Embedded newform 85.4.r.a.12.3

• Label: 85.4.r.a.12.3
• 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			12.3
Character	\(\chi\)	\(=\)	85.12
Dual form			85.4.r.a.78.3

$q$-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 \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.o.a.27.23	✓	200	17.10	odd	16
85.4.o.a.63.23	yes	200	5.3	odd	4
85.4.r.a.12.3	yes	200	1.1	even	1	trivial
85.4.r.a.78.3	yes	200	85.78	even	16	inner