Embedded newform 85.4.r.a.78.3

• Label: 85.4.r.a.78.3
• Level: $85$
• Weight: $4$
• Character: 85.78
• Analytic conductor: $5.015$
• Analytic rank: $0$
• Dimension: $200$
• 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			78.3
Character	\(\chi\)	\(=\)	85.78
Dual form			85.4.r.a.12.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78644 - 4.31285i) q^{2} +(0.459726 + 2.31120i) q^{3} +(-9.75244 + 9.75244i) q^{4} +(0.486014 + 11.1698i) q^{5} +(9.14658 - 6.11155i) q^{6} +(14.4990 - 9.68795i) q^{7} +(24.9802 + 10.3471i) q^{8} +(19.8144 - 8.20741i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 72 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(52\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78644	−	4.31285i	−0.631602	−	1.52482i	−0.837608	−	0.546271i	\(-0.816047\pi\)
							0.206006	−	0.978551i	\(-0.433953\pi\)
\(3\)	0.459726	+	2.31120i	0.0884744	+	0.444791i	0.999475	+	0.0323867i	\(0.0103108\pi\)
							−0.911001	+	0.412404i	\(0.864689\pi\)
\(5\)	0.486014	+	11.1698i	0.0434704	+	0.999055i
							\(7\)	14.4990	−	9.68795i	0.782874	−	0.523100i	−0.0987157	−	0.995116i	\(-0.531473\pi\)
0.881590	+	0.472016i	\(0.156473\pi\)
\(11\)	8.56689	+	12.8213i	0.234820	+	0.351432i	0.930100	−	0.367306i	\(-0.119720\pi\)
							−0.695281	+	0.718738i	\(0.744720\pi\)
\(13\)	35.4414			0.756129			0.378065	−	0.925779i	\(-0.376590\pi\)
							0.378065	+	0.925779i	\(0.376590\pi\)
\(17\)	68.7920	+	13.4409i	0.981442	+	0.191758i
							\(19\)	−101.319	−	41.9677i	−1.22338	−	0.506739i	−0.324895	−	0.945750i	\(-0.605329\pi\)
−0.898482	+	0.439011i	\(0.855329\pi\)
\(23\)	154.125	+	30.6574i	1.39727	+	0.277935i	0.835561	−	0.549397i	\(-0.185143\pi\)
							0.561713	+	0.827332i	\(0.310143\pi\)
\(29\)	224.670	−	44.6896i	1.43862	−	0.286160i	0.586691	−	0.809811i	\(-0.300430\pi\)
							0.851933	+	0.523651i	\(0.175430\pi\)
\(31\)	−16.0437	+	24.0111i	−0.0929528	+	0.139114i	−0.875031	−	0.484066i	\(-0.839159\pi\)
							0.782079	+	0.623180i	\(0.214159\pi\)
\(37\)	−168.360	+	33.4888i	−0.748058	+	0.148798i	−0.554376	−	0.832266i	\(-0.687043\pi\)
							−0.193682	+	0.981064i	\(0.562043\pi\)
\(41\)	−148.320	−	29.5027i	−0.564969	−	0.112379i	−0.0956609	−	0.995414i	\(-0.530496\pi\)
							−0.469308	+	0.883035i	\(0.655496\pi\)
\(43\)	−59.6763	+	144.071i	−0.211641	+	0.510946i	−0.993676	−	0.112289i	\(-0.964182\pi\)
							0.782035	+	0.623235i	\(0.214182\pi\)
\(47\)			66.9708i			0.207845i	0.994585	+	0.103922i	\(0.0331393\pi\)
							−0.994585	+	0.103922i	\(0.966861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.4.r.a.78.3	yes	200
5.2	odd	4		85.4.o.a.27.23	✓	200
17.12	odd	16		85.4.o.a.63.23	yes	200
85.12	even	16	inner	85.4.r.a.12.3	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.o.a.27.23	✓	200	5.2	odd	4
85.4.o.a.63.23	yes	200	17.12	odd	16
85.4.r.a.12.3	yes	200	85.12	even	16	inner
85.4.r.a.78.3	yes	200	1.1	even	1	trivial