.994297i
							\(19\)	−91.5741	+	37.9312i	−1.10571	+	0.458001i	−0.859459	−	0.511204i	\(-0.829200\pi\)
−0.246253	+	0.969205i	\(0.579200\pi\)
\(23\)	8.28755	−	1.64850i	0.0751336	−	0.0149450i	−0.157380	−	0.987538i	\(-0.550305\pi\)
							0.232514	+	0.972593i	\(0.425305\pi\)
\(29\)	−81.4907	−	16.2095i	−0.521809	−	0.103794i	−0.0728433	−	0.997343i	\(-0.523207\pi\)
							−0.448965	+	0.893549i	\(0.648207\pi\)
\(31\)	81.6248	+	122.160i	0.472911	+	0.707762i	0.988858	−	0.148859i	\(-0.0475600\pi\)
							−0.515947	+	0.856620i	\(0.672560\pi\)
\(37\)	−52.8574	−	10.5140i	−0.234857	−	0.0467159i	0.0762586	−	0.997088i	\(-0.475703\pi\)
							−0.311115	+	0.950372i	\(0.600703\pi\)
\(41\)	198.255	−	39.4354i	0.755178	−	0.150214i	0.197536	−	0.980296i	\(-0.436706\pi\)
							0.557642	+	0.830082i	\(0.311706\pi\)
\(43\)	148.184	+	357.747i	0.525530	+	1.26874i	0.934425	+	0.356160i	\(0.115914\pi\)
							−0.408895	+	0.912582i	\(0.634086\pi\)
\(47\)			393.833i			1.22227i	0.791528	+	0.611133i	\(0.209286\pi\)
							−0.791528	+	0.611133i	\(0.790714\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.4.r.a.12.20	yes	200
5.3	odd	4		85.4.o.a.63.6	yes	200
17.10	odd	16		85.4.o.a.27.6	✓	200
85.78	even	16	inner	85.4.r.a.78.20	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.o.a.27.6	✓	200	17.10	odd	16
85.4.o.a.63.6	yes	200	5.3	odd	4
85.4.r.a.12.20	yes	200	1.1	even	1	trivial
85.4.r.a.78.20	yes	200	85.78	even	16	inner